Acta Mechanica (2003) DOI 10.1007/s00707-003-0024-7

Acta Mechanica Printed in Austria

A new method for the orbital modification of a tether connected satellite system A. Djebli, Te´touan, Maroc, and M. Pascal, Evry, France Received March 20, 2003; revised May 23, 2003 Published online: December 30, 2003 Ó Springer-Verlag 2003

Summary. The dynamical behavior of a tethered connected satellite system during tether length variation is considered. The system consists of a space station and a subsatellite moving along a straight massive tether of variable length. The space station and the subsatellite are assumed to be mass points. It is shown that appropriate length variation laws can be used to modify the characteristics of the assumed elliptical orbit of the system mass center. The performances of the method are compared to those obtained in the past for other tether length variation schemes.

1 Introduction The use of tether satellites systems (TSS) has been proposed for solving a large number of dynamical problems, such as performing experiments in the upper atmosphere, construction of long antennas, orbit transfer, etc. Some of these proposals consist of a tether of approximately 100 km length to be deployed from the shuttle orbiter and transporting at its end a subsatellite experimental package [1]. Our main attention in this work is to extend the analytical study of the pumping process or ‘‘central laws’’ to the intermediate scheme which generalises the conventional and the crawler schemes [2]. In the past, a tether length variation proportional to the radius vector (pumping process) was proposed to modify the eccentricity, the argument of perigee, or the energy of a satellite orbiting a spherical planet [3]. It results that the length variation laws considered (‘‘central laws’’) may be effective for orbital modifications. However these laws were exploited in the particular case of the conventional system without tether mass. The aim of this study is to generalize the use of the central laws to the intermediate scheme introduced in [2] and to compare the performances of this method with those of the conventional and crawler schemes.

2 Problem formulation The tether satellites system consists of a space station and a subsatellite moving along a straight tether (inextensible tether) of variable length LðtÞ. The space station and the subsatellite are modeled as two points M1 and M2 of mass m1 and m2 , the distance SðtÞ between M1 and M2 is also variable, with SðtÞ
LðtÞ. Assuming that the subsatellite mass m2 is constant and that the tether is reeled in and out from the space station, the mass m1 of the space station and the mass m3 ¼ q0 L of the tether are time dependent. It results in the following relations:

A. Djebli and M. Pascal

l1 þ l2 þ l3 ¼ 1; l_ 1 þ l_ 3 ¼ 0;

l_ 3 ¼ l3 ðL_=LÞ;

ð1Þ

li ¼ ðmi =mÞ for i ¼ 1  3; m ¼

3 X

mi ;

i¼1

where ð:Þ ¼ ðd=dtÞ; q0 is the tether mass density, and the total mass m is constant. This system describes the so called intermediate system used in [2] as an alternative method for the deployment/retrieval of the tether and extending the previously proposed conventional scheme and the crawler scheme [4]. The conventional system is obtained from (1) when LðtÞ ¼ SðtÞ while for the crawler system L_ðtÞ ¼ 0 (L is constant). The system is submitted to the Earth’s gravitational forces which are assumed to be produced by the central field of Newtonian attraction of a fixed point O located at the centre of the Earth. Since all the external forces are central forces, the overall angular momentum in point O z0 Þ with the ~ x-axis in the remains constant. Let us introduce a fixed reference frame ðO; ~ x; ~ y0 ; ~ direction of this vector. It is assumed that the mass centre G of the system is moving in the plane ~ ¼
! O~ y0 ; O~ z0 in which it is defined by the polar coordinates u and R (R OG ). The position of the tether in the frame ðG; ~ x; ~ y; ~ z Þ located at the mass centre of the system is defined by the angles h (in-plane) and u (out-of-plane), while the ~ y-axis coincides with the local vertical and is directed away from the Earth centre (Fig. 1).The kinetic energy and the gravitational potential energy (to the first order) of the system are respectively given by    2   m _2 2~ 2 2 2 2 2 _ _ _ _ _ _ _ _ ð2Þ T¼ R þ R u þ S K h þ u cos u þ u þ S K þ l3 L ð1  l3 ÞL  2l2 S ; 2

Space station

→

x

j →

→

q

z

y

Local vertical

→

R

G

u O

subsatellite

Fig. 1. Intermediate system model

Orbital modification of a tether connected satellite system

V¼

   mlG S2 K

1 þ 2 3 cos2 u cos2 h  1 ; 2R R

ð3Þ

where



l3 1 L 2 L 1 ; l1 þ l2 þ K ¼ l1 l2 þ l2 l3 3 S S 4 ~ ¼ l ðl þ l Þ, and l is the gravitational constant. K 2 1 3 G Using the classical Lagrangian formulation, the attitude motion of the system is given by  i 3l S2 K d h 2 _ S K h þ u_ cos2 u þ G 3 cos2 u sinð2hÞ ¼ 0; dt 2R ð4Þ   2 d  2  S2 K _ 3lG S2 K 2 S Ku_ þ h þ u_ sinð2uÞ þ cos h sinð2uÞ ¼ 0 dt 2 2R3 and the motion of the mass centre G by l R€  Ru_ 2 ¼  G2 þ CR ; R

 d 2 R u_ ¼ RCu ; dt

ð5Þ

where CR and Cu represent the additional radial and azimuthal accelerations: CR ¼ 

 3lG S2 K

3 cos2 u cos2 h  1 ; 2R4

Cu ¼

 3lG S2 K 2 cos u sinð2hÞ : 2R4

ð6Þ

These perturbing accelerations are small in comparison with the radial acceleration due to the total mass at the mass centre. The overall angular momentum of the system leads to the first integral of motion:   ð7Þ R2 u_ þ S2 K h_ þ u_ cos2 u ¼ const: When CR , Cu are neglected, the mass centre orbit is a Keplerian orbit defined by R¼

f ; R

R ¼ 1 þ e cos m

and e < 1;

ð8Þ

m ¼ u  x;

 where f ¼ a 1  e2 is the orbit parameter while the orbital angular momentum p ffiffiffiffiffiffiffiffi ffi h ¼ R2 m_ ðh ¼ lG f ) and x (argument of perigee) are constant. Subject to CR ; Cu , the mass centre orbit can be viewed as a continuously deformed Keplerian orbit, instantaneously characterised by the eccentricity e, the argument of perigee x, a semimajor axis a; . . . ; evolving in time as [5]: e þ cos m i de f h ¼ CR sin m þ Cu þ cos m ; dt h R 

 dx f 1 ¼ CR cos m þ Cu 1 þ sin m ; ð9Þ dt eh R da 2a2 ¼ ðeCR sin m þ RCu Þ: dt h According to Eqs. (5) and (7) the orbital angular momentum h ¼ R2 m_ is not conserved in the presence of libration and has modulations opposite to those created by libration. This spine-

A. Djebli and M. Pascal

orbit coupling was exploited in [3] for orbital modifications, in the case of the conventional system without tether mass, by defining a tether length variation given by S ¼ constant: R

ð10Þ

For this so called ‘‘central law’’ it can be verified that h ¼ 0 is a solution of the equations of motion, that the orbital angular momentum h remains strictly constant and that the averages over the orbit of de=dt; dw=dt are zero.

3 Selection of tether length variations laws A generalization of the ‘‘central law’’ introduced in [3] can be obtained for the intermediate system by assuming that pffiffiffiffi S K ¼ k ðk constantÞ: R For a radially pointing tether ðh ¼ u ¼ 0Þ; Cu ¼ 0, and the motion of the mass centre is given by l R€  Ru_ 2 ¼  G2 ð1 þ 3k2 Þ; R ð11Þ d 2 ðR u_ Þ ¼ 0: dt It results that the orbital angular momentum h ¼ R2 u_ is constant and the radial perturbation varies with R in the same way as the gravitational acceleration. From Eqs. (9), we deduce: de ¼ 3k2 sin m; dm

dx k2 ¼ 3 cos m; dm e

da 6a2 e ¼  2 lG k2 sin m: dm h

ð12Þ

Therefore, the average values of de dx da ; ; dm dm dm over the orbit are zero.

4 A small deviation from the central law Following the same method as [3] we can analyse the effect of a small deviation of the ‘‘central law’’ for the intermediate scheme by assuming pffiffiffiffi   S~ ¼ S~ ð1 þ ks sin m þ kc cos mÞ; S~ ¼ SR K; ð13Þ where ks ; kc are constants; jks j