Session: MGAT9 – Self-Gravitating Systems
Marco MERAFINA
Department of Physics
University of Rome “La Sapienza”
SPHERICALLY SYMMETRIC RELATIVISTIC
STELLAR CLUSTERS WITH
ANISOTROPIC MOMENTUM DISTRIBUTION
13 July 2009
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Summary
1. Why “relativistic” clusters ?
2. History
3. Anisotropic distribution function
4. Thermodynamic quantities
5. Gravitational equilibrium equations
6. Dimensionless quantities
7. Numerical results: anisotropy and density profiles
8. Conclusions
9. Perspectives
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
1. Why “relativistic” clusters ?
The question of existence of relativistic clusters is open since the discovery of quasars
Are clusters so dense that relativistic corrections to Newtonian theory modify
their structure and influence their evolution ?
- Theoretical calculations
suggest that star clusters might form in the nuclei of some galaxies and quasars
- Astronomical observations
Have yielded no definitive evidence about the existence of relativistic clusters
HST observations
?
Dense stellar clusters are represented by
- Globular Clusters with M ~ 106 M and R ~ 50 ÷ 100 pc
- Active Galactic Nuclei (AGNs) with M ~ 108 M
- Quasars with M ~ 1010 M
A cluster may be considered “relativistic” if the gravitational potential is comparable with c2
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
2. History
(…not exhaustive, sorry for omissions)
1939 - Einstein (first paper on relativistic clusters – stars with circular motion: anisotropy)
1965 - Zel’dovich & Podurets (theory of structure of relativistic clusters by Maxwellian truncated
distribution function: one-parameter models)
1969 - Bisnovatyi-Kogan & Zel’dovich (self-similar relativistic solutions with anisotropy –
problems: infinite density and radius)
1977 - Davoust (Newtonian anisotropic models - different choice of the dependence on
angular momentum in the distribution function)
1989 - Rasio, Shapiro & Teukolsky (numerical simulations of relativistic clusters – isotropic
noncollisional model)
1991 - Ralston & Smith (Anisotropic models of degenerate fermions - “hollow” configurations)
1992 - Ingrosso, Merafina, Ruffini & Strafella (Anisotropic semidegenerate models with
King-Fermi distribution function)
1998 - Bisnovatyi-Kogan, Merafina, Ruffini & Vesperini (generalization of one-parameter models of
Zel’dovich & Podurets – dynamic stability criteria – isotropic systems – different parametrization)
2002 - Chavanis (analysis of dynamical and thermodynamical stability of isothermal gas
spheres and polytropes)
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
3. Anisotropic distribution function
- Spherically symmetric equilibrium  Schwarzschild metric: (r), (r)
for E  Ec
- Distribution function: f (E,L2)
for E  Ec
- Cutoff condition (Zel’dovich & Podurets 1965; Bisnovatyi-Kogan et al. 1998):
- Integrals of motion:
- Components:
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
4. Thermodynamic quantities
- Concentration :
- Energy density :
- Radial pressure :
- Tangential pressure :
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Thermodynamic quantities
- Some simple relations :
- Newton binomial relation :
New form of thermodynamic quantities
(continued)
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
5. Gravitational equilibrium equations
- Equations of equilibrium :
- Boundary conditions :
- Metric coefficients :
Prr(0) = P0 ;
Mr(0) = 0
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
6. Dimensionless quantities
- Some simple relations (Merafina & Ruffini 1989, 1990) :
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Dimensionless quantities
(continued 1)
An important relation for DF :
Then, from considerations of statistical mechanics,
we can define a constant B = A e-1/b for which A e-E/T = B eW-x
Moreover
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Dimensionless quantities
- Dimensionless gravitational equilibrium equations :
- Boundary conditions :
- Dimensionless thermodynamic quantities :
(continued 2)
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
7. Numerical results: mass vs central density
We consider only results for l = 1 (index of distribution function)
b=1
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
7. Numerical results: anisotropy
- Local anisotropy level :
for r = 0
 h = 1 (isotropy)
- For l = 1 :
for r = R
where
Prevalence of tangential motion since index value l > 0
h<1
nevertheless we have a minimum value h = 0.5 for high level of anisotropy
The thickness of the external isotropic region is rapidly decreasing
with increasing of level of anisotropy (small values of a)
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Anisotropy - 1
a = 10–1
b=1
Choice of triad of values of W0: maximum mass (intermediate value); before max; after max
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Anisotropy - 2
a = 10–2
b = 10–5
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Anisotropy - 3
a = 10–3
b=1
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Anisotropy - 4
a = 10–3
b = 10–5
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
7. Numerical results: density profiles
Existence of “hollow” configurations for high levels of anisotropy
(same result obtained by Ralston & Smith in 1991 for a Fermi degenerate gas)
Hollow configuration: the density is increasing to a maximum
in a region different from the center of the configuration !
For increasing levels of anisotropy the central density may be
several order of magnitude smaller than the maximum value
The maximum density is getting far from the center of the configuration,
in progressive way, at smaller values of W0
For a  0 the cluster is approaching the structure of a thick shell
with maximal density close to the border of the configuration
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
7. Numerical results: density profiles
(continued)
Newtonian regime (b«1)
configurations with usual
decreasing density profile
hollow
configurations
Values of anisotropy parameter “a” for which we have hollow configurations (a<a*)
In relativistic regime, hollow configurations exist at
lower levels of anisotropy, for increasing values of b
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Density profiles - 1
a = 10–1
b=1
Choice of triad of values of W0: maximum mass (intermediate value); before max; after max
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Density profiles - 2
a = 10–2
b=1
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Density profiles - 3
a = 10–5
b=1
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Density profiles - 4
a = 10–2
b = 10–5
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Density profiles - 5
a = 10–3
b = 10–5
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Density profiles - 6
a = 10–5
b = 10–5
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
8. Conclusions
- We have used a truncated distribution function with anisotropy in order to construct models of
relativistic selfgravitating spheres with prevalence of tangential motion (l =1).
- The distribution function recovers truncated Maxwellian one for L=0.
- We have obtained hollow configurations for high level of anisotropy and this property is mainly
frequent in more relativistic model (large values of b). Models with low levels of anisotropy
have usual decreasing density profiles.
- Hollowness is independent from choice of kind of anisotropic distribution: we have similar
configurations with degenerate Fermions (Ralston & Smith 1991).
- Maximal density in hollow configurations is closer to surface for decreasing values of W0.
- Mass is generally decreasing for increasing level of anisotropy (at fixed b).
- Equilibrium configurations are isotropic in the center (r = 0) and at the surface (r = R): this is a
characteristics of the distribution function. In the intermediate region the level of anisotropy
increases in correspondence of maximal density.
- For a → 0 (highest level of anisotropy) the cluster is approaching a structure of a thick shell.
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
9. Perspectives
Theory
- Calculations of equilibrium configurations at index l >1 in the distribution function.
- Calculations at l < 0  h > 1 (models with prevalence of radial motion).
- Analysis of dynamic and thermodynamic stability of anisotropic models: calculation of the
binding energy of the cluster may give informations about the onset of dynamic instability.
- Difficulties :
- In drawing conclusions about thermodynamic stability because anisotropy may be sign
that cluster is out of local thermodynamical equilibrium.
- Presence of different kinetic instabilities, especially in clusters with non-monotonic density
profiles (hollow configurations).
Observations
- Density profiles with hollowness may be indicators of presence of high level of anisotropy in
spherical clusters.
- Anisotropic density distribution of dark matter around giant elliptical galaxies may be estimated
by measurements of the orbital velocities of dwarf galaxies around them or by measurements
of a rotational curve in presence of a disk component.
M. Merafina
Spherically symmetric relativistic stellar clusters with
anisotropic momentum distribution
13 July 2009
Thank you
The present work is submitted to The Astrophysical Journal