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Photogravitationally Restricted Three-Body Problem and Coplanar Libration Point

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1993 Chinese Phys. Lett. 10 61

(http://iopscience.iop.org/0256-307X/10/1/017)

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CHINESE PHYS. LETT. Vol.10, No. l ( 1993)61

Photogravitationally Restricted Three-Body Problem and Coplanar Libration Point*

ZHENG Xuetang, YU Lizhong Department of Applied Physics, East China Institute of Technology, Nanjing 210014

(Received 15 June 1992, revision received 21 September 1992)

The photogravitationally restricted three-,body problem in which the mass re- duction factors of two primaries q l , qz E (-co,l] is studied and the calculating formula of the location of coplanar libration points is given in this paper. As an application, the motion of cometary dust in the solar system and dust grain in binary AR Cas is also discussed.

PACS: 96.50.Gn, 97.80.-d

Stars (including the sun) exert not only gravitation, but also radiation pressure on body moving nearby. Since radiation pressure Fp has the same form as that of gravitational Fg, we could express the united action of the both with an equivalent force, called photogravitation. The photogravitation is written as F = qF,, where q = 1 - Fp/Fg is a constant, called mass reduction factor. On the basis of Stefan- Boltzmann’s law, we can easily get

A K P B K T ~ q = 1 - - or q = l - -

aPM apRD ’ where M , R , D , P and T are the mass, radius, density, luminosity and surface tem- perature of star; a and p are the radius and density of moving body; K is the radiation pressure efficiency factor of star; A = &, B = 3A0, and they both are constant, o is Stefan-Boltzmann’s constant. In the system of unit of C. G. S., A=2.9838~10- ’ and B=5.0772x lo-’. For the sun, mass reduction factor

(2) K

qo = 1 - 5.7396 X - . U P

For a general body such as planet, since a is rather big, q N 1; for a small body such as asteroid and satellite, 0 < q < 1; but for a dust grain, since a is very small, it is possible that q < 0. For example, the II kind of comet tail consists of dust grains with their radii being microns, and in this case, -1.2 < q < 0.5. When supernova is in oiitburst, a lot of energy is released in a short time, and then q is able to be smaller. So the value of q could be taken in ( - m , l ] . Since the small body, especially dust grain in the solar system and stellar system kenerally moves under photogravitation, the study of the motion of particle under photogravitation becomes a realistic and important work. Radzievskii et al. have studied photogravitationally restricted three- body problem and found the region in which there exist coplanar libration points. This paper further studies the photogravitationally restricted three-body problem with mass

‘Supported by the National Natural Science Foundation of China.

62 ZHENG Xuetang et al. VOl.10

reduction factors of two primaries q l , q2 E (-oo,l] and gives the calculating formula of the location of coplanar libration points.

Using the method stated in Ref.6 or 7, we can obtain the dimensionless canonical equation of the particle under photogravitation of two primaries in rotation system as

where px = $ - y, p , = $ +- 2, p , = % are generalized momenta; H = i ( p 2 t p i t P:)+YPX-ZP,- [ ’ (’-’) r , t y] is Hamiltonian function, p = ( M 2 5 MI) and M

A group of special solutions of Eq. (3) satisfies

From Eq.(5), we can determine the locations of libration points in photogravi- tationally restricted three-body problem. The libration point corresponding t o y = 0 and z = 0 is the collinear point. We have obtained a method which is able to estimate the number of collinear points and calculate their locations. The libration point cor- responding to y # 0 and t = 0 is the triangular point. Under the conditions of y # 0 and z = 0, we get a solution from the first and second equation of Eq. ( 5 ) :

1 1 or

rl = qlT , 7-2 = q2z (7) 1

From Eqs. (6) and (7) , it is ‘easily obtained that when q1 > 0, q2 > 0 and ql’ t q2’ 2 1, there will be two triangular points Lq and Ls existing in the photogravitational three- body problem. From Eq. (4) , we get their locations in rotating system, they are

When the signs of q1 are opposite t o that of q z , it is possible to get the solutions of y = 0 from the second equation of Eq. ( 5 ) . This indicates that in zz-plane, there exist

No.1 ZHENG Xuetang et al. 63

also libration points called the coplanar libration points. The first and third equations of Eq. ( 5 ) produce

and

From Eq. (9), we have

Since the signs of q1 and q2 are opposite, k is a positive constant. Using Eqs. (4), (10) and ( l l ) , we can get the locations of the coplanar libration points:

where 7-2 is the distance between the coplanar libration point and the smaller primary, and it satisfies

(1 - k 2 ) r , 5 + (2p - 1)r; - 2qzp = 0 . (13) For certain given factor p, q1 and q2, the distance r 2 could be calculated by the numer- ical method.

Since the signs of q1 and q2 should be opposite, we can generally take q1 < 0 and qz > 0. Equation (13) is fifth-order algebraic equation for 7-2 and when k < 1, the signs of the coefficients of Eq. (13) change only once. According to Descartes’s sign rule, we get one and only one positive root r 2 from Eq. (13). Substituting T Z into Eq. (12 ) , we get two coplanar libration points L6 and L7. If q1 > 0, q2 < 0 and k < 1, the signs of the coefficients of Eq. (13) change twice. Therefore it is possible that there are two positive roots ~2 and four coplanar libration points Lg, L7, L8 and Lg. They all are symmetrical to the x-axis. From Eq. ( l l ) , it could be obtained that the existance condition of L6 and L7 is the mass reduction factors of the two primaries have to satisfy -& q2 < q1 < 0. If the smaller primary is an absolute black body, that is qz = 1, the mass reduction factor of the bigger primary has to satisfy -6 < q1 < 0. In such a case, Eq.(13) becomes f ( ~ 2 ) = (1 - k2)r i t (2p - 1 ) ~ ; ’- 2p = 0. Since f ’ ( ~ 2 ) = 2 t 2(1 - k 2 ) r i > 0 and f (1 ) = - I C 2 < 0, the positive real root of f ( 7 - 2 ) = 0 satisfies 7-2 > 1. Therefore, from Eq.(12) we see that the locations of the coplanar libration points have to satisfy the condition of -p < x < 0. It must be pointed out that the coplanar libration point and triangular point will not exist together at the same time in photogravitationally restricted three-body problem. If the relative motion between two primaries is an elliptical orbit, similar results can be obtained by adopting pulsating coordinate system from Ref.8. For example, there is also the coplanar libration point in pulsating coordinate system.

In the following, we will apply the above results to the solar system and binary AR Cas and discuss the motion of cometary dust and dust grain.

64 ZHENG Xuetang et al. Vol.10

1. In the solar system, we have qz = 1 and p = 0.9538 x Besides, we take K = 1. Suppose the radius of some cometary dust is 0.5 x lO-*cm, and its density is 1.1474g/cm3, then Eq. (2) gives q = -0.4532 x From Eq. ( lo) , we have k = 0.7801. Substituting it into Eq. (12), we could calculate the distance between the coplanar libration point and Jupiter, and that is r2=1.597'2 or 8.3099 AU. Substituting them into Eq. ( l l ) , we get the location of L6 and L7: 2 = -0.2341 x or 1.2179 x

AU, and z = f1.2460 or f6 .4825 AU. Since q1 < 0, there is no triangular point. Besides, it is known that there is one collinear point at 2 3 = 1.0295 or 5.3563AU. So, in the motion of this cometary dust, there are only three libration points L3, L6 and L7 in all.

2. In binary AR Cas, the masses of two primaries are 1 1 . 9 M ~ and 3.0ME re- spectively, so p = 0.2013. The distance between them is 34.8 RE or 0.1619 AU. The absolute bolometric magnitude of the two primaries is -4.8" and 0.2" respectively, so their luminosities are PI = 0.6607 x 104P0 and Pl = 0.6607 x 102P@, where Po is the luminosity of the sun. Substituting them into Eq. (1) and taking K = 1, we get q1 = 1 - 3.1867 x & and qz = 1 - 1.2641 x l o v 3 5. Suppose the radius and density of sorne dust grain in binary AR Cas are a = 2 x cm and p = 1.4g/cm3, we have q1 = -0.1381 and qz = 0.9549. From Eqs,(10) and (12), we calculated that k = 0.8310 and r2=1.5129 or 0.2449AU. Substituting them into E q . ( l l ) , we get the locations of the coplanar libration points L6 and L7; z = -0.05551 or 0.8987 x AU, and z = f1.2487 or fO2022AU. Just the same, there is no triangular point, and there is one collinear point at z3 = 1.1911 or 0.1923AU. So, in such a case, there are only three libration points L3, L6 and L7 in all.

The authors are grateful t o Professor V. Szebehely for a helpful discussion.

References

[l] V. V. Radzievskii, Astron. Zh. 27 (1950) 250. f2] Y. A. Chernicov, Astron. J . 47 (1970) 217. [3] R. K. Choudhry, Celest. Mech. 16 (1977) 411. [4] L. G . Lukyanov, Astron. Zh. 61 (1984) 564. [5] V. Kumar and R. K. Choudhry, Celest. Mech. 48 (1990) 299. [6] V. Szebehely, Theory of Orbits (Academic Press, New York and London, 1967) p. 16. [7] Zheng Xuetang and Ni Caixia, Celestial Mechanics and Astrodynamics (Beijing Normal

[8] Zheng Xuetang, Chinese Journal of Space Science, 11 (1991)40 (in Chinese). Univ. Press, Beijing, 1989) p. 112 (in Chinese).