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Available online at www.sciencedirect.com
ScienceDirect
Solar Energy 112 (2015) 339–350

www.elsevier.com/locate/solener

Solar eclipse monitoring for solar energy applications
Ibrahim Reda
National Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, CO, USA Received 7 August 2014; received in revised form 8 December 2014; accepted 11 December 2014
Available online 6 January 2015 Communicated by: Associate Editor David Renne

Abstract
In recent years, the interest in using solar energy as a major contributor to renewable energy applications has increased, and the focus to optimize the use of electrical energy based on demand and resources from diﬀerent locations has strengthened. This article includes a procedure for implementing an algorithm to calculate the Moon’s zenith angle with uncertainty of ±0.001° and azimuth angle with uncertainty of ±0.003°. In conjunction with Solar Position Algorithm, the angular distance between the Sun and the Moon is used to develop a method to instantaneously monitor the partial or total solar eclipse occurrence for solar energy applications. This method can be used in many other applications for observers of the Sun and the Moon positions for applications limited to the stated uncertainty. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Solar eclipse; Moon position; Solar position; Solar energy

1. Introduction
The interest in using solar energy as a major contributor to renewable energy applications has increased, and the focus to optimize the use of electrical energy based on demand and resources from diﬀerent locations has strengthened. We thus need to understand the Moon’s position with respect to the Sun. For example, during a solar eclipse, the Sun might be totally or partially shaded by the Moon at the site of interest, which can aﬀect the irradiance level from the Sun’s disk. Instantaneously predicting and monitoring a solar eclipse can provide solar energy users with instantaneous information about potential total or partial solar eclipse at diﬀerent locations At least ﬁve solar eclipses occur yearly, and can last three hours or more, depending on the location. This rare
E-mail address: Ibrahim.Reda@nrel.gov
http://dx.doi.org/10.1016/j.solener.2014.12.010 0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.

occurrence might have an eﬀect on estimating the solar energy as a resource.
This article includes a procedure for implementing an algorithm (described by Meeus (1998)) to calculate the Moon’s zenith angle with uncertainty of ±0.001° and azimuth angle with uncertainty of ±0.003°. The step-by-step format presented here simpliﬁes the complicated steps Meeus describes to calculate the Moon’s position, and focuses on the Moon instead of the planets and stars. It also introduces some changes to accommodate for solar radiation applications. These include changing the direction of measuring azimuth angle to start from north and eastward instead of from south and eastward, and the direction of measuring the observer’s geographical longitude to be measured as positive eastward from Greenwich meridian instead of negative. In conjunction with the Solar Position Algorithm (SPA) that Reda and Andreas developed in 2004 (Reda and Andreas, 2004), the angular distance between the Sun and the Moon is used to develop a method to instantaneously monitor the partial or total

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I. Reda / Solar Energy 112 (2015) 339–350

solar eclipse occurrence for solar energy and smart grid applications. This method can be used in many other applications for observers of the Sun and the Moon positions for applications limited to the stated uncertainty.
SPA has the details of calculating the solar position, so only the Moon position algorithm (MPA) is included in this report. When the solar position calculation is included in this report, the SPA report will be the source for the SPA calculation.
This article is used to calculate the Moon’s position for solar radiation applications only. It is purely mathematical and not meant to teach astronomy or to describe the complex Moon rotation around the Earth. For more information about the astronomical nomenclature that is used throughout the report, review the deﬁnitions in the Astronomical Almanac (AA) or other astronomy references.
2. Moon position algorithm
2.1. Calculate the Julian and Julian Ephemeris Day, Century, and Millennium
The Julian date starts on January 1, in the year À4712 at 12:00:00 UT. The Julian Day (JD) is calculated using the Universal Time (UT) and the Julian Ephemeris Day (JDE) is calculated using the Terrestrial Time (TT). In the following steps, there is a 10-day gap between the Julian and Gregorian calendars where the Julian calendar ends on October 4, 1582 (JD = 2,299,160), and on the following day the Gregorian calendar starts on October 15, 1582.
2.1.1. Calculate the Julian Day

DT

JDE ¼ JD þ

:

ð2Þ

86; 400

2.1.3. Calculate the Julian Century and the Julian Ephemeris Century for the 2000 standard epoch

JC ¼ JD À 2; 451; 545 ;

ð3Þ

36; 525

JCE ¼ JDE À 2; 451; 545 :

ð4Þ

36; 525

2.1.4. Calculate the Julian Ephemeris Millennium for the 2000 standard epoch

JCE

JME ¼ :

ð5Þ

10

2.2. Calculate the Moon geocentric longitude, latitude, and distance between the centers of Earth and Moon (k, b, and D)
“Geocentric” means that the Moon position is calculated with respect to Earth’s center.
2.2.1. Calculate the Moon’s mean longitude, L0 (in degrees)

L0 ¼ 218:3164477 þ 481267:88123421 Ã T

À 0:0015786 Ã T 2 þ T 3 À

T4

;

ð6Þ

538; 841 65; 194; 000

where T is JCE from Eq. (4).

JD ¼ INTð365:25 Ã ðY þ 4716ÞÞ
þ INTð30:6001 Ã ðM þ 1ÞÞ þ D þ B À 1524:5; ð1Þ
where INT is the integer of the calculated terms (8.7 = 8, 8.2 = 8, and À8.7 = À8, etc.). Y is the year (2001, 2002, etc.). M is the month of the year (1 for January, etc.). If M > 2, then Y and M are not changed, but if M = 1 or 2, then Y = Y À 1 and M = M + 12. D is the day of the month with decimal time (e.g., for the second day of the month at 12:30:30 UT, D = 2.521180556). B is equal to 0, for the Julian calendar {i.e. by using B = 0 in Eq. (1), JD < 2,299,160}, and equal to (2 À A + INT (A/4)) for the Gregorian calendar {i.e. by using B = 0 in Eq. (1), and if JD > 2,299,160}; A = INT(Y/100).

2.2.2. Calculate the mean elongation of the Moon, D (in degrees)

D ¼ 297:8501921 þ 445267:1114034 Ã T

À 0:0018819 Ã T 2 þ T 3 À

T4

:

ð7Þ

545; 868 113; 065; 000

2.2.3. Calculate the Sun’s mean anomaly, M (in degrees)

M ¼ 357:5291092 þ 35999:0502909 Ã T

À 0:0001536 Ã T 2 þ

T3

:

ð8Þ

24; 490; 000

2.1.2. Calculate the Julian Ephemeris Day Determine DT, which is the diﬀerence between the
Earth’s rotation time and TT. It is derived from observation only and reported yearly in the AA (Astronomical Almanac; US Naval Observatory).

2.2.4. Calculate the Moon’s mean anomaly, M0 (in degrees)

M0 ¼ 134:9633964 þ 477198:8675055 Ã T

þ 0:0087414 Ã T 2 þ T 3 À

T4

:

ð9Þ

69699 14; 712; 000

I. Reda / Solar Energy 112 (2015) 339–350

341

2.2.5. Calculate the Moon’s argument of latitude, F (in degrees)

2.2.12. Calculate Dl

F ¼ 93:2720950 þ 483202:0175233 Ã T

Dl ¼ 3958 Ã sinða1Þ þ 1962 Ã sinðL0 À F Þ þ 318 Ã sinða2Þ: ð18Þ

À 0:0036539 Ã T 2 À T 3 þ

T4

: ð10Þ 2.2.13. Calculate Db

3; 526; 000 863; 310; 000

Db ¼ À2235 Ã sinðL0Þ þ 382 Ã sinða3Þ

2.2.6. Use Table A1 to calculate the term l (in 0.000001

þ 175 Ã ða1 À F Þ þ 175 Ã sinða1 þ F Þ

degrees)

þ 127 Ã sinðL0 À M0Þ À 115 Ã sinðL0 þ M0Þ:

ð19Þ

Xn l ¼ li Ã sinðdi Ã D þ mi Ã M þ m0i Ã M0 þ f i Ã F Þ; ð11Þ
i¼1

where li, di, mi, m0i, and fi are the ith term in columns l, d, m, m0, and f in the table.
The terms in column m depend on the decreasing eccentricity of the Earth’s orbit around the Sun; therefore, when the term in column m = (1 or À1) or (2 or À2), multiply li in Eq. (11) by E or E2, respectively, where:

E ¼ 1 À 0:002516 Ã T À 0:0000074 Ã T 2:

ð12Þ

2.2.14. Calculate the Moon’s longitude, k0 (in degrees)

k0 ¼ L0 þ l þ Dl :

ð20Þ

1; 000; 000

2.2.15. Calculate the Moon’s latitude, b (in degrees)

b

¼

b þ Db 1; 000; 000

:

ð21Þ

2.2.7. Use Table A1 to calculate the term r (in 0.001 km)

2.2.16. Limit k0 and b to the range of 0–360° Limit k0 and b to the range of 0–360°.

Xn r ¼ ri Ã cosðdi Ã D þ mi Ã M þ m0i Ã M 0 þ f i Ã F Þ: ð13Þ
i¼1
Similar to step 2.2.6, when m = (1 or À1) and (2 or À2), multiply ri in Eq. (13) by E or E2.
2.2.8. Use Table A2 to calculate the term b (in 0.000001 degrees)

Xn

b ¼ bi Ã sinðdi Ã D þ mi Ã M þ m0i Ã M 0 þ f iF Þ:

ð14Þ

i¼1

Similar to step 2.2.6, when m = (1 or À1) and (2 or À2), multiply bi in Eq. (14) by E or E2.

2.2.9. Calculate a1

a1 ¼ 119:75 þ 131:849 Ã T :

ð15Þ

2.2.10. Calculate a2

a2 ¼ 53:09 þ 479264:29 Ã T :

ð16Þ

2.2.17. Calculate the Moon’s distance from the center of Earth, D (in kilometers)

r

D ¼ 385000:56 þ :

ð22Þ

1000

2.3. Calculate the Moon’s equatorial horizontal parallax, p

p ¼ 6378:14 :

ð23Þ

D

2.4. Calculate the nutation in longitude and obliquity (Dw and De)

2.4.1. Calculate the mean elongation of the Moon from the Sun, X0 (in degrees)

X 0 ¼ 297:85036 þ 445267:111480 Ã JCE

À 0:0019142 Ã JCE2 À JCE3 :

ð24Þ

189; 474

2.4.2. Calculate the mean anomaly of the Sun (Earth), X1 (in degrees)

2.2.11. Calculate a3 a3 ¼ 313:45 þ 481266:484 Ã T :

X 1 ¼ 357:52772 þ 35999:050340 Ã JCE

ð17Þ

À 0:0001603 Ã JCE2 þ JCE3 : 300; 000

ð25Þ

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I. Reda / Solar Energy 112 (2015) 339–350

2.4.3. Calculate the mean anomaly of the Moon, X2 (in degrees)

X 2 ¼ 134:96298 þ 477198:867398 Ã JCE

þ 0:0086972 Ã JCE2 þ JCE3 :

ð26Þ

56; 250

2.4.4. Calculate the Moon’s argument of latitude, X3 (in degrees)

X 3 ¼ 93:27191 þ 483202:017538 Ã JCE

À 0:0036825 Ã JCE2 þ JCE3 :

ð27Þ

327; 270

2.4.5. Calculate the longitude of the ascending node of the Moon’s mean orbit on the ecliptic, measured from the mean equinox of the date, X4 (in degrees)

2.5. Calculate the true obliquity of the ecliptic, e (in degrees)
2.5.1. Calculate the mean obliquity of the ecliptic, e0 (in arc seconds)

e0 ¼ 84381:448 À 4680:93 Ã U À 1:55 Ã U 2 þ 1999:25 Ã U 3 À 51:38 Ã U 4 À 249:67 Ã U 5

À 39:05 Ã U 6 þ 7:12 Ã U 7 þ 27:87 Ã U 8

þ 5:79 Ã U 9 þ 2:45 Ã U 10;

ð33Þ

where U is JME/10.

2.5.2. Calculate the true obliquity of the ecliptic, e (in degrees)

e ¼ e0 þ De:

ð34Þ

3600

2.6. Calculate the apparent Moon longitude, k (in degrees)

X 4 ¼ 125:04452 À 1934:136261 Ã JCE þ 0:0020708 Ã JCE2 þ JCE3 : 450; 000

k ¼ k0 þ Dw:

ð35Þ

ð28Þ 2.7. Calculate the apparent sidereal time at Greenwich at any given time, m (in degrees)

2.4.6. For each row in Table A3, calculate the terms Dwi and Dei (in 0.0001of arc seconds)

Dwi ¼ ðai þ bi Ã JCEÞ Ã sin Dei ¼ ðci þ di Ã JCEÞ Ã cos

! X4
X j Ã Y i;j ;
j¼1
! X4
X j Ã Y i;j ;
j¼1

ð29Þ ð30Þ

where ai, bi, ci, and di are the values listed in the ith row and columns a, b, c, and d in Table A3. Xj is the jth X calculated by using Eqs. (15)–(19). Yi,j is the value listed in ith row and jth Y column in Table A3.

2.4.7. Calculate the nutation in longitude, Dw (in degrees)

Dw ¼

Pn
i¼1

Dwi

;

ð31Þ

36; 000; 000

where n is the number of rows in Table A3 (n equals 63 rows in the table).

2.4.8. Calculate the nutation in obliquity, De (in degrees)

De ¼

Pn
i¼1

Dei

:

ð32Þ

36; 000; 000

2.7.1. Calculate the mean sidereal time at Greenwich, m0 (in degrees)

m0 ¼ 280:46061837 þ 360:98564736629 Ã ðJD À 2; 451; 545Þ

þ 0:000387933 Ã JC2 À JC3 :

ð36Þ

38; 710; 000

2.7.2. Calculate the apparent sidereal time at Greenwich, m (in degrees)

v ¼ v0 þ Dw Ã cosðeÞ:

ð37Þ

2.7.3. Limit m to the range of 0–360° Limit m to the range of 0–360°.

2.8. Calculate the Moon’s geocentric right ascension, a (in degrees)

2.8.1. Calculate the Moon’s right ascension, a (in radians)


a ¼ Arc tan 2

sin k Ã cos e À tan b Ã sin e cos k

;

ð38Þ

where Arc tan 2 is an arctangent function that is applied to the numerator and the denominator (instead of the actual division) to maintain the correct quadrant of a, where a is in the rage of Àp to p.

I. Reda / Solar Energy 112 (2015) 339–350

343

2.8.2. Calculate a in degrees, then limit it to the range of 0–360°
Calculate a in degrees, then limit it to the range of 0–360°.
2.9. Calculate the Moon’s geocentric declination, d (in degrees)

then change Da to degrees.
2.11.5. Calculate the Moon’s topocentric right ascension a0 (in degrees)

a0 ¼ a þ Da:

ð45Þ

d ¼ Arc sinðsin b Ã cos e þ cos b Ã sin e Ã sin kÞ;

ð39Þ

where d is positive or negative if the Sun is north or south of the celestial equator, respectively. Then change d to degrees.

2.10. Calculate the observer local hour angle, H (in degrees)

2.11.6. Calculate the topocentric Moon’s declination, d0 (in degrees)


d0 ¼ Arc tan 2

ðsin d À y Ã sin pÞ Ã cos Da cos d À y Ã sin p Ã cos H

:

ð46Þ

H ¼ v þ r À a;

ð40Þ

where r is the observer geographical longitude, positive or negative for east or west of Greenwich, respectively.
Limit H to the range from 0° to 360° and note that it is measured westward from south in this algorithm.

2.12. Calculate the topocentric local hour angle, H0 (in degrees)

H 0 ¼ H À Da:

ð47Þ

2.11. Calculate the Moon’s topocentric right ascension a0 (in degrees)

“Topocentric” means that the Moon’s position is calculated with respect to the observer local position at the Earth’s surface.

2.11.1. Calculate the term u (in radians)

u ¼ Arc tanð0:99664719 Ã tan uÞ;

ð41Þ

where / is the observer’s geographical latitude, positive or negative if north or south of the equator, respectively. The 0.99664719 number equals (1 À f), where f is the Earth’s ﬂattening.

2.11.2. Calculate the term x

x ¼ cos u þ E Ã cos u;

ð42Þ

6; 378; 140

where E is the observer’s elevation (in m). Note that x equals q * cos /0 where q is the observer’s distance to the center of
the Earth, and /0 is the observer’s geocentric latitude.

2.11.3. Calculate the term y

y

¼

0:99664719

Ã

sin

u

þ

6;

E 378;

140

Ã

sin

u;

ð43Þ

note that y equals q * sin /0.

2.11.4. Calculate the parallax in the Moon’s right ascension, Da (in degrees)


Da ¼ Arc tan 2

Àx Ã sin p Ã sin H cos d À x Ã sin p Ã cos H

:

ð44Þ

2.13. Calculate the Moon’s topocentric zenith angle, hm (in degrees)

2.13.1. Calculate the topocentric elevation angle without atmospheric refraction correction, e0 (in degrees)
e0 ¼ Arc sinðsin u Ã sin d0 þ cos u Ã cos d0 Ã cos H 0Þ: ð48Þ then change e0 to degrees.
2.13.2. Calculate the atmospheric refraction correction, De (in degrees)

P

283

De ¼ Ã

Ã

1:02

;

ð49Þ

1010

273 þ T

60 Ã tan

e0

þ

10:3 e0 þ5:11

where P is the annual average local pressure (in millibars). T is the annual average local temperature (in °C). e0 is in degrees. Calculate the tangent argument in degrees, then convert to radians.

2.13.3. Calculate the topocentric elevation angle, e (in degrees)

e ¼ e0 þ De:

ð50Þ

2.13.4. Calculate the topocentric zenith angle, h (in degrees)

hm ¼ 90 À e:

ð51Þ

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I. Reda / Solar Energy 112 (2015) 339–350

2.14. Calculate the Moon’s topocentric azimuth angle, Um (in degrees)

2.14.1. Calculate the topocentric astronomers’ azimuth

angle, C (in degrees)

C

¼

Arc

tan

 2
cos

H0

Ã

sin

sin H 0 u À tan

d0

Ã

cos

 ;
u

ð52Þ

Change C to degrees, then limit it to the range of 0–360°. C is measured westward from south.

the zenith and azimuth angles of the Sun and the Moon are calculated continuously using SPA described in Reda and Andreas (2004), and the Moon Position Algorithm described above. A copyrighted Solar and Moon Position Algorithm (SAMPA) software and calculator were developed by Andreas and Reda (2012), and then used to monitor the solar eclipse as follows:
4.1. Calculate the local observed, topocentric, angular distance between the Sun and Moon centers, Esm (in degrees)

2.14.2. Calculate the topocentric azimuth angle, Um for navigators and solar radiation users (in degrees)

Um ¼ C þ 180;

ð53Þ

Limit Um to the range from 0° to 360°. Um is measured eastward from north.

3. Moon position algorithm validation

To evaluate the uncertainty of the MPA, arbitrary dates, January 17 and October 17, are chosen from each of the years 2004 to 2010, and 1981, at 0-h TT. Fig. 1 shows that the maximum diﬀerence between the AA and MPA is 0.00055° for the Moon’s declination, Fig. 2 shows that the maximum diﬀerence is 0.00003° for the equatorial Moon parallax, and Fig. 3 shows that the maximum diﬀerence in the calculated zenith or azimuth angles is 0.0003° and 0.00075°, respectively. This implies that the MPA is well within the stated uncertainty of ±0.001° and ±0.003° in the zenith and azimuth angles, respectively.

4. Predicting and monitoring the solar eclipse occurrence

The full astronomical nomenclature for eclipse monitoring is beyond the scope of this report, so only the total and partial solar eclipse nomenclatures are used. In this section,

Ems ¼ cosÀ1½cos hs Ã cos hm þ sin hs Ã sin hm Ã cosð/s À /mÞ; ð54Þ
where hs and /s are the zenith and azimuth angles of the Sun, calculated using SPA, 2003.

4.2. Calculate the radius of the Sun’s disk, rs (in degrees)

959:63

rs ¼ 3600 Ã Rs ;

ð55Þ

where Rs is the Sun’s distance from the center of the Earth, in astronomical units (AUs). This distance is calculated in SPA (Reda and Andreas, 2004).

4.3. Calculate the radius of the Moon’s disk, rm (in degrees)

rm

¼

358;

473;

400 Ã ð1 þ sin 3600 Ã D

e

Ã

sin

pÞ

;

ð56Þ

where e, p, and D are calculated in Section 2.

4.4. Set the boundary conditions for the solar eclipse 4.4.1. No eclipse

Ems > ðrm þ rsÞ;

0.00080

Degrees
Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09

0.00060 0.00040 0.00020 0.00000 -0.00020 -0.00040

-0.00060

-0.00080

Date

Fig. 1. Diﬀerence between the AA and MPA for the Moon’s declination.

I. Reda / Solar Energy 112 (2015) 339–350

345

0.00002

0.00001

0.00000

Degrees
Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09

-0.00001

-0.00002

-0.00003

-0.00004

Date

Fig. 2. Diﬀerence between the AA and MPA for the Moon’s horizontal parallax.

Degrees
Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09

0.00080 0.00070 0.00060 0.00050 0.00040 0.00030 0.00020 0.00010 0.00000 -0.00010 -0.00020 -0.00030 -0.00040 -0.00050 -0.00060 -0.00070 -0.00080

Date

Zenith

Azimuth

Fig. 3. Diﬀerence in the calculated zenith or azimuth angles using the diﬀerences in Figs. 1 and 2.

where rm and rs are the Sun and Moon radii. 4.4.2. Start and end of eclipse
Ems ¼ ðrm þ rsÞ:
4.4.3. Solar eclipse
Ems < ðrm þ rsÞ:
4.4.4. Sun disk area during eclipse If Ems 6 abs(rm À rs), it is a total eclipse where the Sun
and Moon disks (circles) completely overlap; therefore, if

rs > rm, the unshaded Sun area by the Moon will equal the area of the Sun disk minus the area of the Moon disk. Moreover, if rs 6 rm, the unshaded area of the Sun equals zero.
To monitor the solar eclipse, a criterion where Ems equals (rm + rs) is set at the beginning of the eclipse. At this moment the time is noted as Tstart and then Ems is recalculated every second for at least three hours. The distance Ems can then be plotted against time to show the progress of the eclipse. From the plotted data, one might predict the time of a partial or total solar eclipse by calculating the time when minimum Ems occurs, Tmin, then as the eclipse ends, the time when Ems equals (rm + rs), the time Tend is noted. The total duration for the eclipse occurrence will equal Tend À Tstart. Fig. 4 shows Ems versus time for the central

346

I. Reda / Solar Energy 112 (2015) 339–350

solar eclipse on July 22, 2009 (see Table 1 for coordinates). Using this method, the minimum Ems = 0.0001°, which is well within the uncertainty of calculating the Sun and Moon positions. To verify this method, Ems is calculated for some historical total solar eclipses at diﬀerent locations. Table 1 shows that Ems < 0.0011, which is within the stated uncertainty of ±0.003°.

5. Estimating the solar irradiance during a solar eclipse

When a solar eclipse occurs, the Moon’s disk will start shading the Sun’s disk; the shaded area will change as time progresses; therefore, the unshaded area of the Sun disk is called the Sun’s Unshaded Lune (SUL), which will also change with time. The percentage of the SUL, from the total Sun’s disk area, is then calculated. The percentage area might then be multiplied by an estimated direct beam irradiance to calculate the irradiance during the eclipse. A spectacular phenomena occurs during the solar eclipse, when the spectral distribution of the irradiance from the Sun changes. The method described below illustrates how the irradiance might be estimated during the solar eclipse. Users might use other methods or models to achieve smaller uncertainty for such estimates.

5.1. Calculate the area of SUL, ASUL

To calculate this area, draw two intersecting circles, with two diﬀerent radii of the Sun and the Moon, rs and rm. An illustration is shown in Fig. 1.

ASUL ¼ p Ã r2s À Ai;

ð57Þ

where Ai is the area of the Sun’s disk that is shaded by the Moon. A step-by-step method to calculate ASUL is described in Appendix B.

5.2. Calculate the percentage area of the SUL with respect to the area of the Sun’s disk, %ASUL

%ASUL

¼

ASUL Ã 100 p Ã r2s

:

ð58Þ

5.3. Calculate the direct beam irradiance using the appropriate model for the required uncertainty, in W/m2
The Bird and Hulstrom simple model (Bird and Hulstrom, 1981) is used in this article as an illustration.
5.4. Calculate the irradiance (W/m2) during the eclipse, Ie

Ie

¼

I

Ã

%ASUL 100

;

ð59Þ

where I is the direct beam irradiance calculated by the model.
To evaluate the described method, the calculated irradiance is compared against the irradiance measured at the University of Oregon, Eugene, Solar Radiation Monitoring Laboratory. The irradiance was measured during the June 10, 2002 partial eclipse, using a pyrheliometer model NIP, manufactured by the Eppley Laboratory, Inc. Fig. 4 shows the diﬀerence between the measured irradiance and the calculated irradiance using the method described above. The ﬁgure shows that the diﬀerence between the measured irradiance at the University of Oregon and the calculated irradiance by SAMPA is about 8% when the partial eclipse starts, 4% at the maximum eclipse, and 6% as the eclipse ends. These diﬀerences are expected, because the measuring instruments (estimated U95 = ±3%) and the model used above (estimated U95 = ±5%) do not account for the signiﬁcant change in the spectral distribution of the irradiance during the solar eclipse occurrence. In the future, with the advancement in pyrheliometer design and spectral measurement technology, advanced models might be used to improve the uncertainty in measuring such signiﬁcant change in the spectral distribution during the eclipse occurrence.

0.6

120

0.5

100

0.4

80

Ems (°) ASUL(%)

0.3

60

0.2

40

0.1

20

0 0:57:36

1:26:24

1:55:12 Ems

2:24:00 2:52:48 3:21:36 UT
% Area of Sun Unshaded Lune

0 3:50:24

Fig. 4. Distance between the Sun and Moon centers and the percentage of the Sun unshade lune (SUL) for the July 22, 2009 solar eclipse.

I. Reda / Solar Energy 112 (2015) 339–350

Table 1 Historical solar eclipses versus SAMPA eclipse monitor, Ems.

Historical eclipse dates

UT

Observer’s longitude °

7/22/2009 8/1/2008 3/29/2006 4/8/2005 12/4/2002 6/21/2001 2/4/1981

2:33:00 9:47:18 10:33:18 20:15:36 7:38:42 11:57:48 21:57:36

143.3617 34.7417 22.8867 À123.4817 62.8383 0.9867 À145.9033

Observer’s latitude °
24.6117 81.1133 29.6200 À15.7883 À40.5283 À11.5950 À45.8883

347
SAMPA, Ems 0.0001 0.0002 0.0005 0.0011 0.0005 0.0003 0.0004

900

10

Irradiance (W/m2) Measured irradiance - SAMPA irradiance (%)

800

9

700

8

7 600

6 500
5 400
4
300 3

200

2

100

1

0 15:50:24

16:19:12

16:48:00

17:16:48

17:45:36

18:14:24

0 18:43:12

Local Time

Measured Irradiance SAMPA Irradiance Measured irradiance-SAMPA irradiance (%)

Fig. 5. Measured irradiance versus calculated irradiance using SAMPA during the June 10, 2002 partial solar eclipse.

6. Conclusions

Acknowledgments

The MPA achieves uncertainties of ±0.001° and ±0.003° in calculating the zenith and azimuth angles of the Moon (see Figs. 1 and 2). Using MPA in conjunction with the SPA (uncertainty of ±0.0003°) to monitor solar eclipses, is consistent with the historical eclipses to within the stated uncertainty of the MPA, see Table 1. Section 5 and Fig. 5 show that the direct beam irradiance from the Sun during a solar eclipse is estimated within 4–8% from measured irradiance that was collected during the eclipse of June 10, 2002. This implies that solar energy users might be able, with the current technology, to use this information to manage the grid’s solar resources during eclipses to within the model’s limitations. Improved uncertainties might be achieved by developing advanced models that include the change of the spectral distribution during the spectacular solar eclipse. Fig. 4 also shows that the partial solar eclipse of June 10, 2002 in Eugene, Oregon, lasted longer than two hours, which might have an eﬀect on estimating the solar energy when used as a resource.

I would like to thank the Quality Management and Assurance oﬃce, NREL Metrology Laboratory, and DOE-ARM and Solar programs for providing the funds for this publication. I extend special appreciation to my wife Mary Alice and my daughter Lenah for their inspiration about the Moon’s inﬂuence on romance, legends, and art. I also thank the staﬀ from the University of Oregon for providing the measured data during the solar eclipse in Oregon, USA.
Appendix A. Tables
Tables A1–A3.
Appendix B. Sun and Moon lunes
Note that some symbols used in this appendix are independent from those used in the main report.

348

I. Reda / Solar Energy 112 (2015) 339–350

Table A1 Moon’s periodic terms for longitude and distance.

Table A2 Periodic terms for the Moon’s latitude.

d

M

m0

f

l

R

D

m

m0

f

b

0

0

1

0 6,288,774 À20,905,355 0

2

0

À1

0 1,274,027

À3,699,111 0

2

0

0

0

658,314

À2,955,968 0

0

0

2

0

213,618

À569,925 2

0

1

0

0 À185,116

48,888 2

0

0

0

2 À114,332

À3149 2

2

0

À2

0

58,793

246,158 2

2

À1

À1

0

57,066

À152,138 0

2

0

1

0

53,322

À170,733 2

2

À1

0

0

45,758

À204,586 0

0

1

À1

0

À40,923

À129,620 2

1

0

0

0

À34,720

108,743 2

0

1

1

0

À30,383

104,755 2

2

0

0 À2

15,327

10,321 2

0

0

1

2

À12,528

2

0

0

1 À2

10,980

79,661 2

4

0

À1

0

10,675

À34,782 2

0

0

3

0

10,034

À23,210 0

4

0

À2

0

8548

À21,636 4

2

1

À1

0

À7888

24,208 0

2

1

0

0

À6766

30,824 0

1

0

À1

0

À5163

À8379 0

1

1

0

0

4987

À16,675 1

2

À1

1

0

4036

À12,831 0

2

0

2

0

3994

À10,445 0

4

0

0

0

3861

À11,650 0

2

0

À3

0

3665

14,403 1

0

1

À2

0

À2689

À7003 0

2

0

À1

2

À2602

4

2

À1

À2

0

2390

10,056 4

1

0

1

0

À2348

6322 0

2

À2

0

0

2236

À9884 4

0

1

2

0

À2120

5751 2

0

2

0

0

À2069

2

2

À2

À1

0

2048

À4950 2

2

0

1 À2

À1773

4130 2

2

0

0

2

À1595

0

4

À1

À1

0

1215

À3958 2

0

0

2

2

À1110

2

3

0

À1

0

À892

3258 2

2

1

1

0

À810

2616 2

4

À1

À2

0

759

À1897 4

0

2

À1

0

À713

À2117 2

2

2

À1

0

À700

2354 2

2

1

À2

0

691

0

2

À1

0 À2

596

2

4

0

1

0

549

À1423 1

0

0

4

0

537

À1117 1

4

À1

0

0

520

À1571 0

1

0

À2

0

À487

À1739 2

2

1

0 À2

À399

1

0

0

2 À2

À381

À4421 2

1

1

1

0

351

0

3

0

À2

0

À340

4

4

0

À3

0

330

4

2

À1

2

0

327

1

0

2

1

0

À323

1165 4

1

1

À1

0

299

1

2

0

3

0

294

4

2

0

À1 À2

8752 2

0

0

1

5,128,122

0

1

1

280,602

0

1

À1

277,693

0

0

À1

173,237

0

À1

1

55,413

0

À1

À1

46,271

0

0

1

32,573

0

2

1

17,198

0

1

À1

9266

0

2

À1

8822

À1

0

À1

8216

0

À2

À1

4324

0

1

1

4200

1

0

À1

À3359

À1

À1

1

2463

À1

0

1

2211

À1

À1

À1

2065

1

À1

À1

À1870

0

À1

À1

1828

1

0

1

À1794

0

0

3

À1749

1

À1

1

À1565

0

0

1

À1491

1

1

1

À1475

1

1

À1

À1410

1

0

À1

À1344

0

0

À1

À1335

0

3

1

1107

0

0

À1

1021

0

À1

1

833

0

1

À3

777

0

À2

1

671

0

0

À3

607

0

2

À1

596

À1

1

À1

491

0

À2

1

À451

0

3

À1

439

0

2

1

422

0

À3

À1

421

1

À1

1

À366

1

0

1

À351

0

0

1

331

À1

1

1

315

À2

0

À1

302

0

1

3

À283

1

1

À1

À229

1

0

À1

223

1

0

1

223

1

À2

À1

À220

1

À1

À1

À220

0

1

1

À185

À1

À2

À1

181

1

2

1

À177

0

À2

À1

176

À1

À1

À1

166

0

1

À1

À164

0

1

À1

132

0

À1

À1

À119

À1

0

À1

115

À2

0

1

107

I. Reda / Solar Energy 112 (2015) 339–350

349

Table A3 Periodic terms for the nutation in longitude and obliquity.

Y0

Coeﬃcients for sin terms Coeﬃcients for Dw Coeﬃcients for De

Y1

Y2 Y3 Y4 a

b

c

d

0

0

0 0 1 À171,996 À174.2 92,025 8.9

À2

0

0 2 2 À13,187 À1.6 5736 À3.1

0

0

0 22

À2274 À0.2 977 À0.5

0

0

0 02

2062 0.2 À895 0.5

0

1

0 00

1426 À3.4 54 À0.1

0

0

1 00

712 0.1 À7

À2

1

0 22

À517 1.2 224 À0.6

0

0

0 21

À386 À0.4 200

0

0

1 22

À301

129 À0.1

À2

À1

0 22

217 À0.5 À95 0.3

À2

0

1 00

À158

À2

0

0 21

129 0.1 À70

0

0

À1 2 2

123

À53

2

0

0 00

63

0

0

1 01

63 0.1 À33

2

0

À1 2 2

À59

26

0

0

À1 0 1

À58 À0.1 32

0

0

1 21

À51

27

À2

0

2 00

48

0

0

À2 2 1

46

À24

2

0

0 22

À38

16

0

0

2 22

À31

13

0

0

2 00

29

À2

0

1 22

29

À12

0

0

0 20

26

À2

0

0 20

À22

0

0

À1 2 1

21

À10

0

2

0 00

17 À0.1

2

0

À1 0 1

16

À8

À2

2

0 22

À16 0.1

7

0

1

0 01

À15

9

À2

0

1 01

À13

7

0

À1

0 01

À12

6

0

0

2 À2 0

11

2

0

À1 2 1

À10

5

2

0

1 22

À8

3

0

1

0 22

7

À3

À2

1

1 00

À7

0

À1

0 22

À7

3

2

0

0 21

À7

3

2

0

1 00

6

À2

0

2 22

6

À3

À2

0

1 21

6

À3

2

0

À2 0 1

À6

3

2

0

0 01

À6

3

0

À1

1 00

5

À2

À1

0 21

À5

3

À2

0

0 01

À5

3

0

0

2 21

À5

3

À2

0

2 01

4

À2

1

0 21

4

0

0

1 À2 0

4

À1

0

1 00

À4

À2

1

0 00

À4

1

0

0 00

À4

0

0

1 20

3

0

0

À2 2 2

À3

À1

À1

1 00

À3

0

1

1 00

À3

0

À1

1 22

À3

2

À1

À1 2 2

À3

0

0

3 22

À3

2

À1

0 22

À3

B.1. Calculating the areas of the lunes when two circles with diﬀerent diameters intersect
The following steps are intended for calculating the area of the Sun Unshaded Lune, ASUL, during solar eclipses. The Sun and Moon radii are not equal and change with the day of the year. Also, during the solar eclipse, as the Moon starts to shade the Sun disk, the intersection area changes with time. In Fig. B1, the circles with centers Cs and Cm represent the Sun and Moon disks, respectively. Refer to the ﬁgure to calculate ASUL, bounded by sector aebq.
B.2. Calculate the areas of triangles Ts (bounded by abCs) and Tm (bounded by abCm), then calculate ASUL
1. Use the procedure described in this article to calculate the distance between the Sun and Moon centers, Ems.
2. Write the following equation:

Ems ¼ m þ s;

ðB1Þ

where m is the distance cCm, s is the distance cCs. Note that m and s are the heights of the two triangles.

3. Using the Pythagorean theorem:

h2 ¼ r2s À s2 ¼ r2m þ m2;

ðB2Þ

where rs and rm are the Sun and Moon radii, calculated using the procedure described in this report. h is half the
base of the two triangles, Ts and Tm. Therefore:

r2s À s2 ¼ r2m À m2:

ðB3Þ

4. Solve Eqs. (B1) and (B3) with two unknowns to calculate s and m:

s ¼ E2ms þ r2s À r2m ; 2 Ã Ems
and m ¼ E2ms À r2s þ r2m :
2 Ã Ems 5. Use Eqs. (B2) and (B4) to calculate h:
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ 4 Ã E2ms Ã r2s À ðE2ms þ r2s À r2mÞ2 :
2 Ã Ems 6. Calculate the area of triangle abCs, Ts:

ðB4Þ ðB5Þ ðB6Þ

T s ¼ h Ã s:

ðB7Þ

350

I. Reda / Solar Energy 112 (2015) 339–350

Fig. B1. Intersecting circles to calculate the areas of the Sun and Moon lunes.

7. Calculate the area of triangle abCm, Tm:

T m ¼ h Ã m:

ðB8Þ

8. Calculate the area of sector adbCs in the Sun’s circle, As:

As

¼

p

Ã

r2s

Ã

2 Ã xs 360

¼

r2s

Ã

cosÀ1

s rs

;

ðB9Þ

where xs is half the central angle of sector adbCs in the Sun’s circle. 9. Calculate the area of section abd in the Sun’s circle, A1:

A1 ¼ As À T s:

ðB10Þ

10. Similar to Eq. (B9), calculate the area of sector aebCm in the Moon’s circle, Am,

Am

¼

r2m

Ã

cosÀ1

m rm

:

ðB11Þ

11. Calculate the area of section abe in the Moon’s circle, A2:

A2 ¼ Am À T m:

ðB12Þ

12. Calculate the area of the Sun’s circle shaded by the Moon’s circle, Ai:

Ai ¼ A1 þ A2: 13. Calculate the Sun’s Unshaded Lune, ASUL:

ðB13Þ

ASUL ¼ p Ã r2s À Ai:

ðB14Þ

References
Andreas, A., Reda, I., 2012. Solar and Moon Position Algorithm (SAMPA). National Renewable Energy Laboratory, Golden, CO, USA, <http://www.nrel.gov/midc/sampa/>.
Bird, R.E., Hulstrom, R.L., 1981. Simpliﬁed Clear Sky Model for Direct and Diﬀuse Insolation on Horizontal Surfaces, Technical Report No. SERI/TR-642-761. Solar Energy Research Institute, Golden, CO, USA.
Meeus, J., 1998. Astronomical Algorithms, 2nd ed. Willmann-Bell, Inc., Richmond, VA, USA.
Reda, I., Andreas, A., 2004. Solar position algorithm for solar radiation applications. Solar Energy 76 (5), 577–589. http://dx.doi.org/10.1016/ j.solener.2003.12.003, NREL Report No. JA-560-35518.
The Astronomical Almanac <http://asa.usno.navy.mil/>. The U.S. Naval Observatory. Washington, DC, USA <http://www.us-
no.navy.mil/> (accessed 24.07.14).