1(a).%20Milky%20Way%20

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• 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 12
  times of Hubble time (eg. 20Gyr).

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

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• 2(a).
• 2(b).

circular velocity is much smaller than observed
200km/s, which implies dark matter required
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• 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

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• 4.

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gt (0,0,-1.5) at this point, g(r)0
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5.
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6.
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7.
a single Fermion has Phase space density
2 Fermions
maximum phase space density
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8.
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9.
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when rgtgta,
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10.
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• 11.

gtGravity is continuous at rr0 Beyond r0,
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inside r0,
total mass
circular escape velocities at r0,
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• 12. Jeans eq. in spherical isotropic system

The self-gravitating isothermal system, Poisson
eq. is
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Another singular isothermal sphere
trace population
asked to show
isotropic sphere Jeans eq.
plug in
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consider a population,
static
f(E)f(v)Gaussian of zero mean and s dispersion
Gaussian
proved
0
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Angular momentum
Energy change
gt
Tidal radius
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change reduce by a factor of 2 if m-gtm/2
by a factor of 4 if
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pmv, v//p, FFr, r//Fr gt
Angular momentum conserved
if
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C9.4
isotropic Jeans Eq is
equilibrium, thus net velocity 0,
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gt proved
same idea, to prove
we can always choose orthogonal axis of coordinate
when i ! j, dij 0
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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,
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gravity decreases fast when rgt1kpc, and the
stellar component, plays an important role at
small radii but gravity decreases fast when radii
gt core radius
v_cir 141km/s
for an estimate of total mass, at R10kpc,
spherical simplify
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for stellar
R1kpc, zlt0.1kpc, column density estimation ,
z2 ltlt R2
Msun/kpc3
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Black hole
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Roche Lobe
Here galaxies can be treated as point mass
objects
gt gGM/R2
same idea, easily prove
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BH
When giant is close to the supermassive black hole
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Link phase space quantities
r
f(r)
J(r,v)
d?/dt
E(r,v)
EKW-K
Vt
K(v)
vr

AS4021 Gravitational Dynamics

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Link quantities in spheres
Vcir2 (r)
g(r)
M(r)
sr2(r) st2(r)
f(E,L)
f(r)
?(r)
vesc2(r)

AS4021 Gravitational Dynamics