https://doi.org/10.5028/jatm.v11.1048 ORIGINAL PAPER xx/xx A Precise Algorithm for Computing Sun Position on a Satellite Tao Zheng1*, Fei Zheng1, Xi Rui1, Xiang Ji1

How to cite Zheng T https://orcid.org/0000-0002-3619-0884 Zheng T; Zheng F; Rui X; Ji X (2019) A Precise Algorithm For Computing Sun Position on a Satellite. J Aerosp Technol Manag, Zheng F https://orcid.org/0000-0001-5068-2095 11: e3619. https://doi.org/10.5028/jatm.v11.1048 Rui X https://orcid.org/0000-0002-8389-8009 Ji X https://orcid.org/0000-0001-8444-7528

ABSTRACT: To meet the high precision sun tracking needs of a space deployable membrane solar concentrator and other equipment, an existing algorithm for accurately computing the sun position is improved. Firstly, compared with other theories, the VSOP (variation seculaires des orbits planetaires) 87 theory is selected and adopted to obtain the sun position in the second equatorial coordinate system. Comparing the results with data of the astronomical almanac from 2015, it is found that the deviation of the apparent right ascension does not exceed 0.17 arc seconds, while that of the apparent declination does not exceed 1.2 arc seconds. Then, to eliminate the difference in the direction of the sun position with respect to the satellite caused by the size of the satellite’s orbit, a translation transform is introduced in the proposed algorithm. Finally, the proposed algorithm is applied to the orbit of the satellite designated by SJ-4 (shijian-4). Under the condition that both of the existing and improved algorithms adopt the VSOP87 theory to compute sun position in the second equatorial coordinate system, the maximum deviation of the azimuth angle on the SJ-4 is 35.19 arc seconds and the one of pitch angle is 19.93 arc seconds, when the deviation is computed by subtracting the results given by both algorithms. In summary, the proposed algorithm is more accurate than the existing one.

KEYWORDS: Space solar concentrator, Sun tracking, VSOP87, Translation transform, SJ-4 satellite.

INTRODUCTION

Space solar power is the main energy source for a satellite, which is provided by solar arrays at present (Ma 2001; Osipov et al. 2017). The amount of energy provided per unit area of a solar array depends on the angle between the normal direction of the solar array and the sun vector (Lin 2010). Therefore, the higher the tracking accuracy of the solar array with respect to the sun, the greater the power available to the satellite. Meanwhile, as the related technology of space deployable antenna is becoming more and more mature (Miura and Miyazaki 1990; Guest and Pellegrino 1996; Huang 2001; He and Zheng 2018), a space deployable convergent membrane solar concentrator, whose structure and principle of operation are schematically depicted in Fig. 1, is expected to be used as an energy supply equipment for satellites in the future (Zhang et al. 2009; 2016). Compared with a flat solar array, a space deployable convergent membrane solar concentrator concentrates the sunlight received by the reflector onto a small plane of focus, requiring greater solar tracking accuracy. From another perspective, the improvement of the accuracy of sun tracking will reduce the weight of the corresponding power supply equipment and the cost for transportation will be cut down accordingly (Duan 2017). In summary, solar arrays, space deployable convergent membrane solar concentrators and other optical energy supply equipment demand greater sun tracking accuracy, which essentially depends on the accuracy of the algorithms used for computing sun position on the satellite.

1.Xidian University – The College of Mechano-Electronic Engineering – Department of Electronic Mechanism – Xi’an/Shaanxi – China. Correspondence author: [email protected] Received: Feb. 22, 2018 | Accepted: Sep. 27, 2018 Section Editor: Alison Moraes

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At present, the algorithm for computing the sun position with respect to a satellite is divided in four steps: firstly, the current time and the orbital elements of the satellite are obtained; secondly, the sun position in the second equatorial coordinate system is computed; thirdly, the computed sun position is transformed into the orbit coordinate system by a rotation transformation; finally, the sun position in the coordinate system of the satellite body is obtained. In the process of transforming the sun position from the second equatorial coordinate system to the orbit coordinate system, the existing algorithm ignores the error in direction caused by the different position of the satellite in the orbit and its distance from the Earth. That is to say, the line from the satellite to the Sun and the one from the Earth to the Sun are considered to be ideally parallel (Zhai et al. 2009; Lin 2010; Zhang et al. 2011; Zhu 2011). In the above algorithm, the computation accuracy of the second step plays a major role on the whole accuracy. For this step, several high precision methods have been proposed in solar engineering, with both numerical (Pitjeva 2005; Hilton and Hohenkerk 2010) and analytical methods (Simon et al. 2013). Due to their dependence on mass historical observations, numerical methods may include large amounts of data and exhibit instability. In comparison, analytical methods are simpler and more reliable. From the aspect of tracking control, analytical methods are more suitable. In analytical methods, VSOP theories are the most popular. In 1982, Bretagnon published his first planetary theory VSOP (variation seculaires des orbits planetaires) 82, which consists of long series of periodic terms for each of the major planets, from Mercury to Neptune (Bretagnon 1982). The inconvenience of the VSOP82 solution is that one does not know where the different series should be truncated when no full accuracy is required. Fortunately VSOP87, which was an extension of VSOP82, solved the problem well (Bretagnon and Francou 1988). When use is made of the completean improved VSOP87 algorithm theory, ais high proposed, accuracy, which better gives than addi 0.01tional arc second, accuracy. is obtained In the following, (Meeus 1998). firstly, Then, VSOP theories continued to develop further, such as VSOP2000 and VSOP2013, which are more accurate than VSOP87 (Simon et al. 2013). However, thesethe whole methods computation require the process computation of the of improved additional algorithm series of periodicis described terms in while four steps.VSOP87 Secondly, is accurate enough for space engineering. So, VSOP87 is the best choice here. a simulation of the improved algorithm for a large elliptic orbit is carried out to validate the Due to the already discussed weakness of the existing algorithm, based on VSOP87 theory, an improved algorithm is proposed, which gives additional accuracy. In the following, firstly, the whole computation process of the improved algorithm is described proposed algorithm and assess its accuracy. Finally, the work of this paper is summarized. in four steps. Secondly, a simulation of the improved algorithm for a large elliptic orbit is carried out to validate the proposed algorithm and assess its accuracy. Finally, the work of this paper is summarized.

FigureFigure 1. 1The. The space space large large deployable deployable convergent convergent membrane membrane solar solar concentrator. concentrator

COMPUTATION PROCESSCOMPUTATION OF THE PROCESS IMPROVED OF THE IMPROVED ALGORITHM ALGORITHM

The improved algorithmThe improved is divided algorithm into four is steps, divided as shown into four in Fig.steps, 2. asThe shown first stepin Fig. is to 2. obtainThe first the step current is to UTC time and instantaneous orbital elements of the satellite. The second step is to compute the apparent right ascension, declination and the obtain the current UTC time and instantaneous orbital elements of the satellite. The second step earth-sun distance in the second equatorial coordinate system by VSOP87 theory. The third step is to transfer the sun vector to the orbit coordinate system by rotation and translation transforms. Finally, the sun vector is transferred to the coordinate system is to compute the apparent right ascension, declination and the earth-sun distance in the second of the satellite body, according to the attitude information of the satellite. equatorial coordinate system by VSOP87 theory. The third step is to transfer the sun vector to J. Aerosp. Technol. Manag., São José dos Campos, v11, e3619, 2019 the orbit coordinate system by rotation and translation transforms. Finally, the sun vector is

transferred to the coordinate system of the satellite body, according to the attitude information

of the satellite.

A Precise Algorithm For Computing Sun Position on a Satellite 03/12xx/xx

Step 1 Obtain UTC time and orbit roots

Calculate apparent right Step 2 ascension, declination and Earth-Sun distance by VSOP87

Transfer Sun vector to orbit Step 3 coordinate system by rotation and translation transformations

Step 4 Transfer the Sun vector to the coordinate system of the satellite body

Figure 2. Procedure for the improved algorithm.

CORRELATIVE COORDINATE SYSTEMS

Second Equatorial Coordinate System (Inertial Coordinate System) OiXiYiZi

The origin Oi is taken as the center of the Earth. The direction OiZi points to the north along the Earth’s axis, OiXi points to the vernal equinox, OiYi is formed as the cross-product of the two previous vectors (right-hand rule) and OiXiYi is the equatorial plane, as shown in Fig. 3a.

(a) (b) X Oo Z Z o b i Yo

Zo Z Orbit plane o

Oi Xo Yi Ob(Oo )

X Xb Vernal equinox i Equatorial plane Y o Yb

Figure 3. Correlative coordinate systems: (a) second equatorial coordinate system and orbit coordinate system; (b) orbit coordinate system and coordinate system of satellite body.

Orbit Coordinate System OoXoYoZo

The orbit coordinate system is determined by the orbit plane and by the center of the Earth. The origin Oo coincides with the centroid of the satellite. OoZo points to the center of the Earth. OoYo points to the normal direction of the orbit. OoXo is formed as the right-hand rule. When the shape of the orbit is circular, OoXo points to the motion direction of the satellite. The orbit coordinate system is shown in Figs. 3a and 3b.

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Zheng T; Zheng F; Rui X; Ji X 04/12xx/xx

Instead, by selecting important terms from the VSOP87, an error not exceeding 1 arc second Instead, by selecting important terms from the VSOP87, an error not exceeding 1 arc second Coordinate System of Satellite Body ObXbYbZb The origin O between is the centroid the years of the – 2000satellite and and +6000 the three can orthogonalbe obtained axes (Meeus of X , 1998),Y and Zwhich are respectively is accurate parallel enough to the betweenb the years –2000 and +6000 can be obtained (Meeus 1998),b b whichb is accurate enough inertial axes of the satellite, which are fixedly connected with the satellite body. When the attitude of the satellite with respect to the Earth is not changed,to tracking the coordinatecontrol and system not complicatedof the satellite to body implement. coincides with the orbit coordinate system. The coordinate to tracking control and not complicated to implement. system of the satellite body is shown in Fig. 3b. Computation of Apparent Longitude and Latitude of the Sun Computation of Apparent Longitude and Latitude of the Sun COMPUTATION OF THE SUN VECTOR IN SECOND EQUATORIAL COORDINATE SYSTEM As aforementioned, the heliocentric longitude L, heliocentric latitude B, and radius vector As aforementioned, the heliocentric longitude L, heliocentric latitude B, and radius vector VSOP87 Theory R of the Earth contain many periodic terms according to the VSOP87 theory. Appendix II of As aforementioned,R of the Bretagnon Earth contain and Francou many createdperiodic VSOP87 terms planetaccording theory to in the 1987, VSOP87 which gives theory. the periodic Appendix terms II to of compute planetary heliocentric coordinate including heliocentric longitude, latitude and radius vector. When use is made of the complete the literature (Meeus 1998) gives the most important periodic terms from the VSOP87 theory. VSOP87 theory,the a literaturehigh accuracy, (Meeus better 1998) than gives0.01 arc the second, most importantis obtained. periodic For the Earthterms it from contains the VSOP872425 terms, theory. namely 1080 terms for the Earth’s longitude, 348 for the latitude, and 997 for the radius vector. However, this big amount of numerical data is In Chapter 31 of Meeus (1998), the series labelled L0, L1, L2, L3, L4, L5, B0, B1, R0, R1, R2, unfavorable forIn onboard Chapter tracking 31 of Meeuscontrol. (Instead,1998), bythe selecting series label importantled L0, terms L1, fromL2, L the3, LVSOP87,4, L5, B an0, errorB1, R not0, Rexceeding1, R2, 1 arc second between the years –2000 and +6000 can be obtained (Meeus 1998), which is accurate enough to tracking control and not R3, R4 for the Earth are provided. Each horizontal line in the list of Appendix II (Meeus 1998) complicated toR implement.3, R4 for the Earth are provided. Each horizontal line in the list of Appendix II (Meeus 1998) represents one periodic term and contains four numbers: the first number is the label of the COMPUTATIONrepresents OF APPARENT one periodic LONGITUDE term and containsAND LATITUDE four numbers: OF THE the SUN first number is the label of the As aforementioned, the heliocentric longitude L, heliocentric latitude B, and radius vector R of the Earth contain many periodic term in the series and the following three numbers are referred to as A, B, C, respectively. The terms accordingterm to the in VSOP87the series theory. and theAppendix following II of thethree literature numbers (Meeus are referred1998) gives to theas A,most B, important C, respectively. periodic The terms from the VSOP87 theory. In Chapter 31 of Meeus (1998), the series labelled L0, L1, L2, L3, L4, L5, B0, B1, R0, R1, R2, R3, R4 for the value of each term is given by Acos(B + Cτ), where τ is the time measured in Julian millennia Earth are provided.value Eachof each horizontal term is line given in theby listAcos of( BAppendix + Cτ), whereII (Meeus τ is 1998)the time represents measured one inperiodic Julian termmillennia and contains four numbers: the first number is the label of the term in the series and the following three numbers are referred to as A, B, C, from the epoch 2000.0. The required longitude L, latitude B and radius vector R (distance to respectively. Thefrom value the of epoch each term 2000.0. is given The by requiredAcos(B + longitudeCτ), where Lτ ,is latitude the time B measured and radius in Julian vector millennia R (distance from theto epoch 2000.0. The requiredthe longitudeSun in astronomical L, latitude B and units radius) are vector obtained R (distance from toEq. the 1: Sun in astronomical units) are obtained from Eq. 1: the Sun in astronomical units) are obtained from Eq. 1: ì (L0 + L1t + L2t 2 + L3t 3 + L4t 4 + L5t 5 )×180! = 2 3 4 5 ! ì ïL(L0 + L1t + L2t + L3t 8+ L4t + L5t )×180 = 10 ×p ïLï 8 ! 10 ×p ï ï (B0 + B1t )×180 = ! (1) ï íB(B0 + B1t 8)×180 (1) 10 ×p (1) íBï= 8 10 ×p 2 3 4 ï ï (R0 + R1t + R2t + R3t + R4t ) = 2 3 4 ï ïR(R0 + R1t + R2t + 8R3t + R4t ) = 10 ïRî 8 î 10 Then, geocentric longitude Θ and latitude β of the Sun are computed by Eq. 2: Then, geocentric Then, longitude geocentric Θ and latitudelongitudeβ of Θthe and Sun latitude are computed β of the by Eq.Sun 2: are computed by Eq. 2: Q = L +180° , b = -B (2) Q = L +180° , b = -B (2) (2) Conversion to the FK5 System Conversion to the FK5 System Conversion to the FK5 System The Sun’s longitude The Θ and Sun latitude’s longitude β are referred Θ and to the latitude mean dynamical β are referred ecliptic andto the equinox mean of thedynamical date defined ecliptic by the and VSOP87 (Bretagnon et al. 1986).The ThisSun ’references longitude frame Θ differsand latitude very slightly β are from referred the standard to the meanFK5 (Fifth dynamical Fundamental ecliptic Catalogue) and system (Meeus 1998), whichequinox is based of onthe absolute date defined and quasi-absolute by the VSOP catalogues87 (B retagnonwith mean et epochs al. 1986 later). than This 1900 reference (Fricke etframe al. 1988). equinox of the date defined by the VSOP87 (Bretagnon et al. 1986). This reference frame These catalogues consist of about 85 catalogues giving observations from 1900 to about 1980. The observations presented in these catalogues were made with meridian circles, vertical circles, transit instruments, and astrolabes. The conversion of Θ andβ to the FK5 system can be performed by making use of the following equations (Eq. 3) (Meeus 1998):

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differs very slightly from the standard FK5 (Fifth Fundamental Catalogue) system (Meeus

1998), which is based on absolute and quasi-absolute catalogues with mean epochs later than

1900 (Fricke et al. 1988). These catalogues consist of about 85 catalogues giving observations

from 1900 to about 1980. The observations presented in these catalogues were made with

meridian circles, vertical circles, transit instruments, and astrolabes. The conversion of Θ and

β to the FK5 system can be performed by making use of the following equations (Eq. 3) (Meeus A Precise Algorithm For Computing Sun Position on a Satellite 05/12xx/xx 1998):

ìl' = Q -1°.397 T - 0° .00031 T 2 ï ! ï æ 0.09033ö ïDQ = -ç ÷ ï è 3600 ø ° ï æ 0.03916 ö íDb = ç ÷ (cosl' - sin l') (3) (3) ï è 3600 ø ïQ = Q + DQ ï ïb = b + Db ï ì îï ïM = 280°.4665 + 36000 °.7698T ï whereïM T' is= the218 time°.where3165 in centuries +T 481267is the from time°. 8813epoch in centuriesT 2000.0. from epoch 2000.0. ï ï (125.04452-1934.136261 T + 0.0020708 T 2 +T 3 / 450000)×180° ApparentíW = Place of the Sun Apparent Place of the Sun (4) p ïThe Sun’s longitude Θ obtained thus far is the geometric longitude of the Sun. To obtain the apparent longitude λ, the effects ì ° ° ° ° of nutationï andæ 17aberration .20Theö shouldSun’s belongitudeæ 1taken.32 öinto Θ account obtained (Meeusæ 0thus.23 1998).farö is the geometricæ 0.21 longitudeö of the Sun. To obtain ïMïD=y280= -°ç.4665 +÷36000sin W°.-7698ç T ÷ sin 2M - ç ÷ sin 2M' + ç ÷ sin 2W ï For nutation,3600 add the nutation in3600 longitude ΔΨ to Θ according3600 to Eq. 4 (Seidelmann3600 1980): ï °è ø ° è ø è ø è ø ïMï '= 218 .3165the apparent+ 481267 longitude.8813T λ, the effects of nutation and aberration should be taken into account ï îQ = Q + Dy ï (125ì.04452-1934.136261 T + 0.0020708 T 2 +T 3 / 450000)×180° íW = ïM(Meeus= 280 °1998)..4665 + 36000 °.7698T (4) Then,ï to the aberration, apply pthe correction to Θ obtained by Eq. 5 (Meeus 1998), where ï ° ° ï ïM '= 218° .3165 + 481267° .8813T ° ° æ17. 20 öFor nutation,æ 1. 32addö the nutationæ in0 .longitude23 ö ΔΨ to Θæ according0.21 ö to Eq. 4 (Seidelmann 1980): ïDy = -çï ÷ sin W - ç ÷ sin 2M - ç ÷ sin 22 M' +3 ç ÷ sin 2W° R is theè ïearth3600-(125øsun .04452distanceè-36001934 in .136261øastronomical T + è0.3600 0020708unitsø. Following T +T è/ 4500003600this procedureø )×180 , one obtains the ï íW = (4) (4) p îïQ = Q + Dïy ° ° ° ° Sun’s apparentï longitudeæ17.20 ö λ. æ 1.32 ö æ 0.23 ö æ 0.21 ö ïDy = -ç ÷ sin W - ç ÷ sin 2M - ç ÷ sin 2M' + ç ÷ sin 2W Then,ï to the aberration,è 3600 ø apply theè 3600 correctionø to Θ èobtained3600° ø by Eq. 5 è(Meeus3600 ø 1998), where æ 20.4898 ö îïQ = Q + Dy ç ÷ 3600 R is the earth - sun distance in astronomical l = Q -unitsè . Followingø this procedure , one obtains (5) the R Then, to the aberration,Then, to theapply aberration, the correction apply to Θthe obtained correction by Eq. to5 (Meeus Θ obtained 1998), bywhere Eq. R is 5 the (Meeus earth-sun 1998), distance where in Sun’s apparent longitude λ. astronomical Finally, units. the Following apparent this procedure,right ascension one obtains α and the Sun’sdeclination apparent longitudeδ are computed λ. from the apparent R is the earth-sun distance in astronomical units. Following this procedure, one obtains the ° æ 20.4898 ö longitude λ and latitude β by means of Eq. 6ç, which represent÷ a coordinate transformation from Sun’s apparent longitude λ. 3600 (5) l = Q - è ø (5) R the geocentric ecliptical coordinate system to the second equatorial° coordinate system (Meeus æ 20.4898 ö Finally,Finally, the apparent the apparent right ascension right ascension α and declination α and δ aredeclination computedç from δ are the÷ computed apparent longitude from the λ and apparent latitude β by means 3600 of Eq.1998): 6, which represent a coordinate transformation from l the= Q geocentric- è eclipticalø coordinate system to the second equatorial(5) R coordinatelongitude system λ and (Meeus latitude 1998): β by means of Eq. 6, which represent a coordinate transformation from ì sin l cose - tan b sin e Finally, the apparentïtan righta = ascension α and declination δ are computed from the apparent l (6) the geocentric ecliptical coordinateí system to the secondcos equatorial coordinate system (Meeus (6) îïsin d = sin b cose + cos b sin e sin l longitude λ and latitude β by means of Eq. 6, which represent a coordinate transformation from 1998): wherewhere ε stands ε stands for the for true the obliquity true obliquityof the ecliptic of andthe ε eclipticcontains nutationand ε contains in obliquity nutation Δε, which in canobliquity be obtained Δε, by Eq. 7 (Seidelmann the1980; geocentric Laskar 1986): ecliptical coordinate system to the second equatorial coordinate system (Meeus ì sin l cose - tan b sin e which can be obtained by Eq.ï tan7 (Seidelmanna = 1980; Laskar 1986): 1998): í cos l J. Aerosp. Technol. Manag., São José dos Campos, (6) v11, e3619, 2019 îïsin d = sin b cose + cos b sin e sin l ì sin l cose - tan b sin e ïtana = where ε stands for the true obliquity of í the ecliptic and εcos containsl nutation in obliquity Δ ε , (6) îïsin d = sin b cose + cos b sin e sin l which can be obtained by Eq. 7 (Seidelmann 1980; Laskar 1986): where ε stands for the true obliquity of the ecliptic and ε contains nutation in obliquity Δε,

which can be obtained by Eq. 7 (Seidelmann 1980; Laskar 1986):

Zheng T; Zheng F; Rui X; Ji X 06/12xx/xx

ìU = T /100 ï ° ° ° ° ï æ 9.20 ö æ 0.57 ö æ 0.10 ö ' æ 0.09 ö De = ç ÷ cosW + ç ÷ cos2M + ç ÷ cos2M - ç ÷ cos2W ï è 3600 ø è 3600 ø è 3600 ø è 3600 ø ï ïe 0 = 23°.4392911- (4680°.93/3600) U - (1.55° /3600) U 2 ï ° 3 ° 4 ° 5 í (1999 .25/3600) U - (51 .38/3600) U - 249 .67 /3600 U . (7) (7) ï ° 6 ° 7 ° 8 ï - (39 .05/3600) U + (7 .12/3600) U + (27 .87/3600) U ï + (5°.79/3600) U 9 + (2°.45/3600) U 10 ï ïe = e 0 + De ï î

COMPARISON WITH THE CHINESEComparison ASTRONOMICAL with the ALMANAC Chinese Astronomical Almanac To check the accuracy of VSOP87 algorithm, the values of apparent right ascension α and declination δ, which are computed To check the accuracy of VSOP87 algorithm, the values of apparent right ascension α and according to VSOP87 algorithm, are compared with data given by the Chinese astronomical almanac (Purple Mountain Observatory Chinese Academy of Science 2015) at zero hour of the first day each month in 2015. The results of the comparison are shown in declination δ, which are computed according to VSOP87 algorithm, are compared with data Table 1. Deviation results between them are plotted in Fig. 4. Figure 4 shows that the maximum deviation of the apparent longitude α is less than 0.17 arc seconds and that of the apparent declination δ is less than 1.2 arc seconds. Compared with corresponding given by the Chinese astronomical almanac (Purple Mountain Observatory Chinese Academy data of the literature (Zhang et al. 2011), the accuracy has been improved, especially for the apparent right ascension α.

of1.5 Science 2015) at zero hour of the first day each month in 2015. The results of the comparison

are1.0 shown in Table 1. Deviation results between them are plotted in Fig. 4. Figure 4 shows that

the0.5 maximum deviation of the apparent longitude α is less than 0.17 arc seconds and that of the

apparent0.0 declination δ is less than 1.2 arc seconds. Compared with corresponding data of the Derivation result/” Derivation literature-0.5 (Zhang et al. 2011), the accuracy has been improved, especially for the apparent right

-1.0 ascension1 α2. 3 4 5 6 7 8 9 10 11 12 Month in 2015 Apparent right ascension α Apparent declination δ

Figure 4. Comparison between VSOP87 algorithm and the Chinese Astronomical Almanac data for the year of 2015.

Table 1. Values of α and δ given by the VSOP87 algorithm and the Chinese Astronomical Almanac.

Month VSOP87 algorithm Chinese astronomical almanac Deviation 1 18 h 44 m 30.41 s; –23°02’27.12” 18 h 44 m 30.55 s; –23°02’26.3” –0.14”; –0.82” 2 20 h 56 m 59.27 s; –17°14’59.75” 20 h 56 m 59.44 s; –17°14’59.3” –0.17”; –0.45” 3 22 h 46 m 24.27 s; –7°47’26.06” 22 h 46 m 24.35 s; –7°47›26.0” –0.08”; –0.06” 4 0 h 40 m 15.26 s; 4°19’53.0” 0 h 40 m 15.19 s; 4°19›52.7” 0.07”; 0.3” 5 2 h 31 m 37.21 s; 14°54’42.22” 2 h 31 m 37.05 s; 14°54’41.5” 0.16”; 0.72” 6 4 h 34 m 21.44 s; 21°58’35.65” 4 h 34 m 21.32 s; 21°58’34.6” 0.12”; 1.05

7 6 h 38 m 38.10 s; 23°08’17.10” 6h38m38.09s; 23°08’15.9” 0.01”; 1.2” ...continue

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Transform of the Sun Position from Athe Precise Second Algorithm EquatorialFor Computing Sun CoordinatePosition on a Satellite System to the 07/12xx/xx Coordinate System of the Satellite Body

Table 1. Continuation... Since the deviation of the sun position in the second equatorial coordinate system obtained Month VSOP87 algorithm Chinese astronomical almanac Deviation 8 8by h 43 VSOP87 m 31.31 s; 18°08’44.70”algorithm is of about8 h 43 1m 31.42arc seconds; 18°08’43.6” compared with the–0.11”; Chinese 1.1” astronomical 9 10 h 39 m 36.49 s; 8°28’18.86” 10 h 39 m 36.61 s; 8°28›18.1” –0.12”; 0.76” 10 12almanac h 27 m 33.16 data, s; –2°58’33.22” it is not acceptable12 h 27to mignore 33.22 s; the –2°58’33.6” error caused by the–0.06”; size 0.38”of the orbit. So the 11 14 h 23 m 28.05 s; –14°14’59.39” 14 h 23 m 28.00 s; –14°14’59.4” 0.05”; 0.01” 12 16mathematical h 26 m 51.63 s; –21°42’36.18” model of the existing16 h 26 algorithm m 51.53 s; –21°42’35.9” is improved, as shown 0.1”;in Fig. –0.28” 5.

TRANSFORM OF THE SUN POSITION FROM THE SECOND EQUATORIAL COORDINATE SYSTEM TO THE COORDINATE SYSTEM OF THE SATELLITE BODY Since the deviation of the sun position in the second equatorial coordinate system obtained by VSOP87 algorithm is of about 1 arc second compared with the Chinese astronomical almanac data, it is not acceptable to ignore the error caused by the size of the orbit. So the mathematical model of the existing algorithm is improved, as shown in Fig. 5.

Zi Satellite

Yo Oo

X Lo o Zo P Sun γ L i Yi Oi

Figure 5. DeviationXi of sun position on the satellite caused by the size of the orbit. Earth (Vernal equinox)

Figure 5. Deviation of sun position on the satellite caused by the size of the orbit. The Process of the Improved Algorithm THE PROCESS OF THE IMPROVED ALGORITHM Firstly, the position vector of the Sun L in O X YZ is given by Eq. 8: Firstly, the positioni i i vectori i of the Sun Li in OiXiYiZi is given by Eq. 8:

écosd cosaù 8 ê ú = × × d a . (8) (8) Li (1.495978707 10 R)êcos sin ú ëê sin d ûú

Then, Li is transferred from OiXiYiZi to OoXoYoZo by the transform matrix T including Then,L i is transferred from OiXiYiZi to OoXoYoZo by the transform matrix T including rotation matrix R and translation matrix P. The conversion is described by Eqs. 9 and 10: rotation matrix R and translation matrix P. The conversion is described by Eqs. 9 and 10:

R = Ry (- 90° -w)Rx (i - 90°)Rz (W) (9) (9)

(10) Lo = RLi + P (10)

where ω is the argument of perigee, i is the orbital inclination and Ω is the right ascension of J. Aerosp. Technol. Manag., São José dos Campos, v11, e3619, 2019

the ascending node; Rx, Ry, Rz denote the unit rotation matrix around the axis x, y, z, respectively,

and P is the position vector of the satellite in orbit.

The position vector of the Sun in ObXbYbZb, Lb, is obtained by Eq. 11:

Lb = Rx (q )Ry (j)Rz (y )Lo (11)

where φ, θ, Ψ denote the rolling angle, pitch angle and yaw angle of the satellite, respectively.

u The unit position vector of Sun in ObXbYbZb, Lb is given by the modular operation to Lb

(Eq. 12):

u Lb Lb = . (12) Lb

u The azimuth angle ϕ is defined as the angle between the projection of Lb on the plane of

ObXbZb and the negative direction of the axis ObZb in ObXbYbZb, and is given by Eq. 13:

- Lu (z) f = arccos b (13) u 2 u 2 Lb (x) + Lb (z) u ( ) u ( ) u where Lb x and Lb z denote the components of Lb in the direction of axis x and z,

respectively.

u The pitch angle γ is defined as the angle between Lb and the plane of ObXbZb in ObXbYbZb,

which is given by Eq. 14:

u 2 u 2 g = arccos Lb (x) + Lb (z) . (14)

SIMULATION OF THE IMPROVED ALGORITHM FOR A LARGE ELLIPTICAL

ORBIT AND ANALYSIS OF THE ACCURACY

R = Ry (- 90° -w)Rx (i - 90°)Rz (W) (9)

Lo = RLi + P (10)

where ω is the argument of perigee, i is the orbital inclination and Ω is the right ascension of R = Ry (- 90° -w)Rx (i - 90°)Rz (W) (9)

the ascending node; Rx, Ry=, RRz ydenote(- 90 ° -thew )unitRx ( irotation- 90°)R matrixz (W) around the axis x , y , z,(9) respectively , Lo = RLi + P (10) and P is the position vector ofL the= RsatelliteL + P in orbit. (10) where ω is the argument of perigee,o i is the orbital inclination and Ω is the right ascension of R = Ry (- 90° -w)Rx (i - 90°)Rz (W) (9)

Thewhere position ω is vectorthe argument of the Sun of perigee,in ObXb Yi bisZ bthe, Lb ,orbital is obtained inclination by Eq. and 11 : Ω is the right ascension of the ascending node; Rx, Ry, Rz denote the unit rotation matrix around the axis x, y, z, respectively, Lo = RLi + P (10)

the ascending node; R x, R y , RLz bdenote= Rx (q the)R yunit(j) Rrotationz (y )Lo matrix around the axis x , y ,(11) z, respectively , Zheng T; Zhengwhere F; Rui ω X; isJi Xtheand argument P is the positionof perigee, vector i is ofthe the orbital satellite inclination in orbit. and Ω is the right ascension of 08/12xx/xx where andφ, θ ,P Ψ is denote the position the rolling vector angle, of the pitch satellite angle in and orbit. yaw angle of the satellite, respectively. the ascending node;The R positionx, Ry, Rz denote vector theof theunit Sun rotation in Ob matrixXbYbZb ,around Lb, is obtainedthe axis x by, y ,Eq. z, respectively 11: , u The unitThe position position vector vector of of Sun the in Sun Ob Xinb YObbZXb,b YbZb , isL bgiven, is obtained by the modularby Eq. 11 operation: to Lb where ω is the argument of perigee, i is the orbital inclination and Ω Lis the= R right(q )R ascension(j)LRb (y )ofL the ascending node; R , R , R(11) denote and P is the position vector of the satellite in orbit. b x y z o x y z the unit rotation matrix around the axis x, y, z, respectively, and P is the position vector of the satellite in orbit. (Eq. 12): Lb = Rx (q )Ry (j)Rz (y )Lo (11) where φ, θ, Ψ denote the rolling angle, pitch angle and yaw angle of the satellite, respectively. The position vector Theof the position Sun in vectorObXbYb ofZb ,the Lb, Sunis obtained in ObX bbyYb ZEq.b, L 11:b, is obtained by Eq. 11:

where φ, θ, Ψ denoteu L theb rolling angle, pitch angleu and yaw angle of the satellite, respectively. L = . b b b b (12) b The unit position Lbvectorb = R xof(q )SunRy (j in)R Oz (yX)YLoZ , L b is given by the modular (11) operation to(11) L Lb u The unit position vector of Sun in ObXbYbZb, Lb is given by the modular operation to Lb where φ, θ, Ψ(Eq. denote 12): the rolling angle, pitch angle and yaw angle of the satellite, respectively.u where φ, θ, Ψ denote the rolling angle, The pitch azimuth angle angleand yaw ϕ is angle defined of the as thesatellite, angle respectively. between the projection of Lb on the plane of (Eq. 12): u The unit position vector of Sun in O X Y Z , L is givenu Lbyb the modularu operation to L (Eq. 12): The unit positionb vectorb b b ofb Sun =in ObXbYbZb, L is given by the bmodular operation to Lb ObXbZb and the negativeLb direction. of the b axis O b Z b in O b X b Y bZ b, and is (12) given by Eq. 13: Lb u L L = b . (12) (12) (Eq. 12): b - Lu (z) f = arccos Lb b (13) u The azimuth angle ϕ is defined as the angle between the projection of Lb on the plane of Lu (x)2 + Lu (z)2 b ub u Lb u The azimuth angle ϕ is defined as the L bangleThe= azimuthbetween. anglethe projection ϕ is defined of L bas on the the angle plane between of(12) ObX b Ztheb and projection the negative of L directionb on the planeof of ObXbZb andu theL negativeu direction of the axis ObZb in ObXbuYbZb, and is given by Eq. 13 : the axis O Z in O X Y Z , and iswhere given byL Eq.(x) 13:andb L (z) denote the components of L in the direction of axis x and z, b b b b b b b b b ObXbZb and the negative directionu of the axis ObZb in ObXbYbZb, and is given by Eq. 13: - Lb (z) u The azimuth angle ϕ is definedf = asarccos the angle between the projection of L b on the(13) plane of respectively. u 2 u 2 (13) L (x) + L (zu) b -b L (z) f = arccos b (13) ObXbZb and the negative direction of the axis ObZb inu ObX2b+YbZub, and2 uis given by Eq. 13: u u The pitchu angle γu isu defined as the angleLb (x) betweenLb (z) Lb andu the plane of ObXbZb in ObXbYbZb, where Lb (x) and Lb (z) denote the components of Lb in the direction of axis x and z, where Lb (x) and Lb (x) denote the components of Lb in the direction of axis x and z, respectively. u u u - L (zu) u The pitch angle γ is defined as the angle between L andb the plane of O X Z in O X Y Z(13), which is given by Eq. 14: which whereisf given= arccos L byb ( xEq.) and b14: Lb (z) denote theb b componentsb b b b ofb Lb in the direction of axis x and z, respectively. u 2 u 2 Lb (x) + Lb (z) u 2 u 2 u respectively.u g = arccos Lb (x) + Lb (z)u . u (14) (14) ( ) The pitch( ) angle γ is defined as the angle between Lb and the plane of ObXbZb in ObXbYbZb, where Lb x and Lb z denote the components of Lb in the direction of axis x and z, u SIMULATION The pitch OF angle THE γ isIMPROVED defined as the ALGORITHM angle between FORLb and A LARGE the plane ELLIPTICAL of ObXbZb in Ob XbYbZb, respectively. which is given by Eq. 14:

which is given ORBITby Eq. 14 AND: u ANALYSIS2 u u 2 OF THE ACCURACY SIMULATION OF THE IMPROVEDg = ALGORITHMarccos Lb (x) + Lb ( zFOR) . A LARGE ELLIPTICAL (14) The pitch angle γ is defined as the angle between Lb and the plane of ObXbZb in ObXbYbZb, ORBIT AND ANALYSIS OF THE ACCURACY u 2 u 2 g = arccos Lb (x) + Lb (z) . (14) which is given SIMULATION by Eq. 14: OF THE IMPROVED ALGORITHM FOR A LARGE ELLIPTICAL The satellite named SJ-4 has been launchedSIMULATION from China OF withTHE the IMPROVED mission of studyingALGORITHM the environmental FOR A LARGE effect ELLIPTICAL of charged g = arccos Lu (ORBITx)2 + Lu AND(z)2 . ANALYSIS OF THE ACCURACY(14) particles in space (Hu and Chen 1994). Here, the orbitb of SJ-4b is used as a simulation example. The orbital elements are given in Table 2 (Heavens Above 2016). It is assumed that the satelliteORBIT adopts theAND three-axis ANALYSIS stabilization OF THE attitude ACCURACY and O X Y Z coincides SIMULATION OF THE IMPROVED ALGORITHM FOR A LARGE ELLIPTICALb b b b with OoXoYoZo, that is, the satellite has no change of flight pose with respect to the orbit coordinate system.

ORBIT AND ANALYSIS OF THE ACCURACY Table 2. The orbital elements of SJ-4.

Elements Values Eccentricity 0.5675908 Inclination (°) 28.7578 Perigee Altitude (km) 232 Apogee Altitude (km) 17585 RAAN (°) 126.1640 Argument of perigee (°) 288.1275 Initial Mean Anomaly (°) 20.0596

The orbital elements are substituted in the improved algorithm. Then the curves ofϕ and γ in a flight cycle of the satellite are computed and plotted, as shown in Figs. 6 and 7. When the satellite enters the Earth’s shadow area, the value of ϕ or γ are zero, as shown in Figs. 6 and 7.

J. Aerosp. Technol. Manag., São José dos Campos, v11, e3619, 2019 A Precise Algorithm For Computing Sun Position on a Satellite 09/12xx/xx

150

ϕ ( ° ) 100

50 e azimuth angle angle azimuth e

0 0 50 100 150 200 250 300 Time in a ight cycle of satellite (min)

Figure 6. Curve of the azimuth angle ϕ in a flight period.

40

30

20

10 e pitch angle γ ( ° ) angle pitch e

0 0 50 100 150 200 250 300 Time in a ight cycle of satellite (min)

Figure 7. Curve of the pitch angle γ in a flight period.

It is assumed that both the existing and improved algorithms adopt the VSOP87 theory, proposed in this paper, to compute the apparent right ascension α and declination δ in OiXiYiZi in order to assess the accuracy of the improved algorithm. Only the error caused by the size of orbit when SJ-4 is at different positions in its orbit is considered. Then, the azimuth angle ϕ and the pitch angle γ are computed making use of the two algorithms. The difference in azimuth angle given by the two algorithms, σ, is plotted in Fig. 8, and the one corresponding to the pitch angle, τ, is plotted in Fig. 9. Table 3 shows the values of azimuth angle ϕ and their deviation σ at six key points of Fig. 8. At the same time, Table 4 shows the values of pitch angle γ and their deviation τ at six key points of Fig. 9. Figure 8 shows that the maximum absolute value of σ is 35.19 arc seconds at flight time equal to 171 min. Figure 9 shows that the maximum absolute value of τ is 19.93 arc seconds at flight time equal to 231 min.

Table 3. Values of azimuth angle at six key moments.

Azimuth angle (“) Time (min) Improved algorithm Existing algorithm Deviation 0 2.656197e5 2.656160e5 3.67 33 7.461378e4 7.460010e4 13.68 63 2.97365e3 2.97876e3 –5.11 171 1.6798979e5 1.6795460e5 35.19 264 4.0102255e5 4.0101763e5 4.92 314 2.6132750e5 2.613238e5 3.67

J. Aerosp. Technol. Manag., São José dos Campos, v11, e3619, 2019 Zheng T; Zheng F; Rui X; Ji X 10/12xx/xx

Table 4. Values of pitch angle at six key moments.

Pitch angle (“) Time (min) Improved algorithm Existing algorithm Deviation 0 1.3523874e5 1.3524983e5 –11.09 7 1.3524505e5 1.3525832e5 –13.27 94 1.3538215e5 1.3536379e5 18.36 231 1.3550967e5 1.3552960e5 –19.93 280 1.3559584e5 1.3558883e5 7.01 314 1.3561850e5 1.3562990e5 –11.40

40 (171, 35.9)

30

20 (33, 13.68) 10

e deviation of of deviation e (264, 4.92)

azimuth angle σ ( ” ) angle azimuth (0, 3.67) (314, 3.67) 0 (63, -5.11) -10 0 50 100 150 200 250 300 Time in a ight cycle of satellite (min)

Figure 8. Curve of the deviation of the azimuth angle σ in a flight period.

20 (94, 18.36)

10 (280, 7.01)

0 pitch angle τ ( ” ) angle pitch e deviation of of deviation e -10 (0, -11.09) (231, -19.93) (314, -11.40) (7, -13.27) -20 0 50 100 150 200 250 300 Time in a ight cycle of satellite (min)

Figure 9. Curve of the deviation of the pitch angle τ in a flight period.

CONCLUSION

In this paper, a precise method for computing the sun position on a satellite is proposed, improving the accuracy given by other algorithms, as shown by simulation results. The results of this research provide a theoretical basis for spaceborne equipment that need to track the Sun accurately. The improvement in accuracy includes two aspects, which are summarized below: • Compared with the data of 2015 astronomical almanac, the deviation of the apparent right ascension α is not greater than 0.17 arc second and the one of apparent declination δ is not greater than 1.2 arc seconds, which is better than the results from other literature.

J. Aerosp. Technol. Manag., São José dos Campos, v11, e3619, 2019 A Precise Algorithm For Computing Sun Position on a Satellite 11/12xx/xx

• The maximum value of the deviation in azimuth angle σ on the orbit of SJ-4 is 35.19 arc seconds and the one in pitch angle τ is 19.93 arc seconds compared with existing algorithms. The results indicate that the error of the sun position caused by the size of the orbit does exist and reaches the order of the arc second.

AUTHORS’ CONTRIBUTION

Conceptualization, Zheng T and Zheng F; Methodology, Zheng T and Zheng F; Investigation, Zheng T, Rui X and Ji X; Writing – Original Draft, Zheng T and Rui Xi; Writing – Review and Editing, Zheng T and Ji X.

FUNDING

There are no funders to report.

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