Another%20singular%20isothermal%20sphere

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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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