1

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

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)

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

Galaxy Morphology •  Hubble’s “Tuning Fork” classification scheme.

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

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

2

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.

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

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

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

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 "

Size Matters !

size

times

cale

!

LJ

!

tG

=1

G "

!

tS

=L

cS

collapse

oscillate

3

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

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

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

Globular clusters in the Milky Way

Hold the oldest stars. Orbit in the Halo.

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

!

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

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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”