Orbits in Spherical Potentials

Consider the motion of a star in a spherically-symmetric potential, . The orbit of the star remains in a plane perpendicular to the angular momentum vector, and it's natural to adopt a polar coordinate system; call the coordinates and . The system has degrees of freedom, so the phase space has dimensions.

The equations of motion can be derived by starting with the Lagrangian ,

(1) ,

where and . In what follows, we will use a choice of unit mass and will not show in the equations:

(2) .

Differentiating with respect to and yields the momenta conjugate to and ,

(3) ,

(4) ,

where and are velocities in the radial and azimuthal directions. The Hamiltonian (energy) may now be expressed as a function of the coordinates and conjugate momenta:

(5) .

The equations of motions are

(6) ,

(7) ,

(8) ,

(9) .

Here because the conjugate coordinate phi does not appear in ; is called a cyclic coordinate.

The system has two integrals of motion. One, of course, is the total energy , numerically equal to the value of . The other is the angular momentum . These quantities are given by

(10) ,

(11) .

Each of these integrals of motion defines a hypersurface in phase space, and the orbit must remain in the intersection of these hypersurfaces. This can be visualized by ignoring the coordinate and drawing surfaces of constant and J in the three-dimensional space . Surfaces of constant are figures of revolution about the axis, while surfaces of constant are hyperbolas in the plane. The intersection of these surfaces is a closed curve, and an orbiting star travels around this curve.

For an orbit of a given , the system may be reduced to one degree of freedom by defining the effective potential,

(12) ;

the corresponding equations of motion are then just

(13) ,

(14) .

Because diverges as , the star is energetically prohibited from coming too close to the origin, and shuttles back and forth between turning points and .

In addition to its periodic radial motion described by , a star also executes a periodic azimuthal motion as it orbits the center of the potential. If the radial and azimuthal periods are incommensurate, as is usually the case, the resulting orbit never returns to its starting point in phase space; in coordinate space such an orbit is a rosette. The Keplerian potential is a very special case in which the radial and azimuthal periods of all bound orbits are equal. The only other potential in which all orbits are closed is the harmonic potential generated by a uniform sphere; here the radial period is half the azimuthal one and all bound orbits are ellipses centered on the bottom of the potential well. Thus in the Keplerian case all stars advance in azimuth by between successive pericenters, while in the harmonic case they advance by . Galaxies typically have mass distributions intermediate between these extreme cases, so most orbits in spherical galaxies are rosettes advancing by between pericenters.

Any system in which stellar collisions are rare may be idealized as a collection of point-sized bodies, each with mass , position , and velocity . The hamiltonian for such a system is

(15) ,

where depends on all body positions and velocities, the first sum runs over all bodies, the second runs over all pairs of bodies (twice, hence the factor of ), and is the gravitational constant. Then the equations of motion are

(16) ,

(17) ,

where the sum runs over all bodies except body .

The famous Aarseth code in Fortran (Aarseth Fortran code) calculates the general motion of the N-body system in computer simulations.

N-body systems obey several basic conservation laws. A symmetry of the Hamiltonian is a transformation which leaves the physical system unchanged. For example, translation in time, , is a symmetry of Eq. 15 because is not an explicit function of time; consequently the total system energy is conserved. Likewise, symmetry with respect to translation in space , implies conservation of total linear momentum, and symmetry with respect to rotation gives rise to conservation of total angular momentum.

Another general result shown by manipulating Eqs. 16,17 is the scalar virial theorem which states that for a system in equilibrium,

(18)

where and are the total kinetic and potential energy, respectively, and the angle-brackets indicate time-averages. Since , the time-averaged kinetic and potential energies are related to the conserved total energy by

(19) .

The total mass and total energy of an N-body system thus define characteristic velocity and length scales

(20) ,

(21) .

These are sometimes known as the virial velocity and radius, respectively.

The quantity is an estimate of the time a typical star takes to cross the system. This timescale may be expressed in several different ways; for example, in terms of the total mass and energy , it is

(22) .

Note that and are conserved, so is a constant even for systems which are far from dynamical equilibrium. In such cases approximates the time-scale over which the system evolves toward equilibrium.

Another expression for follows from the substitution valid for systems near equilibrium:

(23) .

Here the quantity , which has units of density, appears. In systems with galaxy-like density profiles, the virial radius is approximately proportional to the half-mass radius: . Using this relationship, it follows that

(23) ,

where is the mean density within . Since the crossing time is just supposed to indicate a typical time-scale for orbital motion, it is usual to drop the numerical constant, and define

(24) .