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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Regular and chaotic relative motion of a
dumb--bell--shaped satellite
V. V. Beletsky 1 , M. L. Pivovarov 2 , A. A. Savchenko 3
1 Keldysh Institute of Applied Mathematics, Moscow, Russia
2 Space Research Institute, Moscow, Russia
3 Lomonosov Moscow State University, Moscow, Russia
In this paper an orbital two­body system consisting of two point masses
m 1
,m 2
, connected with an ideally flexible massless inextensible thread is con­
sidered. Equations are obtained for the plane relative motion of the system on
the taut thread (on the assumption that the centre of mass of the system moves
along the Keplerian elliptic orbit). The condition of existence of such motion is
derived. If this condition is not met the system moves freely without being con­
strained. The torques of gravitational forces, aerodynamic pressure, aerodynamic
friction and the aerogradient effect are taken into account. Particular attention
is focused on possible ways of motion chaotization.
First, the problem is investigated in case of taut thread and Keplerian elliptic
motion of the centre of mass. Parameters of aerodynamic pressure a, aerodynamic
friction b and the aerogradient effect k in these formulas are expressed in terms
of aerodynamic coefficients C 1
, C 2
, masses m 1
, m 2
, the focal parameter of the
orbit P , the density of the atmosphere at the perigee ae ъ and the atmosphere scale
height H .
In case of circular orbit (e = 0) and in neglecting the aerodynamic dissipation
(b = 0) the equation of motion has been solved and typical phase portraits have
been considered and classified. It is characteristic of this case that aerogradient
effect and dissipation together may cause the system to spin up with rather high
(but finite) angular velocity ( ё 2deg=sec) corresponding to the so--called limit
cycle of the second type, where the angular velocity is ! = ! 0 R 0 =H , ! 0 and R 0
being the orbital angular velocity and radius of the orbit, respectively. When a
phase trajectory meets zones of leaving the constraints the taut thread becomes
slack and the motion is free which is described by different equations.
In a general case the free motion is bound to become connected. Further
evolution of the motion depends on the character of the impact when the thread
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gets taut. For an absolutely elastic impact the energy of the motion is the same
for all points of the trajectory.
For description of investigation of such kind of motion it is convenient to
use the method of point mapping considering characteristics of the trajectory
only at the time of impact. Examples of phase portraits obtained by this method
are given. Corresponding to chaotic motion are areas filled with points. The
areas are called a chaotic sea in which there are islands of regular (periodic and
conditionally periodic) motion. Fixed points of the mapping correspond to the
periodic motion.
Another way of chaotization (without impacts) is due to the nonautonomous
equations of motion. This is the case when the orbit of the centre of mass is
elliptic including the case of connected motion with a taut thread according to
the previous equation of motion. Chaotization is due to the nonzero value of
the eccentricity (e = 0). The effect of chaotization may be significant even for
quite small values of eccentricity, since the density of the atmosphere has an
exponential growth when the height increases, and the exponent is џ = eR ъ =H ,
the ratio R ъ =H is ё 100. That's why even for e ё 0:01 we have џ ё 1.
Besides known general effects (chaotization appears near separatrices, islands
of regularity) there are ones peculiar to the problem in question: there appear
atolls of regularity (not only `islands'). There is also a tendency of regularization
for large values of aerodynamic parameter.
Chaotization of the connected motion and the limit mode are considered in the
general case. All aerodynamic effects are taken into account including pressure,
friction, aerogradient (and eccentricity of the orbit). Phase portrait was obtained
as above by using the method of point mapping. It was found that the modes of
motion tend to quasi­periodic rotation (in the vicinity of the limit cycle of the
second type, which exists in case of circular orbit).
References
1. Beletsky V. V., Pivovarov M. L. The Effect of the Atmosphere on the At­
titude Motion of a Dumb--Bell--Shaped Artificial Satellite, Applied Mathe­
matics and Mechanics. 2000, 64, 721--731.
2. Beletsky V. V., Levin Ye. M. Dynamics of Space Tethered Systems. M.:
Nauka, 1990 (in Russian).
3. Beletsky V. V., Yanshin A. M. The Effect of Aerodynamic Forces on the
Rotational Motion of Artificial Satellites. Kiev: Naukova Dumka, 1984 (in
Russian).
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