ASTRO504 Extragalactic Astronomy

13. Galaxy Formation Theory

ASTRO504Extragalactic Astronomy

Summary so far

• Before z=1100

• Baryon-photon tightly coupled

• DM perturbation grow logarithmically (RD) and linearly (MD) in a(t).

• After z=1100

• Baryons thermally coupled to CMB till z~150

• Baryons go into the gravitational potential (that DM made), form halos

• Halos (when spherical) collapse at ΔNL~200 [or 18π2!] (when δc=1.686).

From halos to galaxies

1010 1011 1012 1013 1014 1015 1016

Mass [M�]

10�8

10�7

10�6

10�5

10�4

10�3

10�2

10�1

100

Mas

sfu

nct

ion

dn(M

)/d

lnM

[Mpc�

3]

z=0

z=0.5

z=1

z=1.5

z=2

Sheth & Tormen (’99) mass function • How many 1015M halos at z<1?

• Relation between Mhalo to Mgalaxy?

• How many 1014M

galaxies must be there at z<1?

• How many observed?

Galaxy formation inside halos

• CDM(~5/6): no interaction. responsible for growth of structure

• Baryons(~1/6): interaction with photon, can radiate, cool down

I.C. = Virial shock and thermalizationBirnboim & Dekel (2003)

• line= Each shell of spherically collapsed gas in adiabatic case

• Inside = shock-heated gas

• Outside = newly collapsed shell gravitationally infalling

• Shock-heating continuously grow the shock radius, and it follows the viral radius.

• Shock thermalizes the kinetic energy

• At viral radius,

Accretion shock (adiabatic collapse)

Eshell,K =3

2NkBTsh =

1

2Mshellv

2in

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Tsh =1

3

Mshell

N

v2inkB

=µmp

3kBv2in

<latexit sha1_base64="8p3ZKEDAQowzjAFy1nv3Mv2mNbM=">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</latexit><latexit sha1_base64="8p3ZKEDAQowzjAFy1nv3Mv2mNbM=">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</latexit><latexit sha1_base64="8p3ZKEDAQowzjAFy1nv3Mv2mNbM=">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</latexit><latexit sha1_base64="8p3ZKEDAQowzjAFy1nv3Mv2mNbM=">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</latexit>

v2in = ⇣GM

r⌘ ⇣V 2

vir<latexit sha1_base64="5Uol8m87DkL034Mdg3ubrx+wr2g=">AAACJXicbVDLSgMxFM34rPVVdekmWARXZaYIulAouNCFQgX7gE4tmfROG8xkxiRTqMP8jBt/xY0Liwiu/BXTdhbaeiBwOOdcbu7xIs6Utu0va2FxaXllNbeWX9/Y3Nou7OzWVRhLCjUa8lA2PaKAMwE1zTSHZiSBBB6HhvdwMfYbA5CKheJODyNoB6QnmM8o0UbqFM4GncSVAWYivS/jc+w+gSbY9SWhyeVNmsgUu/AYs0Hm1KfxAZMm3ykU7ZI9AZ4nTkaKKEO1Uxi53ZDGAQhNOVGq5diRbidEakY5pHk3VhAR+kB60DJUkABUO5lcmeJDo3SxH0rzhMYT9fdEQgKlhoFnkgHRfTXrjcX/vFas/dN2wkQUaxB0usiPOdYhHleGu0wC1XxoCKGSmb9i2iemIW2KzZsSnNmT50m9XHLsknN7XKxcZ3Xk0D46QEfIQSeogq5QFdUQRc/oFb2jkfVivVkf1uc0umBlM3voD6zvHyvwpRY=</latexit><latexit sha1_base64="5Uol8m87DkL034Mdg3ubrx+wr2g=">AAACJXicbVDLSgMxFM34rPVVdekmWARXZaYIulAouNCFQgX7gE4tmfROG8xkxiRTqMP8jBt/xY0Liwiu/BXTdhbaeiBwOOdcbu7xIs6Utu0va2FxaXllNbeWX9/Y3Nou7OzWVRhLCjUa8lA2PaKAMwE1zTSHZiSBBB6HhvdwMfYbA5CKheJODyNoB6QnmM8o0UbqFM4GncSVAWYivS/jc+w+gSbY9SWhyeVNmsgUu/AYs0Hm1KfxAZMm3ykU7ZI9AZ4nTkaKKEO1Uxi53ZDGAQhNOVGq5diRbidEakY5pHk3VhAR+kB60DJUkABUO5lcmeJDo3SxH0rzhMYT9fdEQgKlhoFnkgHRfTXrjcX/vFas/dN2wkQUaxB0usiPOdYhHleGu0wC1XxoCKGSmb9i2iemIW2KzZsSnNmT50m9XHLsknN7XKxcZ3Xk0D46QEfIQSeogq5QFdUQRc/oFb2jkfVivVkf1uc0umBlM3voD6zvHyvwpRY=</latexit><latexit sha1_base64="5Uol8m87DkL034Mdg3ubrx+wr2g=">AAACJXicbVDLSgMxFM34rPVVdekmWARXZaYIulAouNCFQgX7gE4tmfROG8xkxiRTqMP8jBt/xY0Liwiu/BXTdhbaeiBwOOdcbu7xIs6Utu0va2FxaXllNbeWX9/Y3Nou7OzWVRhLCjUa8lA2PaKAMwE1zTSHZiSBBB6HhvdwMfYbA5CKheJODyNoB6QnmM8o0UbqFM4GncSVAWYivS/jc+w+gSbY9SWhyeVNmsgUu/AYs0Hm1KfxAZMm3ykU7ZI9AZ4nTkaKKEO1Uxi53ZDGAQhNOVGq5diRbidEakY5pHk3VhAR+kB60DJUkABUO5lcmeJDo3SxH0rzhMYT9fdEQgKlhoFnkgHRfTXrjcX/vFas/dN2wkQUaxB0usiPOdYhHleGu0wC1XxoCKGSmb9i2iemIW2KzZsSnNmT50m9XHLsknN7XKxcZ3Xk0D46QEfIQSeogq5QFdUQRc/oFb2jkfVivVkf1uc0umBlM3voD6zvHyvwpRY=</latexit><latexit sha1_base64="5Uol8m87DkL034Mdg3ubrx+wr2g=">AAACJXicbVDLSgMxFM34rPVVdekmWARXZaYIulAouNCFQgX7gE4tmfROG8xkxiRTqMP8jBt/xY0Liwiu/BXTdhbaeiBwOOdcbu7xIs6Utu0va2FxaXllNbeWX9/Y3Nou7OzWVRhLCjUa8lA2PaKAMwE1zTSHZiSBBB6HhvdwMfYbA5CKheJODyNoB6QnmM8o0UbqFM4GncSVAWYivS/jc+w+gSbY9SWhyeVNmsgUu/AYs0Hm1KfxAZMm3ykU7ZI9AZ4nTkaKKEO1Uxi53ZDGAQhNOVGq5diRbidEakY5pHk3VhAR+kB60DJUkABUO5lcmeJDo3SxH0rzhMYT9fdEQgKlhoFnkgHRfTXrjcX/vFas/dN2wkQUaxB0usiPOdYhHleGu0wC1XxoCKGSmb9i2iemIW2KzZsSnNmT50m9XHLsknN7XKxcZ3Xk0D46QEfIQSeogq5QFdUQRc/oFb2jkfVivVkf1uc0umBlM3voD6zvHyvwpRY=</latexit>

Virial temperature of gas• Combining all, we find [ζ and μ?]

• Or, in terms of the collapsed halos (Δ=200) [make sense?]

Of course, in reality, inside the viral radius, temperature gradient builds up to maintain HSE, so virial temperature is a representative temperature.

Tvir = 7.8⇥ 105 K

✓⇣

0.6

◆⇣ µ

0.59

⌘✓⌦mh2

0.145

◆1/3 ✓1 + z

4

◆✓M

1012M�

◆2/3

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Tvir = ⇣µmp

3kBV 2vir =

✓⇣

0.6

◆⇣ µ

0.59

⌘✓Vvir

100 km/s

◆2

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The 106K is too hot!• 106K~86.18eV!

• Jeans mass [make sense?]

• What are the implications?

MJ =4⇡

3⇢̄m�3

J ' 1.87⇥ 1014M�(1 +�)�1/2

✓⌦mh2

0.147

◆�1/2 ✓T

106K

◆3/2

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Radiative Cooling processes• Thermal Bremsstrahlung (free-free)

• Recombination (free-bound)

• Collisional ionization (bound-free)

• Collisional excitation + radiative de-excitation

e- + X+ → X+ + e- + 𝞬

e- + X+ → X + 𝞬

e- + X → X+ + 2e-

e- + X → e- + X`→ e- + X`+ 𝞬

Atomic cooling function (H and He)

• Defined as 𝝠(T,Z) = 𝓒/nH2

• Free-free wins at T>106K (∝T1/2)

• Excitation not important for T>105K . Why?

• Ionization weak at T~104K, Why?

• No cooling at lower temperature T<104K. Why?

Atomic cooling function with metalSutherland & Dopita (1993)

Metal ContributionsWiersma, Schaye & Smith (2009)

Cooling function with MoleculesMaio+(2007)

Metalicity

Cooling time scale• Cooling time = time it takes for a hot, virialized gas to radiate

away its virial temperature:ℇ = specific internal energy per unit mass

tcool ⌘⇢"

C =3nkBT

2⇤(T )n2H

=3.3⇥ 106yrs

✓T

109 K

◆⇣ n

10�3 cm�3

⌘�1✓

⇤(T )

10�23erg cm3/s

◆�1

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Worked example: 1012M at z=3

• Tvir~8×105 K, 𝝠(T)~0.5×10-23 erg cm3/s

• If fully ionized (X = hydrogen mass fraction ~ 0.75),

• Therefore, cooling time is about 109 years. Enough time for cooling! Is that enough to form a galaxy?

n =27

12nH =

27

12

⇢X

mp

' 5.5⇥ 10�3cm�3

✓1 + z

4

◆3

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tcool = 3.3⇥ 109yrs

✓T

106 K

◆⇣ n

10�3 cm�3

⌘�1✓

⇤(T )

10�23erg cm3/s

◆�1

<latexit sha1_base64="wwc2sLcOnnDKGId3Idbs49QEhPo=">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</latexit><latexit sha1_base64="wwc2sLcOnnDKGId3Idbs49QEhPo=">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</latexit><latexit sha1_base64="wwc2sLcOnnDKGId3Idbs49QEhPo=">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</latexit><latexit sha1_base64="wwc2sLcOnnDKGId3Idbs49QEhPo=">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</latexit>

Three time scales• Hubble time (tH) ~ age of the Universe

• Free-fall time (tff) ~ dynamical time to restore HSE (~ tH/10)

• Cooling time (tcool)

• Three cases [discuss!]

• tcool > tH :

• tH > tcool > tff :

• tff > tcool (Rees-Ostriker Criterion) :

Back to the example: 1012M at z=3

• Cooling time is about 109 years. Enough time for cooling!

• But, the free-fall time is (4×108 years) shorter than the cooling time, so the gas will be in the quasi-hydrostatic equilibrium state.

t↵ =

r3⇡

32G⇢'

s3⇡fgas

32Gnµmp' 2.1⇥ 109 yrsf�1/2

gas

⇣ n

10�3cm�3

⌘�1/2

<latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit><latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit><latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit><latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit>

Rees-Ostriker criterion• Cooling time is efficient when tcool < tff

• Therefore, cooling is more efficient at higher redshifts!

t↵ =

r3⇡

32G⇢'

s3⇡fgas

32Gnµmp' 2.1⇥ 109 yrsf�1/2

gas

⇣ n

10�3cm�3

⌘�1/2

<latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit><latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit><latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit><latexit sha1_base64="XQ5dtO2EobqO5LaK32vSV8LRi94=">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</latexit>

tcool = 3.3⇥ 109yrs

✓T

106 K

◆⇣ n

10�3 cm�3

⌘�1✓

⇤(T )

10�23erg cm3/s

◆�1

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Cooling and galaxy formation

Total gas mass

Density (Δ=200)

• Maximum mass of galaxy 1012M !

• Constant density, spherical distribution

• In hierarchical model, smaller Gs formed first.

• Over-cooling problem (White & Rees 1978)

Cooling radius• Relaxing the constant density assumption: ϱ=ϱ0(r/r0)-α

• Cooling is more efficient at the center than outskirt, so

• Cooling time shorter near the center, longer at larger radii.

• Cooling radius = tcool(r) = tage

• For isothermal case [ϱ=ϱ0(r/r0)-2], cooling radius increases in time, called “cooling wave”:

• rcool, Mcool ∝t1/2

Hot & Cold accretion• rcool < rshock : shock heated then cool

• rcool > rshock : “cold mode accretion” [unstable, why?]

• Isothermal sphere:

• rcool = 𝝠1/2t1/2 , rshock = vct

• Shock radius grows faster than cooling radius

Hot mode & Cold mode

• Halos with M<1010M /h always accrete in cold mode.

• Halos with M>1012M /h have a stable accretion shock close to the viral radius (hot mode).

Effect of photoionization• At lower redshifts, we do have UV radiation background that

can photo-ionize the gas.

• Two effects:

• Severely suppress the cooling rate of the low-density gasBecause photoionization reduces the number of neutral atoms and shut off the collisional excitation and collisional ionization

• It heats the gas

M/L ratio

• Lower mass (M<1012M ) :SN feedback is important!

• Higher mass (M>1012M ):AGN feedback is important!

Formation of disk galaxies• While the gas sink

toward the center, it is hard to loose the angular momentum (radiation is isotropic).

• Initial angular momentum from “tidal torque” (see figure).

• Spin parameter:

• Simulation find that distribution of λi = log-normal with mean=0.05, variance=0.5

• Exponential disk : λnow = 0.425, need (0.425/0.05)2~72-fold shrink without DMhalo, would take 43Gyr!

• With halo, only 7-fold, took 1.4 Gyr.

Need dark halo to explain spin!� =

J |E|1/2

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