THERMALIZED NEWTONIAN BOSE STARS HEINZ DEHNEN and RALPH N. GENSHEIMER Universität Konstanz, Fakultät für Physik, Postfach 55 60, D-78457 Konstanz, Germany; E-mail: [email protected]

(Received 11 August, 1997; accepted 4 August, 1998)

Abstract. We consider star models consisting of spin 0 particles interacting only gravitationally, e.g. Higgs-particles as possible dark matter objects. The particle gas of finite temperature is treated according to the Bose-Einstein-statistics in its non-relativistic limit; then the use of Newtonian gravity is sufficient too. The limits of these restrictions are estimated. The local temperature is determined with the use of energy conservation. The mass-radius relation of the stars and their further behaviours as Bose-Einstein-condensation in the outer regions are calculated. We find strong similarities with the features of white dwarfs and neutron stars.

1. Introduction The flat rotation curves of spiral galaxies represent a strong hint for the existence of dark matter in the Universe (for other interpretations see e.g. Bekenstein and Milgrom, 1984; Sanders, 1986; Fahr, 1990; Geßner, 1992). As such are discussed brown or white dwarfs as well as black holes existing in the galactic halos. However in view of the nucleosynthesis in the early Universe and the observed abundance of the light chemical elements the dark matter should not contribute to the early nucleosynthesis and therefore consist of non baryonic particles e.g. neutrinos or Higgs-particles or even more exotic candidates without electromagnetic and strong interactions. Therefore, in a previous paper (Dehnen et al., 1995) we have discussed the possibility, that the dark matter particles consist of only gravitationally interacting Higgs-particles originated in the early Universe, which form a halo surrounding the galaxies. In this way we could explain the flat rotation curves. Another possibility is given by the existence of dark matter stars, e.g. stars consisting of Higgs-particles, which are present in the galactic halo, but observable (beside producing the flat rotation curves) only gravitationally e.g. by gravitational lensing effects. Of course, for forming a star the Higgs-particles must be stable; therefore the standard model of the electroweak interaction must be modified in such a way that the Higgs-particles cannot decay. Such an extension of the standard model exists. In previous papers (Dehnen and Frommert, 1993; van der Bij, 1994) it is proposed in view of Mach’s principle, that the Higgs-field plays simultaneously the role of a variable gravitational constant within the framework of a scalar-tensor theory of gravity. It comes out, that Einstein’s theory of gravity and its Newtonian Astrophysics and Space Science 259: 355–369, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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limit is valid in the lowest order, where the Higgs-field ground state determines the Newtonian gravitational constant beside the masses of the fermions and the gauge bosons; furthermore the differential equation for the excited Higgs-field states takes exactly the form of a homogeneous Klein-Gordon equation coupled only minimally to the gravitational field. In this case the associate massive Higgs-particles are stable and interact only gravitationally, so that they are ideal candidates for the dark matter particles. Because of the only very weak gravitational interaction they cannot be generated in the laboratory but only in the very early stages of the Universe. In the present paper we investigate the existence of stars consisting of such only gravitationally interacting Higgs-particles, which are created in the early Universe. Our interest concerns especially the features of such stars, as radius, mass, temperature and Bose-Einstein condensation of the Higgs-particles within the star. Their detection by lensing effects is not discussed. The investigations of Bose stars are performed usually under the assumption of vanishing temperature and total degeneracy; consequently the condensate of spin 0 particles, e.g. Higgs-particles is described by solutions of the Einstein-KleinGordon equations (see e.g. Kaup, 1968; Liddle and Madsen, 1992 and Schunck, 1995; for a review see e.g. Lee and Pang, 1992 and Jetzer, 1992). In contrast to this we will describe the Bose gas of Higgs-particles (spin 0) by the equations of states according to the Bose-Einstein-statistics with finite temperature. This is also done in the literature (Ingrosso and Ruffini, 1988; for a review see Jetzer, 1992), however under the condition of constant temperature (isothermal spheres of bosons). In contrast to this we determine the temperature locally in a selfconsistent way via energy conservation under the assumption that the Bose particles interact only gravitationally. Then the barometric formulae can be deduced with the help of the hydrostatic equilibrium condition. Subsequently we solve the gravitational field equation, where we restrict ourselves to the non-relativistic Newtonian limit. This is a sufficient approximation as long as the radius of the star is small compared with its Schwarzschild radius. The degeneracy degree in the center or the central temperature and the mass of the star as well as the mass of the Bose particles are free parameters. We find for the Bose stars a homologous behaviour, so that the field equations must be solved only once. The properties, which can then be derived for several configurations, are in the case of degeneracy or semi-degeneracy of the Bose gas similar to those of fermionic systems e.g. white dwarfs or neutron stars. In the Boltzmann limit the radius of the stars diverges. The question of the formation of the Bose stars and their thermalization will not be discussed in detail.

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2. The Bose Star Calculation The equations of state for an ideal Bose gas consisting of spin 0 particles of the mass m are in parametric representation (ρ total mass density, p pressure), see e.g. Cox and Giuli, 1968: √
(2.1) %(γ , β) = 3 2 λ2 β 3/2 F1/2 (γ , β) + βF3/2 (γ , β) ,   √ β 5/2 p(γ , β) = 16 2 λ1 β F3/2 (γ , β) + F5/2 (γ , β) 2

(2.2)

with µ m4 c5 π kT , λ := , , β := 1 kT mc2 6h3 and the Bose integrals:
1/2 Z ∞ n x 1 + βx 2 Fn (γ , β) := dx , n > 0 . eγ +x − 1 0 γ := −

λ2 :=

4m4 c3 π 3h3

(2.3)

Here 0 ≤ γ < ∞ is the degeneracy number (γ → ∞ represents the Boltzmann limit, γ = 0 the totally degenerate Bose gas), µ the chemical potential and T the temperature. In the following we restrict the analysis to the non-relativistic case, β  1; at the end we discuss the conditions for this. Then the equations of state (2.1) and (2.2) read: 2π 3/2 2 √ m 2m (kT )3/2 g3/2 (γ ) , (2.4) % (γ , T ) = h3 2π 3/2 √ m 2m (kT )5/2 g5/2 (γ ) h3 where gn (γ ) is given by: ∗ Z ∞ x n−1 1 gn (γ ) := dx 0(n) 0 eγ +x − 1 p (γ , T ) =

(2.5)

(2.6)

with the asymptotic values g3/2 (γ = 0) = ζ(3/2) = 2.612 ,

g5/2 (γ = 0) = ζ(5/2) = 1.341

g3/2 (γ → ∞) = g5/2 (γ → ∞) = e−γ

(fugacity) .

(2.7)

Herein temperature T and degeneracy number γ , i.e. % and p, are to be considered as functions of the radial coordinate r. In the non-relativistic limit (β << 1) we can restrict ourselves to Newtonian gravity (see (2.42)). In this case we have to solve for the self-gravitating, ideal R ∗ 0(n) := ∞ e−t t n−1 dt (gamma function) 0

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Bose gas the spherical symmetric Poisson equation for the gravitational potential 8 (|r = ∂r∂ ): 2 (2.8) 8|r|r + 8|r = 4π G%, (G = gravitational constant) r with the Bose gas in its hydrostatic equilibrium: %8|r + p|r = 0 . (2.9) Because 8, γ and T are the unknown functions of r to be determined we need beyond (2.8) and (2.9) a third determination equation. For this we deduce – instead of the assumption of constant temperature – an equation for the temperature in the following way: The mean energy balance equation for a single Bose particle underlying only gravity is because of the long free path length similarly as for the photons of a hypothetical pure lightstar the law of energy conservation; this reads: Et ot =

< Ekin (T ) > + Epot (8) = const. , N

Epot = m8

(2.10)

(N mean particle number), wherein < Ekin > /N is the mean kinetic energy of a particle according to the Bose-Einstein-statistics given by:

 Ekin (T ) N −1 = kT Y −1 (γ (r)) ,

Y (γ (r)) :=

2 g3/2 (γ ) . 3 g5/2 (γ )

(2.11)

Normally in the case of convective mixing polytropes are used (see e.g. Chandrasekhar, 1939). However in this case other interactions beside gravity are present reducing the free path length. Assuming, that a particle of mean kinetic energy cannot pass the star surface S (stable star with finite radius), the average velocity of a particle, i.e. its kinetic energy, has to vanish on the surface of the star (r = R), so that there is valid: Et ot = m8(R) = m8s .

(2.12)

After insertion of (2.11) and (2.12) into (2.10) one finds immediately for the temperature: kT (r) = m u(r) ¯ Y (γ (r)) ,

u(r) ¯ := 8s − 8(r) ≥ 0,

r≤R.

(2.13)

Temperature and herewith mass-density (2.4) and pressure (2.5) vanish at the surface. Equations (2.8), (2.9) and (2.13) represent the system for 8, γ and T to be solved, where ρ and p are given by (2.4) and (2.5). In the first step equation (2.13) can be used to eliminate the temperature dependence in (2.4) and (2.5), which results in 4π √ (2.14) % = 3 2m4 u¯ 3/2 A(γ ), h

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Figure 1. Course of A(γ ) according to (2.16).

p=

8π √ 4 5/2 2m u¯ A(γ ) h3

(2.15)

with

1√ π g3/2 Y 3/2 . (2.16) 2 The range of A(γ ) according to (2.16) is given with the use of (2.7) and (2.11) by: A(γ ) :=

0 = A(γ = ∞) ≤ A(γ ) ≤ 3.425 = A(γ = 0)

(2.17)

and drawn in Figure 1 as monotonically decreasing function of γ . When A(γ ) reaches the upper value for γ = 0 Bose-Einstein condensation begins for which with respect to (2.4) and (2.7) it is valid (2.18) n 3t h3 ≥ g3/2 (0) = 2.612 , √ where 3t h := h/ 2π mkT is the thermal de Broglie wavelength and n = mρ the particle number density. During the condensation A(γ ) growth beyond the upper value of (2.17) up to infinity. Insertion of (2.14) and (2.15) into the equilibrium condition (2.9) results in the total differential equation: A|r u¯ |r + =0 (2.19) u¯ A with the integral: B B = const. (2.20) A= , u¯

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Obviously A is a monotonically increasing function of r and therefore γ a monotonically decreasing function in view of (2.17) and Figure 1. Because in direction to the surface u¯ runs to zero, see (2.13), A exeeds at a critical radius rc the upper value in (2.17) and runs to infinity, so that in the outer regions of the star Bose-Einstein condensation happens independently from the central conditions. This behaviour is different from that in the isothermal case, where the condensed phase is in the center and the outer regions are Boltzmannian and spatially unlimited (see Ingrosso and Ruffini, 1988 and the end of this chapter). Putting (2.20) into (2.14) and (2.15) we get the barometric formulae for the self-gravitating √ Bose gas: 4π 2 4 1/2 m B u¯ , (2.21) %= h3 √ 8π 2 4 3/2 p= m B u¯ (2.22) 3h3 according to which the Bose gas obeys a polytropic relation with polytropic index κ = 3: h6 a := . (2.23) p = a%3 , 48π 2 m8 B 2 In contrast to (2.21) and (2.22) it is not possible to represent the temperature (2.13) by the potential u¯ alone because Y (γ ) can not be expressed by A(γ ) only. One gets e.g.:  1/5  2/5 2 u¯ 3/5 4 m , (2.24) B kT = π 3 g5/2 2/5 according to which the temperature decreases with increasing distance from the center. Now we are able to integrate the Poisson equation (2.8) after insertion of (2.21) and the transition 8 → u, ¯ see (2.13); introducing additionally the dimensionless variable x = r/R and the substitution: √ 16 2π 2 m4 BG −2 −4 C := , u≥0 (2.25) u = C R u¯ , h3 we get a homologous form for the Poisson equation independent from any properties of the single star (u dimensionless):
(2.26) x −2 x 2 u|x |x + u1/2 = 0 , x ∈ [0, 1] . This is with respect to (2.23) the Lane-Emden equation belonging to the polytropic index κ = 3. The boundary conditions (no force in the center and continuity of the potential and its radial derivative at the surface) read: (2.27) u|x (0) = u(1) = 0 , and u|x (1) = −

MG , M total mass . C2R5

(2.28)

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The boundary condition problem and its solution is uniquely determinded by (2.26) and (2.27). Condition (2.28) defines the total mass of the star. On the other hand the mass M is given also by the volume integral of the density (2.21); with the substitution (2.25) Z we obtain: C2R5 1 √ 2 ux dx . (2.29) M= G 0 Comparison with (2.28) results in the relation: Z 1 √ 2 u|x (1) = − ux dx , (2.30) 0

which must be fulfilled by the solution of (2.26) and (2.27) and represents a test of the quality of the integration pocedure. Because the potential u as solution of (2.26) and (2.27) is the important variable, it may be useful to represent density, pressure, temperature and the quantity A by u. From (2.20), (2.21), (2.22) and (2.24) it follows with the use of (2.25): A=

h3 , √ 16π 2 2m4 GCR 4 u

C 2 R 2 1/2 u , 4π G C 4 R 6 3/2 u , p= 6π G

(2.31)

ρ=

1 kT = 2π

(2.32) (2.33)

C 8 h6 R 12 2 9G2 m3 g5/2

!1/5 u3/5 .

(2.34)

Herewith the properties of the Bose-star are determined as follows: At first the homologous boundary condition problem (2.26) and (2.27) must be solved which is done in the next chapter. Then u(x) is known uniquely, especially the boundary values: (2.35) u(0), u|x (0); u(1), u|x (1). Secondly we have to choose the value of the degeneracy number γ (r = 0) = γ (0) in the center of the star. Herewith A(γ (0)) can be calculated according to (2.16). With respect to (2.31) it is further valid: A(γ (0)) = or CR 4 =

h3 √ 16π 2 2Gm4 CR 4 u(0)

D h3 = √ 2 4 Gu(0) 16π 2Gm A(γ (0))u(0)

with D=

h3 . √ 16π 2 2m4 A(γ (0))

(2.36)

(2.36a)

(2.37)

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Obviously CR 4 and D are universally given after γ (0) is chosen. With the help of (2.36a) we eliminate C in (2.28) and obtain finally: R3 = −

1 u|x (1) 2 D . MG3 u2 (0)

(2.38)

Equation (2.38) is the mass-radius relation of Bose stars, according to which R ∼ M −1/3 is valid for fixed degeneracy number in the center; it has the same feature as for fermionic systems e.g. white dwarfs or neutron stars, i.e. decreasing radius with increasing mass. From (2.37) and (2.38) one sees furthermore that the radius of the star depends on the mass of the Bose particles as R ∼ m−8/3 , if mass M of the star and degeneracy number γ (0) are fixed. Now all properties of the star are known. If we give beside γ (0) additionally the mass M of the star, the radius is determined by (2.38). Then C is known by (2.36a) (and B by (2.25)). Consequently the density distribution and the pressure within the star is determined according to (2.32) and (2.33), and A(γ ) and herewith γ is given using (2.31) and (2.16). Then also the course of the temperature is known according to (2.34). Thus after choosing γ (0) and M the properties of the Bose-star are determined, if the Bose particle mass m is considered as given. Consequently the Bose star is described by 3 parameters, namely m, γ (0), M. At the end we put together the central values in dependence of γ (0) and M; using (2.36a) and (2.38) we get from (2.32) up to (2.34): ρ(0) = p(0) =

kT (0) =

1 2π

G3 M 2 u5/2 (0) , 4π D 2 u2|x (1)

G5 M 10/3 u25/6 (0)

, 10/3 6π D 8/3u|x (1) !1/5 h6 G2 M 4/3 u5/3 (0) . 2 4/3 9D 16/3g5/2 (γ (0))m3 u|x (1)

(2.39)

(2.40)

(2.41)

Finally we estimate the validity of the assumption of a non-relativistic Bose gas. From (2.41) and (2.38) we find with the use of (2.37): "   # u(0) 4 A(γ (0)) 2/5 kT (0) MG = 2 1. (2.42) √ mc2 c R | u|x (1) | 3 π g5/2 (γ (0)) Because the bracket in (2.42) is of the order of one (also for the asymptotic values of γ , see (2.7), (2.11) and (2.16)) the condition (2.42) is fulfilled as long as the radius is large compared with the Schwarzschild radius. This means that the assumption of a non-relativistic Bose gas and the calculation according to Newtonian gravity is compatible and that the star with special relativistic Bose gas must be calculated simultaneously with the use of general relativity.

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Insertion of R according to (2.38) into (2.42) gives the mass limit for the validity of the non-relativistic calculation:    √ 3/10 hc 3/2 | u|x (1) | 3 π/2g5/2 (γ (0)) 1 m . (2.43) M π m2 G u(0) 29 A8/3 (γ (0)) The order of magnitude of the mass limit on the right hand side of (2.43) is determined by the dimensionless number (hc/m2 G)3/2  1. Obviously, there exist no stability border for the mass in the non relativistic case. However this is to be expected – as for the fermionic systems – within a full relativistic treatment. We finish with an interesting statement. If we assume in the center of the star the situation of a Boltzmann gas, i.e. A(γ (0) = ∞) = 0, then according to (2.37) D = ∞ is valid and the radius of the star diverges (R = ∞) with respect to (2.38). Thus only in the case of degeneracy or semi-degeneracy of the Bose gas a star with finite radius exists. A pure Boltzmann star is impossible under the conditions taken into account. 3. The Integration Procedure Before we integrate the Lane-Emden equation (2.26) we put it into a non singular form by the substitution v = xu (v ≥ 0) . (3.1) Herewith we obtain from (2.26) the differential equation v|x|x + (vx)1/2 = 0

(3.2)

and from (2.27) the boundary conditions: v(0) = v(1) = 0,

v|x (0) = u(0) .

(3.3)

The boundary value problem (3.2) and (3.3) is not solvable exactly. Therefore we perform an iterative analytic and a numerical integration. From (3.3) it follows that for x  1 v = u(0)x

(3.4)

is valid. Insertion of (3.4) into the second term of (3.2) gives after integration: 1p u(0)x 3 + αx + β . (3.5) v(x) = − 6 The integration constants α and β are determined by the boundary condition (3.3) as follows: v(0) = 0 ⇒ β = 0 , (3.6) v|x (0) = u(0) ⇒ α = u(0)

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and u(0) results from 1 . (3.7) 36 Herewith the first iteration solution which fulfills the boundary conditions (3.3) reads: 1 v(x) = x(1 − x 2 ) = u(0)x(1 − x 2 ) . (3.8) 36 In the second iteration step we insert (3.8) into the second term of (3.2), whereby u(0) will be determined anew in this second step. Then the integral of (3.2) takes the general form:  p  1 1 1 1p 2 3/2 2 u(0) x 1 − x + x(1 − x ) + arcsin x + ax + b . (3.9) v(x) = 4 2 3 2 v(1) = 0 ⇒ u(0) =

The boundary conditions (3.6) and (3.7) have the following consequences for the integration constants a and b:   1 π 2 1p u(0), u(0) = . (3.10) − b = 0, a = u(0) − 3 3 16 Herewith the original function u reads with respect to (3.1):   p  1 1 1 1 1 arcsin x π π 2 3/2 2 u(x) = 1 − x + (1 − x ) + − − . (3.11) 4 3 16 2 3 2 x 4 In Figure 2 the solution (3.9), (3.10) and in Figure 3 the solution (3.11) is compared with the numerical ones of (3.1), (3.2) and (3.3). This comparison shows, that (3.11) represents already an acceptable analytic expression for the solution of (2.26), (2.27). The boundary values (2.35) are   1, 742 · 10−2 u(0) = 1, 876 · 10−2 u|x (0) = u(1) = 0 (3.12)   −2 −2 u|x (1) = −2, 689 · 10 −2, 397 · 10 calculated with (3.11) or numerically within the brackets. Finally we have proved the relation (2.30); this is fulfilled by the numerical calculation up to a relative accuracy of 10−4 . The comparison of the iterative results and the numerical ones shows that the iterative solution approximates the exact one up to a maximum relative error of 7%.

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Figure 2. Solution of the boundary condition problem (3.2), (3.3): Solid line represents the numerical course, dashed line the iterative one according to (3.9).

Figure 3. Potential u as function of x. Solid line represents the numerical course, dashed line the iterative one according to (3.11).

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Figure 4. Course of density (solid line) and pressure (dashed line) according to (2.32) and (2.33) calculated numerically.

4. Bose Star Models A)

T OTALLY

DEGENERATE

B OSE

STAR

In view of the statement at the end of Chapter 2, that there does not exist a Bose star with Boltzmann condition in the center, we start with the opposite assumption of total degeneracy in the center, see (2.17) γ (0) = 0,

A(γ (0) = 0) = 3, 425 .

(4.1)

Then Bose-Einstein condensation happens in the whole star. Herewith we find as mass limit for our non relativistic calculation according to (2.43) using (2.7) and (3.12):   hc 3/2 −2 m = 2, 4 · 1033 g (4.2) M  4, 1 · 10 m2 G for m equal the mass of the proton; for u(0) and u|x (1) we use the numerical values, i.e. the brackets in (3.12). Of course the mass of the proton is here taken as a typical mass only; the real mass of the Higgs-bosons is unknown. In this case D has the value D = 4, 86 · 1013 . If we choose with respect to (4.2) as mass of the star M = 10−1 M , then from (2.38) the radius of the star follows as R = 1, 46 · 106 cm

(4.3)

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TABLE I Properties of a Bose star of 10−1 M under variation of the degeneracy number in the center (mass of the Bose particles m = 1, 67 · 10−24 g). The upper mass limit (4.2) for validity of the non relativistic calculation increases with increasing γ (0) (see 2.43) γ (0)

0,2

0,5

1

5

ρ(0) p(0) T (0) R xc

2, 17 · 1012 4, 08 · 1024 3, 00 · 1010 3, 43 · 106 0, 81

5, 83 · 1011 7, 08 · 1023 1, 72 · 1010 5, 32 · 106 0, 90

1, 29 · 1011 9, 48 · 1022 9, 64 · 109 8, 79 · 106 0, 95

2, 53 · 107 1, 08 · 1018 5, 18 · 108 1, 51 · 108 9, 99904 · 10−1

g cm3 bar K cm

i.e. comparable with the radius of a neutron star. Consequently also the central density is of the order of that of a neutron star. From (2.39) up to (2.41) we find ρ(0) = 2, 78 · 1013 g/cm3 , p(0) = 1, 23 · 1026 bar

(4.4)

T (0) = 1, 04 · 1011 K where also the central temperature is of the same order as for a very young neutron star. From (4.4) it follows, that the condition (2.18) for condensation of the Bose particles is just fulfilled in the center of the star. In Figure 4 the course of mass density and pressure is drawn according to (2.32) and (2.33) using the numerical course of u given in Figure 3. B)

S EMI - DEGENERATE B OSE

STARS

In the following we list the properties of several configurations of Bose stars. The calculation is performed as before. The mass of the star is 10−1 M and as mass for the Bose particle the proton mass is taken into account. However the central degeneracy number is considered as variable. The results for γ (0) = 0, 2; 0, 5; 1; 5 are put together in Table I. The course of the mass density and the pressure normalized to the central values is in all cases the same as in Figure 4 because of the homologous behaviour. Beside the central density, pressure, temperature and the radius of the star the critical radius xc = rc /R is listed in Table I, where A(γ ) reaches the value 3, 425 ( =γ ˆ = 0), see (2.17), so that the condensation process begins and increases for x > xc . The radii vary between those of neutron stars and white dwarfs. We have seen, that a pure Boltzmann star does not exist (R → ∞). However in case of γ (0) = 5 the Boltzmann situation is already approximated very well according to

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TABLE II Properties of a Bose star with M = 2, 3 · 1029 g, where the Bose particles have a mass of 5, 35 · 10−23 g ( =30 ˆ GeV). The critical values xc are the same as in Table I because of the homology γ (0)

0

0,2

1

5

ρ(0) p(0) T (0) R

4, 11 · 1019 2, 28 · 1032 4, 19 · 1012 1, 35 · 103

3, 19 · 1018 7, 56 · 1030 1, 20 · 1012 3, 17 · 103

1, 90 · 1017 1, 76 · 1029 3, 88 · 1011 8, 12 · 103

3, 74 · 1013 2, 01 · 1024 2, 08 · 1010 1, 40 · 105

g cm3 bar K cm

Table I: Only a thin skin of 1,45 ·104 cm consists of condensed particles, whereas the whole radius is 1, 51 · 108 cm. It may be of interest to repeat the Table I for the case of more massive Bose particles, because the mass of the Higgs particle may be much larger than that of the proton. If we choose m = 5.35 · 1023 g corresponding 30 GeV the condition (4.2) for the validity of non-relativistic calculations reads M << 2.32 · 1030 g. Therefore we choose as mass of the star M = 2, 3 · 1029 g. Herewith we get the results of Table II calculated in the same way as those in Table I. Obviously, the larger mass of the single Bose particle results in much more compact objects as can be seen by comparison of density and pressure in both tables (compare the remark after Equation (2.38)).

5. Conclusions We have shown, that the existence of Newtonian Bose stars with finite local temperatures and finite radii is possible with masses of the order of the solar mass. An upper mass limit does not exist in the non-relativistic case, but it is to be expected within a full relativistic treatment; such a one is in preparation. All radii are possible for a given mass of the star; however in case of degeneracy or semi-degeneracy of the Bose gas the stars have properties similar to white dwarfs and neutron stars. If the stars consist of only gravitationally interacting spin 0 particles as e.g. Higgs-particles originated in the early Universe they would be candidates for dark matter objects. But the question of their formation remains unsolved; the main problem may be the removal of the binding gravitational energy, which can happen only by gravitational radiation (for further discussions see e.g. Bianchi et al., 1990; Grasso, 1990; Madsen and Liddle, 1990 and Tkachev, 1991). If such stars are formed, they would be optically transparent, but cause optical lensing effects. Perhaps they could be detected in this way.

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