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Congresso del Dipartimento di Fisica
Highlights in Physics 2005
11–14 October 2005, Dipartimento di Fisica, Università di Milano
Structure, formation and dynamical evolution of
elliptical galaxies
S.E. Arena*, G. Bertin*, L. Ciotti†, T.V. Liseikina^,#, F. Pegoraro$, M. Trenti&,
and T.S. van Albada+
* Dipartimento
di Fisica, Università di Milano
† Dipartimento di Astronomia, Università di Bologna
^ Ruhr-Universitaet, Bochum, Germania
# Institute of Computational Technologies, Novosibirsk, Russia
$ Dipartimento di Fisica, Università di Pisa
+ Kapteyn Astronomical Institute, Groningen, Olanda
Elliptical galaxies may be imagined to have formed
from collisionless collapse, reaching dynamical
equilibrium by incomplete violent relaxation. We
have tested this picture by showing that analytical
models constructed under the above scenario and
general statistical considerations, the so-called f()
models, not only match the basic structure of
elliptical galaxies (for a constant mass-to-light
ratio), but are able to fit the density profiles (over
nine orders of magnitude; with relative error within
10%) and the phase space properties ABSTRACT focusing on the slow evolution induced
(predicting the pressure anisotropy profile
by dynamical friction of a host galaxy
on a minority component of “satellites”, in a
with mean error of 5%) of the results of
laboratory of N-body simulations. The basic
collisionless collapse in a variety of N-body
mechanisms have been modeled long ago by
simulations [1]. To our knowledge, this is the first
Chandrasekhar (1943), but are not understood under
time that an analytically simple model constructed
realistic conditions. After a first study [2] of the
from physical arguments is matched successfully to
evolution of an n=3 isotropic polytrope, we now
the results of N-body experiments of galaxy
address a sequence of realistic galaxy models (the
formation. We have then addressed the issue of the
f() models mentioned above),
dynamical evolution of such stellar systems,
1. STRUCTURE AND FORMATION: f() MODELS
1.A Construction and dynamical properties
f() are a family of theoretical collisionless models derived by
extremizing the Boltzmann entropy at fixed values of the total mass,
of the total energy and of an additional third quantity Q, defined as:
1
2
(; )
1.B Comparaison with the products
1.B Comparaison with the products of
collisionless collapse
collisionless collapse
Code: calculates the evolution of a system of simulation
particles interacting with one another via a mean field,
calculated from a spherical harmonic expansion of a smooth
density distribution.
()
As shown in (Stiavelli & Bertin 1987), this leads to the following
distribution function:
where a, A, d and  are positive real constants.
The two-parameter family of models, constructed by solving the
Poisson equation, is described by the  parameter and the
concentration parameter =-a(0), the dimensionless depth of the
central potential well.
Density profiles: Some examples in figure 1.
 r - 4 beahviour, from not too far beyond rM, the half mass radius.
 Increasing  profiles go from a prominent core to an high
central concentration.
 Projected density profiles are well fitted by the R1/n law with the
index n ranging from 2.5 to 8.5.
Pressure anisotropy profiles: Some examples are in figure 2.
(r)=2-(<w2>+<w2>)/<wr2>, w are spherical component of the
particles velocity.
 Core isotropic and outer parts radially anisotropic.
 Higher values of  are associated with a sharper transition
from central isotropy to radial anisotropy.
 The central isotropic region increases with .
finding in general that (i) The role of collective
effects and of inhomogeneities is important; (ii) The
density distribution of the host galaxy tends to relax
to a broader profile, in contrast with the
expectations of adiabatic models; (iii) Satellites
spiraling in on quasi-circular orbits tend to heat the
stellar system preferentially in the tangential
directions. Finally, we are opening the way to the
construction of models characterized by a
significantly non-spherical geometry [3].
Numerical Simulations: initial conditions are clumps
uniformly distributed in space in approximate spherical
symmetry and with a small value of the virial parameter u
(u=-2K/W<0.2). Most simulations have been carried out with 8
105 particles. From such initial conditions, the collisionless
''gravitational plasma'' evolves undergoing incomplete violent
relaxation.
(5/8; 5.4)
Fits: One example is in the figures 3-5 for (; )=(5/8;5.4).
(5/8; 5.4)
3
 Density profiles: the fits are satisfactory not only in the
outer parts, where the density falls under a treshold value
that is nine orders of magnitude smaller than the central
density, but also in the inner regions (relative error within
10%).
4
The end products of
high resolution simulations of
galaxy formation,
where incomplete violent relaxation
of a ''gravitational plasma"
results from a collisionless collapse
process, are well and in detail
described by the f() models, in spite of
their simplicity and
of their spherical symmetry.
(5/8; 5.4)
5
 Anisotropy profiles: are represented extremely well by
the models (relative error within 5%).
 Phase Space: The final energy density distribution N(E)
is in very good agreement with the models, especially for
the strongly bound particles. At the deeper level of
N(E,J2) simulations and models also agree very well.
2. DYNAMICAL EVOLUTION: EFFECTS OF DYNAMICAL FRICTION
2.A The problem of dynamical friction
Problem: the classical theory
(Chandrasekhar 1943): is not suited to describe real systems,
characterized by inhomogeneities and complex orbits. No simple
analytical theory exists able to incorporate these effects. Great help
in understanding the physical processes then derives from numerical
simulations.
Results: Figure 6 and 7.
 The satellite falls slower than expected from classical theory.
 The Coulomb logarithm is position-dependent, lower than
expected at almost all positions and does not depend on Ms, Rs,
r0 and  (in the range explored).
Ms=0.1Mg
Rs=0.3rM
7
To generate new models
we have found it useful to
consider an elementary
property of the asymptotic
expansion for small
flattening of the homeoidal
density-potential pairs.
Numerical Simulations
It is the same code as in
Initial conditions are: an f() galaxy,
box 1, but with the
with given  and , and one satellite
addition of one or more (or a shell of satellites), with given Rs
particles to represent one
and Ms, in circular orbit around the
satellite or a spherical galaxy centre at initial distance r (or a
0
shell of satellites.
range of distances).
These additional
Most simulations have been carried
particles interact directly
out with 2.5 105 particles for the
with the galaxy particles
galaxy and 1, 20, or 100 satellites.
and among them; they
From such initial conditions the
are modeled as Plummer satellite (or shell of satellites) slowly
spheres with radius Rs
sinks toward the centre of the galaxy,
and mass Ms.
because of dynamical friction.
Problem: elliptical galaxies evolve passively (as a stellar population)
but can also be subject to processes of dynamical evolution; one of these
is associated with dynamical friction. These issues are of general
astrophysical interest (e.g., see Nipoti et al. 2004).
Results: the effects of dynamical friction observed on the galaxy are:
 decrease of the central concentration, figure 8,
 increase of the central isotropic region, figure 9,
 for the capture of a single satellite, change in the galaxy shape
from spherical to oblate and gain of systematic rotation.
 The effects on more concentrated galaxies are smaller.
The fall of the satellites is significantly modified by
the collective effects and inhomegeneities associated
with the host galaxy, which, in turn, evolves by
decreasing its density concentration and by changing
the pressure anisotropy in the tangential directions.
6
There are no systematic
procedures to construct
galaxy models with triaxial
geometry. Only a few triaxial
density-potential pairs are
known, one example is that
of the stratified homeoids.
Code
2.B Galaxy evolution induced by dynamical
friction
(3/4; 5.0)
8
(3/4; 5.0)
9
3. GALAXY MODELS WITH NON-SPHERICAL GEOMETRY
r, =3.1
r, =4.9
, =3.1
10
, =4.9
11
Surprisingly, this offers a
device to construct, in a
systematic way, new
density-potential pairs with
finite deviations from
spherical symmetry.
An application of this
method is given in figures 10
and 11, where are illustrated
the isodensity (rR2/r,
>0) and the isopotential
() for two toroidal models.
REFERENCES
 Bertin, G., Liseikina, T.V., Pegoraro, F. 2003, A&A 403, 73
 Ciotti, L., & Bertin, G. 2005, A&A 437,419
 Stiavelli, M., & Bertin, G. 1987, MNRAS 229, 61
 Chandrasekhar, S. 1943, ApJ 97, 225
 Nipoti, C., Treu, T., Ciotti, L., Stiavelli, M. 2004, MNRAS, 355, 1119
 Trenti, M., Bertin, G., van Albada, T.S. 2005, A&A 433, 57.