1. Session: MGAT9 – Self-Gravitating Systems SPHERICALLY SYMMETRIC RELATIVISTIC STELLAR CLUSTERS WITH ANISOTROPIC MOMENTUM DISTRIBUTION Marco MERAFINA Department of Physics University of Rome “La Sapienza” 13 July 2009

2. 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

3. 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 - Astronomical observations Have yielded no definitive evidence about the existence of relativistic clusters 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 HST observations ? Dense stellar clusters are represented by A cluster may be considered “relativistic” if the gravitational potential is comparable with c 2 - Globular Clusters with M ~ 10 6 M  and R ~ 50 ÷ 100 pc - Active Galactic Nuclei (AGNs) with M ~ 10 8 M  - Quasars with M ~ 10 10 M 

4. 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) 1992 - Ingrosso, Merafina, Ruffini & Strafella (Anisotropic semidegenerate models with King-Fermi distribution function) 1965 - Zel’dovich & Podurets (theory of structure of relativistic clusters by Maxwellian truncated distribution function: one-parameter models) 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) 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 gas spheres and polytropes)

5. 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) - Distribution function: f (E,L 2 ) for E  E c for E  E c - Cutoff condition (Zel’dovich & Podurets 1965; Bisnovatyi-Kogan et al. 1998): - Integrals of motion: - Components:

6. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 4. Thermodynamic quantities - Concentration : - Energy density : - Radial pressure : - Tangential pressure :

7. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Thermodynamic quantities (continued) - Some simple relations : - Newton binomial relation : New form of thermodynamic quantities

8. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 5. Gravitational equilibrium equations - Equations of equilibrium : - Boundary conditions : P rr (0) = P 0 ;M r (0) = 0 - Metric coefficients :

9. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 6. Dimensionless quantities - Some simple relations (Merafina & Ruffini 1989, 1990) :

10. 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/  for which A e -E/T = B e W-x Moreover

11. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Dimensionless quantities (continued 2) - Dimensionless gravitational equilibrium equations : - Boundary conditions : - Dimensionless thermodynamic quantities :

12. 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)  = 1

13. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 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 decreasing with increasing of level of anisotropy (small values of a) where

14. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Anisotropy - 1 a = 10 –1  = 1 Choice of triad of values of W 0 : maximum mass (intermediate value); before max; after max

15. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Anisotropy - 2 a = 10 –2  = 10 –5

16. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Anisotropy - 3 a = 10 –3  = 1

17. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Anisotropy - 4 a = 10 –3  = 10 –5

18. 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 W 0 For a  0 the cluster is approaching the structure of a thick shell with maximal density close to the border of the configuration

19. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 7. Numerical results: density profiles (continued) hollow configurations configurations with usual decreasing 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 

20. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Density profiles - 1 Choice of triad of values of W 0 : maximum mass (intermediate value); before max; after max a = 10 –1  = 1

21. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Density profiles - 2 a = 10 –2  = 1

22. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Density profiles - 3 a = 10 –5  = 1

23. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Density profiles - 4 a = 10 –2  = 10 –5

24. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Density profiles - 5 a = 10 –3  = 10 –5

25. M. Merafina Spherically symmetric relativistic stellar clusters with anisotropic momentum distribution 13 July 2009 Density profiles - 6 a = 10 –5  = 10 –5

26. 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  ). 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). - Mass is generally decreasing for increasing level of anisotropy (at fixed  ). - Maximal density in hollow configurations is closer to surface for decreasing values of W 0. - 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.

27. 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 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. 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. - 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).

28. 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