Download Chapter 7 Elliptical Galaxies Chapter 16 Elliptical Galaxies

Chapter 7
Chapter 16
Elliptical Galaxies
Elliptical Galaxies
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Until the late 1970s, it was believed that elliptical galaxies are simple systems: gas-free,
disk-free, rotationally flattened ellipsoids of very old stars. In the last 20 years, most
of these assumptions turned out to be wrong or only crude approximations:
•Massive ellipticals are not flattened by rotation, but are anisotropic.
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7.1 Photometric properties of elliptical galaxies
7.1.1 Stellar Density Profiles of Elliptical Galaxies
The stellar density profiles of elliptical galaxies follow Sersic 1/n profiles (like bulges):
•Ellipticals do have an interstellar medium, but it is hot T > 106K.
•A significant fraction of ellipticals exhibits kinematic peculiarities (like kinematically or
counter-rotating cores) which point to a ‘violent’ formation process
with
, re is the radius that contains half the projected
light, I is the surface brightness, Ie the surface brightness at re. If n=4, then this is called the
de Vaucouleurs law.
•Ellipticals frequently contain faint stellar disks.
•Low mass ellipticals seem to contain intermediate age stars.
•All ellipticals and bulges seem to contain supermassive black holes amounting to about
0.2% of their mass.
Other commonly used surface brightness profiles are the King models (see Binney &
Tremaine 2008) or the Jaffe model. The latter is pretty close to the de Vaucouleurs law but
is fully analytic. It has the density distribution:
Ellipticals are still characterized by their apparent axial ratios:
E0 ... E7, where the number corresponds to 10·(1 − b/a)
where r0 is the radius of the sphere which contains half the light. The cumulative light
distribution is given by:
with b being the projected minor axis and a the projected major axis.
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and the potential is:
Comparison of
different surface
brightness profiles
of galaxies.
The projected brightness distribution of the Jaffe law is:
[
]
[
]
red squares: Jaffe model,
with a = R/r0 and the projected distance from the center being R (Jaffe 1983, MN 202,995).
blue circles: r1/4-model,
cyan crosses: exponential,
The King models are isotropic spherical models which are good first approximations to
describe globular clusters but also ellipticals and galaxy clusters (see Binney/Tremaine).
lines are King models of
different binding energy
(see Binney & Tremaine 2008).
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Photometric scaling relations for elliptical
galaxies, bulges and dwarf spheroidal
dwarf galaxies from Kormendy et al. 2009:
- Ellipticals and bulges form one sequence
- dwarf spheroidals another sequence
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7.1.2 Three-dimensional shapes of ellipticals
•
Isophotes are to the first order elliptical
→ the density is constant on ellipsoids, i.e. the possible shapes are:
oblate (a = b > c, rotationally symmetric ellipsoid, like a flying saucer)
prolate (a > b = c, like a cigar)
triaxial (a ≠ b ≠ c, ellipsoid, like a box with smoothed edges)
•
Projection for the axially symmetric case (oblate or prolate, see Binney/Merrifield):
•
with q = minor axis/major axis = b/a, and i: angle between the minor axis and the line of
sight.
•
•
Projection for the triaxial case is more complicated (see: Ryden ApJ, 1990)
From the observed projected ellipticity distribution as well as from kinematic
measurements one can estimate that elliptical galaxies are in the mean modestly
triaxial (near oblate): a : b : c ≈ 1 : 0.95 : 0.7
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from Binney/Merrifield Galactic Astronomy: An important side effect of a triaxial density
distribution is the so-called isophote twist which is observed in many ellipticals.
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7.1.3 Isophote shapes of elliptical galaxies
Isophotes are generally not exactly elliptical. The “boxiness” or “diskiness” of isophotes
is usually quantified by measuring a quantity denoted a4. First the ellipse Re(φ) is fitted to
the isophote. Then for each angle φ one determines the distance δ(φ) = Ri(φ) − Re(φ)
between the radii of corresponding points on the ellipse and on the isophote. Then one
expresses the function δ(φ) as a Fourier series:
a4 describes the lowest order deviation which is dynamically plausible.
NGC 821: a4/a ~ +0.02, disky
Disky isophotes can be explained by a superposition of an elliptical bulge and a faint edge-on
disk. Isophote shape analysis is perhaps the only possibility to detect weak disks in
ellipticals.
Are disky E’s related to S0’s? Are they forming an intermediate type between boxy E’s
and S0’s? The answer is probably yes. The properties of the disks in terms of radii and
densities smoothly overlap with those of S0 galaxies.
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NGC 2300: a4/a ~ -0.02, boxy
! The majority of ellipticals show either disky or
boxy isophotes
! The isophotes of disky ellipticals can be well
modeled by a superposition of a spheroid and
a faint disk (D/B~ 0.05…0.2).
(Bender et al.1988, 1989; Scorza et al. 1998)
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Kormendy et al. 2009
Kormendy et al. 2009
7.1.4 Central regions of elliptical galaxies and bulges
They are important for the dynamical structure (population of stellar orbits, see below) of
ellipticals and probably all bulges and ellipticals host supermassive black holes (see below).
The structure of the central regions seems intimately linked with the global structure of the
objects. Strong correlations between core- and global properties of ellipticals and bulges
were first found by Lauer 1985 and Kormendy 1985. The Hubble Space Telescope allowed
a break-through in the study of cores (e.g. Faber et al. 1997). Central profiles come in
two variants: cuspy cores (flattening of the density profiles towards the center) and powerlaw
centers (straight power-law to the smallest measurable radii). These profiles can be
described by the so-called Nuker-law or Dehnen-law (see Faber et al. 1997 for references):
Key parameters to describe cores are the central slope γ and the break radius rb.
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Kormendy (1987)
• Ellipticals, dwarfs and GCs
form different sequences
• Core galaxies are mostly
boxy (& slow rotators)
• power-law galaxies are
mostly disky (& fast rotators)
M32
Left: Central density profiles of elliptical galaxies and bulges studied with the Hubble
Faber et al.
(1997);
see also
Nieto et al.
(1991)
Space Telescope. Two distinct classes become apparent: cuspy cores (curved) and
power-law cores (Gebhardt et al. 1996). Two examples are shown on the right: the
power-law core in NGC 4621 and the cuspy core in NGC 720 (courtesy: T. Lauer)
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E
S
dE
GC
7.2 Stellar Dynamics, Boltzmann and Jeans Equations
7.2.1 Relaxation of Stellar Systems
The dynamics of stars in ellipticals and bulges is significantly more complicated than
the motion of fluids or gases in hydrodynamics. This has two basic reasons:
Classical relaxation is based on the redistribution of the orbital energies of stars via two-body
Gases and plasmas (in laboratories, in stars) are collisional where collisions are
dominated by electromagnetic forces which are mostly negligible on scales larger
than a few times the typical separation of the particles.
Stellar systems are collisional where collisions are gravitational and therefore cannot
be shielded. Therefore, stars in galaxies experience accelerations from all other
members in the system.
encounters. After many encounters an equilibrium distribution is established comparable
to the Maxwell-Boltzmann distribution of statistical mechanics.
1. Deflection of a star when passing another star:
The mean free path of particles in most gases is generally small compared to the size
of the system.
In stellar and galaxy systems the mean free path is large compared to the size of the
system (→ few interactions, large relaxation times)
The collisional physics of gases and plasmas is LOCAL
The collisional physics of stellar systems is GLOBAL
Consider passages at large distances first: δv┴ « v
The large scale gravitational potential plays the same role in both.
For a comprehensive overview of stellar dynamics see:
Binney, Tremaine: Galactic Dynamics, Princeton Univ. Press, 2008
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2. Number of interactions experienced by a star when passing through a stellar system
once:
Setting the zero-point in t such that x = v · t gives:
R = Radius of the stellar system
N = Number of stars in the system
Probability P1 that the crossing star will pass one star of the system (e.g. galaxy) in a
distance-interval [b, b+db] :
acceleration x passage time
If the galaxy contains N stars, the total number of interactions for a single crossing is:
With every interaction v changes by the amount of v┴.
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The sum of all interactions will lead to an average change of velocity of <δv┴> ≈ 0
(positive and negative deflections are equally probable).
However the mean square deflection is non-zero:
3. Integration over all impact parameters b:
Plausible values for bmin, bmax:
Using the virial theorem |2T| = |V| gives:
(v = mean velocity of the stars)
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4. Using the virial theorem once more leads to
and thus:
As a matter of fact, interactions with b < bmin are very rare:
The fractional area of a galaxy that corresponds to close passages is given by:
for a single passage through the stellar system.
For relaxation (<δ v┴2> ≈ v2) to occur, a star will have to cross the galaxy Nrelax times:
i.e. for typical stellar systems with N > 105 close interactions are negligible
→ Relaxation is dominated by large distance interactions
The relaxation-time τrelax is:
(τcross = crossing-time)
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7.2.2 The Collisionless Boltzmann Equation
Examples:
N
R
v
τcross
τrelax
age/τrelax
open cluster
102
2 pc
0.5 km/s
4 · 106 yrs
107 yrs
≥1
globular cluster
105
4 pc
10 km/s
4 · 105 yrs
4 · 108 yrs
≥ 10
elliptical galaxy
1012
10 kpc
600 km/s
2·
107
yrs
1017 yrs
10−7
dwarf galaxy
109
1 kpc
50 km/s
2 · 107 yrs
1014 yrs
10−4
galaxy cluster
103
1 Mpc
1000 km/s
109 yrs
2 · 1010 yrs
10−1
Thus, the following applies to almost all stellar dynamical systems:
Stars do not experience significant encounters over their typical orbital time-scale.
The orbit of a star is mostly determined by the smooth gravitational potential of all
other stars and two-body interactions only have effects over many orbital time scales.
Therefore:
The density and velocity distribution of a stellar system can be approximated by a
The time evolution of
i.e. at every location
is determined by Newtonian dynamics.
If we neglect the formation and death of stars, a continuity equation for
exists:
can be true. This corresponds to an ‘anisotropic temperature’.
If two-body interactions cannot be ignored, a collision term would have to be added
on the right-hand side of this equation.
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Collisionless Boltzmann Equation:
∂v j
∑ ∂x
=0
since v j and x j are independent
Basic equation of stellar dynamics = continuity equation for the phase-space density f(x,v).
j
∂ % ∂Φ (
∑ ∂v ' − ∂x * = 0
j &
j )
since Φ is independent of v j for the case of
Important:
gravitational interaction
So far, no assumption has been made as to whether or not the potential Φ is only due
to the particles themselves or has further contributions from other sources. If the
potential is only due to the particles described by f, then self-consistency is fulfilled:
one obtains:
or
m = typical mass of a star
n = number density
which is called the collisionless Boltzmann equation and where we adopted the
convention that summation over repeated indices is implied.
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7.2.3 Stellar Orbits in Ellipticals and Bulges
Stars can move along a large variety of stellar orbits in elliptical galaxies. Generally, the
orbits are not closed and the stars are more or less ‘tumbling’ through the galaxy. However,
even after many orbital time scales, the star will not reach all locations in the galaxy.
Depending on the shape of the volume which contains all locations of the star along its orbit,
one defines three main types of orbits in a triaxial galaxy:
box orbits
short axis tubes
long axis tubes
The orbit types differ with respect to their mean angular momentum (which is not conserved
in a triaxial galaxy!) or approximate symmetry axis. Note that tube orbits along the
intermediate axis do not exist because they are not stable. The envelopes of these three
orbit families are given on the next pages.
Left: Stellar orbits in a two-dimensional barred potential.
Right: Envelopes of stellar orbits in a three-dimenisonal triaxial potential (Binney/Tremaine).
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7.2.4 The Jeans Equations
0th Moment of the Boltzmann Equation in v
and moments of f(x,v):
→
with f (vi = ±∞) = 0 we obtain:
velocity dispersion tensor:
In the solar neighborhood,
ijcan be measured and turns out to be approximately
diagonal in cylindrical coordinates. One obtains:
v
RR
zz
220
50
38
19
km/s
km/s
km/s
km/s
i.e. the dispersion is highly anisotropic which would not be possible if stars collided
frequently (as do the particles in a disk of rotating gas).
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1st Moment of the Boltzmann Equation in v
where we have defined:
With partial integration of the last term:
Subtracting from Jeans equation 2 the continuity equation times
gives:
We now use the velocity dispersion tensor σij2 as defined above:
and
we obtain:
Jeans equation 2
σij2 is the dispersion of the velocities around the mean streaming velocities.
Calculating:
and inserting it in 10.26 we obtain the more frequently used variant of Jeans Eq. 2:
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ellipsoid. In the case of isotropic velocity dispersion σ11= σ22=σ33 and the third Jeans
equation is identical to the Euler equation.
Jeans Equation 3:
In general, the Jeans equations cannot be solved without ambiguities, because for stellar
systems there exists no analogue to the equation of state p = p(ρ) available with gases.
The terms have the following meaning:
n
∂v j
∂t
+ nvi
∂Φ
−n ∂x
−
j
∂v j
: convective (substantive) derivative of v
∂xi
∂( nσ ij2
∂xi
)
: force terms
which can be compared to the hydrodynamical Euler equation:
→ In order to solve a problem of stellar dynamics using the Jeans equations, it is often
necessary to make assumptions concerning σij. Only more recently, improved
observational techniques make it possible to constrain the σij for galaxies.
The projected velocity dispersion along the line-of-sight, which can be measured straightforwardly, is an approximate and simple estimator of the mean velocity of the stars in an
elliptical galaxy.
Note, the difference between the third Jeans equation and the Euler equation:
The pressure term in the Jeans equation is a tensor, whereas the one in the Euler equation
is a scalar and thus isotropic.
The projected velocity distribution contains more information and can be used to constrain
the orbital anisotropy.
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Interlude: measuring velocities and dispersions in elliptical galaxies
To determine the kinematics of an elliptical galaxy one needs to derive velocities,
velocity dispersions (and possibly higher moments) along each line of sight. Because
the spectra of ellipticals are broadened by velocity dispersion, the absorption lines of the
spectra overlap and it is not possible to derive the kinematics from the analysis of single
absorption lines. Therefore, more sophisticated methods are needed.
For example, the velocity v and the projected velocity dispersion σ can be measured
by comparison of the galaxy spectrum with a suitably broadened spectrum of a template star
(e.g. K0III star). For each line-of-sight through the galaxy one approximately has
with:
G = galaxy spectrum as observed,
g = typical unbroadened spectrum of all stars along the line of sight,
B = velocity distribution of the stars along the line of sight
Thus, notionally one obtains B(v,σ,...) by division in Fourier space with a
template star S(λ):
Faber-Jackson relation between central velocity dispersion and total magnitude of elliptical
galaxies (LB ~ σ4 or: log σ = -0.1 ΜΒ + const). This is the equivalent to the Tully-Fisher
relation of spiral galaxies and can be explained by the Virial Theorem (see below).
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provided the template star is chosen appropriately: S(λ) ~ g(λ). Deconvolutions are notoriously
frought with problems (see next page) when applied to noisy data. Therefore, a “forward
process” involving template convolution and comparison is preferred.
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7.3 Dark Matter in elliptical galaxies
In the central parts elliptical galaxies have mass-to-light ratios typical for old stellar
populations: M/L = 5...10MΘ/LB,Θ. There is no need for dark matter. However, do they
also show evidence for dark matter in the outer parts? This question is considerably more
difficult to answer than in the case of spirals. The reasons are the lack of gas on circular
orbits and the presence of anisotropic velocity dispersions within the stellar tracers.
star
star
Still, it is nowadays possible to determine the mass profiles of ellipticals with a variety of
methods: with the analysis of stellar kinematics, from the equilibrium of their X-ray halos
and, in a statistical manner, using gravitational lensing. The results agree (within relatively
large errors) and indicate that ellipticals are surrounded by dark matter halos like
spirals.
galaxy
In the following we discuss the determination of the mass profile of a spherically symmetric
galaxy. In this case the Jeans equation reads:
2
⇤
d ⇥r2
+ 2⇥r2
dr
r
continuum-removed, filtered spectra,
Fourier quotient, and kinematics for a
typical elliptical galaxy (Bender 1990).
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⇥ 2 + ⇥⇥2
⇥⌅
with ρ number density (not necessarily mass density!) and
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d
dr
=
2
=
2
⇥
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k ≈ 3 applies at re (Jaffe model) and so in the case of:
Defining:
= 1
⇥2
⇥r2
with:
β = 0: isotropic velocity dispersion
β < 0: tangential anisotropy
0<β<1: radial anisotropy
the previous equation can be transformed into:
d ln ⇥ d ln ⇤r2
+
+2
d ln r
d ln r
⇥
= r
d
GM (r)
2
= Vcirc
=
dr
r
Locally the stellar density profile can be approximated by a power law (like in the case of a
Jaffe or an r1/n profile):
(r) ⇤
or
k
d ln
=
d ln r
⇥
k
The β-ambiguity can be circumvented using higher moments (kurtosis or 4th momet) of the
velocity distribution:
Then the above equation turns into:
M (r) =
r k
G 1
2
⇥2
r
1
d⇥ 2
dr
r⇥ 2
d(1
)
dr
1
⇥
If β(r) varies with radius, an exact determination of the mass is hardly possible. But if the
velocity dispersion gradient is negligible dσθ ≈ const. and β ≈ const., then:
M (r) =
41
→ tangential orbits cause a box-shaped velocity profile.
→ radial orbits cause triangular-shaped velocity profiles.
This technique requires high signal to noise spectra so that one can better constrain the higher
moments of the velocity distribution.
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Influence of dominantly radial or tangential orbits on the line-of-sight
velocity distribution in elliptical galaxies. The deviations from Gaussian line
profiles can be characterized by Gauss-Hermite moments (h4).
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h4
β
β
h4
σ [km/s]
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⇥2r k
G 1
i.e. a lower limit for the mass can be determined, but the result is generally not robust enough
to conclude about the presence of dark matter.
σ [km/s]
⇤r2
Profiles of velocity dispersion σ and velocity profile shape h4 for
round elliptical galaxies. The inferred anisotropy is given by β
(β > 0 corresponds to radial anisotropy). Kronawitter et al. 2000
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Dark matter scaling relations for ellipticals
! Circular rotation curves
of ellipticals are flat!
Spi
ral
TF
! Vc,max ~ 0.68 σ0.1
! dark matter starts to
dominate around two
effective radii in E’s.
Gerhard et al (2001): points: ellipticals, dashed lines: relations for spirals
At a given stellar mass (and luminosity), dark matter halos of ellipticals
have larger circular velocities and higher densities than spirals.
Gerhard et al. (2001), Thomas et al. (2007)
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7.4 The Virial Equations and Global Velocity Anisotropies
in Elliptical Galaxies and Bulges
For global considerations one generally considers the tensor-virial-theorem. It is obtained
from the first moment of the Jeans equation (2) in the spatial coordinates.
→
Adding 10.52 and 10.53 and dividing the result by 2 gives:
3rd term:
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Thus
1st term:
and
In order to transform this expression, it is necessary to determine the second derivative
in time of the moment of inertia tensor.
using the continuity equation
divergence theorem
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Except for the symmetrization and a factor 2, this expression is identical to term 1.
Term 1 has to be symmetric, too, since the 3rd and the 2nd expression in the virial
equation are symmetric.
Thus:
2nd term:
Using the divergence theorem
expression the second term transforms into:
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Flattening and Anisotropies of Elliptical Galaxies and Bulges
– An application of the Tensor Virial Theorem
Combining the three expressions results in the
Consider an axisymmetric elliptical or bulge in equilibrium (d2I/dt2 = 0) with the z-axis being
the axis of symmetry. In this case we have:
The only remaining non-trivial Tensor Virial Equations are:
or:
with:
It can be shown that Wxx/Wzz for axisymmetric ellipsoids does not depend on their density
profile but just on b/a (see Roberts 1962, Binney/Tremaine).
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Rotational flattening:
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Flattening by anisotropy: Txx = Tyy = Tzz = 0
with the mass-weighted mean-square random velocity of the galaxy σ02
Using:
1
2
Txx + Tyy =
1
2
V¯ d3 x = M Vo2
2
Tzz = 0
with the mass-weighted mean-square rotation speed Vo2 leads to:
Vo
=
o
⇤
a ⇥0.89
b
2
2
E.g:
Note: As σ0 and V0 are not directly observed, but instead often only the major axis
rotation velocity Vm and the mean dispersion σm within half the effective radius, one
often uses the approximation:
Vm
m
⇥ 0.78
Vo
o
⇥
1
b/a
b/a
Obviously, a typical flattening of an elliptical galaxy or bulge of b/a= 0.7 can either
be achieved by a relatively large rotational velocity in the isotropic case or by an
only mild anisotropy and no rotation in the anisotropic case.
where the latter approximation can be found in Kormendy (1982). Thus one needs a
fairly large rotation velocity to obtain significant flattening:
b
= 0.7
a
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⇥
Vo
0.8
o
55
⇥
Vm
0.6
m
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Rotational Properties
of Elliptical Galaxies:
Using the equation for the
isotropic rotator line in the
diagram on can define the
anisotropy parameter:
Vm
V
m
m
⇥
= ⇤
(1
Vm /
m
NGC 821: disky, a4>0
b/a)/(b/a)
! Massive ellipticals are generally
not rotationally supported.
! Low mass ellipticals could be
rotationally supported.
! Bulges could be rotationally
supported (crosses).
= (1
b/a)
see Davies et al. 1983, APJ 266, 41
NGC 2300: boxy, a4<0
Rotation correlates with a4/a, better than with luminosity. Disky ellipticals and bulges show
large rotation velocities. This supports the continuity between disky ellipticals and S0 galaxies.
Boxy ellipticals rotate slowly and also can show minor axis rotation, indicative of triaxiality.
Bender 1988, Bender et al. 1989, Kormendy & Bender 1996
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Modeling Galaxies with Schwarzschild’s method (1979):
(Richstone&Tremaine 1988, van der Marel
et al. 1998, Gebhardt et al. 2003, Thomas
et al. 2004)
- deproject observed surface brightness
profile to derive 3D axisymmetric density
distribution of stars (needs inclination)
- choose a mass-to-light ratio for the stars
and derive the potential from Poisson’s
equation; add the potential of the BH
- calculate several thousand orbits with
different energies, angular momenta
and drop points and derive their
time-averaged density distribution
Further confimation of this conjecture: Core galaxies are boxy and slow rotators, powerlaw galaxies are disky and fast rotators, see Faber et al. 1997.
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- superimpose the orbits such that:
(1) the surface brightness distribution is matched,
(2) the velocity distribution (rotation, dispersion, higher moments) is matched
(3) the phase space distribution of orbits is smooth (e.g. by maximizing the entropy)
- repeat this procedure for a range of inclinations, stellar mass-to-light ratios, black hole masses
and dark halo properties.
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7.5 The ‘Fundamental Plane’ of Elliptical Galaxies
Summary: Kinematics of Elliptical Galaxies
A comprehensive set of global parameters of elliptical galaxies is:
All ellipticals and bulges show anisotropy between the short axis and the
equatorial plane.
Faint ellipticals and bulges rotate fast, bright ellipticals are slow rotators,
often have de-coupled centers and sometimes rotate along the minor axis.
There exists a tight correlation between rotational properties and the shape
of the isophotes and the core properties:
The half light (or effective) radius re
The mean surface brightness Ie (or Σe) within re
The central velocity dispersion σ0
The luminosity L
The mass M
The following two relations relate these quantities:
boxy isophotes, cuspy cores: slow rotation, peculiar velocity fields
disky isophotes, power-law centers: fast rotation, axisymmetry
Kinematically de-coupled centers suggest that ellipticals are not formed by
the collapse of uniformly rotating gas spheres. (→ formation by merging
processes, see later).
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Multiplication yields an expected relation for these parameters:
with the structure parameter c which contains all unknown details about the galaxies’
structure.
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This implies a limitation of the variation of the dynamical structure (σ0), of the M/L of the
stellar population, of the amount of dark matter within re, of the slope of the stellar initial
mass function, and all other possibly varying parameters.
Because neither M/L or c are expected to vary very much, the brackets are nearly constant
and imply that ellipticals should define a plane-like distribution in the 3-space of their global
parameters (re, Σe, σ02).
Astonishingly, this plane is much better defined than naively expected, with very low
dispersion perpendicular to the plane (implying a variance in the product of the brackets less
than 10%) and a small but significant tilt (implying small but significant changes in the
structure of ellipticals as a function of their luminosity or mass), see Djorgovski & Davis 1987,
Dressler et al. 1987.
→ Even though the kinematics of ellipticals can appear to be highly complicated in detail,
the objects must in fact be rather similar with respect to their global structure and their
stellar M/L !
Note: The fundamental plane is an important distance indicator for
elliptical galaxies, like the Tully-Fisher relation for spirals. At higher
redshifts it is a useful indicator for the evolution of elliptical galaxies.
The fundamental plane can conveniently be visualized in the κ-parameter space, using
the parameters (Bender, Burstein & Faber 1992):
The observed so-called ”fundamental plane” relation reads:
This is consistent with the virial expectation, if
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In the κ-parameter space, it is also possible to illustrate the various processes that act on
the galaxies during formation and evolution, i.e. merging, dissipation, winds etc.
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7.6 Hot X-Ray Gas around Elliptical Galaxies
Ellipticals and bulges lie in a ‘fundamental’ plane
" at a given mass, their M/L shows only <15% scatter
" they have homogenous, mostly old stellar populations
X-ray
optical
M 87
Churazov et al. 2008
X-ray
optical
NGC 1399
Dressler et al. 1987, Djorgovski & Davis 1987, Bender, Burstein & Faber 1992,1994
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Churazov et al. 2008:
Massive elliptical galaxies are often surrounded by X-ray coronae. The spectra show that
the emission can be explained by thermal Bremsstrahlung of a thin hot gas with the
following properties:
spectra and P,T,Ne profiles of M87
Heating of the gas is needed because of the short cooling times, the kinetic energy of
the stars and SN Ia are likely to be the main energy source. Gas may flow out of the
galaxy but is replaced by the mass loss of the stars.
The X-ray luminosity is correlated with the optical luminosity. But the dispersion of the
X-ray luminosity may be as high as a factor of ~ 50 for a given optical luminosity (this is
not yet fully understood).
X-ray haloes might only be quasi-stationary phenomena.
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If the X-ray gas is in hydrodynamical equilibrium we can use this to determine the mass of
the elliptical galaxy. Assuming the gas to be spherically distributed, we obtain from hydrostatic
equilibrium (see stellar structure equation):
As it turns out that
Gravitational potentials of
elliptical galaxies derived
from Chandra (red) and
XMM-Newton/MOS (blue)
observations.
Potentials are normalized
to zero at Re. Vertical
lines mark the range of
radii used to approximate
the data by a (vcirc2 ln r+b)
law, and the thin solid
lines are the best-fit
approximations of the
Chandra and XMMNewton data to this law.
The vertical dotted lines
mark the effective radius.
, we can approximate:
This yields similar mass profiles as the dynamical modeling using stars, planetary
nebulae and globular cluster dynamics: → dark matter also exists in ellipticals.
Detailed models with temperature profiles taken into account are shown on the next
page. They are consistent with flat, sometimes even rising circular velocity curves.
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Churazov et al. 2010
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