Galaxies 626

Lecture 2: Galaxy number countsand luminosity functions

How much of the extragalactic background lightcan we identify, and how much is unidentified(unresolved)?

The next step is to count all the galaxies we can findand see how much light they contain

So we now want to form the galaxy number counts atall wavelengths

B, R, z/

15/ x 15/

B (1.7hrs)R (5.2hrs)z’ (3.9hrs)

B (27)R (26.4)z’ (25.4)

OpticalImage

Capak et al. 2003

Measuring Galaxy Luminosities

Galaxies, unlike stars, are not point sources

The Hubble Space Telescope can resolve (i.e. detectthe extended nature of) essentially all galaxies

Even from the ground, most galaxies can easily bedistinguished from stars morphologically andon the basis of their colors

Define the surface brightness of a galaxy I as the amount of light from the galaxy per square arcsecond on the sky

Consider a small square patch, of side D, in a galaxy at distance d:

Dd

Angle patch subtends on sky α = D/d

Measuring Galaxy Luminosities

Finding Galaxies in an Image

Finding galaxies in an image and constructing the numbercounts is the subject of the first student project

Essentially we look for all objects above a limiting surfacebrightness and covering more than a specified area

SExtractor is a commonly used package

20 cm Radio Image from the VLA

Number Counts

We then need to measure the fluxes of all the objects that wefind

The number counts are simply the number of objects we findin a given flux or magnitude bin per unit area of the sky

Measuring Galaxy Luminosities

Consider again the small square patch, of side D, in agalaxy at distance d

Dd

Angle patch subtends on sky α = D/d

If the luminosity of all the stars within the patch is L,then the total flux is

Define surface brightness as:

!

I =L /4"d2

D2/d

2=

L

4"D2

Units of I are mag arcsec-2; i.e., if a galaxy has a surface brightness of 20 mag arcsec-2, then we receive as many photons from one square arcsecond of the galaxy’s imageas from a star of 20th magnitude

Centers of galaxies have IB ~ 18 mag arcsec-2

!

F =L

4"d2

To measure the total amount of light coming from agalaxy, need to integrate the surface brightness acrossthe galaxy image. This leads to a related problem:galaxies do not have sharp well-defined edges

Typically integrate out to some limiting isophote; e.g.,sum up all the light coming from regions with surfacebrightness IB < 25 mag arcsec-2. Alternatively, we canmeasure to a surface brightness level relative tothe central surface brightness (say 1 percent)

This measure of the total galaxy brightness is called anisophotal magnitude---numerous variations are possible

An alternative and commonly used procedure is tomeasure the magnitudes in an aperture that contains mostof the galaxy light and to apply an average correction forthe light outside the aperture

These are usually called corrected aperture magnitudes

The UBL – Unidentified (Unresolved)Background Light

Key Issues:

• We need to know the EBL to high accuracy (removalof spurious foregrounds, calibration issues)

• We need to determine the extragalactic galaxycounts to very faint levels and properly measure theirtotal fluxes (calibration and getting the faintersurface brightness parts of the galaxies)

Counts from U through K extend well into γ<0.4 regime

N(m) deg-2 0.5mag-1 iν cgs Hz-1 sr-1

γ=0.4

Madau & Pozzetti (2000) MNRAS 312, L9

The UBL (above & beyond known counts)

Units

1 Jansky = 10-26 W m-2 Hz-1

1 nW m-2 sr-1 = 3000/λ(µm) MJy sr-1

Multiplying flux of the sources by N(m),the differential counts per unit magnitudeper unit area!

i" (m)=10#0.4(m+const)

N(m)

!

I" = i"-#

#

$ (m)dmUBL= EBL -

where

Contributions to the Optical EBL

Capak et al. 2004Ground-based data

Pozzetti et al. 2000Ground + HST data

Contributions from integrated counts

• Extrapolated counts enable contribution to integrated light fromknown populations to be evaluated as function of wavelength

• Find EBL = ∫ ν Iν dν = 55 nW m-2 sr-1; largest contribution in NIR

• Observational issues: isophotal losses and (1+z)4 redshift dimming

nW m-2 sr-1

CDF-N

1 Ms

370 sources 460 sq arcmin

503 sourcesCDF-N

2 Ms

D. Alexander

RED:0.5-2 keV

GREEN:2-4 keV

BLUE:4-8 keV

CDF-S

COMBINED

CDF-N + HAWAII

COMBINED

CDF-N

CDF-S

SSA13

Moretti et al. 2003

Measurements of the 2-8 keV background

How much unidentified light do we have?

Resolved backgroundMoretti et al. 2003

Hickox & Markevitch 2006

Measured backgrounds

Contribution from counts

The Galaxy Luminosity Function

The luminosities of galaxies span a very wide range---most luminous ellipticals are 107 more luminous than the faintestdwarfs

Luminosity function, Φ(L), describes the relative numberof galaxies of different luminosities

Definition: If we count galaxies in a representative volumeof the Universe, Φ(L)dL is the number of galaxies withluminosities between L and L + dL

Identical to the definition of the stellar luminosity function

Luminosity functions are easiest to measure in clustersof galaxies, where all the galaxies have the same distance

Coma Cluster

The Schecter Luminosity FunctionA convenient approximation to the luminosity function was suggested by Paul Schecter in 1976:

!

"(L)dL = n#L

L#

$

% &

'

( )

*

exp +L

L #

$

% &

'

( ) dL

L#

In this expression:• n* is a normalization factor which defines the overall

density of galaxies (number per cubic Mpc)• L* is a characteristic galaxy luminosity. An L* galaxy

is a bright galaxy, roughly comparable in luminosityto the Milky Way. A galaxy with L < 0.1 L* is a dwarf.

• α defines the `faint-end slope’ of the luminosity function. α is typically negative, implying large numbers of galaxies with low luminosities.

The Schecter Luminosity Function

The Schecter function is:• a fitting formula that does not distinguish between

galaxy types• as with the stellar mass function, parameters must

be determined observationally:n* = 8 x 10-3 Mpc-3

L* = 1.4 x 1010 Lsun

α = −0.7

…where Lsun = 3.9 x 1033 erg s-1 is the Solar luminosity

Illustrativenumbers

- related to mean galaxy density- luminosity of galaxies that dominate light output- lots of faint galaxies

The Schecter function plotted for different faint-end slopes,α = -0.5 (red), α = -0.75 (green), α = -1 (blue):

Properties of the Schecter Luminosity FunctionThe total number of galaxies per unit volume with luminositygreater than L is:

!

n = "(L)dLL

#

$

If α < -1, n diverges as L tends toward zero.Obviously unphysical, so Schecter law must fail for very lowluminosity galaxies (or have a larger alpha).

The luminosity density (units Solar luminosities per cubic Megaparsec) is given by:

!

"(L)LdLL

#

$

Dominated by galaxies with L ~ L* for typical value of α.

This integral is not expressible in terms of elementary functions

Significance of the Schecter Function

Does the Schecter form of the luminosity function have any deeper significance, or is it just a good fitting function?

• Luminous galaxies become exponentially rarer at highenough luminosities. Similar to a general result in cosmology (confusingly, Press-Schecter theory) -the number density of massive objects (e.g. clusters)drops off exponentially at high masses.

• But remember that Schecter function is a composite which includes all types of galaxies. Luminosity function of any individual class of galaxies looks very different - irregular galaxies typically have much lower luminosities than ellipticals for example.

Mass Function of GalaxiesFor stars, measurements of the luminosity function can beused to derive the Initial Mass Function (IMF).

For galaxies, this is more difficult:

• Mass to light ratio (M/L) of the stellar population depends upon the star formation history ofthe galaxy.

• Image of the galaxy tells us nothing about the amountand distribution of the dark matter.

More difficult measurements are needed to try and get at themass function of galaxies.

Local Inventory of StarsRelevant papers: Fukugita et al. 1998, ApJ, 503, 518

Fukugita & Peebles 2004, ApJ, 616, 643

Survey data: Cole et al. 2001, MNRAS, 326, 255 (2dF+2MASS)Kauffmann et al. 2003, MNRAS, 341, 33 (SDSS)

Stellar density: measure local infrared LF, Φ (LK), and scaleby a mean mass/light ratio (M/LK), which depends on theinitial mass function (IMF)

Most of the mass is in lower-mass stars, and most of the lightis from solar mass stars. One empirically tries to determinethe M/LK ratio, but the only information below 1 solar mass isfrom the disk of the Milky Way

One calculates all of the galaxy properties one would expectfor a particular IMF to see if it gives a self-consistentdescription

Some Initial Mass Functions

Mass (M)

Salpeter > 1 M

reproduces colorsand Hα propertiesof spirals

IMF < 1 M

makes minorcontribution tolight but is veryimportant formass inventory(Salpeterdiverges at low-mass end)

where the contributions are coming from

Stellar Mass Function at z=017,000 K-band selected galaxies with K<13.0

2dFGRS redshifts & 2MASS photometry

K Stellarmass

Stellar Mass Density at Present-DayCole et al find Ωstars h = 0.0014 ± 0.00013; M/LK = 0.73 (Miller-Scalo)

Ωstars h = 0.0027 ± 0.00027; M/LK = 1.32 (Salpeter)

Fukugita & Peebles: Ωstars h = 0.0027 ± 0.0005 (5% in brown dwarfs)

Ωgas = 0.00078 ± 0.00016 (H I, He I, H2)

NB: a comparison with WMAP result for the baryon density, only 6% of baryonsare in stars! Baryon density is dominated by gas (WHIM), not stars

Would expect the integral of the past activity to agree with the locally-determinedstellar density (Fukugita & Peebles 2004)

If both the local value and the cumulative star formation arecalibrated with the same IMF, then at least the description isconsistent

However, the IMF may not be invariant; one could beaveraging over a lot of IMFs over time, so the IMFuncertainties are significant

Student Project 2

The evolution of the galaxy luminosityfunctions with redshift will be the topic of thesecond project

Summary

• We have made astonishing progress in understanding theextragalactic populations across the spectrum in the last 10-20 years with EBL measurements and number counts acrossthe spectrum

• The EBL observations remain challenging and controversialbecause of instrumental effects and dominant foregroundsand determining how much light remains unidentified (theUBL) is still a subtle problem

• However, while there may be small (10%-ish) residualexcesses at many wavelengths there is not a large amountof unidentified light

• Local galaxy luminosity functions are also now wellestablished as a consequence of wide field surveys such as2MASS, SDSS and the 2DF and we can use these toestimate the local baryon mass function of galaxies