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2(a). 2(b).

1(a). Milky Way & M31 : a roughly estimation, M31 is on a circular orbit, both MW and M31 are point mass objects, and period of the M31 orbit is 1~2 times of Hubble time (eg. 20Gyr). 1(b). Large Magellanic Cloud (the same idea as 1(a)), period of circular orbit is ~ Hubble time/5 ~ 3Gyr. 2(a).

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2(a). 2(b).

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  1. 1(a). Milky Way & M31 : a roughly estimation, M31 is on a circular orbit, both MW and M31 are point mass objects, and period of the M31 orbit is 1~2 times of Hubble time (eg. 20Gyr).

  2. 1(b). Large Magellanic Cloud (the same idea as 1(a)), period of circular orbit is ~ Hubble time/5 ~ 3Gyr

  3. 2(a). • 2(b). circular velocity is much smaller than observed 200km/s, which implies dark matter required

  4. 3(a). • 3(b). Observational errors. • Real deceleration not accounted: Kuiper belt or dark matter; dust, solar winds, cosmic rays; gas leaks; radiation pressure; electromagnetic forces;... • New physics: clock acceleration between coordinates or Ephemeris time and International Atomic time; MOND;... • http://en.wikipedia.org/wiki/Pioneer_anomaly

  5. 4.

  6. => (0,0,-1.5) at this point, g(r)=0

  7. 5.

  8. 6.

  9. 7. a single Fermion has Phase space density : 2 Fermions: maximum phase space density

  10. 8.

  11. 9.

  12. when r>>a,

  13. 10.

  14. 11. =>Gravity is continuous at r=r0 Beyond r0,

  15. inside r0, total mass: circular & escape velocities at r0,

  16. 12. Jeans eq. in spherical isotropic system The self-gravitating isothermal system, Poisson eq. is

  17. 12 Another singular isothermal sphere trace population asked to show isotropic sphere Jeans eq. plug in

  18. consider a population, static f(E)=f(v)=Gaussian of zero mean and σ = dispersion Gaussian proved =0

  19. Angular momentum: Energy change => Tidal radius

  20. change reduce by a factor of 2 if m->m/2 by a factor of 4 if

  21. p=mv, v//p, F=Fr, r//Fr => Angular momentum conserved if

  22. C9.4 isotropic Jeans Eq is: equilibrium, thus net velocity = 0,

  23. => proved same idea, to prove we can always choose orthogonal axis of coordinate when i != j, δij =0

  24. A toy galaxy first term of potential, the first term of potential predicts flat rotation curve, while the dark matter does the same. second of potential,

  25. gravity decreases fast when r>1kpc, and the stellar component, plays an important role at small radii but gravity decreases fast when radii > core radius v_cir =141km/s for an estimate of total mass, at R=10kpc, spherical simplify

  26. for stellar R=1kpc, z<0.1kpc, column density estimation , z^2 << R^2 Msun/kpc^3

  27. Black hole:

  28. Roche Lobe Here galaxies can be treated as point mass objects => g=GM/R^2 same idea, easily prove

  29. BH When giant is close to the supermassive black hole

  30. Link phase space quantities r φ(r) J(r,v) dθ/dt E(r,v) E=K+W=-K Vt K(v) vr AS4021 Gravitational Dynamics

  31. Link quantities in spheres Vcir2 (r) g(r) M(r) σr2(r) σt2(r) f(E,L) φ(r) ρ(r) vesc2(r) AS4021 Gravitational Dynamics

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