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Hybrid stars and the equation of state
-NUCLEAR PHYSICS B
OCEEDINâs'S SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 24B (1991) 156-159 North-Holland
HYBRID STARS AND THE EQUATION OF STATE A. ROSENHAUER and E.F . STAUBÖ Fysisk Institutt, Universitetet i Bergen, N-5007 Bergen, Norway Gross properties of hybrid stars consisting of a core of strange mattes surrounded by ordinary neutron matter are investigated . We discuss star models based on phenomenological equations of state from nuclear reactions including a phase transition between the hadronic phase and the quark-gluon plasma. For certain parameters, such equations of state support the existence of hybrid stars: While the nuclear equation of state has a minor influence on the gross properties of hybrid stars, the identification o£ such objects could provide us with a rather stringent constraint on the vacuum energy density of the quark gluon plasma. Since the conjecture that strange quark matter might be the true ground state of matter 1 , the possible existence of strange stars has been "widely discussed, remainBesides pure neutron and
ing inconclusive hitherta strange stars, there might also exist so called hybrid stars 2-14 .
characterized by a core of quarks and gluons and an exterior of neutrons .
The equilibrium energy density distribution of slowly rotating spherical stars is determined by the Tolrllan-
Oppenheimer-Volkoffequation (TOV) in geometrical units is _dp
where
- -
dr
(
E =-
rn + 4a r3 P , p) r (r - 2m)
p
rr(r) = J
49r r 'e
dr.
$ere p(r) is the pressuie and e(r) is the total energy density which are related by the specification of an equation
of state p(e) . Up to now, little is known about. the nuclear equation of state (EOS) at densities other than normal nuclear matter density. In contrast to sophisticated parametrizations designed for neutron stars covering several orders of magnitude in
energy density, we want to apply phenomenological EOS, as they are typically used for the description of relativistic heavy ion collisions ls . The the
EOS is chosen to be ass simple as possible retaining only the essential features of nuclear matter in equilibrium and thus cootains only ~, few pkrarneters . We define the hadronic EOS as the compressional 0920-5632/91/$03 .50
energy per particle e,o,,,p(n) in terms of the conserved baryon charge density, n, which is carried by several specïcs of baryons.
Throughout, this paper we will refer to two commonly
used parametrizations for the iadron ;c EOS; namely that of Sierk and Nix 17 eSN(n) -
and the quadratic form
_21t' 9
nn
0
1)
`
,
18
eQ(n) -
K (n - no 18
)2
no
.
The. compression constant. K thus characterizes the propcrties of twclcar chatter at densities ra ; a parametrization
e.omp(n),
,, .
f'ov,-l
a larger /t,' corresponds to
a more repulsive N-N inteiaction . Tho fact that the
quadratic EOS yields an noncausal speed of sound at Lgh densities is not relevant for the discussion of hybrid stars, since the hadronic matter will undergo a phase transition into the quark gluon plasma. (QGP) at ch~ady lower baryon densities. The total energy density then takes the form c(n) = n [ecomp(n) + Wo -{`
where Wo = -16
Mn
+ Waym ]
,
the binding energy per nucleon at normal nuclear matter density no = 0.145 fnl -3 , r~1 MeV is
is the rest mass of a neutron, bV, , = 30 Me V the symmetry energy of neutron matter estimated from the li-
Q 1991 -- Elsevier Science Publishers B.V . All rights reserved .
A . Rosenhauer, E.F. Staubo/Hybri d stars and the equation of state
quid drop model. Within this phenomenological approach, the symmetry and the binding energy determine the properties of matter at saturation, while e,,o,np(n) incorporates all density dependent effects.
157 Quadratic EOS
10
At dcnsit:cs below no, the p:cssure in the star Is no longer determined by the nucleon-nucleon interaction
K=170 K = 240
alone. Within the range 0 < n < n,,,, we apply a polynomial form for e,o,np(n) similarly to the approach used
B'1'=165
in Ref. 2 matching the pressure and energy density of the
1~ K = 3GO
nuclear part of the EOfi at densities around n1D = 1 .2 n o . The quark gluon plasma is taken into account by a "bag model"-EOS of the form
1 P = 3 (( - 43), where the energy density of the QGP is that of a mixture of noninteracting gluons and massless u, d and s-quarks
in first order perturbation theory. The parameter B,
FIGURE 1 Radial density profile n(r) of a hybriJ star
the Lag constant, is the difference between_ }he perturbative vacuum and the "true" vacuum energy density. The
km radius and an outer layer of neutron matter of about 3-4 km thickness. More than 60 % of the total mass is in
by the Gibbs criteria .
criti--al densities both for the hadronic nn and for the QCP phase n,r. Hence, although the radii of the stars are a'..- most the same for our three examples at nc = 10 no
phase boundary of the coexistence region is determined
Depending on the central density of the star, such a
multi-phase EOS will allow for the formation of neutron stars as well as for hybrid stars 2,'4 . Given one of the
the quark matter phase. A stiffer nuclear EOS lowers the
(cf. also Fig. 2), a more repulsive EOS generates a larger
âse and B, the vacuum energy density of the
core of strange matter . The influence of the nuclear EOS on masses and radii of the stars is most clearly revealed by Fig. 2. Without
Due to the perturbatrve , :~_,ritxtious in the QGP-EOS, our calculations also contain a strong coupling constant
densities of 10 r, o, riiaximum masses of M ,~:; 1.8-2 .6 M;, depending on the EOS, minimal radii of R ;~: 10-12 km
al,ove forms of e(n), we are left with two parameters, namely the compression constant K associated with the lia,dronic
p
quark gluon phase.
a phase transition we find stable neutron stars up to
a, = 0.4
supporting a minimal Keplerian rotation period's of P = 0.ri nas. 0.026 (R/km)a/` (M/Mo)'/2 [ms]
2.12
lowel:ng the pressure in the plasma . It has
been shown, however, that the sunning coupling constant
a,(p, T) varies only little within the range of densities considered here ls . r: typical density profile of a hybrid star is depicted in Fig. 1 . At central densities higher than the critical
density of the quark gluon plasma rcyr, toe r ,we of the star will consist of strange quark matter . Once the c:iti-
cal pressure is reached, the density drops discontinuously from nqr to n r , indicating a first order phase transition .
The density profiles shown for nc = 10 no imply that the hybrid star consists of a strange matter core of about 6-7
By comparing different EOS we conclude that neither the functional form of the parametrization nor the variation of the compression constant for a given form
strongly influences the mass-radius relation of stable hybrid stars.
In this context we would like to emphasize that this insensitivity is also recovered when Lquations ^r state desigaed espec;ally fvr neutron matter are applied s° . Within a wide range of compression energies we obtain stable hybrid stars for densities betweei . 4--y 1to,
A. Roser..faaer, F.F. Siaubo f Hybrid stars and the equation of state
158
Ii = 380
2
Quadratic
EOS
x = 240
i
L
Sierk - Nix EOS 10000
1000
100
h=170
C
IM evI.f
nt3 ]
FIGURE 3 The dependence of the star mass on the central energy density at different bag constants. The uppermost curve corresponds to solutions in the absence of a phase transition, i.e . pure neutron stars . 0.2 8
12 R
(krnj
16
FIGURE 2
Mass-radius aelatiun for neutron and hybrid stars masses of M
1 .4 - 1 .7 MW , radii of R
10 - 13 km
and a minimal rotation period of P . 0 .7 ms . Note that the masses of hybrid stars are considerably
lower than those of neutron stars at the same central density because the EOS for the QGP phase is much
softer than. that of neutron matter, whereas the radii
and tl:e rotation periods air, comparable . Stable hybrid stars will only exist at sufficiently low
vacuum energy densities due to the strong increase of the critical density of the quark phase i?,1' with larger B
16 .
Higher drusii.ir,a accompanied by aiglitr pie-ures
are needed to overcome the confining vacuum pressure of the QGP. At B11 ' = 300 MeV one finds that pry 23 no. far above the maximum for stable stars around n` < 10 no .
tu ;+I- B 1 1' reduced to 180 AleV gravity still overtakes
the Fermi pressure which is illustrated in Fig. 3. The critical value for the hag constant, where we fivil stable
. Reducing B hybrid stars is around B11' = 170 IOW stirs 1.owards hybrid further widens the range of stable smaller masses . If B 11 a < 160 MeV, on the other hand, quark gluon plasma becomes the ground state of ordinary
nuclei, i.e . nn' < no. The remaining interval allows only for a small variation of the vacuum energy density. Varying the strong coupling constant effectively widens the allowed range for the vacuum energy density. However, only a small variation is admissible: Increasing c1, decreases the maxinruan value of B which gives sta-
ble hybrid stars. With the original MIT-bag value of Bad' = 145 MeV as a reasonable lower limit we obtain a maximum value of a, = 0.75. Similarly, setting a, = 0
yields an overall maximum value for stable hybrid stars at B11' = 185 MeV . For reference, a, = 0.3 yields a maximum bag pressure B11 ' = 175 MeV. These systematics show that despite the uncertainty in a, there is only a small interval of values for B Wiich allows for stable hybrid stars.
A . Resenhauer, E.n SLaubô ; Hybrid stars and the equation of state
159
In summary, our EOS for two different states of nuclear matter are consistent with the existence of neutron
[5] E. Farhi and R. L. Jaffe, Phys . Rev. D30 (1984) 2379 .
If objects like hybrid stars exist, they would differ in
[7] C. Alcock, E. Farhi and A. Olinto, Astrophys. Journ. 310 (1986) 261.
stars as well as with hybrid stars.
[6] G. Baym et al., Phys . Lett . B160 (1985) 181.
their masses and in their densities. Hybrid stars should have either lower masses or higher central densities as
[8] P. Haensel, J. L. 'Zdunik and R. Schaeffer, Àstron . Astrophys. 160 (1986) 121.
neutron stars which consistently expresses the effective softening of the matter when entering the quark phase
at high densities. Possible experimental signals for the existence of hybrid stars could be an increased neutrino cooling rate 21 .22.23 and a modification of the redshift of photons from the surface 2x .25 .
If hybrid stars could be identified in the future, this could provide us with a rather stringent, constraint on the vacuum energy density of the quart: gluon plasma . ACKNOWLEDGEMENTS
We are indebted to T. Overgkrd and E. Ostgaard for many helpful discussion and comments . '"his work has been st!ppoeted by the Alexander von Hut , ll oldt F ulidation .
REFERENCES [1] E. Witten, Phys Rev . D30 (1984) 272. (2] N. K. Glendenning, Proceedings of NATO Advanced Study Institutes Program, The Nuclear Equation of State, Peniscola, Spain, Mai 21 - June 3, 1989, Plenum Publishing Company, London . N. K. Glendenning, Proceedings of the Rio de Janeiro International Workshop on Relativistic Aspects of Nuclear Physics, Rio de Janeiro, Brazil, August, 28 - 30 ; 1989, World Scientific Publishing Company.
[3] M . Prakash, E. Baron and M. Prakash, Phys . Lett. B243 (1990) 175. [4] E. Alvarez and J . M. Ibanez, Astron . Astrophys. 98 (1981) 390.
[9] H. A. Bethe, G. E. Brown and J. Cooperstein, Nucl . Phys. A462 (1987) 791 . [1(t] A. V. Olinto, Phys . Lett. B192 (1987) 71 .
[111 J. L. Friedman, J. R. Ipser and L. Parker; P,,Js. Rev. Lett . 62 (1989) 3015. [12] J. I. Kapusta and K. A . Olive, Phys . Rev. Lett . 64 (1990) 13 .
[13] J. Ellis, J. I, Kapusta and K. A. Olive, Nucl Phys. B348 (1991) V5 . [14] W-Y. P. Hwang, C. Liu and K. C. Tzeng, Z. Phys . A388 (1991) 223. [15] S. L. Shapiro and S. A. Teukolsky, Wiley (1983) . [16] A . Holme et al ., Phys . Rev. D40 (1989) 3735 .
[17] A. J . Sierk and J. R. Nix, Phys . Rev. C22 (1980) 1920 .
[$] ? H. Stricker and W. Greiner, Phys . Rep. 137 (1986) 277 and references therein.
[19] E. F. Staubo and A. Rosenhauer, University of Bergen Report 242/1990 .
[20] f' . Rosenhauer, E. F. Staubo, l.. P. Csernai, T. 0-,,ergârd and E. Ostgaard, University of Bergen Report 241/1990. [21] N. Iwamoto, Phys . Rev. Lett . 44 (1980) 1637 . [22] B. Datta, S. Rallia and B. Sinha, Mod. Phys . Lett . A3 (1988) 1385. [23] A . Goyal and J . D. Anand, Phys . Lev. D42 (1990) 992.
[24] K. Brecher, Astrophys . Journ. 215 (1977) L17. [25] T. Overgârd and E. Ostgaard, Astron . Astrophys ., in print.
OCEEDINâs'S SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 24B (1991) 156-159 North-Holland
HYBRID STARS AND THE EQUATION OF STATE A. ROSENHAUER and E.F . STAUBÖ Fysisk Institutt, Universitetet i Bergen, N-5007 Bergen, Norway Gross properties of hybrid stars consisting of a core of strange mattes surrounded by ordinary neutron matter are investigated . We discuss star models based on phenomenological equations of state from nuclear reactions including a phase transition between the hadronic phase and the quark-gluon plasma. For certain parameters, such equations of state support the existence of hybrid stars: While the nuclear equation of state has a minor influence on the gross properties of hybrid stars, the identification o£ such objects could provide us with a rather stringent constraint on the vacuum energy density of the quark gluon plasma. Since the conjecture that strange quark matter might be the true ground state of matter 1 , the possible existence of strange stars has been "widely discussed, remainBesides pure neutron and
ing inconclusive hitherta strange stars, there might also exist so called hybrid stars 2-14 .
characterized by a core of quarks and gluons and an exterior of neutrons .
The equilibrium energy density distribution of slowly rotating spherical stars is determined by the Tolrllan-
Oppenheimer-Volkoffequation (TOV) in geometrical units is _dp
where
- -
dr
(
E =-
rn + 4a r3 P , p) r (r - 2m)
p
rr(r) = J
49r r 'e
dr.
$ere p(r) is the pressuie and e(r) is the total energy density which are related by the specification of an equation
of state p(e) . Up to now, little is known about. the nuclear equation of state (EOS) at densities other than normal nuclear matter density. In contrast to sophisticated parametrizations designed for neutron stars covering several orders of magnitude in
energy density, we want to apply phenomenological EOS, as they are typically used for the description of relativistic heavy ion collisions ls . The the
EOS is chosen to be ass simple as possible retaining only the essential features of nuclear matter in equilibrium and thus cootains only ~, few pkrarneters . We define the hadronic EOS as the compressional 0920-5632/91/$03 .50
energy per particle e,o,,,p(n) in terms of the conserved baryon charge density, n, which is carried by several specïcs of baryons.
Throughout, this paper we will refer to two commonly
used parametrizations for the iadron ;c EOS; namely that of Sierk and Nix 17 eSN(n) -
and the quadratic form
_21t' 9
nn
0
1)
`
,
18
eQ(n) -
K (n - no 18
)2
no
.
The. compression constant. K thus characterizes the propcrties of twclcar chatter at densities ra ; a parametrization
e.omp(n),
,, .
f'ov,-l
a larger /t,' corresponds to
a more repulsive N-N inteiaction . Tho fact that the
quadratic EOS yields an noncausal speed of sound at Lgh densities is not relevant for the discussion of hybrid stars, since the hadronic matter will undergo a phase transition into the quark gluon plasma. (QGP) at ch~ady lower baryon densities. The total energy density then takes the form c(n) = n [ecomp(n) + Wo -{`
where Wo = -16
Mn
+ Waym ]
,
the binding energy per nucleon at normal nuclear matter density no = 0.145 fnl -3 , r~1 MeV is
is the rest mass of a neutron, bV, , = 30 Me V the symmetry energy of neutron matter estimated from the li-
Q 1991 -- Elsevier Science Publishers B.V . All rights reserved .
A . Rosenhauer, E.F. Staubo/Hybri d stars and the equation of state
quid drop model. Within this phenomenological approach, the symmetry and the binding energy determine the properties of matter at saturation, while e,,o,np(n) incorporates all density dependent effects.
157 Quadratic EOS
10
At dcnsit:cs below no, the p:cssure in the star Is no longer determined by the nucleon-nucleon interaction
K=170 K = 240
alone. Within the range 0 < n < n,,,, we apply a polynomial form for e,o,np(n) similarly to the approach used
B'1'=165
in Ref. 2 matching the pressure and energy density of the
1~ K = 3GO
nuclear part of the EOfi at densities around n1D = 1 .2 n o . The quark gluon plasma is taken into account by a "bag model"-EOS of the form
1 P = 3 (( - 43), where the energy density of the QGP is that of a mixture of noninteracting gluons and massless u, d and s-quarks
in first order perturbation theory. The parameter B,
FIGURE 1 Radial density profile n(r) of a hybriJ star
the Lag constant, is the difference between_ }he perturbative vacuum and the "true" vacuum energy density. The
km radius and an outer layer of neutron matter of about 3-4 km thickness. More than 60 % of the total mass is in
by the Gibbs criteria .
criti--al densities both for the hadronic nn and for the QCP phase n,r. Hence, although the radii of the stars are a'..- most the same for our three examples at nc = 10 no
phase boundary of the coexistence region is determined
Depending on the central density of the star, such a
multi-phase EOS will allow for the formation of neutron stars as well as for hybrid stars 2,'4 . Given one of the
the quark matter phase. A stiffer nuclear EOS lowers the
(cf. also Fig. 2), a more repulsive EOS generates a larger
âse and B, the vacuum energy density of the
core of strange matter . The influence of the nuclear EOS on masses and radii of the stars is most clearly revealed by Fig. 2. Without
Due to the perturbatrve , :~_,ritxtious in the QGP-EOS, our calculations also contain a strong coupling constant
densities of 10 r, o, riiaximum masses of M ,~:; 1.8-2 .6 M;, depending on the EOS, minimal radii of R ;~: 10-12 km
al,ove forms of e(n), we are left with two parameters, namely the compression constant K associated with the lia,dronic
p
quark gluon phase.
a phase transition we find stable neutron stars up to
a, = 0.4
supporting a minimal Keplerian rotation period's of P = 0.ri nas. 0.026 (R/km)a/` (M/Mo)'/2 [ms]
2.12
lowel:ng the pressure in the plasma . It has
been shown, however, that the sunning coupling constant
a,(p, T) varies only little within the range of densities considered here ls . r: typical density profile of a hybrid star is depicted in Fig. 1 . At central densities higher than the critical
density of the quark gluon plasma rcyr, toe r ,we of the star will consist of strange quark matter . Once the c:iti-
cal pressure is reached, the density drops discontinuously from nqr to n r , indicating a first order phase transition .
The density profiles shown for nc = 10 no imply that the hybrid star consists of a strange matter core of about 6-7
By comparing different EOS we conclude that neither the functional form of the parametrization nor the variation of the compression constant for a given form
strongly influences the mass-radius relation of stable hybrid stars.
In this context we would like to emphasize that this insensitivity is also recovered when Lquations ^r state desigaed espec;ally fvr neutron matter are applied s° . Within a wide range of compression energies we obtain stable hybrid stars for densities betweei . 4--y 1to,
A. Roser..faaer, F.F. Siaubo f Hybrid stars and the equation of state
158
Ii = 380
2
Quadratic
EOS
x = 240
i
L
Sierk - Nix EOS 10000
1000
100
h=170
C
IM evI.f
nt3 ]
FIGURE 3 The dependence of the star mass on the central energy density at different bag constants. The uppermost curve corresponds to solutions in the absence of a phase transition, i.e . pure neutron stars . 0.2 8
12 R
(krnj
16
FIGURE 2
Mass-radius aelatiun for neutron and hybrid stars masses of M
1 .4 - 1 .7 MW , radii of R
10 - 13 km
and a minimal rotation period of P . 0 .7 ms . Note that the masses of hybrid stars are considerably
lower than those of neutron stars at the same central density because the EOS for the QGP phase is much
softer than. that of neutron matter, whereas the radii
and tl:e rotation periods air, comparable . Stable hybrid stars will only exist at sufficiently low
vacuum energy densities due to the strong increase of the critical density of the quark phase i?,1' with larger B
16 .
Higher drusii.ir,a accompanied by aiglitr pie-ures
are needed to overcome the confining vacuum pressure of the QGP. At B11 ' = 300 MeV one finds that pry 23 no. far above the maximum for stable stars around n` < 10 no .
tu ;+I- B 1 1' reduced to 180 AleV gravity still overtakes
the Fermi pressure which is illustrated in Fig. 3. The critical value for the hag constant, where we fivil stable
. Reducing B hybrid stars is around B11' = 170 IOW stirs 1.owards hybrid further widens the range of stable smaller masses . If B 11 a < 160 MeV, on the other hand, quark gluon plasma becomes the ground state of ordinary
nuclei, i.e . nn' < no. The remaining interval allows only for a small variation of the vacuum energy density. Varying the strong coupling constant effectively widens the allowed range for the vacuum energy density. However, only a small variation is admissible: Increasing c1, decreases the maxinruan value of B which gives sta-
ble hybrid stars. With the original MIT-bag value of Bad' = 145 MeV as a reasonable lower limit we obtain a maximum value of a, = 0.75. Similarly, setting a, = 0
yields an overall maximum value for stable hybrid stars at B11' = 185 MeV . For reference, a, = 0.3 yields a maximum bag pressure B11 ' = 175 MeV. These systematics show that despite the uncertainty in a, there is only a small interval of values for B Wiich allows for stable hybrid stars.
A . Resenhauer, E.n SLaubô ; Hybrid stars and the equation of state
159
In summary, our EOS for two different states of nuclear matter are consistent with the existence of neutron
[5] E. Farhi and R. L. Jaffe, Phys . Rev. D30 (1984) 2379 .
If objects like hybrid stars exist, they would differ in
[7] C. Alcock, E. Farhi and A. Olinto, Astrophys. Journ. 310 (1986) 261.
stars as well as with hybrid stars.
[6] G. Baym et al., Phys . Lett . B160 (1985) 181.
their masses and in their densities. Hybrid stars should have either lower masses or higher central densities as
[8] P. Haensel, J. L. 'Zdunik and R. Schaeffer, Àstron . Astrophys. 160 (1986) 121.
neutron stars which consistently expresses the effective softening of the matter when entering the quark phase
at high densities. Possible experimental signals for the existence of hybrid stars could be an increased neutrino cooling rate 21 .22.23 and a modification of the redshift of photons from the surface 2x .25 .
If hybrid stars could be identified in the future, this could provide us with a rather stringent, constraint on the vacuum energy density of the quart: gluon plasma . ACKNOWLEDGEMENTS
We are indebted to T. Overgkrd and E. Ostgaard for many helpful discussion and comments . '"his work has been st!ppoeted by the Alexander von Hut , ll oldt F ulidation .
REFERENCES [1] E. Witten, Phys Rev . D30 (1984) 272. (2] N. K. Glendenning, Proceedings of NATO Advanced Study Institutes Program, The Nuclear Equation of State, Peniscola, Spain, Mai 21 - June 3, 1989, Plenum Publishing Company, London . N. K. Glendenning, Proceedings of the Rio de Janeiro International Workshop on Relativistic Aspects of Nuclear Physics, Rio de Janeiro, Brazil, August, 28 - 30 ; 1989, World Scientific Publishing Company.
[3] M . Prakash, E. Baron and M. Prakash, Phys . Lett. B243 (1990) 175. [4] E. Alvarez and J . M. Ibanez, Astron . Astrophys. 98 (1981) 390.
[9] H. A. Bethe, G. E. Brown and J. Cooperstein, Nucl . Phys. A462 (1987) 791 . [1(t] A. V. Olinto, Phys . Lett. B192 (1987) 71 .
[111 J. L. Friedman, J. R. Ipser and L. Parker; P,,Js. Rev. Lett . 62 (1989) 3015. [12] J. I. Kapusta and K. A . Olive, Phys . Rev. Lett . 64 (1990) 13 .
[13] J. Ellis, J. I, Kapusta and K. A. Olive, Nucl Phys. B348 (1991) V5 . [14] W-Y. P. Hwang, C. Liu and K. C. Tzeng, Z. Phys . A388 (1991) 223. [15] S. L. Shapiro and S. A. Teukolsky, Wiley (1983) . [16] A . Holme et al ., Phys . Rev. D40 (1989) 3735 .
[17] A. J . Sierk and J. R. Nix, Phys . Rev. C22 (1980) 1920 .
[$] ? H. Stricker and W. Greiner, Phys . Rep. 137 (1986) 277 and references therein.
[19] E. F. Staubo and A. Rosenhauer, University of Bergen Report 242/1990 .
[20] f' . Rosenhauer, E. F. Staubo, l.. P. Csernai, T. 0-,,ergârd and E. Ostgaard, University of Bergen Report 241/1990. [21] N. Iwamoto, Phys . Rev. Lett . 44 (1980) 1637 . [22] B. Datta, S. Rallia and B. Sinha, Mod. Phys . Lett . A3 (1988) 1385. [23] A . Goyal and J . D. Anand, Phys . Lev. D42 (1990) 992.
[24] K. Brecher, Astrophys . Journ. 215 (1977) L17. [25] T. Overgârd and E. Ostgaard, Astron . Astrophys ., in print.