Galaxy Luminosity Function (LF) – Φ (yes, we’re using it again!)
- What are the relative numbers of galaxies of different luminosities?
δN = Φ(L) δL δV
δN is the # of galaxies that are located in volume δV with luminosities
between L and L+δL
Φ(L) has units # galaxies/(luminosity interval x volume)
∞
Can also be written in terms of absolute magnitude N = Φ(M) dM
-∞
where N is total number of galaxies per unit volume
Field Galaxy LF
•measure apparent brightness of galaxies in some sample (i.e. field)
•convert apparent to absolute magnitude from estimated distance
•apply k-correction if distance is more than 400 Mpc.
•divide # of galaxies in each magnitude/luminosity interval by volume of space
surveyed
•note that V is greater for intrinsically brighter galaxies in a survey with fixed
apparent magnitude limit
Problems:
•Malmquist Bias in magnitude limited surveys
•Distance estimate uncertainties
•enhancements (clusters/walls) and voids in galaxy distribution (for a given
magnitude, N can depend on spatial distribution as much as Φ)
Field LF reveals
1) # density of galaxies declines w/increasing L
2) decline increases sharply at some characteristic luminosity L*
α
Schechter (1976) found LF well
described by
Φ(L) = Φ*/L* (L/L*)α exp(-L/L*)
this is the SLF
Φ* = 5.5 x 10-3 galaxies/Mpc3
L* = 2 x 1010 Lsun
M*B = -20.4
α = -1.07±0.07
(Efstathiou, Ellis, & Peterson 1988)
In optical bands (from SDSS: Montero-Dorta & Prada 2008)
Move to K-band (IR) to avoid domination of light by star
forming regions and sample more “typical” stars
(Gardner at al. 1997)
Φ* = 5.5 x 10-3 galaxies/Mpc3
L* = 2 x 1010 Lsun
M*K = -23.8
α = -0.9 ± 0.2
2MASS (Kochanek et al. 2001)
Φ* = 4.1 x 10-3 galaxies/Mpc3
L* = 2 x 1010 Lsun
M*K = -23.4
α = -1.09 ± 0.06
 Even though different wavelengths are dominated by different stellar
types, LFs are virtually the same in optical and infrared.
Remarks about SLF
•works for all galaxies lumped together; individual galaxy types not as well fit
•Milky Way and Andromeda are close to or just fainter than L*
•galaxies with L ≥ 3L* are very rare
•SLF predicts too many galaxies at low L (as L  0, N  ∞)
Calculate total # of galaxies in some volume whose luminosity exceeds L
∞
∞
N(>L) = Φ(L’) dL’ = Φ* xα e-x dx = Φ* Γ(1+α)
L’=L
(recall Γ(z) = tz-1 e-t dt)
x=L/L*
for α ≤ -1, as L  0, N(>L)  ∞
Thus, SLF fails at low L with α ≤ -1
Calculate total luminosity of galaxies
∞
∞
Ltot = Φ(L) L dL = Φ* L* xα+1 e-x dx
0
x=0
Ltot = Φ* L* Γ(2+α) ≈ Φ* L*
with α = -1
So, Ltot is finite for α = -1 even though Ntot goes to infinity... phew!
Cluster Galaxy LF
•easier to obtain since all galaxies are at the same distance and together in the sky
•difficult since rich clusters are rare and typically at greater distances
Cluster LFs are reasonably well fit with SLF and thus are similar to the field.
Main differences:
•α is steeper (α ≈ -1.3)
•Φ* is larger due to increased density in clusters compared to field
•excess of bright galaxies (e.g. cD galaxies do not fit well into SLF)
LF divided by Morphological Type
-early and late type galaxies separate in LF
-somewhat similar but note differences
From Las Campanas Redshift
Survey with SLF fit to all galaxies
M*R = -21.1
α = -0.7 ± 0.05
LF divided by Morphological Type
-early and late type galaxies separate in LF
-somewhat similar but note differences
(from SDSS: Montero-Dorta & Prada 2008)
Jerjen & Tammann (1997) decomposed LF into morphological classes (B-band values)
Spirals – Gaussian shape with <M> = -17.6, σ = 1.4 mag
S0’s – Gaussian shape with <M> = -18.3, σ = 1.1 mag
E’s – Gaussian skewed to bright end with <M> = -17.7
Irr – SLF with M* = -15.8, α = -0.3
dE – SLF with M* = -16.7, α = -1.3
High L cut-off occurs at same
place for field and clusters
because of characteristic M* for
Sp’s and E’s and their similarity
Steeper slope for clusters comes
from increased population of
dE’s
Galaxy Groups
•More poorly clustered and containing fewer members than clusters
•Contain mostly disk systems – spirals and Irr
•e.g. Ursa Major group – of brightest 79 galaxies, only 2 are E’s
•Random velocities of galaxies are slower than cluster members
•σ = 700 – 1200 km/s for cluster galaxies
•σ = 100 – 500 km/s for group galaxies
 Thus, group galaxies affect each
other more than do cluster galaxies
– gravity has time to pull at gas and
stars
Stephan’s Quintet
•rare compact group
•109 Msun hot gas (Tx~107K)
•1010 Msun cool gas
•gas primarily intergalactic (galaxies are
mostly stripped of gas)
•gas heated to high temperatures due
to collisions
The Local Group
•about 35 – 40 galaxies within
~1.5 Mpc of MW
•brightest member – Andromeda
(M31)
•other spirals – MW and M33
•No bright ellipticals
•More than half are dE’s and
dSph’s
•Rest are irregulars
(Magellanic Clouds)
•Probably missing some objects
near the galactic plane
•e.g. Sagittarius dwarf just
discovered in 1994 (Ibata,
Milky Way, Andromeda and M33 emit 90% of
Gilmore & Irwin 1995)
visible
light from the Local
Group!
Sb,
Sbc, Sc
•More than 90% of Local Group
MW
& Andromeda
Irr/dIrr
galaxies have Mv fainter than -18 moving toward
dSph
(3 x 109 Lsun)
each other at 120
dE
km/s
Galaxy Collisions and Mergers
In groups, galaxy collisions and mergers can be important
Galaxy with mass M moves
with velocity V past star of
mass m in another galaxy
M/m ~ 108 - 1011
Galaxy M acquires motion perpendicular to path
Applies if galaxy M is small enough that its core radius rc << b. Also must have
 rapid flyby so that M and m do not move
toward one another
(when M=m, rs is the strong encounter radius discussed before)
Star m must gain an equal and opposite momentum, so the total kinetic energy in perpendicular motions is
The smaller mass star acquires most of the energy provided from forward motion of M, which consequently
changes as the velocity of M changes (ΔV parallel). Equating kinetic energy before with after
If ΔV parallel << V, we can drop higher order
terms to find slowing of M due to each star m
DKE^ 2G 2 m(M + m)
-DV|| »
=
MV
b2V 3
DKE^ 2G 2 m(M + m)
-DV|| »
=
MV
b2V 3
• Since the KE transferred to m grows as M2, a larger galaxy is braked more
quickly as galaxy mass increases.
• The faster M flies by (larger V), the less time it has to transfer energy to m
and the less speed it looses.
What is the deceleration of galaxy M due to it passing through a region of another
galaxy with n stars with mass m per pc3? Integrate over all stars to find the average
rate at which it slows:
dV max
2G 2 m(M + m)
4p G 2 (M + m)
= ò nV
2p b db =
nm ln L
dt bmin
b 2V 3
V2
b
-deceleration due to dynamical friction (brakes motion relative to background stars)
Λ=bmax/bmin = bmaxV2/GM
-Slower M moves, larger its deceleration
-For small satellite moving in galaxy’s dark halo, orbit decays and it spirals inward.
Friction force depends on mass density nm, not mass alone.
-A massive satellite will slow more quickly than a small one
LMC will likely merge with MW in several Gyr but most of MW’s globular clusters
(105 times lighter) will not.
Application of Dynamical Friction Formula
Considering a field velocity distribution with dispersion σ, dynamical friction equation can be written
where X = V/ σ √2
from Mark Whittle’s website @ UVa
When two galaxies pass by each other, energy taken from their forward motion is
transferred to random motions of stars.
Spiral or S0  thickens disk
After the interaction, the total KE (rotational and random) will be lower than before
the collision – why?
If they pass at high V (like in rich cluster) will
probably not become bound, but could
cause formation of bar or initiate spiral
structure (e.g M81 2-armed spiral likely
caused by flyby of close companions –
though this example in a group).
If they pass more slowly (like in a group), at speeds not much more than stellar
motions within the galaxies
•disrupts disk motion
•stars and gas can be pushed out to large radii
•if enough energy is drained, they will merge
•collision of gas-rich galaxies may cause a starburst
NGC 4676 (the Mice)
•R-band with HI overlay showing gas in tidal tails (this material
will not fall back into galaxy for several Gyr)
•results of simulation with two disks merged
•velocity of stars in simulations compared with gas velocity
Videos illustrating dynamical interactions of equal-mass disk galaxies (Barnes 1992)
1. Before their collision, two disk galaxies are rotated about their vertical axis. Each galaxy
contains a thin disk (blue), a central bulge (yellow), and a dark halo (red). One disk lies in the
orbit plane, and spins in the same direction that the galaxies circle each other. The other disk is
inclined by an angle of 71 degrees.
2. The galaxies approach each other, tidally interact as they pass each other, and fling out tidal
bridges and tails. Subsequently they reach maximum separation, fall back together, and merge.
The counter in the upper right shows time in units of 250 million years.
3. Just after the merger, the remnant is rotated about the vertical axis to illustrate the shells,
plumes, and tails typical of very young merger remnants.
Josh Barnes - IfA
Encounter and merger of two disk galaxies with
a 3:1 mass ratio. The galaxies orbit each other in
a clockwise direction, and the orbit plane is
viewed face-on. The larger disk lies in the orbit
plane but spins in a counter-clockwise direction,
while the smaller disk is inclined at an angle of
71 degrees.
Each frame shows the dark matter in red, the
disk stars in blue, and the bulge stars in yellow.
The nominal time-span between frames is 250
million years, or roughly one galactic rotation
period.
The main result is that disk galaxies can
sometimes accrete smaller (1/3-mass)
companions without being totally disrupted, in
contrast to the outcome of equal-mass mergers.
Here the large disk survives because it does not
couple strongly to a retrograde perturber.
(Barnes et al 1995)