Transcript Diapositiva 1 - Universidad de Guanajuato

Lecture 4
Statistical Properties of Galaxies
Distribution: Correlation Function of Galaxies
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definition
estimators
angular correlations
results
comparison with simulations
Distribution of magnitudes: the Luminosity Function
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definition
applications
preparation of data
methods of estimation
parametric fits
results
Environmental effects: the Morphology-Density Relation
 morphology-density relation
 possible interpretations
Depto. de Astronomía (UGto)
Astronomía Extragaláctica y Cosmología Observacional
 Correlation Function of Galaxies: definition
Two-point (auto-)correlation function [first order]:
 first used for the distribution of galaxies by Peebles [1980, Large Scale Structures of the Universe]
 widely used because it is the simplest statistical description of a distribution and it is relatively
easy to extract from data
 measures the tendency of the objects under consideration to group together
 may be defined in 3 equivalent ways:
• excess probability, over that expected in a Poisson distribution, of finding a galaxy at
distance r from a galaxy selected at random
dP = <n> [1 + ξ(r)] dV
• joint probability of having 2 objects (a par), in volume elements dV1 and dV2, separated
by r12:
dP12 = <n>2 [1 + ξ(r12)] dV1 dV2
• by density contrasts
dP12 = δ(x) dV1 δ(x+r) dV2
 [1 + ξ(r)] = <δ(x) δ(x+r)> / <n>2
 the value of ξ(r) can be:
• > 0 → means that the distribution is clustered (enhanced probability)
• < 0 → means that distribution is anti-correlated (points avoid each other)
• = 0 → means that the distribution is uniform (Poisson-like)
 Correlation Function of Galaxies: methods of estimation
Estimators for the two-point correlation function:
 [Davis & Peebles 1983, ApJ 267, 465]
ξ(r) = [(nR/n) DD/DR] - 1
 [Landy & Szalay 1993, ApJ 412, 64]
ξ(r) = [(nR/n)2 DD – 2 (nR/n) DR + RR] / RR
 [Hamilton 1993, ApJ 417, 19]
ξ(r) = [(DD * RR) / DR2] - 1
where:
• DD = number of pairs of galaxies with
separations within the interval (r-Δr, r+Δr)
• DR = number of pairs between a galaxy
and a point in the Poisson catalog...
• RR = number of pairs of points in the
Poisson catalog...
+
*
Δ
 Correlation Function of Galaxies: angular
Angular two-point CF:
 the galaxies with measured redshift
are restricted to the Local Universe for
whole sky surveys, or deeper but for
small fractions of the sky
 larger galaxy catalogs for significant
fractions of the sky exist, but only with
projected positions
 the angular two-point CF is defined by:
dP = <nS> [1 + ω(θ)] dΩ
where:
APM Galaxy Survey
• <nS> = average surface density of galaxies
• dΩ = element of solid angle
• ω(θ) = angular correlation function
 ξ(r) can be “deprojected” from ω(θ) by the Limber Equation [Limber 1954, ApJ 119, 655]:
ω(θ) = ∫ y4 φ2(y) dy ∫ ξ(√x2 + y2θ2) dx
where:
• y = comoving distance
• φ(y) = radial selection function
 Correlation Function of Galaxies: real-space
Two-point correlation function:
[Hawkins et al. 2003, MNRAS 346, 78] 
 even with redshifts the correlation function is not directly measured (distortions)
combination
“fingers of God” (random peculiar velocities)
flattening (coherent infall:
underdense to overdense)
 Correlation Function of Galaxies: parametrization
Results for the two-point correlation function of galaxies:
 correlation is larger for smaller separations
 correlation tends towards zero for large r (or θ), satisfying the expected global homogeneity
 correlation can be represented by a power law for a considerable range of r:
ξ(r) = (r/r0)-γ
where:
• γ = inclination
• r0 = amplitude or length [conventionally the value of r for which ξ(r) = 1]
or, for angular correlations:
ω(θ) = A θ1-γ
 observed values, for different catalogs, although depend on limiting magnitude, morphological
type, luminosity, selection criteria, methods of analysis, etc, are in the ranges:
• 1.5 < γ < 1.8
• (3 < r0 < 7) h-1 Mpc
 maximum scales of power law are about 30 h-1 Mpc
 usually galaxy size halos in N-body simulations are able to reproduce the observed values
reasonably well
 correlation function may evolve with redshift
 Correlation Function of Galaxies: results – LCRS
Results for the two-point correlation function of galaxies:
Las Campanas Redshift Survey
r0  6.3 h-1 Mpc, γ  1.5
[see, e.g., Tucker et al. 1997, MNRAS 285, L5]
 Correlation Function of Galaxies: results – 2dFGRS
Results for the two-point correlation function of galaxies:
2dF Galaxy Redshift Survey
r0  5.1 h-1 Mpc, γ  1.7
[Hawkins et al. 2003, MNRAS 346, 78]
 Correlation Function of Galaxies: results – SDSS
Results for the two-point correlation function of galaxies:
Sloan Digital Sky Survey
r0  5.6 h-1 Mpc, γ  1.8
[Zehavi et al. 2005, ApJ 630, 1]
 Correlation Function of Galaxies
Comparisons to simulations:
 simulated galaxies
2dFGRS
- - dark matter particles

[Springel et al. 2005, Nature 435, 3597]
 Correlation Function of Galaxies: dependende on luminosity and color
SDSS (z ~ 0)
[Zehavi et al. 2005,
ApJ 630, 1]
 Correlation Function of Galaxies
DEEP2 (z = 1)
Simulation (z = 0)
[Coil et al. 2006, ApJ 644, 671]
Change of r0 and 
with Luminosity (mass)
LBGs
(z ~ 4)
[Kashikawa et al. 2006,
ApJ 637, 631]
[Springel et al. 2005,
Nature 435, 3597]
 Correlation Function of Galaxies: evolution
Simulation (galaxies)
2QZ
[Croom et al. 2005, MNRAS 356, 415]
[Caretta et al. 2008, A&A 487, 445]
 Luminosity Function of Galaxies: definition
ν(M,r) dM dV = φ(M) D(r) dM dV
where:
• ν(M,r) → number density of galaxies at r, that have absolute mag. between M and M+dM
• φ(M) → luminosity function: fraction of galaxies per unit mag. having absolute mag. in the
interval (M, M+dM)
• D(r) → density function: number density of galaxies (of all mags.) at r
φ(M) and D(r) are probability distributions. Some considerations about φ(M), or LF, are:
 it may be normalized:
∫-∞+∞ φ(M) dM = 1
In practice the normalization if very difficult because the samples are incomplete, specially in the
faint end where the LF grows exponentially (if not normalized it is sometimes called luminosity
distribution)
 it may be differential (as above) or integrated/cumulative:
Ф(M) = ∫-∞M φ(M’) dM’
 it may be represented either by (nonparametric) histograms or by (parametric) analytical forms
 it may depend on environment (field  clusters)
 it depends on the band of the magnitudes
 it depends on the morphological types mixture
 Luminosity Function of Galaxies: applications
 determination of selection effects (Malmquist
bias, p.e.) on particular parameter averages
in samples chosen by apparent mag.
 conversion of the projected (angular) CF to the
spatial (3-D) one
 calculation of the luminosity density (total lum.
of the Universe contained in galaxies) averaged
over a cosmologically interesting volume
 estimate how the galaxy masses are distributed
(in the expectation that all kinds of gals have
about the same M/L)
 studying the evolution of galaxies (history of
mergers, p.e.) by comparing FL at different z’s.
 Luminosity Function of Galaxies: preparation of data
 corrections to magnitudes: Galactic absorption (A), internal absorption, redshift dimming
(K-correction)
 observational selection effects:
• surface brightness effect (both very compact HSB and extended LSB objects can go
undetected, the former by confusing with stars or selection by diameter and the last
by surface brightness limit of detection)
• incorrect faint end (close to the limiting mag., brighter galaxies may be lost and fainter
ones my be included owing to mag. errors
 conversion to absolute magnitudes: require distance information (if by redshift, it is
intrinsically an approximation) for every sample galaxy,
except for clusters since one knows the membership or does statistical corrections for
background
 Luminosity Function of Galaxies: estimators
[Willmer 1997, AJ 114, 898]
Classical:
 galaxy absolute mags. are separated on bins of (Mi – ½ ΔM, Mi + ½ ΔM)
 N(Mi), of each bin, is divided by its respective volume V(Mi)
 since N(Mi) decreases for faint absolute mag., the uncertainties become higher
1/Vmax:
 for each galaxy in the bin, Vmax (the maximum volume for which an object with mag. M
can be detected) is calculated
 φ(Mi) is given by:
φ(Mi) = Σ 1/Vmax(Mk)
 [Felten 1976, ApJ 207, 700] has shown that 1/Vmax is equivalent to the Classical method
since both consider D(r) = cte (surely, an approximation).
φ/Ф:
 proposed by [Turner 1979, ApJ 231, 645] and [Kirshner et al. 1979, AJ 84, 951], and
improved by [Yahil et al. 1991]
consider the ratio between ν(M,r) dm dV and its cumulative integral:
ν(M,r) dM dV
=
φ(M) D(r) dM dV = φ(M) dM  dlog Ф(M)
∫-∞M ν(M’,r) dM’ dV
∫-∞M φ(M’) D(r) dM’ dV
Ф(M)
 integrating φ/Ф gives Ф(M), differentiating Ф(M) back gives φ(M)
 the problem with this method is the large statistical noise
 Luminosity Function of Galaxies: parametrization
Historical:
 [Hubble 1936, ApJ 84, 158]: Gaussian distribution of M, with σ = 0.84 mag
 [Zwicky 1957, Morphological Astronomy]: proposed a monotonic function: φ(M)  100.2 M
 [Kiang 1961, MNRAS 122, 263]: suggested: φ(M) = x3 → 0 < x < 3 ; x = M – M0
100.2x → 2.5 < x < 8 ; M0 = -22 -5 log h
 [Abell 1965, ARAA 3, 1]: proposed: log Ф(<m) = k1 + s1 m → m < m*
k2 + s2 m → m > m*
Schechter Function:
[Schechter 1976, ApJ 203, 297]: loosely based on the Press & Schechter formalism:
φ(M) dM = 0.4 ln(10) φ* 10-0.4 (α+1) (M-M*) exp[-10-0.4(M-M*)] dM
or, for the luminosities:
φ(L) dL = φ* (L/L*)α e-L/L* (dL/L*)
where:
• φ* → normalizing factor (number of galaxies per Mpc)
• M* → characteristic absolute magnitude
(or L*, the characteristic luminosity)
• α → asymptotic slope
 M* (or L*) and α are free parameters (usually fitted by
minimizing χ2)
 LF of different environments and kinds of galaxies can be fitted
by the Schechter function; only cD galaxies violate the LF shape
 it may converge somewhere to avoid the Olber’s paradox...
 Luminosity Function of Galaxies
 Luminosity Function of Galaxies: results for the field LF (optical)
Stromlo-APM [Loveday et al. 92]
LCRS [Lin et al. 96]
ESP [Zucca et al. 97]
CfA2 [Marzke et al. 94]
SSRS2 [da Costa et al. 94]
 Luminosity Function of Galaxies: results for the cluster LF
 Luminosity Functions of Galaxies
Mean Schechter
fit parameters
(and sources of
uncertainties)
<M*> = –20.47 ± 0.2
calibration errors
<> = –1.1 ± 0.1
insufficient statistics
at faint end (lack of
low luminosity galaxies)
<φ*> = (0.006 ± 0.001)
h0.68 Mpc–3
cosmic variance
[Liske et al. 2003,
MNRAS 344, 307]
 Luminosity Function of Galaxies: results for the type LF
[Binggeli, Sandage & Tammann 1988,
ARAA 26, 509] 
 Type Luminosity Function of Galaxies
Stromlo-APM
-- - - CfA2
SSRS2
....... S-APM
ESP [Zucca et al. 97]
LCRS [Lin et al. 96]
 Morphology-density Relation
Environmental effects on morphology:
 although known since the works of Hubble and Humason that early-type galaxies are more
common in clusters than in the field, the first large study of morphology segregation was
carried out by Dressler [1980, ApJ 236, 351]:
 Morphology-density Relation
Environmental effects on morphology:
 Whitmore et al. [1993, ApJ 407, 489] reanalyzed Dressler’s sample (about 6000 galaxies
in 55 clusters) an concluded that the “morphology-clustercentric distance relation” is more
fundamental than the “morphology-local density relation”
 Morphology-density Relation
Possible interpretations:
 heredity: certain types of galaxies do not form in regions that will become, say, rich clusters
 morphology change: certain kinds of galaxies do not survive unaltered in some environments
• S may have their gas stripped either by direct collision (tidal effects) or by ram pressure
of the intracluster gas, including in both cases “galaxy harassment” (high velocity
encounters). Thickening of disc and enhancing of bulge may turn S in S0.
• bulges can be enlarged by absorption of small gas-rich companions and successive
star-bursts
• S could turn into E via mergers.
 the fraction of E [f(E)] was found to correlate with the morphology of the cluster: systems
with large f(E) tend to be more regular, symmetric in appearance, and often with a cD at
its center; systems with small f(E) tend to be less “evolved”.
 Further readings
Papers:
 Bingelli, Sandage & Tammann 1988, ARAA 26, 509
 Dressler 1980, ApJ 236, 351
 Driver 2004, astro-ph 0409461