Hill's Approximation in the Three-Body Problem

Transcription

1 Progress of Theoretical Physics Supplement No. 96, Chapter 15 Hill's Approximation in the Three-Body Problem Kiyoshi N AKAZA W A and Shigeru IDA* Department of Applied Physics, T.okyo Institute of Technology, Tokyo 152 *Geophysical Institute, University of Tokyo, Tokyo 113 (Received September 14, 1988) Hill's equation which describes approximately motions of interacting two bodies orbiting around the Sun is summarized, together with its properties. Hill's equation can be separated into equations of the relative and center of mass motions, and the center of mass motion can be integrated analytically. Further, it can be rewritten in a non-dimensional form scaled by the Hill radius and the heliocentric distance. These characteristics of Hill's equation reduce the degrees of freedom of particle motion and, hence, it is advantageous to use it in studies of gravitational scattering and collision between Keplerian particles, which will be described in later chapters. 1. Introduction Planetesimals which revolve around the protosun experience mutual gravitational scatterings and direct collisions through which they grow, finally, into planets. Mathematical descriptions of such processes (i.e., gravitational scatterings and mutual collisions), which are presented in the later chapters of this Part, have to be formulated, at least, as a three-body problem: the protosun and two interacting planetesimals. Unfortunately, solutions to the general three-body problem are rather complicated and, hence, it is very difficult to treat these problems analytically and, even numerically. However, noticing that masses of planetesimals (and also those of protoplanets) are much smaller than that of the protosun and that they revolve almost along circular orbits, we can make use of the so-called Hill's equation which was originally introduced for the study of the lunar orbit (Hill, 1878). As will be seen later in detail, Hill's equation has many advantages. Namely, (1) the relative and the center of mass motions can be separated and the equation for the motion of the center of mass can be integrated analytically, (2) the equation of relative motion can be scaled by the cube root of the sum of the masses; if we have a solution for the relative motion of planetesimals with certain masses, we can obtain orbits of planetesimals with arbitrary masses, and (3) there exists an energy integral (called the Jacobi integral) in a system described by Hill's equation. Because of the above properties of Hill's equation, we can considerably reduce the degrees of freedom of particle motion. Thus, it is very useful to study gravitational scatterings as well as mutual collisions

2 168 K. N akazawa and S. Ida in the framework of Hill's equation. Prior to the description of the physical processes in the solar gravitational field, we will summarize briefly Hill's equation and its properties for later use. Details have already been presented in the literature (e.g., Szebehely, 1967; Henon and Petit, 1986 ; Petit and Henon, 1987; N akazawa et al., 1988), which readers may refer to. 2. Hill's equation Suppose that a protoplanet with mass m1 and a planetesimal with mass mz (hereafter called particles 1 and 2) are revolving around the protosun. Let us first introduce rotating local Cartesian coordinates (x, y, z) called the Hill coordinates. They are l defined by x=iii-ao, y= ao(8-!jot), z=z, (15 2 1) where ao is a certain reference semimajor axis, (iii, 8, z) are cylindrical coordinates with the z-axis perpendicular to the ecliptic plane and with origin at the center of mass of the system. Further,!2o is the Keplerian angular velocity at iii=ao, given by (15 2 2) where Me is the solar mass. Now we will make the following assumptions, called Hill's approximation (Hill, 1878). The assumptions are and (j=1 and 2) (15 2 3) (15 2 4) where a prime denotes a derivative with respect to time t. In the above, it should be noticed that we need not require the condition IYil~ao. In terms of the heliocentric orbital elements introduced in a later section, the above conditions can also be written as ej, z i~1, lai- aol~ao, (15 2 5) where a;, ej, and ij are the instantaneous heliocentric semimajor axis, eccentricity, and inclination of the j-th particle, respectively. The above conditions (15 2 5) are usually satisfied in our problem. Expanding the equations of motion in a power series of the infinitesimals which appear in Eqs. (15 2 3) and (15 2 4) and retaining only the first order terms, we obtain Hill's equation which is given, for particle j, by

3 Chapter 15 Hill's Approximation in the Three-Body Problem 169 x - " 2.!2> y = ' 3.!2>2 x + -x -x 3J.I; ( ) j J J r3 z j, y/' + 2.!2Jx/ = 2 3J.I;( ) =-.!JJ Zi+r Z;-Zj (15 2 6) with (15 2 7) where r is the distance between two particles given by (15 2 8) In Eq. (15 2 6), the first and second terms on the right-hand side denote the solar tidal force (i.e., the sum of solar gravity and centrifugal force) and the mutual gravity force, respectively. The second terms on the left-hand side of Eq. (15 2 6) come from the Coirolis force. 3. Keplerian motion When the relative distance r is very large and the mutual interaction term can be neglected, Eq. (15 2 6) can be solved exactly (Hen on and Petit, 1986):! Xi(t)= bj-ejaocos(!&.jt- ri), Ylt)=~ ~ ~i(!&.jt- i)+2eiaosin(!&.jt-r;), zlt)=ziaosm(.!&jt-w;), (15 3 1) where the twelve parameters, bj, e;, ii, r.;, ;, and Wi are constants of integration. For the corresponding velocity components, we have! x/(t)=eiao!&.jsin(!&.jt-ri), y/(t)=- ~ bi+2ejaocos(!&.jt-r;)}.!&>, z/(t)= ijao.!&>cos(.!&jt- w;). (15 3 2) We can expect that the above solutions (15 3 1) and (15 3 2) coincide with those of Keplerian motion, because they are obtained under the assumption of J.lj=O. In fact, this can be shown as long as bi/ao, ei, and i; are sufficiently smaller than unity. Here, ao+ bi. ei, and ij are identified with the semimajor axis, eccentricity, and inclination of Keplerian orbit, respectively, and rj, tpi, and w; are the phase angles. It should be noted that the Keplerian motion given by Eq. (15 3 1) is composed of two motions: the motion of the guiding center (xg;, YG;, zgj and the epicyclic motion (xe;, YE;, zd. They are described, respectively, as

4 170 K. N akazawa and S. Ida and l XE;=- eiaocos (!Jot- rj, YE;=~eiao_sin(!Jot- ri), ZE;=zjaosm(!Jot-wi). (15 3 4) The motion of the guiding center comes from the Keplerian shear due to the difference between ao and aj. On the other hand, the epicyclic motion draws an ellipse with axes ei and 2ei in a period of the Keplerian time at ao. Even in the general case where mutual interaction cannot be neglected, it is often convenient to express particle motion in the same form as Eqs. (15 3 1) and (15 3 2). In this case, however, the six elements bj, ei, ii, are no longer constant but are functions of time: They are the instantaneous Keplerian orbital elements. 4. Separation of the relative motion from the center of mass motion As is well known, if particles experience a potential force linear to their position vectors and if the mutual interaction term is a function of only the relative distance between two particles, then their motions can be separated into two parts: the center of mass motion and the relative motion. For the general three-body problem, the potential force GM., rj/r 3 is nonlinear in the position vector and, hence, the separation is impossible. On the other hand, since in Hill's equation (15 2 6) the solar gravity is expressed in a linear form, the motions are separable. Now, we will introduce the relative and center of mass coordinates, which are defined as x=x1-x2 and X= m1x1 + m2x2 m1+m2 ' (15 4 1) respectively. Using the above coordinates, we can rewrite Eq. (15 2 6): for the motion of the center of mass, we have l X"-2 JoY'=3 Jo 2 X, Y" +2 JoX'=O, Z" =-!Jo 2 Z (15 4 2) and, for the relative motion

5 Chapter 15 Hill's Approximation in the Three-Body Problem 171 y" + 2.!J>x' = z" (15 4 3) where f.1. is given by _ m1+mz 3nz f.j.- M. ao ~~. (15 4 4) As seen from Eq. (15 4 2), the center of mass motion is purely Keplerian so that the motion can be analytically expressed in the same form as that of Eq. (15 3 1). On the other hand, the equations of relative motion (15 4 3) are not solvable but have the same form as those of an individual particle (15 2 6). Hence, even in this case, solutions to Eq. (15 4 3) can be expressed in the form of Eqs. (15 3 1) and (15 3 2) with bi> ej, replaced by b, e, which are the instantaneous orbital elements of the relative motion. From Eqs. (15 3 1), (15 3 2) and (15 4 1), it follows that the orbital elements of the relative motion are described in terms of those of individual particles: (15 4 5) 5. Non-dimensional form of Hill's equation As seen in the previous section, Hill's equation can be separated into relative and center of mass motions. This enables us to reduce the degrees of freedom of particle motions because we have a general solution for the center of mass motion. It is also known that Hill's equation has another remarkable advantage (N akazawa et al., 1988 ; Hayashi et al., 1977). That is, Hill's equation can be rewritten in a non-dimensional form independent of mass m1 + mz and the heliocentric distance ao, in which time is normalized by.!jj- 1 and distance by hao: t =t.!jj and r=(x, y, z)=(x, y, z)/aoh' (15 5 1) where h is the reduced Hill radius defined by h=( m1 + m2 ) 1 ' 3 3M., (15 5 2) By the above scaling, Hill's equation for the relative motion (15 4 3) can be expressed as x" -2y'=3x -3x/r 3, y" +2x'= -3y/r 3, z" =- z-3z/r 3, (15 5 3)

6 172 K. N akazawa and S. Ida where a prime denotes a derivative with respect to t. Solutions to the above equations are also given by the same form as before. That is, x ( i) = b- e cos( i- r),.y( i)=- ~ b( i - )+2esin( i- r), z( i)= z sin( i -w)' (15 5 4) x'( i)= esin( i- r),.y'( t - )= b +2 ecos( t- r), z'( i)= z cos( i -w), (15 5 5) where b, e, and z are the instantaneous Keplerian orbital elements of relative motion. These elements are related to those used earlier by b =b/aoh, e=e/h and z =i/h. (15 5 6) By the above normalization, the results of an orbital calculation obtained for a particular mass and heliocentric distance ao are applicable for arbitrary mass and distance. This also reduces the degrees of kinematic freedom. Sometimes, it is useful to introduce the new variables ~. TJ, and (Henon and Petit, 1986), the convenience of which was first pointed out by Hayashi (1980). They are defined by (15 5 7) Using the above variables, solutions given by Eqs. (15 5 4) and (15 5 5) can be re-described as x(t)= b -.;1COS f -.;2sin f, y(f)=- ~ b f + +2.;1sin f -2.;2COS f, z(f)=7}1sin f +1]2COS f, (15 5 8) and '=.;1sin f -,;2COS f, y'(t)=-z-b +2.;1cos t +2.;2sin t, z'(t)= 111 cos i -172sin i. (15 5 9) Note that Eqs. (15 5 8) and (15 5 9) are expressed as linear combinations of b and the new variables. These will be referred to in the next chapter.

7 Chapter 15 Hill's Approximation in the Three-Body Problem The Jacobi integral As is well known, there is an energy integral in a system described by the set of equations (15 5 3), which is called the Jacobi integral: (15 6 1) where U< - - -) x,y,z--2 x 2 z-r 2. (15 6 2) The zero velocity curves U(x, y, z)=e with z=o are shown in Fig.1, together with the potential U(x, 0, 0). The minima of the potential barrier are at (x, y, z) =(±1, 0, 0). These are the liberation points which correspond to the so-called Lagrange points L1 and L2 in the general three-body problem. The constant 9/2 in Eq. (15 6 2) is added so that U = 0 at the liberation points. The Jacobi integral (15 6 1) can be also expressed in terms of the instantaneous Keplerian elements: with the help of Eqs. (15 5 4) and (15 5 5), we have (15 6 3) Note that Eq. (15 6 3) contains none of the phase angles L1 L2 y 0 u X X (a) (b) Fig. L Zero velocity curves in a plane with z=o (a). Dotted regions represent those of E>O and attached numbers show the values of E. The central unfilled region denotes the Hill sphere of a protoplanet P within which mutual gravity overcomes that of the Sun. Further, L1 and L2 denote liberation points which correspond to the Lagrange points. The potential U at y = z = 0 is also illustrated in (b).

8 174 K. Nakazawa and S. Ida As seen from the above mathematical arguments of Hill's equation, it is advantageous to use Hill's framework. In Hill's equation the relative and center of mass motions can be separated, and the center of mass motion can be described analytically by a Keplerian orbit. This reduces the degrees of freedom of orbital motions. Further, we can see that Eq. (15 5 3) does not contain m1 and mz and, hence, the solutions are applicable to any pair of particles, that is, for arbitrary mass ratio. The above-mentioned characteristics of Hill's equation enable us to use a numerical approach for obtaining the collisional probability as well as the scattering rate between Keplerian particles. References Hayashi, C., 1980 : private communication. Hayashi, C., Nakazawa, K. and Adachi, I., 1977: Pub!. Astron. Soc. Jpn. 29, 163. Henon, M. and Petit, J. M., 1986: Celestial Mech. 38, 67. Hill, G. W., 1878: Am. ]. Math. 1, 5; 129 ; 245. Nakazawa, K., Ida, S. and Nakagawa, Y., 1988: submitted to Astron. and Astrophys. Petit, ]. M. and Henon, M., 1987: Astron. Astrophys. 173, 389. Szebehely, V., 1967: Theory of Orbit (Academic Press, New York), p. 16.