Gravitational stability changes predictably with distance from geostationary orbits and Lagrange points
the verdict
INSUFFICIENT LEANING
refutedsupported
the weight of evidence
2 sources for · 0 against
Available astrophysical literature confirms that stability characteristics vary near Lagrangian points and orbits, but the retrieved sources provide only partial coverage regarding predictable stability changes across varying distances from geostationary orbits and Lagrange points.
Qualitative behavior of artificial satellites in the earth- moon-sun system has been discussed in terms of a very restricted four-body problem of masses m1, m2, m3, and m such that m1 m2+m3 m. If the separation between m2 and m3 is very much smaller than the distance of their center of mass from m1, we can first idealize a state of motion (with approximation) of the three bodies m3, m2, and m3 in such a way that m2 and m3 revolve around each other in circular orbits and furthermore the center of mass 0' of m2 and m3 revolves around m, in a circular orbit too. By choosing a coordinate system with the origin at 0' and revolving together with m2 and m3, we derive an integral which is the counterpart to Jacobi's integral in the restricted three-body problem. The position of m1 enters into the integral through the angle subtended at 0' by m1 and m. Therefore no stationary zero-velocity surface can be defined in the present problem. However, we can introduce the conception of osculating zero-velocity surfaces which provide the instantaneous limiting surfaces that m of a given initial condition cannot pass through. Double points of the zero-velocity surfaces have also been studied. In the restricted three-body problem these points are fixed in the rotating frame of reference and are the solution of the problem. It is not so in the very restricted four-body problem studied here. The double points rotates in the xy plane as m3 moves with respect to the m2 - m3 system. A study of the motion of these points leads to an interesting result that under certain conditions depending upon the mass of m1 and its distance from 0', the critical zero-velocity surfaces which pass through two double points become degenerated into one single surface. On both sides of degeneracy, the zero-velocity surfaces have fundamentally different appearances. It can be shown from these surfaces that any artificial satellite around the moon has to be close, in order to be stable.
5
(m2 + m3>(x2 + yz> mlr2 2m2 2m3
(2 9) i- (3 cos 6 - I) t - t - = constant ,
a3 A3 '2 '3
where the term 2m1/A has been absorbed in the constant term.
If a is now taken as the unit of length and m2 t m3 as the unit of mass, Equation 29
reduces to
2(1 -+) mlr
,2 t y2 + - (3 cos20 - 1) + ~ t 21-1. = c, A3 '2 '3
where
and c is a constant of integration. This differs from the zero-velocity surfaces of the
restricted three-body problem only by the addition of a
?
setting ,E = 0. Consider the three points separately: (1) L, between +ffi and x3, (2) L~
between x3 and x,, and (3) L, between x, and -m. 4
(1) Let the distance from m3 in the x-direction to the double point L, be repre-
sented by p. Then Equation 35 becomes, by neglecting the second and higher orders of
P sin 2e0,
Differentiating Equation 39 with respect to P and setting p = 0 give
where
The symbol po at the right side of Equation 41 is the solution of p for Equation 39 with
F = 0. Similarly the change in vahe of c which corresponds to the variation in position
of L, can be computed. Differentiating Equation 33 with respect to p and setting ,E = o
afterwards give
b
P,, in Equation 42 having the same meaning as that in Equation 41.
(2) Let the distance in the x-direction from m3 to the double point L1 be repre-
sented by p . Then by a similar approximation, as before, Equation 35 reduces to
J
By exactly the same procedure as before, we obtain
where
8
c
(3) Let the distance in the x-direction from m2 to the double point L, be repre-
sented by 1 - p. Then Equation 35 reduces by approximation to *
L,(i = 1)
-
P 0.1510
C 3.18843
Ei 0.0741
Fi 0.6053
From this is derived
L,(i = 2) L3(i = 3)
0.1679 0.00709
3.17223 3.01216
-0.1566 0.3326
1.5831 -0.3820
= E3(1 + 3 cos 28,) , (3J=o
where
and
where P, is the solution of Equation 47 with p = 0.
Now if the positions and their corresponding values of c of the three double points
on the x-axis are known for the case p = 0, their positions and the corresponding values
of c for the case of small p may be derived by
-
and
c = c, + 8($) *
a= 0
The change in the ordinates of these points is given by Equation 38.
For the earth-moon system, p = 0.01216. The values for the relevant quantities in
this case are listed in Table 1. The derivative of the coordinates of these three points
e
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(1) Point M: Let its distance from m3 be u. Thus, from Equation 33,
2
c = (1 - ,Ll + v) rl L t (1 + 3 cos 28,)/3] t 2(1 1 +u - i”) + 2. U (53)
Differentiating Equation 53 with respect to P and utilizing the relation given by Equation
46 give
where
in which vo is the solution of Equation 53 with P = 0 and C = C,, corresponding to the
inner contact surface of the restricted three-body problem.
(2) Point N: Let its distance from m2 be u. Following the same procedure as
before, we derive
where
(1 - 2P - Po - c,)(l Po + 00)
1-P
E, =
uo2 (1 + Do)
2[p t u, - - -
.
(57)
where D, is o of N when p = 0. In both Equations 55 and 57, po is the solution of Equa-
tion 43 with P = 0.
For p = 0.01216,
E,,, = 0.6217, and E, = 0.0589. (58)
The changes in position of M and N are illustrated in Figure 1, from which it is seen
that M moves out while L, moves in as ,B( 1 + 3 cos 28,) increases. In other words, the
inner contact surface will eventually meet the outermost contact surface at a certain
value of P( 1 + 3 COS 28,). When this happens, the two surfaces degenerate into one sur-
face. It is evident from Equations 40, 54, 58 and Table 1 that the smallest value of P for
which the critical surfaces become degenerated occurs at 8, = 0. This threshold value
of F (denoted by P, hereafter) can be determined in the following way: First calculate
the two points L, and L, by Equation 35 with 8, = 0. Denote the distances of L1 and L,
from m3 by p1 and P,, from which the corresponding values of C (denoted by Cl and
C,, respectively) can be obtained from Equation 33. The degenerated case is given by
the condition
I