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)