1 : Preliminaries   6 :   Dynamics I 11 : Star Formation  16 : Cosmology
2 : Morphology   7 :   Ellipticals 12 : Interactions 17 : Structure Growth 
3 : Surveys 8 :   Dynamics II 13 : Groups & Clusters  18 : Galaxy Formation 
4 : Lum. Functions  9 :   Gas & Dust   14 : Nuclei & BHs 19 : Reionization & IGM  
5 : Spirals 10 : Populations    15 : AGNs & Quasars 20 : Dark Matter





(1) Introduction

We have, of course, already begun our study of Stellar Dynamics :
Topic 6 considered the highly restricted situation of nearly circular motion in cool galaxy disks.
Here we broaden the discussion considerably to consider motion within more general 3-D systems.
In large part, these notes follow (though simplify) the treatment in B&T.

(a) Gas/Fluid Physics and Stellar Dynamics

To set the stage, lets first compare stellar systems with atomic (or molecular) gases :

(b) A Path Through the Subject

There are a number of themes to cover, and chosing the right sequence isn't straightforward
Here is an outline to help navigate the upcoming (sometimes dense) material.


(2) Potential Theory

(a) Preliminaries

(b) Selected Examples of Density-Potential Pairs

Often, choosing a simple form for (r) [or (r)] yields a complex form for (r) [or (r)]
There are, however, a number of useful illustrative analytic (r) (r) pairs :      

(3) Orbit Classes



(4) Numerical N-Body Methods

Several methods are used :
See B&T-2 § 2.9 and Josh Barnes's nice writeup for more details : [link]


(5) The Virial Theorem

This fundamental result describes how the total energy (E) of a self-gravitating system is
shared between kinetic energy (K) and potential energy (W)
Specifically, we are interested in their ratio :   = K / |W|   (note K is always +ve, W always -ve)

We begin by looking at two illustrative cases and then deal with the general case.

(a) Simple Illustrations

(b) The General Case

The general case comprises an isolated system of self-gravitating masses   (see pdf)
Once again, we ask what is , the ratio of kinetic to potential energies

(c) Mass Determination

(d) Binding Energy : Energy Released During Collapse

(e) Stellar Systems Have Negative Specific Heat

(f) Rotational Flattening


(6) Describing Collisionless Systems

We first consider collisionless dynamics : (in § 8.10 we consider when and how star-star encounters are relevant)

(a) The Distribution Function (DF) : f(r, v, t)

(b) Collisionless Boltzmann (Vlasov) Equation (CBE)

(c) The Jeans Equation(s)

(d) Applications of the Jeans Equation

The Jeans equation, when combined with observations, has a number of applications :

Here we look briefly at the first and second :


(7) Steady State : The DF as   f(E, |L|, Lz)

Taking moments of the CBE lost almost all detailed information from the DF
Rather than working with the full DF, the Jeans equation works with just n, <v> and <v2>
Can we reintroduce the full DF and regain a more complete description of a system ?

The answer is yes, by introducing two new powerful constraints :
  demand that the system is in steady state   ( in equilibrium)
  demand that the DF generate the full potential   (not just act as a tracer population)

We consider these in turn

(a) Integrals of Motion and the Jeans Theorem

(b) Self-Consistency

(c) Spherical Isotropic Systems : DF = f(Er)

(d) Deriving f(Er) from (r) for Non-Rotating Spherical Systems

(e) From   f(Er)d3r d3v   to   N(Er)dE


(8) Model Building Using DFs

We begin with the simplest cases : equilibrium, non-rotating, spherical systems, ie DF f(Er)
With equations 8.28a,b now in hand, we are ready to construct specific models
The process goes as follows :

Here are some examples

(a) Polytropic Sphere:   Power Law f(Er)

(b) Isothermal Sphere:   Exponential f(Er)

(c) Lowered Isothermal (King): Truncated Exponential f(Er)

(d) Other Models


(9) Violent Relaxation


(10) Introducing Star-Star Encounters

So far, we have considered star motion in a perfectly smooth potential
However, in reality, individual stars render this potential bumpy on fine scales
How does this affect the motion of stars --- ie is the "collisionless" assumption valid ?

(a) Estimating Encounter and Relaxation Timescales

(b) Timescales for Real Stellar Systems

(c) Analytic Treatment : The Fokker-Planck Equation

(d) Results : The Effects of Encounters

There are a number of distinct phenomena which result from 2-body encounters :


(11) Further Topics

We defer a few topics of Stellar Dynamics to later sections :