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.

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_0where

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

/ x | (1b) U(x) = - | dx' . f(x') , | / x_0may depend on the exact route taken from points

(2) f(x) = - grad U .Force fields obeying these conditions are

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

**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

**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 / dVwhere

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

- the acceleration inside the shell vanishes, and
- the acceleration outside the shell is
`- G m / r^2`.

/ r / r | | M(x) (6) Phi(r) = - | dx a(x) = G | dx ---- , | | x^2 / r_0 / r_0where the enclosed mass is

/ r | 2 (7) M(r) = 4 pi | dx x rho(x) . | / 0If

A **point** of mass `M`:

M (8) Phi(r) = - G - . rThis is known as a

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 > awhere

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_0In this potential, the circular velocity

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)^2The Plummer (1911) density profile has a finite-density core and falls off as

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.

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

/ (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) . /

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 2The Miyamoto & Nagai (1975) potential includes the Plummer model (if

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.

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^2Just as in the spherical case, the acceleration inside the shell vanishes; thus

x^2 y^2 z^2 (23) m^2 = ----- + --------- + --------- , 1+tau b^2 + tau c^2 + tauwhere

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

- Binney, J. & Tremaine, S. 1987,
*Galactic Dynamics*(BT87). - Jaffe, W. 1983,
*MNRAS***202**, 995. - Kuzmin, G. 1956,
*Astron. Zh.***33**, 27. - Hernquist, L. 1990,
*Ap J.***356**, 359. - Plummer, H.C. 1911,
*MNRAS***71**, 460. - Toomre, A. 1962,
*Ap. J.***138**, 385. - Miyamoto, M. & Nagai, R. 1975,
*PASJ***27**, 533. - Satoh, C. 1980,
*PASJ*,**32**, 41.

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

Last modified: February 5, 1997