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 dtand

d 2 dphi (3) -- (R ----) = 0 , dt dtwhere Eq. 3 follows because

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

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 dtwhere

2 2 (6) Phi_e(R,z) = Phi(R,z) + L_z / 2 R .These equations describe the motion in the

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.

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

_______ _______ (8) v_R v_R = v_z v_z ,Note that this equality does

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 ) , 2where

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_0^2where

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 dzAt each

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

2 ___________ _____ 2 (14) sigma_phi = v_phi v_phi - v_phi .Note that if

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:

- Observe the surface brightness
`Sigma(x',y')`; - Deproject to get the stellar density
`nu(R,z)`, assuming an inclination angle; - Compute the potential
`Phi(R,z)`, assuming a constant mass-to-light ratio; - Solve the Jeans equations for the mean squared velocities;
- Divide the azimuthal motion into streaming and random parts;
- 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')`; - Compare the predicted and observed kinematics.

Some conclusions following from this exercise are that:

- Isotropic oblate rotators (
`k = 1`)*do not fit*; - Some galaxies (
*e.g.*NGC 1052)*are*well-fit by two-integral Jeans equation models; - The models predict major-axis velocity dispersions in excess of those observed in most galaxies;
- Consequently, most of the galaxies must have (and `use') a
third integral, or are in fact
*triaxial*.

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.

- Binney, J.J. 1981,
*M.N.R.A.S.***196**, 455. - Binney. J.J., Davies R.L., & Illingworth, G.D. 1990,
*Ap.J.***361**, 78. - Dejonghe, H. 1986,
*Phys. Rep.***133**, 218. - Evans, N.W. 1993,
*M.N.R.A.S.***260**, 191. - Hunter, C. & Qian, E. 1993,
*M.N.R.A.S.***262**, 401. - Satoh, C. 1980,
*P.A.S.J.***32**, 41. - Toomre, A. 1982,
*Ap.J.***259**, 535. - van der Marel, R. 1991,
*M.N.R.A.S.***253**, 710. - van der Marel, R., Binney, J.J., & Davies, R.L. 1990,
*M.N.R.A.S.***245**, 582.

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

Last modified: February 9, 1995