AY202a Galaxies & Dynamics Lecture 11: Scaling Relations , con’t Luminosity & Mass Functions

1. AY202a Galaxies & DynamicsLecture 11: Scaling Relations , con’tLuminosity & Mass Functions

2. Disk Scaling Relations Observed 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.

3. 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.

4. 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 / 8G, H(z) is the Hubble constant at redshift z

5. From the virial theorem Vvir2 = G Mvir/Rvir Then, setting H(z) = 100 h km/s Mvir= Rvir3 h2 (vir/200) /G Vvir= Rvirh (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!

6. Chemistry [Fe/H] from line indices

7. 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

8. Some Line Indices In typical use today are the modified Lick indices.

9. Fe/H vs L Brodie&Huchra ‘91

10. Dwarf Galaxies Grebel et al 2003

11. Metallicity vs L Globular Cluster Systems (Nantais ‘09)

12. Brodie & Huchra 1991 H0 = 100 km/s/Mpc

13. 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

14. 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

15. 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)

16. 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)

17. Zwicky 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. Hubble Holmberg Guess who was right…

18. 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*

19. 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) = Σ1/Vm , Where Vm is the maximum volume for each galaxy in the bin --- sum inverse volumes individually. N

20. 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

21. 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

22. NED, C. Sarazin

23. 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

24. 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.

25. Felten 1977

26. 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.

27. Malmquist Bias Magnitude 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.

28. Malmquist Bias in LF Eddington derived an analytic correction for the MB. The expected number of galaxies in a magnitude limited sample is ne(L) dL = n*(L/L*)αexp(-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.

29. 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.

30. Non Parametric Estimators Major issue is that we expect density variations along the l.o.s. So far we have assumed • Uniform density • Location independent shape But! 1979 Ed Turner rediscovered LB C-method and re-introduced non-parametric techniques. Variation of φ* with v in CfA

31. 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)

32. d N(L) φ(L) dL = = d ln (L) ∞ N[>L, r ≤ rmax(L)] ∫ φ(L’) dL’ Problems --- little weight to faint galaxies estimates of (L) are not independent do not get φ* unless you normalize somewhere, somehow. L

33. 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)

34. 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.

35. 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

36. 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.

37. CfA1 Marzke et al. (1994)

38. LF in Clusters Smith, Driver & Phillips 1997 α = -1.8 at the faint end….

39. 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

40. SDSS Mass Function vs Density Blanton & Moustakas (2009)

41. SDSS LF in Other Properties From Blanton Galex HI

42. 2MRS

44. NED

45. Caveats & Questions 1. 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.

46. 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.

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

49. Bell et al. 2003 log10 (M/L) = a λ+ bλ (color)