Modeling Damped Lyman-alpha Absorbers

Astronomy 626: Spring 1997

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.

1. Introduction

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 rho_0 is the central cloud density, R_d is the disk's radial scale length, and h is the disk's vertical scale height. The disk is assumed to rotate with constant circular velocity v_rot; in addition, individual clouds are assigned isotropic random velocities with 1-D dispersion sigma_cc.

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.

2. Disk Model

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

  1. the dynamics of the disk are adequately described by the thin-disk and epicyclic approximations,
  2. the circular velocity in the disk plane is constant and equal to v_c,
  3. the disk is vertically self-gravitating, and
  4. the shape of the velocity ellipsoid is constant.
The disk's density profile is similar to Eq. (1), but the vertical density profile is given by the law for an isothermal self-gravitating slab (BT87, Prob. 4-25); thus
                              -R/R_d     2   z
(2)         rho(R,z) = rho_0 e       sech (-----) ;
                                           2 z_0
note that for large z, Eq. (2) falls off exponentially, with scale height z_0. Random velocities of individual clouds are drawn from an anisotropic distribution of the form
(3)         f(v) = g(v_R, sigma_R) g(v_phi-v_d, sigma_phi) g(v_z, sigma_z) ,
where g(x, sigma) is a gaussian with dispersion sigma and mean of zero, sigma_R, sigma_phi, and sigma_z are dispersions in the radial, azimuthal, and vertical directions, and v_d is the net rotation speed of the disk, which differs from v_c due to asymmetric drift.

3. Moment Equations

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:

  1. determine sigma_z from the moment equation (BT87, Eq. (4-29c)), or by solving the 1-D CBE (BT87, Prob. 4-25);
  2. set sigma_R = mu sigma_z, where mu is a constant specified by the user;
  3. determine the ratio sigma_phi/sigma_R from the moment equation (BT87, Eq. (4-52));
  4. determine v_d from the asymmetric drift equation (BT87, Eq. (4-31)).
For the sech^2(z/2z_0) vertical density profile adopted in Eq. (2), the vertical dispersion will be independent of z; thus sigma_z = sigma_z(R); since the velocity ellipsoid has the same shape everywhere, all three dispersions scale together (NB: step #3 above explicitly proves that sigma_phi/sigma_R is a constant). The rotation velocity is also a function of R only.

4. Monte-Carlo Simulation

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:

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

5. Strategy & Tactics

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.


Joshua E. Barnes (

Last modified: April 25, 1997