the local group and galactic evolution

The Local Group and Galactic Evolution

I The Local Group

I Satellite Galaxies

I Cepheid Variables

I Tides and the Roche Limit

I Local Spirals

I Chemical Evolution

I Dwarf Galaxies

I Future of the Local Group

J.M. Lattimer AST 346, Galaxies, Part 5

The Local Group


The Large Magellanic Cloud

HI gas Hα

S Dor

opticalIR - 24µm

⇐= 10 or 8.5 kpc =⇒ ⇐= 7 or 6.0 kpc =⇒

LargeMagellanicCloud


Bulge Large Magellanic Cloud


Magellanic Clouds


Cepheid Variables

MV = −3.525 log(P/d) + 2.88(V − Ic )0−2.80− 1.05AV


Pulsational Frequency

The Euler equation of motion for a surface layer of mass m and radius R

md2R

dt2= −GMm

R2− 1

ρ

∂p

∂r.

Use V = 1/ρ and −Vdp = pdV − pV = pdV = 4πR2dr (p = 0 at thetop of the layer (surface) and V = 0 at the bottom). In equilibrium, onehas GMm/R2

0 = 4πR20 p0. With perturbation R → R0 + δR; p → p0 + δp

md2(R + δR)

dt2= − GMm

(R + δR)2+ 4π(R + δR)2(p + δp)

md2δR

dt2=

(2GMm

R30

+ 8πR0p0

)δR + 4πR2

0δp

d2δR

dt2=

GMR3

0

(4− 3γ)δR

We used pV γ = constant or pR3γ = constant or δp/p0 = −3γδR/R0.This equation has a harmonic solution δR = A sinωt if γ > 4/3.

ω =

√(3γ − 4)

GMR3

0

'

√GMR3

0

∼√ρ.


Pulsational Instability

An instability will occur if opacityincreases with compression(temperature), which is not normallythe case. Normal stellar opacity is ofthe Kramer’s type κ ∝ ρT−3.5 andρ ∝ T 3 in adiabatic matter. Thusκ ∝ ρ−1/6 and decreases uponcompression.

In a partial ionization zone,H↔ H+ + e−,He↔ He+ + e−,He+ ↔ He++ + e−, the temperatureincreases less because of the ionizationenergy sink. This is the key to thekappa mechanism.

V

I A→ B: opacity increases during compression

I B → C : heat absorbed ∆SBC = ∆QBC/Thot

I C → D: opacity decreases during expansion

I D → A: heat released ∆SDA = ∆QDA/Tcold

∆QBC + ∆QDA = ∆SBC (Thot − Tcold )/Thot > 0


Period-Luminosity Relation

I log L = 3.5 logM+ a

I log L = 2 log R + 4 log Teff

I log P = − 12 logM+ 3

2 log R + b

I log Teff ∼ − 120 log L

I log P = ( 34 −

17 + 3

20 ) log L− 3 log Teff + b + a7

I log P = 0.76 log L + b + a7

I MV = − 2.50.76 log P + c = −3.3 log P + c


Dwarf Spheroidal Galaxies

I At least asluminous asglobular clusters

I Surfacebrightness 1% ofLMC

I > 10 in LocalGroup

I No young stars,no gas

I Stars of a widerange of ages3− 10 Gyr

I Low metallicities

Z = 0.02Z

3 Gyr

7

15


Galaxy Encounters

I Weak or distant encountersI Flyby with associated tidesI Satellite orbit decay due to

dynamical frictionI Tidal evaporation of orbiting

satelliteI Tidal or gravitational shocks

I Strong or close encountersI MergersI Global gravitational effects

become important

I Analytic cases (mergers can onlybe treated numerically)

I Dynamical friction (smallsystem moving through a largerone)

I Tidal evaporation(Jacobi/Roche limit)

I Slow encounters (adiabaticapproximation)

I Fast (shocking) encounters(impulse approximation)


Distant Weak Encounters and Drag

Use the impulse approximation, ignoringthe deviation in the stellar paths. Theimpact parameter is b. The perpindicularpull of star m on star M is GmM/r2

times b/r , with r2 = b2 + V 2t2:

F⊥(t) =GmMb

(b2 + V 2t2)3/2=MdV⊥

dt

Deflection angle:

θ =∆V⊥

V=

1

MV

∫ +∞

−∞F⊥dt =

2Gm

bV 2

The encounter has symmetry about thevector of closest approach, the line θ/2backwards from the original perpendicularimpact parameter vector. Newton’s 3rdlaw says m∆v = M∆V .

Drag is caused by the component of theforce parallel and backwards to M’smotion. Then

∆t∆Fdrag = −m∆v|| = −2GMm

bV

θ

2

= −2G 2M2m

b2V 3.

Fdrag =

∫ ∫nV ∆Fdragdt2πbdb

= −4πG 2M2nm ln Λ

V 2

Λ =bmax

bmin


Distant Weak Encounters and DragThe basic idea is that a moving massattracts objects to it, but because ofits motion, the objects tend to gatherbehind the mass, pulling it backwards.

I Open clusters: ln Λ ≈ 6

I Globular clusters: ln Λ ≈ 11

I Large elliptical galaxy: ln Λ ≈ 22

I Galaxy clusters: ln Λ ≈ 7

Allowing for a range of velocities(Chandrasekhar friction formula):

Fdrag = −4πG 2M2m

V 2ln Λ

∫ V

0

f (v)d3v

I V << v , use f (v) ≈ f (0),resembles Stokes law for motionthrough a viscous fluid.

Fdrag = −16

3π2G 2M2mf (0)V ln Λ

I V >> v , all stars contribute,drag decreases with V :

Fdrag = −4πG 2M2nmV−2 ln Λ

I Maxwellian with dispersion σ,X = V /(σ

√2):

Fdrag = −4πG 2M2nmV−2 ln Λ

×[erf (X )− 2Xe−X 2

/√π]

' −4πG 2M2nmV−2 ln Λ[1− e−0.6X 2

]J.M. Lattimer AST 346, Galaxies, Part 5

Applications of Dynamical Friction

I Satellite in circular orbitTake an isothermal galaxy:ρ(r) = V 2

c /(4πGr2), σ = Vc/√

2or X = 1, L =MVc r

Fdrag = −0.43G (M/r)2 lnλ

dL/dT = Fdrag r =−0.43G (M2/r) ln Λ =MVcdr/dt

tinfall = Vc r2i /(0.86GM ln Λ)

Example: globular cluster orbitingMilky WayM = 106M,Vc = 250 km/s,bmax = ri = 2 kpc:

tinfall ' 2.6× 1010 yr

I Massive galaxy encounterM = 1010M, ri = 20 kpc,V = Vc :

tinfall ' 2× 108 yr ∼ orbit period.

I LMCM≈ 2× 1010M, ri = 60 kpc:

tinfall ≈ 3× 109 yr.

Why is LMC still there? Not on acircular orbit; LMC has beenbound to SMC in elliptic(e = 0.2) orbit. Their orbit hasdecayed by factor 2 but willtidally separate when approachingMilky Way within 30 kpc.They will settle in Galactic centerin 1010 yr.


Tidal Limit

Satellite and galaxy are fixed in arotating frame. A star’s energyE = V 2/2 + Φ is not conserved; butEJ = V 2/2 + Φeff is.

Φeff (r) = Φ(r)− |Ω× r|2/2

Along a line connecting m and Mwith origin at m:

Φeff (x) = − GM|D−x| −

Gm|x| = Ω2

2 (x − D)2

Turning points:xJ ' ±D(m/3M)1/3

is the Jacobi (or tidal, Roche, Hill)radius.

For an isothermal potential,xJ ' ±D(m/2M)1/3.


Satellite Evaporation and Destruction

Satellite star with EJ moving awayfrom satellite has decreasing V , andturns around when EJ = Φeff .

If EJ > Φeff (rJ ) the star is lost(evaporated). This differs from slowevaporation caused by scattering ofstars to V > Ve . Even bound stars(E < 0) can have EJ > Φeff (rJ ).

A satellite approaching a galaxy hasdecreasing rJ and Φeff (rJ ) so it losesmore and more stars.Since most stars are only marginallybound (N(E ) peaks near E = 0), asmall decrease in Φeff (rJ ) can result inthe loss of many stars.


Slow and Fast Encounters

Although orbits of many starssignificantly affected by a tidalencounter, the tightly bound ones(Porb << tencounter ) are not. The tidalfield slowly changes and these starsrespond adiabatically.

The opposite extreme is a tidal shock:Porb >> tencounter orVinternal << ∆Vencounter . These starsdon’t move during the encounter, sono change in PE. They feel an impulseand ∆ KE. These stars are thusheated.

System must expand and cool inresponse. A two step process (recallEo,f = - KEo,f ):

I shock: KEi = KEo + ∆KE,Ei =Eo + ∆ KE = - KEo + ∆ KE

I relaxation: Ef = Ei , KEf =KEo −∆KE

Although shock heated system by∆KE, during relaxation it cooled by−2∆KE and expanded.

Some stars received enough energy tounbind and evaporate. Repeatedshocks can disintegrate a cluster.

If the encounter is distant, the systemis left elongated, long axis pointing tothe point of closest approach.


Examples

I Open clusters are shocked bypassage of dense molecular clouds;most evaporated after 5× 108 yr.

I Globular clusters are shocked bypassing through disk.If σ = 5 km/s, r = 10 pc,V⊥ = 170 km/s, R = 3.5 kpc:tdisrupt ≈ 6× 109 yr.

I Galaxies in clusters (galaxyharassment): disks are heated andthicken; spiral arm formation issuppressed; galaxies appear toevolve to earlier Hubble type.Stars and dark matter expand andare lost but join the cluster.Gas loses angular momentum andfalls to center, triggering starformation (starburst).

I Ring galaxies are formed fromtidal shocks. A perturber passesrapidly through center of disk,inducing an inward ∆Vr . Setsup a synchronized epicyclicmotion, results in an expandingcircular density wave or ring.These trigger star formation.


Andromeda Galaxy

I Large than Milky Way: 50% moreluminous, hR is double,V (R) ∼ 260 km/s (about 20-30%larger). Twice the number ofglobular clusters (300).

I Bulge is larger than Milky Way’s,with about 30-40% of theluminosity. Stars are older than afew Gyr and relatively rich inheavy elements.

I Has a compact, semi-stellarnucleus, with two concentrationsof light 2pc (0.5 arcsec) apart.One is a star cluster, one is ablack hole of 4× 106M.Nucleus is free of gas and dust.

I Metal-poor globular clusters havedeeply plunging trajectories.

I Most stars within a few kpc ofdisk are not metal-poor, as if thebulge was overflowing. These areabout 6 Gyrs old and may be theremnant of a merger with ametal-rich (and thereforemassive) galaxy.

I A “ring of fire” circles at 10 kpcand contains most of the youngstars. Marked by HII regions andCO emission.

I Less pronounced, and tightlywound, spiral features makes thisan Sb galaxy, compared to MilkyWay (SBc).

I 4− 6× 106M of HI, 50% morethan Milky Way.


M31


Formation of Local Group

Tidal torques induce slow rotation

Denser clumps collapse, and are already clustered.


Galactic Chemical EvolutionEvolution of total massM(t) =Mg (t) +M∗(t) in terms ofinflows f (t) and outflows o(t):

dM/dt = f − o

In the Closed Box Model, f (t) = o(t).The evolution of the gas mass Mg (t)

dMg/dt = −Ψ + E + f − o

in terms of the Star Formation Rate(SFR, Ψ(t)) and the rate of massejection by dying stars E (t).A star of mass M is created at timet − τM and dies at time t > τM,leaving a remnant of mass CM.

E (t) =

∫ MU

Mt

(M−CM)Ψ(t−τM)ξ(M)dM

The Initial Mass Function (IMF) is ξ.Integrate over stars dying at time t,i.e., from Mt to the upper limit of theIMF MU .The evolution of species i with massMgZi in the gas is

dMgZi/dt = −ΨZi +Ei + fZi,f −oZi,o

The rate of stellar ejection of species i :

Ei (t) =

∫ MU

Mt

Yi (M)Ψ(t−τM)ξ(M)dM

where Yi (M) is the stellar yield andcould depend on t.


Inputs

Stellar lifetimes strongly decreasewith M.Stellar remnants include whitedwarfs, neutron stars and blackholes. Their masses depend onthe extent of mass loss by winds.Ejected mass fractions increasewith M.

Stellar yields are better expressedin terms of net yields

yi (M) = Yi (M)− Zi (M− CM),

the newly created mass of i minuswhat was originally there.


Initial Mass FunctionLuminosity functionf (L) = dN/dL and M− Lrelation for MS starsφ(L) = dM/dL. The present-daymass function (PDMF, F (M)) is

F (M) = dN/dM = f φ.

The IMF can be derived if the SFRis known as it is an integral overtime of the star creation rate ΦΨ.If the IMF doesn’t depend on t:

ξ(M) = F (M)/

∫ T

T−τMΨ(t)dt.

Return Mass Fraction R is massfraction of a stellar generationreturned to ISM

R =

∫ MU

MT

(M− CM)ξ(M)dM

R = 0.28 (S), 0.30 (K + S),0.34 (C + S)

Two simplified cases:

I Short-lived stars

ξ(M) =F (M)

Ψ0τM.

I Eternal stars

ξ(M) =F (M)

〈Ψ〉τM

where 〈Ψ〉 = T−1∫ T

0Ψ(t)dt is

the past average SFR.

Salpeter formula (α = 2.35)

ξ(M) = dN/dM = ξoM−α

1 =

∫ MU

ML

ξ(M)MdM

ξo = ξ0Mα−1


IMF and Its Evolution


Star Formation Rate

Schmidt (1959): Ψ(t) = νMg (t)N

N ∼ 2 theoretically if referring to mass densityN ∼ 1.4 observationally if referring to surface density, more easilymeasured.Kennicutt (1998): Ψ ∝ Σ/τdyn where τdyn = R/V (R).


Analytical Chemical Evolution

Analytical solutions are possible if Instantaneous RecyclingApproximation (IRA) is adopted (Schmidt 1963). Stars are either

I “eternal”, M <M, τM ≥ T ∼ 12 Gyr, orI “dead at birth”, M <M, τM ≈ 0.

With IRA, Ψ(t − τM) = Ψ(t) 6= f (M). Define pi to be the amount of icreated by a stellar generation per unit mass of “eternal” stars andremnants:

pi = (1− R)−1

∫ MU

MT

yi (M)ξ(M)dM,

For a closed box, IRA model

dMg

dt= −(1− R)Ψ(t),

dMgZi

dt= Ψ(t)(1− R)(pi − Zi ).

Eliminating Ψ and t, and setting Mg (t = 0) =M:

MgdZi = −pidMg ; Zi = Zi,0 + pi lnMMg

Assuming a Schmidt Law with N = 1:

Mg =Me−ν(1−R)t , Zi = Zi,0 + piν(1− R)t.


Models

Parameters of the model should be constrained to produce thepresent-day metallicity. Assuming Zi,0 = 0, R ' 0.3, non-IRA modelsimply N = 1 and ν = 1.2 Gyr−1. IRA predicts

Mg (T )/M = e−ν(1−R)T ' 4× 10−5

compared to the observed value Mg (T ) ' 0.2M. Also

p = −(Z − Z0)/ ln(Mg/M) ' 0.6Z ' 0.012


Metallicity Evolution

Nearby F and G stars

b v > 80 km/sr v < 80 km/s


Metallicity Distribution

Mg =Me−[Z(t)−Z(0)]/p.

Where the gas abundance is high relative to stars (outer disk regions) themetallicity is low.

Disk gas in M33


Metal Abundances in Bulge Giant Stars

Mass of stars with metallicity between Z and Z + ∆Z is

dM∗dZ

∆Z =Me−[Z(t)−Z(0)]/p∆Z .

Consistency indicates bulge retained all its gas, turning it into stars.

p = 0.7Z, G and K giant stars in Galactic bulge.

dM∗/dZ dM∗/d ln Z


Metal Abundances in Solar-Neighborhood Dwarf Stars

Closed box model with IRA andZ (0) = 0 suggests for solarneighborhood

Z (T ) ' Z ' p ln(MMg

)

≈p ln(50

13),

p ' 0.74Z.

Fraction of metal-poor stars withZ < Z/4:

M∗(< 0.25Z)

M∗(< 0.7Z)=

1− e−0.25Z/p

1− e−0.7Z/p

' 0.52.

Observed fraction ∼ 25%. Knownas the G-dwarf problem.

If Z (0) = Z0 6= 0, Z0 ∼ 0.15Z:Z − Z0 ' p ln (M/Mg ) and p ' 0.63Z.

M∗(< 0.25Z)

M∗(< 0.7Z)=

1− e(Z0−0.25Z)/p

1− e(Z0−0.7Z)/p

' 0.25.

d[N

(Z<

Z′ )/

N(Z<

ZT

)]/

dln

Z

Pre-enrichment: Z0 = 0.08Z

Infall: exponential τ = 7 Gyr


Evidence For Infall

There is another inconsistency with the closed box model.

Note Ψ ∼ 3− 5M yr−1 in the solar neighborhood.

Present disk gas mass is Mg ∼ 5− 10× 109M.

Remaining epoch of star formation should last no longer thanMg/[Ψ(1− R)] ∼ 1.4− 4.8 Gyr.

Why do we live so near the end of star formation?

Additional gas must be accumulating in the Galactic disk.


Models with Infall

If it is assumed gas infalls with a metallicity Z0 = Z (t = 0) and an infallrate proportional to star formation rate:

f (t) = Λ(1− R)Ψ(t).

The parameter Λ ≤ 1; if Λ = 1 then Mg (t) =Mg (0).

Mg (t)

Mg (0)=

(1− [Z (t)− Z0]

Λ

p

)Λ−1−1

, Z (t) =p

Λ

(1−

[Mg (t)

Mg (0)

]Λ/(1−Λ)).

When Z = Z0 + p/Λ, the maximum metallicity, Mg → 0. Introducing

µ(t) =M(t)−M(0)

M(0)= Λ

1− R

M(0)

∫ t

0

Ψ(t ′)dt ′ =Mg (t)−M(0)

M(0)

Λ

Λ− 1,

Z (t) = Z0 + pΛ−1(

1−[1− (Λ−1 − 1)µ(t)

]Λ/(1−Λ)), Λ 6= 1

Z (t) = Z0 + p[1− e−µ(t)

]. Λ = 1

N(Z < Z ′) =

∫ t′

0

Ψ(t)dt =M(0)

Λ(1− R)µ(t ′),

N(Z < Z ′)

N(Z < Z (T ))=µ(t ′)

µ(T )


Evolution of the Solar Neighborhood (the Thin Disk)

Assume exponentially decreasing infall with timescale τf = 7 Gyr andMg (0) = Z (0) = 0. Use surface densities:

I Infall normalized to observed value:

ΣT =

∫ T

0

f (t)dt = Σg + Σ∗ ' 12 + 38 = 50M pc−2

f (t) = Ae−t/τf =⇒ ΣT = Aτf

(1− e−t/τf

)=⇒ A = (50/7)

(1− e−12/7

)−1

' 8.7M pc−2 Gyr−1

I Assume Schmidt SFR Ψ(t) = νΣg (t) and Σg (T )/ΣT (T ) = 0.24:dΣg

dt= −(1− R)Ψ(t) + Ae−t/τf , Σg (t) =

A

αe−t/τf

(1− e−αt

),

α = (1− R)ν − τ−1f =⇒ ν ' 0.32 Gyr−1, Ψ(T ) ' 3.8M pc−2 Gyr−1

Constrain to solar metallicity Z = Z (t = 7.5 Gyr) ' 0.02:

Z (t) = pν1− R

α

eαt − αt − 1

eαt − 1=⇒ p = 1.17Z

N(Z < Z ′) =

∫ t′

0

Ψdt =Aν

α

[α +

e−t′/τf

τf− (1− R)νe−(1−R)νt′

].


Evolution of the Solar Neighborhood (the Thin Disk)


Oxygen Evolution

Oxygen is characteristic of a primary α−nucleus, originating primarily inType II supernovae.Iron is also primary, produced in a 2:1 proportion by Type I and Type IIsupernovae, but the former don’t ”turn on” until [Fe/H]> −1.5.

halo

thin disk

thick disk


CNO Evolution

Carbon and oxygen are characteristicof primary nuclei, originating primarilyin Type II supernovae.Nitrogen thought to originate fromHot Bottom Burning in AGB stars,but the most massive of these (8M)have lifetimes too long to explain itsprimary character.It is also produced as a secondarynucleus.Now known to be a primary nucleusproduced by rotationally-inducedmixing in massive stars, fromH-burning of C and O produced insidethe star.Requires low-metallicity stars to berotating muich faster (800 km/s) thanhigh-metallicity stars (300 km/s).


Halo Evolution and Hierarchical Galaxy Formation

The metallicity distribution of halofield stars is well fit by a closed box,IRA model dN/d ln Z ∝ (Z/p)e−Z/p.This peaks at Z = p. [Fe/H]= −1.6implies p ' Z/40.

The reduced yield could be explainedby an outflow during halo formation,with a rate 8 times the SFR.

It could also be interpreted as due tohierarchical galaxy formation. Thehalo metallicities are lower by morethan 3 times than those of nearbygalaxies of the same mass as the halo.Those small galaxies show a linearrelation between stellar mass andmetallicity which results from massloss: hot supernova ejecta escape moreeasily from smaller mass galaxies.


The R-Process

Elements heavier than Feare produced by protoncapture (p) or by neutroncapture, either slow (s)or rapid (s).


Element Evolution

Primary α-elements show lessscatter and small trendswith metallicity. Secondary r-processelements show large scatter.


R-Process and Heirarchical Galaxy Formation


Dwarf Spheroidals and Metallicity Distributions


Neutron Star Mergers and the R-Process

Mergers have minimum evolutionlifetimes of 10-100 Myrs.But high r-process abundance canbe made in small metallicity places.Different places have different [Fe/H].[Fe/H] is decoupled from time.

Ishimaru, Wanajo & Prantzos 2010


Dwarf Galaxies

I Types:I Dwarf Spheroidals - diffuse and devoid of

gas and star formationI Dwarf Ellipticals - devoid of gas and star

formationI Dwarf Irregulars -diffuse and abundant

gas and star formation

I Properties:I All have horizontal branch stars, i.e.,

some stars older than 10 Gyr. This ispredicted by cold dark matter.

I M32 is anomalous, with extremely highcentral surface density. Perhaps the coreremnant of a larger galaxy with amassive black hole at its center

I Metallicities in dwarf galaxies correlateswith their masses. This suggestsoutflow, caused by star formation, whichis easier the more shallow is thegravitational potential.

I Gas in dwarf irregularsextends beyond stellar disk

HI gas on R band image of IC10


M33

Wolf-Rayet


History of the Local Group

Treat the Local Group as the galaxiesM31 (M) and the Milky Way (m).They are separated by r = 770 kpcand approaching each otherdr/dt = −120 km/s.

It can be shown that the equation ofmotion for two arbitrarily massiveobjects (m,M) is the same as for asmall mass m moving in the field of amuch larger mass M+ m. In thiscase, the relative positions of the twomasses satisifies

d2r

dt2=

L2z

r3− G (M+ m)

r2

dr

dt= ±

√2G (M+ m)

r− L2

z

r2

Since dr/dt < 0, we use − sign.

Pericenter is reached when dr/dt = 0or rmin = L2

z/[2G (M+ m)].Using rmin = 0 gives M+ m >(dr/dt)2r/(2G ) ∼ 1.3× 1012M.There is a cyclic solution

r = a(1− e cos η)

t =a3/2(η − e sin η)√

G (M+ m)

where a = L2z/[G (M+ m)(1− e)].

Galaxies began moving apart at t = 0(η = 0) and are now approaching.η = π(2π) corresponds to apocenter(pericenter). M+ m, η, a can bedetermined from r , dr/dt, t if e isknown. We find a minimumM+ m >∼ 5.1× 1012M when e = 1,12 times larger than we estimated fromcircular velocities. Pericenter time ist2π = 2πa3/2/

√G (M+ m) = 15 Gyr.


the local group and galactic evolution

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