Session: MGAT9 – Self-Gravitating Systems SPHERICALLY SYMMETRIC RELATIVISTIC STELLAR CLUSTERS WITH...

Session: MGAT9 – Self-Gravitating Systems

SPHERICALLY SYMMETRIC RELATIVISTICSPHERICALLY SYMMETRIC RELATIVISTIC

STELLAR CLUSTERS WITHSTELLAR CLUSTERS WITH

ANISOTROPIC MOMENTUM DISTRIBUTIONANISOTROPIC MOMENTUM DISTRIBUTION

Marco MERAFINAMarco MERAFINADepartment of PhysicsDepartment of Physics

University of Rome “La Sapienza”University of Rome “La Sapienza”

13 July 200913 July 2009

M. MerafinaSpherically symmetric relativistic stellar clusters with

anisotropic momentum distribution13 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



1. Why “relativistic” clusters ?The question of existence of relativistic clusters is open since the discovery of quasars

- Astronomical observationsHave yielded no definitive evidence about the existence of relativistic clusters

Are clusters so dense that relativistic corrections to Newtonian theory modifytheir structure and influence their evolution ?

- Theoretical calculationssuggest that star clusters might form in the nuclei of some galaxies and quasars

HST observations ?

Dense stellar clusters are represented by

A cluster may be considered “relativistic” if the gravitational potential is comparable with c2

- Globular Clusters with M ~ 106 M and R ~ 50 ÷ 100 pc

- Active Galactic Nuclei (AGNs) with M ~ 108 M

- Quasars with M ~ 1010 M



2. History(…not exhaustive, sorry for omissions)

1939 - Einstein (first paper on relativistic clusters – stars with circular motion: anisotropy)

1992 - Ingrosso, Merafina, Ruffini & Strafella (Anisotropic semidegenerate models withKing-Fermi distribution function)

1965 - Zel’dovich & Podurets (theory of structure of relativistic clusters by Maxwellian truncateddistribution function: one-parameter models)

1977 - Davoust (Newtonian anisotropic models - different choice of the dependence onangular momentum in the distribution function)

1989 - Rasio, Shapiro & Teukolsky (numerical simulations of relativistic clusters – isotropicnoncollisional model)

1969 - Bisnovatyi-Kogan & Zel’dovich (self-similar relativistic solutions with anisotropy –problems: infinite density and radius)

1991 - Ralston & Smith (Anisotropic models of degenerate fermions - “hollow” configurations)

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 gasspheres and polytropes)



3. Anisotropic distribution function

- Spherically symmetric equilibrium Schwarzschild metric: (r), (r)

- Distribution function: f (E,L2)

for E Ec

for E Ec

- Cutoff condition (Zel’dovich & Podurets 1965; Bisnovatyi-Kogan et al. 1998):

- Integrals of motion:

- Components:



4. Thermodynamic quantities

- Concentration :

- Energy density :

- Radial pressure :

- Tangential pressure :



Thermodynamic quantities (continued)

- Some simple relations :

- Newton binomial relation :

New form of thermodynamic quantities



5. Gravitational equilibrium equations

- Equations of equilibrium :

- Boundary conditions : Prr(0) = P0 ; Mr(0) = 0

- Metric coefficients :



6. Dimensionless quantities

- Some simple relations (Merafina & Ruffini 1989, 1990) :



Dimensionless quantities (continued 1)

An important relation for DF :

Then, from considerations of statistical mechanics,we can define a constant B = A e-1/ for which A e-E/T = B eW-x

Moreover



Dimensionless quantities (continued 2)

- Dimensionless gravitational equilibrium equations :

- Boundary conditions :

- Dimensionless thermodynamic quantities :



7. Numerical results: mass vs central density

We consider only results for l = 1 (index of distribution function)

= 1



7. Numerical results: anisotropy

- Local anisotropy level :

- For l = 1 : = 1 (isotropy)

for r = 0

for r = R

Prevalence of tangential motion since index value l > 0 < 1

nevertheless we have a minimum value = 0.5 for high level of anisotropy

The thickness of the external isotropic region is rapidly decreasingwith increasing of level of anisotropy (small values of a)

where



Anisotropy - 1

a = 10–1

= 1

Choice of triad of values of W0: maximum mass (intermediate value); before max; after max



Anisotropy - 2

a = 10–2

= 10–5



Anisotropy - 3

a = 10–3

= 1



Anisotropy - 4

a = 10–3

= 10–5



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 maximumin a region different from the center of the configuration !

For increasing levels of anisotropy the central density may beseveral 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 shellwith maximal density close to the border of the configuration



7. Numerical results: density profiles (continued)

hollowconfigurations

configurations with usualdecreasing density profile

Values of anisotropy parameter “a” for which we have hollow configurations (a<a*)

Newtonian regime («1)

In relativistic regime, hollow configurations exist at lower levels of anisotropy, for increasing values of



Density profiles - 1

Choice of triad of values of W0: maximum mass (intermediate value); before max; after max

a = 10–1

= 1




a = 10–2

= 1




a = 10–5

= 1




a = 10–2

= 10–5




a = 10–3

= 10–5




a = 10–5

= 10–5



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 ). Models with low levels of anisotropyhave usual decreasing density profiles.

- Hollowness is independent from choice of kind of anisotropic distribution: we have similar configurations with degenerate Fermions (Ralston & Smith 1991).

- Mass is generally decreasing for increasing level of anisotropy (at fixed ).

- Maximal density in hollow configurations is closer to surface for decreasing values of W0.

- 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.



9. PerspectivesTheory

- Calculations of equilibrium configurations at index l >1 in the distribution function.

- Calculations at l < 0 > 1 (models with prevalence of radial motion).

- Analysis of dynamic and thermodynamic stability of anisotropic models: calculation of thebinding energy of the cluster may give informations about the onset of dynamic instability.

Observations

- Density profiles with hollowness may be indicators of presence of high level of anisotropy inspherical clusters.

- Anisotropic density distribution of dark matter around giant elliptical galaxies may be estimatedby measurements of the orbital velocities of dwarf galaxies around them or by measurementsof a rotational curve in presence of a disk component.

- Difficulties :

- In drawing conclusions about thermodynamic stability because anisotropy may be signthat cluster is out of local thermodynamical equilibrium.- Presence of different kinetic instabilities, especially in clusters with non-monotonic densityprofiles (hollow configurations).



Thank you

The present work is submitted to The Astrophysical Journal

Date post:	17-Dec-2015
Category:	Documents
Upload:	eustace-berry
View:	220 times
Download:	3 times

Session: MGAT9 – Self-Gravitating Systems SPHERICALLY SYMMETRIC RELATIVISTIC STELLAR CLUSTERS WITH...

Documents