Effects of in-track maneuver on Sun illumination conditions of near-circular low Earth orbits

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ScienceDirect Advances in Space Research 53 (2014) 509–517 www.elsevier.com/locate/asr

Effects of in-track maneuver on Sun illumination conditions of near-circular low Earth orbits Jin Zhang ⇑, Ya-zhong Luo, Guo-jin Tang College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, People’s Republic of China Received 28 June 2013; received in revised form 15 November 2013; accepted 16 November 2013 Available online 26 November 2013

Abstract By developing approximate analytical models considering the J2 perturbation, the effects of an in-track maneuver on the orbital Sun illumination conditions of near-circular low Earth orbits are analyzed. First, two approximate models for the variations in orbital sunshine angles are developed, one for variations at a given time and the other for variations at a given argument of latitude. Next, two approximate models for variations in orbital arc in Earth shadow are developed, one considers the small eccentricity and the other uses the zero eccentricity. Finally, the developed approximate models are applied to analyzing the Sun illumination conditions of a typical intrack maneuver mission on a near-circular low Earth orbit. From the results obtained, three major conclusions can be drawn. First, the variations in orbital sunshine angles at a given time may reach tens of degrees when the drifting time reaches hundreds of orbital periods, and the approximate model for that situation cannot effectively approach the numerical results. Second, the variations in orbital sunshine angles for any given argument of latitude are only a couple of degrees even when the drifting time reaches 500 orbital periods, and the approximation model developed can effectively approach the numerical results. Third, for variations in orbital arc in Earth shadow, the approximate model considering the small eccentricity has simple expressions and can effectively approach the numerical results; in contrast, the approximate model using the zero eccentricity has relatively worse precision. Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Orbital maneuver; Sun illumination; Earth shadow; Low Earth orbit; Near-circular orbit; J2 perturbation

1. Introduction Sun illumination conditions are important design factors in the field of space mission analysis and design, and have been the topic of extensive research. Only some contributions in this field are noted here. The effects of Earth shadow have been taken into account in the satellite attitude control model (Dilssner et al., 2011; Zanardi et al., 2005), in the satellite trajectory design using solar-electricity propulsion or solar-sail propulsion (Kluever, 2011), in the precise orbital determination model (Musen, 1960; ⇑ Corresponding author. Tel.: +86 0731 84576316; fax: +86 0731 84512301. E-mail addresses: [email protected], [email protected] (J. Zhang), [email protected] (Y.-z. Luo), [email protected] (G.-j. Tang).

Vallado, 2001), and even in the evolution model of space debris (Hubaux et al., 2012, 2013; Valk and Lemaitre, 2008). The energy subsystem of a satellite has requirements on the orbital Sun angle in order to obtain enough solar energy, and the working of optical sensors on a satellite may not allow the direct incidence of sunlight (Zhang et al., 2012; Zhang and Parks, 2013). In-track maneuvers are usually executed by low-Earthorbit (LEO) satellites to adjust or maintain their orbits, and will cause the variations in Sun illumination conditions. However, few previous studies focused on the properties of these variations. These variations can be obtained by subtracting the Sun illumination conditions of the trajectory with no maneuver from those of the trajectory with maneuvers, and also can be analyzed and explained by developing approximate analytical models.

0273-1177/$36.00 Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2013.11.039

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J. Zhang et al. / Advances in Space Research 53 (2014) 509–517

The purpose of this study is to reveal the effects of an intrack maneuver on the Sun illumination conditions of a near-circular LEO, by developing approximate analytical models considering the J2 perturbation. This paper is organized as follows. First, an orbital dynamics model considering the J2 perturbation is presented. Next, orbital sunshine angles are introduced, and a new simple analytical solution to the cylindrical Earth shadow model for a nearcircular LEO with small eccentricity is developed. After that, the approximate models for variations in sunshine angles and in orbit shadow arc are developed. Finally, a numerical example is employed to analyze the effects of an in-track maneuver on the Sun illumination conditions of a near-circular LEO, and to validate the proposed approximate models. 2. Orbital dynamics and Sun illumination conditions This section presents a near-circular orbital dynamics model considering the J2 perturbation and introduces the sunshine angles and Earth shadow. 2.1. Orbital dynamics considering the J2 perturbation The state of a satellite can be described by classical orbital elements as E ¼ ða; e; i; X; x; mÞ

ð1Þ

where a is the semimajor axis, e is the eccentricity, i is the orbital inclination, X is the right ascension of ascending node (RAAN), x is the argument of perigee, m is the true anomaly. The true anomaly is used to give the instantaneous position of the satellite on the orbit, and has three alternatives: argument of latitude u = x + m, mean anomaly M, and eccentric anomaly. For a near-circular orbit, the mean orbital angular rate, the mean orbital velocity, and the orbital period can be expressed, respectively as sffiffiffiffiffi rffiffiffiffiffi rffiffiffi l l a3 n¼ ð2Þ ; v¼ ; T ¼ 2p 3 a a l where l = 3.986004418  105 km3/s2 is the geocentric gravitation constant (Vallado, 2001). Under the J2 perturbation, orbital elements will drift. Only considering the secular effect of the J2 perturbation, the drifting rates of orbital elements are (Vallado, 2001) 8 a_ J 2 ¼ 0; e_ J 2 ¼ 0; i_J 2 ¼ 0 > > > > > < x_ J 2 ¼ C J 2 ð2  5 sin2 iÞ=ð1  e2 Þ2 2 ð3Þ 2 _ > XJ 2 ¼ C J 2 cos i=ð1  e2 Þ > > > > 3 : _ M J 2 ¼ C J 2 ð1  32 sin2 iÞ=ð1  e2 Þ2 where the2 subscript “J2” stands for the J2 perturbation, 7 3J R pffiffiffi C J 2 ¼ 22  la2 , R ¼ 6378:137 km is the equatorial radius of the Earth, and J2 = 1.0826261  103.When the J2 perturbation is taken into consideration, the mean

orbital angular rate and the orbital period for a circular orbit are ( _ J ¼ n þ C J ð3  4 sin2 iÞ nJ 2 ¼ n þ x_ J 2 þ M 2 2 ð4Þ 2p T J 2 ¼ n2pJ ¼ nþC ð34 2 sin iÞ J2

2

When an in-track maneuver Dv is executed, based on Gauss’s form of variational equations, the variation in semimajor axis caused by that maneuver is 2 Da ¼ Dv n

ð5Þ

and the variation in mean orbital angular rate is rffiffiffiffiffi l Dv Dn ¼ D ð6Þ ¼ 3n a3 v Therefore, the variations in nJ 2 and in X_ J 2 can be given by (Zhang et al., 2012) (

  DnJ 2 ¼ Dn þ ð3  4sin2 iÞDC J 2 ¼  3n þ 7C J 2 ð3  4sin2 iÞ Dv v DX_ J ¼ DC J cos i ¼ 7C J cos i Dv 2

2

ð7Þ

v

2

  7 3J R2 pffiffiffi where DC J 2 ¼ 22  lD a2 ¼ 7C J 2 Dv . From Eq. (7), it v can be found that the variations in argument of latitude and in RAAN will drift linearly as time t: (   Du ¼ DnJ 2 t ¼  3n þ 7C J 2 ð3  4 sin2 iÞ Dv t v ð8Þ Dv _ DX ¼ DXJ t ¼ 7C J cos i t 2

2

v

It should be noted that Eqs. (5)–(8) are only effective for near-circular orbits. 2.2. Orbital sunshine angles The direction of the Sun in the J2000.0 Earth equatorial coordinate system is (Vallado, 2001) R ¼ ðcos k; sin k cos e; sin k sin eÞT

ð9Þ

where k is the ecliptic longitude and e is the obliquity of the ecliptic. Let P = ( cos X, sin X, 0)T be the direction of the ascending node and Q = (  sin X cos i, cos X cos i, sin i) be the direction perpendicular to P in the orbital plane. The cosine values of the angles between R and P and that between R and Q are b1 ¼ R  P ¼ cos k cos X þ sin k cos e sin X

ð10Þ

b2 ¼ R  Q ¼  cos k cos i sin X þ sin k cos e cos i cos X þ sin k sin e sin i

ð11Þ

The orbital coordinate system located at the center of mass of the satellite is defined as follows: x is along the orbital radial direction, y is along the in-track direction, and z is along the orbital normal direction and completes the right-handed system. The direction of the Sun in the orbital coordinate system can be obtained by coordinate transformation, and is given as below:

J. Zhang et al. / Advances in Space Research 53 (2014) 509–517 T

R0 ¼ ðsx ; sy ; sz Þ

ð12Þ

where

 cos k cos i sin X sin u sy ¼ sin k cos e cos i cos X cos u þ sin k sin e sin i cos u  cos k cos X sin u  cos k cos i sin X cos u  sin k cos e sin X sin u sz ¼ cos k sin i sin X  sin k cos e sin i cos X þ sin k sin e cos i Therefore, b angle, the angle between the Sun direction and the orbital plan can be obtained from sz as ð13Þ

b partly represents the maximum ability of a Earth-oriented satellite to obtain solar energy (Zhang et al., 2012).The angles between the Sun direction and the orbital radial and in-track directions are a ¼ cos1 ðsy Þ

ð17Þ

Substituting Eqs. (15) and (17) into Eq. (16) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 b1 cos u þ b2 sin u 6  1  2 R

þ sin k sin e sin i sin u þ sin k cos e sin X cos u

f ¼ cos1 ðsx Þ;

where R = kRk. Moreover, the satellite and the Sun should be in different sides of the Earth: cos f < 0

sx ¼ cos k cos X cos u þ sin k cos e cos i cos X sin u

b ¼ sin1 ðsz Þ

511

ð14Þ

f, a and b can also be expressed as functions of b1 and b2: 8 cos f ¼ b1 cos u þ b2 sin u > > < cos a ¼ b1 sin u þ b2 cos u ð15Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : cos b ¼ b2 þ b2 1 1 2.3. Earth shadow for a circular orbit The cylindrical Earth shadow model is used in this study and is shown in Fig. 1, where O is the Earth center, Rin and Rout are the entry and exit position vectors of the orbit relative to the Earth shadow, and R is the position vector of the satellite (Vallado, 2001). In the shadow part of the orbit, the projections of the position vectors in the plane perpendicular to the Sun direction should be equal to or less than the radius of the Earth: qffiffiffiffiffiffiffiffiffiffiffiffi R sin f ¼ R 1  s2x 6 R ð16Þ

ð18Þ

The equal sign in Eq. (18) corresponds to uin and uout, i.e. the arguments of latitude of Rin and Rout. For a circular orbit, R = a, and the equation part of Eq. (18) can be expressed as quadric equations: 8  qffiffiffiffiffiffiffiffiffiffiffi 2 > R2 2 2 > > 1 b b ð1cos uÞ ¼  cosu 1 < 2 a2  2 ðu ¼ uin or u ¼ uout Þ q ffiffiffiffiffiffiffiffiffiffiffi > > > b2 ð1sin2 uÞ ¼  1 R2 b sinu : 1

a2

2

ð19Þ The sine and cosine values of uin and uout are roots of the quadric equations given in Eq. (19), and then satisfy 8 qffiffiffiffiffiffiffiffiffiffiffiffi2ffi R R2 > > 1  a2 b1 1  a2  b22 > cos u þ cos u in out > > ¼ ; cos uin cos uout ¼ < 2 cos2 b cos2 b : qffiffiffiffiffiffiffiffiffiffiffiffi2ffi > R R2 > 2 > 1  b 1  a2  b1 sin uin þ sin uout > a2 2 > : ¼ ; sin uin sin uout ¼ 2 cos2 b cos2 b ð20Þ

The length of the orbit arc in shadow and the duration of the orbit movement in shadow can be solved from Eq. (20) as qffiffiffiffiffiffiffiffiffiffiffiffiffi  8 R2 < l ¼ u  u ¼ 2 cos1 1  a2 = cos b ; sh out in ð21Þ : dursh ¼ lsh =nJ 2 and the midpoint of the shadow arc, i.e. ush , satisfies cos ush ¼ b1 = cos b ; ð22Þ sin ush ¼ b2 = cos b where ush ¼ ðuin þ uout Þ=2. In consequence, for a circular orbit, the entry and exit arguments of latitude of the shadow arc have closed-form analytical solutions: 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi    > R2 b2 1 1 > >  cos 1  a2 = cos b ; < uin ¼ tan b1   ð23Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi   > R2 > 1 b2 1 > 1  a2 = cos b þ cos : uout ¼ tan b1 where the quadrant of the inverse tangent function can be determined based on Eq. (22). 2.4. Earth shadow for a near-circular orbit

Fig. 1. Geometry of cylindrical Earth shadow.

The simple analytical solution to the cylindrical Earthshadow model of a circular orbit, as given above, can be found more than 50 years ago (Patterson, 1961; Fixler

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1964), but the analytical solution to the cylindrical Earthshadow model of an elliptical orbit is complicated and is inconvenient to use (Vallado, 2001). A new approximate simple analytical solution with considering the small eccentricity is derived as follows. For a near-circular orbit with a small eccentricity, the radius in Eq. (18) should be calculated by (Vallado, 2001) R¼

að1  e2 Þ : 1 þ e cosðu  xÞ

ð24Þ

By ignoring terms of e2, it can be obtained that a(1  e2)  a and [1 + e cos (u  x)]2  1 + 2e cos (u  x), and then the term in right-hand side of Eq. (18) can be approximated by sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 R2 2 1  2 ¼ 1  2 ½1 þ e cosðu  xÞ a R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2  1  2 ½1 þ 2e cosðu  xÞ a sffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 R2 =a2 ffi e cosðu  xÞ  1  2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1  R2 =a2

ð25Þ



rffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < l0 ¼ 2 cos1 1R2 =a2 ; 02 sh b02 1 þb2 : dur0sh ¼ l0sh =nJ 2

ð29Þ

and the midpoint of the orbit arc in shadow, u0sh , satisfies 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 > < cos u0sh ¼ b01 = b02 1 þ b2 ð30Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : > : sin u0 ¼ b0 = b02 þ b02 sh 2 1 2 3. Approximate model for variations of Sun illumination conditions A satellite runs on a near-circular low Earth orbit. Let P1 be a point on the trajectory with no maneuver at time t1, and the corresponding RAAN and argument of latitude are X1 and u1, respectively. An in-track maneuver is executed by the satellite at the initial time t0 = 0, let P2 be a point on the trajectory with the maneuver, the time at P2 is t2, and the corresponding RAAN and argument of latitude are X2 and u2, respectively. The differences in Sun illumination conditions between P1 and P2 present the variations in Sun illumination conditions caused by the in-track maneuver.

Using Eq. (25), the Eq. (18) can be approximated by

3.1. Variations in sunshine angles at a given time

R2 =a2 ffi e cosðu  xÞ b1 cos u þ b2 sin u  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  R2 =a2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 6  1  2 a

In this subsection, the time of the orbit with a maneuver is the same as that with no maneuver, i.e. t1 = t2, and then the variation in ecliptic longitude is zero. Based on Eqs. (10)–(13), the variations in b, b1 and b2 are caused by the drifting of RAAN, and can be expressed as 8 _ Dðsin bÞ @sz DX @sz DXJ 2 > > < Db ¼ cos b ¼ @X cos b ¼ @X cos b t1 1 1 ð31Þ Db1 ¼ @b DX ¼ @b DX_ J 2 t1 @X @X > > : @b2 @b2 Db2 ¼ @X DX ¼ @X DX_ J 2 t1

and then can be further simplified as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 b01 cos u þ b02 sin u 6  1  2 ; a where

R2 =a2

n, b01 ¼ b1  pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1R =a

ð26Þ

ð27Þ

where R2 =a2

b02 ¼ b2  pffiffiffiffiffiffiffiffiffiffiffiffiffi g, 2 2 1R =a

and

n = e cos x and g = e sin x are the nonsingular orbital elements suitable for describing near-circular orbits. Based on Eq. (27), the arguments of latitude for the entry and exit positions of the eclipse arc, i.e. uin and uout, satisfy 8  qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > R2 02 0 2 > > 1   b b ð1  cos uÞ ¼  cos u > 2 1 a2 <  qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > 2 > > : b021 ð1  sin2 uÞ ¼  1  Ra2  b02 sin u

ðu ¼ uin or u ¼ uout Þ;

ð28Þ

@sz ¼ cos k sin i cos X þ sin k cos e sin i sin X; @X @b1 @X

¼ sin k cos e cos X  cos k sin X;

@b2 ¼  cos k cos i cos X  sin k cos e cos i sin X: @X The variations in angles f and a are caused by the drifting of RAAN and the variation in argument of latitude, and are given by 8

 Dsx x x < Df ¼  sin ¼  sin1 f @s DX_ J 2 t1 þ @s DnJ 2 t1 f @X @u   ð32Þ : Da ¼  Dsy ¼  1 @sy DX_ J t1 þ @sy DnJ t1 2 2 sin a @X sin a @u where

Eq. (28) is similar to Eq. (19), and can be solved by using a similar process. Therefore, the length of the orbit arc in shadow and the duration of the orbit movement in shadow are

@sx ¼  cos k sin X cos u  sin k cos e cos i sin X sin u @X þ sin k cos e cos X cos u  cos k cos i cos X sin u;

J. Zhang et al. / Advances in Space Research 53 (2014) 509–517

8 bÞ z DXu z > Db ¼ Dðsin ¼ @s þ @s > @X cos b @k cos b > > < 1 1 Db1 ¼ @b DXu þ @b Dk @X @k > > > > : Db ¼ @b2 DX þ @b2 Dk u 2 @X @k

@sx ¼  cos k cos X sin u þ sin k cos e cos i cos X cos u @u þ sin k sin e sin i cos u  sin k cos e sin X sin u  cos k cos i sin X cos u; @sy ¼  sin k cos e cos i sin X cos u þ cos k sin X sin u @X  cos k cos i cos X cos u  sin k cos e cos X sin u;

513 Dk cos b

ð39Þ

where @sz ¼  sin k sin i sin X  cos k cos e sin i cos X @k þ cos k sin e cos i;

@sy ¼  sin k cos e cos i cos X sin u  sin k sin e sin i sin u @u  cos k cos X cos u þ cos k cos i sin X sin u  sin k cos e sin X cos u:

3.2. Variations in sunshine angles at a given argument of latitude

@b1 ¼  sin k cos X þ cos k cos e sin X; @k @b2 ¼ sin k cos i sin X þ cos k cos e cos i cos X @k þ cos k sin e sin i:

In this situation, the difference in argument of latitude between P1 and P2 is zero, i.e. u2 = u1. However, the variation in argument of latitude actually consists of two parts. The first part is the variation at the same time from t0 caused by different mean orbital angular rate:

The variations in angles f and a are also caused by the variations in RAAN and in ecliptic longitude, and can be expressed as 8

x  x < Df ¼  sin1 f @s DXu þ @s Dk @X @k   ð40Þ : Da ¼  1 @sy DXu þ @sy Dk sin a @X @k

Dulat1 ¼ DnJ 2 t1

where

ð33Þ

The second part is the angular movement of the maneuvered satellite between t1 and t2, and can be given by Dulat2 ¼ ðnJ 2 þ DnJ 2 ÞDt

ð34Þ

where Dt = t2  t1 is the variation in time at a given argument of latitude. The summation of the two parts should be zero: Dulat1 þ Dulat2 ¼ DnJ 2 t1 þ ðnJ 2 þ DnJ 2 ÞDt ¼ 0

ð35Þ

Therefore, the variation in time can be obtained from Eq. (35) as   DnJ 2 DnJ 2 DnJ 2 Dt ¼  t1   1 ð36Þ t1 nJ 2 þ DnJ 2 nJ 2 nJ 2 When the time changes, there is a variation in the Sun’s position. The variation in ecliptic longitude is Dk ¼ x Dt

ð37Þ

where x is the angular rate of the Earth’s movement around the Sun. The variation in RAAN also consists of two parts. The first part is caused by the variation in drifting rate, and the second part is the drifting magnitude between t1 and t2. Therefore, the variation in RAAN is given by DXu ¼ DX_ J 2 t1 þ X_ J 2 Dt

ð38Þ

The variations in b, b1 and b2 are caused by both the drifting of RAAN and the variation in ecliptic longitude, and can be expressed as

@sx @k

¼  sinkcos X cosu þ cos kcos ecosi cosX sinu þ cos ksin esini sinu þcos kcos esin Xcos u þ sink cosi sinX sinu;

@sy @k

¼ cosk cose cosi cosX cosu þ cos ksin esini cosu þ sin kcos Xsin u þsin kcos isin Xcos u  cosk cose sinX sinu:

3.3. Variations in shadow arc for a circular orbit Based on, Eq. (21), the variation in the length of the orbit arc in shadow consists of two parts: Dlsh ¼

@lsh @lsh Da þ Db @a @b

ð41Þ 2

2

2 1  =a pRffiffiffiffiffiffiffiffiffiffiffiffiffi where @l@ash ¼  sinðlsh =2Þ and @l@bsh ¼ 2 cot l2sh cos b 1R2 =a2 a tan b. Based on Eq. (22), the variation in the midpoint of the orbit arc in shadow can be given by   Dðb1 = cos bÞ Db b1 Dush ¼ ¼ 1 tan b Db ð42Þ  sin ush b2 b2

3.4. Variations in shadow arc for a near-circular orbit Based on Eq. (29), the variation in the length of the orbit arc in shadow consists of three parts: Dl0sh ¼ where

@l0sh @l0 @l0 Da þ sh0 Db01 þ sh0 Db02 @a @b1 @b2

ð43Þ

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J. Zhang et al. / Advances in Space Research 53 (2014) 509–517

1

0.5 numerical difference approximate model

0.4

approximate model numerical difference at u1=0 deg

0.6

0.3

numerical difference at u1=90 deg

0.4

0.2

numerical difference at u1=180 deg

Δβ /deg

Δβ /deg

0.8

0.2 0

0.1 0

-0.2

-0.1

-0.4

-0.2

-0.6

-0.3

-0.8

100

0

200

300

400

-0.4

500

0

400

300

200

100

time /TJ2

500

time /TJ2

Fig. 2. Time history of variation in beta angle at a given time.

Fig. 5. Time history of variation in beta angle at a given argument of latitude.

15 numerical difference approximate model

10

150

numerical difference approximate model

100 50

Δζ /deg

Δζ /deg

5 0

0 -50

-5

-100 -10

-150 -15

0

2

4

6

8

10

490

492

494

time /TJ2

496

498

500

498

500

time /TJ2

(a) [0,10TJ 2 ]

(b) [490TJ 2 ,500TJ 2 ]

Fig. 3. Time history of variation in zeta angle a given time. 15 numerical difference approximate model

10

150

numerical difference approximate model

100 50

Δα /deg

Δα /deg

5 0

0 -50

-5

-100

-10 -150

-15

0

2

4

6

8

10

490

492

494

(a) [0,10TJ 2 ]

496

time /TJ2

time /TJ2

(b) [490TJ 2 ,500TJ 2 ]

Fig. 4. Time history of variation in alpha angle a given time.

J. Zhang et al. / Advances in Space Research 53 (2014) 509–517

2

0.04 approximate model numerical difference at u1=0 deg

0.03

1.5

approximate model numerical difference at u1=0 deg numerical difference at u1=90 deg

numerical difference at u1=90 deg

0.02

numerical difference at u1=180 deg

1

Δζ /deg

Δζ /deg

515

0.01 0

numerical difference at u1=180 deg

0.5 0

-0.01

-0.5

-0.02 -0.03

2

0

6

4

-1 490

10

8

492

494

496

498

500

time /TJ2

time /TJ2

(b) [490TJ 2 ,500TJ 2 ]

(a) [0,10TJ 2 ]

Fig. 6. Time history of variation in zeta angle at a given argument of latitude.

0.04

2

approximate model numerical difference at u1=0 deg

0.03 0.02

numerical difference at u1=90 deg

numerical difference at u1=180 deg

1

Δα /deg

Δα /deg

1.5

numerical difference at u1=90 deg

0.01 0

numerical difference at u1=180 deg

0.5 0

-0.01

-0.5

-0.02 -0.03

approximate model numerical difference at u1=0 deg

0

2

4

6

8

10

-1 490

492

494

496

498

500

time /TJ2

time /TJ2

(a) [0,10TJ 2 ]

(b) [490TJ 2 ,500TJ 2 ]

Fig. 7. Time history of variation in alpha angle at a given argument of latitude.

@l0sh @a

¼

R2 =a2 2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; 0 l 02 1  R2 =a2 a sin 2sh b02 1 þ b2

@l0sh l0sh b01 ; 0 ¼ 2 cot 02 2 b02 @b1 1 þ b2

@l0sh l0sh b02 ; 0 ¼ 2 cot 02 2 b02 @b2 1 þ b2 " ! # R2 =a2 1 Da 0 n ; Db1 ¼ Db1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dn þ 1 þ a 1  R2 =a2 1  R2 =a2

and um is the argument of latitude of the maneuver point. Based on Eq. (30), the variation in the midpoint of the orbit arc in shadow can be given by   u0sh Db01 þsin  u0sh Db02 b01 Db0 b0 cos  ffi pffiffiffiffiffiffiffiffiffiffi ffi Du0sh ¼  sin1 u0 D pffiffiffiffiffiffiffiffiffiffi ¼  b0 1  b10 02 02 02 02 sh

b1 þb2

2

2

b1 þb2

ð44Þ



R2 =a2

"

!

Db02 ¼ Db2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dg þ 1  R2 =a2 Dv Dn ¼ 2 cosðum þ x_ J 2 t1 Þ ; v

1þ

#

1 Da g ; 2 2 a 1  R =a

Dv Dg ¼ 2 sinðum þ x_ J 2 t1 Þ ; v

4. Results 4.1. Problem configuration The initial time in Gregorian universal coordinated time (UTCG) format is 1 June 2014 12:00:00.00. Two satellites are used to generate two trajectories, and their initial states

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J. Zhang et al. / Advances in Space Research 53 (2014) 509–517

0.5

variation in shadow arc center /deg

variation in shadow arc length /deg

-0.5 -1 -1.5 -2 -2.5 -3 -3.5 numerical difference approximate model 1 approximate model 2

-4 -4.5

0

100

200

300

400

500

time /TJ2 Fig. 8. Time history of variation in shadow arc length.

are both E ¼ ð6713:960 km;0:0; 42:0 deg;0:0 deg;0:0 deg;0 degÞ, with E given by Eq. (1). The first one runs with no maneuver, and the second one executes an in-track impulse Dv = 10 m/s at the initial time. The maximum drifting time is 500T J 2 , i.e. t1 2 ½0; 500T J 2 , with T J 2 given by Eq. (4). The trajectories of the two satellites are simulated using Eq. (3), their Sun illumination conditions in every time step are both calculated using Eqs. (9)–(18), the exact entry and exit conditions of the orbital Earth shadow and the exact Sun illumination conditions at a given argument of latitude are calculated using the combination of small time step and interpolation based on the trajectory data.

4.2. Variations in sunshine angles at a given time For each point of this analysis, the drifting time of the two satellites are kept the same. The variations in b, f and a are shown in Figs. 2–4 respectively, where the data labeled by “numerical difference” are calculated by directly subtracting the sunshine angles of the first satellite from those of the second satellite, and the data labeled by “approximate model” are calculated by Eqs. (31) and (32). From Fig. 2, it can be found that Db is always less than 1 deg, and Eq. (31) presents a good approximation of the variation in orbital Sun angle b at a given time. From Figs. 3 and 4, it can be found that Df and Da reaches tens of degrees when the drifting time reaches hundreds of orbital periods, and that Eq. (32) can provide an effective approximation of the variations in orbital Sun angles f and a only when the drifting time is short. Eq. (32) is derived on the assumption that the variation in argument of latitude, i.e. Du ¼ DnJ 2 t1 , is a small value, but this assumption fails when the drifting time reaches hundreds of orbital periods.

0

-0.5

-1

-1.5

-2

numerical difference approximate model 1 approximate model 2 0

100

200

300

400

500

time /TJ2 Fig. 9. Time history of variation in shadow arc center.

4.3. Variations in sunshine angles at a given argument of latitude For each point of this analysis, the arguments of latitude of the two satellites are chosen the same. The variations in b, f and a are shown in Figs. 5–7, respectively, where the data labeled by “approximate model” are calculated by Eqs. (39) and (40), and the term “time” stands for the drifting time of the first satellite. Only three typical situations of the numerical difference results are shown there, and their arguments of latitude are u1 = 0°, u1 = 90°, and u1 = 180°, respectively (see Fig. 6). From Figs. 5–7, three major properties can be found. First, the variations in orbital sunshine angles are always less than 1 deg. Second, Df and Da fluctuate over time, the fluctuation periods are both about one orbital period, and the amplitudes of fluctuation also both increase over time. Third, the results obtained by the developed approximate models always effectively approach the numerical differences, and therefore Eqs. (39) and (40) present good approximations of the variations in orbital sunshine angles at a given argument of latitude, even when the drifting time reaches hundreds of orbital periods. 4.4. Variations in shadow arc The variations in shadow arc length and in shadow arc center are calculated and then shown in Figs. 8 and 9 respectively, where the results labeled “approximate model 1” are calculated by Eqs. (41) and (42), and the results labeled “approximate model 2” are calculated by Eqs. (43) and (44). The variations in sunshine angles used in both approximate models are calculated by Eqs. (39) and (40), and then for each point in Figs. 8 and 9, the arguments of latitude of the satellites used to compare shadow arcs are kept the same. From Figs. 8 and 9, two major properties can be found. First, the variations in shadow arc length and in shadow

J. Zhang et al. / Advances in Space Research 53 (2014) 509–517

arc center are both small values at any given argument of latitude. In this example, the magnitudes of these variations are less than 5 deg. Second, the approximate model considering the effects of the small eccentricity has better precision than the model ignoring them.

517

Program (No. 2013CB733100), and the Science Project of the National University of Defense Technology (No. CJ12-01-02).

References 5. Conclusion By developing approximate models considering the J2 perturbation, the effects of an in-track maneuver on the Sun illumination conditions of a near-circular lowEarth-orbit are analyzed. From the simulation results, four major conclusions can be drawn. First, when the drifting time reaches hundreds of orbital periods, the variations in orbital sunshine angles at a given time may reach tens of degrees, and the first-order approximation model developed for that situation cannot effectively approach the numerical results. Second, for any given argument of latitude, the variations in orbital sunshine angles and in the length and center of the orbital arc in Earth shadow are all small values, even when the drifting time reaches hundreds of orbital periods, and the approximation developed can effectively approach the numerical results. Third, for variations in Earth shadow, the approximate model considering the small eccentricity has simple expressions and can effectively approach the numerical results; in contrast, the approximated model using the zero eccentricity has relatively worse precision. The proposed models and the properties identified may help researchers understand the effects of orbital maneuvers on the Sun illumination conditions of a satellite, and may be used in the orbital maneuver mission design with requirements on Sun illumination conditions. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11222215), the 973

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