ITterna. J. Math. & Math. Sci.
(1982)195-202
Vol. 5 No.
195
RESEARCH NOTES
SOME REMARKS ON THE STABILITY ANALYSIS
IN ROBE’S THREE BODY PROBLEM
R. MEIRE
Astronomical Observatory
Ghent State University
Krijgslaan 271- $9
GENT
9000
BELGIUM
(Received October 12, 1979)
ABSTRACT.
An improved technique is presented for the stability analysis of Robe’s
3-body problem which gives more accurate results for the transition curves in the
parameter plane than does Robe’s paper.
A novel property of the system of differential equations describing the motion
is used, which reduces the computer time by more than 50%.
KEY WORDS AND PRASES.
3-body problem, stabily, transition curve, Floquet-theory.
1980 MATHEATICS SUBJECT CLASSIFICATION CDES.
I.
85A05
INTRODUCTION.
In a recent paper Robe [i] presented a new kind of restricted three body pro-
blem, where one body m I is a rigid spherical shell, filled with an homogeneous
incompressible fluid of density
the_
where a second body m
PI’
2
shell, and where m 3 is a small solid sphere of density
is a mass point outside
P3’
inside the shell, its motion determined by the attraction of m
force due to the fluid
restricted to move
the buoyancy
2 and
i"
There exists a solution with m
a Keplerian orbit around it.
3
at the center of the shell while m
2 describes
Robe investigated the stability of this configuration
under the assumption that the mass of m is infinitesimal.
3
The linearized equations
196
R. MEIRE
-
of motion in the neighborhood of this equilibrium are
I+2
I + ecosv
2=
K(1
(I
e2)
3
+ ecosv)
3
+21=
1
+
-
e2)
K(I
(i + ecosv)4
ecosv
2)3
+z
e
K(I
(i + ecosv)
I + ecosv
where
K
m
a
Pl
4
3
(mI +
x
(i,i)
Y
(1.2)
z
(1.3)
3
(I
m21
Pl
3
2
mI + m2)
a
semi-major axis of the Keplerian orbit.
These equations are referred to a coordinate system Oxyz, where 0 is the center of
ml, Ox
If
points to m
2
and Oxy is the plane of the Keplerian orbit.
-
0 (shell empty) or
01
become
with
r
i
+
then K
0 and the equations of motion
01
03,
2#
(I + 2U) r x
(1.4)
r y
(1.5)
r z
(1.6)
9+2
(i
+
(i
z
U
ecosv
Equations (1.4) and (1.5) describe the motion in the orbital plane.
Robe
investigated the stability in the orbital plane by means of the Floquet-theory.
However, one can separate the fourth-order system (1.4), (1.5) into two independent
second-order systems.
2.
THE TRANSFORMATION TO SECOND-ORDER SYSTEMS.
Using
x
[y
and
n
equations (1.4) and (1.5) can be written
#
STABILITY ANALYSIS IN ROBE’S THREE BODY PROBLEM
_
197
(2.1)
rC
2D
0
n
0
i
-i
0
m
where
0
i+2
C
and
O
0
i
Now we make the following transformation (Tschauner [2])
E
E
(2.2)
PI
n
e
P2
and obtain
P2- PI
]
P2PI-rCo-2DPI+PI
p2
-P22+rC0+2DPI-P i
-PIP2+rC0+2DP2-P1/2J
2-rC0-2DP 2 +p
e
Making the nondiagonal elements zero, we obtain
(2.3)
P2
where
PI
and
P2
are two different solutions of the Riccati equation
2DP-
Using
p2
+ rC 0
Pll
PI2
P21
P22
P
(2.4)
198
R. MEIRE
equation (2.4) becomes
2P21 Pll PI2P21
Pll
2
622 -2P12
+ r(l + 2U)
PI2P21
P22
PI2
2P22 PI2(Pll + P22
P21
-2Pll P.21(Pll + P22
+ r(l
)
(2.5)
Now let
P22
+
w
Pll
z
--w- z
PI2
u-v+l
P21
u
+
v
(2.6)
i
Then equation (2.5) becomes
z
2
-i
w
2u
2wz
z
+
2
r
u
3
2
+ v 2 + r( 2 +2
u
U
(2.7)
u =-2z- 2uw
-2vw
The last two equations yield
w=v ()
v
z=- (-)
V
If we use p
1
v
and
q
U
v
we get
(2.8)
STABILITY ANALYSIS IN ROBE’S THREE BODY PROBLEM
199
Substituting (2.8) into the first two equations of (2.7), we have
p.-
i -2
p
+ 4q
+ [2- r(2 + )] p 2
2 =-2(q
2
2
i
+
(2.9)
(2.10)
3rp
Now if we let
k
rp
+ kI
0
(2.11)
cos v,
we find as a solution for (2.10)
3
q
k0
kI
cos v
+ k2
cos
If we substitute the solutions (2.11) and (2.12) into
fication) the values for
For
k0,
k
k0,
and k
I
2
(2.9),
as functions of
(2.12)
v
we obtain (by identi-
and e.
we obtain
with
4e
c
{3#
k0
4
_+-c
+
4e2
4
+ 1)
2
(3# +
This will yield two different solutions,
So c(,e)
which
2
i)2
PI and P2’
( + 3) + #(9#
if c
# 0
and c
8)
(2 13)
2 >
0.
0 will give vs (analytically) a transtion curve in the -e plane,
corresponds to one of the transition curves (IE) that Robe obtained numeric-
ally.
This curve can be written as
4e
4
(3] + i)
The elements
Pll
Pij
of
2
P1
+
4e2
(3 + i)
and
P2
2
(B + 3) + ].(9]
8)
0
(2.14)
are
-(2 + l)e sinv
(3 + i + ecosv) (i + ecosv)
(2.15)
R. MEIRE
200
-( + i + 2ecosv) esinv
(3 + i ecosv) (I + ecosv)
P22
(3 + I)(i
PI2
3
3.
2
+--+ 2e( + l)cosv +
(3 + I + ecosv) (i + ecosv)
-[(3 + I) (i
P21
e
+_ c
(,2.16)
3
e
+ c
e
2
cos 2v
(2.17)
2
2(2 + l)ecosv]
++ ecosv) +--+
+ ecosv)
(3 + i
(2.18)
(i
STABILITY ANALYSIS.
Now we perform a stability analysis using the Floquet theory, on the two
independent second order systems (2.3)
PI
(3.1)
P2 e
(3.2)
Both equations admit solutions for which
u(v + 2)
1,2)
(i
siu(v)
where s. are the roots of the characteristic equation
-i
det [X
(v).X(v + 2)
sE]
(3.3)
0
where X(v) is a fundamental solution matrix of (3.1) (or (3.2)).
Equation (3.3)
can be written
s
2
2s
+ I
(3.4)
0
For stable solutions,
(3.5)
Thus the transition curves in the (-e) plane, separating stable and non-stable
regions, wil+/- be given by
s=+ i
or
Taking X(v
0)
(3.6)
E, equation (3.3) becomes
det [X(2)
sE]
0
is the monodromy-matrix and
a
2
By
It can be shown (R. Meire and A. Vanderbauwhede [3]) that
I.
STABILITY ANALYSIS IN ROBE’S THREE BODY PROBLEM
X(2)
SX
-I
()SX()
(3.7)
0
i
where
201
S
This implies that we only have to integrate the equations over
One can also prove that for s
+ i, one of the elements of
instead of 27.
X(n)
becomes zero
which saves an additional 10% of computer time.
4.
RESULTS.
We applied this method to
yielded two transition curves:
curve HM.
the’
equations of (3.1) and (3.2).
FM and FE.
Equation (3.1)
Equation (3.2) yielded one transition
All results are given in Fig. i.
The curve IE is the analytically obtained curve from equation (2.14).
Along
I
FM
8=0
and
e=-i
FE
y=0
and
=-i
ttN g=5"=O
and
c= 1
The intersection of the curves FE and IE can be obtained
ver accurately
(Robe was
not able to give precise coordinates).
The point E is determined as that point on the curve IE for which the characteristic roots of (3.1) are -i.
u-e plane
The coordinates of the interesting points in the
are
5+ 97
16
0..928053
point F
0.8888
point I
e
8
9
0
0.8596848
point E
e
0 4531741
The stable region consists of the shaded area in Fig. i and is now determined much
R. MEIRE
202
(2.1) was used.
more accurately than in Robe’s paper where the fourth-order system
M
Fig.
Stability-regions in the -e-plane
0.8
O.6
OA
$
I
0.2
C=O
STABLE
STABI_E
P
I
o.85
F
0.90
05
1H
ACKNOWLEDGEMENTS.
for reading
The author thanks Prof. Dr. P. Dingens and Prof. Dr. H. Steyaert
work.
the manuscript and for their valuable suggestions on this
REFERENCES
A new kind of 3-body Problem, Cel. Mech. 1__6 (1978), pp. 343-351.
I.
ROBE, H.
2.
TSCHAUNER, K.
3.
MEIRE, R. and VANDERBAUWHEDE, A.
Die Aufspaltung der Variations-gleichungen des Elliptischen
Eingeschrnkten Dreikrperproblems,
A useful Result for Certain Linear Periodic
_J..Comp. and Applied Math. 5, N I (1979),
Ordinary Differential Equations,
pp. 59-61.
Cel. Mech. 3 (1971), pp. 395-402.