1. phase space deforms but maintains its area.
Incompressible df/dt=0
Nstar identical particles moving in a small bundle in phase space (Vol=Δx Δ p),
phase space deforms but maintains its area.
Likewise for y-py and z-pz.
Phase space density f=Nstars/Δx Δ p ~ const px px x x

2. Stars flow in phase-space
Flow of points in phase space ~
stars moving along their orbits.
phase space coords:

3. Collisionless Boltzmann Equation
Collisionless df/dt=0:
Vector form

4. Jeans theorem
For most stellar systems the DF depends on (x,v) through generally integrals of motion (conserved quantities),
Ii(x,v), i=1..3  f(x,v) = f(I1(x,v), I2(x,v), I3(x,v))
E.g., in Spherical Equilibrium, f is a function of energy E(x,v) and ang. mom. vector L(x,v).’s amplitude and z-component

5. DF & its 0th ,1st , 2nd moments

6. Example: rms speed of air molecules in a box of dx3 :

7. CBE  Moment/Jeans Equations
Phase space incompressible
df(w,t)/dt=0, where w=[x,v]: CBE
taking moments U=1, vj, vjvk by integrating over all possible velocities

8. 0th moment (continuity) eq.
define spatial density of stars n(x)
and the mean stellar velocity v(x)
then the zeroth moment equation becomes

9. 2nd moment Equation similar to the Euler equation for a fluid flow:
last term of RHS represents pressure force

10. Prove Tensor Virial Theorem (p212 of BT)
Many forms of Viral theorem, E.g.

11. Anisotropic Stress Tensor
describes a pressure which is
perhaps stronger in some directions than other
Star cluster, why not collapse into a BH?
random orbital angular momentum of stars!
the tensor is symmetric, can be diagonalized
velocity ellipsoid with semi-major axes given by

12. An anisotropic incompressible spherical fluid f(E,L) =exp(-αE)Lβ
<Vt2>/ <Vr2> =2(1-β)
Along the orbit or flow:
0 for static potential, for spherical potential
So f(E,L) constant along orbit or flow

13. Apply JE & PE to measure Dark Matter
A bright sub-component of observed density n(r) and velocity dispersions <Vr2> , <Vt2>
in spherical potential φ(r) from total (+dark) matter density ρ(r)

14. Spherical Isotropic f(E) Equilibrium Systems
ISOTROPIC:The distribution function f(E) only depends on |V| the modulus of the velocity, same in all velocity directions.
Note:the tangential direction has  and  components

15. Measure (Dark) Matter density r(r)
Substitute JE into PE, ASSUME isotropic velocity dispersion, get
all quantities on the LHS are, in principle, determinable from observations.

16. Non-SELF-GRAVITATING: There are additional gravitating matter
The matter density that creates the potential is NOT equal to the density of stars.
e.g., stars orbiting a black hole is non-self-gravitating.

17. Additive: subcomponents add up to the total gravitational mass