Chin.Astron.Astrophys. 2 (1985) 27 - 34 Act.Astron.Sin. 25 (1984) 255- 264 -

Pergamon Press. Printed in Great Britain 0275-1062/85$10.00+.00

ORBITAL VARIATION OF THE SYNCHRONOUS CALCULATION

SATELLITE AND ITS

LIU Lin* Department of Astronomy, Nanjing University

Received 1983 August 5

ABSTRACT This paper investigates the orbital variation of the 24-hour synchronous satellite and gives a method for calculating the variation. 'Ihe results provide a theoretical basis for the design and calculation of the orbit of the communication satellites.

1. INTRODUCTION

"Synchronous" satellites are those satellites which go around the Earth at the same rate as the Earth spins on its axis and they appear fixed above some points on the equator. The question is whether such orbits can be realized. If the Earth is a uniform sphere and if there are no other forces, then such orbits are unquestionably realizable. But in reality, the forces on the satellite are not like that. Then we must be clear as how the orbit will vary and provide a method for calculating the variation.

2. CHARACTERISTICS OF SYNCHRONOUS SATELLITES

In the following discussion, I shall use the units [l] and the geocentric equatorial coordinates, customary in the work on artificial satellites. First, let us give an order of magnitude estimate for the various perturbing forces acting on a 24-hour synchronous satellite. Such satellites have orbits that are nearly circular, close to the equator and with a semi-major axis of as6.6; considering possible errors in firing and other causes, we may suppose the orbital eccentricity e not to exceed 0.02 and the orbital inclination i not to exceed lo or 0.02 radians. For the surface-to-mass ratio we may take the value log which corresponds to a spheric satellite with a diameter of 4 meters and a mass of 1 tonne. With these values, the various perturbing forces, in units of the Earth's gravitation in the case of the Kepler motion, have roughly the following sizes,

,I‘:lo-', J-:4 x 10-n, J,:4 x 10-9, '.., Sun. Moon: 10-5, J,,a: JO-', IL,: 10-a, .*., Radiation pressure: 10-T,... (1)

The other perturbing forces are so much smaller that they need not be considered. In spherical coordinates, the Earth's gravitational potential is

AL,, - L - .I.,,, ma - ‘g-’ 9, -_I,,, _ (A:,,” + BL”,)“‘. (3)

"l"l n,m

where I is the geocentric radial distance, X and $ are the longitude and latitude, P, and P# are the Legendre polynomials and the associated Legendre polynomials, and An,m and Bn,m are the tesseral coefficients. Under the Earth's gravitation, if we consider only the even-order zonal harmonics Jgk, then Eq. (2) gives immediately the following equilibrium solution

~~-0, I - Aor l.0 4 6.6. (4)

That is, the satellite can remain "fixed" over any point of the equator at a distance 6.6 Earth's radii from the centre. If we include also those tesseral harmonics with even n+ m, then we can deduce without much difficulty from (2) and (3) that we now have equilibrium

* Colleagues LIAO Xin-hao and QIN Peng-gao took part in some of the present study.

28 LIU

solutions only at 4 particular longitudes,

rp-0. r - (ro)i, L* - (a:Ji- (A& - a,,*, (i - l~.*'r 4). (5)

These 4 longitudes are close to O', 90°, 180", 270" (the directions of the longest and shortest axes of the Earth's equator), and the 4 values of ro are all close to 6.6. It can be shown that the equilibrium solutions around 90' and 270° are stable,andthose around 0' and 180' are unstable.

When other perturbations are included, the equilibrium solutions will be disturbed and these particular synchrous orbits will vary. Actually, even if the other perturbations did not exist, errors in the firing and requirements of communication make it impossible that the satellite is placed precisely at the equilibrium position. Thus, in general, synchronous orbits undergo drifts. According to actual requirements, we have to place such satellites at various positions above the equator, hence it is necessary to study their orbital changes in the general case, and to fix their positions in space. Nevertheless, the above conclusion regarding equilibrium solutions is useful in showing that the commensura- bility (appearing in the tesseral harmonics) caused by the orbital period being the same as the Earth's spin has not destroyed the continuity of solution with respect to initial values, so that the calculation of the orbital changes of synchronous satellites will encounter no difficulties of an essential kind, [2].

3. PERTURBATIONS OF SYNCHRONOUS ORBITS

According to the estimates given above, we need only to consider the main zonal and tesseral harmonics in the expansion of the Earth's gravitational potential and the effect of the Sun and the Moon, and the radiation pressure. In the geocentric equatorial coordinate system, the equation of motion of the satellite is

i: - F, + F, (61

F =-I, (7) 7' (8)

F, - grad (R, + Rx + RAY where

RI - m’ ( 1 rcosf+h ---yr- ’ A >

(91

Fg is the central force corresponding to the Earth being a mass-point or a uniform sphere and F, is the perturbing force. R2 is the perturbing function of the Sun or the Moon, the two having the same expression with m’ denoting the mass of the Sun or the Moon in units of the Earth's mass, A, the distance of the Sun or the Moon to the satellite and Y the angle subtended at the Earth's centre by the Sun or the Moon and the satellite, that is

~1 is the geocentric radius vector of the Sun or the Moon. Rg is the perturbing function of radiation pressure, and

s/m is the effective cross-sectional area of the satellite under solar illumination, and the coefficient K is related to the reflecting properties of the satellite surface (determined by the surface material used, the shape etc.), the theoretical range being l- 2, and generally taken to be 1-1.44. PO is the solar radiation intensity; in the vicinity of the Earth, its value is

p,=4.6~10-~ dyn/cm2 =O.3194xlO-17 (12)

Numerical integration of Eq. (6) gives the position and velocity of the satellite. However, for the synchrous orbits, within certain accuracy, it is very convenient to use the

Calculating Synchrous Orbit 29

analytical method, and this has the advantage of revealing certain regularities in the orbital changes and providing important data for the design and control of the orbit.

Because the commensurability difficulty caused by the tesseral harmonics Jn,* is not an essential difficulty, and comes from the averaging of perturbation of a certain kind, it can be removed by a slight improvement of the original method. Here, I shall use a similar method to the one I used previously [3] to remove the difficulty of the critical inclination.

According to the usual requirement on the accuracy of a first order solution (geocentric angle accurate to 2" or 10m5, corresponding to 60 metres on the surface of the Earth), if the total span calculated does not exceed 100 revolutions (i.e. 100 days) then we need only consider J2, J3, Jq, J2,2 (the resonant term), the Sun, the Moon, and radiation pressure, and neglect all Jn with n25 and all Jn,m with n_> 3, and even perhaps J3 and ~4.

1. Choice of Variables

Because both e and i are small, we choose the following singularity-free elements

L - sinicosQ, 4- -sinisinQ, a,

c-ccos(w+Q), ~'-esin(o+Q), z-Mfco+Q. (13)

Note: do not confuse the X introduced here with the longitude introduced earlier.

2. Calculation of the Osculating Elements o(t)

Let o represent any one of the six elements. According to [3], the formal expression for u(t) is

(1(t) - a(r) + 8%). (14)

where us (l)(t) are the short-period terms with period of 1 day, and the mean elements 5(t) have the forms

R(t) _ 2, + [al')(t) - aI”cro>1. b(r) - &cos[Q,(t -to)] + &sid[Q,(r -to)]

+ h,(r - ro) + [/p(t) - Ll’Wl s

d(t) - &OS [Q,(t - r,) ] - ii, sin [Q,(r - d ] + 4h - 10) + [4:“(t) - tj’)(,o) 1,

5(r) = :ocos[(m + Q,)(! - lo)1 + fj osin[(to, f Q,>(r - 6): + 5,(r -lo) + [t:')(t) _E~II(,~)), (15)

l(') - ?Ocosr(W1 +QI)(~ - ro)) - &Gn[(w, +Q,)(r -r,)] + ,,Jr -,0) + [,,1'+> _,,j~,(,~)~,

I(r) - IO + Go + 1, + X,)(1 - r,) + [Ij'l(r) - +'+J]*

ii, _ a;,/'

a, - 0($ - @(lo). (16)

I now give without derivation the formulae for computing all the quantities involved as fellows:

9, ----%cosT *@; 8J Q’ (17)

w, -g&(2 -$-sh’i,), (18)

I,=(ro fQ)fGio 1 I 2.72 (

1-I 2 sin'& J1qE:.

) (1%

(21)

(221

(23)

(24)

30 LIU

(253

(261

(27)

1281

(29)

(30)

p(l) = $2 [4(EcosI - qsinXj1 ‘2

f f $i:[ sin’i’cos(2X - 29’) + (1 + cos’i’) cos(?X - 2P’)cos2rc’

+ 2cosi’sin(2X- 2Q’)sinZu’l~

h(q) _i 351 , 4a” (- ksin2X f ~cosZX), 0

e’(f) - g- (-X&2X - tcosZX>,

(31)

(32)

(331

$l(t) = $$. [ZCOSX + 3(fcos2x - q&2X)1, (34)

~#‘(t) = + [- 2 sin1 - 3(.$&2X + 4cosZX)1, (35) 4nt

.x!‘)(t) - $- (7gsinX + 75jcosX). (36) D

because of the accusacy requirement in computing M, more terms must be taken in as(l) then in the os (I) of the other five elements. In the above expressions, the auxiliary quantities are the following:

e_ ; (t - iJ sin2G - 52.1).

sz 2 is the local sidereal time at the initial epoch t o of the direction of the axis of s&etry of the equator (longitude X2,2) corresponding to the 52,~ term.

Calculating Synchrous Orbit 31

D, - - sin2i’sinQ’ - (2 - 3sin’i’)k - &i’(h dn2Q’ + kc”s2g’),

D, - - sin2i'cosD' -I- (2 - 3sin’i’)h - sin’i’(hcas2P’ - ksi”2Q’),

D, - 2 1 - 1 sin? q ( 2 >

- 5si”‘i’(5si”2M -6 qws2Q)

D, - - 2 (

1 - f sin’?) 6 - 5sinl”(~cos2.@ - q sin2Q),

D,-4co~‘i.+5~i”2i.(h~01~-ksi”0.)~

D6 - sin 2i’ sinP’ - 3sW.Q - (2 - sin’ i’)(h sin29’ + &ms?Q’),

D7 - - 2sini’cosP’f 2cosi’(hcos29’-k&29’),

D, - sin 2i’cosO’ + 3 sidi’h - (2 - sin*i’)(h ws2P’ - k sin 29’))

D9 = 2 sin i’ sinff’ - 2cosi’(h sin 2Q’ + kcos29’).

D,, - 3sin*i’vj - 5(1 -I- a&‘)(~ sin24’ + qcos29’)~

D,, - 10 cos i’(r cos 29’ - q sin 29’))

Du - - 3sin’i’[ - 5(1 -I- cos’i’)(~cos29’ - q sin29’),

Dlt _ - lOcosi’(~sin2Q + qcos24’).

D,, = 7 sini’(h sin8’ f km&‘):

In these, the primed quantities are calculated as follows:

cosi' - cosEc.osJ - sins sinJcosQC, sini’ - J1,

$inLY - sin./ sinQ( c0s.J - cos E CDS i’

sini’ ’ cm Q’ -

sin 6 sin i’ ,

(40)

141)

E and J are the obliquity of the ecliptic and the inclination of the lunar orbit to the ecliptic, and all the other symbols have their usual meanings. For the Sun, we have i'=c, k-l' =O.

For the h, k, 5, rl that appear in Dl- 014, if the time interval t-to is not too long, then they can be replaced by ho, izo, tot ijo; more generally, they should be hi/p, k1/2, Cl/z,

711/z, given by

5 */I - E*COS f -t (W, + .Q,)(l - to) + fj i ] OS” [+G%+P*)(f-r,)], (42)

9111 - siocos $- ((0, -I- Q,)(f -to) - f ‘n 1 1 4 f (0, + Q,)(r - ,o)- * 1 At 1393, HI, X2, ~3 are given two expressions each. When

j3i,+&-?+l <lo+ 143)

the second expressions may be used; otherwise, we use the first expressions. In using the second, t- t, must not be too large, we should ensure that I (Go + 1, - n.)(r - r,)I C lo-‘, ‘IhiS

is satisfied by the above mentioned duration of 100 revolutions (corresponding to t- to%104). In none of the above formulae, was the ~3 term present, this is because all the

corresponding perturbations are accompanied by the small factor e or sin i. In accordance with the accuracy required, we need not allow for the Earth's shadow when

calculating the radiation pressure, this is becausethesetype of satellites enter into the Earth's shadow for only about 23 days around the Spring and Autumn equinoxes, and then only for about 70 minutes in each revolution.

In calculating the solar and lunar perturbations, I do not include the orbital eccentricities, the main variation in the lunar orbit or the P~(COS $J) term in the lunar perturbation expansion. These terms are of the order of 10e7, but they do not contain any secular effects and give rise only to periodic variations of long and short periods.

32 LIU

Obviously, the short-period variations need not be considered; as for the long-period variations, they can also be neglected if the time interval is not too long (10 revolutions,

say), otherwise, they should be included. The reader is referred to Ref. [4] for more details, and simplifications.

In using the above formulae, the following point should be noted. Usually, the given data are the elements ao, eo, io . . . .

. at the initial epoch to, and it is required to find the

position of the satellite at time t. In this case, we first change the variables to to according to (13), then use (31) - (36) to calculate a.s('](to) and hence obtain 30. But if the mean elements ao, eo, ho . . . are given at the beginning, then we get so directly from

(13). We then find the osculating elements u(t) at time t from (14) - (36).

4. COMPARISON WITH NUMERICAL SOLUTION

In this section, Eq. (6) will be solved numerically, standard of comparison for the results given by the

and the results will he.used as a above analytical expressions. It will

thus be verified that, for the synchronous orbits, the simple, analytical method is sufficient for attaining the goaI within the accuracy of a first-order solution.

To avoid some unnecessary calculations we consider only the five perturbations due to Jg,

J3, Jb. J2,2 and the Sun. Since the Moon's perturbation is calculated in the same way as the Sun's, this comparison is no longer necessary; as for the radiation pressure, its value is marginal in any case and so the comparison is not necessary either.

Using the rectangular coordinates

x,=x, X,=-Y, x,- I,

XI=- *, x,--P, X6 = i,

the equation (6) becomes the following set of 6 first-order equations,

(44)

where i, = - $ [x,K, + (I&, - J,K, - J,K,) 1 - p [(-$x1 - z’> + 2’ 1. K* - 1 + J,K, + .!I&+ J,K, + J*.2&,

& = y [C(x: - x:) - S(Zx,x,)], K,, = -$ (CT, + ST,), ,

K, - 3 x,. r2

(45)

(46)

1.1 - x: + x: + x:, A’ = (x, - x’)’ + (x2 - Y’)’ + (X, - 2’)“. (47)

I’= ~~cosL~, y’= r~sinL,cosE, Z’ = 16 sin Lo sin 6,

L, = CL,), + flo(l - rd. (48)

C = c0s2[nc(r -to) + s2.,l, S = sinZ[ne(i -I,> + s2.11. (491

Here, ro' is the radius of the Sun's orbit, and the other quantities are the same as before. The numerical constants used in the calculation are

lo=O. 51.2 = 0, (La), = 0, r; = 2.3455 X lo', p' = 0.258 X IO-', E = 23?44, I, 0 - 1.606420 X lo-', 11~ - 0.588337 x lo-‘,

Calculating Synchrous Orbit 33

Ja - 1.08264 X IO-', J,- - 2.565 X lo-‘, J, - -1.608 x lo-‘,

Jm - -1.7892 X lo-‘.

The set (45) was solved using a 7th (8th) order Runge-Kutta routine with self-adjusting steplengths [S]. First, a comparison was made with the 4th order, fixed steplength RK routine, using the 2h satellite given in Ref. [2], considering only Jp, J3, J4, and continued for 12 revolutions for each of the cases e=O.Ol and e=0.2. In the 4th-order routine, 400 steps per circuit were taken, and in the 7th order routine, 100 steps per circuit. The two methods gave elements that differed b 10m6 -10b7, the difference along the orbit (shown in the element A) was greater, being lo- 5 for e=O.Ol and 3 times larger for e=0.2, which is reasonable.

For the comparison in the present case, the synchronous satellite was taken to have the following initial elements:

(10 - 6.6227861, cg - 0.01, i, - O', s,= 900, 00 - 00, M,-o".

The calculation was carried on for 10 circuits or a time interval of 1070.880382. These initial values were first converted into the initial values ao, ho, kor Co, no, X0 (to be used in the analytical method), and the results of the numerical integration were likewise converted into the osculating elements a, h, k, 5, Q, A. The comparison is shown in TABLE 1.

TABLE 1

Element 0 I h I 4 5 -- \

ri I

1

Numerical 6.622755 2.4x10-’ -0.3X’Y -0.5x10-’ -0.010004 1.574119

Analytical 6.622752 2.3x10-’ -0.3x10-’ 1.3X10_’ -0.010003 1.574111

Difference 3X10mL 10.‘ 0 lo-’ lo-’ lo-’

This result shows that, for the required accuracy, the analytical method given in the present paper is wholly effective. It also removes the difficulties that arise from the singularities e=O, i=O, and from the commensurability, (we took i=O, ;o + A, -_ne 4 lo-', in the calculation and did not encounter any difficulty). As for the speed of calculation, the analytical method is far better, especially in the treatment of a large body of data, it wins hands down over the numerical method.

5. CALCULATION OF THE POSITION where the unit vectors are

When the osculating elements a, h, k, 5, q, X are known, the position and velocity of the satellite can be found as follows.

1. Solving the generalized Kepler equation,

u" -I + (c&i; + ~cosii), (50)

gives B and X - ii,

2. Calculate the geocentric distance,

r = a(1 - ~cosg+~sinu"). (51)

3. Calculate sin u and cos u according to

2 - 5’ + $.

4. Calculate the position and velocity vectors, r- r(cosuP+ oinuQ),

sinu +$)P+(cosu + :)Ql, (53)

where cosi = (1 - ,+’ _ ~)vI. (54)

6. THE TRANSITION ORBIT

Before a 24-hour satellite is placed in its orbit, there is a transition stage, when the perigee height is low (e.g. 400 km) and the apogee is nearly the same as the synchronous orbit. ‘Iha corresponding orbital semi-major axis, eccentricity and period are

0 = 3.8, c = 0.73, T EJ 10:s. (55)

For this eccentric orbit, the various perturbations have the following sizes as regards their mean effect;

J1:2 x lo-‘, J,:2 X lo-', /,:4 X I,-,-‘,... I

J2.>:4 X lo-',..., (56)

34 LIU

SUn: 1.5~10-~, Moon: 3x10-6, Radiation Pressure: 10m7, . . .

all the other perturbations will be less than lo-' in size.

It may appear thesefore that the calculation of such eccentric orbits will present no difficulty in the sense of a first-order solution. But we should remember that the above estimates refer only to the average effect, i.e. only to the secular terms and the long period terms. For the short-period terms, the situation is different. Because of the large eccentricenty and small angular velocity of the average motion, the amplitude in the short-period term arising from ~2 is made greater than the estimate above. For example, for oO- 3.8, ~~-0.73, i- 3o", ~O-co,- M,- (~-we have, at the perigee,

I$'-O.Ol4, c!') - 0.0011. (57)

Similar situation obtains in the short- period terms arising from other causes. Moreover, the initial data provided very often refers to the perigee. In this case, we must calculate the tesseral terms (such as JZ,~) and then the right side of the corresponding equation will be an explicit function of t, thus involving the question

of expanding the disturbing function; but

with e=0.73, the Laplace limit of 0.667 is exceeded and it is impossible to have a series expansion in the mean anomaly M. Therefore, even in the sense of first-order solution, the calculation of the transition orbit cannot be done simply by the analytical method, unless we lower still the requirement on the accuracy. However, in general such transition orbits last only a few or some lo-odd circuits, and numerical methods are convenient; still, the question of large eccentricity must be kept in mind, otherwise, the pre-assigned accuracy may not be attained.

~FERENCES

fl] LIU Lin and ZHAO De-zi. "Theory of the Orbits of Artificial Satellites" (in Chinese), Nanjing University Press (1979).

[Z] LIU Lin and ZHAO De-zi J. Nanjing univ. (Nat. sci.I 2 (1980) 56.

[3] LIU Lin Ckin. Astron. 1 (1977) 63-78. = Act. Astron. Sin. 16 (1975) 65- 80.

[4] LIU Lin and WA0 De-zi J. Nanjing Univ. (Nat. Sci.) 1 (1979) 55.

[S] NASA TR R-287, 52