7. Gravitational Potentials

Astronomy 626: Spring 1997

Analytic calculation of the gravitational potential is easy for spherical systems, but few general results are available for non-spherical mass distributions. Mass models with tractable potentials may however be combined to approximate the potentials of real galaxies.


7.1 Conservative Force Fields

In a one-dimensional system it is always possible to define a potential energy corresponding to any given f(x); let

                     / x
                     |
(1a)        U(x) = - | dx' f(x') ,
                     |
                     / x_0
where x_0 is an arbitrary position at which U = 0. Different choices of x_0 produce potential energies differing by an additive constant; this constant has no influence on the dynamics of the system.

In a space of n > 1 dimensions the analogous path integral,

                     / x
                     |  
(1b)        U(x) = - | dx' . f(x') ,
                     |
                     / x_0
may depend on the exact route taken from points x_0 to x; if it does, a unique potential energy cannot be defined. One condition for this integral to be path-independent is that the integral of the force f(x) around all closed paths vanishes. An equivalent condition is that there is some function U(x) such that
(2)         f(x) = - grad U .
Force fields obeying these conditions are conservative. The gravitational field of a stationary point mass is the simplest example of a conservative field; the energy released in moving from radius r_1 to radius r_2 < r_1 is exactly equal to that consumed in moving back from r_2 to r_1.

7.2 General Results

In astrophysical applications it's natural to work with the path integral of the acceleration rather than the force; this integral is the potential energy per unit mass or gravitational potential, Phi(x), and the potential energy of a test mass m is just U = m Phi(x). For an arbitrary mass density rho(x), the potential is

                         /
                         |  3    rho(x')
(3)         Phi(x) = - G | d x' -------- ,
                         |      |x - x'|
                         /
where G is the gravitational constant and the integral is taken over all space.

Poisson's equation provides another way to express the relationship between density and potential:

(4)         div grad Phi = 4 pi G rho .
Note that this relationship is linear; if rho_1 generates Phi_1 and rho_2 generates Phi_2 then rho_1 + rho_2 generates Phi_1 + Phi_2.

Gauss's theorem relates the mass within some volume V to the gradient of the field on its surface dV:

                   /              /
                   |  3           |  2
(5)         4 pi G | d x rho(x) = | d S . (grad Phi) ,
                   |              |
                   / V            / dV
where d^2 S is an element of surface area with an outward-pointing normal vector.

7.3 Spherical Potentials

Consider a spherical shell of mass m; Newton's first and second theorems (BT87, Ch. 2.1) imply

  1. the acceleration inside the shell vanishes, and
  2. the acceleration outside the shell is - G m / r^2.
From these results, it follows that the potential of an arbitrary spherical mass distribution is
                       / r           / r
                       |             |    M(x)
(6)         Phi(r) = - | dx a(x) = G | dx ---- ,
                       |             |    x^2
                       / r_0         / r_0
where the enclosed mass is
                        / r
                        |     2
(7)         M(r) = 4 pi | dx x  rho(x) .
                        |
                        / 0
If M converges for large r, it's convenient to set r_0 = infinity; then Phi(r) < 0 for all finite r.

Elementary Examples

A point of mass M:

                         M
(8)         Phi(r) = - G - .
                         r
This is known as a Keplerian potential since orbits in this potential obey Kepler's three laws. The velocity of a circular orbit at radius r is v_c(r) = sqrt(M/r).

A uniform sphere of mass M and radius a:

                     { -2 pi G rho (a^2 - r^2 / 3) , r < a
(9)         Phi(r) = {
                     { - G M/r                     , r > a
where rho = M / (4 pi a^3 / 3) is the mass density. Outside the sphere the potential is Keplerian, while inside it has the form of a parabola; both the potential and its derivative are continuous at the surface of the sphere.

A singular isothermal sphere with density profile rho(r) = rho_0 (r/r_0)^2:

                                     2      r
(10)        Phi(r) = 4 pi G rho_0 r_0  ln( --- ) .
                                           r_0
In this potential, the circular velocity v_c = sqrt(4 pi G rho_0 r_0^2) is constant with radius. It's often used to approximate the potentials of galaxies with flat rotation curves, but some outer cut-off must be imposed for a finite total mass.

Potential-Density Pairs

Pairs of functions related by Poisson's equations provide convenient building-blocks for realistic galaxy models. Three such functions often used in the literature are listed here; all describe models characterized by a total mass M and a length scale a:

     Name                Phi(r)                   rho(r)
     =========      ===============     ==========================

                        - G M              3M                 -5/2
(11) Plummer        ---------------     -------- (1 + r^2/a^2)
                    sqrt(r^2 + a^2)     4 pi a^3

                    - G M                  M        a^4
(12) Hernquist      -----               -------- ---------
                    r + a               2 pi a^3 r (r+a)^3

                    G M      a             M         a^4
(13) Jaffe          --- ln( --- )       -------- -----------
                     a      r+a         4 pi a^3 r^2 (r+a)^2
The Plummer (1911) density profile has a finite-density core and falls off as r^-5 at large radii; this is a steeper fall-off than is generally seen in galaxies. Hernquist (1990) and Jaffe (1983) models, on the other hand, both decline like r^-4 at large radii; this power law has a sound theoretical basis in the mechanics of violent relaxation. The Hernquist model has a gentle power-law cusp at small radii, while the Jaffe model has a steeper cusp.

7.4 Axisymmetric Potentials

If the mass distribution is a function of two variables, the cylindrical radius R and the height z, the mathematical problem of calculating the potential becomes a good deal harder. A general expression exists for infinitely thin disks, but only special cases are known for systems with finite thickness.

Disks

An axisymmetric disk is described by a surface mass density Sigma(R). Potential-surface density expressions for several important cases are collected here:

     name                 Phi(R,z)                  Sigma(R)
     =========      =====================     ====================

                           - G M                      a M
(14) Kuzmin         ---------------------     --------------------
                    sqrt(R^2 + (a+|z|)^2)     2 pi (R^2+a^2)^(3/2)

                       d   (n-1)                 d   (n-1)
(15) Toomre         (-----)      Phi_K        (-----)      Sigma_K
                     d a^2                     d a^2

                                                k
(16) Bessel         - exp(-k|z|) J_0(kR)      ------ J_0(kR)
                                              2 pi G

The Kuzmin (1956) disk potential may be constructed by performing surgery on the field of a point of mass M (Eq. 8); let the z coordinate be the axis of the disk, and join the field for z > a > 0 directly to the field for z < -a. The result satisfies Laplace's equation, div grad Phi = 0, everywhere above and below the join, while on the surface where the two halves join together the gradient of the potential is discontinuous. Gauss's theorem (Eq. 5) may be used to evaluate the surface density which generates this discontinuity.

Toomre (1962) devised an infinite sequence of models, parameterized by the integer n. The n=1 Toomre model is identical to Kuzmin's disk, while the n=2 model is found by differentiating the potential and surface density of Kuzmin's disk by the parameter a^2; the linearity of Poisson's equation (Eq. 4) guarantees that the result is a valid potential-surface density pair. In the limit n -> infinity the surface density is a gaussian in R.

The `Bessel' pair, also due to Toomre (1962), describes another field which obeys Laplace's equation everywhere except on the disk plane. The absolute value of z gives this potential a discontinuity at z = 0. Unlike other cases, the surface density -- once again obtained using Gauss's theorem -- is not positive-definite. What good is it? A disk with arbitrary surface density Sigma(R) may be expanded as

                       /
(17)        Sigma(R) = | dk S(k) k J_0(kR)
                       /
where S(k) is the Hankel transform of Sigma(R),
                   /
(18)        S(k) = | dR Sigma(R) R J_0(kR) .
                   /
The potential generated by this disk is then
                                /
(19)        Phi(R,z) = - 2 pi G | dk exp(-k|z|) J_0(kR) S(k) .
                                /

Flattened Systems

Potential-density pairs for a few flattened systems are listed here:

     name                       Phi(R,z)                        rho(R,z)
     =========    ===================================     ==================

                              - G M                         { rho_Pl  if a=0
(20) Miyamoto-    -----------------------------------     = {
     Nagai        sqrt(R^2 + (a^2 + sqrt(b^2+z^2))^2)       { Sigma_K if b=0

                     d   n                                   d   n
(21) Satoh        (-----)  Phi_MN                         (-----)  Sigma_MN
                   d b^2                                   d b^n

                  1  2      2     2   2   2
(22) Log.         - v_0 ln(R_c + R + z / q )              rho_L
                  2
The Miyamoto & Nagai (1975) potential includes the Plummer model (if a=0) and the Kuzmin disk (if b=0); thus it can describe a wide range of shapes, from a spherical system to an infinitely thin disk. Satoh's (1980) models are derived from the Miyamoto-Nagai model by differentiating with respect to the parameter b^2, much as Toomre's models are derived from Kuzmin's.

The logarithmic potential, once again, is often used to describe galaxies with flattened rotation curves; the corresponding mass can be found by inserting this potential in Poisson's equation (see BT87, Eq. 2.54b). This mass distribution is `dimpled'; if q^2 < 1/2 the density must actually be negative along the z axis, which is unphysical. Roughly speaking, the mass distribution is about three times flatter than the potential it generates; this result is likely to hold true for most realistic galaxy models.

A great deal of analytic machinery exists to calculate potentials for systems with densities stratified on ellipsoidal surfaces (BT87, Ch. 2.3). Much of this machinery generalizes to triaxial systems; a few key results are given below.

7.5 Triaxial Potentials

The triaxial generalization of a thin spherical shell is a homoeoid, which has constant density between surfaces m^2 and m^2 + d m^2, where

                        y^2   z^2
(22)        m^2 = x^2 + --- + --- .
                        b^2   c^2
Just as in the spherical case, the acceleration inside the shell vanishes; thus Phi = const. inside the shell and on its surface. Further away the potential is stratified on surfaces defined by
                   x^2       y^2         z^2
(23)        m^2 = ----- + --------- + --------- ,
                  1+tau   b^2 + tau   c^2 + tau
where tau > 0 labels the surface. Notice that for tau -> 0 the isopotential surface coincides with the surface of the homoeoid, while in the limit tau -> infinity the isopotential surfaces become spherical.

The triaxial generalization of the logarithmic potential is

                         1  2       2     2    2   2    2   2
(24)        Phi(x,y,z) = - v_0 ln( R_c + x  + y / b  + z / c ) .
                         2


References


Joshua E. Barnes (barnes@galileo.ifa.hawaii.edu)

Last modified: February 5, 1997