Whittle : EXTRAGALACTIC ASTRONOMY
7. ELLIPTICAL GALAXIES
(1) Introduction
(a) The Myths
Our view of Elliptical galaxies has changed greatly :
In the 1970s, Ellipticals were thought to be :
- Diskless bulges with deVaucouleurs (R1/4) profiles and constant
density (King) cores.
- Oblate spheroids flattened by rotation
- Void of gas and dust
- Contain a single ancient population of stars
- Relaxed dynamically quiescent systems
To a large extent, all of the above are now thought to be wrong.
(b) Subdividing the Elliptical Class
In what follows, it will be useful to consider three classes of
Ellipticals :
- Luminous : L greater than 1-few L*, MB
brighter than about -20
- Midsize (including massive bulges) : L between 0.1 L* & L*,
MB in the range -18 to -20
- Dwarfs : L less than 0.1 L*, or MB
fainter than -18.
Luminous and midsize have somewhat different properties, but form a single
sequence in mass.
dwarf Es are significantly different.
(c) Parameters
Here are a few recurring parameters we need to be familiar with :
- Surface Brightness (SB) : several symbols : IB(R) (flux
units), or
B(R) (mag/ss
units; B = B band)
- Total flux : (a) within projected radius L(<R), or (b)
integrated : Ltot (equivalent to MB)
- Effective Radius : Re defined as the half light
radius : L(<Re) = 0.5 Ltot [also
Ie = I(Re) and
e =
(Re)]
- Stellar Velocity Dispersion :
e =
< (<Re
) >
assumes a Gaussian projected stellar velocity distribution
if possible, aperture includes light out to Re
Also important, are properties of the core :
- Central Surface Brightness :
(0)
- Core Radius : rc or Rc, such that
I(Rc) = 0.5 I(0)
note : sometimes rc (or rb) refers to a
break in the near
nuclear light profile (eg HST Nuker group)
- Central Stellar Velocity Dispersion :
o =
(0)
Remember : I(R) and (R)
are independent of distance ! (for small redshifts)
(d) Deprojection
Note that all the above quantities are projected onto the sky.
Ultimately we want true 3D spatial information. ie we want to derive :
- the luminosity density j(r) from the surface
brightness I(R), where
- R is projected radius
- r is true (3D) radius
- for constant M/L ratio, j(r) and I(R) track the space and projected
mass densities
In general, (see diagram), with z2 = r2 - R2
and dz = r dr / (r2 - R2)1/2, we have
 |
(7.1) |
This is an Abel Integral equation, with solution
 |
(7.2) |
- for certain I(R) functions, j(r) can be expressed algebraically
- for smooth (fitted) profiles, evaluate the integral directly
- for noisy data, use the Richardson-Lucy iterative inversion
Note : if the image is elliptical, a unique inversion is only possible
for an axisymmetric figues viewed from the equatorial plane.
Just to orient ourselves, consider a single power law of index
(typically, 0.5 <
< 1.5)
We have :
These diverge in a number of circumstances :
- I(0), the central surface brightness, diverges for
> 0
- j(0), the central luminosity density, diverges for > -1
- Vc(0), the central circular velocity, diverges for
> 1
- j(infinity), the distant luminosity density, diverges for
< -1
- I(< infinity), the total
light & mass, diverge for < 2
(e) Observational Concerns
There are a number of practical difficulties facing accurate surface photometry
- Sky Subtraction is critical. Typically one aims for I(R) about
5% to 0.5% of sky.
Difficult especially with small CCDs which may not extend far enough
- Seeing affects the central regions : convolving them with
the PSF (typically Gaussian with PL wings) :
- I(R) turns over into a flat core for R < 1
- ellipticity decreases significantly for R < 4
- a4 is affected even further out
- Calibration is often difficult, with typical accuracy only ~5%
(2) Radial Light Profiles : Fitting Functions
Although the light profiles of Ellipticals are quite similar, there are
also subtle but important differences.
Here are some example light profiles [images]
Over the years, a number of analytic expressions have been used to fit
I(R) of Elliptical galaxies.
They each have their strengths and weaknesses :
(a) deVaucouleurs (R1/4) and Sersic (R1/n) Laws
deVaucouleurs noticed that for many ellipticals
R1/4
The fit is usually good over all but the inner and outermost regions
(typically 0.03 - 20 Re) [ images ]
The law is usually written :
 |
(7.3) |
It has the following properties :
- Ltot = 7.22
Re2 Ie
- I(0) = 2000 Ie
- < I(<Re)> = 3.61 Ie (which we abbreviate
to < Ie > and equivalently <
e >)
- Asymptotically, at small R, I(R)
R-0.8
while at large R, I(R)
R-1.7
- in terms of surface brightness :
(R) =
e +
8.325[ (R/Re)1/4 - 1 ] =
(0) + 8.325 (R/Re)1/4
- while originally purely empirical, Binney (1982) has shown that the
R1/4 law arises naturally from a reasonable distribution function.
Unfortunately, deprojection isn't straightforward (however, see Young (1975)
for tables of j(r) and other properties)
The deVaucouleurs law is a special case of a more general,
Sersic, law :
 |
(7.4) |
Where
- b = 1.999 n - 0.327 (N>1) ensures 0.5 Ltot =
I(<Re)
- n=4 gives the deVaucouleurs R1/4 law with b = 7.67
- n=1 gives an exponential profile with b=1.67
- it turns out (see below) that different n's fit the different classes of
Ellipticals
(b) Hubble-Reynolds Law
First Reynolds (1913) and later Hubble (1930) used the following function [ images ]
 |
(7.5) |
- for R < ro, I(R) is well behaved, with finite
(dI/dR)R=0
- for R>>ro we have I(R)
R-2 so
Ltot diverges (though only logarithmically).
- the related Hubble-Oemler profile avoids this by including an
exponential (tidal) cutoff at large R
(c) Modified Hubble Law, Isothermal and King Profiles
The following function avoids some of the above problems, and in addition
has other advantages:
 |
(7.6) |
- at large R >> ro, similar to the Hubble-Reynolds law :
I(R)
R-2 and
Ltot diverges
- however, it has a simple analytic expression for j(r) :
 |
(7.7) |
- at small r < 3ro, this j(r) is similar to isothermal models (though it differs at large r).
Isothermal models :
- are physically grounded : self-gravitating system with
Boltzman distribution in energy (potential + kinetic)
- are physically motivated : violent relaxation (at formation)
can lead to this Bolzmann distribution
- at small R < ro, I(R) turns over in a flat core, which
has known dynamical properties :
- central density
(0) = 9
(0)2/4
Gro2
(where (0) is the
central velocity disperson)
- core M/L ratio = (0)/j(0) where j(0)=0.495 I(0)/ro
- at large R, since j(r)
R-2 we
have a flat rotation curve, Vc ~ const
- however, because of this, at large R, we have I(<R)
R
and mass quickly diverges !
To avoid this divergence King modifed the energy distribution :
a modified Boltzmann distribution with cutoff above some threshold.
These King Models :
- are not analytic, but are solutions to an ODE (solved by simple
integration).
- they are two parameter functions :
- the "core radius" parameter, ro, is related to the
overall binding energy
(NB ro can be larger than
rc, the half-light isophote radius)
- the energy cutoff leads to a truncation in j(r) at a tidal
radius, rt.
- the models form a sequence in concentration c = log10(
rt/ro) [ images ]
- stellar velocity dispersion is ~const across the core, but drops outside
- good fits to Globular clusters are found for c ~ 0.75 - 1.75
- moderate fits to some ellipticals are found for c > 2.2
- c ~ 1.7 is reasonably close to the modified Hubble law.
(d) Dehnen Laws (including Hernquist and Jaffe Laws)
Motivated in part by an observed range in profile gradients, Dehnen (1993)
introduces a 3-parameter law :
 |
(7.8) |
with corresponding light profile :
 |
(7.9) |
Note several things :
A comparison of a number of different models are shown
here and
here.
(e) Central Regions : the "Nuker" Profile
The above functions aren't adequate for the nuclear regions, as imaged by HST
Lauer et al (1995) introduced a new function for these regions, called the
"Nuker" profile :
 |
(7.12) |
- this is a five parameter fit which describes two power laws
- for R >> Rb we have I(R)
R-
describing the outer power law
- for R << Rb we have I(R)
R-
describing the inner cusp or core
- the "break" between the two regions comes at Rb with
I(Rb)
- sets the
sharpness of the transition near Rb.
An example of two such fits to HST data is shown
here.
Notice the significant difference between the two, discussed below in 3b.
(3) Radial Light Profiles : Resulting Fits
In general, one should distinguish between the most nuclear regions and the
overall profile.
(a) Outside the Center
The various 2-parameter functions fit with similar quality (typically ~0.2 mag
over a 6 mag range).
The most commonly used is the R1/4 law.
These fitting functions do not, in general, reproduce the central regions very well
There is some real variation in the outer light profiles.
(i) Variation with Luminosity
Galaxies of different luminosity have somewhat different slopes [ images ]
The R1/4 law fits best near MB~ -21; too steep for
MB~ -22 and too shallow for MB~ -19
Sersic and Dehnen functions are useful with their variable slope parameters.
- Lower luminosity Es have steeper slopes :
(Sersic n< 4 and Dehnen/Jaffe
~ 2 fit well)
- Higher luminosity Es have shallower slopes :
(Sersic n > 4 and Dehnen/Hernquist
~ 1 fit well)
(ii) Variation with Environment
There is some evidence that outer light profiles can be affected by neighbors :
- Ellipticals in dense clusters have profiles that are cutoff
at large radii
likely caused by stars being lost due to tidal evaporation
- Ellipticals with a near neighbor can have a raised outer profile
likely caused by tidal heating which puffs up the outer envelope
(iii) cD galaxies
cD galaxies are well fit by the R1/4 law out to about 20Re
Outside this, their light profiles lie above the fit (eg I(R)
R-1.6),
in an extended halo [ images ]
This halo light may not come from the galaxy but from stars in the cluster
- the stellar velocity dispersion increases with radius, as expected
for cluster stars
(note : velocity dispersion usually drops with radius in normal Es)
- the isophotes can change to match the isopleths of the cluster
galaxy distribution.
(iv) Dwarf Ellipticals
There are two classes of Dwarf Ellipticals :
- compact dEs : these are quite rare, but are clearly a
continuation of their more luminous counterparts
eg M32 which has a reasonable R1/4 profile, perhaps slightly
steeper, as expected.
- diffuse dEs : these are common and are quite different
from the more luminous Ellipticals
The diffuse dEs have exponential light profiles (Sersic with
n ~ 1) [ images ]
Note that these are, however, not disks (which also have exponential
profiles).
This figure compares dSph with dS+Irr profiles
As we shall see, these diffuse dEs should not be thought of as
low luminosity ellipticals.
(b) Central Light Profiles
Before ~1975 the serious influence of seeing,
especially in photographic work, was not appreciated.
The belief in flat King-like cores was shown to
be incorrect with CCD images (eg Kormendy 1977)
Significant progress was only possible using HST (principally, by the
"Nuker" group).
For significant samples, "Nuker" profiles were fitted, and showed :
- There are very few cases where I(R) is flat at the center;
all continue to rise down to 0.1 arcsec.
- On the outskirts of the nucleus, all profiles are quite steep, j(r)
r-2
- Closer in, the profiles divide into two groups [ images ] :
- power laws : profile keeps rising steeply : j(r)
r-1.9
with I(R) diverging at R=0
- cuspy cores : profile breaks to shallower power-law : j(r)
r-0.8
with I(R) finite at R=0
- remarkably, these two types depend on the galaxy's total
luminosity :
- Nuclear power laws are found in Lower Luminosity Ellipticals
and Spiral bulges (L < ~L*)
- Nuclear cores are found in Higher Luminosity Ellipticals
The reasons for this are not yet well understood (see below, 4c).
(4) Correlations Between Parameters
There are many correlations between the various properties of Ellipticals.
The tightness of some are quite remarkable, and point to
an underlying homogeneity of this class of galaxy.
(a) Early 2-Parameter Correlations
(b) The 3-Parameter Fundamental Plane
The above 2-parameter correlations have considerable real scatter
(figure; viewgraph; B&M 4.43)
Furthermore, the residuals in one plot correlate with those in another.
This suggests we look for a tighter correlation among three
parameters :
-
a tilted plane of points in 3-D volume, which
- projects onto 2-D planes as the (looser) correlations seen above
The choice of the 3 parameters is not unique
Three choices have been studied --- they are essentially equivalent.
(i) Log Re, <
e >,
Log e
(Djorgovski & Davis 1987)
Here, Re is in kpc; <
e > is in
B mag/ss; e
is in km/s
[Note : we could have used
e rather
than <
e >
or even Ltot = 2 Le = 2 pi Re2 <
e > ]
Several statistical methods can identify/characterise correlations in n-dimensions :
- Principal Component Analysis (PCA)
- Multiple Linear Regression
- Partial Correlation Analysis
Using these, we find the equation of the Fundamental Plane to be :
-
Log Re = 0.36 <
e >
+ 1.4
Log e
+ const
[normal vector (-0.65, 0.22, 0.86) ]
Viewed edge on, the plane has very little scatter ~15% (see
figure)
(ii) The Dn -
Relation
(Dressler et al 1987 : Seven Samurai)
Before the F-P was found, a very tight 2-parameter correlation was identified
(figure) :
Dn vs
e, at which
Dn = Diameter (in kpc) where <
e > = 20.75
B mag/ss
(the actual value of 20.75 is not important)
It turns out that this choice of parameters renders the F-P essentially edge-on
Here's why :
For R1/4 law, integration gives Dn
Re
< Ie >0.8 or, equivalently
-
Log Dn = Log Re + 0.8 Log
< Ie > = Log Re - 0.32 <
e >
(since <
e >
= -2.5 Log < Ie > )
Substituting for Re in the F-P relation, we get :
- Log Dn + 0.32 <
e >
= 0.36 <
e > +
1.4 Log e
Log Dn = 1.4 Log e - 0.02 <
e >
and we see that the dependency on
< e >
has essentially vanished, leaving
Dn
e 1.4
: a tight 2-parameter correlation
(iii) Kappa Space :
1
2
3
(Bender et al 1993)
A deliberate attempt to render the F-P "edge-on" using more physical
parameters :
- 1
=
2-1/2 Log(
e2 Re )
Log M (M = Mass)
- 2
=
6-1/2 Log( e2 Ie2 / Re )
Log [ Ie (M/L)1/3 ]
- 3
=
3-1/2 Log( e2 / Ie / Re )
Log (M/L)
In this K-space we find tight projections in
1 vs
3 (see
figure 1 and
figure 2)
this suggests a narrow range of M/L which correlates weakly with total mass
(iv) The Physical Basis of the Fundamental Plane
The following gives some insight into the origin of the F-P relation :
Consider :
- < Ie > = ½ Ltot / pi Re2 (just a definition)
- M/Re = c
e2
(virial equilibrium, KE
PE; c = "structure
parameter" containing all details)
Taken together, these give :
- Re = (c/2pi) (M/L)-1
e2
< Ie >-1 or equivalently,
- Log Re = Log [(c/2pi) (M/L)-1]
+ 2 Log e - Log < Ie > or
- Log Re = Log [(c/2pi) (M/L)-1]
+ 2 Log e + 0.4 <
e >
(since <
e >
= -2.5 Log < Ie > )
So, if c and M/L are constants, then we expect
- Log Re = 2 Log
e
+ 0.4 <
e >
+ Log [(c/2pi) (M/L)-1]
Which is close to, but not qute, the F-P relation :
- Log Re = 1.4 Log
e + 0.36 <
e >
+ const
To bring these into agreement, we require :
- (2pi/c) (M/L) M1/5 L1/4
We conclude :
- The F-P is rooted principally in virial equilibrium
- To first order, the M/L ratios and dynamical structures of ellipticals
are very similar
This, in turn, suggests the populations, ages & dark matter properties are
highly uniform
- There is a weak trend for M/L to increase slightly with Mass
(× 3 across 5 magnitudes)
- At any point in this relation, the scatter on M/L is only ~10%
- The actual M/L values, ~10-20 (h=1), are consistent with no
dark matter (within Re)
- The narrow scatter on F-P and Mg2 -
relations place
limits on the ranges of ages and metallicities :
Ages ~ 10 - 13 Gyr; Z ~ 2-4 Z(solar)
- None of these relations seems to depend on environment
: internal properties are relatively robust
- Current work focusses on whether the F-P is different at higher
redshift (implications for evolution)
(v) Use of F-P in Distance Measurement
Much motivation for the above work was to improve methods of distance
measurement
In general, if a
V (km/s) correlates
with a luminosity or size, we have a distance indicator, eg :
- Tully-Fisher :
Vrot
vs MI
- Faber-Jackson :
e
vs MB
Both the F-P and the Dn-
relations yield
a physical length (kpc) from SB &
, with low scatter
(~10-15%)
This has been used to derive distances :
- used in the distance ladder to get Ho
- used with cz to map the peculiar velocity field and large scale flows
(c) Core Parameter Correlations
The previous section considered global parameters, defined on scales
~Re
One can instead consider core parameters, defined on much
smaller scales.
One needs to divide the results into Pre-HST and Post-HST :
(i) Pre-HST Results (eg Kormendy 1985)
This makes use of the best ground based data (CFHT; seeing ~ 0.7 arcsec)
Here, core parameters are defined near ~Rc, where
I(Rc) = ½ I(0)
The principal results are shown here for
bulges and ellipticals;
Also here or
here for wider plots which include
dSph, dS-Irr, and GCs (see also S&G 6.6).
Main results :
- Core parameters [ Rc,
(0),
o ]
are closely tied to global parameters [ MB ]
- more luminous galaxies have larger core radii, with
lower central surface brightness
- this extends the similar result for global parameters into the cores
- Extreme examples :
M32 has Rc = 0.3pc and
(0) = 11 V mag/ss !
M87 has Rc = 800 pc and
(0) = 17 V mag/ss
- there are three different families present, probably with
different formation scenarios :
- ellipticals and bulges (cDs at one end, M32 at the other)
- dwarfs : dSph, dS and Irr galaxies are radiacally different
they have much lower central SB,
they show the opposite trend : more luminous have higher central SB
they have exponential light profiles
We conclude that dSph and diffuse dEs are NOT simply low luminosity ellipticals,
they probably formed from dS or Irr galaxies which lost their gas.
- Globular clusters are different again
they have central densities comparable to the ellipticals,
but they do NOT extend the elliptical relations to low luminosities
- Note that the dwarfs and globular clusters fall well off the
F-P for ellipticals
(ii) Post-HST Results (eg Faber et al 1997, the "Nuker" group)
From 3b above, Nuker profile fits provide somewhat different
"core" parameters :
"break radius", Rb,
"break surface brightness" and Ib
add to these central dispersion,
o; and core
luminosity, Lcore = pi Rb2Ib
Recall that ellipticals and bulges fall into two classes :
- Power Law profiles with no break (Rb is an upper limit)
(lower luminosity galaxies)
- Core profiles with a break at Rb to a shallower
power law (high luminosity galaxies)
To some extent the "core correlations" depend on which group we are talking
about.
Here are the results :
- For core galaxies, core properties correlate with global
propertes
(figure 1 and
figure 2)
as before, less luminous galaxies have denser cores
scaling relations linking core and global properties are shown
here
- core galaxies define a fundamental plane
(figure)
the core F-P is approximately parallel to the global F-P
however, it is 30% thicker (more scattered)
this scatter may be caused by nuclear Black Holes.
- Power Law galaxies (Rb < 10pc, say) can have
densities ~103 higher than core galaxies
- Intermediate luminosities (MV -22 to -20.5) both
types exist, with range ~102 in Rb
- both inside and ouside this range (see below) :
- Core galaxies are boxy and rotate slowly
- Power Law galaxies are disky and rotate faster
These relations shed light on differing histories for the two classes
(see also below) :
-
dissipational or non-dissipational collapse or mergers lead to differing
central densities
- Black holes can alter core properties, especially if aquired by tidal capture
(5) The Shape of Elliptical Galaxies
To first order, isophotes are concentric, aligned, ellipses :
(figure)
ellipticity
= 1 - b/a,
eccentricity e = 1 - (b/a)2 [e =
(2 -
);
= 1 -
(1 - e)½ ]
recall, ellipticals are classified En where n = 10
The apparent ellipticity combines the true shape and
projection effects
Hence, unlike other morphological designations, n (in En) is not intrinsic
(a) 3-D Shapes
We start assuming that surfaces of constant density are ellipsoidal
 |
(7.13) |
where a, b, c may be functions of r
Basic questions :
- Are ellipticals predominently :
- oblate ie a = b > c (flying saucer)
- prolate ie a > b = c (cigar)
- triaxial ie a > b > c (smooth box)
- what is the distribution of true shapes
The distribution of observed axial ratios, N(b/a), is shown
here
Note : it has a rise from E0 to E2, followed by a decline to E7
Can we reproduce this from a random orientation of oblate or prolate ellipsoids?
- projected shape of ellipsoids is more complex than that of disk (eg B&M 4.3.3)
- However, difficult to generate rising distribution from E0 to E2
with just oblate or prolate
- Can be fit by distribution of triaxial, closer to oblate than
prolate :
- b/a ~ 0.95 (close to oblate)
- c/a ~ 0.65 (quite flattened)
- each have Gaussian dispersion ~0.2
The conclusion of triaxiality is supported by the presence of isophote
twists in many ellipticals
- PA of major axis changes with radius (so can
)
- cannot result from projection of oblate or prolate shapes
- intrinsically twisted galaxies are not stable
- can occur if triaxial with axial ratios varying with radius.
(b) Isophote Shapes
- isophotes are not exact ellipses : typical deviations ~few %
- in general, one can express the isophote as a Fourier series :
 |
(7.14) |
where :
a0 is the mean radius
a1, b1 define the ellipse center
a2, b2 define the eccentricity and position angle
a3, b3 are useful diagnostics of dust (asymmetries)
Note : a3 and all bn are zero for 4-fold symmetry
a4 defines the boxiness or diskiness (pointiness is better term)
- parameter a4/a0 typically in the range -0.02 to +0.04
a4 < 0 : boxy
a4 > 0 : disky
examples are shown
here
- a4 is not very sensitive to presence of an exponential
disk; which needs to be
- quite substantial (~40%), or
- viewed close to edge on.
- for sample of Es with a4 > 0, Rix and White (1990) find
data consistent with all having ~20% disk light
The a4 parameters are very important since they correlate with
many other variables (see below, § 8)
(c) Ripples and Shells
About 10% - 20% of Ellipticals have sharp edged "ripples", of amplitude 3-5%
Here is an example (NGC 3923).
These shells indicate recent accretion of disk galaxies (discussed more in
Topic 12)
(6) Kinematics of Elliptical Galaxies
(a) Methods of Analysis
If stars produced single isolated emission lines, their (projected) velocity
field would be easy to find :
the distribution of projected velocities N(v)
F(
) ie the
emission line profile
However, Ellipticals have complex absorption line spectra :
Similar to a K giant, but broadened by Doppler motion of the stars.[ images ]
Consider :
S() =
Stellar Template = a single star spectrum
N(v) = relative (normalised) number of stars of projected velocity v (ie vlos)
note N(v) is usually called the LOSVD (Line Of Sight Velocity
Distribution)
G() = observed (broadened) galaxy spectrum
Loosely speaking, G()
is the same as S()
convolved (smoothed) by N(v)
We observe G()
and S() and try
to obtain N(v).
Details :
- first rebin G()
and S() into pixels
of u = c Ln () space, ie km/s/pix rather than A/pix
our convolution is in fact, therefore : G(u) = S(u) ® N(u)
(where ® is convolution)
- sum several template stars to match the overall galaxy stellar
population
template mismatch is a principle source of error
- subtract a smooth continuum and normalise :
we dont want line strength (ie metallicity) to matter
- remove low frequencies (eg >50A continuum variations) and
high frequencies (noise)
this is achieved in Fourier space : apply filter to FT and transform back
**figure**
Several methods have been devised to extract N(v)
(i) Fourier Quotient (Sargent et al 1977)
Writing Fourier transforms (in k space) in bold face :
Starting with the galaxy spectrum :
|
G(u) = S(u) ® N(u) |
(7.15) |
From the convolution theorem we have :
|
G(k) = S(k) × N(k) |
(7.16) |
giving [ images ]
|
N(k) = G(k) / T(k) |
(7.17) |
We cannot simply inverse transform N(k) because noise
is introduced by the division
instead, we assume N(v) is Gaussian, so N(k) is also
Gaussian
Estimate N(k) by fitting a Gaussian to the quotient
from this fit, we quicky obtain N(v) as a Gaussian
thus, the LOSVD is characterized by just
cz, , and
(effective line
strength)
(ii) Cross Correlation (Tonry and Davis 1979)
It is not difficult to show that :
|
G(u) © S(u) = N(u) ® [S(u) © S(u)] |
(7.18) |
where © is cross-correlation and ® is convolution
(note S(u) © S(u) is also called auto-correlation)
in english : the cross-correlation of the galaxy and template spectra is
just
the cross-correlation of the template with itself convolved by the
broadening function.
In general, cross-correlating the galaxy and template produces an
offset peak
cross-correlating the template with itself produces a narrower peak at
zero offset
[ images ]
- the offset of the peak from zero gives the redshift
- the difference in shape of the two peaks gives the LOSVD
- in practice, only Gaussians are used to model N(u), yielding
(iii) Other Methods (more recent)
A number of related methods have been devised since these originals
Many try to extract more information from the LOSVD (ie deviations from
Gaussian)
These characterisations include : [ images ]
- sums of Gaussians of different velocity, width, and strength
- higher classical moments (ie k>2), eg
k =
µk /
k where
µ is the kth moment
although these are nicely tied to theoretical distribution functions
they weight the noisy wings too much, so alternatives are better
- Gauss-Hermite functions : hk (Gaussians multiplied by
a poynomial)
Ultimately, we really only deal with two extra parameters for the LOSVD :
- Skewness (k=3), ie asymmetric slope to higher/lower velocities
- Kurtosis (k=4), ie stubby/peaky
In practice, these parameters are evaluated as part of an optimized
2 fit
to the observed spectrum of the template convolved by a parameterised LOSVD
(b) Amount of Rotation
(i) Expectations (pre-1975)
Ellipticals should be rotaionally flattened
Stellar Dynamical analysis gives, assuming axisymmetry
and isotropic velocities :
- (Vr / e) ~ [(1-b/a)/(b/a)]½
= [
/ (1 -
]½
to compare with observations, lets define :
-
(Vr / )* =
(Vr / e)observed /
(Vr / e)expected
(ii) Results
The rotation amplitudes results are shown in a few ways :
We conclude :
- for luminous ellipticals, (Vr / )*
< 1
not rotationally flattened
flattened by velocity anisotropy
further evidence for triaxiality
- lower luminosity ellipticals and bulges have (Vr / )*
~ 1
are rotationally flattened
h3 is large and has opposite sign to Vr
this suggests we have a broad non-rotating component plus a narrow
rotating component
in truth, there may be a continuous distribution
Interestingly, (Vr / )* correlates even better with a4/a
as expected :
- disky galaxies rotate
- boxy galaxies dont rotate
(c) Axis of Rotation
Naively, for an axisymmetric rotating galaxy, one expects :
- major axis slit should show maximum rotation
- minor axis slit should show no rotation
- ie kinematic axis = photometric minor axis
While for a triaxial rotating galaxy, one expects :
- the projected photometric minor axis need not align with any true axis
- the rotation axis can be anywhere in the plane defined by the longest and
shortest axes
What do the data suggest ?
Let 
= the projected
kinematic misalignment
= angle between kinematic
and photometric minor axes
For some fiducial radius, Rf, a good estimate of this is :
est ~
arctan [Vr(Rf) minor axis / Vr(Rf)
major axis]
A histogram of est shows [ images ]
- most are ~ 0
- some are 0 - 90
- significant minor-axis rotation occurs in boxy Ellipticals
(figure)
As before : while many Ellipticals are
close to axisymmetric, some (the boxy ones) are clearly triaxial [ images ]
(d) Kinematically Distinct Cores and Dust Lanes
~25 % Ellipticals show a separate, rotating component
in the nuclear regions (~1kpc; 0.1-0.3 Re)
These are called Kinematically Distinct Cores (KDC)
Projection effects and difficult detection suggest maybe
30%-60% Ellipticals have KDCs.
Example is shown
**here** and **here**
The KDCs show the following :
-
rapid rotation (Vr / )* > 1
with a range from "warm" to "cold" : Vr / = 1 to 4.5
- kinematic axis aligned with the photometric axis
- some KDCs even counter-rotate relative to the host
Clearly, they have a different (later?) origin than the main galaxy
- they have higher metallicity than the rest of the galaxy
**figure**
- photometrically, they are difficult to identify (eg not necessarily
disky isophotes) :
they dont contain much mass
kinematically prominent in LOSVD because they have low
(eg large h3)
-
may be related to subtle gas/dust lanes/disks seen in many
(~40%) ellipticals
often randomly aligned at large radii but aligned with minor axis at small radii
extreme example (NGC 5128) shown
**here**
KDCs (and dust lanes) are likely to be a byproduct of dissipational
tidal capture
-
gas and/or star system captured
-
dissipation (loss of orbital energy) occurs :
stellar system decays by dynamical friction
gas settles, loosing energy by line radiation
-
angular momentum (AM) inherited from merger (not from host)
at large radii we have random orientation (of gas/dust)
at center, torques/precession aligns with minor axis
-
gas disk undergoes star formation to generates a stellar disk
- stars age and disk becomes photometrically difficult to identify
Conclusion :
formation of ellipticals via single event is only part of story
ongoing mergers/accretion plays at least some role in construction
of present-day ellipticals
(7) Mass to Light Ratios
In principle :
Stellar velocities & radius give Mass
Photometry gives Light
Together, we get M/L ratios
In practice, not so straightforward :
- ideally, need full velocity distribution function at each location
- now possible to build reasonable models using LOSVD (see Topic 8)
- usually, however, we need to assume nearly isotropic velocity field.
(a) Inner Parts
Simple estimates :
assume approximately isothermal and fit a King profile (see 2c above)
gives for central luminosity density, central mass density and central M/L :
- j(0) = I(0) / 2ro
- (0) =
9 (0)2
/ 4 G ro2
- M/L = 9 (0)2
/ 2 G I(0) ro
Typically, M/L ~ 10 h M
/ LB,
so
dark matter does not dominate in the center
(b) Outer Parts & Halo
For proper analysis, need to consider velocity anisotropy
-
r =
radial velocity dispersion
extreme : radial orbits
-
=
tangential velocity dispersion (assume
=
)
extreme : circular orbits
Define anisotropy parameter : = 1 -
< 2 > /
< r
2 >
We have three cases :
-
= 0
: isotropic
-
< 0
: tangential anisotropy
-
> 0
: radial anisotropy
For Jaffe models with = const, stellar dynamics gives (see Topic 8) :
and so for three extreme cases :
- isotropic ( = 0) : M(r) = 3 2r / G
- fully tangential ( = -
) : M(r) = 2 2r / G
- fully radial ( = 1) : M(r) =
Notice that M(r) is sensitive to if there is strong radial anisotropy.
Can we measure the anisotropy ? Just now able to
affects
h4 : the LOSVD kurtosis
**figure**
-
tangential (
< 0) gives stubby LOSVD with h4 < 0
-
radial (
> 0) gives peaky LOSVD with h4 > 0
These type of measurements have only recently been achieved
**figure**.
Note that if
increases at large R, we know
is increasing
in this case Dark Matter is clearly present.
In practice, the most distant tracers of the potential are GCs and PN.
They do suggest DM halos are present
**figure**
However, better methods exist (see Topic 17)
(8) Two Kinds of Ellipticals : Boxy and Disky
Summary of properties which differ for boxy
(a4 < 0) and disky (a4 > 0) ellipticals
and bulges :
| Property | Boxy (a4 < 0) | Disky
(a4 > 0) |
| Luminosity | high : MB < -22 | low :
MB > -18 |
| Rotation Rate |
slow/zero :
(Vr / )* < 1 |
faster :
(Vr / )* ~ 1 |
| Flattening | velocity anisotropy | rotational |
| Rotation Axis | anywhere | photometric minor axis |
| Velocity Field | anisotropic | nearly isotropic |
| Shape | moderately triaxial | almost oblate |
| Core Profile | cuspy core | steep power law |
| Core Density | low | high |
| Radio Luminosity | radio loud and quiet
1020 - 1025 W/Hz | radio quiet
< 1021 W/Hz |
| X-ray Luminosity | high | low |
Some of these are shown here :
a4 vs V/sig; ellipticity; offset from F-P;
radio luminosity :
**figure** and **figure**
a4 vs V/sig; minor axis rotation :
**figure**
a4 vs radio luminosity; X-ray luminosity; :
viewgraph
Note that to first order : Boxy and Disky galaxies have the same :
- color-magnitude relation
- Mg2 vs
relation
- Fundamental Plane relation
It is still unclear quite how to interpret this dichotomy :
The two types may be closely related, or may have quite different histories
Semi-empirically, Kormendy and Bender suggest a modification to the
Hubble diagram
**figure**
- Disky Ellipticals form an extension of the S0s
- Boxy Ellipticals lie at the extreme left end
they may or may not be related to the other ellipticals and S0s
This all has important implications for Elliptical Formation
(9) Formation of Ellipticals
(Note Mergers and Galaxy Formation discussed more in Topics 12 & 19)
Still unclear -- but we have made progress
(a) Two scenarios discussed
(i) Monolithic Dissipative Collapse
- Early massive gas cloud undergoes dissipative collapse
- Huge starburst during collapse
Note : sub-mm detection of ~1010 M
cold gas at
z ~ 2-3 with high SFR.
- clumpiness during collapse
violent relaxation ~ isothermal
incomplete violent relaxation non-isothermal & non-isotripic
- probably rotate "rapidly" "Disky" Ellipticals ???
(ii) Hierarchical Mergers
- early universe much denser : eg z ~ 2 density ~ 27 times higher than present
- mergers/interactions probably common
- sequence of galactic mergers, starting with pre-galactic substructures
- galaxies continuue to grow during z ~ 1-2
Note : HST finds old ellipticals at z ~ 0.5
- galaxies fall into clusters and merging ceases (encounter velocities too high)
- random accretions low AM & anisotropic "Boxy" Ellipticals ???
(b) Relevant Issues & Problems
(c) Hybrid : Merger Induced Dissipational Collapse
Kormendy & Sanders (1992) combine the two formation scenarios :
Similar to Ultra-Luminous Infra-red Galaxies (ULIRGs; eg Arp 220 etc;
fig_1 and
fig_2), which
- are mergers, showing tidal debris etc
- have high central densities ~102 Mpc-3,
similar to the central stellar densities of ellipticals
- have similar gas mass and stellar mass in the central regions
dissipation has occurred, sending the gas to the center
- there is a huge ongoing starburst, creating a homogeneous metal
rich population
We have, in these systems, a substantial dissipative collapse which is
associated with a major merger
They speculate that the ULIRGs are ellipticals caught in formation
Interestingly, ULIRGs are also thought to be proto-quasars :
- feeding (and possible creation of) nuclear black holes
- star formation luminosity comparable to quasar luminosity
- currently hidden by dust (which re-radiates in the far IR)
- ultimately becomes visible when surrounding gas swept clear
The ULIRGs are locally rare --- maybe they were common at high-z
Elliptical formation may be associated directly with the quasar era ?
