•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)