AY202a Galaxies & Dynamics

Lecture 11: Scaling Relations , con’t

Luminosity & Mass Functions

Disk Scaling RelationsObserved

I band V L0.29 R L0.32 R V1.10

K band V L0.27 R L0.35 R V1.29

Small sigmas, in VL relation imply a good DI

And generally the TF slope of L on V flattens as the bandpass goes bluewards.

Near 4 at K, near 3 at B.

Steeper VL, RL slopes for earlier type spirals.

Color dependence probably due to SFR

VL relation shows essentially no dependence on size or surface brightness in I or K

Scatter in velocity and size probably dominate the VL and RL relations.

Relations broadly understood in terms of disks embedded in dark matter halos.

Scaling relations and galaxy formation?

Gunn & Gott model,define a virial radius Rvir, of a collapsed relaxed gravitational body as the radius inside which the average density is a factor ∆vir times the critical density. The virial mass is then

Mvir = 4/3 Rvir3 vir crit

Where crit has its usual definition

crit = 3 H(z)2 / 8G,

H(z) is the Hubble constant at redshift z

From the virial theorem

Vvir2 = G Mvir/Rvir

Then, setting H(z) = 100 h km/s

Mvir= Rvir3 h2 (vir/200) /G

Vvir= Rvir h (vir/200)1/2

& Mvir= Vvir3 h-1 (vir/200)-1/2 G

With G in units of (km/s)2 kpc Msun-1 and

Rvir in units of kpc.

If Mvir/L, Vvir/V and Rvir/Re are well behaved, there you go!

Chemistry

[Fe/H] from

line indices

Spectral Indices

Some from SDSS papers:

Name C1 Band C2

D(4000) 3855 3950 4000 4100

[O II]3727 3653 3713 3713 3741 3741 3801

H 4030 4082 4082 4122 4122 4170

Index = -2.5 log { 2x[band]/[c1 +c2]}

can also express as an equivalent width

Some Line Indices

In typical use today are the modified Lick indices.

Fe/H vs L Brodie&Huchra ‘91

Dwarf Galaxies Grebel et al 2003

Metallicity

vs L

Globular

Cluster

Systems

(Nantais ‘09)

Brodie & Huchra 1991

H0 = 100 km/s/Mpc

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Galaxy Luminosity Functions

Simple concept: The LF is just the number of galaxies of property X per unit volume per unit {luminosity/magnitude} interval

= (L) or (M)

The LF also defines the “selection function” used in the study of galaxy density distributions, a.k.a. Large-Scale Structure

Galaxy LF is studied for

(A) To derive L, the luminosity density to get Ω from ρm = L <M/L>

as

L = ∫ (L) dL

(B) Aforementioned selection function

(C) Test of galaxy formation models

(D) Input into galaxy count vs evolution analyses, especially as a f(type,color).

∞

0

History

First derivation by Hubble

(1936) By comparing galaxy

magnitudess to brightest stars.

Hubble found a Gaussian

distribution with <MB> ~ -14.2

corrected for Malmquist Bias

with a dispersion σ ~ 1 mag

Also corrected for Galactic

Extinction AB= 0.25(csc b - 1)

N(M) =

e-(M-M0)2/2σ2 /(2π)1/2 σ

Hubble also used the LF to start the study of the velocity-distance relation using 5th ranked galaxies in clusters

(1936)

In the 1940’s+50’s, Erik Holmberg studied galaxies in groups in an attempt to look at volume limited samples.

In the 30’s+40’s Fritz Zwicky discovered dwarf galaxies and predicted an LF that rises steeply at the faint end.

Zwicky

Hubble

Holmberg

Guess who was right…

Zwicky’s 1957 form:

In 1962, George Abell was studying galaxy clusters and proposed a form consisting of two power laws with a break

at a characteristic magnitude M*

Most LF estimates were based on the binning technique. You counted galaxies in some absolute magnitude bin, calculated the distance out to which you could see them, estimated the volume and voila! (M ±ΔM) = N(M ±ΔM)/V

In 1969 Maarten Schmidt introduced a technique for measuring the luminosity function based on the assumption of a uniform (homogeneous) distribution called the V/Vm technique

(M ±ΔM) = Σ 1/ Vm ,

Where Vm is the maximum volume for each galaxy

in the bin --- sum inverse volumes individually.

N

In 1971 Donald Lynden-Bell introduce the non-parametric C-method. Promptly forgotten.

In 1976 Paul Schechter proposed a form of the LF based on a theory for the growth of structure from Gaussian random fluctuations in an expanding medium (Press-Schechter).

In 1976 JPH produced

the first LF as a function

of galaxy color (U-B)

related to SFR

Normal

Markarian

Schechter Function

LF form in terms of a power law + exponential cutoff based on Press & Schechter (1974) self similar stochastic (Gaussian random) galaxy formation.

(L) dL = φ* (L/L*)α e –(L/L*) d(L/L*) φ* = normalization (depends on H0 a lot!)

L* = characteristic luminosity (H0 and color)

α = faint end slope

NED, C. Sarazin

In 1977, Jim Felten examined the effects of extinction on the sample volume and the various LF estimation methods available at the time. For smooth extinction laws that vary with cosec b, the volume surveyed is

effectively an hourglass:

For A = α csc (b)

E2 is the second exponential

integral

we have

The effective volume surveyed, V(m), is given by:

V(m) = 4/3π dex[0.6(ml – M – 25)] x

[E2(0.6 α ln10) –E2(0.6 α ln10 csc bmin)/csc bmin]

where

ml = limiting apparent magnitude of the survey

α = extinction coefficient

bmin = minimum galactic latitude

Felten’s point was that not only did extinction affect the individual magnitudes, it also affected survey volume.

In addition extinction also affects the absolute magnitudes by SB! (he missed that) Note: Felten also found that Schmidt’s V/Vm technique was less statistically biased but also less “efficient” than binning.

Felten 1977

Felten also derived the magnitude form of the Schechter function:

φ(M) dM = 2/5 φ*ln10[dex 2/5(M*-M)]α+1

x exp[ - dex (2/5(M*-M)] dM

from which the luminosity density is

L = Γ(α+2)φ*L*

= Γ(α+2)φ*LSun dex [0.4(MSun-M*)]

and again, be aware of the bandpass issues and Bolometric corrections.

Malmquist BiasMagnitude limited

catalogs suffer from Malmquist Bias. There are several forms of MB which affect the slopes of relations and the counts of objects.

Asymmetry is the key.

Malmquist Bias in LFEddington derived an analytic correction for the MB.The expected number of galaxies in a magnitude

limited sample is

ne(L) dL = n*(L/L*)α exp(-L/L*) d(L/L*)

from which one can derive the LF. The observed ne(L) should be corrected by

nec(L) dL = ne [1 + σ’2 + σ”σ] + 2ne’σ’σ +ne”σ2/2 + ….

where σ(L) is the rms uncertainty in L and ‘ denotes the first derivative w.r.t. L, etc.

So, for example, if the errors are due to

peculiar velocities (or velocity errors --- remember D = v/H and v generally has symmetric errors, leading to aysmmetric errors in L D2. <Δv2>½ = rms

σ(L) = 1.08 (√3) 2 [<Δv2> l L/(4πH02)] ½

where l is the limiting flux.

Non Parametric Estimators

Major issue is that we expect density variations along the l.o.s.

So far we have assumed1. Uniform density2. Location independent shapeBut!1979 Ed Turner rediscoveredLB C-method and re-introducednon-parametric techniques.

Variation of φ* with v in CfA

Define

N(L) = Number of galaxies observed in a sample

with L ± dL/2

φ(L) = differential LF # per luminosity interval

per unit volume

(L) = integral LF = # per unit volume with

LG > L

If N[>L, r ≤rmax(L)] = # of galaxies brighter than L

and inside rmax(L),

Then we can define an integral equation for φ(L)

Problems --- little weight to faint galaxies

estimates of (L) are not independent

do not get φ* unless you normalize somewhere,

somehow.

d N(L) φ(L) dL

N[>L, r ≤ rmax(L)] ∫ φ(L’) dL’= = d ln (L)

L

∞

1979 STY introduced maximum likelihood techniques to fit form of LF (to Schechter):

Calculate the probability that a galaxy of zi & Li is

seen in a sample

Pi φ(Li) / ∫ φ(L) dL

Then the likelihood is

Ł = ∏ Pi and we vary the form of φ

to maximize Ł

∞

Lmin(zi)

1985 SBT studied

Virgo with deep 100” plates. Assumed all galaxies in the same place, few z’s. Deconvolved LF by type and into dwarves vs giants.

Stepwise Maximum Likelihood

1988 Efstathiou, Ellis & Peterson introduced SWML to get the form of the LF w/o “any” assumption about its shape.

Parameterize LF as Np steps φ(L) = φk

Lk – ΔL/2 < L < Lk + ΔL/2

ln L = ∑ W((Li-Lk)ln k -

∑ ln {∑ j L H[Lj-Lmin(zi)]} + C

Where N = number of galaxies in sample

W(x) = 1 for - L/2 < L < +L/2

= 0 otherwise

H(x) = 0 for x - L/2

= x/ L + 1/2 for -L/2 < L < +L/2

= 1 for x > L/2

Normalization via several techniques.

CfA1

Marzke

et al.

(1994)

LF in Clusters

Smith, Driver & Phillips 1997

α = -1.8 at the faint end….

Current State of LF Studies

2dF blue photo ~250,000 gals, AAT Fibers

SDSS red++ CCD, ~650,000 gals SDSS Fibers

2MASS JHK HgCdTe, 40,000 gals one of + 6dF fibers

Stellar Masses from population synthesis

SDSSMass Function

vs Density

Blanton &

Moustakas

(2009)

LF in

Other

Properties

From

BlantonGalex HI

SDSS

2MRS

NED

Caveats & Questions1. How location dependent is (L)?2. What is the real faint end slope?3. Is the Schechter function really a good fit?

At the bright end? At the faint?4. What is (T), φ(L,U-B), φ(L,B-B) …?5. How much trouble are we due to surface

brightness limitations?Galaxies have a large range of SB, color,

morphology, SED, etc.

This week’s paper:

The Optical and Near-Infrared Properties of Galaxies. I. Luminosity and Stellar Mass Functions,

by Bell, Eric F.; McIntosh, Daniel H.; Katz, Neal; Weinberg, Martin D. 2003, ApJS 149, 289.

Bell et al. 2003 M/L versus Color for B- and K-band

Bell et al. 2003

log10 (M/L) = a λ+ bλ (color)