Download Jeans Length

Lecture 6: Jeans mass & length
Anisotropies in the CMB temperature
 density ripples

T
5
~
~ 10

T
at the time of decoupling.
 seeds
These are the
that grow to form galaxies.
Two collapse scenarios:
Initial collapse
(top down)
Hierarchical merging
(bottom up)


q
q
Fragmentation


q

q

q
Merging
q
Jeans’ Analysis
of Gravitational Stability
Which ripples will collapse ?

Gravity pulls matter in.
L
Pressure pushes it back out.
When pressure wins -> stable oscillations (sound waves).
When gravity wins -> collapse.
 triggers collapse.
Cooling lowers pressure,
Applies to both Star Formation and Galaxy Formation.
When does Gravity win?
N molecules of mass m in box of size L at temp T.
• Gravitational Energy: EG ~  G M M
L
• Thermal Energy:
• Ratio:
ET ~ N k T
G  L  m  L 
EG  G M
~
~
  
ET L N k T
L kT
LJ 
3
2
 k T 
LJ ~ 

G  m 
1/ 2
• Jeans Length:

M  N m ~ L3 
• Gravity wins when
L > LJ .
2
Gravity tries to pull material in.
Pressure tries to push it out.
Gravity wins for
L > LJ
----> large regions collapse.
Pressure wins for
L < LJ
----> small regions oscillate.
 k T 
LJ ~ 

G  m 
1/ 2
Jeans Length:
Large cool dense regions collapse.
Collapse Timescale
Ignore Pressure.
Gravitational acceleration:
GM L
g~ 2 ~ 2
L
t
M ~ L3 
Time to collapse:

tG ~
L
~
g
L3
1
~
GM
G
Gravitational timescale, or dynamical timescale.
Note: denser regions collapse faster.

same collapse time for all sizes.
Oscillation Timescale
Ignore Gravity.
Pressure waves travel at sound speed.
P 1/ 2 k T 1/ 2
c S ~   ~  
 m 
 
Aside: before
decoupling,
Sound crossing time:
radiation pressure
>> gas pressure
 m 
L
tS ~
~ L  
cS
k T 
1/ 2
cS ~ 3 c
Small hot regions oscillate more rapidly.


Ratio of Timescales
Collapse time:
1
tG 
G
Sound crossing time:
1/ 2
L
k T 
tS 
c s ~  
cS
 m 
Ratio of timescales:
G  m 

tS L G
L
~
~ L 
 ~
tG
cS
LJ
 k T 
1/ 2
Jeans length (again!)
LJ ~
cS
G
Size Matters!
timescale
tS 
L
cS

oscillate
tG 
collapse

LJ
size
1
G
Jeans Mass and Length
Jeans Length : (smallest size that collapses)
 k T 
LJ ~ 

G  m 
1/ 2
Jeans Mass: (smallest mass that collapses)
 k T 

3
3 / 2 1/ 2
M J ~  LJ ~  
 T 
G  m 
3/2
• Need cool dense regions to collapse stars,
• But galaxy-mass regions can collapse sooner.

Conditions at Decoupling
Today:
 0  1028 kg m -3
T0  2.7 K
Expanding Universe:
1
3
3
T

R


R

T


At decoupling:

T  3000 K
3000 
19
-3

  1.4 10 kg m
 2.7 
3
  10

28
 2 M sun pc3
Size and Mass of first Galaxies
T  3000 K
  1.4 1019 kg m -3  2 M sun pc -3
Jeans Length :
1/ 2
23
1


 k T 
1.4 10 J K 3000K


LJ 
~ 
  
11 3
1 2
19
3
27

G  m  6.7 10 m kg s 1.4 10 kg m 1.7 10 kg

1/ 2
1.6 1018 m

 50 pc
16
3.2 10 m/pc
M J ~  LJ ~ 2 M sun pc -3 50 pc
Jeans Mass:
3
3
 3 10 5 M sun
More than a star, less than a galaxy,
close to a globular cluster mass.

Globular clusters in the Milky Way
Hold the oldest stars.
Orbit in the Halo.

Time to form first galaxies
At decoupling:
  1.4 1019 kg m -3
Collapse timescale:
 t ~
G
1
14
7
 3.3 10 s  10 yr
G
Expect first galaxies to form
~107 yr after decoupling.
Summary
Over-dense regions collapse after decoupling
IF large enough i.e. L > LJ
M > MJ
Large mass
--> Giant Elliptical
Smaller mass
--> Dwarf Galaxy
Smallest that collapse: globular clusters
Tiny regions stable: can’t form stars (yet).