Galaxy Morphology Refresher. Driver et al. 2006

Transcription

1 Galaxy Morphology Refresher Driver et al. 2006

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3 Galaxy Luminosity Functions Luminosity Function is the number density (# per volume) of some population of objects (e.g., galaxies) of a specific luminosity. Here we take the number density of galaxies with absolute magnitude M and M+dM. Φ(M) dm The Total Density of Galaxies is then the integral of the luminosity function: n = Φ(M) dm Rewrite in terms of the # density of galaxies between luminosity L and dl: Φ(L) dl

4 Galaxy Luminosity Functions Measuring Luminosity Functions seems easy, but hard in practice. - Need Luminosities, requires measuring fluxes (magnitudes) and distances. - Need large and representative sample of galaxies. Sufficiently large Volume required (galaxies clustered in structures on ~100 Mpc scales). But, need depth to get low-luminosity galaxies at any given distance. (Always a trade off between depth and area in galaxy surveys.) - Photometric errors are problematic. A well known bias, Malmquist bias. One will always measure an increase in the average luminosity of a sample with distance because the lessluminous objects are missed. (Difference between Flux-limited and Volume Limited samples.)

5 Malmquist bias Introduce the star-count function (star=galaxy=x-ray source=... in this case), A(m), which is the differential number of stars with apparent magnitude, (m+dm, m). A(m) dn dm In practice, there is some limiting magnitude mlim such that A(m) is only measurable for m < mlim. A sample that consists of all objects m < mlim is said to be a Magnitude limited sample. For a magnitude limited sample, the observed mean Absolute magnitude will be brighter than the true mean Absolute magnitude. The Volume (distance) out to which you can see the most luminous object will be larger than for the mean object.

6 Malmquist bias rest-frame B-band luminosity Volume V(<z) [Mpc 3 ] Redshift distribution of galaxies in Hubble Deep Field North Dickinson et al. 2003

7 Malmquist bias For a magnitude limited sample, the observed mean Absolute magnitude will be brighter than the true mean Absolute magnitude. Consider the simpliest case where stars have the same distance (then m=m + constant). measured mean true mean magnitude limit, mlim=21 true distribution

8 Malmquist bias Derive effect of Malmquist bias given that you can observe a sample with measured mean <mm> and variance σm and we will take that the sample has a Gaussian distribution (for Malmquist bias discussion here, not change in notation: s=distance, and ν(s) and n(s) are defined below) Calculate the number of stars in a magnitude-limited sample what have Absolute magnitude in (M+dM, M) and Apparent magnitude (m+dm, m). d 2 N dm dm = Φ(M)dn ds ds dm s(m,m) is the distance to an object in the sample dn is the number of stars that lie within a volume dv = (ω s^2 ds), where ν(s) = number density of objects (number per unit volume). dn ds = ωs2 ν(s) M

9 dn yields A(m) = ds = ωs2 ν(s) dm Malmquist bias d2 N dm dm = ω d 2 N dm dm = Φ(M)dn ds dm Φ(M) s ds m dm change integration variable from M to s, because m - M = 5 log s + 5, and so, ( M/ s) m = ( m/ s) M M M s 2 ν(s) this gives A(m) =ω 0 ds Φ(M) s 2 ν(s)

10 Malmquist bias We can now compute the Mean Absolute magnitude of objects in a sample that have apparent magnitude m M m = dm M dm d2 N dm dm d2 N dm dm Using our expression now for (d 2 N/dmdM), we have M m = 0 0 ds M Φ(M) s 2 ν(s) ds Φ(M) s 2 ν(s) Note that ( M/ m)s = 1, and that so differentiating our expression for A(m) w.r.t. m gives da dm = ω 0 ds dφ dm s2 ν(s)

11 Comparing M m = 0 0 Malmquist bias ds M Φ(M) s 2 ν(s) ds Φ(M) s 2 ν(s) with da dm = ω 0 ds dφ dm s2 ν(s) we can write: 1 da dφ A dm = 1 Φ dm m and also that 1 A d 2 A dm 2 = 1 Φ d 2 Φ dm 2 m

12 Malmquist bias 1 da dφ A dm = 1 Φ dm m Now must have some form for Φ(M), take as a Gaussian: Φ(M) = In which case Φ -1 (dφ/dm)m yields that 1 2πσ 2 exp (M M 0) 2 2σ 2 1 A da dm = M M0 σ 2 m Solving for <M> gives: M m = M 0 σ 2 d ln A dm Therefore, <M> is always lower than M0: <M> is Brighter.

13 Malmquist bias Since <M> is measurable, one can correct for the Malmquist bias iσ is known. In principle, this must be estimated as well. For this, insert into 1 which yields using Φ(M) = A d 2 A dm 2 = 1 Φ 1 A 1 2πσ 2 exp (M M 0) 2 2σ 2 d 2 Φ dm 2 m d 2 A M M0 dm 2 = σ 2 σ 2 m M 2 m M 2 m 2 1 σ 2 with a little algebra... σ 2 m = σ 2 + σ 4 d2 ln A dm 2

14 Malmquist bias Now, consider the simple case that the sample is distributed homogeneously through space, so ν(s) = constant A(m) exp [0.6 log(m M 0 )] Taking the derivatives, we can determine the effect on M m = M 0 σ 2 d ln A dm σ 2 m = σ 2 + σ 4 d2 ln A dm 2 d log A dm = d ln A/ ln 10 dm =0.6 d 2 log A dm 2 =0 For a homogeneously distributed population with a Gaussian distribution of magnitudes, the mean is shifted by 0.6 σ 2 and the variance is unaffected.

15 Galaxy Luminosity Functions The Schecter Luminosity Function - From classic paper on measuring luminosity functions

16 Galaxy Luminosity Functions The Schecter Luminosity Function

17 Galaxy Luminosity Functions The Schecter Luminosity Function Rewrite in terms of Absolute Magnitude. Recall that Φ(L) dl = Φ(M) dm Because dl/dm ~ d(10 0.4M ) / dm = 0.4 log(10) L:

18 Galaxy Luminosity Functions The Schecter Luminosity Function

19 Galaxy Luminosity Functions The total Luminosity Density of a galaxy population described by a Schecter Function is This is formally only defined for α > -2. What happens at lower power law slopes?

20 Galaxy Luminosity Functions Different galaxy populations have different luminosity functions. Driver et al. 2006

21 Galaxy Luminosity Functions Different galaxy populations have different luminosity functions. Are slopes, α > -1 OK? Rudnick et al. 2009

22 Eddington Bias Eddington Bias arises from photometric (flux) errors. In general, there are many, many more fainter objects in your sample than brighter ones. Random, photometric errors will cause relatively few bright sources to enter the fainter bin, but will cause relatively more faint sources to enter your high mass bins. true sample, N ~ (flux) -2 Eddington 1913, MNRAS, 73, 359 A good example of a variant of this is Fontanot et al. (2009, MNRAS, 397, 1776)

23 Eddington Bias Eddington Bias arises from photometric (flux) errors. In general, there are many, many more fainter objects in your sample than brighter ones. Random, photometric errors will cause relatively few bright sources to enter the fainter bin, but will cause relatively more faint sources to enter your high mass bins. true sample, N ~ (flux) -2 d(mag)=0.2 Eddington 1913, MNRAS, 73, 359 A good example of a variant of this is Fontanot et al. (2009, MNRAS, 397, 1776)

24 Eddington Bias Eddington Bias arises from photometric (flux) errors. In general, there are many, many more fainter objects in your sample than brighter ones. true sample, N ~ (flux) -2 d(mag)=0.2 d(mag)=0.5 Random, photometric errors will cause relatively few bright sources to enter the fainter bin, but will cause relatively more faint sources to enter your high mass bins. Eddington 1913, MNRAS, 73, 359 A good example of a variant of this is Fontanot et al. (2009, MNRAS, 397, 1776)

25 Galaxy Luminosity Functions Different galaxy populations have different luminosity functions.

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27 Galaxy Luminosity Functions, Color Dependence Different galaxy populations have different luminosity functions. Driver et al. 2006

28 Galaxy Luminosity Functions, Color Dependence Driver et al. 2006

29 Galaxy Color Distributions See Baldry et al. 2004

30 Galaxy Color Distributions See Baldry et al. 2004

31 Generalized Surface Brightness Distributions Generalized surface brightness profile proposed by Sérsic (1963) I(R) =I 0 exp( kr 1/n ) Where n is the Sersic Index. Can choose n and k to reproduce familiar results for galaxy disks and spheroids: I(R) =I 0 exp( kr), n =1 I(R) =I 0 exp( kr 1/4 ), n =4 Higher n, more compact is surface-brightness profile. Strong relation between sersic indexes and other galaxy properties.

32 Generalized Surface Brightness Distributions

33 Galaxy Color Distributions n μ i-z Blanton et al r-i g-r u-g g-r r-i i-z μ n Mi

34 Galaxy Color Distributions Hogg et al. 2004