In a recent paper, Prochaska & Wolfe (1997) present a kinematic
model of damped Lyman-alpha absorbers as thick rotating disks of
individual clouds. Their model treats the random velocities of the
clouds and the thickness of the disk as independent parameters. The
goal of this project is to develop a simple *dynamical* model for
damped Lyman-alpha absorbers, in which cloud velocities consistent
with equilibrium are determined from the moment equations.

Prochaska & Wolfe's kinematic model for damped Lyman-alpha absorbers is a centrifugally supported thick disk of discrete clouds. The disk's spatial density profile is

-R/R_d -|z|/h (1) rho(R,z) = rho_0 e e ,where

To reproduce the complex profiles and velocity widths of metal
lines in damped Lyman-alpha systems, sight-lines through the disk must
intersect clouds with a wide range of velocities. Such sight-lines are
rare if the disk is thin; thus Prochaska & Wolfe favor thick disk
models with `h = 0.3 R_d`. But a thick disk must be supported
against collapse along the `z` direction. If individual clouds
are assumed to move ballistically then they need vertical velocities
large enough to reach distances of order `h` above the disk's
mid-plane. Thus the disk's vertical scale height and random velocity
dispersion can't be set independently; Prochaska & Wolfe's adopted
dispersion `sigma_cc = 10 km/s` seems too small to support the
thick disks they use to match the observations.

This project will develop a simple dynamical disk model of for damped Lyman-alpha absorbers and implement a Monte-Carlo program to generate metal-line absorption profiles for this model. A consistent relationship between disk thickness and velocity dispersion is the primary goal. Velocity moment equations offer a fairly direct way to approach to this requirement. To keep things simple, a thin-disk approximation will be used, and stretched to treat thicker disks.

The model to be developed is closely based on the one by Prochaska & Wolfe. Some key assumptions, adopted to simplify matters, are that

- the dynamics of the disk are adequately described by the thin-disk and epicyclic approximations,
- the circular velocity in the disk plane is constant and equal
to
`v_c`, - the disk is vertically self-gravitating, and
- the shape of the velocity ellipsoid is constant.

-R/R_d 2 z (2) rho(R,z) = rho_0 e sech (-----) ; 2 z_0note that for large

(3) f(v) = g(v_R, sigma_R) g(v_phi-v_d, sigma_phi) g(v_z, sigma_z) ,where

`sigma_R`, `sigma_phi`, `sigma_z`, and
`v_d` are functions of position, to be determined from velocity
moments of the Collisionless Boltzmann Equation. The steps are:

- determine
`sigma_z`from the moment equation (BT87, Eq. (4-29c)), or by solving the 1-D CBE (BT87, Prob. 4-25); - set
`sigma_R = mu sigma_z`, where`mu`is a constant specified by the user; - determine the ratio
`sigma_phi/sigma_R`from the moment equation (BT87, Eq. (4-52)); - determine
`v_d`from the asymmetric drift equation (BT87, Eq. (4-31)).

Given expressions for `sigma_R`, `sigma_phi`,
`sigma_z`, and `v_d` as functions of `R`
simulates the line profile due to a population of clouds distributed
according to Eqs. (2) and (3) along a given sight-line through the
disk. Let the sight-line be specified by the radius `R_s` at
which it crosses the disk mid-plane and a unit vector ` n`
which points in the viewing direction. The steps are:

- chose cloud positions along the sight-line by treating
`rho(R,z)`as a probability density; - for each cloud, chose velocity components by treating
`f(`as a probability density;**v**) - project each cloud's velocity
along**v**to determine its line-of-sight velocity;**n** - combine gaussian absorption profiles for individual clouds to obtain a curve showing total optical depth as a function of line-of-sight velocity.

The moment equation solutions and the Monte-Carlo implementation are largely independent of one another; thus two teams of three each can work on these aspects of the project simultaneously, combining their work at the end. If possible, each team should include someone taking the QSO absorption-line seminar. The moment equation team will need good analytical skills, while the Monte-Carlo team will require some programming expertise. However, the makeup of the teams is left to the class.

While the moment equation team is working on their part of the
problem, the Monte-Carlo team can design and test the simulation
program, using dummy functions to compute `sigma_R(R)`,
`sigma_phi(R)`, `sigma_z(R)`, and `v_d(R)`. When
the actual expressions for these functions are available, they can be
inserted in the simulation program.

Each team will turn in a written description of their work, organized as a section to be included in a paper. This description need not spell out every detail but should enable a reader to reconstruct what has been done. Also, the moment equation team should turn in the actual calculations leading up to their result, while the Monte-Carlo team should turn in a commented listing of the program. Finally, sample results for various choices of the input parameters should be provided to illustrate the operation of the program.

The due date for this project is the last day of exam period, May 16.

- Binney, J. & Tremaine, S. 1987,
*Galactic Dynamics*(BT87). - Prochaska, J.X. & Wolfe, A.M. 1997, astro-ph/9704169.

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

Last modified: April 25, 1997