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Incompressible df/dt=0 N star identical particles moving in a small bundle in phase space (Vol= Δ x Δ p ),

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

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Incompressible df/dt=0 N star identical particles moving in a small bundle in phase space (Vol= Δ x Δ p ),

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  1. 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 fluidf(E,L) =exp(-αE)Lβ • <Vt2>/ <Vr2> =2(1-β) • 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 <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 upto the total gravitational mass

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