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On the stability of the inner planets, satellites and close binaries
Chin.Astron.Astrophys. 6 (1982) 267-272 Act.Astrophys.Sin. 2 (1582) 215-223
ON
THE
STABILITY
CLOSE
BINARIES
ZHENG
Xue- tang
Received
OF
THE
Department
1981 November
Pergamon
INNER
of
PLANETS,
Press. Printed in Great Britain 0275-1062/82/030267-06$07.50/O
SATELLITES
Astronomy,
Beijing
AND
Normal
University
14
Abstract. I consider the range of Hill stability in the restricted circular problem of three bodies when the larger one of the two principal bodies has a finite oblateness. I show that the range P satisfies the equation I--P
r=
c..-3p+P”--Y!1--)r-‘T(I+3u)J[,_p)(*+~)r
’
where u is the mass parameter and v is an oblateness parameter. This result is It applied to the solar system, the Earth-Moon system and binary star systems. is then shown that, all the inner planets of the solar system, the great majority of asteroids and some short-period comets are Hill stable, that direct artificial satellites of the Earth are more stable than retrograde ones, and that contact binaries possess cores between which no mass exchange takes place.
1.
INTRODUCTION
In his study of the motion of the Moon, Hill (1878) gave one definition of stability in terms of the Jacobi constant in the restricted problem of three bodies, [l]. According to Hill, if the small body always moves inside the zero-velocity surface of one of the larger bodies, then its motion around that body is said to be stable. This kind of stability is now known as stability in the sense of Hill. Szebehely (1976), when discussing the Hill stability of the Moon in the Sun-Earth-Moon system, introduced the quantity s = (caCc~_)/c,, as a measure of this stability, [2]. Here Cat is the actual value of the Jacobi constant of the small body and Co, is the critical value corresponding to the Lagrangian point L2 for satellites and inner planets, or the Lagrangian point Lo for outer planets. When S>O, that is, when C,o>C,,, then the motion is stable; when .S1+2.4p1i3; in both expressions p= ml/(ml +m2), where ma is the mass of the Sun, and ml is the mass of the planet or Jupiter. If the respective condition is satisfied, the satellite will never evolve into an independent planet and an outer planetwillnever become a satellite of Jupiter or an inner planet. Their results are: all the natural satellites apart from J8, J9, Jll and 512, and all the outer planets are stable, but the S-values for the Moon, S9 and 57 are very small. We know that many celestial bodies such as the Earth, but more especially Jupiter, Saturn and the component stars in binary systems, have considerable rotation and their shape departs significantly from a sphere. Hence there is a need to extend the usual three-body problem to include the case where the larger bodies have a finite oblateness. The inclusion of the oblateness will conceivably alter the region of stability. Also, when calculating C,, Szebehely et al. considered only the values at two particular points of the ciruclar orbit, but it seems safer to use the minimum value over the circle, especially where the measure In this paper, I have derived a more accurate region of stability for of stability is low. an inner planet type system by including the oblateness of the larger primary and by using the minimum value of Cat actually attained by the small body. I have applied my results to the solar system, the Earth-Moon system, and binaries in a discussion of the stability of the inner planets, short-period comets, asteroids and artificial satellites, and also the region of no mass exchange in binary systems and their dynamical evolution. In my derivation, the following simplifying assumptions are made:
ZHENG
268
Of tbe two larger bodies, the lesser one is spherical and the greater one is oblate, 1. whose equator is inclined at a small angle to the mutual orbit, 2. The mutual orbit is approximately circular. 3. The small body movea in an approximate circular orbit in the orbital plane of the two larger bodies, which can be continued analytically into a Paincark's periodic orbit of the first kind, [6]. 2.
EQUATIONS OF MOTION AND POSITIONS OF THE LIBBATION POINTS
As a first approximation, the potential including the oblateness effect under the above assumptions is
v = JzR2f2 is a small positive parameter. rotating coordinates are
where
The equations of mation of the small body in
where (3) $ fCl--P)t:Cpt:1c a-(1 -+ +I* rl r: rt Here, uf< l/2) and 1-u are the masses of the tmo major bodies, rl and rs are the distances of tbo small body to them. Let the distance between the two major bodies be 1. Then P-
t:-(X-rY+P,
(4)
+:I fx - p + 1y i-yz.
n is the angular velocity of the rotating coordinate system. It is easy to show that d -
1 f
(51
3v.
Eqns, (2) have the Jacobi integral
(6)
c-29-V' where V is the velocity of the small body in the rotating system. The equations aQ.!ax=ara,Latj-0 give a set of particular solutions of (21, hence the positions of the libration points. Using (3) and (41, we have
Fig. 1
17) arid(8) give five particular solutions, the three collinear solutions LX, Zn, LS and the two triangular solutions LII~Ls. Referring to Fig. 1, the locations of the five libration points are
Hill Stability
XI -
c -
1 - 51, y, - 0
x3 -
P -
f -+
x3 -
r + ED
x, -
p -
+
-
Y +
O(v’h
y, _
x4 -
P -
+
-
v +
O(d),
y5 _
51,
y3 -
0
269
I
(9)
Y3- 0 I
JT J1;y+ O(lq
:ix+3$zv _t 1 I
O(a?)
(10)
l
2
Here, & = @' - $ vA(1 - $A)+- O(G)
(11)
~~-b~~)--~vn(l+fh)+o(~~~) &I
1 ?f,(O) f $ Y& + O(@)
i
In the above, superfix (*) refers CO the correspondingvalues in the usual restricted planar circular problem, whose well-known developments in terms of u are given in texts such as [7] and PI,
(13)
3.
CALCDLATION OF Cao AND STABILITY CONDITION FOR INNER PLANET TYPE SYSTEMS
To fix the region of stability of the small body, we first suppose it to move in a circular orbit around the larger principal body, as shown in Fig. 1. We shall later use the method of analytical continuation to generate from it a PoincarB's periodic orbit of the first kind. Inserting (3) in (6), we find that the Jacobi constant when the small body is at point P on the circular orbit is C -ti[(l- p>t:+ &&I+ =+(l where r2*=1+r,Z+2rlcose,
+ ')+ r:
*71
(14)
V'
8 being as shown in Fig. 1.
The velocity at P is
It is easy to show that the velocity V in the rotating system at any point P of this circular orbit depends only on r1 and not on 8, and V'- [nr,TdQ+(l
+ $1'
(15)
where the negative sign applies to direct orbits, the positive sign, to retrograde orbits. After substituting (15) into (14), we have c - +I(1 -p),:+pr:1+1--c l--Y +* ?I ( t:) r) --n’rt*2+
-4
+3r‘,
(16)
270
ZHRNG
Because r2 varies with e, the different points of the circular orbit give different C-values, This is because such an orbit is not a rigorous solution of the problem.considered. To ensure that the small body is Hill stable in the region derived, we must use the minimum value of c over the circular orbit. In (16), since rl does not depend on 8, dC/dB =0 gives I*=,-~/~. Since, here,
C assumes minimum values cmin at the two points
e - rfcos
-f:+;)
which are different for different system with different values of u and v. As mentioned, we should identify Cs, with Cmin, obtained by substituting r*=n -2/3 in (16), that is, C,-3p--pt:+
I--IL(l -$*(t+ rl
3$/(l
-&(I
t+
(17)
+ O(PY)
on using (5). As we are discussing the inner planet case, C,, should be identified with the value of C at the libration point Lz. From (9) we have ri(L~) =l-<~,ra(L~) =F,= and, of course, V=O at Lz. Substituting these in (14) and using (11) we have
(18) Here 52(o), hence Ccr "' ' can be easily looked up in Ref.[6] or calculated from (12). Thus, ccr is easily found for any given p and v. The critical distance r of r1 defining the region of stability is given by equating (17) and (18), that is, by l-
tC,-338+fW~+Y(1-~)r
-'+(2 + 3Y)J(l-&(l+
(19) 3y/?)r *
This equation for r can be solved by iteration. For starting values, we can take 1 for direct orbits and l/4 for retrograte orbits. Since p-5l/2, (17) shows that dCa,/dri ~0, hence when rl < r, Cac>Ccr. Thus, if the small body moves approximately in a circular orbit around the bigger primary, then as long as its distance from the latter is less than r it is Hill stable. For any known rr we can calculate Cao from (17), and hence the measure of Hill stability, s-c,-_ Cc?
!2G)
If S is appreciably greater than zero, then even if the small body is subject to other perturbative forces, it will remain stable.
4.
APPLICATIONS
I have applied the foregoing result separately to the Sun-Jupiter system, Earth-Moon system and binary systems. Since the orbital eccentricity of Jupiter is only 0.05, we can regard the Sun, Jupiter, an inner planet (or an asteroid or a short-period comet) to form the circular problem of three bodies. If we take the oblateness of the Sun to be a=5 ~10~~ and put, as an approximation, Ja= (Z/S)=, we shall find v~O.8~10-~~. For this system, u=0.954x10-3 and we find Ccr~ccr(o)= 3.0397. Inserting these values in (19) gives r(direct orbit)=0.7828= 3.0397 AU and r(retro orbit)=0.2480=1.290 AU. Of the inner planets, Mars is the furthest out at about 1.52 AU; using (17), this gives C,,=4.5022, hence ~=0.4811. This shows that Mars is in a very stable state which will not be destroyed even when the perturbations by Earth, Saturn or other planets are taken into
Hill Stability
271
account. Thus, all the four inner planets of the solar system are Hill stable. Over 952 of the asteroids are located at solar distances between 2.2 and 3.7 AU. Hence the great majority of the asteroids are Hill stable, but for some individuals, the degree of stability is low and, when the orbital eccentricity of Jupiter and the perturbationsby Saturn are taken into account, these could well be captured by Jupiter, or even escape beyond Jupiter. Asteroid No, 10 is a case in point. It has e=0.099, i=3:8, as3.151 AU, hence we can find C,, = 3.2076 and S=O.O552. While this object is basically Hill stable, its degree of stability is low, and its stability character could possibly change. Some of the short-period comets have distances from the Sun less than 3 or 4 AU, and again some may have a low degree of stability. For example, for P/Oterma, with emO.14, i=4', a=3.96 AU, we find Cao =3.0929 and S=O,O175 only. The orbital eccentricity of the Moon about the Earth is about 0.05, hence, the Earth, the Moon and an artificial satellite of the Earth approximately satisfy the conditions of the circular restricted problem of three bodies. Taking for the Earth, Jp=l.O8~10-~, then w=1*49x10-7. Also, for the Earth-Moon system, we have n=O.O1215, hence from the tables, cor = &f") =3.19178 * Hence, we obtain, for the limit of stability of an artificial. satellite, r(dir. orbit) =0.6144=236 171 km, and r(retro orbit)=0.2390=91 872 km. Take a circular orbit with radius 105 km as an example. If the orbit is direct, then ~~~~4.8474, and ~=0.5185, showing that it is highly stable. On the other band, if the orbit is retrograde, then Ca,=2.8198
For W UMa, we have ~=2r/n=0.33 d and e=O.O. Substituting these in (21) and using the approximate relation between w and 32, [9] gives v=6.47~10-~. Also for this system, we have n=1/3 and hence Co, f")14.1677 from the tables. Eqn. (18) then gives C,r=4.1870 and Eqn. (19) then gives r(dir. orbit)=0.2702=0.6754 Ro. Recall that the primary has a radius of 1.11 R,. We can therefore divide the primary into two portions:acondensed core within 0.68 Re of the centre and a surface layer of thickness 0.43 R,. Matter from the surface layer can be captured by the secondary, and some may even escape the system altogether. A similar division of the secondary component into two portions can be made. The two surface layers can exchange matter, but not the two condensed cores. Exchange of matter will alter the values of p and v and may increase the content of one component at the expense of the other, but the system will always remain a binary.
1. When discussing the limit of the Kill stability of a planet in the solar system, the effect of the oblateness of the Sun can be neglected altogether. When dealing with artificial satellites, the effect of the Earth's oblateness should be included for closer satellites. In close binaries, the effect of the oblateness should be taken into account in general, otherwise sensible errors will result. For example, if oblateness is neglected in the W UMa case, then r(dir. orbit)=0,7lR,, instead of 0,68R,. 2. The range of stability is smaller for a retrograde orbit than for a direct orbit of the same size. This is because the retrograde orbit has a much larger velocity in the rotating coordinates. If we wish to have a satellite that will remain in a stable orbit for a long time, then we should send it into a direct orbit rather than a retrograde orbit. 3. If we set v=O in (18) and (191, then, on using (12) and (13), we can express the stability limit r as a power series in p1/3 (22) r(dir. orbit) - 1-Z2.402~'fl+ 3.181~'3-11.17p+O(C""'> r(retro. orbit) - l/4(1-- 0.9618~1"-0.5556~>+O(p"> Accurate to u1i3, (22) is r(dir. orbit)=l-2.402
n 113, which is the range estimated by
(23)
272
ZHENG
Szebehely et al., [S]. We should point out that this approxi~tion may give large errors unless n is very small. It is only when u is small that r decreases with increasing n, 4. Set v=O in (16) and calculate the C-values at the two points A and B of Fig. 1 for a direct orbit. Their differences with Cmin are I,) Cn - Cd. - rl:(3 -I1 + I,. rr: (3 - fl). (1 - r,>
CB - C& -
Since r1 ~1, we have C* > C'minand CB > Cmin, showing that the Cao-values used by Szebehely et al. are not the minumum values and consequently their stability limits are not the smallest. The errors are the greater, the greater n is. 5. Generalization of the present result into three-dimensionsis possible under certain conditions. Also, it is in principle possible to consider the case where the lesser one of the two principal bodies has a finite oblateness, (the satellite case), and the ease where both components of a binary are oblate, which will be closer still to reality. In this latter case, all we need to do is to alter slightly n and s2, ACKNOWLEDGEMENT. I thank Colleague LIU Xue-fu for helpful discussions concerning binary systems.
REFERENCES [ll r21 [31 I41 151 ISI 171 rs1
Hill, 0.
W.,
dm. .7. Yofh.,
1 (X378),
129, 246.
Szebebely, V. and Mckenzie, R., dsfrm.
J., 82
Szebehely, V.,
C&r.
333.
Baebehhely. V.,
Natural
f&b&&, Poine&,
Mesh.,
18 (1978),
and Artificial
(1977),
Satellite Motion
V. and McKenzie, R., Celea. Tech., 23 (1980). If., Lea MQthodes Nouvfles
Szebehely, V. and Wilisme, Elzebehely, V.
(19791,
(19s1),
175.
3.
de fa Mdeanique C&&
C.. dafmtr. J., 69
Theory of Orbits
303.
(1339).
46%
(1967).
[91 ZHENGXue-tang, TONG Yi, Journal of Beijing i~ormalUniversity (Nat. Sci. Ser.) 4 (1981) Z.,Close binary Systems (1959). WI sopal,
ON
THE
STABILITY
CLOSE
BINARIES
ZHENG
Xue- tang
Received
OF
THE
Department
1981 November
Pergamon
INNER
of
PLANETS,
Press. Printed in Great Britain 0275-1062/82/030267-06$07.50/O
SATELLITES
Astronomy,
Beijing
AND
Normal
University
14
Abstract. I consider the range of Hill stability in the restricted circular problem of three bodies when the larger one of the two principal bodies has a finite oblateness. I show that the range P satisfies the equation I--P
r=
c..-3p+P”--Y!1--)r-‘T(I+3u)J[,_p)(*+~)r
’
where u is the mass parameter and v is an oblateness parameter. This result is It applied to the solar system, the Earth-Moon system and binary star systems. is then shown that, all the inner planets of the solar system, the great majority of asteroids and some short-period comets are Hill stable, that direct artificial satellites of the Earth are more stable than retrograde ones, and that contact binaries possess cores between which no mass exchange takes place.
1.
INTRODUCTION
In his study of the motion of the Moon, Hill (1878) gave one definition of stability in terms of the Jacobi constant in the restricted problem of three bodies, [l]. According to Hill, if the small body always moves inside the zero-velocity surface of one of the larger bodies, then its motion around that body is said to be stable. This kind of stability is now known as stability in the sense of Hill. Szebehely (1976), when discussing the Hill stability of the Moon in the Sun-Earth-Moon system, introduced the quantity s = (caCc~_)/c,, as a measure of this stability, [2]. Here Cat is the actual value of the Jacobi constant of the small body and Co, is the critical value corresponding to the Lagrangian point L2 for satellites and inner planets, or the Lagrangian point Lo for outer planets. When S>O, that is, when C,o>C,,, then the motion is stable; when .S
ZHENG
268
Of tbe two larger bodies, the lesser one is spherical and the greater one is oblate, 1. whose equator is inclined at a small angle to the mutual orbit, 2. The mutual orbit is approximately circular. 3. The small body movea in an approximate circular orbit in the orbital plane of the two larger bodies, which can be continued analytically into a Paincark's periodic orbit of the first kind, [6]. 2.
EQUATIONS OF MOTION AND POSITIONS OF THE LIBBATION POINTS
As a first approximation, the potential including the oblateness effect under the above assumptions is
v = JzR2f2 is a small positive parameter. rotating coordinates are
where
The equations of mation of the small body in
where (3) $ fCl--P)t:Cpt:1c a-(1 -+ +I* rl r: rt Here, uf< l/2) and 1-u are the masses of the tmo major bodies, rl and rs are the distances of tbo small body to them. Let the distance between the two major bodies be 1. Then P-
t:-(X-rY+P,
(4)
+:I fx - p + 1y i-yz.
n is the angular velocity of the rotating coordinate system. It is easy to show that d -
1 f
(51
3v.
Eqns, (2) have the Jacobi integral
(6)
c-29-V' where V is the velocity of the small body in the rotating system. The equations aQ.!ax=ara,Latj-0 give a set of particular solutions of (21, hence the positions of the libration points. Using (3) and (41, we have
Fig. 1
17) arid(8) give five particular solutions, the three collinear solutions LX, Zn, LS and the two triangular solutions LII~Ls. Referring to Fig. 1, the locations of the five libration points are
Hill Stability
XI -
c -
1 - 51, y, - 0
x3 -
P -
f -+
x3 -
r + ED
x, -
p -
+
-
Y +
O(v’h
y, _
x4 -
P -
+
-
v +
O(d),
y5 _
51,
y3 -
0
269
I
(9)
Y3- 0 I
JT J1;y+ O(lq
:ix+3$zv _t 1 I
O(a?)
(10)
l
2
Here, & = @' - $ vA(1 - $A)+- O(G)
(11)
~~-b~~)--~vn(l+fh)+o(~~~) &I
1 ?f,(O) f $ Y& + O(@)
i
In the above, superfix (*) refers CO the correspondingvalues in the usual restricted planar circular problem, whose well-known developments in terms of u are given in texts such as [7] and PI,
(13)
3.
CALCDLATION OF Cao AND STABILITY CONDITION FOR INNER PLANET TYPE SYSTEMS
To fix the region of stability of the small body, we first suppose it to move in a circular orbit around the larger principal body, as shown in Fig. 1. We shall later use the method of analytical continuation to generate from it a PoincarB's periodic orbit of the first kind. Inserting (3) in (6), we find that the Jacobi constant when the small body is at point P on the circular orbit is C -ti[(l- p>t:+ &&I+ =+(l where r2*=1+r,Z+2rlcose,
+ ')+ r:
*71
(14)
V'
8 being as shown in Fig. 1.
The velocity at P is
It is easy to show that the velocity V in the rotating system at any point P of this circular orbit depends only on r1 and not on 8, and V'- [nr,TdQ+(l
+ $1'
(15)
where the negative sign applies to direct orbits, the positive sign, to retrograde orbits. After substituting (15) into (14), we have c - +I(1 -p),:+pr:1+1--c l--Y +* ?I ( t:) r) --n’rt*2+
-4
+3r‘,
(16)
270
ZHRNG
Because r2 varies with e, the different points of the circular orbit give different C-values, This is because such an orbit is not a rigorous solution of the problem.considered. To ensure that the small body is Hill stable in the region derived, we must use the minimum value of c over the circular orbit. In (16), since rl does not depend on 8, dC/dB =0 gives I*=,-~/~. Since, here,
C assumes minimum values cmin at the two points
e - rfcos
-f:+;)
which are different for different system with different values of u and v. As mentioned, we should identify Cs, with Cmin, obtained by substituting r*=n -2/3 in (16), that is, C,-3p--pt:+
I--IL(l -$*(t+ rl
3$/(l
-&(I
t+
(17)
+ O(PY)
on using (5). As we are discussing the inner planet case, C,, should be identified with the value of C at the libration point Lz. From (9) we have ri(L~) =l-<~,ra(L~) =F,= and, of course, V=O at Lz. Substituting these in (14) and using (11) we have
(18) Here 52(o), hence Ccr "' ' can be easily looked up in Ref.[6] or calculated from (12). Thus, ccr is easily found for any given p and v. The critical distance r of r1 defining the region of stability is given by equating (17) and (18), that is, by l-
tC,-338+fW~+Y(1-~)r
-'+(2 + 3Y)J(l-&(l+
(19) 3y/?)r *
This equation for r can be solved by iteration. For starting values, we can take 1 for direct orbits and l/4 for retrograte orbits. Since p-5l/2, (17) shows that dCa,/dri ~0, hence when rl < r, Cac>Ccr. Thus, if the small body moves approximately in a circular orbit around the bigger primary, then as long as its distance from the latter is less than r it is Hill stable. For any known rr we can calculate Cao from (17), and hence the measure of Hill stability, s-c,-_ Cc?
!2G)
If S is appreciably greater than zero, then even if the small body is subject to other perturbative forces, it will remain stable.
4.
APPLICATIONS
I have applied the foregoing result separately to the Sun-Jupiter system, Earth-Moon system and binary systems. Since the orbital eccentricity of Jupiter is only 0.05, we can regard the Sun, Jupiter, an inner planet (or an asteroid or a short-period comet) to form the circular problem of three bodies. If we take the oblateness of the Sun to be a=5 ~10~~ and put, as an approximation, Ja= (Z/S)=, we shall find v~O.8~10-~~. For this system, u=0.954x10-3 and we find Ccr~ccr(o)= 3.0397. Inserting these values in (19) gives r(direct orbit)=0.7828= 3.0397 AU and r(retro orbit)=0.2480=1.290 AU. Of the inner planets, Mars is the furthest out at about 1.52 AU; using (17), this gives C,,=4.5022, hence ~=0.4811. This shows that Mars is in a very stable state which will not be destroyed even when the perturbations by Earth, Saturn or other planets are taken into
Hill Stability
271
account. Thus, all the four inner planets of the solar system are Hill stable. Over 952 of the asteroids are located at solar distances between 2.2 and 3.7 AU. Hence the great majority of the asteroids are Hill stable, but for some individuals, the degree of stability is low and, when the orbital eccentricity of Jupiter and the perturbationsby Saturn are taken into account, these could well be captured by Jupiter, or even escape beyond Jupiter. Asteroid No, 10 is a case in point. It has e=0.099, i=3:8, as3.151 AU, hence we can find C,, = 3.2076 and S=O.O552. While this object is basically Hill stable, its degree of stability is low, and its stability character could possibly change. Some of the short-period comets have distances from the Sun less than 3 or 4 AU, and again some may have a low degree of stability. For example, for P/Oterma, with emO.14, i=4', a=3.96 AU, we find Cao =3.0929 and S=O,O175 only. The orbital eccentricity of the Moon about the Earth is about 0.05, hence, the Earth, the Moon and an artificial satellite of the Earth approximately satisfy the conditions of the circular restricted problem of three bodies. Taking for the Earth, Jp=l.O8~10-~, then w=1*49x10-7. Also, for the Earth-Moon system, we have n=O.O1215, hence from the tables, cor = &f") =3.19178 * Hence, we obtain, for the limit of stability of an artificial. satellite, r(dir. orbit) =0.6144=236 171 km, and r(retro orbit)=0.2390=91 872 km. Take a circular orbit with radius 105 km as an example. If the orbit is direct, then ~~~~4.8474, and ~=0.5185, showing that it is highly stable. On the other band, if the orbit is retrograde, then Ca,=2.8198
For W UMa, we have ~=2r/n=0.33 d and e=O.O. Substituting these in (21) and using the approximate relation between w and 32, [9] gives v=6.47~10-~. Also for this system, we have n=1/3 and hence Co, f")14.1677 from the tables. Eqn. (18) then gives C,r=4.1870 and Eqn. (19) then gives r(dir. orbit)=0.2702=0.6754 Ro. Recall that the primary has a radius of 1.11 R,. We can therefore divide the primary into two portions:acondensed core within 0.68 Re of the centre and a surface layer of thickness 0.43 R,. Matter from the surface layer can be captured by the secondary, and some may even escape the system altogether. A similar division of the secondary component into two portions can be made. The two surface layers can exchange matter, but not the two condensed cores. Exchange of matter will alter the values of p and v and may increase the content of one component at the expense of the other, but the system will always remain a binary.
1. When discussing the limit of the Kill stability of a planet in the solar system, the effect of the oblateness of the Sun can be neglected altogether. When dealing with artificial satellites, the effect of the Earth's oblateness should be included for closer satellites. In close binaries, the effect of the oblateness should be taken into account in general, otherwise sensible errors will result. For example, if oblateness is neglected in the W UMa case, then r(dir. orbit)=0,7lR,, instead of 0,68R,. 2. The range of stability is smaller for a retrograde orbit than for a direct orbit of the same size. This is because the retrograde orbit has a much larger velocity in the rotating coordinates. If we wish to have a satellite that will remain in a stable orbit for a long time, then we should send it into a direct orbit rather than a retrograde orbit. 3. If we set v=O in (18) and (191, then, on using (12) and (13), we can express the stability limit r as a power series in p1/3 (22) r(dir. orbit) - 1-Z2.402~'fl+ 3.181~'3-11.17p+O(C""'> r(retro. orbit) - l/4(1-- 0.9618~1"-0.5556~>+O(p"> Accurate to u1i3, (22) is r(dir. orbit)=l-2.402
n 113, which is the range estimated by
(23)
272
ZHENG
Szebehely et al., [S]. We should point out that this approxi~tion may give large errors unless n is very small. It is only when u is small that r decreases with increasing n, 4. Set v=O in (16) and calculate the C-values at the two points A and B of Fig. 1 for a direct orbit. Their differences with Cmin are I,) Cn - Cd. - rl:(3 -I1 + I,. rr: (3 - fl). (1 - r,>
CB - C& -
Since r1 ~1, we have C* > C'minand CB > Cmin, showing that the Cao-values used by Szebehely et al. are not the minumum values and consequently their stability limits are not the smallest. The errors are the greater, the greater n is. 5. Generalization of the present result into three-dimensionsis possible under certain conditions. Also, it is in principle possible to consider the case where the lesser one of the two principal bodies has a finite oblateness, (the satellite case), and the ease where both components of a binary are oblate, which will be closer still to reality. In this latter case, all we need to do is to alter slightly n and s2, ACKNOWLEDGEMENT. I thank Colleague LIU Xue-fu for helpful discussions concerning binary systems.
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