zotero/storage/JED3ASG6/.zotero-ft-cache

NATIONAL AERONAUTICS A N D SPACE A D M I N I S T R A T I O N
Theodore D. Moyer

LSI

L

T Y

CALIFORNIA INSTITUTE OF TECHNOLOGY

P A S A D E N A, C Ab1 F O R N 1 A

May 15, 197'1

N A T I O N A L AERONAUTICS A N D SPACE ADMlNlSTRATiOh

Theodore D.

f

ay 15, 1971

Prepared Under Contract No. NAS 7- 100 National Aeronautics and Space Administration

The work described in this report was performed by the Mission Analysis sion of the Jet Propulsion Laboratory.
7

This report documents the complete mathematical model for the DoublePrecision Orbit Determination Program (DPODP), a third-generation program which has recently been completed at the Jet PropulsionLaboratory. The DPQDP processes earth-based doppler, range, and angular observables of the spacecraft to determine values of the parameters that specify the spacecraft trajectory for lunar and planetary missions. The program was developed from 1964 to 1968; it was first used operationally for the Mariner VI and VI1 spacecraft which encountered Mars in August of 1969.
The DPQDP has more accurate mathematical models, a significant increase in numerical precision, and more flexibility than the second-generation SinglePrecision Orbit Determination Program (SPODP). Doppler and range observables are computed to accuracies of m/s and 0.1 m, respectively, exclusive of errors in the tropospheric, ionospheric, and space plasma corrections.
7

1. I

n........................ 1

............. 3

. . . . . . . . . . . . . . . . . . . . 19 A. Systems of Time . . . . . . . . . . . . . . . . . . . . . . 19
1. Ephemeris Time (ET) . . . . . . . . . . . . . . . . . . . 19 2. Atomic Time (Al) . . . . . . . . . . . . . . . . . . . . 19 .3 Universal Time (UT) . . . . . . . . . . . . . . . . . . . 19 4. Broadcast Universal Time (UTC) . . . . . . . . . . . . . . . 19 5. Station Time (ST) . . . . . . . . . . . . . . . . . . . . 20 B. Transformations Between Time Scales . . . . . . . . . . . . . . 20

emerides . . . . . . . . . . . . . . . . . . . . 22 .A Description of Precomputed n-Body Ephemerides . . . . . . . . . . 22 .B Obtaining Corrected Position. Velocity. Acceleration. and
Jerk From Each Ephemeris . . . . . . . . . . . . . . . . . . 25
1. Uncorrected position and velocity . . . . . . . . . . . . . . . 25 2. Solar and lunar constraints . . . . . . . . . . . . . . . . . 25
3. Corrected position and velocity . . . . . . . . . . . . . . . 27 .4 Partial derivatives of position and velocity with respect to
. orbital elements . . . . . . . . . . . . . . . . . . . . 28 .5 Acceleration and jerk . . . . . . . . . . . . . . . . . . . 29
C. Position. Velocity. Acceleration. and Jerk of One Celestial Body
. Relative to Another . . . . . . . . . . . . . . . . . . . . 29

jeetory . . . . . . . . . . . . . . . . . . . . 30

.A General Description . . . . . . . . . . . . . . . . . . . . 30

B. Spacecraft Acceleration . . . . . . . . . . . . . . . . . . . 31

1. Point-mass gravitational acceleration . . . . . . . . . . . . . 31

.2 . Direct acceleration of spacecraft due to oblateness . . . . . . . . 31

. . . . . . 3 Indirect acceleration of center of integration due to oblateness

34

4. Acceleration of spacecraft due to solar radiation pressure and
small forces originating in spacecraft . . . . . . . . . . . . . 35

5. Acceleration due to motor burn . . . . . . . . . . . . . . . 37

@

BB . . . . . . . . . . . . . . . . . . . . . 38

.A Introduction . . . . . . . . . . . . . . . . . . . . . . . 38

7

.B Formulation . . . . . . . . . . . . . . . . . . . . . . . 38
. . C Procedure . . . . . . . . . . . . . . . . . . . . . . . 40
II.
. . . . . 40 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . 40 . B. Body-Fixed Rectangular Coordinates . . . . . . . . . . . . . . 41
. . 1 Fixed tracking station or landed spacecraft . . . . . . . . . . . 41 2. Moving tracking ship . . . . . . . . . . . . . . . . . . . 42 .C Transformation of Body-Fixed Rectangular Coordinates to 1950.0 Position, Velocity. Acceleration. and Jerk . . . . . . . . . . . . . 43 D. Body-Fixed to Space-Fixed Transformation for the Earth . . . . . . . . 44

bservables . . . . . . . . . . . . . . . . . . . . 46 .A Introduction . . . . . . . . . . . . . . . . . . . . . . . 46
B. General Expressions . . . . . . . . . . . . . . . . . . . . 47
e
C. Doppler Frequency Shift . . . . . . . . . . . . . . . . . . . 53
.D Second Derivative of Doppler Frequency Shift . . . . . . . . . . . 56

bservables . . . . . . . . . . . . . . . . . . . . . 59 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . 59 8. Formulation . . . . . . . . . . . . . . . . . . . . . . . 60 C. Ranging Systems . . . . . . . . . . . . . . . . . . . . . 62

servables . . . . . . . . . . . . . . . . . . . . 63

A. Coordinate Systems and Unit Vectors . . . . . . . . . . . . . . 63

. . . . 1 Right ascension. hour angle. and declination . . . . . . . .

64

2. The north-east-zenith coordinate system . . . . . . . . . . . . 64

3. Azimuth and elevation . . . . . . . . . . . . . . . . . . . 65

. . . . 4. The X and Y angles for MSFN stations with 6-m (20-ft)antenna .

66

. . 5 The X’ and Y’ angles for MSFN stations with 26-m (8%) antenna . . . 66

B. Computation of Angular Observables . . . . . . . . . . . . . . 67

1. Computation of unit vector . . . . . . . . . . .

. . . 67

.2 Computation of observed angles . . . . . . . . . . . . . . . 69

C. Corrections Due to Small Rotations of Reference Coordinate System at
. Tracking Station . . . . . . . . . . . . . . . . . . . . . 70 .1 Hour angle-declination . . . . . . . . . . . . . . . . . 70

7

2. Azimuth-elevation . . . . . . . . . . . . . . . . . . . . 70 3. Angles X. Y . . . . . . . . . . . . . . . . . . . . . . 71
.4 AnglesX’, Y’ . . . . . . . . . . . . . . . . . . . . . . 71
D. Partial Derivatives of Angular Observables With Respect to Heliocentric 1950.0 Position Vectors of Spacecraft and Tracking Station . . . . . . . . 71
ler . . . . . . . . . . . . . . . . . . 72 A. Introduction . . . . . . . . . . . . . . . . . . . . . . . 72
B. Equivalence of Doppler Observables and Differenced Range Observables . . . . . . . . . . . . . . . . . . . . . . . 73
.C Modified Range Observable Formulation . . . . . . . . . . . . . 75
1. Numerical considerations . . . . . . . . . . . . . . . . . 75 2. Formulation . . . . . . . . . . . . . . . . . . . . . . 76

ntenna. .ro.o$..ere.
bservables . . . . . . . . . . . . . . . . . . . . . . . 80

A. Definition of Correction Terms . . . . . . . . . . . . . . . . . 80
.1 Range observables . . . . . . . . . . . . . . . . . . . . 80 2. Doppler observables . . . . . . . . . . . . . . . . . . . 80

3. Angular observables . . . . . . . . . . . . . . . . . . . 81

B. Evaluation of One-Leg Range Corrections . . . . . . . . . . . . . 81
.1 Antenna correction . . . . . . . . . . . . . . . . . . . . 81

2. Troposphere and ionosphere corrections . . . . . . . . . . . . . 83

tions . . . . . . . . . . . . . . . . . . . . 87

A. Variational Equations and Method of Integration . . . . . . . . . . 87

B. Computation of A Matrix . . . . . . . . . . . . . . . . . . . 89

. . 1 Contribution from Newtonian point mass acceleration . . . . . . . 89
.2 . Contribution from oblateness acceleration . . . . . . . . . . . 90

. . . . . 3 Contribution from solar radiation pressure and small force models

92

. . C Computation of B Matrix . . . . . . . . . . . . . . . . . . 93

D. C Matrix and Integration Tables for Each Parameter . . . . . . . . . 93

1. injection parameters . . . . . . . . . . . . . . . . . . . 93 . . 2 Reference parameters . . . . . . . . . . . . . . . . . . 93

.3 Gravitational constants pj . . . . . . . . . . . . . . . . . 94

4. Harmonic coefficients,,J .,C, ,.S . . . . . . . . . . . . . . 95

. . . . . 5. Coefficientsof solar radiation pressure and small force model

95

Contents (conid)

6. Coefficients of finite burn motor model . . . . . . . . . . . . . 95

. . . . . . . . . . 7 Parameters for instantaneous burn motor model

96

.8 Parameters affecting transformationfrom atomic . time to ephemeris time . . . . . . . . . . . . . . . . . . 96

E.Summary . . . . . . . . . . . . . . . . . . . . . . . . 97

. XIV . Regression Partial Derivatives . . . . . . . . . . . . . . . . 98

.A General Expression for Partial Derivatives of Doppler and Angular Observables With Respect to q . . . . . . . . . . . . . 98

B. Partial Derivativesof Doppler and Angular ObservablesWith
Respect to State Vectors of Each Direct Participant . . . . . . . . . . 100 1. Doppler observables . . . . . . . . . . . . . . . . . . . 100 2. Angular observables . . . . . . . . . . . . . . . . . . . 101

C. Partial Derivativesof Body-CenteredState Vector of Tracking Station

. . . . . or Ship or Landed Spacecraft With Respect to Parameter Vector

102

1. General formulas . . . . . . . . . . . . . . . . . . . . 102

2. Partial derivatives of body-fixed position and velocity
vectors with respect to parameter vector . . . . . . . . . . . . 102

D. Partial Derivativesof Doppler and Angular Observables With Respect

. . . . . to Speed of Light and Parameters Affecting Time Transformations

104

. . 1 Speed of light c . . . . . . . . . . . . . . . . . . . . 104

.2 Parameters affecting (ET - A l ) time transformation: ATlss8 and Afcesium . . . . . . . . . . . . . . . . . . . 104

. . . . . 3 Parameters affecting (UTC - ST) time transformation: a,b, and c

105

E. Partial Derivatives of Doppler and Angular ObservablesWith

. . . . . . . Respect to Parameter Vector, Holding State Vectors Fixed

106

.1 Speed of light c . . . . . . . . . . . . . . . . . . . . . 106

.2 Polynomial coefficients b and c of (UTC - ST) . . . . . . . . . . 106

.3 Polynomial coefficientsfro,fT1, and f T z of 1-way doppler transmitter frequency . . . . . . . . . . . . . . . . 107

.4 Rotations T’, E, 5’ or 7, E, 5 of referencecoordinate . system for angular observables . . . . . . . . . . . . . . . 107

.5 Parameter y of the Brans-Dicke . theory of relativity . . . . . . . . 107

.F Partial Derivatives of Range Observables With Respect to Parameter Vector . . . . . . . . . . . . . . . . . 107

.G Partial Derivativesof Differenced-Range Doppler . Observables With Respect to Parameter Vector . . . . . . . . . . 108

viii

JPL TECHNICAL REPORT 32-1527

Contents (conid)

.XV . Normal-Equations Form of Estimation Formulas . . . . . . . . . 109
.A Introduction . . . . . . . . . . . . . . . . . . . . . . . 109

B. Categorization of Parameters and Constraints . . . . . . . . . . . 110
.C Error Function . . . . . . . . . . . . . . . . . . . . . . 112
.D Parameter Estimation Formula . . . . . . . . . . . . . . . . . 113
.E Covariance Matrix . . . . . . . . . . . . . . . . . . . . . 114

F. Mapping Covariance Matrix to New Epoch . . . . . . . . . . . . 117

. . . . . . . 1 Mapped covariance matrix relative to center of integration

117

.2 Mapped covariance matrix relative to body other . than center of integration . . . . . . . . . . . . . . . . . 119

. XVI . Square-Root Form of Estimation Formulas . . . . . . . . . . . 120
.A Introduction . . . . . . . . . . . . . . . . . . . . . . . 120
.6 Parameter Estimation Formula . . . . . . . . . . . . . . . . . 120 .1 Equations . . . . . . . . . . . . . . . . . . . . . . . 120 2. Numerical techniques . . . . . . . . . . . . . . . . . . . 124
.C Covariance Matrix . . . . . . . . . . . . . . . . . . . . . 126
.D Mapping Covariance Matrix to New Epoch . . . . . . . . . . . . 130 . . 1 General mapping formulas . . . . . . . . . . . . . . . . 130 2. Mapping . matrix . . . . . . . . . . . . . . . . . . . . 131 .3 Multiplication of matrices . . . . . . . . . . . . . . . . . 132

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

. . . . . . . Appendix A: Derivation of n-Body Relativistic Equations of Motion

137

Appendix B: Derivation of ET .A1 Time Transformation . . . . . . . . . . 141 . Appendix C: Derivation of Light Time Equation . . . . . . . . . . . . . 152

References . . . . . . . . . . . . . . . . . . . . . . . . . . 156

. . . . . . 1. Conversion factors and modulo numbers for ranging systems

62

. B.1 . Values of a and b . . . . . . . . . . . . . . . . . . . . 143

Figures
.1 Spherical and rectangular coordinates . . . . . . . . . . . . . . 4 . . 2 Light path . . . . . . . . . . . . . . . . . . . . . . . 17

JPL TECHNICAL REPORT 32-1527

ix

3. x’. y‘. and z‘ axes relative to body-fixed xb. yb. and 21,axes . . . . . . 32 .4 Latitude and longitude relative to mean pole of 1903.0
. and true pole of date . . . . . . . . . . . . . . . . . . . 41 . . 5 Right ascension. hour angle. and declination . . . . . . . . . . . 64 .6 The north-east-zenith coordinate system . . . . . . . . . . . . . 65 7. Azimuth and elevation . . . . . . . . . . . . . . . . . . . 65 8. X and Y angles . . . . . . . . . . . . . . . . . . . . . . 67 9. X’ and Y’ angles . . . . . . . . . . . . . . . . . . . . . 57 . . 10 Antenna correction . . . . . . . . . . . . . . . . . . . . 82 . C.1 . Geometry for straight line motion . . . . . . . . . . . . . . .154
X

ucai
This technical report documents the mathematical model for the Double-PrecisionOrbit Determination Program (DPODP), a third-generation program that has recently been completed at the Jet Propulsion Laboratory (JPL).The DPODP will be used to determinevalues of the parameters that specify the spacecraft trajectory for lunar and planetary missions; it will be used for both real-time and post-flight reduction of tracking data. The DPODP dgerentially correctsa priori estimatesof injectionparameters, physicalconstants,maneuver parameters, and station locations to minimize the sum of weighted squares of residual errors between observed and computed quantities.
The analysis began in 1964, and coding for the IB 7094 computer was initiated the next year. The program was completed and fully checked out at the end of 1968; it was first used operationally for the Mariner VI and VI1 spacecraft, which encountered ars in August, 1969. Conversion of the program to the Univac 1108 computer was completed early in 1970.
PODP has more accurate mathematical models, significantlymore numericalprecision, and more flexibility

than the second-generation Single-Precision Orbit Determination Program (SPODP). The basic limitations on the accuracy of computed observables are the inaccuracies in the troposphere and ionosphere corrections. Before these corrections are added, ,thecomputed values of the doppler and range observables have accuracies of m/s and 0.1 m, respectively. The parameters whose values may be estimated by the DPODP are:
Injection parameters. Rectangular components of the spacecraft position and velocity vectors at the injection epoch.
Reference parameters. Parameters that affect the relative position and velocity of the sun, planets, and the moon:
AE= the number of kilometers per astronomical unit (AU). converts the precomputed heliocentric ephemerides of eight planets and the earth-moon barycenter from astronomical units to kilometers.

RE = scaling factor for lunar ephemeris, km/fictitious earth radius. This factor converts the precomputed geocentric lunar ephemeris from fictitious earth radii to kilometers.

(11)Speed of light. An adopted constant which defines the light-second as the basic length unit; it is not normally included in the solution vector.
(12) Constant bias for range observables.

= osculating orbital elements for the precomputed ephemeris of a planet, earth-moon barycenter, or the moon. The estimated correction ~ 3 3 is used to differentiallycorrect position and velocity obtained from the precomputed ephemeris.
p E , p M = gravitational constants for the earth and moon, km3/s2. These parameters affect the location of the earthmoon barycenter.
(3) Gravitationalconstants. The constant p i is the gravitational constant for body i, such as the sun, a planet, or the moon. (Note that p E and pM are also listed under reference parameters.)
(4)Harmonic coefficients. The harmonic coefficients J,, C,,,, S,, along with the gravitational constant p , describe the gravitational field of a planet or the moon.

(13) Spacecraft transmitter frequency for one-way doppler.
(14) Biases affecting observed angles.
+ + (15) Relativity parameter y. This parameter will be added to the program. It is equal to (1 0)/(2 W) where w is the coupling constant of the scalar field, a free parameter of the Brans-Dicketheory of gravitation.
Given the a priori estimate of the parameter vector q, the program integrates the spacecraft acceleration using the second-sum numerical integration method to give position and velocity at any desired time. Using the spacecraft ephemeris along with the precomputed ephemerides for the other bodies within the solar system, and the parameter vector q, the program computes values for each observed quantity (normally doppler, range, or angles) and forms the obseroed minus computed (0- C) residuals.

(5) Parameters affecting the acceleration of the spacecraft due to solar radiation pressure.
(6) Coefficients of quadratic for small acceleration acting along each spacecraft axis. These quadratics are used to represent gas leaks and small forces arisingfrom operation of the attitude controlsystem.
(7) Parameters affecting spacecraft motor burns.
(8) Parameters affecting the transformation from universal time to ephemeris time.
(9) Coefficients of quadratics which represent the departure of atomic time at each tracking station from broadcast UTC time.
(10) Station parameters. (1) Radius, (2) latitude, and (3)longitude or (1)distancefrom spin axis, (2)height above equator, and (3) longitude for each tracking station and a landed spacecraft on a planet or the moon. For a tracking ship: (1)spherical coordinates at an epoch, (27 velocity, and (3)azimuth.

In addition to integrating the acceleration of the spacecraft to obtain the spacecraft ephemeris, the program integrates the partial derivative of the spacecraft acceleration with respect to (wrt) the parameter vector q using the second-sum numerical integration procedure to give
the partial derivative of the spacecraft state vector X
(position and velocity components) wrt the parameter vector q, aX/aq. Using aX/aq, the program computes the partial derivative of each computed observable quantity
. Given the 0 - C residuals, az/aq, and the
weights applied to each residual along with the a priori parameter vector and its covariance matrix, the program
+ computes the differential correction Aq to the parameter
vector. Starting with q Aq, the program computes a new spacecraft ephemeris, residuals, and partial derivatives and obtains a second differential correction Aq. This process is repeated until convergence is obtained and the sum of weighted squares of residual errors between observed and computed quantities is minimized.
The DPQDP formulation was heavily influencedby the general theory of relativity. Section I1 gives the equations from general relativity, which are the basis of the DPODP

T

3 7

formulation, and also the principal relativistic equations contained in the formulation. The derivations of three of these equations are given in Appendixes A, B, and C.

The time transformations used throughout the program and the formulation for computing the relative position, velocity, acceleration, and jerk of any two celestial bodies (sun, moon, or planets) are described in Sections I11 and IV, respectively. The equations for the acceleration of the spacecraft relative to the center of integration (any planet, the sun, or the moon) are given in Section V.

The first step in the computation of all observable quantities is the light time solution, which is described in Section VI. The formulation for computing the geocentric inertial position and velocity of a tracking station is presented in Section VII. The computation of doppler, range, and angular observables is described in Sections VIII-X.

A forthcoming change to the formulation will be to compute doppler observables from differenced range observables divided by the count time, with partial derivatives of the doppler observables with respect to estimated parameters obtained from differencedrange partial derivatives. The formulation necessary to implement this change is given in Section XI.

Corrections to the observables due to antenna motion, the troposphere, and the ionosphere are described in Section XII. The variational equations for the spacecraft trajectory and the partial derivatives of the observables with respect to the estimated parameters are described in Sections XI11 and XIV.

In the original version of the DPODP, the parameter

estimate was obtained from the normal equations, which

are documented in Section XV. In the latest version of

the program, this formulation has been replaced by the

square root form of the normal equations, which is de-

scribed in Section XVI. The square root formulation is

theoretically equivalent to the normal equations but is

numerically superior. superior numerical techniques

of the square root fo

applied to

linear least squares problem by R. nson and C.

Lawsod (Ref. 1).

of the symmetrical metric tensor gpg:
g2l g22 g23 g24 gP, =
g31 g3Z g33 g34
The subscripts 1,2,3, and 4 correspond to the space-time coordinates xl, x2, x3, and x4, which are associated with a particular space-time frame of reference. Usually the frame of reference is nonrotating and centered at the barycenter of the system of masses considered. Then xl, 9, and x3 are position coordinates and x4 = ct,where c is the speed of light and t is coordinate time, a uniform system of time which exists throughout the frame of reference; it is synonymous with ephemeris time. The components of the metric tensor g,, are obtained from a solution of Einstein’s field equations. The solution depends upon the distribution of matter and the system of coordinates selected.
The invariant interval ds between two points with differences in their space and time coordinates of dx’, dx2,
dx3,and dx4is given by

where, using the Einstein convention, the repeated indices p and q are summed over the integers 1through 4.

In an inhitesimally small region surrounding an observer, the components of the metric tensor are constant
and the expression for the interval ds can be transformed
to the special relativity form

dS2 = c2dT2- dX2- dY2- dZ2

(3)

where is proper time recorded on the observer’s atomic clock and X, Y, and Z are components of observed position referred to the observer’s local frame of reference. Since the atomic clock is fixed relative to the observer, the interval ds corresponding to an observed interval of proper time dr is

1Jet Propulsion Laboratory, Computation and Analysis Section.

or

d7 = -ds C

(5)

ence Eq. (2) relates an observed interval of proper time d7 to the changes in the space and time coordinates of the clock.

The space-time coordinates are used to represent the motion of particles, bodies, and light. The coordinates have no physical significance and are not observable. Furthermore, the choice of coordinates is completely arbitrary. The solution of &e field equations for g,, varies with the coordinates selected in such a manner that the value of an observed interval of proper time computed from Eqs. (2) and (5) is independent of the coordinates selected to represent the motion of the atomic clock.

The field equations have been solved exactly for the case of a massless particle moving under the influence of a single spherically symmetric massive body located at the origin of a nonrotating system of coordinates. The solution of this 1-body problem was first obtained by Schwarzschild and is given in Ref. 2, p. 85, Eq. (38.8). A simple transformation in the radial coordinate gives the “1-body” solution in isotropic spherical coordinates (Ref. 2, p. 93,Eq. 43.2):

Expanding and retaining all terms of order 1/c2 gives

(1 g) + + + -

(dr2 r2de2 r2sin2ea+*) (7)

In isotropic rectangular coordinates,

( +1 -

-&--)4

+ + (dr’ r2de2 r2sin2

(6) where

p = gravitational constant of nonrotating spherically symmetric massive body located at origin of nonrotating frame of reference, kms/s2. The constant p is equal to the product of the uni-
versal gravitational constant C and the rest
mass m of the body.

c = speed of light
r, (6, e = spherical coordinates. he spherical and rec-
tangular coordinates of a particle P are shown
in Fig. 1.

t = coordinate time

(1 2) + + + -

(dx2 dy2 dz2)

where

+ + r = [x2 y2 z2]%

(9)

Fock (Ref. 3) and Yilmaz (Ref. 4)differ from Einstein and
obtain metrics that differ from Eq. (6). However, when expanded, their metrics are identical with Eq. (8) to order 1/c2. The small departures of the components of the metric tensor in Eq. (8) from the unity values of special relativity in Eq. (3) represent the ‘ccurvature” of space-time due to the mass of the central body.

The trajectory of a massless particle in the gravitational field of a massive body is a geodesic curve which extremizes the integral of ds between two points:

7

n order to obtain the equations of motion with coordinate time t as independent variable, Eq. (10) is written as

s8 Ldt=O

(11)

where the Lagrangian L is given by

L = -ds dt

t 12)

and L2 is obtained from the expression for ds2by replac-
ing differentials of the space coordinates by derivatives of the coordinates with respect to t multiplied by dt. The Lagrangian L may be obtained by expanding the square
root of L2in powers of 1/c2. Given L, the equations of
motion that extremize the integral (11) are the EulerLagrange equations:

(") -a - --aL-- 0
dt a2 ax

x+y,z

The derivatives E (aL/ax) and L (aL/%) are obtained by direct differentiation of L2.For the usual situation where only the 1/c2 terms of the relativistic perturbative acceleration are required,

r: LI: L t
-=-Al-
- L L2 c2

(19)

where L2has been replaced by its leading term c2and L i is obtained by differentiating a simplified expression for L2 containing terms to l/co only. Computation of the equations of motion from Eqs. (18) and (19) is simpler
than taking the square root of L2and using Eq. (13).

From Eqs. (8), (12), (18), and (19), the relativistic perturbative acceleration of a massless particle moving in the gravitational field of one body is given by

where
(I4)
A simpler procedure for obtaining the equations of
motion directly from derivatives of L2 is developed as
follows. The Euler-Lagrange equations (Eq. 13) are un-
changed by multiplying both terms by L:

where the dots indicate differentiation with respect to coordinate time t. The position, velocity, and acceleration vectors are given by
..

z

.z .

Differentiating L (aL/Z&)with respect to t gives

where

Ee = - -dL dt
of motion are obtained by substituting
. (16) into Eq. (15):

B = magnitudeof i
An approximate solution to the field equations for the case of a massless particle moving in the gravitational field of n massive bodies was first obtained by J. Droste in 1916 (Ref. 5). In that same year, W. deSitter extended the work of Droste to account for the mass of the body whose motion is desired (Ref. 6). However, he made a theoretical error in the calculation of one of his terms, which was corrected by Eddington and Clark in
ef. 7). The components of the Droste/deSitter/ Eddington and Clark metric are given by (Ref. 7, Eqs. 3.1, 3.2, and 3.6)

gll = gzz = 9 3 3 = -

(22)

i#i

g34 = g43 = -c43

- Pi%
Ti i

Since terms of order greater than 1/c2 will not be retained in the expression for the acceleration of body i, the accel-
q. (30) may be evaluated from Newtonian theory:

g44 = 1 - -2 C2

- Pi rii

i#i

3#i

k#j

The summation over k # j includes body i. The four space-time coordinates associated with the n-body metric are
x1 = xi
x2 = yi
x3 = zi
x4 = ct

i#i

where the indices j and k refer to the n bodies and k
includes body i whose motion is desired. Also,

pi = gravitational constant for body j

= Gmi, where G is the universal gravitational
constant and mi is the rest mass of body j.

5 Y 7 2;
.. ixa-,, tyj*.,,ix;-- rectangular components of position, velocity,
and acceleration (4= dx/dt, etc.) relative to a
nonrotating frame of reference centered at the barycenter of the system of n bodies. The position, velocity, and acceleration vectors are given by Eq. (21); they and their components
are identified by the subscript i, j , or k.

rii = coordinate distance between bodies i and i

+ + - = [(xi - xj)2 (yi

(xi -

+ + 3 = square of velocity = 2 c” i2

The second partial derivative of rii with respect to t in Eq. (27) is obtained holding ri k e d :

Hence, from Eq. (2) and Eqs. (22-27), the expression for ds2 is
+ + + ds* = c2944 at2 gll (dxt dyB dxB) + + + 2cgl4 dXi dt 2cg24 dyi dt 2cg34 dZi dt

(33) Dividing by dt2gives the expression for L2:

+ + + L2 = c2g*4 gll (?H f/? if)

+ + + 2cg1,2i 2cg2,$i 2cg34ii

(34)

The equations of motion for body i are obtained from Eqs. (18) and (19) with x and 2 replaced by xi and 3Ei.
owever, in carrying out the required differentiations of Eq. (34), the contribution to the field from the mass of body i must be held fixed.

Specifkally, the Newtonian potential at each perturb-
ing body j in the fifth term of g44(Eq. 27) must be con-
sidered to be a function of time only. The potential at
body j due to body i, prc/rjkwith k set equal to i, must not
be differentiated with respect to xi, yi, and xi.

The last term of g44is evaluated with
contains the acceleration of body i given by Eq. (31).The
acceleration of body j , “r;., must also be considered to be a
function of coordinate time t only. The contribution fram
the mass of body i must not be differentiated with sespect

7

to xi, yi, and zi.(I am indebted to two relativists at JPL, Dr. Frank B. Estabrook and Dr. Hugo Wahlquist, for pointing out these special conditions.)
The details of the derivation of the expression for the acceleration of body i are given in Appendix A. The h a 1expression for the acceleration of body i relative to the barycenter of the system of n bodies with rectangular components referred to a nonrotating coordinate system is given by

+ - 2c12 (rj - ri)e Y j

iti

where "r; is computed from Eq. (31) and the summation over k # i in Eqs. (31) and (35) includes body i. Note that the first term of Eq. (35)is the Newtonian acceleration of body d. The effect of the mass of body i on its own accel-
eration is contained in its contribution to the Newtonian
potential at each perturbing body i (term 3) and in its
contribution to the Newtonian acceleration of each body j (terms 8 and 10).
A method for obtaining the motion of a system of n heavy bodies directly from the field equations, without recourse to additional equations such as those of a geodesic, was obtained for the first time by Einstein, Infeld, and Hoffman in 1938 (Ref. 8). The method, referred to as
approximation method, was subsequently perm the mathematical viewpoint in Refs. 9 and 10. The EIH method is illustrated in Ref. 8 by obtaining the equations of motion for two bodies. The equations for the motion of a system of n bodies were obtained from a later (1960) work of Infeld and P to Baiahski (Ref. 12), the E

in principle, the only tool in the problem of the motion of heavy bodies in the general theory of relativity.
After deriving the n-body relativistic equations of motion, Infeld and Plebahki noticed that these equations could be put into the form of a Lagrangian L with the equations of motion following from the Euler-Lagrange equations :
where i refers to the body whose motion is desired. The Infeld Lagrangian is given in ef. 11, p. 112, Eq. (3.3.37) or p. 128, Eq. (4.2.25). The same Lagrangian may be found on p. 149 of Ref. 13.
In the notation used for the de Sitter n-body metric
(except that the index i, as well as i and k, now refers to
the n bodies), the Lagrangian is given by

where
5 = position vector with components 5' = x, t2= y, and #3 = x. A repeated superscript implies a summation over
the values 1,2, and 3.
Carrying out the partial derivatives in term five gives two terms, one of which combines with term four. Also, the last term contains three identical subterms; two of them may be deleted and the coefficientof the remaining term multiplied by three. With these changes, the expression for L becomes

L=1z

+ piif - 81c2

+ -3
pi ($)2 4c2

i

i

i

Pi& -(BH
Tij j#i

+ 9)- -7 4c2

- Pi& ri i j
Tij

i j#i

This equation, expressed in a slightly different form, may be found in Ref. 14,p. 372.
The equations of motion (Eq. 36) involve the partial derivatives of the Lagrangian L with respect to the position and velocity coordinates of the particular body whose motion is desired. Hence, Eq. (38) will be rewritten with the
index i referring to the particular body (body i) whose motion is desired and the indices i and k referring to the n
other bodies (perturbing bodies). For the single-summation terms of Eq. (38), the transformation consists simply of
removal of the i summation. Since all double-summation terms are unchanged by interchanging the indices i and i,
they are transformed by removing the i summation and multiplying by two. Terms of the triple summation with the index i or k referring to the specific body i are transformed to the original triple-summation term multiplied by two
with the i summation removed. Terms with the index i referring to the specifk body i are transformed to
After transformation, the gravitational constant pi may be deleted from each term. Thus, with i now referring to
the specific body i whose motion is desired, and i and k referring to the other bodies, the Lagrangian L is given by

The expression for the acceleration of body i is obtained by applying the Euler-Lagrange equation, Eq. (36), to Eq. (39) for E. The details are given in Appendix A. The resulting n-body equations of motion, derived from the Infeld
agrangian, are identical to Eqs. (35)derived from the Droste/de Sitter/Eddington and Clark metric.

The equations of motion for a massless particle moving in the field of one massive body may be obtained by simplifying the n-body equations of motion (Eq. 35). With one perturbing body, its position, velocity, and acceleration are zero. Also, with the mass of body i, whose motion is desired, set equal to zero, the Newtonian potential at
the perturbing body i is zero. With these simplifications,
the n-body acceleration (Eq. 35) reduces exactly to the acceleration (Eq. 20) obtained from the 1-body isotropic metric (Eq. 8). Of course, the components of the n-body metric tensor (Eqs. 22-27) reduce to those of the 1-body isotropic metric (Eq. 8). Some of the relativity terms of the DPODP formulation are derived from the 1-body metric, whereas others are obtained from the n-body metric. The 1-body isotropic metric was selected since it is a special case of the n-body de Sitter metric, or equivalently the n-body Infeld Lagrangian. The choice of coordinates in general re€ativity is arbitrary, but the same coordinates must be used in all computations.
The general theory of relativity has been generalized by C. Brans and R. H. Dicke (Ref. 15).Supposedly, their theory is more in accord with Mach‘s principle than the general theory of relativity. According to Mach‘s principle, the inertial forces experienced in an accelerated laboratory are gravitational, having their origin in the distant matter of the universe accelerated relative to the laboratory. Brans and Dicke (Ref. 15) state that “locally observed inertial reactions depend upon the mass distribution of the universe about the point of observation and consequently the quantitative aspects of locally observed physical laws (as expressed in the physical “constants”) are position dependent.”
The Eotvos experimentwas recently repeated at Princeton University by Dicke et al. and showed that all bodies fall with the same acceleration to an accuracy of 1part in loll. Braqs and Dicke concluded from this result that the only physical “constant” of their theory (Ref. 15) whose value needs to vary with position in the universe
is the universal constant of gravitation G (see Ref. 16, p. 7-8). In order to obtain this variation, they added a
scalar gravitational field to the tensor field of general
relativity. The gravitational constant G varies with the
strength of the scalar field. owever, it can be considered constant in the small region of the universe known as the solar system.
icke scalar-tensor theory of gravity, the two particles of matter is due partly
to the tensor field and partly to the scalar field.

tion of the gravitational attraction due to the scalar field is given by
1 4+20
where w is the coupling constant of the scalar field, a free parameter of their theory. It is shown below that 01-6. For w = 6, 1/16 of the force of gravity is derived from the scalar field and 15/16 is due to the tensor field.

Because of the expansion of the universe, the strength of the scalar field (if it exists) is changing, and G should decrease by roughly 1-3 parts in 10l1 per year (Ref. 16, p. 107).The variation in G is inconsistent with the strong principle of equivalence, which is one of the postulates of the general theory of relativity. According to this principle, in a freely falling, nonrotating laboratory, the form of the locally determined laws of physics and the values of the dimensionless physical constants appearing therein do not vary with the position of the laboratory in space and time.

Nutku (Ref. 17) has obtained the post-Newtonian equations of hydrodynamics for a nonviscous fluid in the scalar-tensor theory of Brans and Dicke. From these equations, Estabrook (Ref. 18) has obtained the n-body metric tensor, the n-body Lagrangian, and the resulting n-body equations of motion. These equations contain exactly the same terms as the corresponding equations of general relativity; however, the coefficientsof these terms, which were constant in general relativity, are functions of the free parameter a,the couplingconstant of the scalar field. The value of w must be positive, and, as the value of w approaches infinity, the equations of the Brans-Dicke theory revert to the corresponding equations of general relativity.

From Ref. 15, the relativistic perihelion rotation rate e’
of a planetary orbit is

[ ] e* =

4+3
__+ _

W_3o_

x [value from general relativity]

(40)

ercury, the predicted value from general relativity

prad (43 arc-seconds)/century, which agrees with

the observed value.

the solar oblateness re-

f. 19) would produce an

of 16 p a d (3.4 arc sec-

onds)/century, leaving only 192 prad (39.6 arc se

century to be attributed to relativity. The Bran

theory will produce this perihelion rotation rate for a value

7

roximatelgr equal to 6. Since the true solar oblate-

ness lies somewhe

en zero and approximately the

value observed by

6 (approximately).

The basic equations of the given below with coefficients e the parameter Y. where

icke theory are as functions of

1+0
y=K
As o increases from zero to infinity, y increases from 1/2 to unity (its general relativity value).

The DPODP will be modified so that the value of the parameter y may be estimated. The constant coefficients of all existing DPODP relativity terms, derived from the general theory of relativity, will be changed to the functions of y specified in this report. Also, the partial derivatives of the observables with respect to y specified in Section XIV will be added to the program. This will enable the value of y to be obtained by fitting the theory to observation. Given y, the corresponding value of o is given by (see Eq. 41)

+ g34 =g43 = 7 2 2y &

(47)

Tij

j#i

g44 = 1 --2 CZ 3#i

+ --1 2 y

- pjdf

c4

Ti j

j#i

+-2 c4

i#i

k#i

3#i
where a2rij/at2is given by Eqs. (30) and (31). The coeffi-
+ + cients 2y, 2 2y, and 1 2y appearing in Eqs. (43-48) + + + + above appear as ( 2 20)/(2 0), (6 40)/(2 0), and + + (4 30)/(2 0), respectively, in Ref. 18. With y equal to
unity (its general relativity value), the equations above are identical to Eqs. (22-27), derived from general relativity.

It will be S t ~ nthat the relativity terms of the DPODP formulation which are functions of y vary linearly with y. Also, it will be seen that the only components of the 1-body isotropic metric tensor that are functions of 0 are
+ + gll = gaZ= gSs.The departure of this coefficient from
unity is proportional to the function (1 w)/(2 o). is the source of the change of variable to y (Eq. 41). parameter y was first used at JPL by Anderson (
ef. 18),the components of the n-body here as functions of y ) are

If the mass of body i is reduced to zero and the number of perturbing bodies is reduced to one, the n-body metric tensor reduces to the following diagonal 1-body metric:

g11 = gzz = g33 = -

+ - g44 = 1- 2 c2PT

2P2
C4T2

using the notation listed after q. (6).In spherical coo nates, the expression for the interval is

g11 = gzz = g83 = -
i+i

+ 7 2 2 y
g14 = g41 =
T ij i#i

(45) Setting y equal to unity gives the general relativity sion (Eq. 7).

L

7

ef. 18) also gives an expression for the n-body Lagrangian L in the Brans-Dicke theory. Changing the coefficients to functions of y (using Eq. 42)and also changing the form of his equation slightly gives

i

i

i j#i

i j#i

i j#i k#j,i
The corresponding equation from the general theory of relativity is Eq. (38);for y = 1, the two expressions are identical.
TransformingEq. (52)so that the index i refers to the particular body i whose motion is desired and the indicesj and k
refer to the n other bodies gives

j#+
The corresponding equation from general relativity is Eq. (39). In Appendix A, the n-body equations of motion are derived from the n-body metric tensor (Eqs. 43-48) and from the agrangian (Eq. 53). The result (also given in Ref. 18) is

q. (31) and the summation over k+ j in qs. (31) and (54)includes body i. With y = 1, Eq. (54) ), derived from general relativity. SimplrfylngEq. (54)to the case of a massless particle moving in
the field of one massive body gives the following relativistic perturbative acceleration:
5 @] + + + [z (1 y) - r 2(1 y)( 8 4 ) i

or y = 1, this equation is identical to Eq. (ZO),derived from general relativity.

7

11

The ephemerides of the moon, sun, and planets could be obtained by a simultaneous numerical integration using Eq. (54). Using these precomputed n-body ephemer-
P could generate the spacecraft ephemeris using Eq. (54) to calculate the point-mass gravitational accelerations of the spacecraft and the body which is the center of integration.
However, a number of the relativistic perturbative acceleration terms (the 1/c2terms) would be insignificant. For instance, for the heliocentric ephemeris of a planet other than the earth, only the perturbative acceleration of the planet due to the mass of the sun, computed from Eq. ( 5 3 , need be considered. Equation (54) is required only when a planet or moon is nearby; that is, when one is computing the acceleration of the earth, the moon, or the spacecraft when it is near the earth and moon or a planet.
The relativisticperturbative acceleration terms required are specified in Sections IV and V, which describe the precomputed n-body ephemerides and the spacecraft ephemeris. A more detailed discussion of the required terms and their effect on the various ephemerides may be found in Refs. 21 and 22.
A brief summary of the effect of general relativity on the various ephemerides is as follows. For the orbit of a planet, the mean distance a is about 1.5km less than the Newtonian value. Periodic variations in position are proportional to the eccentricity and range from about 0.2 km for Venus and Neptune to about 6 km for Mercury and Pluto. Periodic variations in velocity are proportional to the product of the mean motion and the eccentricity. The largest variation is 4 mm/s for Mercury; the variations for the remaining planets are less than 0.25 mm/s, which is the value for Mars.

position and velocity are less than 10m and m/s. differential solar relativistic acceleration produces a secular variation in the moon’s perigee of 10 prad (2 arc seconds)/century.
For the Pioneer VI, Mariner IV, and Mariner V space-
craft, the periodic variations in position and velocity are in the ranges of 3to 5km and 0.7 to 1.1mm/s. The major terms of these variations have periods equal to the orbital period and one-third of the orbital period. For an earth orbiter with a perigee of 7000 km and an eccentricity of 0.2,the advance of perigee is 39 prad (8arc seconds)/year.

The ephemerides for the planets, the earth-moon barycenter, the moon, and the spacecraft give the position coordinates (and their derivatives with respect to coordinate time) as a function of coordinate time t. For a given proper time T at some point on earth, the time transformation t - T is thus required to interpolate the ephemerides.

The time transformation may be derived from the expressionfor the interval which relates an observedinterval of proper time T to the changes in the space and time coordinates of the atomic clock. Substituting the components of the n-body metric tensor (Eqs. 43 to 48) into Eq. (33) for the interval and retaining terms to order ( l / c ) O gives

$) + + ds2 = ( 1 - c2dt2- (d? dy2 dx2)

(56)

where x, y, and z may be interpreted as heliocentric coor-
+ dinates of the atomic clock, although strictly speaking
they are referred to the barycenter, and is the Newtonian potential at the clock given by

The primary terms of the periodic Variations in position and velocity have periods equal to the orbital period and one-third the orbital period. The only significant secular variation in the orbital elements is the advance of perihelion, which amounts to the well known value of 208 prad (43 arc seconds)/century for Mercury.
For the orbit of the moon relative to the earth, the mean distance is about 8 m less than the Newtonian value (using the same values for the gravitational constants of the earth and moon). Maximum values of the periodic variations in

(57)
j
where ri is the coordinate distance from the clock to
body i. Expressing the second term of Eq. (56) as the square of the heliocentric velocity of the clock i multi-
plied by dt2 and using Eq. (5) gives

H

dt

(58)

7%

T

7

Since l/c4 terms are ignored,

-d.rwl---dt

(59)

Equation (59) relates an interval of proper time (obtained from the observer’s atomic clock) to the corresponding interval of coordinate time t, the Newtonian potential at the clock, and the heliocentric velocity of
the clock.
Coordinate time t may be considered to be a uniform system of time that pervades the nonrotating heliocentric
+ frame of reference. For a fixed atomic clock at infinite
distance from the sun, = i = 0 and dr = dt. That is, the atomic clock runs at the rate of a coordinate clock (a clock yielding coordinate t h e t). This condition and the length of the coordinate time second fixes the conversion factor (n cycles/second) used to convert cycles or ticks from the observer’s atomic clock to seconds of proper time 7. From Eq. (59), the rate of an atomic clock decreases as the Newtonian potential at the clock and the heliocentric velocity of the clock increase.

For a fixed atomic clock on earth, dr < dt, and proper
time falls behind coordinate time t. simple expedient of choosing a different number of cycles
from the observer’s atomic clock per second of proper time, the latter can be made to agree on the average with coordinate time t. Equation (59) may be written as

Note that dr is obtained as dN cycles from the observer’s atomic clock divided by the conversion factor n cycles/s. If the conversion factor is changed to n*, where

ct ( ;:) n*=n I-----

(622)

and proper time is obtained as dN/n* and denoted by dr*, then Eq. (61) may be written as

-- -d=r* 1 - -+-- F 1 b 2 - 8 2-

dt

C2

2 c2

(63)

Thus proper time r* obtained from the observer’s atomic clock using the conversion factor n* cycles/s will, on the
+ average, agree with coordinate time t. Periodic variations
in r* from t are due to variations in and 2 from their
average values.

Coordinate time t is the independent variable for the equations of motion and is commonly referred to as ephemeris time ET. The A1 atomic time scale on earth is based upon oscillations of a cesium atomic clock. The adopted length of the A1 second is fcesium = 9,192,631,770 cycles of cesium, which is the current experimentally determined average length of the ET second.2 In the
+ DPODP, the true average length of the ET second is
represented by fcesium L\fcesium cycles of cesium. The quantity Afcesium iS a solve-for parameter; its value is probably no more than two or three cycles. The quantity
fcesillm f Afcesium is the length of the T* second and hence

-d=A1 dr*

+ f c e s i u m

Afcesium = 1 + - Afcesium

fcesium

fcesium

where

+ 6 = time average of

The quantity dAl/dET is the product of this equation and Eq. (63), which is given to sufficient accuracy by

3 = time average of $2

Ignoring l/c4 terms, this may be written as

a7

- -- 1 - -+-- F -- 1 ; 2 - ; 2

C2

2 c2

(61)

+ where is the Newtonian potential at a particular A1
atomic clock and is the heliocentric velocity of the atomic clock.
2Interpolationof the lunar ephemeris with an observed longitude of the moon gives the value of the independent variable, ET. The value of dfCeslUgimven above was determined by counting cycles of a cesium atomic clock between two obsewations of the moon separated by 10 years and dividing the observed number of cycles by the “observed” ET interval.

+ Appendix B, equations are generated for the
of and 2 from their average values, and E
is integrated to give an expression for T - Al. The ini-

tial conditions were evaluated by considering the method

by which the A1 atomic time scale was set up. The master

A

was set equal to UT2s on January 1, 1958,

oh

A1 clock. at other locations are synchro-

nized with the masten clock by means of radio signals,

accounting for the propagation delay, or by means of a

traveling clock, or by other methods. Hence, the average

offset between A1 time and ET is the same for all A1

clocks. The resulting expression for ET - A1 (in units of

seconds) is

PQDP; its current nominal value IS zero
to January 1,1958, Oh
M = mean anomaly of heliocentric orbit
of earth-moon barycenter E = eccentric anomaly of heliocentric
orbit of earth-moon barycenter L = geometric mean longitude of the
sun, referred to mean equinox and ecliptic of date

- ( t - 252,460,800) Afcesium

+ 1.658 X

foesium
sin E

+ + 0.317679 X usin (UT A)

+ + 5.341X 10-l' u sin (UT h - M )

+ + 1.01 x 10-13u sin (UT A - 2 ~ )

D = C - L = mean elongation of the
moon from the sun, where
a = mean longitude of the moon,
measured in the ecliptic from the mean equinox of dateto the mean ascending node of the lunar orbit, and then along the orbit

+ + - 1.3640 X 1O-I' u sin (UT h 2L)

- 2.27 X

+ + u sin (UT h 2L -I-M)

+ 1.672 X 10-6sinD

u = distance of atomic clock from earth's spin axis, km
h = east longitude of atomic clock

+ + 1.38 x 10-13u sin (UT A - D) (a)

UT = universal time, hours past midnight, converted to radians. I t is

where

AT195=8 ET - UT2 on January 1,1958,
Oh Om 0" UT2 minus the periodic

computed from
[-]UT1
UT = %r 86,400 decimal part

(66)

terms of Eq. (65) evaluated at this epoch using u and of the master A1 clock. The master A1 clock was set equal to UT2 on this date. The parameter AT1958 may be estimated by the DPODP
fcesfum = 9,192,631,770cycles of cesium

where UT1 = seconds of UT14time past January 1,1950,
OhUT1.The angles N,L, and D in radians are given by

+ M = 6.248291 1.99096871 X lo-?t

(67)

+ L = 4.888339 1.99106383X lo-?t

(68)

+ D = 2.518410 2.462600818 X

(69)

atomic clock per second of A1 time
(definition).This adopted length of
the A1 second is the current

To a sufficient degree of accuracy, the eccentric anomaly E is given by

experimentally determined average

E z M S esinM

(70)

+ fcesium Affeesium

length of the ET second cycles of cesium atomic clock per

where

ephemeris second. The parameter Afces may be estimated by the

e = eccentricity of heliocentric orbit of earth-moon barycenter = 0.01672

3The UT2 time scale is described in Section 111.

4The UT1 time scale is described in Section 111.

7

Term 4 of Eq. (65)is the sum of two terms with coefficients of 0.318549 and -0.000870. The larger term arises
from the dairy variation in the heliocentric velocity of the atomic clock, while the smaller term accounts for the diurnal variation in potential. The expression for ET - A1 used in the current version of the DPO first three terms of Eq. (65) and the following term derived by J. D. Anderson (Ref. 20):
+ 2.03 X le6cos 4 sin (UT A)
where 4 is the latitude of the atomic clock. Anderson’s
term is the fourth term Of Eq*(”1 with the coefficient
+. of 0.318549 mentioned above and r8 set equal to 6372
km cos
Changing Anderson’s diurnal term to the fourth term of Eq. (65) and addition of the last six terms of Eq. (65) is required to implement the change to the current version of the program specified in Section XI, namely, the computation of doppler observables from differenced range observables divided by the count time. The contribution to “differenced-range” doppler from a term of ET - A1 is approximately equal to the second time derivative of the term multiplied by the spacecraft range. All terms affecting “differenced-range” doppler by more than 2 X 10-~m/s per astronomical unit of distance from the tracking station to the spacecraft were retained in Eq. (65). Terms of ET - A1 which could be derived from the I/c4 terms of dr/dt would be at least eight orders of magnitude smaller than the terms of Eq. (65). Their contribution to differenced-range doppler would be several orders of magnitude less than the criterion above. Hence, there is no requirement for l/c4 terms in the expression for ET -Al.
In order to compute doppler, range, and angular observables, the time for light to travel from the transmitting station on earth to the spacecraft, and from there to the receiving station on earth, must be computed. Thus, an equation is required which relates the position coordinates of two points to the coordinate time t for light to travel from one of the points to the other. This equation will be referred to as the light time equation. It will be derived from Eq. (51), the 1-body expression for the interval in
icke theory. Thus, the effects of the masses of the planets and the moon on the propagation time are neglected.

coordinates selected. This is also true for light with the additional condition that ds = 0. Thus, light moves along
a null geodesic. The equations of a geodesic are the Eul
equations which extremize the integral of ds between two
points. When Eq. (10) is written as Eqs. (11)and (121, the Euler-Lagrange Eq. (13) or (18) gives the secondorder differential equations for the three position coordinates with coordinate time t as independent variable. However, if proper time s is taken as the independent
are obtained for the three position coordinates and also for coordinate time t. The equation for the fourth coordinate is required in the derivation of
the light time equation. Eq. (10) may be expressed as
8/2ds =0
where
From Eq. (51),
dt
+, The Euler-Lagrange equations for q = r, 0, or t are
The equation for e is

A massless particle moves on a geodesic curve in the four-dimensional geometry of spap+time, which is determined by the distribution of matter and the system of

If coordinates are chosen so that a particle moves initially in the plane 0 = n/2, then de/& = 0 and Eq. (75) gives the result that dze/ds2= 0. s, in the %-body

problem, the motion of particles and of light is planar, ntegrating between limits of (r,.p) and (R, 0) and ignor-

and the equations may be simplified by setting

ing l/c4 terms gives

e =4 2

+, Since 2 is explicitly independent of t and first

integrals of Eq. (74) for q = t and .p are given by

a 2 / a (dt/ds) = constant, and

(&/ds) = constant.

= +cos-1 E[ ,

(1 + Y) P

c2 - (1+ Y I P

r

CZR

Differentiating Eq. (73) accordingly with e = r/2 and
making use of the fact that Q = 1gives

-dd=st

constant

1 - - +c22P-r

2P2 c4r2

and

where the plus sign applies for increasing r and the minus
+ sign applies for decreasing r. When T approaches a in
Eq. (82), the angle will approach one of the two asymptotic values:
The angle between the incoming and outgoing asymptotes is thus

-d+-- constant

Dividing Eq. (77) by Eq. (78) and ignoring l/c4 terms gives
Setting ds = 0 and e = z/2 in Eq. (51)gives

For general relativity, y = 1and A+ = 4p/c2R, which has a maximum value of 8.48 prad (1.75 arc seconds) when R is set equal to the radius of the sun, 695,500 km. Figure 2 shows the curved path of a photon passing the sun S. Light is moving in the positive y direction and the point of closest approach occurs at x = R, y =0. The polar coordinates (r, +) and rectangular coordinates (x, y) of two points on the light path are shown along with the straight line path (of length r12)joining these two points. The y intercept was obtained from Eq. (82) by setting cos+ equal to zero; the x intercept of the asymptotes follows from the y intercept and the angle of the asymptote.

Substituting dt from Eq. (79) into Eq. (80), setting dr/d+ = 0 when r = R (the minimum value of r on the light path), and ignoring l/c4 terms gives

Given that light moves in a plane along the curved path (Eq. 82), the light time equation may be derived by two alternative methods. The first method consists of
substituting d+ from Eq. (79) into Eq. (
tion between dr and dt. Integration equation. The second method is a direct integration of the differential of coordinate distance divided by the coordinate speed of light ocalong the light path between two points. For planar motion, the space coordinates of a photon change by dr and d+ in coordinate time dt.
ence, an expression for the square of the coordinate velocity ucis

:)4 =

+ geodesic path is the bending, 2(1 y ) p/c2R. If the nom-
inal length of the light path is 1, the difference in length
between the curved and straight line paths will satisfy
the inequality

which is of order l/c4.Thus, the direct effect of the bending of light on the light time is an additional term of order 1 p .

/

The indirect effect of the bending of light is to alter the value of T used in Eq. (86) by a term of order 1/c2.

X The coordinate velocity divided by c along the curved

geodesic path will differ from the corresponding value

along the straight line path by a term of order 1/c4.Thus,

the indirect effect of the bending of light upon the light

time is the same order as the direct effect, namely l/c5.

Since all terms of order l/c5 and greater are ignored in the light time equation, it is obtained by integrating the differential of coordinate distance divided by B, along the straight line path joining two points.

Both of the above-mentioned derivations of the light time equation are given in Appendix C. In either case, the resulting light time equation is

Dividing Eq. (80) by dt2,substituting Eq. (S)an,d igBor-
ing l/c4 terms gives
The coordinate speed of light v, decreases slightly as the photon approaches the sun. The Newtonian light time between two points is the straight-line Coordinate distance between them, divided by the speed of light c. However, since v, < c y the actual light time will be longer; the additional time is of order l/c3.
The direct effect of the bending of light upon the light time is the increase in the path length divided by the nominal velocity c. The maximum angle between the straight line path between two points and the curved

where light travels from point i at coordinate time (ephem-
eris time) ti to point i at coordinate time ti, and
e ~ i=j 11 ry (ti)- (ti)I[
= 11 ~7(ti)11 ri = II r;(ti) I1
e 6 (ti), (ti)= heliocentric position vectors of point i at transmission time ti (ET) and point i at reception time ti (ET), respectively, with rectangular components referred to a nonrotating frame of reference
p = gravitational constant of sun, km3/s2
is form for the relativistic perturbation of the light was derived independ ulsion Laboratory by

er, it had been f. 24, Eq. 6-105). Appendix C.

ed a year earlier by two alternative forms, see

As discussed in detail in Section X, the relativistic correction to the light time becomes as large as 36 km/c when the spacecraft approaches superior conjunction and the minimum distance from the light path to the surface of the sun becomes very small. This effect is seen directly in range observables and is the only really large effect of general relativity on earth-based tracking data.

and its heliocentric position and velocity at t2.Similarly, solution of the light time equation for the up leg of the light path gives the transmission time tl and the heliocentric position and velocity of the tracking station at tl.
For the range observable, Eq. (65) is used to convert the round-trip light time from an accurate value of the coordinate time interval (t3- t l ) to the observed proper time interval 7 3 - rl. The doppler observable is

The most accurate observables computed by the DPODP and observed by the Deep Space Network are round-trip range and two-way doppler data. The remainder of this section wiIl summarize briefly the procedure for computation of these observables from a relativistic point of view.
The observables are defined as follows,A signal is transmitted from the tracking station at coordinate time tl (proper time rl),received and retransmitted by the spacecraft at coordinate time t z , and received by the tracking station at coordinate time t3 (proper time r3).The range observable is the elapsed round-trip proper time r3- rl. For purposes of this discussion, two-way doppler may be considered to be the ratio of the received frequency f R to the transmitted frequency f T . In actuality, it is the average value of - 1 ( f R / f T ) over a period of time called the count time.
As previously mentioned, the precomputed ephemerides for the planets, the earth-moon barycenter, and the moon are obtained, in principle, by a simultaneous numerical integration using Eq. (54). Given the estimated values of the spacecraft injection conditions and other parameters, the spacecraft ephemeris is integrated numerically using Eq. (54) to compute the point mass gravitational accelerations, These ephemerides give the position coordinates and their derivatives with respect to coordinate time as a function of coordinate time t. Given the ephemerides, the first step in the computation of each observable quantity is the solution of the light time problem. Equation (65) is used to convert the reception time r3 for each observable to coordinate time (ephemeris time) t3,and the heliocentric position and velocity of the tracking station are computed at this epoch.
Solution of the light time equation (Eq. 88) for the down leg of the light path gives the spacecraft time tz

where dn cycles are transmitted in the interval of proper time dr, and received in the interval dr3. The ratio of received to transmitted frequency is computed from
_f R --
fT ($)3
The ratios dtl/dt2 and dt2/dt3are obtained by differentiation of the light time equations for the up and down legs, respectively, of the light path. The dr/dt ratio is evaluated at tl and t3 from Eq. (59).
All observable quantities are functions of intervals of the observer’s proper coordinates associated with his local space-time frame of reference. The range and two-way doppler observables are functions of intervals of proper time r only, namely 73-71 and dTl/d73, respectively. Thus, the computation of observables will always involve a transformation from the space and time coordinates of the frame of reference in which the motion of bodies and of light is represented mathematically to the observer’s proper coordinates.
Theoretically, the frame of reference and the coordinates selected are arbitrary. The relativistic terms in the equations of motion (Eq. 54), the light time equation (Eq. 88), and the transformation from coordinate time to proper time (Eq. 65) will vary with the frame of reference and system of coordinates selected. In general, the numerical values of the various constants, obtained by fitting the theory to observations, will also vary. the- numericaf values of the computed observables are independent of the frame of reference and system of coordinates selected.

ewe, UT is a function only of eM:

section describes the systems of time used in the P and gives the formulas for transforming between
these time scales.

+ 6 p = UT Ru(UT) - 12h O L 6 , , U
(92)

]rSk

e

The DPODP uses the five systems of time discussed below.

* EP

.This is a uniform measure of

time which is synonymous with coordinate time t of the

general theory of relativity. It is the independent variable

for the motion of bodies and of light rays in the bary-

centric space-time frame of reference. The represented

motion is strictly mathematical in the sense that the

three position coordinates and their independent variable

(coordinate time) are not observable. However, the values

of observable quantities computed using these coordi-

nates are invariant with the selection of coordinates. Thus,

the selection is arbitrary. Ephemeris time differs from

the other four time scales of the DPODP since it is an

abstract, unobservable time scale.

2. Atomic time (AI).This is derived from oscillations
of a cesium atomic clock. The value of A1 was set equal to UT2 on January 1, 1958, OhOmOS UT2. The adopted length of the A1 second is 9,192,631,770 cycles of cesium, which is the current experimentally determined average length of the ET second.

3. Universal time (UT) (specifically UTO, UTI, OT UT2). This is the measure of time which is the basis for all civil time-keeping. Universal time is defined in Ref. 25, p. 73 (the differences between WTO, UT1, and UT2 will be described below) as 12 h plus the Greenwich hour angle of a point on the true equator whose right ascension measured from the mean equinox of date is:
+ + RU(UT) = 18h38m45!836 8,640,184S542TU 0%929T&
(91)
where
Tu = number of Julian centuries of 36,525 days of UT elapsed since January 0,1900, 12hUT
The Greenwich hour angle of this point is e, - RU(U
where
Bp = Greenwich mean sidereal time, the Greenwich hour angle of the mean equinox of date

(Note that any integer multiple of 24 h may be added to

the right-hand side, and hence the

term could also

be written as +E..)

Universal time is obtained from meridian transits of stars, observed in practice with a photographic zenith tube (PZT). At the instant of meridian transit, the right ascension of the observing station is equal to that of the observed star, relative to the true equator and equinox of date. Subtracting the east longitude of the observing station givesthe true Greenwich sidereal time 6 at the instant of observation:
6 = true Greenwich sidereal time, the Greenwich hour angle of the true equinox of date
Subtracting the nutation in right ascension (Ref. 25, p. 43) gives Greenwich mean sidereal time BM. Solving Eq. (92) gives the value of UT at the instant of observation. Each observing station has a nominal value of longitude used for computing UT; if this nominal value is used, the resulting UT is labeled UTO. Because the pole wanders, the latitude and longitude of a fixed point on the earth are a function of time.5 Using the true longitude of the observing station at the observation time, the resulting UT is labeled UT1. There are fairly predictable seasonal fluctuations in UT1; if the adopted seasonal correction is added to UT1, the resulting time is labeled UT2.

The DPODP uses only U 1.It takes the value of UT1 supplied by the U.S. Naval Observatory and computes By from Eq. (92). Adding the nutation in right ascension gives 8, which is used to compute the position of a tracking station relative to the true equator and equinox of the date of observation.

a& Uniuersal time (UTC).This is Greenwich
civil time, which is an approximation of WT2; UTC is derived from oscillations of a cesium atomic clock. It is broadcast from several stations of the National Bureau of Standards such as
onds pulses are the length of 9,192,631,770 (1- S ) cycles
of cesium.
5See Subsection VII-B-1.

7

1

e value of the frequency offset S is adopted annually by international agreement. Since 1964, the value of S must be a positive or negative integral multiple of
. 26, p. 306). For the years 1960 to 1969,
the annual values of S were -150, -150, -130, -130, -150, -150, -300, -300, -300, and -300 X 10-lo, respectively. At Oh UTC on the first day of any month, UT@ may be advanced or retarded by exactly 0.100 s (Ref. 26, p. 307). These step adjustments to broadcast UTC are announced in advance. The frequency offsets and step adjustments are selected so that broadcast UTC will deviate from UT2 by no more than a few tenths of a second.
5. S t a h a h e (ST).This is the operational time scale
at each tracking station derived from oscillations of a rubidium atomic clock. The ST second is ideally equal to the UTC second. Also, the ST clocks are stepped along with the step adjustments in UTC. Currently, ST at each tracking station departs from UTC by less than 100 ps and is known to 10-20 ps. The value of the UTC-ST offset is determined by using a traveling UTC clock (previously synchronized with the National Bureau of Standards) or by transmitting a timing signal (derived from the master UTC clock of the DSN) from the Deep Space Communications Complex at Goldstone, Calif., to a particular tracking station via moon bounce (accounting for the fairly well known propagation delay). The traveling clock provides UTGST to 5ps or better, while the moon bounce currently provides an accuracy of about 20 ps.
In the DPQDP, time is represented as double-precision seconds past January 1, 1950, Oh. On the IBM 7094 computer, double precision is 54 bits or slightly more than 16 decimal digits; from 1967to 1984, time is represented to 0.6 X s. If UTC is 600,000,000s past January 1, 1950,
Oh UTC, and ET - UTC = 40 s, then ET is 600,000,040 s past January 1, 1950, Oh ET.
e5
The complete transformation between A1 time and is given by Eq. (65). The terms of Eq. (65) are defined in detail after that equation. The first term, AT1958,is the constant part of the offset between A1 time and ET. The
+ second term accounts a possible difference in the aver-
age length of the E second (9,192,631,470 Afcesium cycles of cesium) and the length of the A1 second (9,192,631,770cycles of cesium). The nominal values of
AT^^^^ and Afcesiumare 32.15 s and 0, respectively; both
are solve-for parameters.

The remaining terms of Eq. (65) arise horn general relativity; they represent periodic variations in proper time on earth (namely the Al, U ,and ST atomic time scales) relative to uniform coordinate time t (ephemeris time ET). These variations in proper time relative to coordinate time are due to variations in the Newtonian potential at the atomic clock and in the heliocentric velocity of the atomic clock (see Eq. 64).

In the computation of the range observables used to compute differenced-range doppler (see Section XI), the complete expression for ET - A1 (Eq. 65) is required to accurately transform round-trip ephemeris time from the light time solution to observed round-trip station time.
However, in the general time transformation subroutine of the DPODP, only the annual relativity term of ET- A1 has been retained. The expression, giving ET - A1 in seconds, is

ET

-

A1

=

AT1958

-

(t

-

Affcesium
252746078009) ,192,631,770

+ 1.658 X sin E

(93)

where E is computed from Eqs. (67) and (70).

The largest terms of ET - A1 neglected in Eq. (93) are the 2-ps daily term (the fourth term of Eq. 65) and the 1.7-ps monthly term. Also, there are long period variations of the same approximatemagnitudedue to periodic variations in the heliocentric orbital elements of the earthmoon barycenter arising from perturbationsfrom the other planets. Thus, the accuracy of ET - A1 computed from Eq. (93) in the general time transformation subroutine is about s.

The remaining transformations between the various time scales are specified by linear or quadratic functions of time t . The coefficients of these polynomials are specified by time block and the argument t is seconds past the start of the time block. Thus

+ + UTG - ST = a bt ct2

(94)

(95)

+ + A1 - UT1 = f g t ht'

(96)

Equations (93-96) are used to transform in either direction, the right-hand side being evaluated with the known time. For instance, Eq. (95) is evaluated with UTC when transforming from a UTC epoch to the corresponding A1

7

epoch. Alternatively, it is evaluated with A1 time when transforming from an A1 epoch to a UT@ epoch.
As previously indicated, observed values of UTC - ST
are available for each tracking statioh. Values of a, b,
and c may be obtained by fitting to these data. The value of UTC - ST is typically less than 100 ps and is known to 10-20 ps. The coefficients a, by and c are solve-for parameters; however, it is doubtful if the estimated values
of a, by and c would yield UTC - ST more accurately
than the observed accuracy of 10-20 ps.
The U.S. Naval Observatory supplies values of A1 - UTC and A 1 - UT1 to the nearest 0.1 ms. Curvefitting techniques are used to obtain the polynomial coefficients d through h by time block, normally of 1month's duration. Real-time reduction of tracking data is accomplished by using extrapolated polynomials for the current month.

5 X s. For a typical range rate of 30 km/s, the error in computed range is 1.5 m, which is close to the desired accuracy of 0.1 m. The largest conceivable range rate is about 1000 km/s, which can occur for the spacecraft on a hyperbola grazing the solar surface. For this extreme case, the error in computed range is an acceptable 50 m. Thus, an accuracy of about s in the individual time transformations is acceptable for the accurate computation of range observables.
The maximum error in the computed value of a doppler observable due to an error of 5 X s in the ET epoch at which it is evaluated is the acceleration of the spacecraft relative to the tracking station multiplied by
5 X lo-" s. During heliocentric cruise, this acceleration
is less than 0.1 m/s2, and the error in computed doppler is less than 5 X m/s. This compares favorably with the desired accuracy of m/s.

The fitted expressions for A1 - UTC are probably accurate to about 2 X s. A more accurate expression could be obtained by fitting to the data published by the National Bureau of Standards (to the nearest ps) or, better yet, by computing the expression directly from the known frequency offsets and step adjustments. The published data are obtained in this manner.
A small error is incurred in the evaluation of Eqs. (93) to (96) since each may be evaluated with either of the two time scales which it relates. The largest error occurs in the evaluation of Eq. (95) or (96) where e and g are about 0.3 X h is about 10-15,and t may be as large
as 3 X lo6 s. Since 1 vanes by about 8 s, depending upon
whether it is evaluated with A1 or UT, the resulting
uncertainty in A1 - UTC or A1 - UT1 is about 2 to 3 x 10-7 s.
e observables are recorded in ST. In order to obtain the computed values of the observables, the ephemerides of the spacecraft, planets, and moon which affect the observables must be interpolated at the ET value of the
epoch of observation, obtained from the ST epoch by
using Eqs. (93-95). Since Eq. (93) could be in error by 10 p and each of Eqs. (94) and (95) could be in error by 20 ps, the ET value of the epoch of observation could be in error by as much as 5 X s.
The error in the computed value of a range observable due to an error of 5 X s in the ET epoch at which it is evaluated is the spacecraft range rate multiplied by

However, for a grazing encounter with Venus or Jupiter, or an approach to within 1solar radius of the sun's SUIface, the accelerations are 9 m/s2, 25 m/s2, and 70 m/s2, respectively. For a 5 X s timing error, the errors in computed doppler observables are 5 X m/s, 1.3X m/s, and 3.5 X m/s, respectively. These doppler residuals are one to two orders of magnitude larger than desired. With good tracking data, doppler residuals are often obtained with a maximum value of about m/s. Thus, during heliocentric cruise, a timing accuracy of 5 X s is adequate for the accurate computation of doppler observables. But, when the spacecraft is near a planet or the sun, this timing accuracy is only marginally acceptable.
When the offset from UTC to ST at each tracking sta-
tion is known to significantly better than the current accuracy of 10-20 ps, one of the two previously indicated methods for increasing the accuracy of the A1 - UTC time transformation should be implemented. The next step in increasing the accuracy of the time transformations would be to add the 2-ps daily term and the 1.7-ps monthly term to the expression for ET - A1 used in the general time transformation subroutine. Evaluation of the daily term would require that each A1 and UTG epoch be associated with a particular tracking station and that the location of the station be input to the subroutine.
owever, there is no point in attempting to obtain time transformations much more accurate than the microsecond level, because of the unknown long period fluctuations of order s in ET - Al.

e value of A1 - UT1 computed from Eq. (96) at any instant defines the location of the Oo meridian on earth
at that instant. Over a short period of time from this
epoch (a few weeks or months), the angular position of this meridian computed from Eqs. (96) and (92) will depart in a random manner from its actual position by an
angle equivalent to an error of 5-8 m (1sigma) in UT1.
n addition to this random error in U 1,there may be a secular error of a few milliseconds per year. The geocentric velocity of a tracking station on the equator is 465 m/s. Hence, the random error in UT1 of 5-8 ms (1 sigma) produces fluctuations in the computed right ascensions of tracking stations of 2-4 m (1sigma).6A secular error in UT1 of 2 ms per year would cause the estimated station longitudes to drift by about 1m per year.gThese errors are large in relation to the current goal of obtaining station locations to an accuracy of 1 m. Currently, the uncertainties in the estimated tracking station locations are about 5 m (see Mottinger, Ref. 27).
For further details on the subject of timing, see Trask and Muller (Ref. 28) and Ref. 29, Sections 11-E and 11-F.
Section IV-A describesthe precomputed n-body ephemerides for the celestial bodies of the solar system and the manner in which they were generated. Section IV-B describes the method by which these ephemerides are differentially corrected within the DPODP and gives the formulation for obtaining corrected position, velocity, acceleration, and jerk from any ephemeris. Section C gives the formulas for combining these quantities to obtain the relative position, velocity, acceleration, and jerk between any two celestial bodies of the solar system.
Acceleration and jerk are required to compute doppler observables. Acceleration is also used in the computation of partial derivatives of the observables with respect to the estimated parameters.
ses the following precomputed position and velocity ephemerides for the celestial bodies of the solar system: (1)heliocentric ephemerides for eight planets and the earth-moon barycenter and (2) the geocentric lunar ephemeris. The lunar ephemeris is obtained by a numerical integration fit to a corrected version of the Improved Brown Lunar Theory, as will be described
GThe angular error multiplied by the distance of the tracking station from the earth’s spin axis.

in detail below. Given the precomputed ephemeris of the moon, the planetary ephemerides are obtained by a simul-
a1 integration performed by the SS a Processing System).
Values of a number of parameters are differentially corrected to produce a least-squares fit to observed angular data for all of the planets and the sun, radar range
ercury, Venus, and Mars, and ranging data to a spacecraft when it is in the vicinity of a planet. The parameters whose values may be estimated are (1)osculating orbital elements for each ephemeris, (2) osculating orbital elements for the trajectory of the spacecraft relative to the planet it is passing, (3) masses of the planets, (4)radii of planets which have been tracked by radar ranging, (5) right ascension and declination limb biases for Mercury and Venus, and (6) the astronomical unit.
The equations of motion are Newton’s equations plus relativistic perturbative accelerations derived from the 1-body metric of the Brans-Dicke theory. When the
solve-for parameter y approaches unity, this metric re-
duces to the 1-body isotropic metric of general relativity. Development Ephemeris 69 (DE69), which is the latest export ephemeris produced at JPL, is the first to be based upon isotropic relativistic coordinates. Previous ephemerides were based upon the Schwarzschild coordinates of general relativity. This permanent change was made so that the precomputed n-body ephemerides would be compatible with the DPODP, which is based upon isotropic coordinates.
The ephemeris DE69 is based upon a 60-year back-
ward integration from the epoch of August 2, 1970, Oh ET
to 1910. The observations consist of over 34,000 optical observations of the planets (except Pluto) and the sun obtained from the 150-mm and 230-mm transit circles of the U.S. Naval Observatory for 1910-1968, radar range data for Mercury, Venus, and Mars for 1964-1968, and range observables for the Muriner V spacecraft near its encounter with Venus (data for June 21-November 12, 1967). After being fitted to these data, the ephemerides were integrated forward from the 1970 epoch to 1976.
E69 consists of the latter portion of the 60-year integration from October 28, 1961, to the 1970 epoch and the forward integration from this epoch to January 23,1976. The lunar ephemeris contained in is Lunar Ephemeris 16 (LE16), described below; BE69 is described in Ref. 30.
An easy way to describe E16 is to consider the evolution of LE4 (Ref. 31) through LE6 (

(Ref. 33).The Improved is the result of removing certain deficiencies in the orig-
(Refs. 35 and 36). Brown’s solution for the motion of the moon was obtained in rotating rectangular coordinates and then transformed to spherical coordinates. Because precise observations were not available in his time, Brown evaluated this coordinate transformation with less accuracy than he used in his solution for the moon’s motion.
‘Xlhese coordinate transformations have recently been recomputed to a higher precision by Eckert, Walker, and Eckert (Ref. 34). Eckert and Smith (Ref. 38) have obtained a numerical general theory for the motion of the moon that is independent of the Brown Lunar Theory. From a comparison of the two theories, Eckert has recommended that the ILE be augmented by the longitude correction
0!‘072sin (2F - 21)
Positions for LE4 were obtained by evaluating the ILE with aberration terms removed to make the ephemeris strictly geometric, addition of the transformation corrections of Eckert et al. (Ref. 37) and the longitudecorrection of Eckert and Smith (Ref. 38), and addition of corrections to effectively change the constants of the theory to those adopted by the International Astronomical Union (IAU) in 1964 (Ref. 26, pp. 594.4, except for the value of the
second zonal harmonic J, for the earth. Numerical H e r entiation sf these positions gave the velocities for LE4.
Addition to LE4 of correction terms to account for the
modern value of J, gave LE6.
Van Flandem has obtained corrections to certain constants of the ILE from a reduction of meridian circle observations of the moon and a few grazing occultations in the period 1956-1966 (Refs. 39 and 40). The latter observations are particularly accurate in declination. The observations were referred to the moon’s center of mass by the use of Watts’ limb corrections ( charts indicate that the geometric center moves relative to the center of mass with a maximum amplitude of 7.3 prad (1.5 arc seconds) (Ref. 39).
Van Flandem’s corrections to the constants of the ILE
essentially change the equinox from Brown’s equinox (close to Newcomb‘s equinox) to the F is the basis of modern observations and the planetary ephemerides. Correction terms were added to LE6 to change certain of the constants in the theory to those
. obtained by Van Flandern ( 40).A numerically inte-

grated lunar ephemeris was obtained by fitting to this
version of the lunar theory. Addition of corrections to account for certain observable but currently unmodelable terms of the lunar motion gave LE16.
In Refs. 21 and 22, it is shown that the significant part of the relativistic perturbative acceleration for the heliocentric ephemeris of a planet or the earth-moon barycenter is the direct perturbative acceleration due to the sun, the indirect perturbative acceleration of the sun due to the other bodies of the solar system being negligible.
In the general theory of relativity, the perturbative acceleration of a body i due to the sun i s given by Eq, (35)
with the Newtonian term and the i summation removed and the index i referring to the sun. In Ref. 21, pp. 4951,
it is shown that all terms containing the sun’s barycentric velocity, the sun’s acceleration, or the Newtonian potential at the sun are insignificant and hence that the relativistic inertial acceleration (relative to the barycenter of the solar system) of a body due to the sun, denoted i.’(S), may be computed from
..r(S) = $ [ ( 4 + - i P ) r + 4 ( r * i ) i ]
where
p8 = gravitational constant of sun, km3/s2
c = speed of light
r,; = heliocentric position and velocity vectors of body, with rectangular components referred to the mean earth equator and equinox of 1950.0
f,; = magnitudes of r and t, respectively
4 = Newtonian potential at body (positive sign convention)
cke theory, Eqs. (35) and (97)are replaced by Eq. (54) and the following equation:
+ + + r } F(S) ={.” [2(1 y ) - yP] r 2 (1 y ) (re5) C2T3 (98)
where y (or O; see Eq. 41) is the free parameter of the ram-Dicke theory whose value is to be estimated by
fitting the theory to observation.
As y approaches unity, its general relativity value, Eq. (98) approaches Eq. (97) of general relativity.

in Eqs. (97) and (98) were replaced by the potential due to the sun, pS/r, these equations would be identical to the corresponding 1-body equations, namely Eqs. (20) and (55),respectively.

6 m. The relativistic acceleration of the earth-moon barycenter computed from Eq. (99) should also contain the terms

For the heliocentric ephemeris of a planet, the relativistic perturbative acceleration is given by Eq. (98).
+ owever, the only signifcant term of is ps/r and thus,
for this application, Eq. (98) reduces to the corresponding 1-body equation, Eq. (55).For the heliocentric ephemeris of the earth-moon barycenter, the perturbative acceleration is computed from7

(99)

where and

PE
P=-
PM

PE, ppar = gravitational constants of the earth and moon, respectively, km3/s2

The perturbative accelerations of the earth and moon due to the sun are computed from Eq. (98)with the potentials at these two bodies given by

where ri j is the coordinate distance from body i to body j . The formulas above are used in the SSDPS to compute the relativistic perturbative acceleration for each planetary ephemeris.
From Ref. 22, Table 3, the maximum amplitude of the periodic variations in position for a planetary ephemeris,
arising from Eq. (98),is about 6 km.It is shown in Ref. 21,
p. 51, that the ratio of terms of Eq. (54) not included in Eq. (98) to the acceleration computed from Eq. (98) has a maximum value of Thus the above-mentionedposition variations are computed to an accuracy of at least
The notation 3 ( i ) is the relativistic perturbative acceleration of body i due to body i.

where the mutual accelerations of the earth and moon are computed from Eq. (54). owever, it is shown in Ref. 21, p. 53, that the periodic variations in the position of the earth-moon barycenter due to these terms are more than three orders of magnitude smaller than the relativistic variations due to the sun, which, from Table 3 of Ref. 22, have a magnitude of about 400 m. Thus, the
variations in position of the earth-moon barycenter due
to the mutual accelerations of the earth and moon have an amplitude of less than 1m. The errors in the planetary ephemerides due to neglecting the contribution to the
+ Newtonian potential in Eq. (98) from the other planets
are less than 10 m for the inner planets and 100 m for the outer planets.
The relativistic accelerationdue to a planet or the moon is significant, relative to the solar relativistic acceleration, in only a small region surrounding the body (small in relation to the scale of the solar system). For simplicity, this region is taken to be a sphere, termed the relativity sphere, whose center is at the center of mass of the body. The relativistic acceleration due to a planet or the moon should be computed only within that body's relativity sphere. The radius of the relativity sphere for each body of the solar system is given in Ref. 21, Table 5. Since no planet is within the relativity sphere of another planet, the relativistic acceleration of a planet or the earth-moon barycenter due to another planet is negligible. It has been estimated (Ref. 21, p. 53) that neglecting the indirect relativistic acceleration of the sun produces periodic errors in position of less than 1m for the inner planets and less than 1km for the outer planets.
Considering all of the errors mentioned above, the planetary ephemerides produced by the SSDPS contain periodic errors of up to 20 m for the inner planets and up to 1km for the outer planets due to neglected terms in the specified formulation for the relativistic perturbative acceleration.
Fragmentary evidence indicates that LE16 may be as accurate as 100 m. The maximum effect of general relativity on the geocentric lunar ephemeris is less than 10 m in position and m/s in velocity ( ef. 22, p. 4). Thus, it

is not important which relativity terms were included in the numerical integration fitted to the lunar theory, which produced LE16.

constraints, respectively. These constraints and the recommended values of the scaling factors are given in the following section.

owever, in the future when the lunar ephemeris is obtained by a numerical integration fitted to observations, as is currently done for the planetary ephemerides and the spacecraft ephemeris, the relativistic perturbative acceleration of the moon relative to the earth should be computed from

2. Solar and lwmr constraints. The solar constraint is an exact relation between the estimated value of
AH = the number of kilometers per astronomical unit
and the estimated value of

The first two terms are evaluated with Eqs. (98), (101), and (102). The last two terms are evaluated with Eq. (54),
with the Newtonian term and the i summation removed and the index i referring to the body producing the accel-
eration. All velocitiesappearing in Eq. (54)are barycentric but may be evaluated with heliocentric values. The acceleration of the perturbing body may be evaluated with Newtonian theory, Eq. (31). The Newtonian potentials
at bodies i and i may be evaluated with Eqs. (101) and
(102).The sum of terms 1 and 2 of Eq. (103)is about km/s2, whereas the individual terms are one order of magnitude larger. The magnitudes of terms 3 and 4 are about 10-13and 10-15km/s2, respectively. The total acceleration computed from Eq. (103) is accurate to three or four figures.
btaining Corrected Position, Velocity, ~eeeleration, and Jerk From Eaeh Ephemeris
1. Uncorrected position and velocity. As previously mentioned, the n-body ephemeris consists of (1) heliocentric ephemerides for eight planets and the earth-moon barycenter and (2) the geocentric lunar ephemeris. These ephemerides are in the so-called type-50 format; they contain modified second and fourth central differences of position and velocity. Interpolation with the fifth-order Everett’s formula gives rectangular components of position and velocity referred to the mean earth equator and equinox of 1950.0(commonly referred to as 1950.0 coordinates). Positions and velocitiesfrom the planetary ephemerides are expressed in astronomical units AU and AU/day, respectively, while data from the lunar ephemeris are expressed in “fictitious earth radii” and “fictitious earth radii”/day.
The conversion factors used to convert the length units
to lom meters are A, km per AU and REkm per fictitious
earth radius. The scaling factors A, and REare related to other solve-forparameters by the so-called solar and lunar

,ux = gravitational constant of the sun, km3/s2

The relation is where

k2A; ” (S6,400)2

k = the Gaussian gravitational constant = 0.01720209895AU3I2/day(exactly)

The gravitational constant of the sun k2expressedin astro-
nomical units cubed per day squared is a mathematical constant which d e h e s the length of 1 AU. The solar constraint is simply a conversion of the sun’s gravitational constant from AU3/day2 to km3/s2.

From Ref. 29, p. 35, Table 17, the values of ,u8 and Ag currently adopted by JPL are

ps = 1.32712499 X loll km3/s2 A B = 149,597,893km/AU

These values satisfy the solar constraint (Eq. 104) to the stated accuracy of nine figures. The value of AEis the recommended scaling factor for the planetary ephemerides of DE69.

One of the constants of the lunar theory is

sin r C = the constant of sine parallax for the moon = the ratio of a fictitious mean equatorialradius of the earth (the length unit of the lunar ephemeris) to the perturbed mean distance of the moon. The constant sinnC is dimensionless; however, it is usually expressed in seconds of arc by multiplying by the number of seconds of arc in one radian.

The value of sin?rcadopted by the and used in the construction of 4 and succeeding lunar ephemerides is 3,422.451 arc seconds. The mean distance to the moon in terms of fictitious earth radii is given by

, ax = sinrr

1

- 206,264.80625

(dimensionless) - sinrr (arc seconds)

where a, = perturbed mean distance of moon (the perturbation is due to the sun), fictitious earth radii
The value of uarin kilometers is

where
R E = scaling factor for the lunar ephemeris, km/fictitious earth radius
The value of REaxis obtained from a modified version of Kepler’s third law:

Equation (107) is the so-called lunar constraint. The value of uM in Eq. (108) is computed from the value of
sin^, used to generate the lunar ephemeris. Either aM
or sin w , may be considered to be a defined constant of
+ the lunar theory. Hence, the accuracy of C is that of n,
namely about 10 figures. On the other hand, pa p x is known to only about seven figures. Hence, for all practical purposes, the lunar constraint, Eq. (107), is an exad relation which must be satisfied by the estimated values of PE, P M , and R E .
The lunar ephemeris LE16 is based upon values of p E and par adopted by the IAU in 1964, namely

and which gives

,UE = 398,603km3/s2 P = p E / p , = 81.30 p a r = 4,902.87 km3/s2

Substituting these values into Eq. (107) gives

RE = 6,378.160km/fictitious earth radius

where
nx = sidereal mean motion of moon (1900) = 2.661699489 X rad/s
F2 = 0.999093141975298 (as computed by E. W. Brown in 1897) ratio of perturbed mean distance of moon to 2-body mean distance (sun not present and mean motion remains constant) gravitational constants of earth and moon, respectively, km3/s2.
Solving for R E gives
+ RE = C ( p E
where
For sin,, = 3,422.451 arc seconds, the numerical value
of C is 86.3135017.

which is the value of the mean equatorial radius of the earth adopted by the IAU in 1964.

However, since 1964, more accurate values of pE and p , have been adopted by JPL (Ref. 29, p. 35, Table 16):

and which gives

PE = 398,601.2km3/s2 p = 81.3010
p.x = 4,902.78km3/s2

The corresponding value of R E is

R E = 6,378.1492km/fictitious earth radius

Strictly speaking, the lunar ephemeris should be corrected for these more modern values of pE and px as was done in the generation of LE4 where Brown’s constants were corrected to those adopted by the IAU in 1964. the major part of this correction can be obtained by scaling the lunar ephemeris with R E = 6,378.1492km rather than the value of 6,378.160km.

7

3. Corrected position an velocity. Each of the precomputed ephemerides may be differentially corrected with conic formulas. Position and velocity are interpolated from the ephemeris at an epoch of osculation specified by the user and are converted to orbital elements, specifically the Brouwer and Glemence Set 111 (Ref. 42, pp. 241-242). The elliptical orbit with these elements agrees exactly with the precomputed ephemeris at the osculation epoch and approximately at other epochs. The orbital elements of the precomputed ephemeris at the osculation epoch are solve-for parameters. Partial derivatives of position and velocity from the ephemeris with respect to these orbital elements are approximated by those from the osculating elliptical orbit. These partial derivatives are used to determine corrections in the osculating orbital elements and, given these corrections, to apply a linear differential correction to the ephemeris.

The actual parameters whose values are estimated are sixparameters which represent corrections AEto the osculating orbital elements E. The corrections are
-

where

+ AE= AM, AW
1 AP
- eAAqw
a = semimajor axis of osculating elliptical orbit
e = eccentricity

M o = value of mean anomaly at osculation epoch, to(ET)

Ap, Aq, Aw = right-handed rotations of the orbit about axes, respectively, where

consists of the accumulated correction obtained from the
previous n - 1iterations:

1
If the correction process is convergent, AE, will be less than AE,+ and the accumulated correction win approach a limit.
Given 1950.0 position r (AU) and velocity i (AU/s) obtained from a planetary ephemeris (at any time) in units of AU and AU/s (the interpolated value in AU/day divided by 86,400), corrected position and velocity for the nth iteration, expressed in km and km/s, are computed from
+ r, (km)= AEr(AU) ar AE(n) r +i (111)
where AE ( n ) is given by Eq. (110). For the lunar ephemeris,
r, (km) = REr(fictitiousearth radii)

In these equations,

-ar -

[6m’y aE -

ar

a

ar (AM, 4-

Aw)

,-a,-,(a-Arp)

ar
a (Aq)

ar
a (eAw)

r-+ 3

(113)

Let AE, equal the estimated value of AE obtained from
the &st iteration of the orbit determination process (see e second iteration will produce an addi-
al correction AE, or a total correction
the contribution to AE obtained from the
be denoted as A E ~W. ith this notation, the correction
AE (n) used to correct the ephemeris for the nth iteration

where

I

where x, y, and x are the rectangular components of r referred to the mean earth equator and equinox of 1950.0.
The formulation for computing aqaE and $-/a
in the next section.

The mean motion n is computed from
n = - P%
a3/2

artial derivatives of position and velocity with respect to orbital elements. In order to compute ar/aE and ai./aE for any of the precomputed ephemerides, position
osculation epoch must be converted to

and the follo-g

computations are made: ecos E, = 1 - - TaO

ro= 1950.0 position interpolated from ephemeris at osculation epoch to(ET) in AU or fictitious earth
radii and converted to km by multiplying by AE
or RE.
io= 1950.0 velocity interpolated from ephemeris at
osculation epoch to(ET) in AU/day or fictitious earth radii/day and converted to km/s by multiplying by AHor REand dividing by 86,400.
For the heliocentric ephemeris of a planet, the parameter p is computed from

esinE --r, *eo
O - (Pa)%
+ e = [ ( e c o ~ E , ) ~(esinE,)2]%

cosEo=

( e cos e

E,)

sin E,

=

(e sin E,) e

The unit vectors P, Q, and W are computed from

For the heliocentric ephemeris of the earth-moon barycenter, p is given by
+ + p (earth-moon barycenter) = ps pE pM (116)
For the geocentric lunar ephemeris,
where

=WXP

(128)

The partial derivatives &/aE and ai/aE are computed
from the orbital elements a, e, n, By
computed once, and from the following quantities, which are computed at each timet that the partials are evaluated:

ps, pE, p x , pP = gravitational constants for the sun, the earth, the moon, and a planet, kms/s2
Given ro(km),io(km/s), and p, the required orbital ele-
ments are computed as follows:

r, i = 1950.0position and velocity interpolated from the ephemeris at time t (ephemeris time) and converted to units of km and km/s as indicated previously for r, and io.

The semimajor axis a is given by

T

7

f. 42, p. 241, the partial e t with respect to each element
in Eq. (113),are given by

K 2 = e ( l - e2)( 1 - - ar

(144

where the quantities Hland K,, which are functions of t,
are given by (Ref. 42, p. 237)

- r-a(1+e2) 1 - ae (1 - e2)

(135)

(137)

--ar a(Ap) - P X r

--ar a(Aq) - Q x r

-=aLr (wx~-:)

(140)

a(eAw) e

Differentiating Eqs. (133-140) with respect to ephemeris time gives the partial derivatives of C at ephemeris
time t with respect to each element of AE:*

5. Acceleration and jerk. Acceleration and jerk vectors

from each ephemeris are computed from corrected posi-

tion and velociw vectors using 2-body formulas. Given the

corrected position and velocity vectors, denoted here as

r and g, compute a corrected value of T from Eq. (129), the

acceleration vector 2 from Eq. (132), and the jerk vector

Y from

... . r =

3p(r-k) p
r5 r - - Tr3

(149)

where p is given by Eq. (115), (116), or (117).

C. Position, Velocity, Acceleration, and Jerk of Celestial Body Relative to Another
Section IV-B gave the formulation for computing the corrected position, velocity, acceleration, and jerk of a planet P or the earth-moon barycenter B relative to the sun S or of the moon M relative to the earth E:

where

+ a i
a(ae)- -H2r K2i

The position, velocity, acceleration, and jerk of the moon relative to the earth-moon barycenter and of the barycenter relative to the earth are computed from
(142)
and

*The velocity partials were first derived by P. R. Peabody, formerly of the Jet Propulsion Laboratory.
7

PE
P=Par

Listed below are sums of the above-mentioned position vectors which give the position vectors of the earth, moon, sun, and a planet relative to each of the following bodies:
*(1)Earth = reference body = r; e=r;--l$ + Ir"p = r; - g r"p (2) Moon = reference body
I$= -r;
= -G-l$
e =-c-e+r"p
(3) Sun = reference body <=G-rZ
$=e+%
$=I$
(4) Planet = reference body
I$= -I$+$---;
e =- $ + e + %
I$= -$
I$,= -I=$+$,
where P and P' represent two different planets. All of the sums above apply when r is replaced by i, F, or iE:
The solve-for parameters which affect the relative position and velocity between two celestial bodies are the scaling factor A E for the heliocentric ephemerides; the scaling factor RE for the lunar ephemeris; corrections to osculating orbital elements hE for any of the ephemerides; and the gravitational constants of the earth and moon, p E and pM. These are known as reference parameters.
The acceleration of the spacecraftrelative to the center of integration consists of:
The Newtonian point mass acceleration relative to the center of integration. The perturbative acceleration from general relativity.

The direct acceleration of the spacecraftdue to the oblateness of a near planet or the moon.
The indirect acceleration of the center of integration (if it is the earth or the moon) due to the oblateness of the earth and the moon.
The acceleration due to solar radiation pressure.
The acceleration due to small forces originating in the spacecraft, such as from operation of the attitude control system and from gas leaks.
The acceleration due to motor bums.
Section V-B contains the formulation for computation of each of these terms of the spacecraft acceleration.
The total acceleration is integrated numerically to give the spacecraft ephemeris, with ephemeristime (ET) as &e independent variable. The acceleration is computed at each integration step and is used to produce three sum and difference (s. a. d.) arrays (one for each rectangular component of position). Each s. a. d. array contains two sums and ten differences of an acceleration component. The arrays may be interpolated at any ET epoch to give the rectangular components of position, velocity, acceleration, and jerk of the spacecraft relative to the current center of integration. The rectangular components are referred to the mean earth equator and equinox of 1950.0. The x axis is directed along the mean equinox of 1950.0, the x axis is normal to the mean earth equator of 1950.0, directed north, and the y axis completes the right-handed sys tem.
The center of integration is located at the center of mass of the sun, the moon, or one of the nine planets. It may be specaed as one of these bodies, or it may be allowed to change as the spacecraft passes through the sphere of influence of a planet (relative to the sun) or of the moon (relative to the earth). For this case, the center of integration will be that body within whose sphere of influence the spacecraft lies. At a change in center of integration, the position and velocity of the spacecraft relative to the old center of integration are incremented by the position and velocity, respectively, of the old center relative to the new center (computed from the formulation of Section IV).
The 1950.0rectangular components of the spacecraft position and velocity vectors at the injection epoch are solve-for parameters and may be referenced to any body (not necessarily the center of integration). The injection epoch must be specified in the AI, UT@,or ST time scales

and transformed to E

me transformation and the

ET value of the epoch

ry from iteration to iteration

of the orbit determination process if ATlg5* or Afeeaiumis

an estimated parameter. The injection position and veloc-

ity vectors are transformed to values relative to the initial

center of integration (using the formulation of Section

and are used to start the s. a. d. arrays.

A motor burn of short duration or a spring separation may be represented as an instantaneous change in the position and velocity vectors of the spacecraft. The estimated parameters are the bum time t a and the rectangular components of the velocity increment Ai. At the epoch of the motor burn, the velocity is incremented by Ai-and the position is incremented by

Ar

=

1 ---2&a

The equations for computing each term of the total spacecraft acceleration relative to the center of integration are given below.

1. Point-mass gravitational acceleration. The point-

mass gravitational acceleration of the spacecraft (S/C)

relative to the center of integration (C) includes all gravi-

tational accelerations except those arising from the oblate-

ness of the various bodies. The point-mass acceleration

is given by

.. .. .. T = rN/c - Tc

(153)

..where rN/C,

= inertial gravitational acceleration of spacecraft and center of integration, respectively, computed by treating each body of the solar system as a point mass. These inertial accelerations are relative to the barycenter of the solar system and have rectangular components referred to the mean earth equator and equinox of 1950.0.

Each of these accelerations is computed from Eq. (54).

The l/co term is the Newtonian acceleration and the

remaining l/c2 terms a ativistic perturbative acceler-

ations derived from th

s-Dicke theory (these terms

. rt to those of general relativity, (35),when y + 1).

summation over j # i includes sun, the nine plan-

ets, and the moon. or each of these perturbing bodies,

the user has the option of

(I) Computing the Newtonian acceleration and the
relativistic perturbative acceleration.

(2) Computing the Newtonian acceleration only.
(3) Ignoring the acceleration due to that body.
The acceleration Fj of each perturbing body in Eq. (54) is computed from the Newtonian expression,Eq. (31).The
summation over k# j in Eqs. (31) and (54) and over Z # i
in Eq. (54) includes all bodies of the solar system which are “turned on” (treated as (1)or (2) above and included
in the i summation of Eq. 54). The velocities in Eq. (54)
are heliocentric.
ct acceleration of spacecraft due to oblateness. The acceleration of the spacecraft relative to the center of integration due to the oblateness of the bodies of the solar system consists of the direct acceleration of the spacecraft minus the indirect acceleration of the center of integration. Currently, the oblateness for only the earth, the moon, and Mars is considered. However, the capability for accounting for the oblateness of the remaining planets and the sun will be added in the near future. The direct acceleration of the spacecraft due to the oblateness of a body is computed only when the spacecraft is within the so-called harmonic sphere for the body. The radii of the harmonic spheres may be changed by input; the nominal
values for the earth, Mars, and the moon are 2.5 X lo8km, 1.0 X lo6km,and 2X lo5km,respectively. The formula-
tion for computing the direct acceleration of the spacecraft due to the oblateness of a body is given in this section. The indirect acceleration of the center of integration due to oblateness, computed only when the center of integration is the earth or the moon, accounts for the oblateness of each of these two bodies. The formulation is given in Section V-B-3.
The direct acceleration of the spacecraft due to the oblateness of a body is derived from the generalized potential function (Ref. 43, pp. 173-174) for that body:

+ ] x (C, cosmh s,, sin mA)

(1%)

where

,p = gravitational constant of body, km3/s2
+,r, h = radius, latitude, and longitude (positive
east of prime meridian) of spacecraft relative to body

7

ap= mean equatorial radius of body (an adopted constant used for U )
PF (sin+) = associatedLegendre function of the first kind. The argumentsin 4 willbe omitted
here.
Cnm and Snm = numerical coefficients(tesseralharmonic coefficients). The values may be estimated by the DPODP.

PRIME

z‘ ‘b

The associated Legendre function F’E is defined by

dm F’: = cos’*”+( d sin+)m p n

(1%)

where
+ P n = Legendre polynomial of degree n in sin

The zonal harmonic coefficient I n is defhed as

z’ axes relative to

PLANE

In = -Cno

(156)

Equation (154)may be written as the sum of three terms
corresponding to the potential of a point mass, zonal har-
monics In, and tesseral harmonics C S m and Snm (m#0):

+ + u =: U ( J ) U ( C , S )

where

where
x cos #J cos
[ -sin h
= -sin #J cos h

+ cos sinh
cos x -sin (p sin x

(161)
cos +

The position vector of the spacecraft relative to the body
(denoted as body i) with rectangular components referred
to the mean earth equator and equinox of 1950.0 is r - rp
where

u (C, S ) =

I = position vector of spacecraft relative to center of integration with rectangular components referred to the mean earth equator and equinox of 1950.0, Le., the “1950.0”position vector

(159)
The inertial acceleration of the spacecraft is computed in a rectangular coordinate system (x’y’z’) with the x’ axis directed outward along the instantaneous radius to the spacecraft,the y‘ axis directed east, andthe z’ axis directed north. Figure 3 shows these axes relative to body-fixed axes xbyb%,, where xb is along the intersection of the prime meridian and equator of the body, z b is directed north along the axis of rotation of the body, and Yb completes the right-handed system. The transformation from body-
fixed coordinates rb = (xb,&,zb)T to r’ = (x’,y’, z’)* co-
ordinates is given by

r: = 1950.0 position vector of body i relative to the center of integration C
The transformation from these 1950.0body-centered coordinates to body-fixed coordinates r b is denoted as
The overall transformation from (r - r:) to r’ is thus
The inverse transformation is

3

Using rb from Eq. (162), the sines and and the angle are computed from

F'J) =-- I
rcos+

aavxo)

( J ) + (6,S)

(171)

Carrying out these differentiations gives

5 nl
;"( = I)'[')$(nJ
n=l

] n + 1)P, 0 -cos 4 P:,

(173)

The transformation T is currently specified in the DPODP for the earth, the moon, and Mars. These and most of the other coordinate transformations of the DPODP were specifiedby F. M. Sturms. The formulation for T for the earth is specified in Section VII. Sturms' formulation for T for the moon and for Mars are specified in JPL internal publication^.^^^^ He has speciiied modifications to the existingtransformationsand specifiedtransformations for the remaining planets and for the sun in another internal publication.ll Sturms also plans to publish this formulation in a JPL Technical Report.
Let 2' denote the inertial acceleration of the spacecraft due to the oblateness of any body with rectangular components along the instantaneous directions of the x', y', and z' axes. This acceleration can be broken down into Y' (J) due to the zonal harmonics J , and Y(C, S) due to the tesseral harmonics ,C, and S,,. Given these terms, the direct acceleration of the spacecraft due to the oblateness of any body, with rectangular components referred to the mean earth equator and equinox of 1950.0, is given by
The components of P(J) and Y' (C, S) are given by
% = '( ar I)(J)+ (C,S)

n = 1 ,=I
+ + -(n 1)P; {Cnmcosmh S,,sinmX}
msec+P~{-C,,sinmh+S,,cosmX}
+ I cos+PZ' {c,,, cosmx s, sinmX} (174)

where the primes indicate derivatives with respect to sin+. Currently, n, has a maximum value of 15 and n2 has a maximum value of 8. These limits will undoubtedly be increased in a future version of the program.

The Legendre polynomial P, is computed recursively from (Ref. 44,p. 308, Eq. 11)

P, = - 2nn- 1

n-1

07-59

starting with

P, = 1

(176)

P, = sin+

(177)

The derivative of P, with respect to sin+, denoted Pk, is given by (Ref. 44,p. 308, Eq-I)

+ + PL = sin PL-, n P,-,

(178)

starting with

P: = 1

(179)

BWamer, M. R., et al., Double Precision Orbit Determination Program, Vol. 111, TRAJ Segment, EPD 426 (JPL Internal Report), June 15, 1967. lOWitt, J., User's Guide for TRIC, 900-168 (JPL Internal Report), Oct. 20, 1968. 1%turms, F. M., New Coordinute Transformationsfor DPTRAJ, RFP 392-16 (JPLInternal Report), Dec. 16, 1969.

The function sec.+P; is computed by first generating

+ + + sec 9;= (2n - 1)cos (sec P;::)

(180)

starting with

sec+Pi = 1

(181)

7

and continuing until n = n,, and then generating

+e (-) + + sec

=

2n - 1
n-m

sin

(sec

PF-~)

+ For each value of m between 1and n,, n is varied from
m 1to %. The general term Pfcis zero if b > a. Equa-
tion (180) may be obtained by successive differentiation of Eq. (175) with respect to sin+ and substitution into Eq. (155). Equation (182) was obtained from Ref. 45,
+ +. p. 161, Eq. 12. The function Pz is obtained by multiply-
ing (sec PE) by cos
+, The function cos+P;', where P;' is the derivative of
PE with respect to sin is computed from (Ref. 45,p. 161, Eq. 19)
+ + + + + cos+ P;' = -n sin (sec PE) ( n m) (sec Pa,)
(183)

YE ( M )= inertial acceleration of point-mass earth due
to the oblateness of the moon
These accelerations, with rectangular components referred to the mean earth equator and equinox of 1950.0, may be computed from the formulation of Subsection V-B-2. In the computation of & ( E ) , the moon is treated as the
e . spacecraft of Subsection V- -2, and r - r: in Eq. (162) is
replaced by Similarly,in the computation of YE (M)t,he
earth is treated as the spacecraft and r - r%is replaced by
rf .
Consider the force of attraction between the earth and moon due to the oblateness of the earth, assuming the
moon to be a point mass. This force produces B( E ) and
also
?E' ( E ) = inertial acceleration of the earth due to the force of attraction between the oblate part of the earth and the point-mass moon
Since these two accelerations are derived from equal and opposite forces,

3. Indirect acceleration of center of integration due to oblateness. As previously mentioned, the indirect oblateness acceleration of the spacecraft relative to the center of integration is the negative of the acceleration of the center of integration due to oblateness. It is computed only when the center of integration is the earth or moon and accounts for the oblateness of both of these bodies.
The force of attraction between the earth and moon consists of
The force of attraction between the point-mass earth and point-mass moon.
The force of attraction between the oblate part of the earth and the point-mass moon.
The force of attraction between the oblate part of the moon and the point-mass earth.
The force of attraction between the oblate part of the earth and the oblate part of the moon.
The force (1)is accounted for in Subsection V-B-1. The formulation of this section will account for the forces (2) and (3), but will ignore the force (4).
Let
% ( E ) = inertial acceleration of point-mass moon due
to the oblateness of the earth

Similarly, consider the force of attraction between the earth and moon due to the oblateness of the moon, considering the earth to be a point mass. This force produces
&(M)and ?SO
Yaf (M)= inertial acceleration of the moon due to the
force of attraction between the oblate part of the moon and the point-mass earth
Since these two accelerations are derived from equal and opposite forces,

Y J f ( M )= - EYE((M)
PJf

(185)

e acceleration of the earth due to the oblateness of the earth and moon is

Similarly,

?E = "rE ( M )f?E ( E ) =".;( M )- ""&(E)
PE

527

) is proportional to pB and Yx (E) is pro-
portional to p E . The contribution to the spacecraft acceleration relative to the center of integration is the negative of the acceleration of the center of integration, or
where
If earth = center of integration, --+pi = i - p x
f moon = center of integration, &pi = -pB
Sturms’ algorithm for computation of this acceleration accounts for J,, C,,, and S,, of the earth and moon. Equation (188), evaluated with these harmonic coefficients, is equivalent to Sturms’ formulation. An earlier version of his formulation, which is based upon the principal moments of inertia A, B, and C for the earth and moon, is given in Ref. 4.6.
4. Acceleration of spacecraft due to solar radiation pressure and small forces originating in spacecraft.This section gives the model for representing the acceleration of the spacecraft due to solar radiation pressure and to small forces originating in the spacecraft, such as those from operation of the attitude control system (particularly if it uses uncoupled attitude control jets) and from gas

leaks. The model applies to any spacecraft which has one axis (the roll axis) continuously oriented toward the sun and utilizes a star or planet tracker to orient the spacecraft about the roll axis. The various Mariner spacecraft are of this type.
The solar radiation pressure model accounts for the acceleration of the spacecraft due to solar radiation pressure acting along three mutually perpendicular spacecraft axes, one of which is the roll axis. Normally, the solar panels are oriented normal to the roll axis so that the largest component of the force due to solar radiation pressure is along the roll axis. However, the model can also account for the small forces acting along the other two spacecraft axes and arising from departures of the spacecraft shape from rotational symmetry about the roll axis.
The small force model accounts in a crude fashion for the acceleration arising from small forces originating in the spacecraft. The component of this acceleration along each spacecraft axis is represented as a quadratic. This model is currently being expanded to allow this acceleration to be represented alternatively as an exponential decay with components along each spacecraft axis.
The acceleration of the spacecraft due to solar radiation pressure and small forces originating in the spacecraft is represented by

e terms in this equation are defined as 8 p = unit vector from sun to spacecraft ong spacecraft x and y = Usp)(defined below)
ai7bi, ci where i = r, X, or y = solve-for coefficients of acceleration polynomials, km/s2,km/s3, km/s4
t = ephemeris time
7

c2 = epochs at which the acceleration polynomials are turned on and off, respectively. The epochs may be
specified in the UTC, ST,or A1 time
scales. They must be transformed to ET for use in Eq. (189).The transformation will be different for each iteration of the orbit determination process
if the values of AT^^^^ or Afcesiumare
estimated.

A a , Aa,, Aa, = input acceleration (not solve-for), km/s2. The value for each ?Aais obtained by linear interpolation between input points specifiedin any time scale. The acceleration is started at the epoch of the first point and ended at the epoch of the last point.

c, = J-AxJ
c

- 1lk06mmz2- 1.010x 10sk- sm2m3k2g

where

J = solar radiation constant = 1.3525 X lo3W/m2 (Ref. 47)12
= 1.3525 X lo3kg/s3
A, = 1.496 X 10' km

G, = solve-foreffectivearea for acceleration of spacecraft in the direction of its positivex axis (along divided by A,
G, = solve-for effective area for acceleration of spacecraft in the direction of its positive y axis (along divided by A,
Gi, Gk, G: = solve-for derivatives of G,, G,, G, with respect to earth-spacecraft-sun angle, EP S
EPS = earth-spacecraft-sun angle, rad
AG,,AG,, AG, = incrementsto G,, G,, and G, obtained by linear interpolation of input points specified in any time scale. The value of AGi is computed at each integration step contained between the epoch of the first point and the epoch of the last point.

c = 2.997925 X lo5km/s
4 = nominal area of spacecraft projected
onto plane normal to sun-spacecraft line, m2
rn = instantaneous mass of spacecraft, kg
rSp= distance from sun to spacecraft, km
TSRp = epoch at which acceleration due to solar radiation pressure is turned on (epoch of solar panel unfolding). The epoch may be specified in the UTC, ST, or A1 time scales and must be transformed to ET for use in Eq. (189).

The term G!,(EPS) along each spacecraft axis was included so that the model would be compatible with the Mariner I1 spacecraft, which contained a high-gain antenna that moved continuouslywith respect to the spacecraft axes and always pointed toward the earth. These terms account for the variation in G,., G,, and G, due to this moving antenna.
The Mariner IV spacecraft contained movable attitude control vanes situated at the end of each solar panel. Movement of these vanes caused G,, G,, and G, to fluctuate with time. The AGi terms account for these fluctuations.
The unit sun-spacecraft vector US, is computed from

u* (t - TSEP=) 1 for t 1-TsRpif spacecraft in sunlight,
0 for t < TSRpor if spacecraft in
shadow of a planet or the moon
Gr = solve-foreffectivearea for acceleration of spacecraftin radial direction due to solar radiation pressure, divided by nominal area A,
~~
12On July 20, 1970, the author of Ref. 47 stated that a more accu-
rate reduction of the data gave a value of 1.348 X 103 W/mZ.

where

r - rg
U S P = I! r - rs"II

r = position vector of spacecraft relative to center of integration with rectangular components referred to the mean earth equator and equinox of 1950.0
I$ = 1950.0 position vector of sun relative to center of
integration C

3

&T

the angle K :

The unit tangential vector (tangent to sun-spacecraftreference body plane) is
, the vectors X* and
Eqs. (191). The angle may be selected to achieve a
spec& orientation of and Y* relative to the space-
craft.
The EPS angle may be computed from

cosK sinK

[::]=[-sinK

cosK][E]

(lgl)

The angle K is an input (non-solve-for) constant. Computation of the unit vectors T and N requires the unit vector WR

where
Wk is computed from Eq. (193)using B = earth.
5. Acceleration due to motor burn. The acceleration of the spacecraft due to a motor bum is represented by
i:= UW [U (t - To)- u (t - TI)] km/s2 (197)

W R= unit vector from spacecraft to reference body
which orients the spacecraft about the roll axis (sun-spacecraft line). The reference body may be a star, a planet, or the moon. If the reference body is a star,
where the right ascension (Y and declination 6 of the star are referred to the mean earth equator and equinox of 1950.0. If the reference body B is a planet or the moon (normally the earth),
(193)

where a = magnitude of 'r'
W = unit vector in direction of i:
To= effectivestart time of motor, the ET value of the solve-for epoch, which may be specified in the UTC, ST, or A1 time scales
T f = effective stop time of motor, ET t = ephemeris time 1f o r t hTo
u (t - To)= Ofort < To To+ T f
The effective stop time T I is given by

where :-I = 1950.0 position vector of reference body B relative to center of integration C
The unit normal vector (normal to sun-spacecraftreference body plane) is computed from

where T = solve-for burn time of motor,
The acceleration magnitude a is given by

(194)
7

(199)
7

where F (t)= magnitude of thrust at time t.The polynomial coefficients of F ( t )are solve-for parameters
t = t - To,seconds
rn ( t )= spacecraft mass at time t m, = spacecraft mass at To
GoM, I ,i02,n%, = polynomial coefficients of propellant
+ + mass flow rate (positivg)at time t :
G(t)= iOo &,f+ M22 ni3F
(not solve-for parameters) C = 0.001for F in newtons and m in kg.
For F in lb and m in lbm, C = 0.00980665
The unit vector W in the direction of thrust is given by
a,8 = right ascension and declination, respectively, of
U,referred to the mean earth equator and equi-
nox of 1950.0
given by

transmitted directly by the spacecraft at time t,. All observables are related to characteristics of this electromagnetic radiation, i.e., the angle of the incoming ray, the ratio of received to transmitted frequency, or the round-trip transit time. The transmitting station, the spacecraft, and the receiving station are referred to as direct participants, and tl,t,, and t3,respectively, are their epochs of participation. The solution of the light time problem consists of these epochs of participation and the heliocentric position, velocity, acceleration, and jerk of each direct participant evaluated at its epoch of participation. The rectangular components of these vectors are referred to the mean earth equator and equinox of 1950.0. Sections VIII-XI give the formulations for computing doppler, range, and angular observables, starting with the solution to the light time problem.
The solution to the light time problem is obtained by solving the light time equation for each leg of the path of electromagnetic radiation from the transmitting to the receiving station. The light time equation relates the light time between two points to the heliocentric positions of each of the two participants evaluated at their epochs of participation. Starting with the known reception time t3, the light time equation is solved by an iterative technique for the down leg of the light path to give the epoch of participation for the spacecraft, t,. Given t,, the light time equation is solved iteratively for the up leg of the light path to give the transmission time tl.
Section VI-B gives the formulation for solution of the light time problem; the detailed procedure is given in Section VI-C.

where the polynomial coefficients of Eqs. (201) and (202) are solve-for parameters.
This section gives the formulation and procedure for solution of the light time problem, which is the first step in the computation of all observable quantities.
An electromagneticsignal is transmitted from a tracking station on earth at time tl. This signal is received by the spacecraft (either a free spacecraft or a landed spacecraft on the moon or on one of the planets) and retransmitted at time t,, arriving at the same or a different tracking station on earth at time t3.Alternatively, the signal may be

Let the subscripts i or i equal 1, 2, or 3 where
1 refers to the transmitting station on earth at the transmission time tl
2 refers to the spacecraft (free or landed) at the reflection time t,
3 refers to the receiving station on earth at the reception time t3
The time for light to travel from point i at ephemeris time
(coordinate time) ti to point i at ephemeris time t j is
given by Eq. (88), repeated here:

3%

where
rii = 11 rij 11 = 11 $(ti)- c(ti)ll
e = II (ti)II
Ti = I1 r: (ti)II
(ti)$, (ti)= heliocentric position vector of point i at transmissiontime ti and point i at recep-
tion time ti, with rectangular components referred to the mean earth equator and equinox of 1950.0
c = speed of light, km/s
pLS= graviiational constant of sun, km3/s2
y = solve-forfree parameter of Brans-Dicke theory of relativity. The parameter y is related to w, the coupling constant of the scalar field, through Eq. (41).
Equation (203), which is referred to as the light time equation, relates the light time in ephemeris time for a given leg of the light path to the heliocentric position vectors of the two participants evaluated at their epochs of participation. The light time equation applies to the
down leg of the light path when i = 2 and i = 3; when i = 1 and i = 2, it applies to the up leg.

Each of these 1950.0 vector sums applies with r replaced by 2, Y, and *E The heliocentric position, velocity, acceleration, and jerk of the earth, as well as the center of integration or the body upon which the spacecraft has landed are obtained as indicated in Section IV. The position, velocity, acceleration and jerk of the spacecraft relative to the center of integration are obtained by interpolation of the spacecraft ephemeris s u m and difference arrays. The formulationfor computing the 1950.0position, velocity, acceleration, and jerk of a tracking station relative to the earth or of a landed spacecraft relative to the body B on which it is located is given in Section VII. The geocentric 1950.0 position and higher derivatives for a tracking station are primarily functions of the UT1 value of the epoch, although the ET value is also required.
Solutionof the light time equation (Eq. 203) for a given leg of the light path gives the transmission time ti for that leg. The time ti is used to compute rf(ti)in the evaluation of the right-hand side of the light time equation and also appears explicitly in the left-hand side. The light time equation must be solved for ti by an iterative technique. The DPODP uses the Newton-Raphson method. Let the function f whose value is to be minimized be the left-hand side of the light time equation minus the right-hand side:

Let
r$ = rii = position vector of point i relative to point i,
with rectangular components referred to the mean earth equator and equinox of 1950.0.
With this notation, the heliocentric position vectors of the transmitter, spacecraft, and receiver at their epochs of participation are computed from the following equations. For the transmitter,

When the relativity term is ignored, the partial derivative of f with respect to ti is

where S = sun and E = earth. Similarly,for the receiver,

Let A(ti)equal the linear differential correction to the estimate of ti. Then

For a free spacecraft S/C, with center of integration C, Substituting Eqs. (208) and (209) into Eq. (210) gives

For a landed spacecraft on body B,

1-

P

The procedure for using this iterative formula for obtaining the transmission time t, for the down leg and the transmission time tl for the up leg is given in the folbwing section.

cedure
The procedure is as follows:
Convert the observation time t3(ST) to t3(UTC),
e t3(Al), t3(UTl), and t3(ET) using the time trans-
formations of Section 111. Compute (t3)from
Eq. (205). Compute also tt (t3),q(t3),.ji:(t3).

This section gives the formulation for computation of the position, velocity, acceleration, and jerk of a tracking station relative to the center of the earth or of a landed spacecraft relative to the center of the body on which it is located, with rectangular components referred to the mean earth equator and equinox of 1950.0. In addition to a fixed tracking station, a model is included for representing the motion of a tracking ship.

Obtain the first estimate for t, (ET) as:
(a) For the first observation of the spacecraft on a pass of the spacecraft relative to the receiving station, tz = t3.
(b) For the remaining observations of the pass, t, = t3minus the converged light time for the down leg of the previous observation.
..Given the estimatefor tz(ET),compute ri (tz)2:, (t2), + ri(t,), and'Yt(t,) from Eq. (206) or (207) and A(t,)
from Eq. (211).The next estimate fort, is tz A (t2).
Repeat step 3 until A (t,) < s. (On the IBM 7094
computer, time is represented as double-precision'
seconds past January 1, 1950, Oh to a precision of
0.6 X lo-? s from 1967 to 1984.)

The first step in the computation of 1950.0 position, velocity, acceleration, and jerk is to obtain the 'bodyfixed" position rb(and also velocity, acceleration, and jerk in the case of a tracking ship), where xg is along the intersection of the prime meridian (passing through the instantaneous axis of rotation) and the instantaneous equator, where zb is along the instantaneous axis of rotation, directed north, and where Y b completes the right-handed rectangular coordinate system.
Given r b (and higher derivatives for a tracking ship), the 1950.0 position, velocity, acceleration, and jerk are
T obtained from the transformation matrix (which relates
these two coordinate systems) and from T,!t?, and "i.: As
mentioned in Section V, these transformations are currently specified for the earth, the moon, and Mars. The transformations for the remaining planets and for the sun have been specified by F. M. Sturms and will be added to the program in the near future.

Obtain the first estimate for tl (ET) as t, minus the converged light time for the down leg of the current observable.
Convert the estimate for tl (ET)to tl (Al), tl (UTC),
e e tl (UTl), and tl (ST). Compute (tl), (tl), (tl),
and":: (tl) from Eq. (204) and A (tl) from Eq. (211). The next estimate for tl is tl -I-A (tl).Repeat step 5
until A (tl)< lo-' s.
ost of the intermediate quantities used in the computation of the heliocentric position, velocity, acceleration, and jerk of each participant at its epoch of participation are saved and used in the computation of the observable and the partial derivatives of the observable with respect to the estimated parameters.

The location of a fixed tracking station on earth is specified by its spherical or cylindrical coordinates relative to the mean pole, equator, and prime meridian of 1903.0.These station coordinates are solve-forparameters. Because the pole (axis of rotation) wanders relative to the earth, the "body-fixed" coordinate system moves relative to the earth and the "body-fixed"position q,of a fixed tracking station on earth is a variable quantity. puted from the time-varying coordinates of the true pole of date relative to the mean pole of 1903.0supplied by the B.I.H.13The location of a landed spacecraft on a planet or the moon is specified by constant spherical or cylindrical coordinates (solve-for parameters) relative to the body-fixed coordinate system. The body-fixed position of a tracking ship is specified by its spherical coordinates
13Bureau International de l'Heure.

at an arbitrary epoch, and by its azimuth and velocity; the values of these five parameters may be estimated. The value of the geocentric radius to the ship is constant.
Section VII-B gives the formulation for computing body-fixed position (and higher derivatives for a tracking ship). Section VII-C gives the general formulation for transforming these quantities to 1950.0position, velocity, acc$eFFtion, apd jerk using the transformation matrices
T, T , T,and T. These matrices are specified for the earth
in Section VII-D.

For a landed spacecraft on a planet or the moon, the spherical or cylindrical coordinates are constant and are solve-for parameters. For a tracking station on earth, the solve-for parameters are the spherical or cylindrical coordinates relative to the mean pole, equator, and prime meridian of 1903.0.The spherical coordinates are denoted by r, 40,and ho;the cylindricalcoordinates are denoted by uo,vo,and io The.transformations from these 1903.0coordinates to those referred to the “body-fixed” coordinate system are

4 = $0 4-A+

(5314)

1. Fixed tracking station or landed spacecraft. For a tracking station on earth or a landed spacecraft on the moon or a planet, the spherical coordinates referred to the x7,ybzb “body-fixed” coordinate system are
T = radius from center of body, km
4 = body-centered latitude measured from true equator
(plane normal to instantaneous axis of rotation and containing center of mass)
h = longitude measured east from prime meridian (passing through instantaneous axis of rotation)
The cylindrical coordinates are u = distance from spin axis (instantaneous axis of rotation), km
= TCOS4
o = height above true equator, km
= rsin4
X = longitude measured east from prime meridian (passing through instantaneous axis of rotation)

The formulas for computing the corrections A+, Ah, Au, and Au are derived below. Given the body-fixed spherical or cylindrical coordinates, the rectangular components of rb are computed from Eq. (212) or (213).
Figure 4 shows the latitude +o and longitude ,io of a tracking station S relative to the mean pole of 1903.0(Po), and the instantaneous latitude 4 and longitude h relative to the true pole of date (P). The pole Po and associated grid of equator and meridians is rotated through the angle o carrying Po to P. The angular coordinates of P

GREENWICH MERIDIAN

PO
I . -v

90’E MERIDIAN

For spherical coordinates, the body-ked rectangular coordinates are

For cylindrical coordinates,

TRUE EQUATOR
U@

relative to Po are x measured south along the Greenwich meridian of 1903.0 (strictly the 1903.0 meridian of zero longitude) and y measured south along the WOW meridian of 1903.0. Values of x and y are obtained from the
. They are represented by h e a r polynomials:

From Eq. (221),

+ tan& -Y

tanao =

X

1- -Ytanko

X

(227)

Substituting Eqs. (220) and (227) into Eq. (226) gives

AX = tan +o (x sin A. 4-y cos A,)

(228)

The coefficients I, m, p, and q are specified by time block, usually of one month’s duration, and t is in seconds past
the start of the time block. Since the angles x and y correspond to a displacement along the earth’s surface of
+ only a few meters (to date the maximum value has been
about 10 m), an approximate expression for A+ = - +o is

The cylindrical coordinatesrelative to the -pole of 1903.0 and the -wepole of date are

= UO

TCOS $0

+ + u = TCOS = TCOS (+o A+)

vo = Tsin4, v = r s h + = T S ~ ( +4-~ A+)

A+ =xcosXo - ysinA,

(220) Solving for Au = u - uo and Av = v - o0 gives

Noting a. and a on Fig. 4, one obtains

AU = -uo.A+

(+) + ho = a0 tan-1 (+) + = a tan-1
Thus,

AV = Uo A+

where A 4 is given by Eq. (220).Using cylindrical coordinates, ~h is computed from

+ AA = - uVOo (x sin A. y cos A,)

(233)

From the spherical triangle P PoS,
-si-nao - sina cos+ cos+o

(224)

The “body-fixed” position n,of a fixed tracking station
on earth varies with the motion of the pole, and hence the body-fixed velocity i b is non-zero. However, its maximum magnitude is about 2 X m/s, which is less than the desired accuracy of m/s for computed doppler observables. Hence i b is taken to be zero.

Cross multiplying and using Eqs. (223) and (214) gives

For a description of the wandering of the earth’s axis of rotation, see Ref. 48.

sin a0cos +o = sin (ao4- Ah) cos (+o 4-A+)

(225)

Expanding, noting that Ah and A+ are very small angles, and ignoring the higher-order term containing AA A+ gives

AA = tan a. tan +o A+

(226)

e The ship is assumed to move on a sphere of radius T at constant azimuth A measured east of north, and at constant speed v. The ship passes through the point with latitude ,+o and longitude ho at time
to(UTC).All quantities are referenced to the xb Yb Xb bodyfixed coordinate system defined in Section VII-A. The parameters T, (p0, Lo, u, and A are solve-for parameters.

The velocity along the meridian is given by
~4=VCOSA
Thus the latitude may be expressed as
+ + = .+o - v cos A T

(234)

This expression is indeterminate for A = 90 or 270 deg. For these cases, compute

X = h +- 2, [t(UTG) - to(UTG)] O- TcoS+O + forA = 90deg
- forA = 270 deg

(238)

The velocity normal to the meridian is given by

Given from Eq. (235) and from Eq. (237) or (238), rb is given by Eq. (212), repeated here:

a Equation (236) can be integrated by replacing dt in
the integral of dt by rd+/v cosA from Eq. (234). The result is

(2 +) + tan
X = A, tanAln

+

A # 90 deg, 270 deg

(237)

(239)
Differentiation with respect to time using Eqs. (234) and (236) gives
-cosAsin+cosh - sinAsinX
+ -cosAsin+sinX sinAcosX
+ cos A cos

Similarly, differentiation of this equation gives
+ ") + .+ - (COS~ACOS+- cos cos X (sinAcosAtan +) sinX

+ + - (cos2Acos+

sinzA cos

sin X - (sin A cos A tan 4) cos X

-V2

~

T

+ -COS A sin

Equation (241) would be simpler if the tracking ship were moving along a great circle (at varying azimuth A). The transformation from body-fixed position, velocity, and acceleration to 1950.0position, velocity, acceleration, and jerk is given in the next section. The body-fixed jerk @,) is ignored since its maximum contribution of about
m/s to computed doppler is considerably smaller than the accuracy of tracking-ship data.

which it is located be denoted by r50, i50, ?50, and .C0.
The transformation from the body-fixed position vector R, to the 1950.0 position vector r50 is given by
where Ti is the 3 X 3 transformation matrix for the body i in question.

or a fixed tracking station on earth or a landed space-
craft on a planet or the moon, & is negligibly small and
is taken to be zero. Thus,

Let the 1950.0 position, velocity, acceleration, and jerk of a fixed tracking station, a moving tracking ship, or a landed spacecraft relative to the center of the body i on

For a moving tracking ship, i b and %, are nonzero and '5 is ignored. Thus,

the instantaneous equator, Zb is along the instantaneous axis of rotation, directed north, and Y b completes the right-handed rectangular coordinate system.
The matrix B is given by

where T.& has been ignored in Eq. (249).

where
e = apparent (true) sidereal time = Greenwich hour
angle of true equinox of date

The formu!atign for computing the transformation matrices Ti, Ti, Ti, and 'i;i for the earth (i = E) is given in the next section.
.Body-Fixed to Space-Fixed ~ r a n s f o r ~ a t i o n
for the Earth
For the earth, the transformation T is given by the product of three 3 X 3 matrices:
Substituting Eq. (250) into Eq. (242) gives

The derivative of TE with respect to ephemeris time Tn is given by

+ + fE= (6NA BfiA BNA)T

(254)

The formulation for computation of the precession matrix A, the nutation matrix N , and their derivatives with
A respect to ephemeris time, and i?, is given in a JPL
internal p~blicati0n.lD~ifferentiation of B with respect
to ephemeris time gives

i ] -sin0 cos 8

00

(255)

or

r b = Tg rso= BNA r,,,

(252)

The matrices A, N , and B are defined as
A = precession matrix, transforming from coordinates referred to the mean earth equator and equinox of 1950.0 to coordinates referred to the mean earth equator and equinox of date
N = nutation matrix, transforming from coordinates referred to the mean earth equator and equinox of date to coordinates referred to the true earth equator and equinox of date
B = rotation from coordinates referred to the true earth equator and equinox of date to body-fixed coordinates Fb = (xb, Yb, zb)T, where is along the intersection of the prime meridian (passing through the instantaneous axis of rotation) and

where 4 is the derivative of 6 with respect to ephemeris
time.
The contribution to the "space-fixed" velocity of the tracking station relative to the center of the earth, is0, from the precession and nutation rates is a maximum of about lo-* m/s. Since doppler observables are computed to an accuracy of m/s, these terms are included in Eq. (254). The computation of doppler observables also requires the acceleration and jerk of each participant;
however, only approximate values are needyd. Thys, YE
and TE are obtained by differentiation of TE z (BNA)= holding N and A constant:
14Warner, M. R., et al., Double Precision Orbit Determination PTO-
gram, Vol. 111, TRAJSegment, EPD 426 (JPL Internal Report), June 15, 1967.

The second and third derivatives of B with respect to ephemeris time are obtained by successivf:differentiation
owever, the sidereal rate 6 in Eq. (255)is
an extremely constant quantity and is held fixed during this differentiation. The resulting expressions are:

--cos6 --sine

B = [ sine -cos6

0

0

sine -cos6

B = [ .I( cos6 sin6

0

0

o 0 ]i2
0
] 0
0 i3
0

(258) (259)

Y E YE The neglected terms of and contribute less than
m/s to the computed doppler observables.

The true sidereal time 6 and true sidereal rate are computed from the following formulation (where dots indicate differentiation with respect to ephemeris time). Let
Bar = mean sidereal time = Greenwich hour angle of mean equinox of date
S$ = nutation in longitude = longitude of mean equinox relative to true equinox
SE = nutation in obliquity
E = true obliquity of ecliptic
T = mean obliquity of ecliptic

rom Ref. 25, p. 98,

+ + - A B T + CT2 DT3

E==

206,26480625

(rad)

where A = 23O27'8126 = 84,428'.%6 B = -461845
G = -0'!0059
= 010018B

(264)

ulian centuries of 36,525 ephemeris days elapsed since January 0,1900, lZh ET

The quantity T is computed from

+ T =

JED

- 241 36,525

5020

=

0.5

ET 86,400 X 36,525

where

(265)

JED = Julian ephemeris date

ET = seconds of ephemeris time from January 1,
1950, Oh ET

Differentiation of Eq. (264) with respect to ET gives

6

+ +B 2CT 3DT2

E = 86,400 X 36,525 X 206,264.80625 (rad/s)

(266)

The nutations S$ and SE and their derivatives $$ and $E are contained on the n-body ephemeris tapes (described in Section IV). The nutations S$ and SE are based upon the theory of E. W. Woolard (Ref. 49). The derivatives 8 3 and S'E are obtained by numerical differentiation.

Mean sidereal time 1 9 i~s a function of universal time. The expression for 6, is obtained by substituting Rv (UT) from Eq. (91) into Eq. (92). Since 6, is the hour angle of the mean equinox of date measured from the Oo meridian passing through the instantaneous axis of rotation, it should be computed specifically from UT1 (see Section 111).Thus, from Eqs. (91) and (92),

+ + 6r = UT1 J KTUf LT; (angular seconds, ")

whereI6

(267)

UT1 = seconds of UT1 time past January 1, 1950, oh UT1

J = 6h38m45%3=6 23,9258836
K = 8,640,1848542
L = 0:0929 Tu = number of Julian centuries of 36,525 days of
UT1 elapsed since January 0, 1900, 12h UT1

15Note that 1 second of UT1 time is the time for the angle UT1 (see Section 111) to change by 1 angular second (86,400 angular seconds = 271 radians ).

e quantity Tu is computed from

TU =

36,525

+ = 0.5

UT1

86,400 367525

(268)

where (UT1) = Julian date computed from UT1
Substituting q. (267) into Eq. (260),and removing multi-
a, ples of 2~ io tha; 0 < 6 < &es

*> + + + + UT1 J KTO LT8
6=[

12, (rad)

86,400

2T decimal part

The quantities UT1/86,400 and KTu/86,400 currently have magnitudes of about 7,OOOrevolutions (1revolution of 6 = % radians of 8) and 70 revolutions, respectively. Thus, when taking the decimal part of 6 expressed as revolutions, four decimal digits are lost. Since double precision on the IBM 7094 is about 16 decimal digits, 6 is represented to a precision
of about 12 figures or 2~ X 10-l2rad. For a tracking station with spin axis distance u of 6 X lo6m, its longitudinal posi-
tion is represented to a precision of about 4 X m.

Differentiating Eq. (267) with respect to ET gives

dET

3 6 , ~ 2 ~ ~ > &0%06) (radian/ephemeris second)

From Section 111, and where

UT1 = ET - (ET - A l ) - (A1 - UT1)

-dU-Tl dET -

-I-

Affcesium
9,192,631,770

-

g

-

2ht

t = seconds past start of current time block for polynomial coefficientsf , g, and h of Eq. (96).

Substituting Eq. (272) into Eq. (270) gives

) - - g - 2ht 4 3 b (radian/ephemeris second)

(273)

Given il, e" is computed from Eq. (261).
+ The term g 2ht in q. (273) has a typical magnitude
of 3 X lo-* and affects the geocentric tracking station
velocity by about m/s, which is the accuracy of computed doppler observables. Since Afcesiumis probably no more than 5, the term AfcesiUm/9,192,631,77is0 probably not significant. In the derivation of Eq. (272), the annual relativity term of - A1 (Eq. 93) was not differentiated. The derivative of this term has a maximum magnitude of about 3 X 10-lo,which is not significant. Equation (65) is a more accurate expression for (E - A1) than Eq. (93) used in the general time transformation subroutine. The time derivatives of the additional relativity terms of Eq. (65) are 1.5X 10-loor smaller.

This section gives the formulation for computation of doppler observables, namely, 1-way doppler, 2-way doppler, and 3-way doppler.
For 1-way doppler, an electromagnetic signal is transmitted continuously from the spacecraft and received by a tracking station on earth. For 2-way doppler, the signal is transmitted continuously from a tracking station on earth, received and retransmitted by the spacecraft, and
received continuousIy by the same tracking station. The
signal may also be received by a different tracking sta-

7

tion; in this case, the resulting observable is 3-way doppler. For each of these cases, the frequency of the received signal differs from that of the transmitted signal because of the doppler shift. The observable is the average value of this frequency shift over a period of time called the count time or count interval T,. It is proportional to the average range rate along the light path from the transmit-
ter to the receiver during T,or, more accurately, to the
change in range along this light path during T,. The count intervals for successive observables are contiguous.

An intermediate output from the electronic equipment at the receiving station on earth is a signal whose frequency in cycles per second of station time (ST) is denoted by f. This signal contains the doppler frequency shift16 and a bias frequency whose primary purpose is to keep f positive when the spacecraft range rate is negative. For 1-way, e-way, and 3-way doppler, the expressions for f are

The expression for computing each of these observables is obtained by expressing the frequency shift in a Taylor series, with coefficients evaluated at the midpoint of the count interval, and integrating term by term. The odd derivatives of the frequency shift vanish and the fourth and higher even derivatives are ignored. Thus, doppler observables are computed from the frequency shift and its second time derivative evaluated along the light path whose reception time at the receiving station is the midpoint of the count interval.
For observables computed to an accuracy of m/s, truncation of the Taylor series limits the count time to values as low as 1-10 s when the spacecraft is very near the earth or another planet. When the spacecraft is in heliocentric cruise, count times as large as 1,000 s may be used. In each of these cases, however, larger count times may be used if the observable is computed from the subinterval doppler formulation. For this case, the count interval is divided into m subintervals, each of which is short enough so that the Taylor series truncation error is negligible. The observable is the sum of the observables computed for each subinterval divided by m.
In a future version of the DPODP, the Taylor-series doppler formulation will be replaced or supplemented by the differenced-range doppler formulation described in Section XI. The primary advantage of differenced-range doppler is that there is no upper limit to the count time.

(274)
(275)
where C, to C, are constants, defined below, and fs/c = spacecraft auxiliary transponder oscillator frequency, cycles per UTC second [9,192,631,770 (1-S ) cycles17of imaginary cesium atomic clock carried by spacecraft]
The quantity fsIc is the frequency of the signal transmitted by the spacecraft for 1-way doppler. It is represented by
where fT,,= nominal value of fsIc
AfTo,fT1,fT2 = solve-for parameters, specified by time block
to= UTC epoch at start of time block tz= UTC value of spacecraft transmission
time

The formulation for computation of 1-way, 2-way, and 3-way doppler from the frequency shift and its second time derivative is given in Section VIII-B, and the fonnulas for computing these two quantities are given in Sections VIII-C and -D. The equation for computing each doppler observable contains a correction term A, which accounts for the effects of the troposphere, the ionosphere, and the motion of the tracking point on the antenna during the count time. e computation of A is described in Section XII.

The remaining quantities in Eqs. (274-276) are defined as
f R / f T = ratio of received to transmitted frequency (for unity frequency multiplication at spacecraft). The received frequency f E is measured in cycles per second of station time ST derived from the
16The transmitted frequency minus d e received frequency. 17See Subsection 111-A-4.

7

7

atomic frequencystandard at the receiving station. For 2-way or 3-way doppler, the transmitted frequency fT is measured in cycles per second of ST derived from the atomic frequency standard at the transmitting station. For 1-way doppler, f T is measured in cycles per UTC second (9,192,631,770 (1 - S) cyc1eP of imaginary cesium atomic clock at spacecraft).

L-band L-S band S-band
L-band S-band

f q (t,),f q (t3)= reference oscillator frequency at transmitting station, cycles per second of ST (derived from transmitter atomic fre-

105 6, =
106

quency standard), evaluated at trans-

mission time t, and reception time t3,

respeotively.lsThe frequency f q is reset

periodically but remains constant be-

tween settings. The doppler formula-

tion presumes that f q (t3i)s constant over

the reception interval T, for 2-way dop-

pler and that f q (t,)is constant over the

transmission interval. If these intervals

overlap for 2-way doppler, f q ( t l )must

equal f q (t3).

where

L-band S-band L-band
L-S band
S-band

The doppler tracking equipment originally operated in the L-band frequency range.I9 Later, the system was changed to operate in the S-band range.2oIn the interim period, some tracking data were obtained in the so-called
G S configuration (modified L-band tracking stations with an S-band transponder on the spacecraft). The DPODP
has the capability of processing doppler tracking data from each of these frequency bands. The only change in the doppler formulation due to changing the frequency band is the change in the values of the coefficients C, through C,:

930.15 X lo6

L-band

+ c, = 9.375 X lo6 30K1(t3) L-S band

(E) + 96 - K , (t3) lo6 S-band

lsNote that f q ( t 3 )applies only for 2-way doppler. *9390-1550 MHz. 201550-5200 MHz.

K , (t3)= receiver reference oscillator (synthesizer)frequency at reception time t3 for L-S band doppler. The frequency K , (t3)is different from f q ( t l ) .
Ks (t3)= receiver reference oscillator frequency at reception time t3 for S-band doppler. The receiver and transmitter reference oscillators are physically the same and operate at the same nominal frequency.
As with f q , both of these frequencies are reset periodically but remain constant between settings. The doppler formulation presumes that K , (t3)and Ks (t3)are constant over the reception interval T,. Two-way L S band doppler is computed from the 3-way formulation. Hence, L-S band values of C3 and C, do not exist.
The second term of Eqs. (274), (245), and (276) is the frequency of the received signal (relative to ST at the receiving station). The first term (plus C, for 2-way doppler) is the frequency of a reference signal derived from the receiver atomic frequency standard.

or 2-way doppler, the reference frequency and received frequency are derived from the same atomic frequency standard. Hence 2-way doppler gives the most accurate measure of the doppler frequency shift and thus the range rate from the tracking station to the spacecraft.
For 1-way and 3-way doppler, the reference signal and received signal are derived from different atomic frequency standards. Hence, these data types are less accurate than 2-way doppler. Furthermore, for 1-way doppler, the signal transmitted from the spacecraft is currently derived from a crystal oscillator. Because of the large drift in frequency of this type of oscillator, 1-way doppler is very inaccurate and is rarely used in the determination of accurate spacecraft trajectories.

+ ( t ) c - f3 = Cl C5fq(t1) 5fq (t1) 1 - -

(280)

Part or all of the constant part of each expression for f is designated as fbias:

For fR/fT = 1, that is, for a spacecraft range-rate of
zero, the values of fl, fz, and f3 are lo5 Hz for L-band and G S band operation and lo6 HZ for S-band opera-

tion. These biases are included so that the frequency f

will remain positive for negative spacecraft range rates

> ( f R / f T

l)*

( 3 fl - = flbias Czfs,c 1- -

For the existingS-band doppler system, the transmitted frequency is 96 times the transmitter reference oscillator frequency. The spacecraft transponder multiplies the frequency of the received signal by 240/221 before retransmitting. The reference oscillatorfrequencyis approximately 22 MHz and hence the frequency of the signal received at the tracking station on earth is about 2300 MHz plus the effect of the doppler frequency shift. For 1-way doppler, the frequency of the signal transmitted by the spacecraft is also about 2300 MNz. For 1-way, e-way, or 3-way doppler, the frequency of the reference signal at the receiving station is 96 (240/221) times the receiver reference oscillator frequency plus the 1-MNz bias. For 2-way doppler, of course, the receiver reference oscillatoris the transmitter reference oscillator.

Noting the S-band values for C1, Cs, C4, and C5,one can see that the expressions for fz and f3are identical for S-band operation. The only differencesare that two physically different atomic frequency standards are used for 3-way doppler and that the frequency shifts are based upon different light paths.

Equations (274276) for f may be written as
( k + fl = Cl - Czfs,c CZf8,C 1- -

(278)

The signal with frequency f is input to an electronic counter whose register is incremented by 1each time the magnitude of the signal changes from minus to plus. A total of N cycles are counted during the count time Tc. The doppler observable F which the data editing program passes on to the orbit determination program is:21
2% addition to the integer cycle count, the time from the start of the count interval to the first positive zero crossing is observed. Multiplying this time by N cycles per T,seconds gives an estimate of the fraction of one cycle not counted at the beginning of T.. One minus this quantity for the next observable is the fraction
of one cycle not counted at the end of T,.Adding these two frac-
tions of 1 cycle to the integer cycle count gives N used in Eq. (287).

where f b i a s is computed from q. (281), (282), or (283).
Since N is the integral of f over the count time T,,
where t3(ST) = station time (ST) at receiving station, derived from station atomic frequency standard t3,(ST) = epoch at midpoint of count interval T ,
Equations (284), (285), and (286) for f - fbias are substituted into Eq. (288). For 1-way doppler, the variations in fsl0 and the second term of Eq. (284) over the count interval are ignored. In each of these three equations, the quantity - [l ( f R / f T ) ] is expanded in a Taylor series, with the reception time t3(ST) minus the epoch t,, (ST) as the argument. The coefficients of each Taylor series are the derivatives of [1- ( f R / f T ) ] with respect to t3(ST), evaluated along the light path with reception time t,, (ST).A term-by-termintegration of each of these equations gives the desired expressions for the computation of 1-way doppler (Fl), 2-way doppler (F2), and 3-way doppler (F3).
In carrying out the integrations, the odd derivatives of
[I - ( f R / f T ) ] with respect to t3(ST)vanish, and the
fourth and higher even derivatives are ignored. The resulting expressions are
ty F1 = CzfsI(c1 -
k)" F2 = C,fq(tl)(1 -
i;
(291)

the receiving station, t3(ST), is the midpoint t,, ( S the count interval T, (station time). The quantity
- [l ( f R / f T ) ] " is the second derivative of [I- ( f R / f T ) ]
with respect to t3(ST).The first term that has been truncated in Eq. (292) is (1/1,920) (2':) [l- ( f ~ / f ~ ) ] ' ~ where iv indicates the fourth derivative with respect to t,(ST). For 1-way doppler, fsIa and the second term of
Eq. (289) are evaluated with the spacecraft transmission time tz for the above-mentioned light path.

For 2-way or &way doppler, the definition of f R / f T is

---.-=- f R - dn dtl (ST) dt, (ST)

f T dt,(ST) dn

dt3(S

(293)

where

dn = infinitesimal number of cycles transmitted at time tl. The dn cycles travel at constant phase from the transmitter to the receiver and are received at time t,. The propagation speed is the phase velocity, which is greater than c in the presence of charged particles.
dt,(ST)= infinitesimal period (of station time ST) for transmission of dn cycles from transmitting station at time tl.
dt,(ST) = infinitesimal period (of station time ST)for
reception of dn cycles at receiving station at time t,.

Equation (293) may be written as

(-dST-l) - d y ( d r )

-f _R - dUTC

dt 1 - dt1 - dt,

fT

- dST
(dUTC),

- dUdTTC (ddt~),

dt,

dt3

where

(294)

dt,, dt,, dt3 = ephemeris time (ET) value of transmission interval [dt, ( S T ) ]r,eflectioninterval at the spacecraft, and reception interval
[dt3(ST)].

where

$)* 2(I. (1 - =(I - f )+

-f)**

(292)

The quantities [l - ( f R / f T ) ] , - [l ( f R / f T ) ] " , and tl are
evaluated along the light path whose reception time at

The ratios dtl/dtz and dt,/dt, will be obtained by differentiation of the light time equations for the up and down legs of the light path. The factors (dT/dt) at tl and t3 transform dt, and dt, from ephemeris time to proper time P2 obtained from imaginary ideal atomic
Z Z T h i s time scale was defined in Section I1 after Eq. (58).

7

clocks at the transmitting and receiving stations; they are computed from q. (58) using the Newtonian potential at each tracking station and the heliocentric velocity of each tracking station.

The factor dUTC/dr converts the transmission and reception intervals from seconds of atomic time T to seconds of UTC atomic time. These i n 7 0 atomic time scales differ only in the length of the second (the number of cycles defined equal to 1s).

The factors dST/dU at t, and t3convert the trans-

mission and reception intervals from UTC seconds ob-

tained from ideal atomic clocks to seconds of station time

ST obtained from the actual atomic clocks at the trans-

mitting and receiving stations (the same station and clock

for 2-way doppler). The transformation from UTC to ST

at each tracking station is specified by Eq. (94), repeated

here:

+ + UTC - ST = a bt ct2

(295)

where a, b, and c are specified by time block and t is in seconds past the start of the time block. Let the coefficients of Eq. (295) which apply for the receiving station at t3 and for the transmitting station at t, be denoted by subscripts R and T respectively. Also, define F by:

The effect on 2-way doppler of the variation in F during the count interval T , is about m/s, which is completely negligible. The corresponding effect on 3-way doppler is about m/s, which is the desired accuracy for computed doppler observables. owever, the error in 3-way doppler due to the unknown difference in frequency of the two atomic frequency standards (Af/f 2 X lO-ll) is a few mm/s, which probably cannot be reduced to the 10-5-m/slevel by estimating the b and c coefficients of UTC - ST for the transmitting and receiving stations. Thus, the variation in F during the count interval T , is ignored and

Substituting Eqs. (299) and (300) into Eq. (292) gives

where

(301)

Substitution of Eqs. (301) and (302) into Eqs. (290) and
(291) gives 2-way and 3-way doppler as a function of
- (FR/FT)],[I - (FR/FT)I* ', and F.

dUTC

For 1-way doppler, the definition of f R / f T is

Then, since dST/dUTC is extremely close to unity,

where the transmission and reception times t, and t3are expressed as seconds past the start of the time blocks for a, b, and c used at t, and t3, respectively. Also, define FR/FT by

since fsIc is referenced to an imaginary UTC atomic clock on board the spacecraft. This equation may be written as
.(m),dT\dt),

Then, substituting gs. (296) and (298) into Eq. (294)

gives

( ( 2 ) l - &f T > = ( l + F ) 1 - - - F

(299)

As in Eq. (296), define Fl by
( 1
+ =
Then,

(305)

J

where t3is expressed as seconds past the start of the time block for a, b, and c used at t3.Also, define F R / F T €or 1-way doppler by

-_-- F R - (%)z dt,

(307)

the midpoint t3,n(ST) of the count interval T,. Expres-
sions for these quantities are derived in Sections VIII-C and -D respectively, starting from q. (298) for F R / F T for 2-way and 3-way doppler and Eq. (307) €or 1-way doppler. The quantities fs/c, F , and Fl are computed from Eqs. (277), (297), and (306), respectively. The quantities
- [1 ( F R / F T ) l , [1 - (FR/FT)I.*7F , Fz, fs/c, t z , ti, and A
are evaluated with quantities obtained from the light time
solution for the above-mentioned light path (see Sec-
tion VI).

Substituting Eqs. (305) and (307) into Eq. (304) gives Eqs. (299-302) with F replaced by F, and F R / F T defined by Eq. (307).
Substituting Eq. (301) into Eqs. (289),23(290),and (291) gives the final expressions for the computation of 1-way doppler (Fl),2-way doppler (F2),and 3-way doppler (F3). Each of these expressions contains an additive correction A, which accounts for the effects of the troposphere, the ionosphere, and the motion of the tracking point on the transmitting and receiving antennas during T,. The computation of A is described in Section XII. The expressions for F1, F2, and F3 are

Equations (308), (309), and (310) are used to compute 1-way,2-way, and 3-way doppler using either the L-band, L-S band, or S-band values of the coefficientsCz,C3,and
C,.In the G S band configuration, the so-called 2-way
doppler observable is actually 3-way doppler (from the electronics point of view) obtained using the same tracking station as the transmitter and the receiver. This data type is computed from the 3-way formula, Eq. (310).
Another data type not previouslymentioned is coherent 3-way doppler, which is essentially 2-way doppler obtained from two different tracking stations. The two stations are only a few kilometers apart and the reference frequency f q ( t 3is) beamed from the transmitter to the receiver via microwave link. Coherent 3-way doppler is computed from the 2-way formula, Eq. (309).

+ - cz [AfT, f f~~( t z - t o ) f~~( t z - to)']

(308)

(( )* [ ( )*]

F 2 = C 3 f q ( t 1 ) 1 - - FR - F 1- 1- - FR + A

FT

FT

(309)

The term in Eq. (308) containing F, and the term in Eq. (309) containing F are not included in the current DPODP formulation. The latter wiIl be added at the
earliest convenience,and the former will be added when fs,c is derived from an atomic frequency standard on board the spacecraft instead of the currently used crystal oscillator.

l - 5 )F~T~ - F [ l - ( l - ~ ) * ] + A
(310)
where [l- ( F R / F T ) ] *is given by Eq. (302) in terms of [1 - (FR/FT)a]nd its second derivative with respect to
t3(ST),[I - (FR/FT)]", evaluated along the light path
whose reception time at the receiving station, t3(ST),is
z3With F replaced by F,.

Because of truncation of the fourth and higher even derivatives of [l - (FR/FT)] in Eq. (302), the doppler observables are limited to count times as low as 1-10 s when the spacecraft is near a planet and no more than
roughly 1,000 s in heliocentric cruise. However, larger count times may be used if the subinterval doppler formulation is utilized. With this method, the count time T , is divided into m subintervals of length T,/m. For each subinterval, a light time solution is obtained for the light
path with reception time t 3 (ST) equal to the midpoint of the subinterval, and a doppler observable F (1-way, e-way, or 3-way doppler) is computed using T,/m in place of T , in Eq. (302).

J

7

Let the observable computed for subinterval i be denoted as Fi. Then, the observable for the overall count interval T, is given by
1m
This follows directly from Eq. (287).
Predicted values of the number of cycles N which a station will observe in a given count interval T, are computed from

Sohtion of these equations (see Section VI) gives the following quantities:
tl,t,, t3= ephemeris time (ET) values of transmission time at tracking station on earth, reflection time at spacecraft (or transmission time for 1-way doppler), and reception time at tracking station on earth, respectively. The station
time (ST)value of t3is the midpoint of the
count interval T,.
rl, r,, r3 = heliocentric position vectors of transmitting station on earth at tl, spacecraft at t,, and receiving station on earth at t3,respectively, with rectangular components referred to the mean earth equator and equinox of 1950.0.

where F =FlyF2, or F3 and fbias is the correspondingbias frequency from Eq. (281), (282), or (283). Equation (312) follows directly from Eq. (287).
. Doppler Frequency Shift
The expression for [1 - (FR/FT)]used to compute 2-way and 3-way doppler and also the expression used to compute 1-way doppler are derived in this section. The definitions of F R / F T are Eq. (298) for 2-way and 3-way doppler and Eq. (307)for 1-way doppler, evaluated along the light path whose reception time at the receiving sta-
tion, t3(ST), is the midpoint of the count interval T,.The
expressions for [I - (FR/F~a)r]e obtained as expansions in powers of l/c. In order to obtain the desired accuracy of lC5m/s for computed doppler, all terms to order l/c3 are retained.

ti,Y;,Yj = heliocentric velocity, acceleration, and jerk
vectors of participant i at its epoch of participation ti (i = 1,2, or 3). The dots indicate differentiation of ri with respect to ephemeris time.
The quantities on the right-hand sides of Eqs. (313) and (314) are

r1 = (rl * rl>'h

(317)

The terms atl/&, and dt,/dt3 are obtained by differentiation of the light time equations for the up and down legs of the light path. The light time equation for a given leg of the light path is Eq. (88) or (203). For the up and down legs, it is given by

c = speed of light, km/s

/L# = gravitational constant of sun, km3/s2

and

y = solve-for free parameter of the

theory of relativity. The parameter y is related

to W, the coupling constant of the scalar field,

through Eq. (41).

q. (313) with respect to t2gives

. +-+-+-- + - dr, - dtl dr, arI2 ar12dt,
(1 7 )pa dt, dt, dt, at, at, dt2

+

c3

+ + r1 r2 r12

c3

+ r1 - T2 Tl2

The derivative of Eq. (314)with respect to t3is obtained from Eq. (320) by replacing the subscripts 1 and 2 by 2 and 3, respectively. The expression for dtl/dt2obtained from Eq. (320).is unity plus terms of order l/c and greater arising from the l/c (Newtonian) term of Eq. (313) plus a term of order l/c3 arising from the l/c3 (relativity) term of Eq. (313).

(327)

Since terms of order greater than l/c3 are not retained in dtl/dt2from Eq. (320), the factor dt,/dt2 appearing in the l/c3 terms may be approximated by unity. The derivatives appearing in Eq. (320) and combinations of them are given by

(329)
Substituting these expressions into Eq. (320) and using dt,/dt, = 1 in the l/c3 terms gives

Using the notation the remaining terms are

(323)

where

+ + + ;, - i, - i,, i, i, i,,

+ + + €12 = r, T , - T,a

9.1 T , T i ,

1 32 2+ 3

(331)

(324)

The first term of Eq. (331) approaches 0 f 0 as the distance from the light path to the center of the sun ap-
owever, because of the finite radius of
the sun (700,000 km),the limiting indeterminacy will not
occur. For a light ray grazing the surface of the sun and
+ rl = r, = 50 AU, the sum T , r, - T,, is about 65 km. + Since (T, T,) and r12are 100 AU, which is represented

7

to km on the 16-decimal-digit BM 7094 computer,

SubstitutingEq. (332) into the reciprocal of the denomi-

the 65-kmdifference is represented to 7 decimal digits.

nator of Eq. (330) and expanding gives

+ For any case where the light path grazes the surface of
the sun and rl r2z r12,the contribution to the spacecraft range rate from the first term of Eq. (331) is a maximum of about 0.5 m/s (for a spacecraft velocity of 100 km/s). Since the denomination of this term is represented to at least 7 decimal digits, the contribution of 0.5 m/s is accurate to at least m/s, which is smaller than the desired accuracy of m/s for computed doppler. Thus, the numerical difficulties associated with the first term of Eq. (331) are not significant.

(334)
Multiplying by the numerator of Eq. (330) and retaining terms to order l/c3 gives

Let

and, for the down leg,

l+ 2 2+3

(335)

From Eq. (58), the quantities (ds/dt),, ( d ~ / d t ) ~an, d (ds/dt), are given by
=[l-F-($)w2]i=1,2,or3
(337)
where
+; = Newtonian potential at participant i at its epoch
of participation ti
i; = heliocentric velocity of participant i at its epoch
of participation ti
+; The potential is given by

where the summation over i includes the sun, all of the
planets, and the moon, and rij is the coordinate distance
from the participant i to the center of the body i. The velocity & is obtained from

Since terms of order greater than 1/c3 are not retained, Eq. (337) may be approximated by

For 2-way or 3-way doppler,

(340)

(338)

3

where terms of order 1/c4 have been ignored. Similarly, for 1-way doppler,

SubstitutingEqs. (341) and (336) into Eq. (298)and retaining all terms to order l/c3 gives the desired expression for

- 1 [

(FR/FT)]

for 2-way or 3-way doppler:

(343)
This quantity is used in Eq. (302), which is substituted into Eq. (309) or (310) to compute 2-way or 3-way doppler. Similarly, substituting Eq. (342) and dt2/dt3obtained from Eq. (335)into Eq. (307) gives the expression for [1 - ( F R / F T ) ] for 1-way doppler:

Equation (344) is used in the computation of 1-way doppler from Eqs. (302)and (308). Note that setting all up-leg factors equal to zero in Eq. (343) and changing $1 and 8, to +2 and 8, gives Eq. (344).

For 1-way doppler, +z and $3 are computed from Eq. (338) as indicated after that equation.

For 2-way or,3-way doppler, +l is very nearly equal to $3. The contribution to (& - $3) from the other planets and from the moon affects the observable by less than
m/s and hence can be ignored. Thus, only the potential from the sun and from the earth needs to be considered, and and $3 are given accordingly by
(345)
(346)
e where and are the geocentric radii of the transmit-
ting and receiving stations, respectively. The second terms of Eqs. (345) and (346) are required for the computation
of 3-way doppler but cancel in (+1 -43) used for 2-way
doppler.

The computation of doppler observables requires an
expression for [1- ( F R / F T ) ] * ,which is the second derivative of [ l - ( F R / F T ) ]with respect to the reception time
t3(ST),evaluated along the light path whose reception time is the midpoint of the count interval T,. The expression for [l- ( F R / F T ) ] *f'or 2-way and 3-way doppler and also the expression for 1-way doppler are derived in this section. They are obtained by differentiation of the corresponding expressions for [1- (??R/FT)] obtained from Section VIII-C.
In order to limit the doppler truncation error (due to ignoring the fourth and higher even derivatives of the frequency shift in Eq. 302) to m/s or less, count times as low as 1-10 s must be used when the spacecraft is very near one of the celestial bodies of the solar system;

7

alternatively, when the spacecraft is in heliocentric cruise, count times as large as 1,000 s may be used.
For either of these situations, the l/c3 terms of - [1 ( F R / F T ) ] affect doppler observables by less than
m/s. Hence, the expressions for - [1 ( F R / F T ) ] * * are
obtained by differentiating Eqs. (343) and (344), ignoring the l/c3 terms. For 2-way or 3-way doppler, the variations in (+1 - +,)/c2 and (i?- i3/2c2 over the count interval affect the observable by less than m/s; hence these terms are also ignored. For 1-way doppler, the corresponding terms and their variations are quite large. However, they have not been included in the expression that is differentiated because of the inaccuracy of 1-way doppler obtained by using a crystal oscillator on board the spacecraft.

The second term of Eq. (349) affects doppler observables by less than 10-lo m/s and hence can be ignored. The
second derivative of [I - ( F R / F T ) ] with respect to t3(ET)
contains l/c and l/cz terms and hence is accurate to about 8 figures. The multiplicative factor in the first term of Eq. (349) is unity to about this many figures; hence, it may be ignored. Thus, [1- ( F R / F T ) ]* * is computed from
In terms of first and second derivatives of the terms of Eq. (347) with respect to t, (ET), denoted as t,,

In the future, when 1-way doppler derived from an
atomic frequency standard becomes available, it will be mandatory that [1- (F R / F T ) ]* * include the derivatives of (& - +,)/c2 and (B; - i3/2cz. For 2-way or 3-way doppler, [1 - F R / F T ] * is obtained from

For 1-way doppler, the corresponding expression is

which applies for 2-way or 3-way doppler. Similarly, from Eq. (34%

The terms in Eqs. (347) and (348) are functions of the heliocentric position and velocity vectors of the transmitter, spacecraft, and receiver at their epochs of participation. Since the time unit for the velocity, acceleration, and jerk vectors of each participant is ephemeris time (ET), the derivatives of [l- ( F R / F T ) ] ,which are obtained naturally, are the first and second derivatives with respect to t3(ET). Given these quantities, the second derivative with respect to t,(ST) is

(352)
which applies for 1-way doppler.
The quantities i12i ,2 3 , Ijl2, and l j 2 3 are functions of tl(ET), &(ET), and &(ET) which will be denoted as tl, tz,and t,, respectively, in the remainder of this section.

d2 dt, (ST)z

[ ( 2)][ ] [ $)] = dt, &')2

-

dt3 (ET) dt, (ST) +

d
(I -

d't, (ET) dt3(ST)2

(349)

In order to obtain derivatives of these quantities wi respect to t3,the following subpartiall derivatives are required:

Substituting Eqs. (356)

The terms above are derived from Eqs. (313) and (314), ignoring the 1/c3relativity terms.

fi12, The first and second derivatives of ;12, ;23,

and 6 2 3

with respect to t, are functions of the following partial

derivatives, whose s u m s are denoted as:

The following derivatives are required to order l/co: (364)

. + r,,

=

- ar,,
at,

- al;,
at,

1-2 2+ 3

..r,, = -aa+ht2, - aait,,,

...
TI,

=

-aa;+.t;,,

- aa?,t,

1+2 2+3

PI2 = - - aa~t,, ,

1+2 2+ 3

.. -+ p,, = a h z - a h 2 - --a;,,

at, at,

at,

1+2 2-3

.p..,,

=

- a i +~- afi12 -
at, at,

--a&
at,

1+2 2+ 3

(355)

where the previously d e h e d quantities i,, and $, have been included for completeness. Equation (358) follows from Eqs. (332), (333j, (325), and (328). 'Substituting Eq. (358) into the first form of Eq. (359), changing the order of differentiation in the second mixed partial derivative, and substituting Eq. (355)gives the second form of Eq. (359).Similarly, the second form of Eq. (360) follows from the second form of Eq. (359)and from Eq. (356).

Differentiating Eq. (362) with respect to t3,using Eqs.

(353) and (354),gives

( a> [ ;,)I+ -d=2h-,
dtg

a?,
at,

1--

+ - a?,
at,

- -1 (ilz C

Since 1/c2 terms are ignored, the l/c terms were differentiated by inspection using Eqs. (353) and (354) equal to unity. Substituting Eqs. (357) and (360) gives

-d='i.1, dtg

... C + Ti2 f

1 [2(+125z

- +a%,)

%z (&z - V m ) ]

(367)

Similarly,

+ + -d=2i2,
dtg

.*r. 23

71 (2+23?;3

;23&3)

(368)

Using (353)and (354),one obtains
+-- -d;-,, --ai-,, dt, a& dt,
dt3 at, dt, at, dt3

and, to order l/co,

Substituting Eqs. (362-365) and qs. (367470) into Eqs. (351) and (352) gives, for 2-way and 3-way doppler,

The quantities in Eqs. (371) and (372) are defined in Eqs. (355360). They are computed from:

where

r,, = - rlZ i,, rlz
p,, = - rl2 * i.,
TlZ
rij = rj - ri

. .....
r + r,x, r

1+2 2+3
1+2 2+ 3
1+2 2+ 3
l+2 2+ 3
1+ 2 2+ 3
1+2 2+3

Equations (373) and (376) are Eqs. (326), (329), (332), and (333). The remaining equations follow by successive differentiation according to Eqs. (356), (357), (359), and (360).

This section gives the formulation for computation of range observables.

same tracking station at time t3.The mathematical representation of the range observable p is
p = (t3- t1),* F , modulo M

There are several different range tracking systems. owever, all of them are conceptually the same. For each system, an electromagnetic signal is transmitted from a tracking station on earth at time t,, received and retransmitted by the spacecraft at time tz,and received by the

where
(t3- tl)sT = round-trip time of the signal in seconds of
station time ST (derived from the atomic
frequency standard at the tracking station)

7

F = conversion factor from seconds of station time ST to the units of the range observable
M = modulo number. The largest integer mul-
tiple of M which is less than (t3- ti),* F is removed from this quantity, leaving the
observable p, which is less than M. This operation on a number n will be referred
to as “modding” n by M.
The conversion factor F and modulo number M for each ranging system are given in Section IX-C.
The first step in obtaining the computed value of a range observable is to solve the light time equations for the down and up legs of the light path, whose reception time t3(ST) is the observation time. This light time solution, described in Section VI, gives the quantities used to compute a precision value of the round-trip light time in seconds of ephemeris time. This precision value is converted to seconds of station time by using the time trans-

formations of Sections I1 and 111. Corrections are added to account for the effects of the troposphere, the ionosphere, and the offset of the tracking point on the antenna from the earth-fixed “station location” on the antenna mount. In addition, the estimated value of a range bias is added. This sum for the round-trip station time is multiplied by F and modded by M , as indicated above. The expression for computing the range observable p is given in Section IX-B. The computation of the troposphere, ionosphere, and antenna corrections is described in Section XII.
Section XI contains the formulation for computation of doppler observables from differenced range observ-
ables divided by the count time T,.The required changes
to the range observable formulation of this section, which are minor, are described in Section XI.
.~ o ~ m M ~ a t i o ~
The range observable p, obtained from any of the tracking systems described in Section IX-C, is computed from:

Equation (379) is evaluated with quantities obtained from the light time solution for the observable, listed after Eq. (314).The epochs of participation ti,tz,and t3 are available in the ET, Al, UTC, UT1, and ST time scales. The quantities ri2, r23, rl, rz, and r3 are computed from Eqs. (315-319). The definitions of c, pN,and y follow Eq. (319). The time transformations (ET - Al), (A1 - UTC), and (UTC - ST) are given by Eqs. (93),
(95), and (94), respectively. The quantity 8 (ET - Al), to
be discussedbelow, represents additional relativity terms of (ET - A l ) not contained in Eq. (93), which is used in the general time transformation subroutine of the DPODP.
Each of the four time transformations of Eq. (379) is evaluated with the transmission time tiand with the reception time t3,expressed in one of the two time scales related by the transformation. Either time scale may be

used, but the same time scale must be used at both tl and t3.The remaining terms of Eq. (379) are
R, = estimated constant range bias (specified by time block for each station)
Aip ( j ) = range correction in meters due to i = A (antenna offset), T (troposphere), or I (ionosphere) for down leg ( j = t3)or for up leg
(i = ti)
The sum of the first two terms of Eq. (379) is the righthand side of the light time equation for the up leg of the light path (Eq. 313). Similarly, the sum of terms 3 and 4 is the right-hand side of the light time equation for the down leg of the light path (Eq. 314). The sum of these

four terms is an accurate expression for the round-trip ephemeris time. The largest error in the computation of this quantity arises from truncation of the epochs of participation beyond a precisionz4of s.
The maximum conceivable heliocentric velocity of the spacecraft is 1,000 km/s. For this velocity, the maximum error in the computed round-trip ephemeris time due to truncation of the epochs of participation is 1.4X s. The corresponding error in range is 0.4 m round trip or 0.2 m one way. The typical errors are at least one order of magnitude lower than these figures.
An alternative method for obtaining the round-trip ephemeris time would be to subtract the ET values of the epochs of participation t3and tl. However, this difference
could be in error by as much as 2 X lo-‘ s. The corre-
sponding range error would be 60 m round trip or 30 m one way, which would be unacceptable.
The time transformations of Eq. (379) convert the precision round-trip light time from an interval of ephemeris time to an interval of station time ST. The remaining terms of Eq. (379) account for the effects of the troposphere and the ionosphere, the offset of the tracking point on the antenna from the earth-fixed “station location,” and a constant range bias R,, whose value may be estimated.
Section XI contains the differenced-range doppler formulation, Le., the formulation for computing doppler observables from differenced range observables divided by the count time. The required analytical change to the range observable formulation consists of a more accurate expression for the (ET - Al) time transformation used to transform the round-trip light time from ephemeris time to station time. The required expression is Eq. (65), which is derived in Appendix B.
The (ET - Al) time transformations in Eq. (379) are evaluated with the general time transformation subroutine of the DPODP. This subroutine computes (ET - Al) from Eq. (93), which consists of the first three terms of Eq. (65). Currently, S(ET -Al) in Eq. (379) consists only of term 4 of Eq. (65). The following listing gives
24On the 16-decimal digit IBM 7094 computer, time is represented as seconds past January 1, 1950, 0”to a precision of 0.6 X 10-7 s from 1967 to 1984.

the maximum contributions to 1-way range (p/2) from each of terms 3-10 of Eq. (65):

Term No.
3 4 5 6 7 8 9 10

Contributionto 1-wayrange (m/AU of 1-way range)
50 22 0.4 0.007 1 0.02 0.6 0.01

The observables obtained from the Tau or Mu ranging systems described in Section IX-C have a potential accuracy of about 1m or slightly better. In order to obtain the maximum benefits from these accurate data types, the computed range observables should have an accuracy of about 0.1 m. For the forthcoming Grand Tour missions to the outer planets, the range to the spacecraft will be several tens of AUs, and all of the relativity terms of Eq. (65) will contribute more than 0.1 m to it (see the listing above). Therefore, terms 5 through 10 of Eq. (65) should be added to 8 (ET - Al). There is a smallmonthly variation in (ET - Al), which is not included in Eq. (65) since it does not significantly affect differenced-range doppler. However, it does affect 1-way range by about
0.05 m/AU. Hence, an expression for computing this term should be derived and added to 8 (ET - Al).
The second and fourth terms of Eq. (379) are the relativistic corrections to the light time for the up and down legs of the light path. These terms become very large when the spacecraft approaches superior conjunction and the minimum distance from the up and down legs of the light path to the surface of the sun becomes very small. For this situation with the light ray grazing the sun of radius R and with the earth and the spacecraft at the same distance r from the center of the sun, the relativistic correction to the light time for each leg of the light path is given approximately by

With y = 1, its general relativity value, r = 1AU =
150X lo6 km, and R =0.4X lo6 km, =&is 1-way light
time correction amounts to 36 km/c. The round-trip range
observable is affected by 72 km/c or 240 ,ps. This is the

L

7

only really large effect of general relativity on earthbased tracking data. Fitting to tracking data obtained from a spacecraft which is in the vicinity of superior conjunction provides this so-called fourth check of general relativity. Presuming that the observed minus computed range residuals will be vastly smaller when the second and fourth terms of Eq. (379) are turned on, fitting to these tracking data should provide an estimate of the
parameter y and hence, from Eq. (a)t,he coupling
constant o of the scalar field of the Brans-Dicke theory of gravitation.

Ranging system AFETR Mark 1 Mark 1A

- c
2
- ;yf, (11)
96 X 1,496,500
Id
lo6

None

785,762,200

785,762,208

1.00947x 1.0002

loo

64 x zn x ,os
3 f , (fl)

To date, range tracking data have been obtained from five different range tracking systems: the Air Force Eastern Test Range (AFETR) pulse-radar ranging system, the Mark 1 and Mark 1A lunar ranging systems, and the Tau and Mu planetary ranging systems. The latter four systems have operated at tracking stations of the Deep Space Network. The lunar ranging systems are used for lunar missions and during the early phases of planetary missions. The planetary ranging systems are used for all deep space applications. The Mark 1 system has been replaced by d e Mark 1A system. The Mu system is the latest research and development planetary ranging system. Both the Tau and Mu ranging systems have a potential accuracy of a few meters and possibly as low as 1m or slightly better. Table 1gives d e values of the conver-
sion factor F and the modulo number M for each of these
systems, where
c = speed of light, km/s
fq ( t l )= reference oscillator frequency at transmitting
station, cycles per second of station time ST
(derived from transmitter atomic frequency standard), evaluated at transmission time tl
n = number of components of ranging code used with Mu ranging system
The frequency fq(t,) is the same quantity used in the computation of doppler observables. The number n associated with the Mu system varies from a typical value of 10 to the maximum system capability of 18.
servables are referred
station to the spacecraft. 1.04 m. Using the nomigives approximately the
The units of the Tau

and Mu observables are round-trip nanoseconds and microseconds, respectively.
The Mark 1and 1A range observables are modded by
approximately 800,OOO km in 1-way range to the space-
craft. The corresponding figure for the Tau system is 150,000 km. Using the maximum value of n = 18 and
fq ( t l )= 22 X lo6 Hz, the Mu range observables are
modded by about 38,000 km,one way.
The AFETR range observables computed by the DPODP are expressed in one-way kilometers and are not modded; they are used primarily for study purposes.
For all practical purposes, all of the range tracking systems except the Mu system provide a continuous measure of the range observable p given by Eq. (379). The Mu system provides one range observable each time the ranging system is initialized during the pass of the spacecraft over the tracking station. Also, it provides a direct measure of the correction to all 2-way doppler observables obtained during the pass due to charged particles of the ionosphere and interplanetary medium.
The output from the Mu ranging system at time t is
Po 01,given by
The ranging system is initialized at some epoch to during the pass of the spacecraft over the tracking station. The first term is the range observable p obtained at time t. The second term is counted doppler from d e epoch toto t. It is the 2-way doppler observable F2 of Section VIIH multi-
plied by the count time T,,which extends from toto t, and with the units converted from those of F2 to those of p.

In the absence of charged particles, counted doppler is equivalent to differenced range:25

correction Acp(to)for a range observable. Thus, the output from the Mu ranging system is a range observable p corresponding to Eq. (379)only at an initialization epoch.

and the output from the Mu ranging system would be

The output p o ( t ) of the Mu ranging system evaluated at an epoch tzminus the value at an epoch tl is

That is, the output would be constant and equal to the range p at the initialization epoch to.With charged particles present,
p (t)corrected = p ( t ) Acp ( t ) and
where A,p (t) is the correction to p (t)due to charged particles. The effect of charged particles on counted doppler is the negative of the correction to the corresponding differenced range observables. Thus, when doppler observables are computed from differenced range observables, the sign of the charged particle correction to each range observable must be changed. From the two equations above, the output of the Mu ranging system with charged particles present is

This quantity is an observed value of twice the charged particle correction to the 2-way doppler observable whose count time T , extends from tl to tz.
This section gives the formulation for computing angular observables, which are of two types: (1)directly observed angles of the incoming radiation relative to the tracking station’s earth-fixed reference coordinate system, and (2) optical angles-topocentric right ascension (Y and declination &obtained from reduction of photographic plates. As opposed to directly observed angles, optical angles do not contain effects due to stellar aberration and atmospheric refraction (to f i s t order).
The directly observed angle pairs are: (1)hour angle HA and declination 6-most DSN stations; (2) azimuth u and elevation y-AFETR stations and some DSN stations; (3) X, Y angles-Manned Space Flight Network (MSFN stations); and (4)X’, Y’ angles-MSFN stations.

The output at t = tois
This quantity is equal to p computed from Eq. (379), with the reception time t3 equal to the initialization epoch
+ to.The charged particle correction aCp(tois) the round-
trip ionospheric correction [AI,p(t3) Alp (tJ]F/103c of
Eq. (379). The output p o ( t ) for t > tois not a true range
observable with reception time tobecause the charged particle correction 2A,p (t)- A,p (to)does not equal the
25However, differences can arise from sources other than charged particles, such as from variations in the electrical path length through the range tracking system which differfrom those of the doppler tracking system.

The topocentric coordinate systems and unit vectors associated with each directly observed angle pair are described in Section X-A. The formulation for computing the direction of the incoming radiation and each pair of angular observables is given in Section X-B. Corrections to the directly observed angles due to small solve-for rotations of the earth-fixed reference coordinate system are given in Section X-C. Partial derivatives of the angular observables with respect to the heliocentric positions of the spacecraft and the tracking station are given in Section X-D. These will be used in Section XIV to form the partial derivatives of the angular observables with respect to the solve-for parameters.

yste

kS

The reference coordinate system at each tracking station is rigidly fixed to the earth, and its orientation relative to the true pole, equator, and prime meridian varies with

the motion of the pole (see Section VII). The maximum excursion of the earth‘s axis of rotation from its mean position is about 10 m, and since the latitudes of all tracking stations are low (less than 45 deg), the maximum change in the orientation of the reference coordinate system from its mean orientation relative to the true pole, equator, and prime meridian is about 1arc second. The maximum attainable accuracy for directly observed angles is about 0.002-0.003 deg or 7-11 arc seconds, and thus the 1-arc second variations due to polar motion may be neglected. Therefore, the computation of directly observed angles is based upon a fixed orientation of the reference coordinate system relative to the true pole, equator, and prime meridian,
1. Right ascension, hour angle, and declination. Figure 5 shows a rectangular coordinate system centered at the tracking station on earth. The x- and y-axes are parallel to the earth’s true equator; the x-axis is toward the true vernal equinox, and the z-axis is parallel to the true axis of rotation of the earth, directed north.

+ observer’s meridian contains the unit vectors
makes an angle ( e h) with the vernal e
e = true sidereal time = Greenwich hour angle of true
equinox at reception time t3
x = east longitude of tracking station, relative to true
pole
The sidereal time 19is computed from Eq. (269) and associated equations, using t3(UT1) and t3(ET). The unit
vector E is normal to P and . The angle HA is the hour
angle of the spacecraft.
Nominal computed values of directly observed HA and 6 are based upon the geometry of Fig. 5. However, the reference coordinate system QEB may be rotated through
the small angles 5’ about .Q, E about E, and v’ about P,
thus changing the angle HA in the QE plane and the angle 6 normal to it. Corrections to the nominal computed values of HA and 6 due to the solve-for rotations
c‘, E, and 7’ are given in Section X-C.

The unit vector E is directed from the tracking station at the reception time t3 to the spacecraft (a free spacecraft or a station on some celestial body other than earth) at its transmission time t?. The angles CY and 6 are the right ascension and declination of the spacecraft. The
z

The unit vectors D and A in the directions of increasing declination and right ascension are used in computing the partial derivatives of CY,6, and HA with respect to the estimated parameters. The vector A is normal to L and The rectangular components of D and A along x, y, and z are

2. The north-east-zenith coordinate system. Figure 6

shows a rectangular coordinate system whose origin coin-

cides with the center of the earth. The x- and y-axes are

in the earth’s true equator with the x-axis directed toward

the true vernal equinox and the z-axis along the instan-

taneous axis of rotation, directed north. The unit vectors

+ ginate at the tracking station S, whose
an angle (e h) with the x-axis. The

s contained in the meridian plane and

makes an angle +g with the true equatorial plane, whe

+g is the comput

. The north vector

/

and east vector

and are directed to

X

the north and to the east, respectively. The angle ’

a-y, X-Y, and X’-Y’ are referred to the rectangular

r = solve-for geocen.tric radius of tracking station
Ue = mean equatorial radius of earth = 6,378.160 km
The eccentricity e can be computed from the flattening f, using a nominal value of 1/298.25, as

e2 = 2f - f"

(387)

3.Azimuth and elevation.Figure 7 shows the unit veo

coordinate system centered at the track-

ing station S. The angles u and y are the azimuth and

elevation, respectively. The reference coordinate system

may be rotated through the smaII angles 7 about N, E

about E, and g about Z. Section X-C gives corrections to

x/

the computed values of u and y as a function of the solve-

for rotations 7,E, and 5'.

coordinate system at the tracking station. The rectangular components of these unit vectors along x, y, and x are

The unit vectors 6 and (normal to L) in the directions
of increasing y and U, respectively, are used in computing
the partial derivatives. The components of 5 and along
N, E, and Z are

+ cos & cos (e A)
(384)

-sinycoso

-sin y sin u

[ ] =[ cosY -sic;nu]u

,.,

A=

(389)

+, The geodetic latitude of the tracking station is com-
puted from

90 = 9 + (98 - 9 )

(385)

where

4 = solve-for geocentric latitude of tracking station, re€erred to true pole and equator

and @, - 4 ) is computed from

N

B = eccentricity of reference spheroid 7

Using Eqs. (382,-384), the rectangular components of

referred to the true earth equator and equinox are

+ sinasin+,cos(O X) - cosUsin(0 + A ) + sinusin+,sin(6 + A ) cosucos(B + A )
-sin u cos (bg

(391)

4. The 21:and Y angles for MSFN stations with 6-m (20-fi) antenna. Figure 8 shows the angles X and Y referred to the NEZ reference coordinate system at the tracking station.

[z][ The unit vectors D' and A' (normal to k)are in the directions of increasing Y and X, respectively. The components
of D' and A' along the ,E, and Z axes are

' =

= -sinYcossiYnx]

-sinY cosx

(392)

A'=[:]=[

-sXc:i]nx

(393)

Using Eqs. (382-384), the rectangular components of and A' referred to the true earth equator and equinox are

+ ] ~ i n Y [ ~ i n x ~ i n ( e + x ) - ~+~A~) ] x-c~osY~s~in++,c,o~s(d~ ~+ A()e
-sinY [sinXcos(e + A ) c o s x ~ ~ + , s i n +( Ae) ] - cosYsin+,sin(e + A )
+, +, cos Y cos - sinY cos X sin

(394)

+ - ~ i n x c o ~ + , ~ s+(Ae)- cosxsin(e A) + -sinXcos+,sin(d + A ) cosXcos(B + A )
-sinxsin+,

(395)

5.

an

the

co

tations With26-m(85 antenna. Figure 9 shows the angles X' and Y' referred to at the tracking stati

The unit vectors

in the directions of increasing Y' and X', respectively. The components

D; [&I='

] sinY' sinX'
-sinYC'cOoSsYX' '

(396)

The rectangular components of

’ referred to the true earth equator and equinox are

The computation of each pair of angular observables requires the following quantities from the light time solution (see Section VI):
r3,i., = heliocentric position and velocity vectors of tracking station at reception time t3,with rectangular components referred to mean earth equator and equinox of 1950.0
rE = heliocentric position vector of earth at reception time t3, with rectangular components referred to mean earth equator and equinox of 1950.0

X-Y, and X’-Y’) and will be computed by a second procedure for optical right ascension-declination obtained from the reduction of photographic plates.
a. Directly observed angles. The unit vector E is directed from the heliocentric position of the tracking station at the reception time t3to the heliocentric position of the spacecraft (a free spacecraft or a station on some celestial body other than the earth) at its transmission time t,. This vector, with rectangular components referred to the mean earth equator and equinox of 1950.0, is denoted as I L50. t is computed from

r, = heliocentric position vector of spacecraft at transmission time t,, with rectangular components referred to mean earth equator and equinox of 1950.0

where

rZ3= r3- r,

(401)

t 3 (ET), t3(UT1) = ET and UT1 values of reception time t3.

The true sidereal time 0 at the reception time t3is com-

z

puted from Eq. (269) and associated equations, using

t3(UTI) and t3(ET).

5

1. Computation of unit vector L. The unit vector E will be computed by one procedure for the directly observed angles (hour angle-declination, azimuth-elevation,

c

E

.x

les

7

The unit vector L50 is directed from the station to the spacecraft in the heliocentric space-time frame of reference. In the observer's topocentric space-time frame of reference, the direction to the spacecraft is where AL50 can be derived from the Lorentz transformation of special relativity. The following first-order expres-
50 is the same as that due to the stellar aberration of light, the change in the direction of incoming light due to the heliocentric motion of the tracking station:

y are required. They are obtained f r o d Eqs. (423-425)
q. (404). Given u and y, the rectangular referred to the true earth equator and computed from Eq. (390). The refraction correction is computed as a function of th tion angle y from the formulation of D. Cain ( pp. 21-22):
y < 0.17 rad

where c = speed of light

(403)

The unit vector L with rectangular componentsreferred to the true equator and equinox of the reception time t3is denoted as LtrueI.t is given by

where

N s = surface refractivity at tracking station (see Subsections XII-B-2-a and -b).

b, = 1.0 - (1.216 X lo5b3 yrad)

- (51.0 - 300.0 yrad) (b3)%

(408)

precession matrix, transforming rectangular components of a vector referred to the mean earth equator and equinox of 1950.0 to components referred to the mean earth equator and equinox of t3.
nutation matrix, transforming rectangular components of a vector referred to the mean earth equator and equinox of t3 to components referred to the true earth equator and equinox of t 3 .
The A and M matrices are a function of ephemeris time and hence are computed from t3(ET).
true from Eq. (404)does not account for e incoming ray due to atmospheric refraction, which increases the elevation angle y of the incoming ray by &y. Referring to Fig,7, the change in L due to atmospheric refraction is Ary vector from the observer outward along the incoming ray is given by

1 b3 = 103( r - a,) pad= elevation angle, rad
a, = mean equatorial radius of earth = 6378.160km
r = geocentric radius to the spacecraft, km = llrz - rml
b. Optical right ascension and declination. Optical right ascension and declination obtained from the reduction of photographic plates are referred to the mean or true earth equator and equinox of a date t R , which generally is not equal to the observation time (the reception time t3).The unit vector L with rectangular components referred to the mean or true equator and equinox of t B is computed from
or

Lopt (true) =

This vector has been normalized since the value of the vector in the numgrator is slightly greater than unity. In order to compute and Ary, the azimuth u and elevation

(412)
where A (tR) and N (tR)are the precession and nutation matrices evaluated at the reference time t R . The vector

is computed from Eq. u and y computed from

and compute right ascension from
sina = - cLos,6 Odeg La:& 360deg

(420)

The right ascension and declination of a star obtained from the reduction of photographic plates are free from the effects of stellar aberration and refraction at least to first order. If a second-order plate reduction method is used, the effects of refraction can be removed completely.
owever, the right ascension and declination of a spacecraft obtained from the reduction of photographic plates axe affected to a small extent by refraction because the spacecraft is much nearer than the background stars. The expression for the correction to the computed elevation angle ~ , ydue to this effect has been derived by D. Cain (Ref. 50, p. 22). However, the sign of the correction is wrong and should be negative. The corrected expression is

L, Cosa:=- cos 6
b. Hour angle and declination. Compute a: and 6 from Eqs. (419-421). Compute HA from (see Fig. 5)
H A = ( e + . X ) - a , OdegLHAL36Odeg (422)
where 8 = true sidereal time at reception time t3 X = east longitude of tracking station, relative to true pole

(414)

,where

+0.00211
b4 = (yrad 0.0598)2.42

(415)

b, = (bt - a$cos2y)" - a, sin y
+ b6 = a, 51.2064

(416) (417)

The right ascension and declination of a spacecraft or star obtained from the reduction of photographic plates are
not affected by stellar aberration; hence, a,d,oe, s not
appear in Eqs. (411-413).

2. Computation of observed angles. The directly observed angles are computed from L given by Eq. (405). Optical right ascension and declination are computed from L given by Eq. (411) or (412). In either case, the rectangular earth equatorial components of L are denoted below by

c. Azimuth and elevation. Compute the unit vectors N, E, and Z for the reception time t3from Eqs. (38-84). Compute the elevation angle y from (see Fig. 7)

and compute the azimuth u from
sing = - L * E 0 deg -Lu4360deg cosy
cos0 = - L * N
cos Y Note that u is indeterminate for y = 90deg.

(424)

d. X and Y angles for MSFN stations with a 9-m (30-ft) antenna. Referring to Fig. 8, compute the angle Y from

and compute the angle X from

-90 deg . L X L90 deg

(427)

a. Right ascension and declination. eferring to Fig. 5, compute declination 6 from26

sin 6 = L,, -90 deg 46 . 4 90 deg

(419)

Note that X is indeterminate for Y = +90 deg, which can occur only when the spacecraft is on the horizon.
e. X' and Y' angles for MSFN stations with a 26-m antenna. Referring to Fig. 9, compute the angle Y' from

20The angular observables are measured in degrees.

sinY' = L-E, -90deg4YY'.L90deg (428)

7

and compute the angle X' from

-90deg LX' L90deg (429)
Note that X' is indeterminate for I" = _t90 deg, which
can occur only when the spacecraft is on the horizon.

The variation in 6 due to the variation in from Eq. (432) as
(cos6)as =LOA
Substituting Eqs. (435),(430),and (431) gives
+ AS = ['sin HA ECOS HA

(436) (437)

The computed angles may not agree with the observed angles because the mathematical representation of the orientation of the reference coordinatesystem at the tracking station differs from the actual orientation of the coordinate system. The difference in orientation is due to two errors: (1) errors in the mathematical model (primarily the difference between the actual plumb bob direction and the geodetic plumb bob direction computed from a reference ellipsoid of revolution), and (2) errors in orientation of the instrument axes (e.g., alignment of the vertical axis with the plumb bob direction for the azimuthelevation system).
Formulas 'are developed for corrections to the computed angles as linear functions of the small rotations of the computed reference coordinate system about each of its three mutually perpendicular axes.
This type of correction does not apply for right ascension and declination obtained from the reduction of photographic plates.
ur angle-declination. Referring to Fig. 5. the the rotations are and 9' about the rection, using the
right-hand rule.

From Eq. (430), the variation in HA is given by

(COS 6 sin HA) AHA = - L e a - (sin 6 cosHA) AS (438)

Substituting Eqs. (433),(431),(432), and (437) and simplifying gives

+ AHA= 9' tan 6 ( E sin HA - ['COS HA)

(439)

This same equation may be obtained by differentiating Eq. (431).

The meridian plane is determined by the vector P to the pole and by the plumb bob line. If the plumb bob is displaced to the west through the angle d, the meridian plane is displaced to the east through the angle

If 8," is known, this equation provides an a priori estimate of 9'.
2. Azimuth-elevation. Referring to Fig. 7, the reference coordinate system is and the rotations are 9 about

The dot products of

(430)

LeE = -cosSsinHA

(431)

=sins

(432)

In terms of the rotations, the variations in the unit vectors are
(433)

(434)

The variations in the unit vectors due to the rotations are
The variations in E :vation y and azimul u due to the variations in the unit vectors are obtained from Eqs. (423425). The results are

7

7

+ Ao = 5 - tany(7cosu Esino)

(444)

3.

.Referring to Fig. 8, the reference coordi-

nate

rotations are the same as for the azimuth-

elevation system. Using Eqs. (426), (427), (U)a,nd (441),

and Fig. 8,

A Y = - - s ~ h xf EWSX

(445)

+ + Ax = -9 tanY (esinX CcosX)

(W

4. Angles X’, F.“he azimuth-elevation reference co-
ordinate system and rotations are also used for the X’Y’ system. Using Eqs. (428),(429), (M),and (441), and

-

-3=,

T
50

ar, rZ3COS

ax2 r23
_ax _-- 4:
3r2 T,~COSY

’dr, r23
-a=x’ AZT
ar2 T23cosY’

Ax’= -c+tanY’(bcosX’- ?,7SinX’)

(447) ( 4 8 ) For any of these angles,

erivatives of Angular Respect to Heliocentric 1950.0PositionVectors of Spacecraft and Tracking Station
This section gives the partial derivativesof each angular observable with respect to the rectangular components of the heliocentric position vectors of the spacecraft and tracking station, referred to the mean equator and equi-
nox of 1950.0. These subpartial derivatives will be used in Section XIV to form the partial derivative of each
angular observable with respect to the total parameter vector q.

For the directly obsezed- angles, compute D , A from Eqs. (380) and (381), D , A from Eqs. (390) and (391), D’,A’ from Eqs. (394) and (395), and D”,A” from Eqs. (398) and (399). These unit vectors all have rectangular components referred to the true equator and equinox of the reception time t3.Transform the rectangular components of each ol these vectors to the mean equator and equinox of 1950.0 as

The partial derivatives of the observed angles with respect to r,, obtained from an examination of Figs. 5 and 7-9 are given below. In these expressions, a subscript 50 after a unit vector indicates that the rectangular components of the vector are referred to the mean earth equator and equinox of 1950.0.

For optical right ascension and declination, compute from Eqs. (380) and (381). For angles referred to the true equator and equinox the date t R , transform the rectangular components of and A to the mean earth equator and equinox of 1950.0 as

(- - -

, A050

&50

,

) rZ3cos 6 rZ3cos 6 rZ3cos 6

ar2 r23

(449) For angles referred to the mean equator and equinox of the date tR,

(450)

Note that the partial derivatives are computed using angles affected by refraction. Strictly, these angles should not include refraction, and the refraction correction should

also be differentiated with respect to the position of the spacecraft. Because of the approximationsmade, the partial derivatives of the angular observables with respect to the positions of the spacecraft and tracking station are accurate to roughly five sigdicant figures for near the zenith and three sigdcant figures for toward the horizon. These figures apply for directly observed angles. For optical angles obtained from the reduction of photographic plates, the secondary refraction correction and hence the error in the partial derivatives approaches zero with increasing range.
ler
This section gives the formulation for the computation of 1-way, 2-way, and 3-way doppler observables from the difference of two range observables whose reception times are the end and start of the count interval T,. The computation of accurate doppler observables with this differenced-range doppler formulation requires a computer with a large word length. On the Univac 1108 computer with a double-precision word length of 60 bits or 18 decimal digits, the formulation for the computation of 2-way and 3-way differenced-range doppler is accurate to about m/s for all count times above a lower limit which varies from about 0.1 to 1.5 s. This formulation was made possible by the derivation (in Appendix B) of an accurate expression (Eq. 65) for the transformation from coordinate time (ephemeris time ET) to proper time on earth (atomic time Al). The computation of accurate 1-waydifferenced-rangedoppler requires a similarexpression for ET minus A1 obtained from an atomic clock on board the spacecraft. This expression does not exist and the resulting 1-way formulation is accurate to only about
m/s for count times ranging from about 10 s when the spacecraft passes by a planet or the moon at very low altitude to about 1,000 s when the spacecraft is in heliocentric cruise.

The computation of doppler observables to an accuracy of m/s with the Taylor series formulation thus requires the computation of 43 observables for a 1/2-day pass of the spacecraft over a tracking station during heliocentric cruise. However, preliminary considerations indicate that the information content of a pass of tracking data during heliocentric cruise is not significantlyreduced if the count time is increased to about 8,640 s, which requires the computation of only five observables. The use of the differenced-range doppler formulation will allow these very large count times to be used and greatly reduce the number of observables which must be computed and hence the running time of the DPODP. Furthermore, the formulation is much simpler, which further reduces the running time and also decreases the size of the program. The differenced-range doppler formulation will be added to the Univac 1108version of the DPODP, either as a replacement for or alternative option to the existing Taylor series formulation.
Reference 51 demonstrates the m/s accuracy of 2-way differenced-range doppler. However, in order to obtain this accuracy for 2-way and also for 3-way doppler, a number of changes to the range observable formulation of Section IX are required. The primary analytical change is the use of the more accurate expression (Eq. 65) for the relativistic transformation from coordinate time (ephemeris time ET) to proper time (atomic time Al). Currently, only the first four terms of this equation are used. The increase in numerical precision from the 16 decimal digits of the IBM 7094 to the 18 decimal digits of the Univac 1108 is required; also, the precision of representation of time must be increased from double- to triple-precision seconds past January 1, 1950,0h.Alternatively, time could be represented as one single-precision word (8-decimal digits) for the Julian day number plus one doubleprecision word (18-decimal digits) for seconds past the beginning of the day. It is also recommended that the current type-50 n-body ephemeris be replaced by the more accurate type-66 ephemeris or the equivalent.

The primary advantage of the differenced-range doppler formulation is that there is no upper limit to the count time for 2-way or 3-way doppler, whereas count times used with the current Taylor series formulation (Section VIII) are limited due to truncation of the fourth and higher even derivatives of the doppler frequency shift in the Taylor series expansion. For an accuracy of m/s, the maximum allowable count time for the Taylor series formulation varies from 1-10 s when the spacecraft passes by a planet or the moon at very low altitude to about 1,000s when the spacecraft is in heliocentric cruise.

The expressions for the computation d 1-way, 2-way,
and 3-way doppler observables from differenced 1-way, 2-way, and 3-way range observables are derived in Section XI-B. SectionXI-C gives the numerical and analytical modifications to the 2-way range observable formulation of Section IX required for the computation of %way and 3-way differenced-range doppler. Also, the formulation is modified for an approximate computation of the change in 1-way range during the count time, used to compute 1-way differenced-range doppler.

7

The doppler observables are defined by Eq. (288), repeated here:

tl,(ST) = end of transmission interval T', tl, (ST) = start of transmission interval T',
Also, define 2-way range p2 and 3-way range p3 as

1 tl,(m)+(1/Z)Tc

F=-/

(f - fbias) d t 3 (ST)

tam(8T)-(1/2)Tc

(462)

The notation is that of Section VIII. Equations (284-286) give the expressions for f - fbias for 1-way doppler (Fl), %way doppler (F2),and 3-way doppler (F3),respectively. Substituting these equations into Eq. (462) gives

- + + F1= czfs/c 1 - 6 2 [.afT, f T 1 ( t 2 - t o ) f T z ( t Z - to)'] Tc (463)

pz t 3 (ST) - tl (ST) 2 4 3

(469)

where
tl (s ) = transmission time of the crest of a wave at
the transmitting station (station time at transmitter )
t3(ST) = reception time of same crest at receiving station (station time at receiver)

Then, the range p with reception time equal to the end of T , is

- F3= I c5f'Z(tl)
TC
where

(465)

and the range p with reception time equal to the start of T , is

pz, = t 3 , (ST)- t,, (ST) 2 + 3

(471)

For 2-way or 3-way doppler, f R / f T is given by Eq. (293) and

[ 1- t3,(ST)+(1/2)Tc
I=

1 -

dt3(ST) (467)

I = T , - T', = (ST) - t 3 8 (ST)] - [tl,(ST) - tl, (ST)]

-- Pze - Pz8

2+ 3

(472)

For 1-way doppler, f R / f T is given by Eq. (303) and

The count time T , is an interval of reception time; the corresponding transmission interval is denoted by TC and has midpoint tImT. hus,

t.3, ( S T ) + ( 1 / 2 ) T o

I =

[' ] d t z (UTG) - dt, (ST) dt, (ST)

,n(sT)-(1/2)Tc

(473)

The transmission interval at the spacecraft in UTG seconds (9,192,631,770 (1- S) cycles2' of an imaginary
cesium atomic clock at the spacecraft) is denoted by TE
and has midpoint tzm.Thus,

The epochs corresponding to the start and end of the
reception and transmission intervals T , and Ycare de-
noted as
t3e(ST) = end of reception interval T,
t,, (ST) = start of reception interval T ,

2 7 S e e Subsection111-A-4.

The epochs corresponding to the start and end of the transmission interval TE are denoted as
tze(UTC) = end of transmission interval TE t,, (UTC) = start of transmissioninterval TE
Also, define 1-way range pl as

The resulting error in differenced-range doppler (DRD) expressed in 1-way meters/second is

1 8 (ET - A1)g3, - S (ET - Al)t3e
SDRDZ/!l T,

L

-I

Then,

p l e = t3e (ST) - t,, (UTC)

(476)

= t 3 , (ST)- t z , (UTC)

(477)

Thus,

I = T, - TE = (ST) - t3,(ST)] - [he(UTC) - t z , (UTG)]
-- Ple - P l s

(478)

Substituting Eq. (478) into Eq. (463), and Eq. (472) evaluated with pz and p3 into Eqs. (464) and (465),respectively, gives

where i is the 1-way tracking-station-to-spacecraftrange-

..prait.se

evaluated at the midpoint of the count interval and
the time derivative of ,; assumed constant over T,.

The second term of Eq. (482) has been discussed in Section 111. It represents the time derivative of the observable multiplied by the error in the time at which it is evaluated. The largest terms of 6 (ET - Al) are the 2-ps daily term and the 1.7-psmonthly term. Furthermore, there are unknown long-period variations in (ET - Al) of the same approximate magnitude due to periodic variations in the heliocentric orbital elements of the earthmoon barycenter arising from perturbations from the other planets. Hence, Eq. (93) for (ET - A l ) used in the general time transformation subroutine may be in error by as much as s. For a spacecraft acceleration of 25 m/s2 in the vicinity of Jupiter, the resulting error in doppler observables can be as large as 2.5 X m/s.

In the computation of differenced-range doppler, the epochs at the end and start of the count interval T, are converted from ST to ET and used to start the light time solutions for pe and ps. This conversion is accomplished using the general time transformation subroutine of the
. This subroutine evaluates (ET- A l ) from
Eq. (93), which consists of the fist three terms of the complete expression for E T - A1 (Eq. 65). The converted epochs t3,(E ) and t3,(ET) are in error by
-6 (ET -Al)",, and -8 (ET -Al)t3s,respectively,where
6 (ET - A l ) = the last seven terms of Eq. (65).That is, S (ET - A l ) consists of the terms of (ET - A l ) not included in the general time transformation subroutine of the BPODP.

The first term of Eq. (482) is due primarily to neglecting the 2-ps daily term of (ET - A l ) in the general time transformation subroutine and has a typical value of about m/s. It can be eliminated in favor of a much smaller error by a simple modification of T , used in Eqs. (479-481). If the epochs t3e(ET)and t3,(ET), obtained using Eq. (93), are transformed back to ST using Eq. (65) and subtracted, the result is a computed count time given by
+ T, (computed) = T, S (ET - Al)",, - 6 (ET - Al)",,
(483)
The computation of differenced-range doppler using T,(computed) rather than T, in Eqs. (479-481) eliminates the error given by the first term of Eq. (482). However, the computed observableis based upon a count time of T, (computed)rather than the correct value of T,. Fortunately, doppler observables vary slowly with T, and
the maximum error is about le7m/s, which is negligible.

Thus, differenced-range doppler observables are computed from
(485)

differenced-range doppler is a maximum of 3 X m divided by the count time (Ref. 51). However, the differenced-range doppler formulation will be added to the Univac 1108 version of the DPODP, which has a double-precision word length of 18 decimal digits (60 bits). The increase in the word length from 54 to 60 bits increases the precision of representation of time from
0.6 X lo-?s to s in the interval 1967-1984. This should
decrease the time truncation error of difFerenced-range doppler to about 5 X m divided by the count time.

For the desired accuracy of m/s, the minimum

allowable count time is 5 s. Since count times as low as

0.1 s are sometimes used, it is recommended that the rep-

resentation of time be changed from double-precision to

where T , (computed) is given by Eq. (483).The formulation for computing the 1-way, 2-way, and 3-way range observables at the end and start of the count interval is given in SectionXI-C. As in the Taylor series formulation, the variation in fsIc over the transmission interval TE for 1-way doppler is ignored. It is computed from Eq. (277)

triple-precision seconds past January 1,1950, Oh or doubleprecision seconds past midnight with one single-precision word used for the Julian day number. This will, for all practical purposes, completely eliminate the time truncation error, and allow count times as low as 0.1 s to be used.

using tz equal to the average of tZe(UTC) and t2,(UTC)

obtained from the light time solutions for ple and pl,, re-

In order to utilize the increased precision for represen-

spectively. This value of tzis also used in the second term tation of time, the accuracy of the light time solution for

of Eq. (484). As in Section VIII, the doppler formulation the epochs of participation of the transmitter and the

is valid only when fq(tl)is constant over TE and f q (t3), spacecraft must be increased from the current value of
K , (t3)a,nd Ks (t3)are constant over T,. Also, if TE over- lo-' s to s. For the maximum conceivablespacecraft

laps T,, fq(t3m) ust equal fq(tl).It is recalled that the velocity of 1,000 km/s, the maximum error in computed

doppler observablewhich the data editing program passes range due to an error of s in the epoch of participa-
on to the orbit determination program is given by tion of the spacecraft is le6m. The maximum correspond-

Eq. (287), which uses fbias computed from fq(tl),fq(t3), ing error in differenced-range doppler is 2 X 10-6m/T,,

Kl(t3)a,nd Ks (t3)using Eqs. (281-283).

allowing an accuracy of m/s to be obtained for all

count times above 0.2 s.

odifie

erical considerations. Each of the computed ables used to form differenced-rangedoppler
containsrandom errors due to truncation of time and position beyond the double-precisionword length of the computer being used.

On the forthcoming Grand Tour missions to the outer planets, the tracking-station-to-spacecraft range will approach the 50-AU radius of the solar system. For ranges of 29-57 AU, the computed round-trip range (p2 or p3) of 57-114 AU willbe represented to a preci on the 60-bit Univac 1108 computer.

The range observables computed by the XBM 7094 version of the DPODP contain a random error of a few millimeters due to truncation of time (seconds past 1950) beyond 16 decimal digits.28The corresponding error in
28Timeis represented as double-precision(54 bits on the IBM 7094
computer) seconds past January 1, 1950,o". From 1967 to 1984,
the value of the last bit is 0.6 x io-' s. The transmission time,
reflection time at the spacecraft, and reception time (in ephemeris time) obtained from the light time solution may be in error by about this amount. Hence, for a spacecraft range rate of 30 h / s , the error in computed range will be about 30 h / s x IO6 mm/km X 0.6 X lO-'s = 1.8 mm.

doppler may be in error by 3 X 10-5m/T, (round-trip) or 1.5 X 10-5m/T, (one way), allowing the desired accuracy of m/s to be obtained for all count times above 1.5 s. For ranges of 3.5-7 AU, the round-trip range of
4-14 AU is re-presented to 2 X le6m, and differenced-
range doppler may be in error by as much as 2X le6m/Tc (one way). For the desired accuracy of le5m/s7 count times as low as 0.2 s may be used.
The precomputed n-body ephemeris tapes used by the DPODP are of the so-called m e 5 0 format. They contain modified second and fourth central differencesof position

E

and velocity. Interpolation is obtained by the fifth-order Observable is used twice: once as pe for the preceding

Everett’s formula. Both the velocity interpolation error, doppler observable and the second time as p8 for the

which affects doppler observables computed from the succeeding doppler observable.

Taylor series formulation, and the differenced position

interpolation error divided by the count time, which affects differenced-range doppler, can approach m/s. This small error could be eliminated by converting to the type-66n-body ephemeris tape format, which containsthe full sum and difference array (on acceleration) used to generate the ephemeris. The heliocentric velocity of the spacecraft is affected by errors in interpolation of the heliocentric ephemeris of the center of integration for

a. Two-wayrange pz and three-way range p,. The 2-way range observables of Section IX are computed from Eq. (379). Considering this equation and the definition (Eq. 469) for 2-way range pz and Sway range p3 used to compute differenced-range doppler, it is evident that pz and p3 may be computed from Eq. (379) using F = 1and
M = co.

the spacecraft trajectory, while errors in interpolation of the heliocentric ephemeris of the earth-moon barycenter affect the heliocentric velocity of the tracking station.

The (ET - Al) time transformation in Eq. (379) is evaluated with the general time transformation subroutine of the DPODP using Eq. (93), which consists of the first

2. Formulation. This section gives the modifications to the 2-way range observable formulation of Section IX which are necessary for the computation of l-way range pl, 2-way range p2, and 3-way range p3 used in Eqs. (484 486), respectively, to compute l-way, e-way, and 3-way differenced-range doppler.

three terms Of Eq* (65).

(ET - in

Eq. (379)consistsof an approximationof term 4 of Eq. (65)

(See Section 11 after Eq*70). In order to compute accu-

rate differenced-rangedoppler, 6 (ET - A l ) must be com-

puted from the last seven terms of Eq. (65) so that

(ET-Al)+S (ET-A1) will equal Eq. (65)for (ET-Al).

This expression was derived in Appendix B specifically

The range observable pi, (where i = 1,2, or 3) is corn- for the Purpose of computing XXXELte differenced-range

puted from a light time solution with reception be doppler. However, it was shown in Section IX that all of

t3 (ST) equal to

the terms of Eq. (65) are also required in order to compute the range observables to the desired accuracy of

+ t 3 (ST) = t,, (ST) y1 T,

(487) 0.1 m.

where

I

t3,(ST) = “time tag” for doppler observable

= midpoint of count interval T,, station time

Similarly, the range observable pi, (where i = 1, 2, or 3) is computed from a light time solution with reception time equal to

1

In the computation of p3 from Eq. (379), evaluation of S (ET - Al) at tl and t , is accomplishedusing the longitude and spin axis distance of the transmitting and receiving stations, respectively. Similarly, (UTC - ST) is evaluated at tl and t3using coefficientswhich apply for the transmitter and receiver, respectively. Since the constant range bias R , cannot affect differenced-range doppler, it is set equal to zero in the computation of p2 or p3.

The l-way range observables are based upon a l-leg light time solution, and the 2-way and 3-way range observablesare based upon a 2-leg light time solution.As indicated in Subsection XI-C-1, the iteration for the epochs of participation for the spacecraft and transmitter must be continued until the indicated correction to the epoch is less than 10-l2s. Aside from this change, the light time solution for each range observable is identical to that described in Section VI.
Since the count intervals for successivedoppler observables are contiguous, each light time solution and range

The range observables of Section IX represent the time for a signal to travel from the transmitter to the receiver at the group velocity ( Ac). On the other hand, the range observables used to compute differenced-range doppler represent the time for the crest of a wave to travel from the transmitter to the receiver at the phase velocity (hc). In the presence of charged particles, the departure of each of these velocities from c is equal in magnitude but opposite in sign. Hence the ionospheric range corrections Arp(tl)and AIp(t3)in Eq. (379) for true range observables will be equal in magnitude but opposite in sign to those for range observables used to compute differencedrange doppler. The corrections for the true range observables will be positive.

Each periodic relativity term of (ET - Al) is evaluated at t3e and tle in the computation of pze or p3e from Eq. (379) and also at t3sand tIs in the computation of p2, or p3s from Eq. (379). The effectof these four values of a periodic term of (ET - Al) on 2-way differencedrange doppler computed from Eq. (485) is
where
8; = effect on F2, expressedas 1-waym/s
M = amplitude of periodic term of (ET - Al), s
c = speed of light, m/s
T,= count time, s
P = period of periodic term of (ET - Al), s p = one-way range to spacecraft, m
The periodic terms of (ET - A l ) have periods of 1day, 1 month, and 1 year. Since the minimum value of P is 1day and the maximum possible value of Tcis normally about 1/2 day, the argument of the first sine term of Eq. (489) will rarely exceed ~ / 2H.ence, a rough approximation for this term is its small angle approximation, which gives
(490)
For a daily term of (ET- Al) and a count time of 1/2 day, the right-hand side of Eq. (490) is 57%greater than that of Eq. (489).However, for a count time of about 1/10 day, which probably will be used with differenced-rangedoppler, the difference between Eqs. (490) and (489) is negligible.
Equation (490) gives the contribution to 2-way differenced-range doppler from a daily, monthly, or annual term of (ET - Al). It also gives the contribution to 3-way differenced-range doppler from a monthly or annual term of (ET - A l ) . The contribution from a daily term is given by

For a daily term of (ET - Al), the argument of the sine term of Eq. (490) approaches ~ / 2as p approaches the 40-50 AU radius of the solar system. The argument of the sine term of Eq. (491) can also approach ~ / 2 .However, the range at which this occurs depends upon the separation in longitude A i of the receiving and transmitting stations. The maximum effect of a diurnal term of (ET - A l ) on %way or 3-way differenced-range doppler is thus

(492)

The maximum effect from the 2-ps daily term of ET - A1
(term 4 of Eq. 65) is 0.05 m/s.

For a monthly or annual term of (ET - Al), the argument of the sine term in Eq. (490) is very small. Hence, this term may be replaced by its small angle approximation, and Eq. (490) becomes

(+y Si <M

p

(493)

For a range of 50 AU, the maximum effect of the monthly term of Eq. (65) (term 9) on 2-way or 3-way differencedrange doppler is about 7.5 X m/s; the annual term (term 3) contributes about 5 X 10-4m/s. The contribution from the 2-ps daily term of Eq. (65), computed from Eq. (493), is 0.08 m/s, whereas the actual upper limit computed from Eq. (492)is 0.05 m/s. The ratio 0.05/0.08 is (sin x ) / x evaluated at x = ~ / 2F.or a range of 10 AU or less, Eq. (493) is a fairly accurate representation of the contribution from a daily term of (ET - A l ) to 2-way differenced-range doppler.
In Appendix B, Eq. (493) is used to determine which terms should be retained in the final expression for ET - A1 (Eq. 65). All terms affecting 2-way differencedrange doppler by more than 2 X lo-?m/s/AU of range to the spacecraft are retained. Several terms of this magnitude are neglected, and the resulting error in differencedrange doppler is no more than mjs/AU or 5 X lCb5 m/s at 50 AU (using Eq. 493).

where
Ah = east longitude of receiving station minus that of transmitting station

b. One-way range pl. From the definition (Eq. 475)for
1-way range pl, it may be obtained from Eq. (379) (used to compute the range observables of Section IX) by removing the terms associated with the up leg of the light path, evaluating the time transformations with subscript

tl at the spacecraft transmission time t2, deleting the resulting term (UTC - ST)t2a, nd by setting R, = 0, F = 1,
and M = 00. The result is

+ - (ET - Al)t3 (ET - Al)t,
- 6 (ET - Al)t,
+ -(A1 - UTC)t3 (A1 - U T Q ,
- (UTC - ST)t,
+ + AAP(t3) ATP(t3) AIP (t3)
103~

(494)

these conditions, the term (ET - Al)t,e of ple will M e r significantly from the term (ET - Al)", of pls. The remainder of this section gives an approximate formulation for computing the difference between these two terms and also the range change ple - plg used in Eq. (484) for 1-way differenced-range doppler.
Define a modified 1-way range p: as
It is computed from Eq. (494) for pl with the term (ET - Al)t, omitted. Then,

+ The (ET- Al) time transformation at the reception
time ts, i.e., (ET - Al)t, 6 (ET - Al)t3, relates A1 time at the tracking station to ET. It is evaluated with Eq. (65), which applies for A1 time derived from any fixed atomic clock on earth. However, an expression is not available for evaluating (ET - Al)t2,which relates A1 time obtained from an atomic clock on board the spacecraft (9,192,631,770cycles from a cesium atomic clock equals one A 1 second) to ET. The differential equation relating these two time scales is Eq. (64). With a slight change in notation,

-d=AIl dET

+s/o - $E c2

(495)

where
+s/o = Newtonian potential at spacecraft
&/a = heliocentric velocity of spacecraft
I&= average value of Newtonian potential at a fixed - point on earth
if = average value of square of heliocentric velocity of a fixed point on earth
It would be extremely difficult to integrate Eq. (495) to obtain an expression for ET - A 1 obtained from the spacecraft atomic clock which would be valid for the trajectory of any spacecraft. From Eq. (64), the average rate of an A1 clock on earth is equal to the rate of an ET clock (if Afeesium= 0). However, from Eq. (495),the rate of an A1 clock on board a spacecraft will be sigdicantly different from the rate of an ET clock if the heliocentric distance and velocity of the spacecraft are significantly different from 1 AU and 30 km/s, respectively. Under

pie - pi, = de- PT, + (ET - A1)tze- (ET - Al)tz8
(497) or
The last two terms represent the transmission interval T', at the spacecraft in the ET and A1 time scales, respectively. The last term is evaluated as the product of the next-to-last term and an approximation to the average
value of dAl/dET from Eq. (495) over Ti.The light time
solutions for p:, and p:g allow the computation of the Newtonian potential at the spacecraft and the square of the heliocentric velocity of the spacecraft at the epochs t2, and t2,.The potential +s/c is computed from Eq. (338) as indicated after that equation. Assuming a linear variation in these quantities over T',,their average values are
(499)
Substituting these quantities into Eq. (495) gives the approximation to the average value of dAl/dET over T',.
T as indicated above to evaluate the last term of Eq. (498) gives

7

7

Substituting Eq. (495) gives
Since the mean distance of an inner planet from the earth is about 1 AU and the mean distance of an outer planet from the earth is approximately equal to the semi-major axis of its heliocentric orbit, the average value of (PB is given approximately by

where
Ps, W e , P.v; &?!a, PJ,
psa, pu, ~ LpN~ l,p,E = gravitational constants for the sun, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, the moon, and the earth, respectively,
km3/s2:
ps = 1,327.1250X 10' p.ne = 0.0002 X 10' pv = 0.0032 X 10' para = 0,0004 X 10'
pj = 1.267 X 10' p,ya = 0.349 X 10' pu = 0.058 X 10'
pN = 0.069 x 10'
pill = 4,902.78 p~ = 398,601.2

From Eq. (B-14)and associated equations of Appendix B,
the average value of 4is given approximately by

where

(505)

u = distance of tracking station from earth's spin axis, km
,8 = mean sidereal rate (see Eq. 273)
= 0.729,212X rad/s

Substituting numerical values gives
+ -
Bg z 887.131 0.532 X lo-'uz

km2/s2

(506)

Dividing Eq. (506)by 2 and adding the result to Eq. (504) gives

+ + &.t

-21 4- z1330.90 7 398601

+ 0.266 X 10-'u2 km2/sz

(507)

A B = the number of kilometers per astronomical unit AU
= 149,597,900km
r = geocentric radius of tracking station, km

In Eq. (503), the gravitational constant of each outer planet is divided by the semimajor axis of its heliocentric orbit expressed in AU, and the gravitational constant of the moon is divided by the mean distance to the moon. Substituting numerical values gives

+ & z884.336 - 398601 kmZ/S2 T

(504)

This equation is accurate to 0.01 km2/s2, a value that affects the spacecraft range rate by 3 X m/s.

The range change ple - pls used in Eq. (484)to compute 1-way digerenced-range doppler is given by Eq. (502)
using p:, and pTS computed from Eq. (494) with the term
+ (ET - Al)$, omitted, (psIc from Eq. (499), from
Eq. (500), and ( A M g ) from Eq. (507). The times
&,(ET) and tZs(ET)are available from the light time
solutions for p:, and p;sy respectively.

+ The 1-way differenced-range doppler formulation is

based upon the assumption that

%&31c) varies

linearly over the transmission interval 2% The resulting

+ error in the observable varies directly with the departure
from linearity (the second derivative of +SIC W $Io) and

with the square of TE.An accuracy of at least m/s can be achieved if the count time T,does not exceed approxi-
mately 10 s when the spacecraft passes by a planet or the moon at extremely small altitudes or 1000s in heliocentric cruise. This is approximately the range of count times used with the Taylor series formulation of Section VIII. The 1-mm/s accuracy for computed 1-way doppler is acceptable, since this data type is currently derived from a crystal oscillator on board the spacecraft rather than an atomic frequency standard.
Section XII-A defines the correction terms for the range, doppler, and angular observables which account for the effects of (1)the offset of the tracking point on the moving antenna from the earth-fixed “station location” (see Section VU), (2) the troposphere, and (3)the ionosphere. The evaluation of these corrections is described in Section XII-B. Expressions are given for the antenna and the troposphere corrections. The general procedure for obtaining the ionosphere corrections is summari~ed.~~
ange obseruables. The range observables (see Section IX) are computed from Eq. (379). The quantity in braces represents the time for the signal (ranging code) to travel from the tracking station to the spacecraft and return, in seconds of station time. In the presence of charged particles, this signal travels at the group velocity
( < c ) . The range corrections Aap, ATP, and AI^ in meters divided by 103c(where c is the speed of light in h / s )
represent the time delay in seconds due to the antenna offset, the troposphere, and the ionosphere, respectively. Each type of correction Aip has a value dip(t3) for the down leg of the light path and a value &p(t1) for the up leg.
The antenna corrections AAp(tl) and Aap (t3)represent the distance along the light path from the “station location” to the actual tracking point on the antenna at the transmission time tl and reception time tg,respectively. Addition of these corrections changes the round-trip light time based upon transmission and reception at the station location to the light time based upon transmission and reception at the actual tracking point on the antenna.
The troposphere corrections ATP(&) and ATP(&) account for the increase in round-trip light time due to the
29Details are available in Ref. 59.

reduction in propagation speed below c and the increase in path length due to bending when passing through the troposphere.
The ionosphere corrections Arp (tl) and Arp (t3)account for the increase in light time due to propagation through the charged particles of the ionosphere at the group velocity, which is less than c.
oppler obseruables. Equations (308), (309), and (310) for 1-way, 2-way, and 3-way doppler observables contain a term A which accounts for the effects of antenna offsets, the troposphere, and the ionosphere. The expression for A is obtained by comparing these equations to the equivalent differenced-rangedoppler formulation of S e e tion XI, which contains correction terms for these effects.
Differenced-range doppler is computed from the dif-
ference of two range observables whose reception times are the end and start of the count interval T,. Each of these range observables represents the time for the crest of a wave to travel from the transmitter to the receiver. In the presence of charged particles, the propagation speed for the crest of a wave is the phase velocity, which is greater than c.
As with the true range observables of Section IX, the range corrections A A ~A, TP, and Arp in meters divided by 103c represent the time delay in seconds due to the antenna offset, the troposphere, and the ionosphere, respectively. For 2-way and 3-way range used to compute 2-way and 3-way differenced-range doppler, respectively, each of these corrections has a value Aip (t3)for the down leg of the light path and a value Aip (tl) for the up leg. For 1-way range used to compute 1-way differenced-range doppler, there are no up-leg corrections.
The antenna and troposphere corrections are the same as those described in Subsection XII-A-1 above for the true range observables of Section IX. The ionosphere corrections have the same magnitude as those for true range observables but with the opposite sign, because charged particles cause the phase velocity to increase above c by the same amount that the group velocity decreases below c. Hence, charged particles of the ionosphere cause
+ the range code for true range observables to arrive late
by [AIp (t3) Arp (t1)]/(103cse)conds and the crest of a wave transmitted and received by the doppler tracking equipment to arrive early by the same amount. Thus, the ionosphere corrections for range observables used to compute differenced-range doppler are negative.

7

Comparing the correction terms of the differencedrange doppler formulation (Section XI) to the correction term A of the Taylorseries doppler formulation (Eqs. 308310) gives, for 2-way or 3-way doppler,

ception time t,, and a transmission time t , , which is the
midpoint of the transmissioninterval a’,.Given tlmin any
time scale, tle and t,, in the same time scale are given approximately by

+ tl, ==:t,, -21T,

(513)

where
c = speed of light, km/s T, = count interval, s t3,= epoch at end of reception interval T , t3s= epoch at start of reception interval T, t,, = epoch at end of transmissioninterval T’, tlg= epoch at start of transmissioninterval T’,

(514)
3. Angular observables. The formulation of Section X for computing directly observed angles containsan expression for the increase in the elevation angle A,.y of the incoming ray due to bending of the ray by the troposphere. Specihally, Ary is the elevation angle of the incoming ray minus the elevation angle of the straight line path from the tracking station to the spacecraft.

and Ap ( t )= sum of range corrections in meters due to the antenna offset, the troposphere, and the ionosphere for up leg with transmission time t or for down leg with reception time t
That is,
As mentioned above, the antenna and troposphere corrections are the same as those used for a range observable; the ionosphere correction has the same magnitude but the opposite sign (negative in Eqs. 508-509) as that used for a range observable. For 1-way doppler, the light path consists of a down leg only and
Given the midpoint t,, of the reception interval T , in any time scale, the epochs t,, and t38in the same time scale are given to sufficient accuracy by
tal=l t,, - -21 T,

B. Evaluation of One-beg Range Corrections
This section gives the formulation for computation of corrections to the 1-way range from the tracking station to the spacecraft due to (1)the offset of the tracking point on the antenna from the station location, AAP; (2) the troposphere, ATp; and (3) the ionosphere, Alp. As descibed in Subsection XII-A-l, the range observable formulation includes these corrections for the up and down legs of the light path. From Subsection XII-A-2, the doppler observable formulation includes these corrections for the up and down legs of the light paths whose reception times are the end and start of the reception interval T,.
1. Antenna correction. The antennas at the tracking stations of the DSN, MSFN, and AFETR have four different types of mounts: (1) hour angle and declination
(HA-dec); (2) azimuth and elevation (az-el); (3) X and Y
angles (MSFN); and (4) X’ and Y’ angles (MSFN). These angles are defined in Section X, Figs. 5-9. For the 26m (85-ft) HA-dec, az-el, and X’-Y’ antennas, the two mutually perpendicular axes do not intersect. The offset between the two axes (the perpendicular distance between them) is denoted by b and ranges from about 1to 7 m. The axis which has a fixed position relative to the earth
will be denoted as the primary axis (the HA, az, or X‘
axis). Due to the offset b between the two axes, rotation of the antenna about the primary axis causes the secondary axis to move relative to the earth.

where T , is given in seconds of station time (ST). The light time solution for the doppler observable has a re- of

10 shows the two mutually perpendicular axes -dec, az-el, or X’-Y’ antenna. The primary axis

of p“ - p is due to the component of b along the direction
to the spacecraft. Since b < 10 m and p> lo5m,

ANTENNA-

/

to an accuracy of better than m le3 and

PRIMARY AXIS
STATION LOCATION
(HA, az, or X’) is in the plane of the paper, and the secondary axis (dec, el, or Y’) is normal to it. The offset between the two axes is b. The positions of the station location and spacecraft are indicated. The secondary angle (dec, el, or Y’) is indicated by 8.
Each range tracking system is calibrated so that the tracking point lies on the secondary (moving) axis. That is, the calibrated range observable obtained from the tracking station corresponds to a 1-way range p^ measured from the secondary axis to the spacecraft. However, the computed range observable is based upon the 1-way range p (i.e.,rlz or rZ3of Eq. 379) measured from a specific point on the antenna which is fixed relative to the earth. This point is called the station location. From Section VII, its geocentric position is represented by spherical or cylindrical coordinates, which are solve-for parameters. For all antennas, the station location is the intersection of the primary axis with the plane perpendicular to it which contains the secondary axis.

From Eq. (508), the doppler observable formulation includes antenna corrections for the up and down legs of the light paths which have reception times equal to the end and start of the reception interval T,. The tracking point for doppler observables is located along the spacecraft to secondary axis line at a constant distance r, from this axis. Hence, each of the four antenna corrections is given by Eq. (517) plus the constant r,. However, since the round-trip range correction at the beginning of the count interval T , is subtracted from the corresponding correction at the end of T,, the effect of r, on A and hence on doppler observables is zero. Hence, Eq. (517) applies also for doppler observables.
For the 26-m HA-dec antennas of the DSN,
where 6 is the observed declination of the spacecraft and b = 6.706 m. These antennas are located at DSN Deep Space Stations 11, 12, 41, 42, 51, 61, and 62.
For the 26-m az-el antenna at Deep Space Station 13,
where y is the observed elevation of the spacecraft and b = 0.9144m.
For the 26-m X’-Y’ antennas of the MSFN,

From Eq. (379),the computed range for the up or down
leg of the light path is rlz or rZ3(denoted as p in Fig. 10)
PIUS *AP for that leg. The sum FJ 4-Amp must equal 8.
ence, the antenna correction AAP is given by

AAp = 8 - p

(515)

AAP= - b COS Y’

(520)

where Y‘ is the observed angle Y’ to the spacecraft and b = 1.2192m. These antennas are located at station MAD at Madrid, Spain; DRA at Canberra, Australia; and 0 at Goldstone, California.

The maximum displacement of the secondary axis from the tracking station to spacecraft line is less than 10m. The maximum effect of this transverse displacement upon
p” - p is about 0.5 X m (for a spacecraft range of
lo5 m) which is insignificant. Thus, the significant part

The axis offset b is zero for the 64-m (210-ft) az-el antenna at Deep Space Station 14, the 9-m (30-ft) X-Y antennas of the MSFN, and all antennas of the AFETR (station numbers 73-77, 79-84, and 87). Hence there are no antenna corrections for these stations.

7

The antenna correction for the up leg of a light path is based upon the antenna type of the transmitting station and the value of the angle 6, y, or Y' to the spacecraft at the transmission time for that leg. Similarly, the antenna correction for the down leg of a light path is based upon the antenna type of the receiving station and the value of 6, y, or Y' at the reception time for that leg. For 3-way doppler, the antenna types at the transmitter and receiver may be different.

The maximum transverse displacement of the secondary axis from the tracking station to spacecraft line is less than 10 mywhich affects directly observed angles by less than 20 arc seconds at the minimum spacecraft range of 100 km. Since such small ranges are rarely encountered and the maximum attainable accuracy for directly observed angles is only 7-11 arc seconds, the computed angular observables are not corrected for this effect.

2. Troposphere and ionosphere corrections. Discussed below are ray path equations, troposphere corrections, and ionosphere corrections.

a. Ray path equations. The speed of propagation of the doppler or ranging signal through the troposphere is given by

0,

= -C
n

where c = speed of light in vacuum n = index of refraction of troposphere

From Ref. 52, p. 9, or

where

h = altitude above mean sea level, h

The speed of propagation through the ionosphere is given by Eq. (521)using the following index of refraction:

n = 1 -t -4Nf0".e3

(524)

where

N e = electron density
= number of electrons/m3
f = transmitted frequency for up or down leg of light path (see Section VIII), Hz

For range observables, the range code travels at the group velocity, which is less than cyand hence the positive sign of Eq. (524)applies. For doppler observables, the doppler signal (the crest of a wave) travels at the phase velocity, which is greater than cy and hence the negative sign applies. The electron density vs altitude profile is assumed to be that of the Chapman model:

where Nmax = maximum value of N e = ( h- hmax)/B h = altitude above mean sea level, km h,,, = altitude of N,,, = scale height of ionosphere, km
The doppler and ranging signals travel on a curved path C through the troposphere and ionosphere. The time for the signal to travel between the tracking station and spacecraft along C is given by

N = refractivity

given by

N =N,cBh

(523)

where

N o = refractivity at mean sea level B = reciprocal of scale height of troposphere, km-'

where ds is an increment of distance along C. The path C follows from the condition that the propagation time T is
a minimum (Fermat's principle). Since n is a function of altitude only, the path is planar and may be described by its geocentric radius r and geocentric angle from the tracking station. ence Eq. (526) can be written as

where n is indicated as a function of T. The differential equation of the path which extremizes the integral (Eq. 527) is the Euler-Eagrange equation of the calculus of variations applied to the integrand of Eq. (527).

The range correction was assumed to be of the form

+ A
' T P = (siny B)U

(5533)

The equations for the path C were developed by D. L. Cain and A. Liu and were documented by A. Liu in Ref. 55. Equation (14) gives the total bending of the path and Eqs. (17) and (18) give the range correction. Use of the index of refraction givenby Eqs. (522) and (523) gives the bending and range correction ATPdue to the troposphere. Use of n given by Eqs. (524) and (525) gives the bending and range correction Arp due to the ionosphere.

where A, B, and C are constants. Fitting this expression to the tabular data above gave

+1.8958m
ATP= (siny 0.06483)1.4

(529)

which was originally obtained by D. L. Cain.

Given the observed value of the elevation angle, these corrections are obtained by a quadrature integration from the position of the tracking station to that of the spacecraft (assumed at i&nite distance from the earth). Equations (14), (17), and (18) give computed minus observed values of the corrections. However, observed minus computed corrections are added to the computed values of the angular, range, and doppler observables.For this purpose, the sign of Eq. (14) and of each term of Eq. (17) must be changed. Furthermore the factor 1/C, must be added to Eq. (18). In the derivation of Eq. (14), the term -Eo was omitted in Eqs. (11) and (12).
Given N o and B for the troposphere and N,,,, La,,
and B for the ionosphere in the vicinity of a tracking station, Eqs. (14), (17), and (18) of Ref. 55, as modified above, givethe elevationangle correctionused in the computation of directly observed angles and the tropospheric and ionospheric range corrections used in the computation of range and doppler observables.

Let Ns = surface refractivity at tracking station

which ideally could be computed from Eq. (523) using the altitude h of the tracking station. The range correction ATp varies directly with N s and since Eq. (529) was obtained using Ns = 340.0, the general result is

+1.8958m

NS

0-

ATP = (sin y 0.06483)1.4 340.0

(530)

Recommended values of Ns for the various,tracking stations are given in Ref. 56.

For elevation angles above 15deg, where most tracking data are taken, the maximum difference between the model (Eq. 530) and the tabular data obtained from the ray tracing formulation is 1-2 my which is quite large. Hence Eq. (528) was fitted to the tabular data for
15 < y < 90 deg, giving

b. Troposphere corrections. The expression that will be given below for the tropospheric range correction ATp was obtained by a procedure equivalent to the following: For selected values of the observed elevation angle y o between 0 and x/2 rad, the ray tracing formulation described in Subsection XII-B-2-a above was used to compute the elevation angle correction A.7 and the range correction ATP for a spacecraft at infinite distance from the earth. Subtraction of Ary from yo gave the corresponding computed elevation angle y based upon a straight-line light path from the tracking station to the spacecraft. The corrections were computed using a sea level refractivity N o of 340.0 and a scale height of 7 km or inverse scale height B of 0.142 km-l. The range corrections ATP were plotted vs the computed elevation angle y.

+ ATP=

2.6 m sin y 0.015

0-

NS
340.0

(5331)

For y > 15 deg, the maximum difference between this
model and the tabular ray tracing data is less than W m.

The models (Eqs. 530 and 531) are based upon an average value of the surface refractivity Ns at each tracking station and a global average value of the scale height. The daily departures of these parameters from the constant values used are currently not accounted for. The resulting errors in ATp from Eq. (530) or (531) are less than 10%for about 90%of the time, with a maximum possible error of about 15%.The following listing shows the

approximate range corrections for elevation angles of 90, 15, and 0 deg and the corresponding 10%errors:

magnetic latitude. It varies from essentially zero at a magnetic latitude of k-90 deg to a maximum in the general vicinity of the magnetic equator.

The time for the doppler or ranging signal to travel

between the tracking station and the spacecraft is given

90

2.5

0.25

by Eq. (526). Since-the index of refraction of the iono-

15

9.5

0.95

sphere is given by Eq. (524), the effect of the ionosphere

0

87

8.7

on the propagation time is given by

Reference 57 describes the daily variations in the parameters of the troposphere and the resulting variations in the range correction ATp.
Equations (530) and (531) are based upon the assumption that the spacecraft is at an infinite distance from the earth. The error due to this assumption increases as the topocentric range to the spacecraft decreases. The maximum error is about 5 m, which occurs at an altitude of 100 km and an elevation angle of 0 deg.
An expression should be found which approximates with negligible error for all elevation angles the range correction ATpobtained from the ray tracing formulation. Furthermore, a correction factor should be added which accounts for the noninfinite range to the spacecraft.

(532)
Since ST is expressed as the so-called ionospheric range
correction Alp divided by c,
(533)
For the ionosphere, the effect of the bending is negligible and the integral can be evaluated along the straight line path from the tracking station to the spacecraft. The propagation speed for the doppler signal is the phase velocity, which is greater than c, and hence the negative signs of Eqs. (532) and (533) apply. The ranging signal propagates at the group velocity, which is less than c, and hence the positive signs apply. The integral

The change in the elevation angle due to tropospheric refraction, Ary, which affects directly observed angles, is computed from Eq. (406) or (407) of Section X. Equation (407), which applies for high elevation angles, is &e standard flat-earth textbook equation (see Ref. 58, p. 61, Eq. 6). Equation (406), which applies for low elevation angles, was obtained by D. L. Cain by fitting to values obtained from the ray tracing formulas. The factor b, is the total bending of the path, which equals Ary for a spacecraft at infinite distance from the earth. The factor b, accounts for the noninfinite range to the spacecraft.
c. Ionosphere corrections. The earth‘s ionosphere is caused by ultraviolet light from the sun ionizing the upper atmosphere. The maximum electron density is in the general direction of the sun. ence the density of charged particles above a tracking station increases and decreases with a diurnal period. A given tracking station passes under the point of maximum ellecbon density between 12 p.m. and 3 p.m. local time (1:30 p.m., average). The electron density is a minimum and fairly constant throughout the night. The electron density also varies with the

evaluated along a Particular (straight-line) light Path is referred to as the electron content for that path. It is a function Of:
(1) Time of day. For an elevation angle y of 90 deg, the maximum electron content occurs between 12 p.m. and 3 p.m. local time. The minimum electron content occurs at night.
(2) Elevation angle. As the elevation angle decreases, the path length through the ionosphere and the electron content increase.
(3) Geomagnetic latitude. The electron density approaches zero as the geomagnetic latitude approaches 290 deg.
Unfortunately, the ionosphere is a very dynamic entity. There are models that describe the properties of the ionosphere, but the parameters of the model vary greatly with the position in the ionosphere and with time for a fixed position in the ionosphere. The large and unpredictable

variations in these parameters preclude the computation of Alp from a model. Thus, the only way of determining Arp is by making direct measurements of the ionosphere.
These measurements may be obtained from a measuring station which is within a few hundred kilometers of the tracking station. wever, ideally, they would be made at the tracking station along the actual ray path to the spacecraft. Reference 59 describes the computation of Arp for the Mariner Mars 1969 mission and discusses the various types of ionospheric measurements and the procedure used to map measurements obtained from stations near a tracking station to the actual ray path to the spacecraft. The mapping is also discussed in Ref. 60.
Some types of ionospheric measurements that can be made are:
(1) Dual frequency. Two different frequencies (one of which is an exact integer multiple of the other) are transmitted in phase. Since the phase velocity for the charged particles of the ionosphere and space plasma is frequencydependent, the two carrier signals will be out of phase when received. This phase shift gives the total electron content along the ray path, using Eq. (532).
(2) Group velocity vs phase velocity. As discussed in Section XI, doppler observables are equivalent to differenced range observables whose reception times are the end and start of the count interval. However, the ionospheric corrections for these pseudo-range-observablesare the negative of the corrections for true range observables at the same epochs. Thus, a comparison of doppler observables with differenced true range observables yields twice the correction to doppler observables due to the charged particles of the ionosphere and space plasma. The doppler correction represents the change in the electron content along the roundtrip light path during the count interval and is used to correct the computed values of doppler observables. This Differenced-Range Versus Integrated Doppler ( ID) technique does not provide the absolute value of the electron content necessary to correct range observables.
araday rotation. If the radio wave is linearly polarized, the plane of polarization will rotate as the signal passes through the earth's ionosphere because of the presence of the earth's magnetic field. This is the Faraday effect. Since the earth's magnetic field is known, the polarization of the received sig-

nal minus that of the transmitted signal can be used to compute the electron content along the ray path due to the ionosphere.
(4)Ionosonde. A radio signal is transmitted vertically,
reflected by the ionosphere, and received by the transmitting station. The height of reflection h is the observed round-trip time multiplied by c/2. The electron density at this height is given by (Ref. 59)

where

N e = 1.24x 10-2f2

(534)

N e = electron density, electrons/m3

f = transmitted frequency,

As the frequency is increased, the density Ne and altitude h increase until the critical frequency is reached where the signal pierces the ionosphere. Substituting this frequency into Eq. (534)gives the maximum electron density Nma. The plot of h vs f
gives the corresponding altitude ha,A.ssuming
that the electron density Ne vs altitude profile is given by the Chapman model (Eq. 525), the vertical electron content E , is given by

A comparison of E , from Eq. (535)with Faraday rotation data gives the scale height B. Since B is fairly constant, a constant value is usually used at each ionosonde station to compute E , from Eq. (535).
Currently, there is no means for directly measuring the electron content along the ray path from any tracking station to the spacecraft. However, plans for converting the tracking stations to the dual frequency (S-band and X-band) mode of operation are under consideration. Unfortunately, it will probably be for only the down leg of the light path. Implementation of such a system for both the up and down legs of the light path would provide a direct measure of the round-trip electron content, which would be used to compute the charged particle (ionosphere and space plasma) corrections for computed doppler and range observables.
RVID technique is currently being used at Deep Space Station 14 (Goldstone) to provide the chargedparticle corrections for doppler observables. It will be available at other tracking stations when the Mu ranging
system (see Section IX) is installed. Faraday rotation

equipment is also available at Goldstone. However, most of the spacecraft to date (and probably those forthcoming) have not had the linearly polarized antennas that are required in order to use this equipment. Furthermore, this equipment does not measure the electron content due to space plasma.

The approximate correction is based upon a uniform electron distribution between the altitudes of 206.5 and 441.5 km. The elevation angle correction factor is the straight-line distance through this uniform ionosphere at the spacecraft elevation angle yo divided by the distance at the measurement elevation angle yM.

For the Mariner Mars 1969spacecraft, Faraday rotation data from tracking of a geostationary satellite and/or ionosonde data were obtained from measuring stations which were within a few hundred kilometers of some of the tracking stations. These measurements gave the electron content along ray paths differing from that of the spacecraft.
However, the physical separation between these paths and the difference in the measurement times were small enough so that the parameters of the ionosphere could be presumed to be the same for both paths. This enabled the electron content to be mapped from the measured ray path to that of the spacecraft, accounting for the differences in time of day, elevation angle, and geomagnetic latitude. The details of this mapping are given in Ref. 59 and are summarized in the following paragraphs.
The Faraday rotation and ionosonde data were taken at a constant elevation angle y (90deg for the ionosonde data). The ray path for each of these measurements pierces
the ionosphere (assumed to be at an altitude of 400 km) at east longitude Ax. At an observation time t (UTC),the
ray path to the spacecraft pierces the ionosphere at east longitude ho.Then the electron content for the spacecraft ray path must be obtained by correcting the electron content for the measured ray path that has the measurement time

Finally, the electron content must be multiplied by a correction factor which accounts for the difference in the geomagnetic latitudes of the points where the spacecraft and measurement ray paths pierce the ionosphere (+o and +M respectively). This correction factor is (90deg - +o)/ (90deg - +M).
Given the corrected electron content, the ionospheric range correction AIp is given by Eq. (533).
riutional Equutions
This section gives the formulation for the solution of the variational equations. The partial derivative of the spacecraft acceleration vector with respect to the solvefor parameter vector q is integrated numerically by the second-sum procedure to give the partial derivatives of the spacecraft velocity and position vectors with respect to q. These subpartial derivatives are used in Section XIV to form the partial derivatives of the doppler, range, and angular observables with respect to q.
The partial derivatives specified in this section are obtained by differentiation of the formulation of Section V for the acceleration of the spacecraft relative to the center of integration. However, the relativity terms and the indirect acceleration of the center of integration due to the oblateness of the earth and moon are ignored. The notation is that of Section V.

tio

tegr

owever, the spacecraft ray path has a different elevation angle (yo) than the measurement ray path (yM). Hence, the measurement must be multiplied by the ratio of the electron content at elevation angle yo to the electron content at elevation angle yM. This correction factor is computed from an approximate formula which agrees very well with the ratio obtained from the ray tracing formulation (Ref. 55 and Subsection XII-B-2-a above) using h,, = 300 km and a scale height B of 39 km.

(536)
where
I,i,F= position, velocity, and acceleration vectors of
spacecraft relative to center of integration with rectangular components x, y, and x referred to the mean earth equator and equinox of 1950.0. The argument is ephemeris time

7

= solve-for parameter vector

en, Z=AZ+Bi+G

(545)

where

[$1 =

= state vector (position

tors) Of spacecraft

to

bot

the center Of

C) at injection epoch to

a = dynamic constants affecting spacecraft trajectory

The state vector of the spacecraft relative to the center of integration at the injection epoch, Xo, is given by

x, = x: + (xg)o

(537)

where

(Xg)o = state vector of reference body B relative to center of integration C at injection epoch

Differentiating Eq. (535) with respect to q gives

where the first six columns of G corresponding to the injection conditions

The variational equation (Eq.545)is integrateg numeri-

cally by the second-sum method to give Z and 2 as func-

tions of ephemeris time t. The partial derivative of the

spacecraft state vector.

x = [-;I

(545)

with respect to q at any time t is

ax (547)

where

u=-aa-x:x --aa-xx((tt,))- -u(t,to)

(548)

V = - ax aa

(549)

For each parameter qi, three sum and difference numer-

Let

ical integration arrays, having two sums and 10differences

of a$aq i, a@/i3qi,and a2/aqi, respectively, are generated.

Thesethree sum and differencearrays may be interpolated

at any time t to give ax,3E/aqi, ay,G/aqi, and az,;/aqi,

respectively, which are the elements of the qi column
(539) of u or V.

= -a'J ai

When the injection conditions are referred to the center of integration, the initial value of aX/aq at the injection epoch is
(550)

(541) where Z is a 6 X 5 identity matrix. When the injection conditions are referred to a body B other than the center of integration C,

z= ai

(543) (544)

(551)
e eighteen sum and difference arrays for the six arameters are started at the injection epoch to, = Z(5 X 5) as initial conditions. For rea-
sons that will become evident later, these sum and differ-

7

ence arrays are restarted with initial values I at a number of intermediate epochs tl, tz,t3, * * * ,tn. The U matrix of
q. (548) is then formed by the chain rule as:
...

= U(t,tn) U(tn,tn-,) * U(t1,tO)

(552)

Eqs. 553 and 554). If the discontinuity occurs during the

integration of the sum and differencearrays, they must be

restarted using the increm
tial values. For ta < t < ti,,

a1 derivatives as inii may be obtained di-

rectly by interpolation. Othe the accumulated value

aai at the last discontinuity epoch or stop time for

sum and difference arrays is mapped to the current time,

using Eq. (554).

Similarly, a U matrix from any intermediate epoch ti to any time t is formed by

u u u u (t7ti) = (t7tn) (tn7 tn-1) * ' ti) (ti+17

(553)

For a dynamic parameter ai, if the corresponding col-
umn of the C matrix is always nonzero, the three sum and difference arrays are started at the injection epoch and continued for the duration of the mission. For each
Of these parameters, the elements of the column of the v
matrix can be obtained by interpolation of the three sum and difference arrays at the desired time t.

For certain other parameters ai, the column of the C
matrix is nonzero only for t, < t < ti,, and the sum and
difference arrays are generated only for this interval of
time. For t < ta, aX/aai = 0. For ta < t < ti,, the column
of V is obtained by interpolation of the sum and difference
arrays. For t >ti,,

-a-x (t)
aai

ax(t) ax(ti,)

ax(ti,) aai

- -

U

(6tb)

- axa(atbi )

(554)

(ti,)/aaiis obtained from the sum and difference
at the stop tirne ti,, and u (t,ti,) is obtained from
Eq. (553) using ti = ta.

Some parameters have an initial value aX(tb)/aai at a discontinuity epoch ti,, and the column of the C matrix is zero for all time. For this case, aX(t)/aai is computed directly from Eq. (554); no sum and difference arrays are generated for t h i s type of parameter.

Some parameters are a combination of the two previous
cases. A period of time ta < t < ti, exists when the column
of C is nonzero and sum and difference arrays are generated; also there are several epochs where discontinuities to the partial derivatives occur. At each discontinuity epoch or stop time for sum and difference arrays, the increment to the partial derivative is added to the accumulated partial and mapped to the next discontinuity epoch or start time for sum and difference arrays (using

The A matrix is defined by Eq. (539). The terms of the spacecraft acceleration vector (considered in t h i s section) which are a function of the spacecraft position vector are:

(1) The direct Newtonian point mass acceleration due to each celestial body i (nine planets, sun, and moon).

(2) The direct Newtonian acceleration due to oblateness for each oblate body j .

(3) The acceleration due to the solar radiation pressure (SRP) and small force (SF) models.

The A matrix is computed from the following sum of

terms :

A=

.. . + arge'wartonian

+ %.{)late a'r'(SRP- SF)

ar

ar

(555)

The formulation for computing each of these terms is given in the following Sections. The notation used iS that of Section V. All vectors appearing in the formulation are column vectors.

1. Contribution from Newtonian point mass acceleration. The direct Newtonian acceleration of the spacecraft due to body treated as a point is given by

(556)
where
r = position vector of spacecraft relative to center of integration with rectangular components x, y, and x referred to the mean earth equator and eq. uinox of 1950.0
E$= 1950.0 position vector of body i relative to center
of integration
p i = gravitational constant of body i, km3/s2

7