Axisymmetric Models

Astronomy 626: Spring 1995


Orbits in Axisymmetric Potentials

In describing axisymmetric galaxy models we may use cylindrical coordinates (R,phi,z), where R and phi are polar coordinates in the equatorial plane, and z is the coordinate perpendicular to that plane. If the mass distribution is axisymmetric, the potential is also axisymmetric:

(1)     Phi = Phi(R,z) .
The equations of motion in cylindrical coordinates are
         2                             2
        d R      dphi 2     dPhi      d z     dPhi
(2)     --- - R (----)  = - ---- ,    --- = - ---- ,
          2       dt         dR         2      dz
        dt                            dt
and
        d    2 dphi
(3)     -- (R  ----) = 0 ,
        dt      dt
where Eq. 3 follows because Phi does not depend on phi; thus the azimuthal force is always zero.

Classical integrals

Motion in a time-independent axisymmetric potential conserves two classical integrals of motion: the total energy and the z-component of the angular momentum

                  1  2             2 dphi
(4)     E = Phi + - v  ,    L_z = R  ---- .
                  2                   dt

The meridional plane

In terms of L_z, the equations of motion may be rewritten:

         2                   2
        d R     dPhi_e      d z     dPhi_e
(5)     --- = - ------ ,    --- = - ------ ,
          2       dR          2       dz
        dt                  dt
where Phi_e is the effective potential, given by
                                   2     2
(6)     Phi_e(R,z) = Phi(R,z) + L_z / 2 R  .
These equations describe the motion in the meridional plane, which rotates about the z-axis at an angular rate of L_z/R^2.

Numerical calculations show that most orbits in `plausible' axisymmetric potentials are not completely characterized by the energy and the z-component of angular momentum, implying that most orbits have a third integral! No general expressions for this non-classical integral are known, although for nearly spherical systems it is approximated by |L|, the magnitude of the total angular momentum.

Two-Integral Models

Despite the existence of third integrals in most axisymmetric potentials, it is reasonable to ask if models based on just two integrals can possibly describe real galaxies. In such models the distribution function has the form

(7)     f = f(E,L_z) .
One immediate result is that the distribution function depends on the R and z components of the velocity only through the combination v_R^2+v_z^2; thus in all two-integral models the velocity dispersions in the R and z directions must be equal:
       _______   _______  
(8)    v_R v_R = v_z v_z ,
Note that this equality does not hold for the Milky Way; thus our galaxy cannot be described by a two-integral model. For other galaxies, however, the situation is not so clear, and a two-integral model may suffice.

Distribution functions

Calculating rho(R,z) from f(E,L_z): Much as in the spherically symmetric case described last time, one may adopt a plausible guess for f(E,L_z), derive the corresponding density rho(R,Phi), and solve Poisson's equation for the gravitational potential. Perhaps the most interesting example of this approach is a series of scale-free models with r^-2 density profiles (Toomre 1982); however, these models are somewhat implausible in that the density vanishes along the minor axis.

Calculating f(E,L_z) from rho(R,z): Conversely, one may try to find a distribution function which generates a given rho(R,z). This problem is severely underconstrained because a star contributes equally to the total density regardless of its sense of motion about the z-axis; formally, if f(E,L_z) yields the desired rho(R,z), then so does f(E,L_z)+f_o(E,L_z), where f_o(E,L_z) = -f_o(E,-L_z) is any odd function of L_z. The odd part of the distribution function can be found from the kinematics since it determines the net streaming motion in the phi direction (BT87, Chapter 4.5.2(a)).

Even if kinematic data is available, this approach is not practical for modeling observed galaxies. The reason is that the transformation from density (and streaming velocity) to distribution function is unstable; small errors in the input data can produce huge variations in the results (e.g. Dejonghe 1986, BT87). A few two-integral distribution functions are known for analytic density distributions, and recent developments have removed some mathematical obstacles to the construction of more models (Hunter & Qian 1993).

An `unbelievably simple' and analytic distribution function exists for the mass distribution which generates the axisymmetric logarithmic potential (Evans 1993). This potential, introduced to describe the halos of galaxies (Binney 1981, BT87, Chapter 2.2.2), has the form

              1  2      2     2   2    2
(9)     Phi = - v_0 ln(R_c + R + z  / q ) ,
              2
where v_0 is the velocity scale, R_c is the core scale radius, and q is the flattening of the potential (the mass distribution is even flatter). The corresponding distribution function has the form
                      2       4 E            4 E            2 E
(10)    f(E,L_z) = A L_z exp(-----) + B exp(-----) + C exp(-----) ,
                             v_0^2          v_0^2          v_0^2
where A, B, and C are constants. Evans also divides this distribution function up into `luminous' and `dark' components to obtain models of luminous galaxies embedded in massive dark halos; his results illustrate a number of important points, including the non-gaussian line profiles which result when the luminous distribution function is anisotropic.

Jeans-equation models

Since we cannot (yet) construct distribution functions for real galaxies, consider the simpler problem of modeling observed systems using the Jeans equations. If we assume that the underlying distribution function depends only on E and L_z we can simplify the Jeans equations, since the radial and vertical dispersions must be everywhere equal; thus

        d     _______   nu  _______   ___________         dPhi
(11)    -- nu v_R v_R + -- (v_R v_R - v_phi v_phi) = - nu ---- ,
        dR              R                                  dR

        d     _______                                     dPhi
(12)    -- nu v_R v_R                              = - nu ---- .
        dz                                                 dz
At each R one can calculate the mean squared velocity in the R direction by integrating Eq. 12 inward from z = infinity; the mean squared velocity in the phi then follows from Eq. 11.

The Jeans equations do not tell how to divide up the azimuthal velocities into streaming and random components. One choice which is popular, although lacking a physical basis, is

         _____ 2      ___________   _______
(13)     v_phi   = k (v_phi v_phi - v_R v_R) ,
where k is a free parameter (Satoh 1980). The dispersion in the phi direction is then
              2      ___________   _____ 2
(14)     sigma_phi = v_phi v_phi - v_phi   .
Note that if k = 1 the velocity dispersion is isotropic and the excess azimuthal motion is entirely due to rotation, while for k < 1 the azimuthal dispersion exceeds the radial dispersion.

Application to elliptical galaxies

The Jeans equations have been used to construct models of a number of elliptical galaxies (Binney et al. 1990, van der Marel et al. 1990, van der Marel 1991). In practice, the procedure is to:

  1. Observe the surface brightness Sigma(x',y');
  2. Deproject to get the stellar density nu(R,z), assuming an inclination angle;
  3. Compute the potential Phi(R,z), assuming a constant mass-to-light ratio;
  4. Solve the Jeans equations for the mean squared velocities;
  5. Divide the azimuthal motion into streaming and random parts;
  6. Project the velocities back on to the plane of the sky to get the line-of-sight velocity and dispersion v_los(x',y') and sigma_los(x',y');
  7. Compare the predicted and observed kinematics.
The inclination angle, mass-to-light ratio, and rotation parameter k are unknowns to be determined by trial and error.

Some conclusions following from this exercise are that:

Alas, without an analytic expression for the third integral the machinery developed so far cannot be extended to model real galaxies in more detail. The best available methods are very similar to those used for triaxial systems, the subject of the next lecture.


References


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

Last modified: February 9, 1995