Download Mass/Luminosity Dark Matter so far

Dark Matter
so far
Mass/Luminosity
• Local stellar luminosity function:
• Our Galaxy, at larger scales:
• Local motions  disk (Oort limit):
• MW Rotation curve
• Escape speed
• Pop II dynamics (glob. clusters, etc.)
• Flat rotation curves in other spirals
• E galaxy virial theorem
M/L = 0.67
M/L ~ 3-5
> 30
> 30
~ 27
> 20
9
Virial Theorem for Clusters
•
Galaxy clusters – “fair samples” of the universe.
•
•
Coma is closest relaxed cluster
Original mass measurement was by Zwicky (1933).
5R r
 3  1015
G
2
M virial 
•
Virial Theorem
2K = -U
[CO 2.4], and
pp. 959-962
Measure n(r) , v(r)
Determining membership
•
Fit to models based on
collisionless Boltzmann eq.
~ isothermal, non-spherical.
•
•
M = 2x1015 M
M/L = 360h (+0, -180h)
Perseus: M/L = 600h
Coma:
F. Zwicky
Radial Velocity 
n(r) = # of galaxies,
v(r) = vel. Dispersion
Radius (arcmin) 
1
X-ray emitting gas in clusters
Wien’s Displacement Law
maxT = .003 m K [CO eq. 3.15]
10-10m  3x107K
(x-ray)
Hydra A - Chandra
Hydra A - Optical
X-ray emitting gas in clusters
[CO fig. 27.17]
T ~ 107K gas is important mass component of cluster
• emission by thermal bremsstrahlung (free-free).
(5x1044 erg/s for Coma)
• LX ~ 1043 - 1045 erg/s
freq.
distr.
[CO eq. 27.18]
4
4
Ltotal  R 3   d  R 3 1.42 10  40 ne2T 1/ 2 W
3
3
[CO eq. 27.19]
Flux 
amplitude
Measure LX, R, T
Solve for ne = electron density (electrons m-3)
= H nuclei m-3
Mass = ne  mH  volume
• Mgas = (4/3) R3 nemH
• Mstars = (M/L)Local LV
Energy 
= h
= 1x1014 M
= 1.5x1013 M
10x more baryons in hot
intergalactic gas than in stars
But still factor of ~10 short…
Hydra A - Chandra
Hydra A - Optical
2
Gravitational Lensing
•
•
•
Foreground cluster distorts images of numerous background galaxies.
Use to determine total mass of foreground cluster.
Shows that 85% of mass is Dark Matter.
Gravitational Lensing
Robert Frost
3
Gravitational Lensing
The Schwarzschild metric:
For light: ds = 0
dr
 2GM 
 c1 

dt
rc 2 

[CO 17.28]
Wavefront is retarded near a massive object.
 path of light is bent.
Gravitational Lenses
1938+666
HST
radio
The “Einstein Cross”
Galaxy at center causes 4
images of same quasar.
4
Gravitational Lens Simulator
Blandford & Narayan 1986 ApJ, 310, 568
Source
Observer
Reflective, stretchy membrane
Lensing Mass
Gravitational Lensing by a Point Mass
[CO Sect. 28.4]
(28.20)
From Schw.
Metric.
(28.21)
The Quadratic Eqn.
(28.24)
5
Point mass
forms
two images
(or ring)

See Refsdal (1964) MNRAS 128, 295
Effect
of
Lensing
on
Flux
Source
Fn = no lensing
 = E
F /FN 
FT /FN
F1 /FN
Max amplification when  = 0
~ E/S
 /E F1 /FN
Lens
F2 /FN
FT /FN
Not
aligned
F2 /FN
 = E
 / E 
F = FN
Earth
Close
alignment
6
See Refsdal (1964) MNRAS 128, 295
Effect
of
Lensing
on
Flux
 /E F1 /FN
Max amplification when  = 0
~ E/S
F2 /FN
FT /FN
Not
aligned
 = E
F = FN
Close
alignment
Caustics & Catastrophes
7
Lensing by a Transparent Mass Distribution
Wavefront retarded by
gravitational field:
For a transparent
mass distribution
with Grav Pot = :

dr
 2GM 
 c 1 

dt
rc 2 

 
2
( ds ) 2  c dt 1  2 / c 2  1  2 / c 2
 dx
2
2
 dy 2  dz 2 
Rays:
Caustic Surfaces:
Number of images
changes by 2 each time
a caustic is crossed
always an odd
number
(if lens is transparent).
Wave fronts:
Caustic
Lensing by a Transparent Mass Distribution
Rays:
Caustic Surfaces:
Number of images
changes by 2 each time
a caustic is crossed
always an odd
number
(if lens is transparent).
Wave fronts:
Caustic
8
Conjugate Caustic Surfaces
Building blocks are
“elementary catastrophes”
Location of
source
relative to
conjugate
caustics
If source object is on one of these surfaces,
observer is on a caustic.
Resulting
images &
“critical
curves”
Transparent
elliptical lens
from Blandford & Narayan
Conjugate Caustic Surfaces
Building blocks are
“elementary catastrophes”
Location of
source
relative to
conjugate
caustics
Resulting
images &
“critical
curves”
If source object is on one of these surfaces,
observer is on a caustic.
Extended sources
from Blandford & Narayan
9
Conjugate Caustic Surfaces
radio
Building blocks are
“elementary catastrophes”
If source object is on one of these surfaces,
observer is on a caustic.
1938+666
The “Einstein Cross”
Location of
source
relative to
conjugate
caustics
HST
Resulting
images &
“critical
curves”
Extended sources
x
x
x
x
x
x
from Blandford & Narayan
 >> E
 < E
1938+666
HST
radio
The “Einstein Cross”
10