latex version of disk setup post

Nicolas Strobel (
29 Aug 1994 10:16:35 -0500

SAR moderator: here's a latex version of my disk setup post. Some were wanting
to have a prettier version of the equations. A postscript version would take
up a lot of bandwidth, but interested people could latex this to get the
nice-looking postscript version if they want.


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I would like to set up a galactic disk made up of one isothermal component of
stars of a given scale height and integrate their motion in a background
galactic potential made up of a bulge, thin disk, and dark matter halo. The
stars will not interact with each other and the background potential has no
time dependence. The stars should start off with velocity dispersions such that
the scale height is independent of galactic radius. What I would like is for
the scale height $h_z$ of the stars to remain constant with time (and
independent of galactic radius).

What I've tried to set this up is an initial model with these features:
1) Have the stars start out in the plane (zero height).

2) Give each star a random vertical velocity drawn from a gaussian with a
dispersion $\sigma_z^2 = h_z^2 8 \pi G \rho$
where $\rho$ is the background mass density producing the background potential
($\rho = \rho_{disk} + \rho_{bulge} + \rho_{DM halo}$ ).

3) The ratio of vertical velocity dispersion to radial dispersion,
$\sigma_z / \sigma_R =$ some value between 0.5 and 0.78. The ratio is independent
of galactic radius. The radial velocities are random ones drawn from a gaussian
of the given dispersion.

4) The velocity dispersion in the direction of galactic rotation,
$\sigma_{\phi}$, is found from $\sigma_{\phi} = \sigma_R \kappa / (2 \Omega)$,
where $\kappa$ is the radial epicyclic frequency and $\Omega$ is the angular
rotation rate of the disk ($\kappa$ and $\Omega$ come from the rotation curve
of the background potential). The phi velocities are random ones drawn from a
gaussian of the given dispersion about a mean = circular velocity so that the
stars will start out at their guiding centers.

Unfortunately, this setup produces a disk after integrating for 1--2 Gyr that is
thinner in the inner part than the outer part. I want the thickness to be
independent of radius.

The background potential includes a disk with
$\rho_{disk}(R,z) \propto \exp{-R/h_r} sech^2 (z/z_0)$. This vertical profile
is for an isothermal disk and $z_0$ not necessarily equal to the scale height
$h_z$ of the stars.

Any ideas of what I'm doing wrong in this setup? Please email me any
suggestions. Thanks!