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Periodic orbits of the first sort in the restricted problem of three bodies with an ellipsoidal primary
Pergamon Press.
Ghin.Astron.Astrophys. 8 (1984) 255-259 Act.ktin.S-in. -25 (198?) 64-70
Printed in Great Britain 0275-1062/84$10.00+.00
PERIODIC ORBITS OF THE FIRST SORT IN THE RESTRICTED PROBLEM OF THREE BODIES WITH
AN ELLIPSOIDAL PRIMARY
DING Hua, Department of Astronomy, Nanjing University TONG Fu, Purple Mountain Observatory,Academia Sinica Received
1983 February 4
ABSTRACT We discuss the existence of periodic orbits of the first sort in the restricted problem of three bodies in which one of the main bodies The circular orbits in the two-body problem are taken is an ellipsoid, as generating orbits and are analytically continued into a set of symmetric, periodic orbits around the ellipsoidal body. The initial conditions for these periodic orbits differ from those in the case where oblatenesses are neglected by quantities of the order of the perturbations due to the figure. Therefore, when the oblatenesses are appreciable, the shape must be takenintalaccount in discussing the periodic orbits, especially for those orbits that are close to the main body. 1.
INTRODUCTION
There is a vast literature on the periodic orbits of the planar, circular, restricted problem of three bodies, [l]. In 1967, S.S. Huang [2] proposed to use these periodic orbits in a determination of the mass-ratios of the two components of a binary system possessing a gaseous ring. It was already discovered in the 40’s that certain binary systems have gaseous rings revolving around the primaries. If we identify the two components of a binary with the two finite mass bodies and the mass points in the ring with the massless body of the restricted problem of three bodies, then the gaseous ring will be the nearly circular periodic Huang used the solutions of that problem. method of series expansion to derive the periodic orbits and for different values of the mass-ratio derived different relations between the ring velocity and its distance from the primary then from the observed relation, the mass-ratio was inferred. In general,the components of a binary system are not spherical, and sometimes they are even rather oblate ellipsoids. In such cases the above identification can only be approximate. We have therefore undertaken to study the case where the primary is a triaxial ellipsoid, and to investigate the periodic orbits in such a case. If the mass-ratio between the two components is a small quantity and if the oblateness of the primary are also small quantities, then the existence of periodic orbits can be proved by analytic continuation. We shall use numerical means to derive a set of periodic
orbits in our model. We begin by neglecting the effect of the secondary and find a set of circular orbits around the primary, then using these as generating orbits, we continue them into a set of symmetric, nearly circular, periodic orbits of our model. We shall also continue them into another set of periodic orbits in the usual restricted model of three bodies which does not include oblateness. The initial conditions we found for this set are in total agreement with what Huang derived by analytical means. When we compare the initial conditions for the two sets of orbits, we shall find that their differences are of the order of the perturbations due to the of the figure . Therefore, when oblatenesses primary are negligibly small, or when the gaseous ring is at some considerable distance from the primary, Huang’s method of determining the mass-ratio is valid; but if the oblatenesses arenotnegligible and if the ring is at a small distance from the primary, then Nuang’s method will not be precise. 2.
EQUATIONS OF MOTION
We denotetheellipsoidal primary by Pl, the secondary by P2 and the massless body by P3. We assume Pl and P2 to revolve around each other in a circular orbit with the same period as the spin period of Pl, so that the long axis of Pl always points to P2 and we assume P3 to move in the plane of the mutual orbit of Pl and P2. We take a rotating coordinate system with the mass centre of Pl as origin, the common orbital plane as the XY plane and the X-axis along PlP2. We take
256
DING & TONG
the distance between Pl and P2 as unit distance, the sum of the masses of Pl and P2 as unit mass, and the reciprocal of the angular vel,oaitybetween Pl and P2 as unit time. In this system of units, the gravitational constant G= 1. The longest axis of Pl is along the X-axis, and the three coordinate axes are the three principal axes of Pl. In this system of coordinates, the potential function of Pl is P-z! +
t++c
-2A) -+((A
- B)yZ-f-(A- C)sV.
Hence, ml is the mass of Pl, A, B, C are its principal moments of inertia, and x, Y, z, are the coordinates of PT. All terms of 0((o/r14) have been omitted from V, P being the distance of a volume element Gf Pl from its centre. 'Ihemoments of inertia, A, B, C can be expressed in terms of the three radii a, b, c of the ellipsoid Pl: B - y (0’ + c'), c - Y'U + 63).
A-~(b'Cc'),
The meridional oblateness is a= (a I c)/a, and the equatorial oblateness is B= (a - b)/a. Since we assume P3 to move in the XY-plane, we have z= 0. Hence gmd V - - !!$- :? -Q&ft' 2 9
c-z/f> r-~~(n-n)y4
+ -$cn - B>Yi.
Here j is the unit vector in the Y-direction. The equations of motion of P3 are
- y f (4 -
E)y% -
z
(x - I) -
m,,
fl) J--
yj,,.,-2i.-
-""(n-t-C-Z&p 2 1.5
-~~(/i--)g'+~(n-L()y--m'y. r:, where rn2 is the mass of P2 and ~3 is the distance between Pz and Pg. For the equations (l), we have the generalized energy integral $59 + Yl)- $
- mr)'+ y'l-F
- -&(D + c - 2.4) - *5(^ - n)yl- z - CO"% (21
3.
EXISTENCE OF PERIODIC ORBITS
We begin with a brief description of Poincare's method of analytic continuationin connection with the existence of periodic solutions [3]. Consider a system of differential equations, *t-fk(r,a),
(k-
I,-.*,m)
(31
containing a parameter Q, for which there exists an integral not containing t explicitly: rL(r,a) - consr.
(41
Let us denote the solution of (3) by x(i,5,a>, x(0,5ra)-5. If, for ~=a*, system (3) has a known periodic solution x(t,c*,o*), with period r*, then, for value a in the neighbourhood of a*, the necessary and sufficient condition for (3) to have a periodic solution with the same period is xq(T*,5tn)-&-o.
k-!.a--,m
If we write &(e,d) -x4(?*, g, a> - 54,then (5) becomes
(51
257
Periodic Orbit Around Oblate Primary
Since x(t,c*,a*) is a periodic solution of (3) with period T*, we have ‘bk(E*,a*) According to the theorem of the existence of implicit functions, if
=O.
then we can solve (6) for Sk-&*, expressed as a power series in a- a*. That is, for value a in the neighbourhood of a*, system (3) also possess period solutions of period T*, or the periodic solutions can be analytically continued. If system (3) possesses the inte$ral (4), and if L_,(t*,#)+ 0, and if, further, we assume fn(<*) #O, then the condition (7) can be changed to,
%!being ekSl with its (n- I)-the row and n-th column removed, If (8) holds, then for value cuin the neighbourhood of a*, system (3) also have periodic solutionsof‘period T*, x(t,c,a) and ~2-51" (2=1 ,..a) are expressible as power series in a-a*; at the same time, for any fixed a, system (3) also has periodic solutions of period T, T being a value in the neighbourhood of T* and 5 is expressible as a power series in T -T*. We think that if the system of differentialI equations (3) contains more than one parameter, that is, if a is a vector, then the above conclusions should still hold. For the condition (7) or (8) is deduced from Eqs. (6) by virtue of the theorem of the existence of implicit functions, so if we appeal to the theorem of the existence of implicit functions of many parameters, we should be able to say that, in the case of many parameters, the condition (7) or (8) is still the condition for the continuation of periodic functions. The theorem of implicit functions is as follows: [4] Let ~t(~‘,*..,o*;x’,.*.,~“)):R‘X R*-,R” be continuous and differentPBbfe%nan open set . . and let H denote the contamng (J, a:r.. . ,a:; x:,&.+*, x;) and f,(t+,a:,**., at; xi,**., xl;) - O, m xmmatrix
If det M# 0, then there must exist an open set ACR” containing (&.*.,u:) and an open set containing(&,..-.&such that for every ja')6d, there exists a unique {g,(d,...,a=)]6 B, such that
BCR*
fda’,**.,
a”;
k?,(a),
gm,***,
da>)
-
0
i-
l,.",M
and the functions {g,(a',-.*,a")] are differentiable. Now, in the equations (l), when the oblatenesses a, 6 are zero, m.2=O, exists the following periodic solutions (circular orbits) +- (ICOSOI,y-a&to;,
A=B=C,
there
f - -ufosiaf4:, P - oLDws(D;. (9) a'(w+ I)'- 1
X0- II 7,) - 0 4 - 0 %-aw In virtue of (Z), we have 9(x,Y1 &(n, 3(a,O,
2, 3> ml, a, a) -
o, o, 00, 0, ow> 0,
o, 0,O)
o,o>-
cam. -
IIW # -f-l-
0, (1 -I- 2aw -
-cm*
#
0,
Re-write the equation (1) as a set of first-order differential equations, and delete the 4th row and 3rd column of the Jacobi matrix to get the matrix*. When m2 "0, A=B=C, Eqns. (1) are equivalent to the case u= 0 in the usual restricted model, [3]. Therefore, the Q,! here is the same %! as in the proof of the existence of periodic solutions in the usual restriced model, and we have, [3],
Thus, the condition for the existence of periodic solutions in our dynamic model is the same as in the usual restricted model, namely, w+ -2,o,g --I(g- *1,*2,-.-)
DING & TONG
258
4.
NUMERICAL EVALUATION OF PERIODICSOLUTIONS
We used numerical methods and found a set of symmetric, periodic orbits in the rotating The initial conditions for the periodic orbits were chosen to be coordinate system. io=yo=o* x0 was taken as the parameter of the periodic orbit, and 40 was to be determined by the condition of periodicity. At the initial position, these orbits are perpendicular to the X-axis. If we make the coordinate changes, x=x1, y= -91, t= -tl, then Eqns. (1) are not changed and the values of the initial conditions are not changed, so we have differentiation with respect to tI, hence, the solution will also be - y(r), showing that the the same: r,(s) - r(g), y&r,) - rft) in other words =(-t) - x(r),--7(-r) orbits are symmetrical about the X-axis. In virtue of this property, we need only integrate over half the circuit, and the condition of periodicity is simplified to read, at the second at the passage across the X-axis, 3 - 0, or, P3 again crosses the X-axis perpendicularly half-way point. Starting with the circular orbits of the two body problem, the initial conditions for the For the circular orbits, we have periodic orbits in our model were found by iteration. We use prime to denote
5.
RESULTSOF THENUMERICAL EVALUATION
We took P2 to have a mass * = 0.1, Pl to have a longest radius a= 0.01, a meridional Our calculated periodic orbits are oblateness a=O.l and an equatorial oblateness B=O.OS. suffix f refers to the values at the shown in TABLE1. T/2 represents the half-period, midway point. Suffix c refers to the generating (circular) orbit. Our periodic orbits are obviously very nearly circular. In order to discuss the effect of the oblateness on the initial conditions, we calculated a second set of periodic orbits for a= 0, B= 0, and the same values of the other parameters. by Huang are indicated by the ‘Ihe results are in TABLE2. Values obtained analytically suffix H, they can be seen to be in complete agreement with our values. TABLE 1 9
i. -0.45 -0.40 -0.35 r;o.30 -0.25 -0.28 -0.15 -0.10 -0.09 -0.08 -0.07
-
1.03i97
-1.13768
-1.27522 -1.44498 -1.65504 -1.92571 -2.30195 -2.90185 -3.07426 -3.27639 -3.51857
1.80712 1.26096 0.920643 0.674461 0.482615 0.328881 0.205387 0.108379 0.0920488 0.0767594 0.0625342
h
0.449503 0.410678 0.356974 0.303817 0.251880 0.200809 0.150281 0.100067 0.0900469 0.0800323 0.0700225
Fig.
1
I .05859 1.10469 I.24584 1.42392
1.64112 1.9172G 2.29739 2.89984 3.07261 3.27504 3.51741
-0.96421 -1.1000 -1.2536 -1.4320 -1.6474 -1.9213 -2.2995 -2.9000 -3.0723 -33.2741 -3.5157
1.466 1.142 0.8771 0.6581 0.4768 0.3270 0.2049 0.1083 0.09203 0.07676 0.06255
PeriodicOrbit AroundOblatePrimary
259
TABLE 2 if
h -0.45 -0.40 -0.35 -il.30 -0.2s -0.20 -0.15 -0.10 -0.09 -0.08 -0.07
-1.03495 -1.13765 -1.27517 -1.44490 -1.65493 -f.92553 -2.30158 -2.90082 -3.07293 -3.27461 -3.51609
I.80716 1.26099 0.920668 0.674485 0.482639 0.328906 0.205414 0.108410 0.0920812 0.0767935 0.0625701
0.449499
1.05858 0.410676 1.10467 0.35G972 1.24580 0.303816 1.42386 0.251879 1.641U3 0.200808 1.91708 O.f50280 2.29703 0.100066 2.89885 0.0900457 3.u7133 0.0800311)3.27332 O.Q7QO211 3.51501
3n. -1.035Q -1.1376 -1.2752 -1.4449 -1.1549 -1.9255 -2.3016 -2.9001
--0.%421-f.fQOO -1.2536 -1.4320 -f-6474 -1.9213 -2.2995 -2.9000 -3.0723 -3.2741 -3.5157
1.466 1.142 0.8771 U.6581 0.4768 0.3270 0.2049 0.1083 0.09203 0.07676 0.06255
6. CONCLUSION Our numericalresultsshow that,when the primary Pl is an ellipsoid,
so long as the
oblatenesses are small,periodicsolutionsof the restrictedthree-bodymodel still
exist. We can use the circular orbits of the two-body problem as generating orbits, and continue them to the nearly circular, periodic orbits of our model with an ellipsoidal primar . Write “$0 for the difference (go -I&I, given in TABLE2. We note that when ~AXO 20.25, Al\jo=O(lO- ); when 0.095 IAx 50.20, Ajl0=0(10-~); when [Axol cO.09, A~0=0(10’~). The perturbation by P2 on P3 is of the order of (m;!/m~)q*, and this agrees with the order of magnitude of Ajo. This shows that the difference between the initial conditions of periodic orbits in the restricted model and their generating orbits is of the order of the perturbations by P2. Now write $0 for differences between the fill,-,of TABLE1 and the j, of TABLE 2. We find that, when (x0( ~0.15, @0=0(10’~); when
and this agrees with the order of magnitude of ~$0. Mis shows that the difference between the initial conditions of periodic orbits of the models including and not including the oblatenesses is of the order of the perturbation due to the body fi re. Comparing Ajo and I&, we see that, when lxol ;? 0.1s hj Y< IAGoP. when 0.09< 1x01eO.15, G$J and AGo are of the same order; when 1x01 < 0.09, 120 I > PA$01 We recall that the relation between ring velocity and distance proposed by Huang to determine the mass ratio contains only m2 and not the oblatenesses a,B . From the above comparison, it followsthat Huang’s method is valid for /xe\ z-0.15, but becomes increasingly imprecise as we consider smaller and smaller [x01.
Ghin.Astron.Astrophys. 8 (1984) 255-259 Act.ktin.S-in. -25 (198?) 64-70
Printed in Great Britain 0275-1062/84$10.00+.00
PERIODIC ORBITS OF THE FIRST SORT IN THE RESTRICTED PROBLEM OF THREE BODIES WITH
AN ELLIPSOIDAL PRIMARY
DING Hua, Department of Astronomy, Nanjing University TONG Fu, Purple Mountain Observatory,Academia Sinica Received
1983 February 4
ABSTRACT We discuss the existence of periodic orbits of the first sort in the restricted problem of three bodies in which one of the main bodies The circular orbits in the two-body problem are taken is an ellipsoid, as generating orbits and are analytically continued into a set of symmetric, periodic orbits around the ellipsoidal body. The initial conditions for these periodic orbits differ from those in the case where oblatenesses are neglected by quantities of the order of the perturbations due to the figure. Therefore, when the oblatenesses are appreciable, the shape must be takenintalaccount in discussing the periodic orbits, especially for those orbits that are close to the main body. 1.
INTRODUCTION
There is a vast literature on the periodic orbits of the planar, circular, restricted problem of three bodies, [l]. In 1967, S.S. Huang [2] proposed to use these periodic orbits in a determination of the mass-ratios of the two components of a binary system possessing a gaseous ring. It was already discovered in the 40’s that certain binary systems have gaseous rings revolving around the primaries. If we identify the two components of a binary with the two finite mass bodies and the mass points in the ring with the massless body of the restricted problem of three bodies, then the gaseous ring will be the nearly circular periodic Huang used the solutions of that problem. method of series expansion to derive the periodic orbits and for different values of the mass-ratio derived different relations between the ring velocity and its distance from the primary then from the observed relation, the mass-ratio was inferred. In general,the components of a binary system are not spherical, and sometimes they are even rather oblate ellipsoids. In such cases the above identification can only be approximate. We have therefore undertaken to study the case where the primary is a triaxial ellipsoid, and to investigate the periodic orbits in such a case. If the mass-ratio between the two components is a small quantity and if the oblateness of the primary are also small quantities, then the existence of periodic orbits can be proved by analytic continuation. We shall use numerical means to derive a set of periodic
orbits in our model. We begin by neglecting the effect of the secondary and find a set of circular orbits around the primary, then using these as generating orbits, we continue them into a set of symmetric, nearly circular, periodic orbits of our model. We shall also continue them into another set of periodic orbits in the usual restricted model of three bodies which does not include oblateness. The initial conditions we found for this set are in total agreement with what Huang derived by analytical means. When we compare the initial conditions for the two sets of orbits, we shall find that their differences are of the order of the perturbations due to the of the figure . Therefore, when oblatenesses primary are negligibly small, or when the gaseous ring is at some considerable distance from the primary, Huang’s method of determining the mass-ratio is valid; but if the oblatenesses arenotnegligible and if the ring is at a small distance from the primary, then Nuang’s method will not be precise. 2.
EQUATIONS OF MOTION
We denotetheellipsoidal primary by Pl, the secondary by P2 and the massless body by P3. We assume Pl and P2 to revolve around each other in a circular orbit with the same period as the spin period of Pl, so that the long axis of Pl always points to P2 and we assume P3 to move in the plane of the mutual orbit of Pl and P2. We take a rotating coordinate system with the mass centre of Pl as origin, the common orbital plane as the XY plane and the X-axis along PlP2. We take
256
DING & TONG
the distance between Pl and P2 as unit distance, the sum of the masses of Pl and P2 as unit mass, and the reciprocal of the angular vel,oaitybetween Pl and P2 as unit time. In this system of units, the gravitational constant G= 1. The longest axis of Pl is along the X-axis, and the three coordinate axes are the three principal axes of Pl. In this system of coordinates, the potential function of Pl is P-z! +
t++c
-2A) -+((A
- B)yZ-f-(A- C)sV.
Hence, ml is the mass of Pl, A, B, C are its principal moments of inertia, and x, Y, z, are the coordinates of PT. All terms of 0((o/r14) have been omitted from V, P being the distance of a volume element Gf Pl from its centre. 'Ihemoments of inertia, A, B, C can be expressed in terms of the three radii a, b, c of the ellipsoid Pl: B - y (0’ + c'), c - Y'U + 63).
A-~(b'Cc'),
The meridional oblateness is a= (a I c)/a, and the equatorial oblateness is B= (a - b)/a. Since we assume P3 to move in the XY-plane, we have z= 0. Hence gmd V - - !!$- :? -Q&ft' 2 9
c-z/f> r-~~(n-n)y4
+ -$cn - B>Yi.
Here j is the unit vector in the Y-direction. The equations of motion of P3 are
- y f (4 -
E)y% -
z
(x - I) -
m,,
fl) J--
yj,,.,-2i.-
-""(n-t-C-Z&p 2 1.5
-~~(/i--)g'+~(n-L()y--m'y. r:, where rn2 is the mass of P2 and ~3 is the distance between Pz and Pg. For the equations (l), we have the generalized energy integral $59 + Yl)- $
- mr)'+ y'l-F
- -&(D + c - 2.4) - *5(^ - n)yl- z - CO"% (21
3.
EXISTENCE OF PERIODIC ORBITS
We begin with a brief description of Poincare's method of analytic continuationin connection with the existence of periodic solutions [3]. Consider a system of differential equations, *t-fk(r,a),
(k-
I,-.*,m)
(31
containing a parameter Q, for which there exists an integral not containing t explicitly: rL(r,a) - consr.
(41
Let us denote the solution of (3) by x(i,5,a>, x(0,5ra)-5. If, for ~=a*, system (3) has a known periodic solution x(t,c*,o*), with period r*, then, for value a in the neighbourhood of a*, the necessary and sufficient condition for (3) to have a periodic solution with the same period is xq(T*,5tn)-&-o.
k-!.a--,m
If we write &(e,d) -x4(?*, g, a> - 54,then (5) becomes
(51
257
Periodic Orbit Around Oblate Primary
Since x(t,c*,a*) is a periodic solution of (3) with period T*, we have ‘bk(E*,a*) According to the theorem of the existence of implicit functions, if
=O.
then we can solve (6) for Sk-&*, expressed as a power series in a- a*. That is, for value a in the neighbourhood of a*, system (3) also possess period solutions of period T*, or the periodic solutions can be analytically continued. If system (3) possesses the inte$ral (4), and if L_,(t*,#)+ 0, and if, further, we assume fn(<*) #O, then the condition (7) can be changed to,
%!being ekSl with its (n- I)-the row and n-th column removed, If (8) holds, then for value cuin the neighbourhood of a*, system (3) also have periodic solutionsof‘period T*, x(t,c,a) and ~2-51" (2=1 ,..a) are expressible as power series in a-a*; at the same time, for any fixed a, system (3) also has periodic solutions of period T, T being a value in the neighbourhood of T* and 5 is expressible as a power series in T -T*. We think that if the system of differentialI equations (3) contains more than one parameter, that is, if a is a vector, then the above conclusions should still hold. For the condition (7) or (8) is deduced from Eqs. (6) by virtue of the theorem of the existence of implicit functions, so if we appeal to the theorem of the existence of implicit functions of many parameters, we should be able to say that, in the case of many parameters, the condition (7) or (8) is still the condition for the continuation of periodic functions. The theorem of implicit functions is as follows: [4] Let ~t(~‘,*..,o*;x’,.*.,~“)):R‘X R*-,R” be continuous and differentPBbfe%nan open set . . and let H denote the contamng (J, a:r.. . ,a:; x:,&.+*, x;) and f,(t+,a:,**., at; xi,**., xl;) - O, m xmmatrix
If det M# 0, then there must exist an open set ACR” containing (&.*.,u:) and an open set containing(&,..-.&such that for every ja')6d, there exists a unique {g,(d,...,a=)]6 B, such that
BCR*
fda’,**.,
a”;
k?,(a),
gm,***,
da>)
-
0
i-
l,.",M
and the functions {g,(a',-.*,a")] are differentiable. Now, in the equations (l), when the oblatenesses a, 6 are zero, m.2=O, exists the following periodic solutions (circular orbits) +- (ICOSOI,y-a&to;,
A=B=C,
there
f - -ufosiaf4:, P - oLDws(D;. (9) a'(w+ I)'- 1
X0- II 7,) - 0 4 - 0 %-aw In virtue of (Z), we have 9(x,Y1 &(n, 3(a,O,
2, 3> ml, a, a) -
o, o, 00, 0, ow> 0,
o, 0,O)
o,o>-
cam. -
IIW # -f-l-
0, (1 -I- 2aw -
-cm*
#
0,
Re-write the equation (1) as a set of first-order differential equations, and delete the 4th row and 3rd column of the Jacobi matrix to get the matrix*. When m2 "0, A=B=C, Eqns. (1) are equivalent to the case u= 0 in the usual restricted model, [3]. Therefore, the Q,! here is the same %! as in the proof of the existence of periodic solutions in the usual restriced model, and we have, [3],
Thus, the condition for the existence of periodic solutions in our dynamic model is the same as in the usual restricted model, namely, w+ -2,o,g --I(g- *1,*2,-.-)
DING & TONG
258
4.
NUMERICAL EVALUATION OF PERIODICSOLUTIONS
We used numerical methods and found a set of symmetric, periodic orbits in the rotating The initial conditions for the periodic orbits were chosen to be coordinate system. io=yo=o* x0 was taken as the parameter of the periodic orbit, and 40 was to be determined by the condition of periodicity. At the initial position, these orbits are perpendicular to the X-axis. If we make the coordinate changes, x=x1, y= -91, t= -tl, then Eqns. (1) are not changed and the values of the initial conditions are not changed, so we have differentiation with respect to tI, hence, the solution will also be - y(r), showing that the the same: r,(s) - r(g), y&r,) - rft) in other words =(-t) - x(r),--7(-r) orbits are symmetrical about the X-axis. In virtue of this property, we need only integrate over half the circuit, and the condition of periodicity is simplified to read, at the second at the passage across the X-axis, 3 - 0, or, P3 again crosses the X-axis perpendicularly half-way point. Starting with the circular orbits of the two body problem, the initial conditions for the For the circular orbits, we have periodic orbits in our model were found by iteration. We use prime to denote
5.
RESULTSOF THENUMERICAL EVALUATION
We took P2 to have a mass * = 0.1, Pl to have a longest radius a= 0.01, a meridional Our calculated periodic orbits are oblateness a=O.l and an equatorial oblateness B=O.OS. suffix f refers to the values at the shown in TABLE1. T/2 represents the half-period, midway point. Suffix c refers to the generating (circular) orbit. Our periodic orbits are obviously very nearly circular. In order to discuss the effect of the oblateness on the initial conditions, we calculated a second set of periodic orbits for a= 0, B= 0, and the same values of the other parameters. by Huang are indicated by the ‘Ihe results are in TABLE2. Values obtained analytically suffix H, they can be seen to be in complete agreement with our values. TABLE 1 9
i. -0.45 -0.40 -0.35 r;o.30 -0.25 -0.28 -0.15 -0.10 -0.09 -0.08 -0.07
-
1.03i97
-1.13768
-1.27522 -1.44498 -1.65504 -1.92571 -2.30195 -2.90185 -3.07426 -3.27639 -3.51857
1.80712 1.26096 0.920643 0.674461 0.482615 0.328881 0.205387 0.108379 0.0920488 0.0767594 0.0625342
h
0.449503 0.410678 0.356974 0.303817 0.251880 0.200809 0.150281 0.100067 0.0900469 0.0800323 0.0700225
Fig.
1
I .05859 1.10469 I.24584 1.42392
1.64112 1.9172G 2.29739 2.89984 3.07261 3.27504 3.51741
-0.96421 -1.1000 -1.2536 -1.4320 -1.6474 -1.9213 -2.2995 -2.9000 -3.0723 -33.2741 -3.5157
1.466 1.142 0.8771 0.6581 0.4768 0.3270 0.2049 0.1083 0.09203 0.07676 0.06255
PeriodicOrbit AroundOblatePrimary
259
TABLE 2 if
h -0.45 -0.40 -0.35 -il.30 -0.2s -0.20 -0.15 -0.10 -0.09 -0.08 -0.07
-1.03495 -1.13765 -1.27517 -1.44490 -1.65493 -f.92553 -2.30158 -2.90082 -3.07293 -3.27461 -3.51609
I.80716 1.26099 0.920668 0.674485 0.482639 0.328906 0.205414 0.108410 0.0920812 0.0767935 0.0625701
0.449499
1.05858 0.410676 1.10467 0.35G972 1.24580 0.303816 1.42386 0.251879 1.641U3 0.200808 1.91708 O.f50280 2.29703 0.100066 2.89885 0.0900457 3.u7133 0.0800311)3.27332 O.Q7QO211 3.51501
3n. -1.035Q -1.1376 -1.2752 -1.4449 -1.1549 -1.9255 -2.3016 -2.9001
--0.%421-f.fQOO -1.2536 -1.4320 -f-6474 -1.9213 -2.2995 -2.9000 -3.0723 -3.2741 -3.5157
1.466 1.142 0.8771 U.6581 0.4768 0.3270 0.2049 0.1083 0.09203 0.07676 0.06255
6. CONCLUSION Our numericalresultsshow that,when the primary Pl is an ellipsoid,
so long as the
oblatenesses are small,periodicsolutionsof the restrictedthree-bodymodel still
exist. We can use the circular orbits of the two-body problem as generating orbits, and continue them to the nearly circular, periodic orbits of our model with an ellipsoidal primar . Write “$0 for the difference (go -I&I, given in TABLE2. We note that when ~AXO 20.25, Al\jo=O(lO- ); when 0.095 IAx 50.20, Ajl0=0(10-~); when [Axol cO.09, A~0=0(10’~). The perturbation by P2 on P3 is of the order of (m;!/m~)q*, and this agrees with the order of magnitude of Ajo. This shows that the difference between the initial conditions of periodic orbits in the restricted model and their generating orbits is of the order of the perturbations by P2. Now write $0 for differences between the fill,-,of TABLE1 and the j, of TABLE 2. We find that, when (x0( ~0.15, @0=0(10’~); when
and this agrees with the order of magnitude of ~$0. Mis shows that the difference between the initial conditions of periodic orbits of the models including and not including the oblatenesses is of the order of the perturbation due to the body fi re. Comparing Ajo and I&, we see that, when lxol ;? 0.1s hj Y< IAGoP. when 0.09< 1x01eO.15, G$J and AGo are of the same order; when 1x01 < 0.09, 120 I > PA$01 We recall that the relation between ring velocity and distance proposed by Huang to determine the mass ratio contains only m2 and not the oblatenesses a,B . From the above comparison, it followsthat Huang’s method is valid for /xe\ z-0.15, but becomes increasingly imprecise as we consider smaller and smaller [x01.