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 
(8.1a)
(8.1b) 
(8.2a) (8.2b) (8.2c) 
8.2b is Poisson's equation, for locations within the mass distribution
8.2c is Laplace's equation, for locations outside the mass distribution
(8.3a)
(8.3b) 
(8.4) 
Note that, with this definition, potential energy is always negative
(8.5) 
(8.6a)
(8.6b) 
(8.7) 
(8.8a) (8.8b) 
Here are two examples :
(8.9) 
where y = R / 2R_{d}, and I_{n} K_{n} are Bessel functions
or the 1st and 2nd kind
see [Topic 5.6a] for an analytic approximation and rotation curve.
(8.11a)
(8.11b)
(8.11c) 
We begin by looking at two illustrative cases and then deal with the general case.
(8.12) 
(8.13) 
where the five tensors are :
(8.14) (a,b,c,d,e) 
where _{i,j} arises from the expansion: <v_{i} v_{j}> = <v_{i}><v_{j}> + _{i,j}^{2}
(8.15a) 
the Kinetic and potential energies are related for each tensor element
^{ }
for example, they are related separately along each axis^{ }
(8.15b) 
(8.15c) 
So the total energy is negative : the system is bound !
its value is equal to either
Knowing R_{g} and measuring <v^{2}> allows us to
determine M, the system mass.
What to use for R_{g} isn't obvious for most stellar systems
with no clear "edge" or "size" ^{ }
However, we can make use of the median radius : R_{m} which
encloses half the mass
For many stellar systems, it turns out that R_{g} R_{m} / 0.4
(note R_{m} is written r_{h} in B&T)
We then have :
(8.16) 
which resembles the circular orbit relation: M = V^{2} R / G, but applies to a general selfgravitating system.
Here are diagrams to illustrate the situation : [image]
(8.17a) 
B&T1 fig 4.5 shows this relation for several ,
including projection corrections [ images ]
for isotropic velocities, = 0,
and we get, for small :
(8.17b) 
However, there are other constraints :
(8.18a) 
the net flow due to the velocity gradient is
(8.18b) 
the sum of these equals the net change to f in the region, ie at x, v_{x} of size dx dv_{x}
(8.18c) 
or, dividing by dx dv_{x} dt, we get
(8.19a) 
but since
(8.19b) 
we have
(8.19c) 
adding the y and z dimensions, which are independent, we finally have
 (8.19d) 
This is the collisionless Boltzmann equation (CBE)

(8.20) 
Clearly, the phase space density (f) along the star's orbit is constant
ie the flow is "incompressible" in phasespace
for example
If we take moments of the CBE, we transform it into equations in these
new variables.
Lets look in more detail at these first two moments in v (see B&T2 §4.8) :
(8.21) 
where n n(x,t) is the space density and
<v_{x}> is the mean drift velocity along x
This is a simple continuity equation for the number of stars along the
x axis.
(8.22a) 
where
_{x}^{2} is the velocity
dispersion about the mean velocity,
it arises from <v_{x}^{2}>
= <v_{x}>^{2} + _{x}^{2}
(8.22b) 
where the summation convention applies (sum over repeated indices)
here, i=1,2,3 and j=1,2,3 refer to x,y,z, eg x_{2}
y and v_{2}
v_{y}
(8.23) 
which is clearly analogous.
Here we look briefly at the first and second :
(8.24a) 
Introducing anisotropy parameters :
_{} =
1  _{}
^{2} / _{r}^{2}
and _{} =
1  _{}
^{2} / _{r}^{2}
and writing 2_{} for _{} + _{}
^{ } and V_{rot} for
<v_{} > this becomes
(8.24b) 
which is equivalent to the equation of hydrostatic support :_{ }
dp /dr + anisotropic correction ^{ } + centrifugal correction = F_{grav}
(8.24c) 
This parallels the equation for hydrostatic support of an ideal gas, where
p = nkT
the equivalences are :
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)

(8.25) 
Any function of integrals of motion f (I_{1}, I_{2}, I_{3}, ..... ) is also a solution of the steady state CBE 
(8.26a) ^{ } (8.26b) 
where f here is the mass DF (ie we've multiplied f by the mean stellar mass)
(8.27) 
This is now a fundamental equation describing spherical equilibrium systems.
Solutions not only have self consistent and f, but
f also satisfies the steady state CBE.
Such a solution now describes a selfconsistent, physically plausible stellar
dynamical system.
(8.28a)
(8.28b) 
These now describe a spherical, nonrotating, isotropic velocity dispersion
system.
They will be our starting point in constructing specific spherical models in
§ 8.8
(8.29) 
(8.30) 
where r_{m} = largest radius out to which a star with E_{r} can be found i.e. v=0 at (r_{m}) = E_{r}
Integrate f(E_{r}) over velocity to find the density in terms of (eq 8.26a) :
(8.31) 
after substituing v = (2_{})^{½}cos,
we find _{}(_{}) = c_{n
}^{n} (_{} > 0)
where c_{n} is a constant depending on n and^{ }F.
(8.32) 
(8.33) 
Recall, more +ve & E_{r} means more bound.
Also, note f(E_{r}) > 0 for E_{r} < 0: there are unbound stars! .... we anticipate problems at large radii.
OK, substituting  ½v^{2} for E_{r} and integrating f(E_{r}) over v gives = _{1} exp ( / ^{2})
(8.34) 
This is, in fact, the equation for a hydrostatic
sphere of isothermal gas, with ^{2} =
kT/m
Why is this^{ } ?
At every point, N(v) exp(½v^{2}/^{2}), for both
the stellar system and a gas of atoms
it is irrelevant, therefore, whether the stars are collisionless or not, they
mimic a gas of atoms.^{ }
This method is called "core fitting" or "King's method"
Typical values for ellipticals cores are 1020 h M_{}/ L_{}
suggesting minimal/no dark matter
To rectify this problem, we attempt to modify things slightly by removing the unbound stars:
(8.35) 
where _{o} is a (dispersion like) parameter.
(8.36) 
Solve this by integration, choosing boundary conditions at r = 0 :
If the initial distribution is hotter
less concentrated
If the initial distribution is rotating slowly less concentrated & rotating oblate figure
If the initial distribution is rotating faster even less concentrated & prolate/bar figure
If the initial distribution is ellipsoidal
rotating ellipsoid, anisotropic everywhere
(8.37a)
(8.37b) 
where = b_{max} / b_{min}
(8.38a)
(8.38b) 
where _{10} has units 10 km/s, m has
units of M_{}, and
_{3} has units 10^{3} M_{} / pc^{3}
Substituing, we get
(8.38c) 
System  N  R (pc)  V (km/s)  t_{cross}  t_{relax}  t_{age}  age/relax 
Open Cluster  10^{2}  2  0.5  10^{6}  10^{7}  10^{8}  10 
Globular Cluster  10^{5}  4  10  5 ×10^{5}  4 ×10^{8}  10^{10}  20 
Dwarf Galaxy  10^{9}  10^{3}  50  2 ×10^{7}  10^{14}  10^{10}  10^{4} 
Elliptical  10^{11}  10^{4.5}  250  10^{8}  4 ×10^{16}  10^{10}  10^{7} 
Spiral Disk  10^{11}  10^{4.5}  20  1.5 ×10^{9}  6 ×10^{17}  10^{10}  10^{8} 
MW Nucleus  10^{6}  1  150  10^{4}  10^{8}  10^{10}  100 
Luminous Nucleus  10^{8}  10  500  2 ×10^{4}  10^{10}  10^{10}  1 
(Galaxy Cluster)  10^{2}  5 ×10^{5}  500  10^{9}  (3 ×10^{9})  10^{10}  (3) 