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
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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 à sinðdi à D þ mi à M þ m0i à M0 þ f i à 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 à cosðdi à D þ mi à M þ m0i à M 0 þ f i à 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 à sinðdi à D þ mi à M þ m0i à 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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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 à JCEÞ Ã sin Dei ¼ ðci þ di à JCEÞ Ã cos
! X4
X j à Y i;j ;
j¼1
! X4
X j à 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 à 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 à cos e À tan b à 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.
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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 à cos e þ cos b à sin e à 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 à sin pÞ Ã cos Da cos d À y à sin p à 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 à sin p à sin H cos d À x à sin p à 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 à sin d0 þ cos u à cos d0 à 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 à cos hm þ sin hs à sin hm à 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
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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
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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 à 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 à 100 p à 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 à E2ms à 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 à 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 à 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 à r2s À Ai:
ðB14Þ
References
Andreas, A., Reda, I., 2012. Solar and Moon Position Algorithm (SAMPA). National Renewable Energy Laboratory, Golden, CO, USA, .
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 . The U.S. Naval Observatory. Washington, DC, USA (accessed 24.07.14).