Astrophysics, Vol. 42, No. 4, 1999

S T R A N G E S T A R S A N D T H E I R C O O L I N G W I T H P O S I T R O N E L E C T R I C A L N E U T R A L -

I Z A T I O N O F Q U A R K M A T T E R

G. S. Hajyan, Yu. L. Vartanyan, and A. K. Grigoryan UDC 524.354-823

Questions of the equilibrium, stability, and observational manifestations o f strange stars are considered, in which electrical neutralization o f the quark matter is provided by positrons, as occurs for some sets o f bag parameters resulting in a stiffer equation o f state. Such models consist entirely o f self-contained, strange quark matter and their maximum mass reaches 2.4-2.5 M~ with a radius o f 13-14 km. The cooling of such strange quark stars both in the absence and in the presence o f mass accretion is investigated. It is shown that in the absence o f mass accretion onto the strange star, the dependence o f temperature (T, K) on age (t, yr) depends very little on the mass o f the configuration and has the form T ~ 2.3.10st 1/5. I f the star's initial temperature is sufficiently high (T O > 2.101~ K), then the total number of electron-positron pairs emitted does not depend on it and is determined only by the total mass o f the configuration. In the case o f accretion, the annihilation o f electrons of the infalling fatter with positrons of the strange quark matter results in the emis- sion of y-rays with an energy o f -0 .5 MeV, by observing which one can distinguish candidates for strange stars. The maximum temperature o f strange stars with mass accretion is calculated.

1. Introduction

The investigation of strange quark matter, consisting of approximately equal amounts of u, d, and s quarks with

a small admixure of electrons or positrons to provide electrical neutrality, was begun in [1]. Such matter is assumed to

be an absolutely stable state of cold superdense matter and may form self-confining, bound states in the form of so-called

"strange stars" (SSs). This hypothesis was considered by Farhi and Jaffe [2], who investigated the dependence of the stability

of strange quark matter on phenomenological parameters of the bag model that are not known sufficiently accurately [3]:

the bag constant B, the quark-gluon interaction constant ctc, and the mass m r of a strange quark.

In neutron stars the total positive charge of the baryons is neutralized by the negative charge of electrons and [a-

mesons. Quark matter, depending on the numerical values of the parameters of the bag model, can have either a positive

or a negative excess electric charge. In the first case, therefore, electrical neutrality is provided, as in the case of neutron

stars, by electrons, and in the second case by positrons.

In the case of electron neutralization, an electrostatic barrier is formed at the free surface of the SS due to the partial

escape of electrons, which prevents the inward transfer of ordinary matter. In this variant there can exist both bare SSs,

consisting entirely of strange quark matter, and SSs with a crust consisting of atomic nuclei and degenerate electrons (the

Ae phase). The Aen phase, in which matter consists of atomic nuclei and degenerate electrons and neutrons, is excluded

because of the unimpeded transfer of free neutrons into the strange quark matter [4]. In the case of electrical neutralization

1999. Yerevan State University, Armenia. Translated from Astrofizika, Vol. 42, No. 4, pp. 617-630, October-December, Original article submitted April 23, 1999; accepted for publication May 25, 1999.

0571-7256/99/4204-0467522.00 �9 Kluwer Academic/Plenum Publishers 467

by positrons, only bare SSs can exist, since ordinary matter in contact with strange quark matter is inevitably swallowed

by it.

The basic properties of SSs have been considered in [4-6]. In [7, 8] the parameters of SSs were compared with

observational data and the problem of the parallel existence o f strange and neutron stars was investigated.

The case of electron electrical neutralization has been considered in an absolute majority of papers on SSs. The

existence of model SSs in the presence of positrons is not ruled out, although it is less likely, considering the values of

the parameters of the bag model required for this. In [9], by analogy with a Thorne-Zytkow object (a red giant with a neutron

star as its core), we constructed a model of a red giant inside of which is a SS in the presence of positrons instead of a

neutron star.

The purpose of the present work is to investigate SS models with positron electrical neutralization of the strange

quark matter. Observational manifestations of such SSs both in the absence and in the presence of mass accretion are

considered. The results obtained are compared with observational data.

2. Basic Parameters of Strange Stars

The equation of state of strange quark matter is determined by phenomenological parameters of the bag model that

are not known sufficiently accurately: B, ct c, and m s (we refer to [2] for details). For small B and m s and large r c the strange

quark matter has a negative electric charge, neutralized by positrons.

In the present work we consider two SS models, corresponding to two sets of bag parameters (1. B = 30 MeV/fm 3,

m = 150 MeV, c t = 0.9; 2. B = 34 MeV/fm 3, m s --- 150 MeV, ct c = 0.9). For both models the curve of average energy e

per baryon [e = (pc2/n) - moc z, where m 0 = M(56Fe)/56] has a negative minimum at a certain value nmi n of the baryon density.

For our models we have the following: 1. er,~, = -28.5 MeV, nmt,/n o = 1.05; 2. emi, = -3.8 MeV, n f fn o = 1.15, where

n o = 0.15 fm 3 is the nuclear density. This accounts for the binding of SSs having a clearly defined surface and with a density

n . We also give values of the chemical potentials of quarks and positrons at the SS surface for our models: 1. ~ t /= i.t

= 301 MeV, I.t = ~t s - ~t = 1.2 MeV; 2. ~ta = las = 309.7 MeV, ~t = ~ - [.t = 2.7 MeV.

The presence of positrons results in stiffer equations of state, with the threshold density for the appearance of strange

quark matter, which occurs at the SS surface, being close to the nuclear density in this case.

For the two equations of state under consideration we used the standard method to integrate the system of relativistic

equations of stellar equilibrium (the Tolman-Oppenheimer-Volkov system of equations) and obtained the SS integrated

parameters. In Tables 1 and 2 we give values of the total mass M, the rest mass M o, the proper mass Mp, the stellar radius

R, the gravitational redshift from the stellar surface Z , and the relativistic moment of inertia I as functions of the central

density Pc"

Stable SSs have a dependence of mass on radius that differs from that for neutron stars. The radius increases with

increasing mass over almost the entire curve, except for configurations close to the maximum, where gravitation begins

to dominate. This is because SSs are bound by the strong interaction and can exist even in the absence of self-gravitation.

For the models that we considered we obtained the maximum masses M ~ x = (2.38-2.51) M| with their corresponding

radii Rmax = (13.1-13.8) km and central densities (P~)m~x = (1.2-1.3)" 10 j5 g/cm 3. We note that the following results were

obtained in [7] with electrical neutralization by electrons for a realistic range of parameters of the bag model: Mm, x = (1.75-

1.86) Mo, R x = (9.8-10.4) km, and ( p ) , ~ = (2.2-2.5)- 101 s g/cmL The presence of a crust, supported by the electrostatic

barrier, results in an increase in radius by only 170-180 m [10].

The presence of positrons results in a considerable increase in SS mass and radius, occurring at lower central

densities. We note that a mass of 2.5 M~ is not reached even for the extensive set of realistic equations of state of the material

of neutron stars considered in [11], and is obtained only for extremely stiff equations of state, of the type considered in

[12, 13].

The calculation of configurations of the maximum masses is due to the importance of comparing the parameters

of theoretical models with the observed parameters of stellar objects, by means o f which one decides whether one or another

468

TABLE 1. Main Parameters of Cold Strange Stars (B = 30 MeV/fm 3, m s = 150 MeV, a c = 0.9)

Pc, M, M o, Mp , R, Z s 1,

1014 g/cm ~ M. M. M| km 1045 g.cm -~

2.568

2.627

2.685

2.773

2.918

3.062

3.205

3.347

3.631

4.052

4.471

5.298

6.117

6.660

7.470

8.544

11.343

12.006

12.667

15.965

0.067

0.136

0.212

0.328

0.523

0.701

0.867

1.013

1.265

1.556

1.786

2.049

2.219

2.295

2.377

2.439

2.504

2.508

2.505

2.448

0.085

0.166

0.252

0.390

0.589

0.788

1.011

1.149

1.439

1.854

2.125

2.448

2.723

2.829

2.952

2.996

3.129

3.135

3.tY93

3.097

0.067

0.138

0.218

0.340

0.550

0.747

0.934

1.103

1.401

1.759

2.031

2.410

2.656

2.772

2.904

3.014

3.160

3.176

3.183

3.202

4.835

6.170

7.188

8.297

9.734

10.696

11.356

11.972

12.781

13.432

13.852

14.296

14.376

14.389

14.354

14.287

13.913

13.831

13.778

13.381

0.021

0.034

0.047

0.064

0.090

0.113

0.136

0.154

0.188

0.232

0.266

0.316

0.355

0.374

0.398

0.419

0.460

0.466

0.468

0.488

0.013

0.044

0.093

0.193

0.421

0.688

0.978

1.268

1.826

2.551

3.120

3.884

4.324

4.499

4.660

4.721

4.619

4.567

4.497

4.200

theoretical model occurs. The SS models are consistent with current observational data for pulsars and compact x-ray

sources, which yield masses mainly in the range of 1.1-1.8 M~ [14]. Of particular interest for models with higher masses

is the massive x-ray pulsar 4U0900-40, better known as Vela X- l , with a mass M = (1.56-1.98) M o [15]. A more accurate

determination of the mass of 4U0900-40 may considerably limit the possible range of bag parameters.

We also note that the presence of positrons leads to a considerable increase in the moment of inertia, and there is

also no contradiction with observational data in this respect.

The gravitational redshift from the stellar surface is also a directly observable parameter, in principle. From an

analysis of radiation from the source of the March 1979 g-ray burst, identified with the object SNR N49, a gravitational

redshift Z = 0.23 + 0.05 was obtained [11]. The extensive set of equations of state of the material of neutron stars considered

in [11] yielded masses M = (1.1-1.6) M o and radii R = (10-14) km for configurations with Z = 0.23, with M = (1.4-1.6)

M| and R = (12.5-14) km for stiff equations of state. Our values of M = (1.45-1.55) M| and R = (12.7-13.4) km are

comparable to the results for stiff equations of state of neutron stars. We note that M = (1.11-1.17) M| and R = (9.7-10.2)

km were obtained for electrical neutralization by electrons for bare SSs with Z = 0.23 [6], while M = (1.17-1.23) M~ and

R = (10.1-10.7) km were found for SSs with a crust [10]. Determining the mass of SNR N49 would undoubtedly contribute

to choosing between soft and stiff equations of state for superdense matter.

The next important parameter for observational differentiation between strange and neutron stars is the rotational

period of superdense stars. The absolute upper limit on the angular frequency of uniform rotation is the Kepler frequency,

corresponding to the limiting orbital velocity of a particle at the star's equator. When this frequency is exceeded, centrifugal

force prevails and matter starts to escape from the star's equator. The maximum Kepler frequency of rotation can be

469

TABLE 2.

% = 0.9)

Main Parameters of Cold Strange Stars (B = 34 MeV]fm 3, m s = 150 MeV,

~,, M, M, M, R, L t,

1014 g/cm ~ M| M| Mo km 10 J5 g.cm-"

2.876

2.933

2.992

3.078

3.222

3.365

3.648

3.790

4.210

4.627

5.042

5.591

6.680

7.220

7.356

8.297

10.301

i 2.952

13.217

16.250

0.054

0.110

0.172

0.269

0.434

0.590

0.868

0.988

1.291

1.516

1.686

1.859

2.081

2.151

2.167

2.248

2.338

2.375

2.374

2.367

0.065

0.120

0.194

0.299

0.480

0.645

0.940

1.084

1.462

1.743

1.951

2.174

2.477

2.568

2.594

2.703

2.831

2.883

2.878

2.862

0.055

0.112

0.176

0.278

0.455

0.625

0.938

1.078

1.439

1.720

1.940

2.170

2.486

2.591

2.616

2.746

2.909

3.003

3.006

3.033

4.376

5.638

6.484

7.537

8.807

9.734

11.001

11.424

12.292

12.810

13.136

13.393

13.597

13.619

13.620

13.589

13.419

13.140

13.112

12.810

0.019

0.030

0.042

0.057

0.082

0.103

0.142

0.159

0.204

0.239

0.269

0.301

0.350

0.368

0.373

0.397

0.434

0.463

0.465

0.483

0.009

0.029

0.061

0.129

0.286

0.477

0.908

1.126

1.744

2.259

2.669

3.092

3.622

3.770

3.804

3.945

4.017

3.906

3.885

3.818

approximated fairly accurately by the expression [16] t'~nax = C4(M/M| km) 3 , where M and R are the mass and

radius of the static configurations of maximum mass and C = 7200 sec -~ is an empirical coefficient, obtained by comparing

with the results of numerical integration within the framework of general relativity for rotating models. For the models

that we considered, the minimum rotational period is Prom = 2rC/fZm~x = (0.79-0.84) msec. For SSs with electrical neutral-

ization by electrons, we have Pmi, = (0.6-0.64) msec, whereas for model neutron stars P u, > 0.7 msec. The discovery of

submillisecond pulsars would undoubtedly favor SSs, and those neutralized by electrons.

In Table 3 we give a comparison of central density, Kepler rotational period, moment of inertia, and redshift for

model neutron stars and strange stars with a mass of 1.44 M~, corresponding to the currently most accurate determination

of the mass of the pulsar PSR 1913-16 [17]. The choice of this mass is also due to the fact that an analysis of observational

data in [18] indicates a concentration of the probable masses of superdense stars in the range of 1.3-1.6 Mo. The results

of model calculations using 16 equations of state considered in [11] were used for neutron stars, while the results for 12

equations of state from [7, 10] were used for SSs with electron neutralization. Given the great similarity of the results for

neutron stars and strange stars, we note that the gravitational redshift and the rotational period are perhaps the determining

parameters for their observational differentiation.

470

TABLE 3.

to a Mass of 1.44 M|

Comparison of Models of Neutron Stars and Strange Stars Corresponding

O/Oh" Prnin, I, Z Model msec 10 45 g.cm 2

2+5

3.5+4.5

1.6+1.8

0.8+1.1

0.7+0.75

0.96+ 1.04

0.9+ 1.6

1.4+ 1.5

2.1+2.3

Neutron stars

Strange stars

(electron neutralization)

Strange stars

(positron neutralization)

3. Cool ing o f a S trange Star

0.23+0.45

0.29+0.31

0.22+0.23

Using the results of the calculation of the intemal structure of SSs considered above, let us investigate their cooling.

Intense thermal emission from the surface of a SS does not occur, in all probability, since the plasma frequency COp of the

quark matter is so high (hCOp > 10 Mev) [4] that most photons of thermal radiation, even at T = 10 t~ K (kT z 1 MeV) cannot

propagate in such a medium. The surface of a bare SS will be an ideal mirror for photons with an energy e < 1 MeV. So

losses of thermal energy by a bare SS will occur by only two channels: a) neutrino emission from the entire volume of

the star; b) emission of electron-positron pairs (e-e + pairs) from the star's surface [19].

Neutrino production is due to URCA processes on quarks, the power of which was obtained in [20]. Using those

results, for the neutrino luminosity L V we can obtain

R 2 L,. = 5.1-1032 Tl 7 2 7~( ~tq ] e_SV/2eX/2r2dr =_ lvT7 erg/sec, (1)

,t.5oo)

where ~:(1-2GM/Rc2)I /2 , TIo :T /10 l0 is the surface temperature in units of 10 '~ K, ~tq is the chemical potential of a

quark in MeV [the chemical potentials of all quarks can be considered to be the same, to within the chemical potential

of electrons (positrons), which is justified by the relatively low e- (e § density], and R is the star's radius in centimeters.

In (1) e V and e x are the temporal and radial components, respectively, of the Schwarzschild metric. Because of the high

thermal conductivity of quark matter, a SS can be assumed to be "isothermal," i.e., T(r).e q2 = T = cons t , where T is the

temperature from the standpoint of an infmitely distant observer.

For a nonzero temperature, the occupation number for a degenerate electron (positron) gas is less than unity, i.e.,

there are unfilled levels. If these levels are sufficiently deep in the charged layer at the SS surface, they can be filled due

to the Dirac sea of electrons with the subsequent production of positrons (electrons) [21]. The e-e + pair ultimately escapes

the star, as a result of which the latter loses thermal energy at a rate [19]

Le_e+/L| = 2.38" 1012 (1 +0.843 TIO )T130 exp( - 1.182/T10 )It e R 2 A XeJ(y ) �9 ~, (2)

where IX e is the Fermi energy of positrons (electrons) in the outer charged layer at the SS surface in MeV, AX is a quantity

on the order of the thickness of that layer, equal to 10 ~~ cm, R is the star's radius in cm, y = 2~t e ~r~7~/kT, oL = e2/hc is

the fine-structure constant, and

5 1 - 995/y+ 15100/y 2, y > 20,

j(y) = t0.23, yl.2, 1 _< y _< 20,

[(1/3)y 3 ln(2/y), y < 1.

Here we have corrected an error (or misprint) in Eq. (10) of [19]: instead of ct there should be ,f~'~, as can easily be

ascertained from Eq. (8.64) from [22].

471

TABLE 4. Main Parameters of Hot Strange Stars

M/M~ R,

km

0.110 5.64

0.988 11.42

2.375 13.14

I r. 10 "3~

erg- K--"

0.057

0.573

1.700

/ . 102o,

erg- see~- K-7

0.117

1.020

3.150

A- 10 -8,

yr ~:5. K

2.28

2.34

2.33

If there is no energy release in the SS, then the above losses occur due to the star's thermal energy U r,

R UT =4r~fETev/2,.2(1 2Gm~ -1/2

o t. ---c-~-r) dr, (3)

where e r is the total thermal energy of all types of particles per unit volume. Expressions for the thermal energies of

positrons (electrons) and quarks can be found in [22] and [23], respectively. From these equations we find Ur= IrT:, where

I r is is an integral, independent of temperature, over the volume of the star. Numerical values of I v and I r for three

configurations are given in Table 4 (the case of B = 34 MeV/fm3).

Calculations show that neutrino losses dominate in the entire temperature range. The ratio Le_e./L v for all the

configurations reaches a maximum at Tjo ~ 0.2. With increasing mass of the configuration, it varies from 0.03 to 0.008.

Taking the star's temperature at birth to be very high, from energy balance dUr/dt = L v (from the standpoint of a

distant observer) we obtain for the surface temperature as a function of the age of a bare SS

= t - I/5 T ~ 51 v - At-I/5. (4)

It is seen from Table 4 that the ratio (lvllv) 1/5 depends little on mass. We can therefore assume that the temperature

of a bare SS depends only on the star's age, and for all masses we can approximately take

T = 2.3.10st -I 5K .yr I 5 (5)

For t ~ 1000 yr we obtain T = 5.8.107K. At such a temperature for a SS with a mass of 1 M| the neutrino luminosity is

about 6 L o. In Fig. 1 we give the time dependences of the surface temperature, neutrino luminosity, and power of energy

losses by the emission of e-e* pairs.

Both neutrino emission and the emission of e-e+ pairs are significant only right at the birth of a SS. If, as in [19],

the star's initial temperature is taken to be 10" K, then calculations show that in the first second the total energy emission

will be about 5.7.105~ erg. Although in our case at T > 10 l~ K the positrons cannot be considered to be highly degenerate

(laJkT .~ 2), the cooling law (4) is valid if the inequality L_~§ v is satisfied. As seen from Fig. 1, after the first second

following the star's formation, positrons can already be considered to be highly degenerate.

In Fig. 2 for three configurations we give the time dependences of the neutrino luminosity, the rate of emission of

e-e + pairs he-e§ and the total number of e-e + pairs Ne_e+ emitted from the time of formation of the star. The calculations

show that if the star's initial temperature is taken to be T o > 101~ K, then the total number of e-e* pairs emitted already

at t > 1 sec depends very weakly on this chosen value of T 0. In fact, at T > 101~ K the rate of emission of e-e + pairs is

he_e§ ~ In (T) [19], so we have

, r0) Ne-e+ (t)= ~ne-e~ (T)dt = I ~ - / ~ - d T = ~ 3 - / l n T + - / - - ~ 3 - / l n T o + - / .

o 7o ~o r 5 r ~. 5 ; 5 T ~ . 5,)

Hence it is seen that even at T(t) = T0/2, the second term is more than an order of magnitude smaller than the first, i.e.,

472

[-

1 0 2 0 I 1 I I I I I I I I ! I I I

101~

10~ - - L - L

v

10 ~

L 4+

10 -5 10 "~~ 10 "s 10 .6 10 .4 10-: 10 ~ 102 104

t, yr Fig. 1. Time dependences of the surface temperature T, neutr ino luminosity L v, and power of energy losses through the emission of e-e + pairs for a strange star with a mass of 0.988 M e.

~d

3' 10 53 L - - l0 s,- ~ N .

10 ~J 10 s~ 1049

104~ 1 (}47

1046

1 0 44

1 0 4 3 ~ L 1 0 4 - ' . . . . . . . . i . , . . . . . . I �9 . . , t , , , . l . . . . . . . . i i . . . . . . . i �9

10 "1 10 ~ 1() I 102 103 1 0 4

l, See

Fig. 2. Time dependences of neutrino luminosity L v and rate of emission ne_~+ and total number N + of emitted e-e + pairs for three configurations of strange stars (1. M = 0.11 M| 2. M = 0.988 M e, and 3. M = 2.375 Me).

at T(t) < To, Ne-e§ does not depend on the initial temperature. It is easy to show that this is because we have Le_~§ v

and due to the cooling law (4). As seen from Fig. 2, most of the e-e § pairs are emitted during the first several hundred seconds of the star's life and,

depending on its mass, comprise IlY2+IlY 3 pairs. The emission of e-e + pairs thus has the character of a flare.

If the emitted e-e § pairs are annihilated near the star's surface, then SS candidates may be identified from the y-ray

emission with an energy of -0.5 MeV [19]. This may be facilitated, it seems to us, by the presence of a strong magnetic

field near the SS surface. Pairs may be trapped by the field, as a result of which radiation belts, similar to the earth's, develop

around the star. In these zones the opposite fluxes of electrons and positrons either annihilate directly or, losing energy

through magnetic bremsstrahlung, "fall" onto the star's surface and annihilate there. The strong emission of e-e + pairs does

not continue for long (~100 sec), but the y-ray emission will last far longer due to the accumulation in radiation belts. The

details of these questions require a separate investigation.

473

4. Temperature of a Strange Star During Steady Mass Accretion

If a SS is in a close binary system or is surrounded by ordinary matter, then accretion will be an additional energy

source. For the chemical composition of the falling material below we take X = 0.7, Y = 0.27, and Z = 0.03.

In the configurations that we have considered, the excess electric charge is neutralized by positrons. The

annihilation of electrons of the falling material with positrons of the quark matter occurs immediately above the SS surface.

The generated y-rays with an energy of --.0.5 MeV, even those that initially had momenta toward the center of the star, being

reflected, escape it. The power of this radiation for an infinitely distant observer will be

Lpai r =(l+~)l(.lc2 m e 1 mp v e ' (5)

where h~/ is the accretion rate for the distant observer, l/v e = (1 + X)/2 is the number of electrons per baryon of falling

material, and m and mp are the electron and proton masses, respectively.

Atomic nuclei penetrate freely through the charged surface layer into the quark matter, since the electric field is

directed into the star, and immediately go into the quark phase. Although it is complicated to determine the complete

picture of the process of energy release and energy emission during accretion, we can give some limiting estimates.

If the energy of the falling material is converted weakly into radiation from the star, then from the standpoint of

a distant observer the power W of energy release from the surface layer of the SS is

W = (~l +[32) 1('1c2, (6)

where 131 = ~( f~qm-Yfne-Z f ) /mpc 2 and 132 = 1 - ~ . Here fne, f, andf~q,,, are the binding energies of nucleons in a helium

nucleus, heavy nuclei, and strange quark matter, respectively. This energy can escape the star by different channels

(neutrinos, the emission of e-e* pairs, bremsstrahlung both of y-rays and of elementary particles, etc.).

The star's temperature for a given M will be highest when this energy goes entirely into heating. Although the

energy is released in a thin surface layer, because of the high thermal conductivity of quark matter the SS can again be

considered to be isothermal. As before, therefore, all the energy released due to accretion escapes the star through the

neutrino channel. Equating (6) and (1), for the temperature we obtain

�9 ,~ 1/7 r, =[(13, + l~)Mc- / / , , ] . (7)

The star's temperature for a given M will be the lowest if all the gravitational energy of the falling material is

converted to radiation before reaching the surface. For this case we have

=[~l" 2" 11/7 T2 Mc /1,.J , (8)

from which we have T 2/T 1 = [13,/(~1 + ~2 )] 1/7, values of which for the configurations under consideration are given in Table

5. The weak dependence of the star's maximum temperature for a given mass accretion rate on the assumptions made and

on the mass is due to the strong temperature dependence of the neutrino luminosity.

If the SS is in a close binary system or is surrotmded by a fairly dense cloud, then the maximum mass accretion

rate "~/max is determined from equality of gravitational attraction and light pressure on the falling material. Light pressure

is produced only by Lr, ,, which, in turn, is due exclusively to mass accretion. Accretion therefore cannot be stopped by

radiation: a critical accretion regime is established. Since for ),-rays with an energy -mecZ the cross section for scattering

from an electron is half the Thomson cross section a 0, we obtain

/~/max(M / y r ) = 4 n G M 9~ m e , 2 m p ~ v2 ( M I - - V e =8"14'10-61-'~" ~ e~--'~')" (9) C l + ~ m p ~o

474

TABLE 5. Main Parameters of Hot Strange Stars in the Critical Accretion Regime

M/M|

0.110

0.988

2.375

( 13, )"'

0.83

0.68

0.58

/•max ~

10 -6.MJyr

0.478

4.042

8.514

Tax

108K

3.40

4.42

4.70

RL )

3.82

4.72

5.00

RL.)

7.15

8.05

8.33

Numerical values of /~max are given in Table 5. There we also give values of the maximum temperature Tax and

of the luminosities L~ r and L v at that temperature. In such a mass accretion regime, the age of the SS cannot be more than 10 ~ years.

5. Conclusion

The presence of positrons in strange quark matter, providing for electrical neutrality for certain sets of parameters

of the bag model, thus results in stiffer equations of state. The maximum SS mass in this case reaches 2.5 M~ with central

densities that exceed the nuclear density by a factor of five. A comparative analysis was made with the known observational

data, which does not yet enable us to choose between soft and stiff equations of state for superdense matter.

The calculations of SS cooling made here show that, in the absence of accretion, it occurs mainly due to neutrino

energy losses. The temperature-age dependence is a very weak function of the mass of the configuration and has the form

T ~ 2.3.10~r ~/~. The total number of e-e + pairs emitted from the time of the star's formation up to t > 1 sec hardly depends

on the star's initial temperature and is determined only by the mass of the configuration. Therefore, if a T-ray burst is

associated with the formation of a bare SS, then from the integrated flux of T-rays with an energy of ~0.5 MeV one can

obtain information about the mass of such objects. For a given accretion rate it was shown that a star's maximum and

minimum possible temperaturers differ by less than a factor of two. In the critical accretion regime the star's temperature

can reach 5.10 x K and depends little on mass. The results obtained for the cooling of strange stars in the absence of accretion

remain qualitatively valid for the case of electron neutralization of strange quark matter.

This work was carried out under topic N96-857, supported by the Ministry of Education and Science of the Republic

of Armenia.

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