Incompressibledfdt0

1

• Incompressible df/dt0
• Nstar identical particles moving in a small
  bundle in phase space (Vol?x ? p),
• phase space deforms but maintains its area.
• Likewise for y-py and z-pz.
• Phase space density fNstars/?x ? 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/dt0
• 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), i1..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)/dt0, where wx,v CBE
• taking moments U1, 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
fluidf(E,L) exp(-aE)Lß

• ltVt2gt/ ltVr2gt 2(1-ß)
• Along the orbit or flow

0 for static potential, 0 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 ltVr2gt , ltVt2gt
• in spherical potential f(r) from total (dark)
  matter density ?(r)

14
Spherical Isotropic f(E) Equilibrium Systems

• ISOTROPICThe distribution function f(E) only
  depends on V the modulus of the velocity, same
  in all velocity directions.

Notethe tangential direction has ? and ?
components
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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 upto the total
gravitational mass