Whittle : EXTRAGALACTIC ASTRONOMY

 
     
 

1 : Preliminaries	6 :   Dynamics I	11 : Star Formation	16 : Cosmology
2 : Morphology	7 :   Ellipticals	12 : Interactions	17 : Structure Growth
3 : Surveys	8 :   Dynamics II	13 : Groups & Clusters	18 : Galaxy Formation
4 : Lum. Functions	9 :   Gas & Dust	14 : Nuclei & BHs	19 : Reionization & IGM
5 : Spirals	10 : Populations	15 : AGNs & Quasars	20 : Dark Matter

(1) Introduction

• Diskless bulges with deVaucouleurs (R^1/4) 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

• Luminous : L greater than 1-few L_*, M_B brighter than about -20
• Midsize (including massive bulges) : L between 0.1 L_* & L_*, M_B in the range -18 to -20
• Dwarfs (dE & dSph) : L less than 0.1 L_*, or M_B fainter than -18.

• Surface Brightness (SB) : I(R) (flux units), µ(R) (mag/ss units). I_B is for B band, etc
• Total flux : (a) within projected radius L(<R), or (b) integrated: L_tot (equivalent to M_B)
• Effective Radius : R_e defined as the half light radius: L(<R_e) = 0.5 L_tot   [c.f. I_e I(R_e) ]
  
• Stellar Velocity Dispersion: (R)
  Assumes a Gaussian projected stellar velocity distribution
  Usually single central aperture [ideally, includes light out to R_e, giving _e = < (<R_e ) >   ]

• Central Surface Brightness: I₀ = I(0)
• Core Radius: r_c or R_c, such that I(R_c) = 0.5 I(0)
  Sometimes r_c (or r_b) refers to a break in the near nuclear light profile.
• Central Stellar Velocity Dispersion: _o = (0)

• The luminosity density j(r) from the surface brightness I(R), where
• r is true (3D) radius and R is projected radius
• For constant M/L ratio, j(r) and I(R) track the space and projected mass densities [i.e. (r) and µ(R) ]

(7.1)

This is an Abel Integral equation, with solution

(7.2)

• There are a few useful I(R) & j(r) pairs that can both 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)

• I(R)    R ^-         Note: I(0) diverges for > 0
• j(r)     r ^{- - 1}      Note: j(0) diverges for > -1
• < I(<R) > R ^{2 -}      
• L(<r) r ^{2 -}         Note: L_tot diverges for < 2
• V_c     r ^{(1 - )/2}     Note: V_c(0) diverges for > 1

• Sky Subtraction is critical. Typically one aims for I(R) about 5% to 0.5% of sky. [image]
  Difficult especially with small CCDs which may not extend far enough

• Seeing affects the central regions: convolving with the PSF (e.g. Gaussian + 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

(2) Radial Light Profiles: Outside The Core

There is a long history of trying to fit these brightness profiles.
Reynolds (1913) and Hubble (1930) used: I(R) = I(0) / (1 + R/R₀)²

A modified form has the benefit of having an analytic deprojected density:
I(R) = I(0) / [1 + (R/r₀)²]   with   j(r) = I(0) / 2r₀[1 + (r/r₀)²]^3/2

Today's preferred fitting functions are a little different.

(a) Fitting Functions

(i) deVaucouleurs (R^1/4) and Sersic (R^1/n) Laws

deVaucouleurs noticed (1948) that for many ellipticals µ R^1/4
The fit is usually good over all but the inner and outermost regions (typically 0.03 - 20 R_e) [image]
The law is usually written :

(7.3)

It has the following properties :


• L_tot = 7.22 R_e² I_e
  
• I(0) = 2000 I_e
  
• < I(<R_e)> = 3.61 I_e   (which we abbreviate to < I_e > and equivalently < µ_e >)
• Asymptotically, at small R, I(R) R^-0.8 while at large R, I(R) R^-1.7
  
• In terms of surface brightness: µ(R) = µ_e + 8.325[ (R/R_e)^1/4 - 1 ]   =   µ(0) + 8.325 (R/R_e)^1/4
  
• While originally purely empirical, Binney (1982) has shown that the R^1/4 law arises naturally from a reasonable distribution function.

The deVaucouleurs law is a special case of a more general, Sersic (1963,1968), law:

(7.4)

Where


• b = 1.999 n - 0.327 (n>1) ensures 0.5 L_tot = L(<R_e)
• n=4 gives the deVaucouleurs R^1/4 law with b = 7.67
• n=1 gives an exponential profile with b=1.67

• A key feature is that the slope in µ(R) vs log R is a steadily increasing function of R: [image]
      d logI / d logR   = -(b/n) (R/R_e)^1/n     -2 (R/R_e)^1/n  
    higher n means slower roll-over in the gradient.
    inside R_e higher n means steeper profile; outside R_e higher n means shallower profile.
    higher n is more concentrated (e.g. meaning higher values of R90/R50).
  (Note: at R_e all Sersic profiles have logarithmic slope of -2.)

• Unfortunately, deprojection isn't straightforward, although Ciotti (1991 o-link) give approximations.
  A comprehensive analysis of the Sersic Law is given by Graham & Driver (2005 o-link)

It turns out (see below) that different n's fit the different classes of Ellipticals

(ii) Double Power-Law (Dehnen) Laws

An alternative approach is to:
    (a) Choose a double power law to match inner and outer slopes
    (b) Specify the space luminosity density, j(r), rather than the projected light I(R).

One well-studied example is:

(7.5)

Clearly, for r << r₀ we have j(r) r ^- and for r >> r₀ we have j(r) r ^-

Specific models include:

• = 4   as a group were introduced by Dhenen (1993), with nice analytic properties.
• = 4   = 1   is called the Hernquist law
• = 4   = 2   is called the Jaffe law.
• = 4   = 3/2   is closest to the deVaucouleurs R^1/4 law
  
• = 3   = 1   is the NFW profile used for dark matter halos (where (r) replaces j(r) ).

For example the Jaffe model has the following properties:
• r_o contains half the unprojected light
• R_e = 0.76 r_o contains half the projected light
• Expressions for the integrated (aperture) light, and gravitational potential are:

(7.6a)     (7.6b)

• The projected light profile has an analytic form (a=R/r_o) :

(7.7)



• An example of a Jaffe model fit to NGC 3379 is shown here [image]

(b) Resulting Fits

(i) Variation with Luminosity

Galaxies of different luminosity have different concentrations (n-index) [image]
The R^1/4 law fits best near M_B~ -21.
Fitting Sersic profiles to find n-index gives:
• Lower luminosity Es less concentrated, n < 4   (Jaffe ~ 2)
• Higher luminosity Es more concentrated n > 4   (Hernquist ~ 1)

(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 R^1/4 law out to about 20R_e
Outside this, their light profiles lie above the fit (eg I(R) R^-1.6), in an extended halo [image]
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) Ripples and Shells

About 10% - 20% of Ellipticals have "ripples", of amplitude 3-5% [image]
These shells indicate recent accretion of disk galaxies (discussed more in Topic 12)
Clearly, mergers play a role in the formation of at least some ellipticals.

(v) Dwarf Ellipticals & Dwarf Spheroidals

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 R^1/4 profile, perhaps slightly steeper, as expected.
• Diffuse dEs : these are common and are quite different from the more luminous Ellipticals
  Kormendy calls these "Spheroidals" to keep the distinction clear
  They extend in luminosity down to include the Milky Way dwarf spheroidals, like Leo I etc.
These Spheroidals have exponential light profiles (Sersic with n ~ 1) [image]

(3) Nuclear Regions

Early (pre-1975) work suggested that I(R) turned over in a flat core of constant density
This was naturally understood in terms of isothermal and King models (see below)
However, the serious influence of seeing, especially in photographic work, had not been appreciated.
The existence of flat cores was shown to be incorrect with CCD images (eg Kormendy 1977)
Significant progress was only possible using HST.

In general, the above functions fail to match the nuclear regions very well.
Within a "break radius" there can be deviations both below and above the best fit to the outer parts.

(a) Fitting Functions

(i) Isothermal and King Profiles

Historically, isothermal and King models were used to understand "flat cores": [image]
Because they yield simple physical results, it is worth looking at them briefly.
• They assume a self-gravitating spherical system with isotropic velocity dispersion.
• They have a Boltzmann distribution in star energy (potential + kinetic) (see Topic 8.8b,c)
  
• Their projected light, I(R), turns over in a flat core, with known dynamical properties:
  • Within 3r₀ it approximates the modified Hubble law: I(R) = I(0) / [ 1 + (R/r₀)²]
  • It has a central density (0) = 9 (0)²/4 Gr_o²
    
  • It has core M/L ratio = (0)/j(0) where j(0)=0.495 I(0)/r_o

• At large R, isothermal models have j(r) R^-2 with divergent mass.
  
• King models introduce a cutoff in energy that truncates the outer profile yielding finite mass.

Important Note: These models in general do not make good fits to Elliptical galaxies.
They are included here (i) for historical reasons, and (ii) they yield quick estimates of (0) and M/L.

(ii) The Core Sersic Profile

One can combine an inner power law and a Sersic outer profile: [image]

An alternative approach is simply to measure the inner deviation from a best fit Sersic law.
This was the approach of Kormendy et al (2009 o-link)

(iii) The Two-Power-Law "Nuker" Profile

Lauer et al (1995 o-link) introduced a broken power-law function to fit around the break radius.
They call it the "Nuker Profile" (referring to their team's name):

(7.8)

• This is a five parameter fit which describes two power laws: [image]
• For R >> R_b we have I(R) R^- describing the outer power law
• For R << R_b we have I(R) R^- describing the inner cusp or core
• The "break" between the two regions comes at R_b with I(R_b)
• sets the sharpness of the transition near R_b.

(b) Results: Two Types, Cores & Power-laws

For significant samples, "Nuker" profiles were fitted, and showed : [image]


• There are no cases where I(R) is flat at the center; all continue to rise down to 0.1 arcsec.
• The inner profiles divide into two groups:
  • 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

• 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

There now seems to be three types of "Elliptical" galaxies:
[image] In order of increasing total luminosity:

Spheroidals: (diffuse dEs and dSph).

• Luminosity profiles more like disks (n ~ 1).
• They follow a sequence in M_V vs µ(0) which is similar to disks.
• In the cube R_e, M_V, µ_e, they are distinct from true ellipticals.
• Local group examples have multiple age populations (e.g. 10 Gyr + 6 Gyr).

    They are gas stripped dIrr and dS galaxies.

Coreless Ellipticals:

• Have central profiles with no break, a steep inner power law to the smallest radii.
• They are lower luminosity Ellipticals, fainter than M_V ~ -20.5.
• They have high central brightness, decreasing to lower luminosity galaxies.

    They may have formed from wet mergers of other ellipticals.
    Gas moves to center to form new stars, leaving a denser core.

Core Ellipticals:

• Have central profiles with a break to shallower slope -- a "cuspy (not flat) core"
• They are luminous Ellipticals, brighter than M_V ~ -20.5.
• They have relatively low central brightness, increasing to lower luminosity galaxies.

    They may have formed from dry mergers of other ellipticals (no gas disippation).
    Maybe binary black holes "scour out" central regions.

(4) Scaling Relations

(i) Color-Magnitude and Similar Correlations

Several correlations exist between :
• Parameters tracking metallicity (and/or age): (B-V) and Mg₂ strength
• Parameters of total galaxy mass: M_B and _e

Color-Magnitude Relation

• More luminous ellipticals are slightly redder [image]

Mg₂ vs Velocity Dispersion
• Galaxies with deeper potentials have stronger Mg₂ [image].

We conclude: more massive galaxies are more metal rich (stronger Mg₂ & UV blanketing)
The reason: deeper potentials hold ISM longer allowing metals to build up
Note: there are similar correlations within individual galaxies
Suggests metallicity in fact correlates with escape velocity [image].
 

(ii) Size & Luminosity vs Surface Brightness (Kormendy) Relation

A couple of correlations suggest larger, more luminous galaxies have lower surface brightness

< I_e > correlates with R_e : Kormendy Relation [image]


• R_e < I_e > ^{-0.83 +/- 0.08}

< I_e > correlates with L_tot :

• L_tot < I_e > ^-2/3
• this follows from the above relation, given L_e = 1/2 L_tot = pi < I_e > R_e²

We conclude: larger and more luminous galaxies are fluffier with lower densities
An interpretation is not yet too clear, though galaxy formation models must explain it.
One inference: low-luminosity ellipticals formed with more gaseous dissipation than giant ellipticals.

(iii) Luminosity vs Velocity Dispersion (Faber-Jackson) Relation

Faber & Jackson (1976) first found that more luminous ellipticals and bulges have deeper potentials

_e correlates with L_tot :   [image]


• L _*ⁿ with 3 < n < 5  
• The scatter is ~0.6 mag (greater than measurement errors)
• A very rough argument shows why this might apply :
  V² GM/R and SB L/R² (independent of distance)
  Squaring the first and substituting the second, we get L x SB x (M/L)² V⁴
  If we assume M/L and SB for ellipticals doesn't vary much, then we have L V⁴
  This underlies both the Tully-Fisher and Faber-Jackson relations.

• A tilted plane of points in 3-D volume, which
• Projects onto 2-D planes as the (looser) correlations seen above
• One example is: Log R_e = a Log + b Log I_e + c

(i)  Log R_e,   < µ_e >,   Log _e     (Djorgovski & Davis 1987)

Here, R_e is in kpc; < µ_e > is in B mag/ss; _e is in km/s
[Note : we could have used µ_e rather than < µ_e > or even L_tot = 2 L_e = 2 pi R_e² < µ_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 R_e = 0.36 < µ_e >   +   1.4 Log _e   +   const       [normal vector (-0.65, 0.22, 0.86) ]

Viewed edge on, the plane has very little scatter ~15%   [image]

(ii)  The D_n - Relation     (Dressler et al 1987 : Seven Samurai)

Before the F-P was found, a very tight 2-parameter correlation was identified: [image]
D_n vs _e,   where
D_n = Diameter (in kpc) where < µ > = 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 R^1/4 law, integration gives D_n     R_e < I_e >^0.8   or, equivalently
• Log D_n   =   Log R_e   +  0.8 Log < I_e >  =   Log R_e   -   0.32 < µ_e >
  (since < µ_e >   =   -2.5 Log < I_e > )
Substituting for R_e in the F-P relation, we get :
• Log D_n   +   0.32 < µ_e >   =   0.36 < µ_e >   +   1.4 Log _e
  Log D_n   =   1.4 Log _e   -   0.02 < µ_e >
and we see that the dependency on < µ_e > has essentially vanished, leaving
D_{n     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 R_e )           Log M     (M = Mass)
• ₂   =   6^-1/2 Log( _e² I_e² / R_e )     Log [ I_e (M/L)^1/3 ]
• ₃   =   3^-1/2 Log( _e² / I_e / R_e )     Log (M/L)

In this K-space we find tight projections in ₁ vs ₃ [image]
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:
• < I_e >   =   ½ L_tot / R_e²     (just a definition)
• M/R_e   =   c _e²     (virial equilibrium, KE PE; c = "structure parameter" containing all details)
  
Taken together, these give:
• R_e   =   (c/2) (M/L)^-1   _e²   < I_e >^-1     or equivalently,
  
• Log R_e   =   Log [(c/2) (M/L)^-1]   +   2 Log _e   -   Log < I_e >     or
  
• Log R_e   =   Log [(c/2) (M/L)^-1]   +   2 Log _e   +   0.4 < µ_e >     (since < µ_e >   =   -2.5 Log < I_e > )

So, if c and M/L are constants, then we expect
• Log R_e   =   2 Log _e     +   0.4 < µ_e >   +   Log [(c/2) (M/L)^-1]

Which is close to, but not quite, the F-P relation:
• Log R_e   =   1.4 Log _e   +   0.36 < µ_e >   +   const

To bring these into agreement, we require:
• (2/c) (M/L)     M^1/5     L^1/4

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)
• The actual M/L values, ~10-20 (h=1), are consistent with no dark matter (within R_e)
• 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

(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   :   V_rot vs M_I
• Faber-Jackson   :   _e vs M_B

Both the F-P and the D_n- 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 H_o
• Used with cz to map the peculiar velocity field and large scale flows

(5) The Shape of Elliptical Galaxies

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

(7.9)

where a, b, c may be functions of r

Basic questions :

• Are ellipticals predominently: [image]
  • 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: [image]
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 ~ 0.98 (close to oblate)
  • c/a ~ 0.69 (quite flattened)
  • each have Gaussian dispersion ~0.11

The conclusion of triaxiality is supported by the presence of isophote twists in many ellipticals: [image]
• 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.

• Isophotes are not exact ellipses : typical deviations ~few %
• In general, one can express the isophote as a Fourier series :

  (7.10)

  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 b_n 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   [image]
  a₄ < 0 : boxy
  a₄ > 0 : disky
  

The a₄ parameters are very important since they correlate with many other variables (see below, § 8)

(6) Kinematics of Elliptical Galaxies

If stars produced single isolated emission lines, their (projected) velocity field would be easy to find:
    The distribution of projected velocities N(v) F( )   i.e. the emission line profile
However, Ellipticals have complex absorption line spectra:
    Similar to a K giant, but broadened by Doppler motion of the stars:   [image]

Consider :
S() = Stellar Template = a single star spectrum
N(v) = relative (normalised) number of stars of projected velocity v (ie v_los)
    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 u = c Log_e
  so pixels u = c Log_e = c/ = v, i.e. km/s/pix rather than A/pix
  our convolution is, in fact: 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
• Remove low frequencies (e.g. >50A continuum variations) and high frequencies (noise)
  do this in Fourier space: apply filter to FT and transform back   [image]

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)

(7.11)

From the convolution theorem we have :

G(k)   =   S(k)   ×   N(k)

(7.12)

giving

N(k)   =   G(k)   /   S(k)

(7.13)

We cannot simply inverse transform N(k) because noise is introduced by the division.
    Instead, assume N(v) is Gaussian, so N(k) is also Gaussian
Estimate N(k) by fitting a Gaussian to the quotient   [image]
    from this fit, we quicky obtain N(v) as a Gaussian
    so 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)]

(7.14)

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 [image]
Cross-correlating the template with itself produces a narrower peak at zero offset

• 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 (i.e. deviations from Gaussian)
These characterisations include:
• Sums of Gaussians of different velocity, width, and strength
• Higher classical moments (ie k>2), eg _k = µ^k / ^k where µ is the k^th moment
  however, they weight the noisy wings too much, so alternatives are better
• Gauss-Hermite functions: h_k   (Gaussians multiplied by a poynomial)
Ultimately, we really only deal with two extra parameters for the LOSVD: [image]
• Skewness (k=3), i.e. asymmetric slope to higher/lower velocities
• Kurtosis (k=4), i.e. stubby/peaky

In practice, these parameters are evaluated as part of an optimized ² fit [image]
to the observed spectrum of the template convolved by a parameterised LOSVD

(i)   Expectations (pre-1975)

Ellipticals should be rotationally flattened
Assuming axisymmetry and isotropic velocities, stellar dynamicals gives (Topic 8.5f):

(7.14)

To compare with observations, lets define:

(7.16)

(ii)   Results

The rotation amplitudes results are shown in a few ways: [image]
• as V_r / _e     vs            
• as (V_r / )^*   vs   M_B    
• as (V_r / )^*   vs   a₄      

We conclude:
• Lower luminosity ellipticals and bulges have (V_r / )^*   ~   1
    Are rotationally flattened
  

• For luminous ellipticals, (V_r / )^*   <   1
    Not rotationally flattened
    Flattened by velocity anisotropy   (i.e. _{x y z} and R_{x x} etc.)
    Further evidence for triaxiality

Interestingly, (V_r / )^* correlates even better with a₄/a   as expected:

• Disky galaxies rotate
• Boxy galaxies don't rotate

Naively, for an axisymmetric rotating galaxy, one expects:
• Major axis slit should show maximum rotation
• Minor axis slit should show no rotation
• i.e. 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, R_f, a good estimate of this is:

(7.17)

A histogram of _est shows: [image]

• Most are   ~   0
• Some are 0 - 90
• Significant minor-axis rotation occurs in boxy Ellipticals, suggesting triaxiality.

~25 % Ellipticals show a separate, rotating component in the nuclear regions (~1kpc; 0.1-0.3 R_e)
These are called Kinematically Distinct Cores (KDC)
Projection effects and difficult detection suggest maybe 30%-60% Ellipticals have KDCs.

Here are some examples: [image]

The KDCs show the following :

• Rapid rotation (V_r / )^*  >   1, with a range from "warm" to "cold": V_r /  =  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 [image]
• Photometrically, they are difficult to identify (e.g. not necessarily disky isophotes) :
  they don't contain much mass
  kinematically prominent in LOSVD because they have low (e.g. 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: [image]

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

Stellar velocities & radius give Mass
Photometry gives Light
Together, we get M/L ratios.

• 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.

Simple estimates:
Assume approximately isothermal and fit a King profile (sec 3a above)
We find a central luminosity density, central mass density, and central M/L:


• j(0)   =   I(0) / 2r_o
• (0)   =   9 (0)² / 4 G r_o²
• M/L   =   9 (0)² / 2 G I(0) r_o

Typically, M/L   ~   10 h M / L_B,     so dark matter does not dominate in the center.

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) :

• M(r) = (3-2)/(1-)   ²r / G

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 possible to do this:
affects h₄ : the LOSVD kurtosis   [image]

• tangential ( < 0) gives stubby LOSVD with h₄ < 0
• radial ( > 0) gives peaky LOSVD with h₄ > 0
These type of measurements can yield the mass profile
or equivalently, the circular rotation curve: [image].

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 [image]

(8) Two Kinds of Ellipticals : Boxy and Disky

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:   [image]
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 modified Hubble diagram   [image]

• 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

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 detections 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: e.g. z ~ 2 density ~ 27 times higher than today.
    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 ???

(i) The Need for Dissipation (loss of energy)

• High stellar densities require dissipation during formation
  • eg collapse factors of ~10 needed to convert spiral disks to elliptical cores
  • eg at M_B ~ -16, Dwarf Spiral vs M32 we have r_c ~ 700pc vs 1pc and µ(0) ~ 23 vs 11
• Unlikely/impossible by mergers of stellar systems alone
        such mergers cannot increase the (phase-space) density
  
• Dissiptation requires gaseous collapse/merger & subsequent star formation

  (ii) Observational Support for Importance of Mergers

• Ongoing mergers are seen (eg NGC 7252, NGC 6240, Arp 220)
  • They have ~ R^¼ profiles (in K light)
  • Have high central (gas) densities ~10² Mpc^-3
    comparable to Ellipticals of similar total mass
  • Lie close to the F-P for Ellipticals ( from CO; photometry in K)
• Brightest cluster galaxies are clearly accreting (eg multiple nuclei)
• Ongoing accretion is common in ellipticals:
  • KDCs
  • gas/dust lanes in non-equilibrium configurations
  • ripples & shells
    (however, don't confuse minor accretion with major merger formation).

  (iii) Theoretical Support for Importance of Mergers

• Compact groups have expected lifetimes ~10^8-9yr     must merge
• N-body simulations of major mergers of two bulge/disk/halo galaxies     R^¼
  however, highest densities inherited from pre-existing bulges
• N-body simulations of mergers including gas (in SPH formalism) and star formation (SFR (gas)^1.5) yields:
        rapid gas collapse to center (before merger complete) with nuclear starburst
        final profile: dense core with "break" in power-law

  (iv) Problems to Overcome

• Dense nuclei survive mergers  -  so why do giant Es have larger lower density cores if built from denser bulges and Es?
• Mergers of Es &/or bulges should destroy the F-P
• GC frequency (per unit luminosity) is ~25 times higher in Es than Spirals
  (maybe GCs form during merger  -  e.g. as in NGC 1275)
• difficult to generate metallicity vs luminosity correlation by merging lower luminosity systems
  (unless metals made during merging starburst)

Kormendy & Sanders (1992) combine the two formation scenarios:
Similar to Ultra-Luminous Infra-red Galaxies (ULIRGs; e.g. Arp 220 etc; [image] )

• They are mergers, showing tidal debris etc
• They have high central densities ~10² Mpc^-3, similar to the central stellar densities of ellipticals
• They 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 ?