Download - 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).
•2(b).
circular velocity is much smaller than observed 200km/s, which implies dark
matter required
•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
•4.
=> (0,0,-1.5) at this point, g(r)=0
5.
6.
7.a single Fermion has Phase space density :
2 Fermions:
maximum phase space density
8.
9.
when r>>a,
10.
•11.
=>Gravity is continuous at r=r0
Beyond r0,
inside r0,
total mass:
circular & escape velocities at r0,
•12. Jeans eq. in spherical isotropic system
The self-gravitating isothermal system, Poisson eq. is
Another singular isothermal sphere
trace population
asked to show
isotropic sphere Jeans eq.
plug in
12
consider a population, static
f(E)=f(v)=Gaussian of zero mean and σ = dispersion
Gaussian
proved
=0
Angular momentum:
Energy change
=>
Tidal radius
change reduce by a factor of 2 if m->m/2
by a factor of 4 if
p=mv, v//p, F=Fr, r//Fr =>
Angular momentum conserved
if
C9.4isotropic Jeans Eq is:
equilibrium, thus net velocity = 0,
=> proved
same idea, to prove
we can always choose orthogonal axis of coordinate
when i != j, δij =0
first term of potential,
A toy galaxy
the first term of potential predicts flat rotation curve, while the dark matter does
the same.second of potential,
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/sfor an estimate of total mass, at
R=10kpc,
spherical simplify
for stellar
R=1kpc, z<0.1kpc, column density estimation ,
z^2 << R^2
Msun/kpc^3
Black hole:
Roche Lobe
Here galaxies can be treated as point mass objects
=> g=GM/R^2
same idea, easily prove
BH
When giant is close to the supermassive black hole
AS4021 Gravitational Dynamics
Link phase space quantities
r
J(r,v)
K(v)
φ(r)
Vt
E(r,v)dθ/dt
vr
E=K+W=-K
AS4021 Gravitational Dynamics
Link quantities in spheres
g(r)
φ(r) ρ(r)vesc
2(r)
M(r)Vcir2 (r)
σr2(r)
σt2(r)
f(E,L)