Jeans Length
Download
Report
Transcript 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:
2
3
2
G L m L
EG
GM
~
~
ET L N k T
L kT
LJ
• Jeans Length:
ET ~ N k T
M N m ~ L3
k T 1/ 2
LJ ~
G m
• Gravity wins when
L > LJ .
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
k T 1/ 2
c S ~ ~
m
Aside: before
decoupling,
Sound crossing time:
radiation pressure
>> gas pressure
1/ 2
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 1/ 2
LJ ~
G m
Jeans Mass: (smallest mass that collapses)
3/2
kT
3
3 / 2 1/ 2
M J ~ LJ ~
T
G m
• Need cool dense regions to collapse stars,
• But galaxy-mass regions can collapse sooner.
Conditions at Decoupling
Today:
0 1028 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 pc3
Size and Mass of first Galaxies
T 3000 K
1.4 1019 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 3 ~ 2 M sun pc -3 50 pc
Jeans Mass:
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 1019 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).