Astrophysics, Vol. 42, No. 3, 1999

M O D E L S O F S T R A N G E S T A R S H A V I N G A C R U S T

Yu. L. Vartanyan and A. K. Grigoryan UDC 524.3-335.3

Models of strange quark stars with a crust consisting of atomic nuclei and degenerate electrons, maintained by an electrostatic barrier at the smface of the strange quark mattel, are investigated for a realistic range of parameters of the MIT bag model. The density at which neutrons escape ~'om nuclei, p = Pdril,' is taken as the maximum possible boundary density of the crust. Series of strange stars are calculated as a function of central density. Configurations with masses of 1.44 and 1.77 M e and a gravitational redsh(ft Z s = 0.23, corresponding to the best-known observational data, are investigated. The presence o.f a crust results in the existence of a minimum mass for strange stars, and also helps to explain the glitch phenomenon of pulsars within the framework of the e.ristence of strange quark matter.

I. Introduction

Witten [1] has proposed that strange quark matter, consisting of approximately equal amounts of u, d, and s quarks

with a small admixture of electrons or positrons to provide electrical neutrality, is absolutely stable, cool matter. This

hypothesis was investigated by Farhi and Jaffe [2], who showed, using the MIT bag model [3], that the stability of strange

quark matter depends on insufficiently accurately known phenomenological parameters of the model: the bag constant B, the

quark-gluon interaction constant a c, and the mass m s of a strange quark. Different sets of these constants can result in the

realization both of self-contained strange stars, considered in the present paper, and neutron stars with a quark core.

If the version of strange quark matter in which the excess electric charge of the quarks is neutralized by electrons

is realized, an electrostatic barrier, which prevents the inward transfer of ordinary matter, is formed at the free surface of

a strange star due to the partial escape of the electrons. The probability of the tunneling of atomic nuclei is so low that the

two phases can coexist almost indefinitely [4]. Both bare strange stars, consisting entirely of strange quark matter, and

strange stars with a crust consisting of atomic nuclei and degenerate electrons (the Ae phase), can exist in this version. 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. In the case of electrical neutralization by positrons, only

bare strange stars can exist, since ordinary matter in contact with strange quark matter is inevitably swallowed by it.

If it turns out that Witten's hypothesis is correct, it will be of decisive importance for the physics of superdense stars.

Much attention has therefore been paid to the investigation of strange stars in recent years. The main properties and structure

of strange stars were considered in [4-7]. In [8, 9] the parameters of models of bare strange stars were compared with

observational data and the problem of the parallel existence of strange and neutron stars was investigated. The the present

work has the purpose of extending this analysis to strange stars with a crust.

The MIT bag model is used for the equation of state of strange quark matter in the present work. We considered

12 sets of bag parameters corresponding to the most likely range of their variation. The equation of state from [10] is used

to describe the crust. The maximum possible density,/gma x = P a r i p = 4"3x10t~ g/cm3 (the density at which neutrons escape from

nuclei), is taken as the boundary density of the crust.

Yerevan State University, Armenia. Translated from Astroftzika, Vol. 42, No. 3, pp. 439-448, July-September, 1999. Original article submitted November 19, 1998; accepted for publication March 10, 1999.

330 0571-7256/99/4203-0330522.00 �9 1999 Kluwer Academic/Plenum Publishers

For the calculated equations of state we integrated the system of equations of stellar equilibrium (the TOV equations)

and obtained integrated parameters of strange stars with a crust. Models with the maximum masses, as well as models with

masses of 1.44 and 1.77 M , and a gravitational redshift Z s = 0.23, corresponding to the best-known observational data, are

investigated in detail. The minimum possible mass of a strange star with a crust is calculated.

2. Equation of State

Because the mass of a strange quark far exceeds the masses of u and d quarks, strange quark matter has a small deficit

of s quarks. Since quarks are bound by the strong interaction, a strange quark star has a sharply defined surface. The excess

positive charge of the quark plasma is neutralized by electrons which, being bound only by the Coulomb force, can partially

escape from the quark surface, propagating for hundreds of fermis. For this reason, a thin charged layer is formed at the

surface of a strange quark star in which the electric field strength reaches 1017-1018 V/cm [4].

The electrostatic barrier at the surface of a strange star can support the crust of ordinary matter. The crust is not

in chemical equilibrium with the strange quark matter and is held to the quark core only by gravity. Since free neutrons,

having no charge, can pass freely through the barrier and be absorbed by the strange quark matter, the maximum density

of the crust must be limited to the density of neutron escape from nuclei, ponp = 4.3x10 H g/cm 3. A strange star may acquire

its crust at the time of formation or due to accretion from interstellar space [7, 11].

Neglecting the gap of several hundred fermis between the strange quark matter and the crust, we use an equation

of state consisting of two parts. The first part describes the normal matter of the Ae phase with a maximum density Pdnp"

We use tabular data from [10]. The second part corresponds to the strange quark matter, for which we use the MIT bag

model. In this model the equation of state is determined by insufficiently exactly known phenomenological parameters: the

bag constant B, the quark-gluon interaction constant a c, and the mass m s of a strange quark. In the presence of a crust, the

pressure at the boundary of the quark core does not go to zero but will be on the order of Pa~p. In contrast to [7, 11 ], in

which a simplified equation of state of strange quark matter was used ( m = a s = 0), we use an inverted form (we refer to

[8, 9] for details).

In the present work, as in [8, 9], we use equations of state for 12 sets of bag parameters, corresponding to a realistic

range of variation of the parameters B and m s (B = 55-60 MeV/fm 3, m s = 175-200 MeV) [12]. The energy per baryon as

a function of baryon density has a negative minimum for all the models, which ensures the binding of the strange quark

matter. In Table 1 we give values of the bag parameters for the models considered, grouped into four series by values of

B and m. Values of the threshold density for the occurrence of strange quark matter for each model, as well as the binding

energy per baryon corresponding to that model, are given in [8, 9].

3. Results of the Calculation and Observational Data

The main parameters of strange stars were calculated by numerical integration of the relativistic equations of stellar

equilibrium of [13], supplemented by two equations for determining the relativistic moment of inertia [14]. We note that

the same result for the relativistic moment of inertia can be obtained by numerical integration of the one differential equation

obtained in [15]. The calculations were made using the equations of state of Table 1. For several series of configurations

we calculated values of the stellar radius R, the total mass M, the rest mass M 0, the proper mass Mp, the relativistic moment

of inertia I, the redshift Z s from the stellar surface, and the m a s s Mcr and thickness tcr of the crust as functions of the central

density Pc" The data for one such series (model 1.1) are given in Table 2.

The presence of a crust has virtually no influence on the total mass of a strange star ( M = I0 -s Me). For all the

models considered, the binding energy has the normal sign, M < M 0, which is a necessary condition for stability. The

influence of the crust is negligible for the mass defect, a detailed analysis of which for bare strange stars was made in [16].

331

TABLE 1. Values of Bag Parameters for the Models under Con-

sideration

Series B m,

MeV/fm 3 MeV

55

55

60

175

200

175

0.05

0.17

0.26

a Model

m , ,

0.05 1.1

0.17 1.2

0.3 1.3

0.38 1.4

0.05 2.1

0.17 2.2

0.31 2.3

3.1

3.2

3.3

4.1

4.2

4 60 200 0.05

0.18

TABLE 2. Integrated Parameters of Stellar Configurations for Model 1.1

p~ M M o M e M R t . Z I

1014 g]sm 3 M, M| M~ M|215 -5 km km 104Sg.sm2

4.431 0.021 0.022 0.021 4 24.71 21.873 0.001 0.001

4.443 0.025 0.027 0.026 3 13.52 10.497 0.003 0.002

4.449 0.028 0.029 0.028 2.8 11.58 8.475 0.004 0.002

4.466 0.035 0.037 0.035 2.5 8.93 5.571 0.006 0.003

4.525 0.062 0.066 0.063 2.3 6.99 2.934 0.013 0.008

4.643 0.123 0.132 0.126 2.3 6.84 1.761 0.028 0.026

4.760 0.189 0.205 0.195 2.4 7.19 1.358 0.041 0.053

4.936 0.290 0.318 0.302 2.5 7.76 1.050 0.060 O. 109

5.111 0.389 0.430 0.409 2.5 8.24 0.875 0.078 O. 177

5.374 0.529 0.592 0.563 2.6 8.83 0.730 0.102 0.295

5.810 0.732 0.834 0.795 2.6 9.53 0.587 0.137 0.508

6.388 0.951 1.103 1.053 2.5 10.12 0.487 0.176 0.780

6.820 1.083 1.270 1.214 2.4 10.41 0.435 0.201 0.963

7.251 1.193 1.411 1.351 2.3 10.62 0.400 0.223 1.123

8.110 1.363 1.635 1.570 2.1 10.88 0.349 0.259 1.381

9.249 1.518 1.846 1.781 1.9 11.04 0.305 0.297 1.620

10.949 1.662 2.047 1.989 1.7 11.10 0.263 0.338 1.827

14.601 1.803 2.252 2.216 1.3 10.97 0.214 0.393 1.970

17.392 1.843 2.311 2.296 1.2 10.82 0.201 0.417 1.963

21.971 1.860 2.337 2.352 1 10.56 0.182 0.442 1.884

24.045 1.859 2.334 2.362 0.9 10.45 0.172 0.450 1.839

332

The radius as a function of the mass of a strange star with a crust for model 1.1 is given in Fig. 1. In contrast to

the case of neutron stars, for our models, like those of bare strange stars, the radius increases with increasing mass over

almost the entire curve. Only near the very maximum, i.e., before stability loss, is this dependence like that of stable neutron

stars. Such a difference may be used for distinguishing observationally between strange and neutron stars. Li et al. [17]

proposed a semiempirical dependence of mass on radius for the binary x-ray source Hercules X- 1 and argued that this object

is a strange rather than a neutron star. This analysis is subject to criticism, however, both in connection with a refinement

of the mass of Hercules X-I [18] and in connection with an incorrect choice of bag parameters [19]. Further refinement

of the observational data, in a context with realistic theoretical calculations, may answer the question of whether one or

another model is realized.

The crust is thinnest for configurations o f the maximum mass and increases with decreasing mass. For strange stars

with masses of 1. I-1.8 M~, which are typical for observed superdense stars [20], the thickness of the crust is on the order

of 180-500 m. Note that such a crust thickness for this mass range is obtained for model neutron stars calculated using the

Grigoryan-Sahakian equation of state, in which the pionization effect and the quark structure of hadrons are taken into

account (for details see [15, 21]). A sharp increase in crust thickness is observed for strange stars o f low masses, which

become gravitationally unstable (dM/dp, < 0) at some minimum mass o f the strange star. We note that the existence of

arbitrarily small masses is possible for bare strange stars. In Fig. 2 we give the radius of low-mass strange stars with a crust

as a function of mass. For model 1.1, Mm~" ~ 0.018 M| at a radius of 450 km. A detailed investigation of low-mass strange

stars, which are very sensitive to the value of the maximum boundary density of the crust, will be made in future papers.

In Table 3 we give the main parameters of stellar configurations of the maximum masses. For the range o f bag

parameters that we considered, we obtained maximum masses Mma ~ = 1.75-1.86 M~ with their corresponding radii Rr,~, = 10-

10.6 km and central densities (Pc)r,,x = (2"2-2"5) xI0~5 g/cm3" The contribution o f the crust is very slight: Rcr = 170-180 m

and M r ~- 10 -~ M~. The configurations of the maximum masses were calculated because o f the importance of comparing

the parameters of theoretical models with the observed parameters o f stellar objects. The values that we obtained are

consistent both with the most accurately determined masses o f superdense stars at present and with observational data on

20

15

25

10 .

5

0

0 0 5 1 1,5

]V!/M|

Fig. 1. Radius of a strange star with a crust as a function of mass for model 1.1. The configuration with the maximum mass is indi- cated by a dot, Ran corresponds to the radius o f the stellar surface, and R c corresponds to the radius o f the quark core.

333

500

=r

400

300

200

IC,0

0.0175 0.018 0.0185 0.019 00195 0.02 0.0205

M/M~

Fig. 2. Radius of a low-mass strange star with a crust as a

function of mass for model 1.1. The configuration of mini- mum mass is indicated by a dot.

TABLE 3. Integrated Parameters of Stellar Configurations of the Maxi-

mum Mass

Model Mm= x Pc R

M. 10Jag/sm 3 km

1.1 1.860 21.971 10.562 10.380

1.2 1.860 22.061 10.544 10.363

1.3 1.860 22.271 10.511 10.340

1.4 1.861 22.358 10.512 10.330

2.1 1.829 22.590 10.444 10.264

2.2 1.827 22.680 10.420 10.238

2.3 1.825 22.743 10.390 10.211

3.1 1.786 23.861 10.135 9.957

3.2 1.786 23.943 10.121 9.944

3.3 1.787 24.030 10.106 9.934

4.1 1.757 24.502 10.020 9.846

4.2 1.755 24.576 9.995 9.823

R~ Mr Z km M|215 10 -5

1

1

0.9

1

0.9

0.442

0.444

0.446

0.447

0.438

1 0.439

1 0.441

1 0.443

0.9 0.444

1 0.446

1 0.439

0.9 0.440

334

the moment of inertia.

In Table 4 we give data on strange stars with masses M = 1.44 M| and M = 1.77 M~, which are the most likely

observed masses of the pulsars PSR 1913+16 [22] and 4U 0900-40 [23], respectively. A mass of 1.77 M| is not reached

for our models of the fourth series. Refinement of the mass of 4U 0900-40 may considerably restrict the possible range

of bag parameters.

The gravitational redshift Z s is an important parameter for distinguishing between strange and neutron stars. From

an analysis of radiation from the March 1979 gamma-ray burst, identified with the object SNR N49, a gravitational redshift

Z s = 0.23 + 0.05 was obtained [23]. An extensive set of equations of state of the matter of neutron stars considered in [23]

yielded masses M = 1.1-1.6 M| and radii R = 10-14km for configurations with Z s = 0.23, with M = 1.4-1.6 M~ and

R = 12.5-14 km for relativistic equations of state. In Table 5 we give data on configurations with Z s = 0.23 for the models

of strange stars with a crust that we have considered. The ranges of values obtained for them, M = 1.17-1.23 M~ and

R = 10.1-10.7 km, just like the data for bare strange stars, M = 1.11-1.17 M~ and R = 9.7-10.2 km [8], can contribute to the

choice of an equation of state for superdense matter if the mass of SNR N49 is determined.

The rotational period of a superdense star is also an important parameter for observational distinguishing between

strange and neutron stars. The Keplerian frequency, corresponding to the orbital velocity of a particle at the star's equator,

becomes the absolute upper limit of the angular frequency of uniform rotation. The minimum rotational period of a

configuration with the maximum mass can be approximated well by the expression Pmin = O ' 0 2 7 6 [ ( R / k m ) 3 / ( M / M | )1 lp. msec

[24]. For a realistic range of parameters of the bag model we obtain Pro+, = 0.66-0.69 msec for strange stars with a crust

and Pmin = 0.64-0.68 msec for bare strange stars [8]. In the case of model neutron stars we have Pmin ~ 0.7 msec, with a

few models yielding submillisecond rotational periods. The fastest pulsar yet recorded, PSR 1937+21, has a rotational period

P = 1.558 msec [25]. The discovery of submillisecond pulsars would contribute to solving the problem of choosing

theoretical models.

Models of bare strange stars with no crust experience major difficulties in the standard explanation of a pulsar glitch.

It was shown in [7] that a rigid nuclear crust, which is supported by the electrostatic barrier at the surface of a strange star,

has a sufficient moment of inertia to provide the observed sizes of pulsar glitches within the framework of the "starquake"

model. The ratio of the moment of inertia of the crust to the star's total moment of inertia, l J/o,,,], decreases monotonically

with increasing mass. The values of l f l ,o t , , I ~ 10-3-10 -5 that we obtained for a realistic range of bag parameters are consistent

with the results of [7].

4. Conclusion

Models of strange stars with a crust are investigated in the present paper for the most probable range of variation

of the bag parameters in the context of their comparison with observational data. The authors believe that the possibility

that strange stars exist is consistent with presently existing observational data, while the presence of a crust contributes to

the standard explanation for a pulsar glitch. It is important to consider the possibility of observational distinguishing of these

objects from neutron stars in order to confirm or refute the hypothesis that strange quark matter is absolutely stable.

First, strange stars with a crust, like bare strange stars, have a mass-radius dependence that is different from that of

neutron stars. The radius increases with increasing mass over almost the entire curve, except for configurations near the

maximum mass, as well as near the minimum mass when a crust is present. I f the observed masses and radii are refined,

the difference in the mass-radius dependence may serve as one of the main criteria for observational differentiation.

Second, the difference in the mass-radius dependence results in shorter rotational periods for strange stars. For our

models the Keplerian rotational frequency for strange stars lies in the range of 0.64-0.69 msec, whereas for neutron stars

Pk > 0.7 msec. No presently observed pulsar has a submillisecond rotational period, and this comparison cannot yet be used.

Third, a difference in the mass-radius dependence leads to different gravitational redshifts for strange and neutron

stars. For the redshifl determined for the object SNR N49, Z s ~- 0.23, in particular, the ranges of masses and radii obtained

for the two types of superdense stars have a very small range of overlap. Determining the mass of SNR N49 is thus very

335

TABLE 4. Integrated Parameters of Stellar Configurations with Total Masses M = 1.44

M| and M = 1.77 M|

Model M = 1.44M| M = 1.77M|

R R, M Z, R R c M . Z

km km Mq~x 10 5 km km M x l 0

1.1 10.977 10.650 2 0.277 11.040 10.812 0.377

1.2 10.959 10.634 2 0.278 11.027 10.796 0.378

1.3 10.937 10.617 1.8 0.279 11.011 10.7813 0.379

1.4 10.930 10.610 2 0.279 11.003 10.772 0.379

2.1 10.902 10.582 2 0.280 10.881 10.661 0.386

2.2 10.875 10.556 2 0.281 10.845 10.626 0.389

2.3 10.839 10.525 1.9 0.282 10.799 10.591 0.391

3.1 10.581 10.285 1.8 0.292 10.379 10.185 0,419

3.2 10.568 10.271 1.8 0.293 10.363 10.169 0.420

3.3 10.554 10.263 1.7 0.294 10.353 10.164 0.421

4.1 10.502 10.212 1.7 0.296 Does not occur

4.2 10.471 10.186 1.7 0.297 Does not occur

TABLE 5. Integrated Parameters of Stellar Configurations

with a Gravitational Redshift Z s = 0.23

Model M Pc R R c Mcr

M| 10t4g/sm3 km km M|215 10 .5

1.1 1.235 7.395 10.679 10.290 2.3

1.2 1.223 7.464 10.676 10.291 2.3

1.3 1.228 7.474 10.651 10.265 2.2

1.4 1.226 7.480 10.637 10.252 2.2

2.1 1.222 7.588 10.610 10.229 2.2

2.2 1.220 7.639 10.583 10.202 2.2

2.3 1.218 7.693 10.550 10.169 2.2

3.1 1.180 8.060 10.243 9.875 2

3.2 1.176 8.070 10.223 9.855 2

3.3 1.173 8.076 10.209 9.840 2

4.1 1.169 8.235 10.167 9.799 1.9

4.2 1.167 8.287 10.141 9.773 2

336

urgent for future research.

The criteria considered above for observational differentiation from neutron stars pertained both to bare strange stars

and to strange stars with a crust. Only bare strange stars can have arbitrarily small masses, however. The presence of a

crust results in the existence of a minimum mass for a strange star on the order of 0.018 M| with a radius of up to 450 krn

for the maximum boundary crust density (Pm,x =/gar ip = 4.3x10H g/cm3) �9 The minimum mass for stable neutron stars reaches

~0.1 M| [26, 27] with a radius of 200 kin. We note that reducing the maximum boundary density of the crust results in lower

values of the minimum mass. Huang and Lu [28] argued that the maximum density of a crust that can be supported by the

electrostatic barrier at the surface of a strange star is on the order of 8.3x 10 H~ g/cm 3. A detailed investigation of low-mass

strange stars with a crust will be made in future papers.

The present work was carried out within the framework of topic N96-857, supported by the Ministry of Education

and Science of the Republic of Armenia.

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