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11 th Lec

11 th Lec. Phase Space. Collisionless Systems. We showed collisions or deflections are rare Collisionless : stellar motions under influence of mean gravitational potential! Rational : Gravity is a long-distance force, decreases as r -2

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11 th Lec

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  1. 11th Lec • Phase Space

  2. Collisionless Systems • We showed collisions or deflections are rare • Collisionless: stellar motions under influence of mean gravitational potential! • Rational: • Gravity is a long-distance force, decreases as r-2 • as opposed to the statistical mechanics of molecules in a box

  3. Collisionless Systems • stars move under influence of a smooth gravitational potential • determined by overall structure of system • Statistical treatment of motions • collisionless Boltzman equation • Jeans equations

  4. provide link between theoretical models (potentials) and observable quantities. • instead of following individual orbits • study motions as a function of position in system • Use CBE, Jeans eqs. to determine mass distributions and total masses

  5. Fluid approach:Phase Space Density PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v) (all called Distribution Function DF). The total number of particles per unit volume is given by:

  6. E.g., air particles with Gaussian velocity (rms velocity = σ in x,y,z directions): • The distribution function f(x,v) is defined by: mdN=f(x,v)d3xd3v where dN is the number of particles per unit volume with a given range of velocities.

  7. The total mass is then given by the integral of the mass distribution function over space and velocity volume: • Note:in spherical symmetry d3x=4πr2dr, • for isotropic systems d3v=4πv2dv • The total momentum is given by:

  8. Example:mean speed of air molecules in a box of dx3 : These are gamma functions

  9. Gamma Functions:

  10. How to calculate dx3 and dv3

  11. DF and its moments

  12. Additive: subcomponents add upto the total gravitational mass

  13. Full Notes online • http://www-star.st-and.ac.uk/~hz4/gravdyn/GraviDynFinal3.ppt GraviDynFinal3.pdf

  14. Liouvilles Theorem We previously introduced the concept of phase space density. The concept of phase space density is useful because it has the nice property that it is incompessible for collisionless systems. A COLLISIONLESS SYSTEM is one where there are no collisions. All the constituent particles move under the influence of the mean potential generated by all the other particles. INCOMPRESSIBLE means that the phase-space density doesn’t change with time.

  15. Consider Nstar identical particles moving in a small bundle through spacetime on neighbouring paths. If you measure the bundles volume in phase space (Vol=Δx Δ p) as a function of a parameter λ (e.g., time t) along the central path of the bundle. It can be shown that: It can be seen that the region of phase space occupied by the particle deforms but maintains its area. The same is true for y-py and z-pz. This is equivalent to saying that the phase space density f=Nstars/Vol is constant. df/dt=0! px px x x

  16. motions in phase-space • Flow of points in phase space corresponding to stars moving along their orbits. • phase space coords: • and the velocity of the flow is then: • where wdot is the 6-D vector related to w as the 3-D velocity vector v relates to x

  17. stars are conserved in this flow, with no encounters, stars do not jump from one point to another in phase space. • they drift slowly through phase space • In the COMBINED potential of stars and dark matter

  18. fluid analogy • regard stars as making up a fluid in phase space with a phase space density • assume that f is a smooth function, continuous and differentiable • good for N >105

  19. as in a fluid, we have a continuity equation • fluid in box of volume V, density r, and velocity v, the change in mass is then: • Used the divergence theorem

  20. continuity equation • must hold for any volume V, hence: • in same manner, density of stars in phase space obeys a continuity equation: If we integrate over a volume of phase space V, then 1st term is the rate of change of the stars in V, while 2nd term is the rate of outflow/inflow of stars from/into V. 0

  21. Collisionless Boltzmann Equation • Hence, we can simplify the continuity equation to the CBE: • Vector form

  22. in the event of stellar encounters, no longer collisionless • require additional terms to rhs of equation

  23. CBE cont. • can define a Lagrangian derivative • Lagrangian flows are where the coordinates travel along with the motions (flow) • hence x= x0 = constant for a given star • then we have: • and • rate of change of phase space density seen by observer travelling with star • the flow of stellar phase points through phase space is incompressible • f around the phase point of a given star remains the same

  24. incompressible flow • example of incompressible flow • idealised marathon race: each runner runs at constant speed • At start: the number density of runners is large, but they travel at wide variety of speeds • At finish: the number density is low, but at any given time the runners going past have nearly the same speed

  25. DF & Integrals of motion • If some quantity I(x,v) is conserved i.e. • Assume f(x,v) depends on (x,v) through the function I(x,v), so f=f(I(x,v)). • Such phase space density is incompressible, i.e

  26. Jeans theorem • For most stellar systems the DF depends on (x,v) through generally three 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

  27. 3D Analogy of 6D Phase space • If DF(x,v) is analogous to density(x,y,z), • Then DF(E,L,Lz) is ~ density(r,theta,phi), • Integrals analogous to spherical coordinates • E(x,v) analogous to r(x,y,z) • Isotropic DF(E) ~ spherical density(r) • Normalization dM=f(E)dx3dv3 ~ dM=density(r)dr3 • Have non-self-gravitating subcomponents: DF1+DF2, like rho1+rho2 to make up total gravity.

  28. 12th Lec • Phase Space

  29. Tensor Virial Theorem • Equation of motion: This is Tensor Virial Theorem

  30. E.g. • So the time averaged value of v2 is equal to the time averaged value of the circular velocity squared.

  31. Scalar Virial Theorem • the kinetic energy of a system with mass M is just where <v2> is the mean-squared speed of the system’s stars. • Hence the virial theorem states that Virial

  32. Stress Tensor • describes a pressure which is anisotropic • not the same in all directions • and we can refer to a “pressure supported” system • the tensor is symmetric. • can chose a set of orthogonal axes such that the tensor is diagonal • Velocity ellipsoid with semi-major axes given by

  33. Subcomponents in Spherical Equilibrium Potential • Described by spherical potential φ(r) • SPHERICAL subcomponent density ρ(r) depends on modulus of r. • EQUILIBRIUM:Properties do not evolve with time.

  34. In a spherical potential So <xy>=0 since the average value of xy will be zero. <vxvy>=0

  35. Spherical Isotropic f(E) Equilibrium Systems • ISOTROPIC:The distribution function f(E) only depends on the modulus of the velocity rather than the direction. Note:the tangential direction has  and  components

  36. Anisotropic DF f(E,L) in spherical potential. • Energy E is conserved as: • Angular Momentum Vector L is conserved as: • DF depends on Velocity Direction through L=r X v • Hence anisotropic

  37. e.g., f(E,L) is an incompressible fluid • The total energy of an orbit is given by: 0 for static potential, 0 for spherical potential So f(E,L) constant along orbit or flow

  38. spherical Jeans eq. of a tracer density rho(r) • Proof :

  39. Jeans eq. Proof cont.

  40. SELF GRAVITATING:The masses are kept together by their mutual gravity. • In non-self gravitating systems the density that creates the potential is not equal to the density of stars. e.g a black hole with stars orbiting about it is NOT self gravitating.

  41. 13th Lec • Phase Space

  42. Velocity dispersions of a subcomponent in spherical potential • For a spherically symmetric system we have • a non-rotating galaxy has • and the velocity ellipsoids are spheroids with their symmetry axes pointing towards the galactic centre • Define anisotropy

  43. Spherical mass profile from velocity dispersions. • Get M(r) or Vcir from: • RHS: observations of dispersion and b as a function of radius r for a stellar population.

  44. Isotropic Spherical system, β=0 • This is the isotropicJEANS EQUATION, relating the pressure gradient to the gravitational force. Note: 2=P Above Solution to Isotropic Jeans Eq: negative sign has gone since we reversed the limits.

  45. Hydrostatic equilibrium  Isotropic spherical Jeans equation • Conservation of momentum gives:

  46. Tutorial g M 2 vesc (E) (r) (r)

  47. Tutorial Question 3 • Question: Show dispersion sigma is constant in potential Phi=V02ln(r). What might be the reason that this model is called Singular Isothermal Sphere?

  49. • Since the circular velocity is independent of radius then so is the velocity dispersionIsothermal.

  50. Flattened Disks • Here the potential is of the form (R,z). • No longer spherically symmetric. • Now it is Axisymmetric

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