2(a). 2(b)
DESCRIPTION
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). - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/1.jpg)
•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).
![Page 2: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/2.jpg)
•1(b). Large Magellanic Cloud (the same idea as 1(a)), period of circular orbit is ~ Hubble time/5 ~ 3Gyr
![Page 3: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/3.jpg)
•2(a).
•2(b).
circular velocity is much smaller than observed 200km/s, which implies dark
matter required
![Page 4: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/4.jpg)
•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
![Page 5: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/5.jpg)
•4.
![Page 6: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/6.jpg)
=> (0,0,-1.5) at this point, g(r)=0
![Page 7: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/7.jpg)
5.
![Page 8: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/8.jpg)
6.
![Page 9: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/9.jpg)
![Page 10: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/10.jpg)
7.a single Fermion has Phase space density :
2 Fermions:
maximum phase space density
![Page 11: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/11.jpg)
8.
![Page 12: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/12.jpg)
9.
![Page 13: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/13.jpg)
when r>>a,
![Page 14: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/14.jpg)
10.
![Page 15: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/15.jpg)
![Page 16: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/16.jpg)
•11.
=>Gravity is continuous at r=r0
Beyond r0,
![Page 17: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/17.jpg)
inside r0,
total mass:
circular & escape velocities at r0,
![Page 18: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/18.jpg)
![Page 19: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/19.jpg)
•12. Jeans eq. in spherical isotropic system
The self-gravitating isothermal system, Poisson eq. is
![Page 20: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/20.jpg)
![Page 21: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/21.jpg)
Another singular isothermal sphere
trace population
asked to show
isotropic sphere Jeans eq.
plug in
12
![Page 22: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/22.jpg)
consider a population, static
f(E)=f(v)=Gaussian of zero mean and σ = dispersion
Gaussian
proved
=0
![Page 23: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/23.jpg)
Angular momentum:
Energy change
=>
Tidal radius
![Page 24: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/24.jpg)
change reduce by a factor of 2 if m->m/2
by a factor of 4 if
![Page 25: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/25.jpg)
![Page 26: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/26.jpg)
p=mv, v//p, F=Fr, r//Fr =>
Angular momentum conserved
if
![Page 27: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/27.jpg)
C9.4isotropic Jeans Eq is:
equilibrium, thus net velocity = 0,
![Page 28: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/28.jpg)
=> proved
same idea, to prove
we can always choose orthogonal axis of coordinate
when i != j, δij =0
![Page 29: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/29.jpg)
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,
![Page 30: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/30.jpg)
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
![Page 31: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/31.jpg)
for stellar
R=1kpc, z<0.1kpc, column density estimation ,
z^2 << R^2
Msun/kpc^3
![Page 32: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/32.jpg)
Black hole:
![Page 33: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/33.jpg)
![Page 34: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/34.jpg)
Roche Lobe
Here galaxies can be treated as point mass objects
=> g=GM/R^2
same idea, easily prove
![Page 35: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/35.jpg)
BH
When giant is close to the supermassive black hole
![Page 36: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/36.jpg)
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
![Page 37: 2(a). 2(b)](https://reader035.vdocuments.site/reader035/viewer/2022062719/5681323d550346895d98a8fa/html5/thumbnails/37.jpg)
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)