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Printed in the U.S.A.. SPHERICALLY SYMMETRIC STELLAR CLUSTERS WITH ANISOTROPY AND CUTOFF ENERGY IN. MOMENTUM DISTRIBUTION. II.
The Astrophysical Journal, 709:1174–1182, 2010 February 1  C 2010.

doi:10.1088/0004-637X/709/2/1174

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

SPHERICALLY SYMMETRIC STELLAR CLUSTERS WITH ANISOTROPY AND CUTOFF ENERGY IN MOMENTUM DISTRIBUTION. II. THE RELATIVISTIC REGIME Gennady S. Bisnovatyi-Kogan1 , Marco Merafina2 , and Maria Rosaria Vaccarelli2 2

1 Space Research Institute (IKI), Profsoyuznaya 84/32, Moscow 117997, Russia; [email protected] Department of Physics, University of Rome “La Sapienza,” Piazzale Aldo Moro 2, I-00185 Rome, Italy; [email protected], [email protected] Received 2009 June 24; accepted 2009 December 9; published 2010 January 12

ABSTRACT We numerically construct models of spherically symmetric relativistic stellar clusters with anisotropic distribution functions. Newtonian solutions obtained in Paper I are generalized as isotropic Maxwellian ones with energy cutoff in their distribution function. We consider distributions with different levels of anisotropy and discuss some general characteristics of the models. Key words: dense matter – galaxies: star clusters: general – hydrodynamics – relativistic processes

such systems (see, e.g., Heller & Shlosman 1994, and references therein). Numerical calculations of the evolution of Newtonian stellar clusters due to close encounters among stars are performed by directly solving the non-relativistic Fokker–Planck equation or using Monte Carlo simulations (see Hut et al. 2007; Freitag 2008a, 2008b, and references therein). The qualitative picture of the evolution of dense stellar clusters of different masses taking into account quite different physical processes had been analyzed by Bisnovatyi-Kogan (1978): it was shown that, due to star evaporation from the cluster during its evolution, the mass of relativistically collapsing core does not exceed approximately 0.1% of its whole initial mass. In most papers mentioned above, where solutions of RnCBE had been obtained, isotropic distribution functions f (E) had been considered. Nevertheless, Einstein (1939) had already constructed a highly relativistic cluster with circular orbits. Analytic self-similar solutions for RSC with an arbitrary level of anisotropy had been found by Bisnovatyi-Kogan & Zel’dovich (1969a, 1969b); however, all formal solutions had infinite central densities and infinite radii, so they could not be applied directly to reality. Anisotropic stellar clusters with larger transverse momentum (the extreme case is a cluster with purely circular orbits) are more stable against relativistic collapse than the isotropic clusters and lose stability in a general relativistic regime at larger mass concentration. Anisotropy in the momentum space appears always on the stage of a rapid contraction of the cluster due to the preservation of the angular momentum. Strong anisotropy in the momentum distribution is expected in dense clusters with a supermassive black hole at the center, where in the vicinity of the last stable orbit only stars with circular orbits can survive. It is also important to mention here papers with solutions for anisotropic stellar clusters obtained in the Newtonian approximation (Bisnovatyi-Kogan & Zel’dovich 1969b; Dejonghe 1987; Ingrosso et al. 1992). Because a large part of the matter in the universe consists of the “dark” non-collisional components, we can argue that it forms equilibrium configurations well described by RnCBE. The evolution of dark matter clusters and the formation of a large-scale structure can lead to the formation of systems with an arbitrary level of anisotropy which could be described by distribution functions far from the equilibrium isotropic Maxwell–Boltzmann one.

1. INTRODUCTION One of the possible ways that supermassive black holes develop in quasars and galactic nuclei is through the collapse of a dense stellar cluster after loss of dynamic stability. This suggestion was first proposed in the pioneering work of Zel’dovich & Podurets (1965), which is considered the starting point of the study of the problem of the structure and stability of relativistic stellar clusters (RSCs). On the basis of this hypothesis, in the papers of Ipser (1969, 1980) and Suffern & Fackerell (1976), and, more recently, in the works of Bisnovatyi-Kogan et al. (1993, 1998, hereafter BKMRV93 and BKMRV98, respectively; see also references therein), the structure and stability of RSC was studied by using the solution of the general relativistic non-collisional Boltzmann equation (RnCBE), similar to that introduced by Zel’dovich & Podurets (1965). The principal motivation for these papers arises from the scenario introduced by Zel’dovich & Podurets in which the combined effects of the core contraction due to gravothermal instabilities and collisions among stars and coalescence inevitably drive a star cluster to states of increasing central density until the appearance of the onset of dynamic instability in a relativistic regime and the subsequent collapse of the system to a supermassive black hole. Even if this scenario can be considered correct, and numerical integrations of the dynamical evolution of RSC (see Shapiro & Teukolsky 1985a, 1985b, 1985c) as well as numerical simulations of RSC consisting of non-collisional point masses in a common gravitational field performed by Rasio et al. (1989) confirm the original idea of Zel’dovich & Podurets that clusters become relativistically unstable at a sufficiently high central redshift, zc  0.5 (see also BKMRV93; BKMRV98; BisnovatyiKogan & Merafina 2006), recent Fokker–Planck and general relativistic dynamic calculations lead us to reconsider the suggestion proposed by Zel’dovich & Podurets, by introducing the possibility that the endpoint of the coalescence phase may be a cluster composed of neutron stars or black holes with masses on the order of solar mass (Begelman & Rees 1978). On the other hand, the possible formation of black holes during the cosmological galaxy formation could also follow different evolutionary paths, due probably to star formation, bar instabilities, or gas dynamical processes concentrating matter in such ways as to form favorable conditions for the origin of 1174

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In this paper, we generalize Newtonian models described in Paper I (Bisnovatyi-Kogan et al. 2009) to RSC with an anisotropic distribution in the momentum space. We use the same approach for obtaining the solutions as in BKMRV93, BKMRV98, and Bisnovatyi-Kogan & Merafina (2006). Here, we restrict ourselves to analysis of equilibrium configurations, while in a future paper we shall address the investigation of the stability of these clusters.

2GM , (9) Rc2 where Mr is the mass inside a given Lagrangian radius r. The solutions of equilibrium equations describing RSC have both density and components of pressure vanishing at the outer boundary r = R.

2. MAIN EQUATIONS

Thermodynamic quantities such as the concentration n, connected with the rest-mass density ρ0 = nm, the total energy density ρc2 , and the components of the stress tensor (Prr , Pt ), are expressed by integrals containing the distribution function f of Equation (3). We have  n = 2π fpt dpt dpr , (10)

Let us consider spherically symmetric equilibrium configurations with a Schwarzschild-like metric (see Landau & Lifshitz 1962) ds 2 = eν c2 dt 2 − eλ dr 2 − r 2 (dθ 2 + sin2 θ dϕ 2 ).

(1)

The exact solution of RnCBE must depend on integrals of motion which are represented by energy E and angular momentum L, expressed by the relations and L = pt r, (2)  where m is the stellar mass and p = pr2 + pt2 is the stellar momentum, with pr and pt being the radial and transversal components, respectively. We search for a solution depending only on E and L in the form

eνR = e−λR = 1 −

3. THERMODYNAMIC QUANTITIES

 ρc = 2π 2

E = eν/2 (p2 c2 + m2 c4 )1/2



L2 f =A 1+ 2 Lc

l e

−E/T

f = 0 for

for

E  Ec ,

(3)

 Prr = 2π

f (p2 c2 + m2 c4 )−1/2 c2 pr2 pt dpt dpr , 

f (p 2 c2 + m2 c4 )−1/2 c2 pt3 dpt dpr .

Pt = π

where T (constant) is the temperature in energy units measured by an infinitely remote observer and Lc = mcra is a constant depending on the anisotropy radius ra that represents the value of the radius beyond which the orbits begin to be tangential in prevalence. The cutoff energy is chosen, following Zel’dovich & Podurets (1965), in the form Ec = mc − αT /2. 2

dPrr G (Prr + ρc2 )(Mr c2 + 4π Prr r 3 ) 2 − (Prr − Pt ), =− 2 dr rc rc2 − 2GM r r (5) dM r = 4π r 2 ρ, (6) dr with boundary conditions Prr (0) = P0 and Mr (0) = 0. The metric coefficients are determined by relations   2GM r −1 λ e = 1− , (7) rc2 2G Mr c2 + 4π Prr r 3 dν = 2 , dr c r(rc2 − 2GM r )

(8)

(12) (13)

pt = p sin θ,

with 0  θ  π.

(14)

Taking into account that L = rpt = rp sin θ and the Newton binomial relation l   l    2 k l L L2 1+ 2 = , k Lc L2c k=0

(15)

we can write the integrals (10)–(13) in the form

(4)

In order to find the equilibrium configurations representing different models of RSC, we solve the relativistic equilibrium equations (see, e.g., Bisnovatyi-Kogan & Zel’dovich 1969a) that generalize the well-known Oppenheimer–Volkoff equations in the presence of an anisotropic pressure tensor with Prr = Pt . The equations are

(11)

To simplify the integrations we introduce the angle θ in the plane (pr , pt ), so that pr = p cos θ,

E > Ec ,

f (p2 c2 + m2 c4 )1/2 pt dpt dpr ,

n = 2π A

   pc l    l r 2k π (sin θ )2k+1 dθ p2k+2 e−E/T dp k L c 0 0 k=0

  l    l r 2k π ρc = 2π A (sin θ )2k+1 dθ k L c 0 k=0  pc p2k+2 (p2 c2 + m2 c4 )1/2 e−E/T dp × 2

(16)

0

Prr = 2π Ac2 

pc

×

  l    l r 2k π (sin θ )2k+1 cos2 θ dθ k L c 0 k=0 p2k+4 (p2 c2 + m2 c4 )−1/2 e−E/T dp

0

  l    l r 2k π Pt = π Ac (sin θ )2k+3 dθ k L c 0 k=0  pc × p2k+4 (p2 c2 + m2 c4 )−1/2 e−E/T dp. 2

0

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BISNOVATYI-KOGAN, MERAFINA, & VACCARELLI

For solving Equations (5) and (6) with thermodynamic functions (16), it is convenient to introduce the following variables: ε = (p2 c2 + m2 c4 )1/2 − mc2 (current “particle” kinetic energy),

From Equations (19) we also have (17) e−E/T = e−(ε+mc

TR β= mc2 (nondimensional temperature, in energy units, at the boundary of the cluster), W =

Moreover, we have the following relations: εc (R) + mc2 Ec mc2 1 = = = , (18) T TR TR β

mc2 εc mc2 mc2 1 = − = − , Tr TR Tr β Tr

(19)

mc2 1 − βW . = Tr β If we introduce the nondimensional variables x = ε/Tr and y = ε/mc2 and consider Equations (19), we obtain (Merafina & Ruffini 1989) x=

ε ε mc2 1 − βW y, = = Tr mc2 Tr β

= e−(x−W +1/β) = eW −x e−1/β , (22)

 2k  k+3/2 l    √ l r β 2 2k+3 mc ( 2mc) Ak ρc = π B k Lc 1 − βW k=0    2  W βx/2 k+1/2 βx W −x 1+ 1+ × e x k+1/2 dx, 1 − βW 1 − βW 0 (24) 2

where we have taken into account that the ratio Ec /T is a constant all over the cluster and the stellar velocities at the outer boundary are zero, so that εc (R) = 0. By using relations (4), (9), (17), and (18) we obtain the following formulae:   2GM 1/2 Ec = mc2 − αT /2 = mc2 eνR /2 = mc2 1 − , Rc2 W ≡

)/Tr

 2k  k+3/2 l    l √ r β 2k+3 n = πB ( 2mc) Ak k Lc 1 − βW k=0  k+1/2   W βx/2 βx 1+ x k+1/2 dx, × eW −x 1 + 1 − βW 1 − βW 0 (23)

εc (0) T0 (the value of W at the center of the cluster).

E ε + mc2 = , T Tr

2

and therefore, from considerations of statistical mechanics, we can define the constant B = Ae−1/β . Finally, due to these transformations, we obtain the following form of the integrals of Equations (16):

εc Tr (the nondimensional ratio of maximal kinetic energy to the local temperature),

W0 =

  1/2   √ βx βx/2 1/2 p = y(y + 2) = 2 1 + , mc 1 − βW 1 − βW (21) βx  p 1 + 1−βW  y+1  = d y(y + 2) = √ dy = d βx/2 mc y(y + 2) 1 + 1−βW

dx β × √ . 1 − βW 2x

εc = (pc2 c2 + m2 c4 )1/2 − mc2 (maximal kinetic energy at a given radius r), Tr = T e−ν/2 (local temperature at a given radius r),

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y=

βx . (20) 1 − βW

As Merafina & Ruffini (1989, 1990) showed, it is more convenient to use variables x instead of p/mc in the calculations of nondimensional integrals (16) and W in the equilibrium equations (5) and (6). In order to transform the integrals (16) in a more suitable form, it is useful to introduce the following relations:  p 2 c 2 + m2 c 4 ε = + 1 = y + 1, 2 mc mc2

 2k l    √ l r 2 2k+3 2mc ( 2mc) Prr = π B k L c k=0 k+5/2  β × (Ak − Ak+1 ) 1 − βW    W βx/2 k+3/2 k+3/2 × eW −x 1 + x dx, (25) 1 − βW 0  2k  k+5/2 l    √ l r β mc2 ( 2mc)2k+3 Ak+1 k Lc 1 − βW k=0  k+3/2  W βx/2 × eW −x 1 + x k+3/2 dx. (26) 1 − βW 0

Pt = π B

The coefficients Ak are represented by integrals over the trigonometric functions, which lead to analytic expressions (Bronstein & Semendyaev 1985) 

π

Ak ≡

(sin θ ) 0

2k+1

dθ = 2

k    k (−1)i i=0

i 2i + 1

  k k! = , i!(k − i)! i

,

0! = 1

(27)

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SPHERICALLY SYMMETRIC STELLAR CLUSTERS. II. 

π

(sin θ )2k+1 cos2 θ dθ = Ak − Ak+1 .

0

In particular, the first four values of Ak are given by A0 = 2,

A1 =

4 , 3

A2 =

16 , 15

A3 =

32 . 35

(28)

In order to quantitatively evaluate the effects of anisotropy on RSC, it may be useful to introduce the local anisotropy level, defined as η = Prr /Pt (see also Bisnovatyi-Kogan et al. 2009). Due to the form of the distribution function (3), the value of η is equal to unity (isotropic function) at the center and edge of the cluster, like the Newtonian case. Except for these two points, η < 1 all over the cluster, since we have considered only positive values of l (see details in Section 5). 4. MAIN EQUATIONS IN NONDIMENSIONAL VARIABLES From relations (17) and (19), we can write 1 − βW =

−νR /2

2

ν−νR βmc TR Te = = =e 2 −ν/2 Tr Tr Te

(29)

and −β

1 ν−νR dν 1 dν dW = e 2 = (1 − βW ) . dr 2 dr 2 dr

(30)

Substituting relation (30) into Equation (8) we obtain, instead of Equation (5), the equilibrium equation in the following form:   dW G 1 − βW Mr c2 + 4π Prr r 3 =− 2 , (31) dr c β r(rc2 − 2GM r ) which must be solved together with Equation (6). The solution of the system (6) and (31) is uniquely determined by the boundary conditions at the center of the equilibrium configuration, W (0) = W0

and

Mr (0) = 0,

(32)

with condition W (R) = 0 which arises from the condition εc (R) = 0 at the edge of the configuration and uniquely determines the radius R and the mass M of the cluster. Let us now introduce the nondimensional variables r = ξ r˜ , ρc2 =

c4 ρ, ˜ Gξ 2

c2 n= n, ˜ Gmξ 2

Prr =

c4 ˜ Prr , Gξ 2

  k+5/2  l    l k+5/2 r˜ 2k β 2 (Ak − Ak+1 ) k a 1 − βW k=0 k+3/2   W βx/2 × eW −x 1 + x k+3/2 dx, 1 − βW 0    k+5/2 l    l k+3/2 r˜ 2k β ˜ 2 Pt = π Ak+1 k a 1 − βW k=0  k+3/2  W βx/2 × eW −x 1 + x k+3/2 dx. 1 − βW 0

P˜rr = π

The equilibrium equations (31) and (6) can be rewritten in the nondimensional form as   dW 1 − βW M˜ r + 4π P˜rr r˜ 3 , (35) =− d r˜ β r˜ (˜r − 2M˜ r ) d M˜ r = 4π ρ˜ r˜ 2 , (36) d r˜ with the boundary conditions W (0) = W0 and M˜ r (0) = 0. This system of equations gives a unique solution for fixed parameters l, a, β, W0 . The parameters (β, W0 ) are connected with the original nondimensional physical parameters of the problem (α, T˜ ), where T˜ = T /mc2 (see Equation (4) and BKMRV93). We have therefore Ec = eν/2 (εc + mc2 ) = mc2 − αT /2 and, in particular, at the boundary of the equilibrium configuration where r = R, from Equations (33) and (9) we obtain α T˜ 1− = 2

c4 ˜ Pt , Gξ 2

1−

β = T˜ e−νR /2 = ra = ξ a,

(37)

T˜ 1 − α T˜ /2

(38)

and, also using relations (19), we obtain W =

(33) where

2M˜ , r˜

being εc (R) = 0. Moreover, from definitions (17), we have

c2 ξ ˜ Mr = Mr , G Pt =

1177

k+3/2    l    l k+3/2 r˜ 2k β 2 Ak ρ˜ = π a 1 − βW k k=0  k+1/2  2  W βx/2 βx 1+ × eW −x 1 + x k+1/2 dx, 1 − βW 1 − βW 0 (34)

1 eν/2 α 1 − eν/2 α − − = − , ˜ ˜ ˜ 2 2 T T T

so that ξ = (m GBc) 4

−1/2

.

Then the thermodynamic functions (23)–(26) are written in the nondimensional form as    k+3/2 l    l k+3/2 r˜ 2k β n˜ = π 2 Ak k a 1 − βW k=0 k+1/2     W βx/2 βx W −x x k+1/2 dx, 1+ 1+ × e 1 − βW 1 − βW 0

W0 =

α 1 − eν0 /2 − . ˜ 2 T

(39)

Equations (38) and (39) relate parameters β and W0 , used in Merafina & Ruffini (1989, 1990), to parameters α and T˜ , used in BKMRV93 and BKMRV98. Thus, at fixed l and a, the variety of models can be considered by different parameter spaces (α, T˜ ) or (β, W0 ). As in Bisnovatyi-Kogan & Merafina (2006), we shall use the space (β, W0 ) for identifying the different families of equilibrium configurations.

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Figure 1. Mass of the cluster as a function of the central density at different values of β, for l = 1 and a = 10−2 . Increasing values of the central density correspond to larger values of the parameter W0 . The quantities are dimensionless.

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Figure 3. Same as Figure 1, for l = 1 and a = 10−5 .

Figure 4. Same as Figure 1, for l = 2 and a = 10−5 . Figure 2. Same as Figure 1, for l = 2 and a = 10−2 .

5. NUMERICAL RESULTS The calculations have been performed for values l = 1, 2 of the index of the distribution function (3). The physical implications on the choice of other values of l, also negative ones, will be considered in a future paper, in which the main characteristics of such models will be described. In Figures 1–4, we represent the nondimensional mass of the cluster as a function of its central density, at different values of the temperature parameter β, for indices l = 1, 2 and anisotropy parameters a = 10−2 , 10−5 . In these figures, it is simple to note that the effects of the presence of anisotropy (at increasing l and/or decreasing a) manifest a general decreasing of masses of the equilibrium configurations. We can also note that the curves converge for β  1. By considering a different point of view, in Figure 5 we have represented the same curves described in the first four figures by varying the anisotropy parameter from a = 10−1 to a = 10−5 , for fixed values l = 1 and β = 1. The effects of anisotropy on masses of equilibrium configurations previously indicated are explicit; moreover, at smaller levels of anisotropy (a  0.1), the curves approach the ones describing isotropic clusters, shown

Figure 5. Mass of the cluster as a function of the central density at different values of a, for l = 1 and β = 1. The quantities are dimensionless.

in the diagram (b) of Figure 5 in BKMRV98. Note that, unlike the figure in BKMRV98, we have chosen to fix the parameter β instead of T˜ .

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SPHERICALLY SYMMETRIC STELLAR CLUSTERS. II.

Figure 6. Values of the ratio of the pressure components η = Prr /Pt as a function of the relative radius r/R along the cluster, for l = 1, a = 10−1 , β = 1, and W0 = 0.001; 0.216; 0.4.

Figure 7. Same as Figure 6, for l = 1, a = 10−2 , β = 10−5 , and W0 = 1; 2.50; 5.

The level of the anisotropy in the distribution function (3) depends on the value of the parameter a. The local anisotropy level may be represented by the ratio between the components of the stress tensor (see Bisnovatyi-Kogan et al. 2009) 2 vr2 P˜rr Prr η = 2 = = . (40) Pt vt P˜t The quantities vr2  and vt2  are the radial and tangential mean square velocities of stars, respectively. The value of η approaches to unity for isotropic functions. In Figures 6–9, we have represented the quantity η as a function of the relative radius r/R for l = 1 and selected values of the parameters a, β, and W0 . Due to the choice of a positive value of index l in the distribution function (3), the anisotropy evidences a prevalence of tangential motion over the radial one, which increases with the decrease of the parameter a. The triad of values of W0 chosen in each figure refers to the equilibrium configuration whose mass corresponds to the maximum value in M versus ρ0 diagrams of Figures 1–5 and to values before and after this maximum. The nondimensional radius of the cluster R˜ depending on the parameters a, W0 , and β is represented in Table 1.

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Figure 8. Same as Figure 6, for l = 1, a = 10−3 , β = 1, and W0 = 0.001; 0.176; 0.4.

Figure 9. Same as Figure 6, for l = 1, a = 10−3 , β = 10−5 , and W0 = 1; 3.12; 5.

It follows from the calculations that clusters described by the distribution (3) are isotropic not only at the center but also at the edge of the configuration; in fact, it is easy to see from the expressions of the components of pressure (25) and (26) that Prr /Pt → 1 at the boundary of the cluster, where W → 0 and r → R. As we may see in Figures 6–9, the thickness of the external isotropic region rapidly decreases with the decrease of the anisotropy parameter a which corresponds to higher levels of anisotropy. Moreover, like in the Newtonian regime, we can express the quantity η by Equations (34). We have η=

1+

4 5

1+

8 5

r˜ 2 a

I2 I1

a

I2 I1

r˜ 2

,

(41)

where  I1 =  I2 =

β 1 − βW β 1 − βW

5/2 

W

0

7/2 

  βx/2 3/2 3/2 eW −x 1 + x dx, 1 − βW

W

e 0

W −x

  βx/2 5/2 5/2 1+ x dx. 1 − βW

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Figure 10. Relative density ρ/ρ0 as a function of the relative radius r/R along the cluster, for an index l = 1. The values of the parameters are a = 10−1 , β = 1, and W0 = 0.001; 0.216; 0.4. Table 1 Some Numerical Characteristics of the Anisotropic Clusters with Different Values of a, W0 , and β W0

β



r˜m

rm /R

ρm /ρ0

10−5 10−5 10−5

1 3.64 5

10−5 10−5 10−5

1.98 × 10−1 1.09 × 10−1 9.20 × 10−2

9.40 × 10−2 4.10 × 10−2 2.90 × 10−2

4.74 × 10−1 3.76 × 10−1 3.15 × 10−1

3.66 × 102 1.64 × 102 9.05 × 101

10−5 10−5 10−5

0.001 0.174 0.4

1 1 1

3.67 × 10−2 4.19 × 10−3 2.37 × 10−3

1.80 × 10−2 1.89 × 10−3 8.70 × 10−4

4.90 × 10−1 4.51 × 10−1 3.67 × 10−1

1.61 × 103 4.08 × 103 3.10 × 103

10−3 10−3 10−3

1 3.12 5

10−5 10−5 10−5

1.95 × 100 1.18 × 100 9.75 × 10−1

8.40 × 10−1 3.80 × 10−1 9.90 × 10−2

4.32 × 10−1 3.21 × 10−1 1.02 × 10−1

3.50 × 100 1.91 × 100 1.02 × 100

10−3 10−3 10−3

0.001 0.176 0.4

1 1 1

3.65 × 10−1 4.16 × 10−2 2.38 × 10−2

1.80 × 10−1 1.90 × 10−2 8.65 × 10−3

4.93 × 10−1 4.57 × 10−1 3.64 × 10−1

1.59 × 101 4.07 × 101 3.09 × 101

10−2 10−2 10−2

1 2.50 5

10−5 10−5 10−5

4.38 × 100 2.96 × 100 2.13 × 100

0 0 0

0 0 0

1 1 1

10−2 10−2 10−2

0.001 0.184 0.4

1 1 1

1.07 × 100 1.27 × 10−1 7.53 × 10−2

4.30 × 10−1 5.40 × 10−2 2.50 × 10−2

4.01 × 10−1 4.26 × 10−1 3.32 × 10−1

1.61 × 100 3.98 × 100 3.09 × 100

10−1 10−1 10−1

1 2.48 5

10−5 10−5 10−5

4.74 × 100 3.20 × 100 2.23 × 100

0 0 0

0 0 0

1 1 1

10−1 10−1 10−1

0.001 0.216 0.4

1 1 1

1.66 × 100 2.77 × 10−1 1.84 × 10−1

0 0 0

0 0 0

1 1 1

a

Notes. Numerical characteristics: the nondimensional radius of the configura˜ the nondimensional radius of maximal density r˜m , the relative radius of tion R, maximal density rm /R, and the ratio of the maximal to central density ρm /ρ0 . The value of the index of the distribution function is l = 1.

˜ we have I2 /I1 → 0, At the edge of the cluster, where r˜ → R, so that η → 1. At large levels of anisotropy, when a is smaller, the terms with I2 /I1 in Equation (41) are much larger than 1 and the equilibrium configuration reaches the maximal anisotropy corresponding to ηm = 0.5. This is the same result that obtained in the Newtonian regime (see Figures 8 and 9). 6. DENSITY PROFILES OF THE EQUILIBRIUM CONFIGURATIONS It is interesting to note that density profiles of the equilibrium configurations, described by the distribution function (3), have

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Figure 11. Same as Figure 10, for l = 1, a = 10−2 , β = 1, and W0 = 0.001; 0.184; 0.4.

Figure 12. Same as Figure 10, for l = 1, a = 10−5 , β = 1, and W0 = 0.001; 0.174; 0.4. Due to the large scale used in the values of ρ/ρ0 , it seems that the relative density approaches to zero at the center of the configuration, but the actual value is 1.

an increasing behavior in the central region, due to the presence of a sufficiently large level of anisotropy, and therefore the maximum density is achieved out of the center. Thus, also in the relativistic regime, the existence of “hollow” configurations is clearly a feature of anisotropy and confirms the results obtained by Ralston & Smith (1991) with a different anisotropic distribution function. In Figures 10–15, we have represented the behavior of density in clusters with different values of a, β, and W0 , for an index l = 1. The triads of values of W0 in such figures are chosen with the same criterion considered in Figures 6–9. At very large levels of anisotropy, the central density may be several orders of magnitude smaller than the maximum value and the maximal density is situated far from the central region of the configuration, especially for the decreasing values of W0 (see, in particular, Figures 12 and 15). Moreover, for a → 0, the cluster is approaching a structure of a thick shell. This situation becomes different at larger values of the anisotropy parameter a, when the configuration begins to be closer to the isotropic one; in these cases the maxima of density disappear and the profiles recover the usual monotonic decreasing behavior from the center to the boundary of the equilibrium configuration. The presence of the maxima of density in a region different from the center of the configuration is clearly related to the

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structure if β = 10−5 . For more relativistic models (β = 1) the critical value of η increases to 0.7–0.8, allowing one to obtain hollow configurations at lower levels of anisotropy (see, in particular, Figure 11). The results concerning the configurations of Figures 10–15 are summarized in Table 1, where we have represented the relative rm /R and the nondimensional r˜m radius corresponding to the maximum density for different values of a, β, and W0 . 7. CONCLUSIONS

Figure 13. Same as Figure 10, for l = 1, a = 10−2 , β = 10−5 , and W0 = 1; 2.50; 5.

Figure 14. Same as Figure 10, for l = 1, a = 10−3 , β = 10−5 , and W0 = 1; 3.12; 5.

We have constructed models of anisotropic relativistic clusters with a distribution function which generalize the quasiequilibrium Maxwellian distribution function with the energy cutoff. The presence of anisotropy leads to density profiles which can exhibit a maximum lying far from the center. This particular feature, called “hollowness,” is mainly present in more relativistic configurations and for smaller values of the anisotropy parameter a corresponding to higher levels of anisotropy. A similar type of anisotropy is created around supermassive black holes due to the existence of a loss cone. Self-gravitation of the cluster is important when the mass of the bulge is comparable to the black hole mass. Nevertheless, in principle, we can also produce anisotropy with prevalence of the radial component, for example, in a spherically symmetric collapse, where radial bulk motion can be transferred into radial velocity dispersion. By analyzing the behavior of the mass of the equilibrium configurations as a function of the central density, we have seen that the effect of the presence of anisotropy leads to a general decreasing of equilibrium mass at fixed values of the parameter β. The masses of the equilibrium configurations at increasing values of the central density also show a nonmonotonic behavior for fixed values of the parameters β and a. It was shown by Antonov (1962) and Lynden-Bell & Wood (1968) that the first maximum on the curve M(ρ0 ) for a Newtonian cluster in a box corresponds to loss of the thermodynamic stability of the cluster and the beginning of its rapid contraction. In Bisnovatyi-Kogan & Merafina (2006), it has been concluded that the behavior of a cluster with a cutoff is very similar to the one in a box, both qualitatively and quantitatively. This result was also extended to the relativistic regime. However, analyzing thermodynamic stability in the presence of anisotropy may be difficult, being that the cluster is out of local thermodynamic equilibrium. Nevertheless, studies on gravothermal catastrophe in these systems have been performed by Spurzem (1991) and Magliocchetti et al. (1998). Conclusions about dynamical and thermodynamical stabilities of relativistic clusters in the presence of anisotropy have not been drawn in the present work and the problem will be systematically discussed in a future paper. REFERENCES

Figure 15. Same as Figure 10, for l = 1, a = 10−5 , β = 10−5 , and W0 = 1; 3.64; 5. As in Figure 12, the actual value of ρ/ρ0 at the center of the configuration is 1.

behavior of the parameter η as a function of the relative radius r/R previously discussed. It seems, in fact, that models with η  0.6 all over the configuration cannot have a “hollow”

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