![Page 1: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/1.jpg)
Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling ( z = 1100 ) . These are the seeds that evolve (gravitational collapse)
to form the structured distribution of galaxies we see around us today:
voids, walls, filaments, clusters, galaxies, …
!
"#
#~"T
T~ 10
$5
![Page 2: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/2.jpg)
How did Galaxies Form ?
Initial Collapse: Hierarchical Merging: TWO COMPETING SCENARIOS
Elliptical Spiral Irregular
(rotation => slower collapse) (many small high-z galaxies)
(fewer and larger low-z galaxies)
![Page 3: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/3.jpg)
Did over-dense regions collapse directly to form galaxies ? or
Did small “building blocks” form first and then merge?
Both processes clearly occur.
Initial conditions important:
Mass and Angular Momentum conserved during collapse.
How did Galaxies Form ?
Elliptical Spiral Globular Irregular
Mass
Angular Momentum
![Page 4: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/4.jpg)
Galaxy Morphology • Hubble’s “Tuning Fork” classification scheme.
![Page 5: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/5.jpg)
Ellipticals Spirals Irregulars Globular Clusters
Galaxy formation is a topic of active research. ( We don’t yet have a complete understanding. )
low a.m. high a.m.
building blocks?
Extended Hubble Sequence
![Page 6: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/6.jpg)
Which ripples will collapse ? Gravity pulls matter in. Pressure pushes it back out. When pressure wins -> oscillations (sound waves). When gravity wins -> collapse. Cooling lowers pressure, triggers collapse. Applies to both Star Formation and Galaxy Formation.
ρ
Jeans’ Analysis of Gravitational Stability
!
L
![Page 7: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/7.jpg)
When does Gravity win?
• Gravitational Energy:
• Thermal Energy:
• Ratio:
• Jeans Length:
• Gravity wins when L > LJ .
!
EG~ "
G M M
L
!
ET~ N k T
N molecules of mass m in box of size L at temp T.
!
EG
ET
~G M
2
L N k T~G " L3( ) mL k T
=L
LJ
#
$ %
&
' (
2
!
LJ~
k T
G " m
#
$ %
&
' (
1/ 2
!
M = Nm
~ L3 "
End up with L2 on top. For units to balance
the bottom must have same units, call this LJ
2. To collapse top must be larger than
bottom.
![Page 8: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/8.jpg)
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. Jeans Length:
Large cool dense regions collapse.
!
LJ~
k T
G " m
#
$ %
&
' (
1/ 2
![Page 9: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/9.jpg)
Collapse Timescale Ignore Pressure. Time to collapse = free fall time, tG.
Gravitational acceleration:
Time to collapse:
Gravitational timescale, or dynamical timescale.
Note: denser regions collapse faster.
same collapse time for all sizes.
!
M ~ L3 "
!
tG ~L
g~
L3
G M~
1
G "
g ~GM
L2~L
tG2
![Page 10: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/10.jpg)
Oscillation Timescale Ignore Gravity.
Pressure waves travel at sound speed.
Sound crossing time:
Small hot regions oscillate more rapidly.
!
cS~
P
"
#
$ %
&
' (
1/ 2
~k T
m
#
$ %
&
' (
1/ 2
!
tS~L
cS
~ Lm
k T
"
# $
%
& '
1/ 2
Aside: before decoupling,
radiation pressure >> gas pressure
Ideal Gas
!
cS~1
3c
![Page 11: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/11.jpg)
Ratio of Timescales Collapse time: Sound crossing time:
Ratio of timescales:
Jeans length (again!)
!
cs~
k T
m
"
# $
%
& '
1/ 2
!
tS
=L
cS
!
tS
tG
~L G "
cS
~ LG " m
k T
#
$ %
&
' (
1/ 2
~L
LJ
!
tG
=1
G "
!
LJ~
cS
G "
![Page 12: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/12.jpg)
Size Matters !
size
times
cale
!
LJ
!
tG
=1
G "
!
tS
=L
cS
collapse
oscillate
![Page 13: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/13.jpg)
Jeans Mass and Length Jeans Length : (smallest size that collapses) Jeans Mass: (smallest mass that collapses) • Need cool dense regions to collapse stars, • But galaxy-mass regions can collapse sooner.
!
LJ~
k T
G " m
#
$ %
&
' (
1/ 2
!
MJ~ " L
J
3
~ "k T
G " m
#
$ %
&
' (
3 / 2
)T 3 / 2"*1/ 2
![Page 14: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/14.jpg)
Today:
Expanding Universe (matter dominated):
At decoupling:
!
" =10#28 3000
2.7
$
% &
'
( )
3
=1.4 *10#19 kg m-3
+ 2 Msun pc#3
!
T "R#1 $ "R#3 "T 3
!
T0
= 2.7 K
Conditions at Decoupling
!
"0 =10#28 kg m-3
!
T = 3000 K
![Page 15: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/15.jpg)
Size and Mass of first Galaxies
Jeans Length : Jeans Mass: More than a star, less than a galaxy,
close to a globular cluster mass.
!
T = 3000 K
" =1.4 #10$19 kg m-3 % 2 Msun pc-3
!
LJ
~k T
G " m
#
$ %
&
' (
1/ 2
=1.4 )10*23 J K*1( ) 3000K( )
6.7 )10*11m3 kg*1 s*2( ) 1.4 )10*19kg m*3( ) 1.7 )10*27kg( )
#
$ % %
&
' ( (
1/ 2
=1.6 )1018 m
3.2 )1016 m/pc= 50 pc
!
MJ
~ " LJ
3~ 2 Msun pc-3( ) 50 pc( )
3
= 3#105 Msun
![Page 16: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/16.jpg)
Globular clusters in the Milky Way
Hold the oldest stars. Orbit in the Halo.
![Page 17: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/17.jpg)
At decoupling:
Collapse timescale:
Expect first “galaxies” ( M > 3x105 Msun) to form ~107 yr after decoupling.
!
" =1.4 #10$19 kg m-3
!
tG
~1
G"= 3.3#1014 s = 107 yr
Time to form first galaxies
![Page 18: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/18.jpg)
!
T "R#1 $ "R#3 "T 3 R" t 2/3
Jeans mass : MJ"
T3
$
%
& '
(
) *
1/ 2
"R0
Jeans length : LJ"
T
$
%
& ' (
) *
1/ 2
"R+1 " t 2 / 3
Collapse time : tJ
=1
G $
%
& '
(
) *
1/ 2
"R+3 / 2 " t
Jeans Analysis: Scaling with Expansion factor
Collapse time scales with expansion time, so actual collapse takes longer.
From earlier lectures
![Page 19: Lecture 6: Jeans mass & length - ASTRONOMY GROUPstar-kdh1/ce/ce06.pdf · Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling](https://reader033.vdocument.in/reader033/viewer/2022042221/5ec7223ef4da2540802589ec/html5/thumbnails/19.jpg)
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).
Summary
We enter the “Dark Ages”