The Orbits of Stars

3 The Orbits of Stars In this chapter we examine the orbits of individual stars in gravitational fields such as those found in stellar systems. Thus we ask the questions, “What kinds of orbits are possible in a spherically symmetric, or an axially symmetric potential? How are these orbits modified if we distort the potential into a bar-like form?” We shall obtain analytic results for the simpler potentials, and use these results to develop an intuitive understanding of how stars move in more general potentials. In 3.1 to 3.3 we examine orbits of growing complexity in force fields of decreasing§§ symmetry. The less symmetrical a potential is the less likely it is that we can obtain analytic results, so in 3.4 we review techniques for integrating orbits in both a given gravitational§ field, and the gravitational field of a system of orbiting masses. Even numerically integrated orbits in gravitational fields of low symmetry often display a high degree of regularity in their phase-space structures. In 3.5 we study this structure using analytic models, and develop analytic to§ols of considerable power, including the idea of adiabatic invariance, which we apply to some astronomical problems in 3.6. In 3.7 we develop Hamiltonian perturbation theory, and use it to study§ the phenomenon§ of orbital resonance and the role it plays in generat- ing orbital chaos. In 3.8 we draw on techniques developed throughout the chapter to understand§how elliptical galaxies are affected by the existence of central stellar cusps and massive black holes at their centers. All of the work in this chapter is based on a fundamental approximation: 3.1 Orbits in spherical potentials 143 although galaxies are composed of stars, we shall neglect the forces from individual stars and consider only the large-scale forces from the overall mass distribution, which is made up of thousands of millions of stars. In other words, we assume that the gravitational fields of galaxies are smooth, neglecting small-scale irregularities due to individual stars or larger objects like globular clusters or molecular clouds. As we saw in 1.2, the gravitational fields of galaxies are sufficiently smooth that these irregularities§ can affect the orbits of stars only after many crossing times. Since we are dealing only with gravitational forces, the trajectory of a star in a given field does not depend on its mass. Hence, we examine the dynamics of a particle of unit mass, and quantities such as momentum, angular momentum, and energy, and functions such as the Lagrangian and Hamiltonian, are normally written per unit mass. 3.1 Orbits in static spherical potentials We first consider orbits in a static, spherically symmetric gravitational field. Such fields are appropriate for globular clusters, which are usually nearly spherical, but, more important, the results we obtain provide an indispens- able guide to the behavior of orbits in more general fields. The motion of a star in a centrally directed gravitational field is greatly simplified by the familiar law of conservation of angular momentum (see Appendix D.1). Thus if r = reˆr (3.1) denotes the position vector of the star with respect to the center, and the radial acceleration is g = g(r)eˆr, (3.2) the equation of motion of the star is d2r = g(r)eˆ . (3.3) dt2 r If we remember that the cross product of any vector with itself is zero, we have d dr dr dr d2r r = + r = g(r)r eˆ = 0. (3.4) dt × dt dt × dt × dt2 × r ! " Equation (3.4) says that r r˙ is some constant vector, say L: × dr r = L. (3.5) × dt Of course, L is simply the angular momentum per unit mass, a vector perpendicular to the plane defined by the star’s instantaneous position and 144 Chapter 3: The Orbits of Stars velocity vectors. Since this vector is constant, we conclude that the star moves in a plane, the orbital plane. This finding greatly simplifies the determination of the star’s orbit, for now that we have established that the star moves in a plane, we may simply use plane polar coordinates (r, ψ) in which the center of attraction is at r = 0 and ψ is the azimuthal angle in the orbital plane. In terms of these coordinates, the Lagrangian per unit mass (Appendix D.3) is = 1 r˙2 + (rψ˙)2 Φ(r), (3.6) L 2 − where Φ is the gravitational poten# tial and$ g(r) = dΦ/dr. The equations of motion are − d ∂ ∂ dΦ 0 = L L = r¨ rψ˙2 + , (3.7a) dt ∂r˙ − ∂r − dr d ∂ ∂ d 0 = L L = r2ψ˙ . (3.7b) dt ∂ψ˙ − ∂ψ dt % & The second of these equations implies that r2ψ˙ = constant L. (3.8) ≡ It is not hard to show that L is actually the length of the vector r r˙, and hence that (3.8) is just a restatement of the conservation of angular× momentum. Geometrically, L is equal to twice the rate at which the radius vector sweeps out area. To proceed further we use equation (3.8) to replace time t by angle ψ as the independent variable in equation (3.7a). Since (3.8) implies d L d = , (3.9) dt r2 dψ equation (3.7a) becomes L2 d 1 dr L2 dΦ = . (3.10) r2 dψ r2 dψ − r3 − dr ! " This equation can be simplified by the substitution 1 u , (3.11a) ≡ r which puts (3.10) into the form d2u 1 dΦ + u = 1/u . (3.11b) dψ2 L2u2 dr % & 3.1 Orbits in spherical potentials 145 The solutions of this equation are of two types: along unbound orbits r and hence u 0, while on bound orbits r and u oscillate between finite→ limits.∞ Thus eac→h bound orbit is associated with a periodic solution of this equation. We give several analytic examples later in this section, but in general the solutions of equation (3.11b) must be obtained numerically. Some additional insight is gained by deriving a “radial energy” equation from equation (3.11b) in much the same way as we derive the conservation of kinetic plus potential energy in Appendix D; we multiply (3.11b) by du/dψ and integrate over ψ to obtain du 2 2Φ 2E + + u2 = constant , (3.12) dψ L2 ≡ L2 ! " where we have used the relation dΦ/dr = u2(dΦ/du). − This result can also be derived using Hamiltonians (Appendix D.4). From (3.6) we have that the momenta are p = ∂ /∂r˙ = r˙ and p = r L ψ ∂ /∂ψ˙ = r2ψ˙, so with equation (D.50) we find that the Hamiltonian per unitL mass is H(r, p , p ) = p r˙ + p ψ˙ r ψ r ψ − L p2 = 1 p2 + ψ + Φ(r) 2 r r2 (3.13) ' dr 2 ( dψ 2 = 1 + 1 r + Φ(r). 2 dt 2 dt ! " ! " When we multiply (3.12) by L2/2 and exploit (3.9), we find that the constant E in equation (3.12) is simply the numerical value of the Hamiltonian, which we refer to as the energy of that orbit. For bound orbits the equation du/dψ = 0 or, from equation (3.12) 2[Φ(1/u) E] u2 + − = 0 (3.14) L2 will normally have two roots u1 and u2 between which the star oscillates radially as it revolves in ψ (see Problem 3.7). Thus the orbit is confined 1 between an inner radius r1 = u1− , known as the pericenter distance, and 1 an outer radius r2 = u2− , called the apocenter distance. The pericenter and apocenter are equal for a circular orbit. When the apocenter is nearly equal to the pericenter, we say that the orbit has small eccentricity, while if the apocenter is much larger than the pericenter, the eccentricity is said to be near unity. The term “eccentricity” also has a mathematical definition, but only for Kepler orbits—see equation (3.25a). 146 Chapter 3: The Orbits of Stars Figure 3.1 A typical orbit in a spherical potential (the isochrone, eq. 2.47) forms a rosette. The radial period Tr is the time required for the star to travel from apocenter to pericenter and back. To determine Tr we use equation (3.8) to eliminate ψ˙ from equation (3.13). We find dr 2 L2 = 2(E Φ) , (3.15) dt − − r2 ! " which may be rewritten dr L2 = 2[E Φ(r)] 2 . (3.16) dt ±) − − r The two possible signs arise because the star moves alternately in and out. Comparing (3.16) with (3.14) we see that r˙ = 0 at the pericenter and apocenter distances r1 and r2, as of course it must. From equation (3.16) it follows that the radial period is r2 dr Tr = 2 . (3.17) 2 2 r 2[E Φ(r)] L /r * 1 − − In traveling from pericenter+to apocenter and back, the azimuthal angle ψ increases by an amount r2 dψ r2 L dt ∆ψ = 2 dr = 2 dr. (3.18a) dr r2 dr *r1 *r1 Substituting for dt/dr from (3.16) this becomes r2 dr ∆ψ = 2L . (3.18b) 2 2 2 r r 2[E Φ(r)] L /r * 1 − − + 3.1 Orbits in spherical potentials 147 The azimuthal period is 2π T = T ; (3.19) ψ ∆ψ r | | in other words, the mean angular speed of the particle is 2π/Tψ. In general ∆ψ/2π will not be a rational number. Hence the orbit will not be closed: a typical orbit resembles a rosette and eventually passes close to every point in the annulus between the circles of radii r1 and r2 (see Figure 3.1 and Problem 3.13).

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