THE CONVERSION FROM NEUTRON STARS TO STRANGE STARS

AND ITS I M P L I C A T I O N S

ZIGAO DAI Department of Astronomy, Nanjing University, Nanjing 210093, P.R. China

XUEJUN WU Department of Physics, Nanjing Normal University, Nanjing 210024, P.R. China

and

TAN LU CCAST (Worm Laboratory), P.O. Box 8730, Beijing 100080; Department of Astronomy, Nanjing

University, Nanjing 210093, PR. China

(Received 14 May 1995; accepted 31 May 1995)

Abstract. The conversion from neutron stars with different equation of states (EOSs) for neutron matter into strange stars with different EOSs for strange quark matter has been studied in a general relativistic numerical calculation in this paper. For hot neutron stars, their conversion may lead to great variations in their rotation periods, of which the magnitude would be greatly dependent upon the EOS for neutron matter, and of which the timescale would be greatly determined by the EOS for strange matter. This phenomenon appears as giant glitches, which might provide a probe of EOSs for both neutron matter and strange matter. But for cold neutron stars, their conversion may result in a population of gamma-ray bursts.

1. I n t r o d u c t i o n

Neutrons and protons are composed of quarks. In principle, at sufficiently high densities, neutron-star matter can deconfine, leading to two-flavor quark matter, which can subsequently convert into three-flavor quark matter (strange matter). Since strange matter was conjectured to be more stable than hadronic matter (Witten, 1984), and was shown to exist reasonably within uncertainties inherent in a strong-interaction calculation (Farhi and Jaffe, 1984), many authors (e.g. Alcock et al., 1986; Haensel et al., 1986; Alcock, 1991; Colpi and Miller, 1992; Glendenning and Weber, 1992) have studied the physics of strange stars. Strange stars are dense objects analogical to neutron stars in the following aspects. Firstly, they have similar radii for the observed mass near 1 .4Mo, so their observational quantities such as gravitational redshifts are nearly equal. Secondly, strange stars with crusts have the same magnetospheres as neutron stars do at the stage of radio pulsars, so they give same properties and structures of radio pulses. Thirdly, neutrino cooling of strange stars with thin crusts might provide a definite observational signature for such exotic objects (Pizzochero, 1991), but this importance has been greatly reduced by the revival of direct URCA processes (Lattimer et al., 1991). Thus,

it seems to be difficult to make a distinction between strange stars and neutron stars through X-ray observation. In brief, one has not so far found an efficient

Astrophysics and Space Science 232: 131-138, 1995. (F) 1995 Kluwer Academic Publishers. Printed in Belgium.

132 ZIGAO DAI ET AL.

physical mechanism to distinguish strange stars from neutron stars by means of astronomical observation.

Neutron stars result from supernova explosions. What do strange stars originate from? Witten (1984) suggested that neutron stars may convert into strange stars. This conversion may have two cases, viz., the transition from protoneutron stars formed during the collapse of supernova cores and the transition from ordinary neutron stars. The former may result in a supernova explosion and enhancement of the energy of a shock wave (Gentile et al., 1993; Dai et al., 1995), while the latter has been studied by several authors (e.g., Olinto, 1991; Olesen and Madsen, 1991). It is assumed that a strange-matter seed exists in the interior of a neutron star. Since neutrons are neutral, they can diffuse freely into the seed, and convert to strange matter. Or equivalently we can say that the seed expands outward to the stellar surface, and therefore the neutron star convert to a strange star. Olinto (1987) and Olesen and Madsen (1991) have investigated the speed at which the conversion of a neutron star occurs, which is strongly related to the nonleptonic weak reaction rate. However, they both neglected the influence of the strong interactions in quark matter on the reaction rate, and didn't study the conversion of neutron stars with different equations of state (EOSs) for neutron matter into strange stars with different EOSs for strange quark matter in detail. Here we will overcome these two shortcomings, and study the conversion based on a general relativistic numerical calculation. We will see that, for hot neutron stars, the conversion may lead to great variations in the rotation periods of pulsars, of which the magnitude would be greatly dependent upon the EOS for neutron matter, and of which the timescale would be greatly determined by the EOS for strange matter. This phenomenon appears as giant glitches, which might provide an astrophysical probe of EOSs for both neutron matter and strange matter. But for cold neutron stars, the conversion may produce a population of gamma-ray bursts.

2. Model and Calculation

We construct a series of hybrid stars of which each has total baryon mass kept to be 1.4M e. Its interior is a strange-matter core, of which the radius is set to be from zero to Rs (where Rs is the radius of a strange star with 1.4M o bary- on mass), and its exterior is a neutron-matter crust. Assuming that the angular momenta of these hybrid stars are identical, we study their properties such as gravitational masses, radii and moments of inertia. This is done by integrating the general relativistic equation of hydrostatic balance of a spherical compact star, the Tolman-Oppenheimer-Volkoff (TOV) equation (e.g., also see Shapiro and Teukol- sky, 1983):

dp [M(r) + 47rr3P](p + P) - ( 1 )

dr f i r - 2M(r)]

CONVERSION FROM NEUTRON STARS TO STRANGE STARS 133

with

d M - 47rr2p, (2)

dr

where P and p are the pressure and the energy density respectively. We will neglect the effect of rotation, because the rotation frequencies of the observed pulsars are much smaller than the Keplerian limit corresponding to some EOS, so that the effect of rotation on the structures of the pulsars is not significant (Wu et al., 1991).

It is very clear that integrating the TOV equation needs EOS(s) to be given. For neutron matter, we use three different EOSs, which may be considered as soft, medium stiff, and stiff, respectively. The first EOS (soft) is BPS (Baym et al., 1971a), nuclei constructed from a compressible liquid-drop model. For densities between 4.3 x 10 lI and 5 × 1014 g cm -3 the EOS is taken from the calculation of (Baym et al., 1971 b). To still higher density, it includes the effects of hyperons. The second EOS (medium stiff) is chosen from Tsuruta and Cameron (1966) (hereafter VT), where some of the hyperons were included and a phenomenological baryon- baryon potential was used to approximate the effect of the strong interactions. The third EOS (stiff) is taken from Waldauer et al. (1987) (hereafter Wb). This EOS satisfies the known bulk properties of nuclear matter like binding energy per nucleon E / A = -16MeV at the nuclear saturation density with a compression modulus of 300MeV. These EOSs have been tabulated by Wu (1992). For strange matter, its EOS is determined by the thermodynamical potential which is dependent upon three physical quantities, viz., the strange-quark mass (ms), the vacuum energy density (B), and the strong coupling constant (o~) (Farhi and Jaffe, 1984). The typical values of ms and B are picked to be 200MeV and 57MeV fm -3 respectively. The coupling constant is set to be two values of c~ = 0 (hereafter EOS S1) and oz~ = 0.6 (EOS $2).

After giving EOSs for neutron matter and strange matter, we can integrate the TOV equation numerically. Tables I and II give the properties of hybrid stars with different EOSs. In these tables, radii of strange-matter cores, and center densities, radii, gravitational masses and moments of inertia of these stars are shown at the ith (i = 1,2, 3, 4 and 5) columns respectively. The last column gives timescales for the conversion of pure neutron stars to these hybrid stars at temperature T = 108K (see next section).

It can be seen from the tables that, for a soft EOS (e.g., BPS), the radius and moment of inertia of a hybrid star increase with the radius of its strange-matter core, but for a hard EOS (e.g., Wb), we can draw a conclusion just opposite to the above. Comparing Table I with Table II, we find, the moment of inertia of a hybrid star increases with the coupling constant, no matter whether the EOS of neutron matter is BPS or V 7 (or Wb).

134 ZIGAO DAI ET AL.

TABLE I

Properties of hybrid stars

EOS S1 + BPS

rs (kin) pc (1015g cm -3) R(km) M(M®) 1(1045g cm 2) t(s)

0.0 ...... 2.998 8.089 1.2476 0.4459 0.0 2.0 ...... 2.820 8.121 1.2484 0.4507 0.090 4.0 ...... 1.840 8.657 1.2549 0.5389 0.170

6.0 ...... 1.065 9.891 1.2566 0.7441 0.235 8.0 ...... 0.816 11.011 1.2303 0.9106 0.293

10.01... 0.794 10.010 1.1969 0.8933 0.351

EOS S1 + V7

0.0 ...... 0.8488 12.103 1.2865 1.0250 0.0 1.0 ...... 0.8347 12.121 1.2864 1.0252 0.029

3.0 ...... 0.8194 12.127 1.2835 1.0309 0.087 5.0 ...... 0.7910 12.241 1.2739 1.0445 0.145 7.0 ...... 0.7682 12.289 1.2518 1.0378 0.201

8.0 ...... 0.7672 12.240 1.2362 1.0088 0.230 9.0 ...... 0.7734 11.977 1.2182 0.9595 0.258 10.01... 0.7936 10.010 1.1969 0.8933 0.287

EOS S 1 + Wb

0.0 ...... 0.3686 16.378 1.3152 1.8753 0.0 1.0 ...... 0.5631 16.369 1.3150 1.8733 0.019

3.0 ...... 0.5795 16.243 1.3119 1.8287 0.068 5.0 ...... 0.6151 15.886 1.2989 1.6725 0.118

7.0 ...... 0.6738 15.031 1.2712 1.3834 0.172

9.0 ...... 0.7523 13.171 1.2228 1.023Y 0.228 10.01... 0.7936 10.010 1.1969 0.8933 0.257

3. T h e C o n v e r s i o n P r o c e s s

If a series of hybr id stars above is descr ibed as a d y n a m i c a l process , namely , that

a s t range-mat te r seed is formed at the center of a neu t ron star and then it absorbes

neut rons , l ead ing to new strange matter, then we can study the conve r s ion of the

neu t ron star into a s t range star. The condi t ion for this is that the t imesca le for the

conve r s ion is m u c h larger than that for sound to travel through a neu t ron star. The

speed at which this conve r s ion occurs is de t e rmined by the process for quarks

towards equ i l i b r ium via weak interact ions and the d i f fus ion of s t range quarks

towards the c o n v e r s i o n front. Thus , the speed is

CONVERSION FROM NEUTRON STARS TO STRANGE STARS 135

TABLE II

Properties of hybrid stars

EOS $2 + BPS

r , (km) pc (1015g cm -3) R(km) M(Mo) /(1045g cm 2) t(s)

0.0 ...... 2.998 8.089 1.2476 0.4459 0.0

2.0 ...... 2.770 8.123 1.2514 0.4520 2.25

4.0 ...... 1.880 8.653 1.2659 0.5429 10.6

6.0 ...... 1.080 9.810 1.2684 0.7624 30.7

8.0 ...... 0.887 10.410 1.2864 0.9461 35.7

10.11... 0.869 10.111 1.2989 0.9810 40.6

EOS $2 + V. r

0.0 ...... 0.8488 12.103 1.2865 1.0250 0.0

1.0 ...... 0.8446 12.106 1.2866 1.0254 2.05

2.0 ...... 0.8386 12.108 1.2867 1.0273 4.05

4.0 ...... 0.8232 12.151 1.2873 1.0406 7.72

6.0 ...... 0.8095 12.250 1.2885 1.0589 11.1

7.0 ...... 0.8100 12.245 1.2893 1.0595 12.8

8.0 ...... 0.8192 12.172 1.2906 1.0466 14.6

9.0 ...... 0.8384 11.929 1.2932 1.0176 16.6

10.11... 0.8690 10.1l l 1.2989 0.9810 19.2

EOS $2 + Wb

0.0 ...... 0.3686 16.378 1.3152 1.8753 0.0

1.0 ...... 0,5765 16.363 1.3149 1.8723 0.546

3.0 ...... 0.5945 16.217 1.3144 1.8300 1.74

6.0 ...... 0.6652 15.409 1.3092 1.5607 4.24

8.0 ...... 0.7550 14.145 1.3058 1.2536 6.82

9.0 ...... 0.8120 13.125 1.3002 1.0916 8.54

10. l l . . . 0.8690 10.111 1.2989 0.9810 11.1

1/2

(3)

( O l i n t o , 1 9 8 7 ) w h e r e 7-~ is t h e n o n l e p t o n i c w e a k r e a c t i o n t i m e s c a l e d e f i n e d b y

O l i n t o a n d a0 is h a l f o f t h e d i f f e r e n c e b e t w e e n t h e d o w n - a n d s t r a n g e q u a r k

c o n c e n t r a t i o n s in s t a b l e s t r a n g e m a t t e r . D is t h e d i f f u s i o n c o e f f i c i e n t o f s t r a n g e

q u a r k s

D ~ 10 - 3 c m 2 s - 1 (4)

1 3 6 ZIGAO DAI ET AL.

(Olinto, 1987; Baym et al., 1985) where # is the chemical potential of down (or strange) quarks and T is the temperature inside a neutron star.

Several authors (Wang and Lu, 1984; Sawyer, 1989; Heiselberg, 1992; Madsen, 1993; Dai et al., 1995) neglected the influence of the strong interactions between quarks on the nonleptonic weak reaction rate. However, this influence was recently found to be significant (Dai and Lu, 1995). After the strong interactions are taken into account, the nonleptonic weak reaction timescale turns out to be

"rw = 3.36 x 103 (1 - 2c~c~-2 #-5 s. (5) \ r e /

For o~c = 0, this result is consistent with that of Olesen and Madsen (1991). From Equations (3)-(5), the speed at which the conversion from a neutron star

into a strange star occurs can be calculated. This conversion can also be represented a s

t ( r ) = f dr' (6/ J v o

The numerical result has been shown at the last column in Tables I and II, where the temperature in hot neutron stars is 10SK. We can see, for different EOSs for either neutron matter or strange matter, there is an obvious difference in the timescale for the conversion of a neutron star to a strange star. When the EOS for strange matter is fixed, the timescale decreases as the EOS for neutron matter becomes hard, but the difference in the timescale is quite small. When the EOS for neutron matter is fixed, however, the timescale increases as the EOS for strange matter becomes hard (that is, the coupling constant increases), and the difference may be as large as two orders.

Since the speed at which the conversion front moves is inversely proportional to the temperature inside a neutron star (seen from Equations (3) and (4)). The timescale for the conversion of a neutron star with T --- 106K is of two orders smaller than the result studied above (T = 108K).

4. Discussion

In the above two sections we have studied the conversion from a neutron star into a strange star, based on an assumption of the existence of a strange-matter seed inside the star. Alcock et al. (1986) have discussed five mechanisms by which a strange-matter seed could be formed in a neutron star. Here we suggest that accretion of a neutron star in a close binary be likely to result in formation of a strange-matter seed. It is recently thought (Kuschera and Kotlorz, 1993) that the center density of a massive neutron star (perhaps > 1.43//o) could be higher than the density at which neutron matter deconfines into quark matter. The mass of the

CONVERSION FROM NEUTRON STARS TO STRANGE STARS 137

(massive) neutron star in a close binary increases as it accretes from its companion, leading to the increase of its center density, and when this density exceeds the deconfinement density of neutron matter, a strange-matter seed could be formed. Thus, the conversion front moves outward with a timescale, by which the neutron star converts to a strange star.

During the conversion, the moment of inertia of a hybrid star changes sig- nificantly. The conservation law of angular momentum requires that the angular velocity should also change. Since the relative variation in the rotation frequency of this star (viz., Af~/f~) may be the order of 0.1 (seen from Tables I and II), the conversion of the neutron star into a strange star could give rise to a great variation in the period of a pulsar, hereafter referred to as a slow 'giant glitch'. The glitch is strongly dependent upon EOSs for both neutron matter and strange matter. On one hand, if the glitch behaves as a spin-down of a pulsar, the EOS for neutron matter must be soft; if the glitch behaves as a spin-up, the EOS must be stiff. So one could study the EOS in the interior of a neutron star in terms of the magnitude of the glitch. On the other hand, the timescale of the glitch is greatly determined by the EOS for strange quark matter. For example, the timescale for c~ = 0.6 is of two order of magnitude larger than that for ozc = 0. Therefore, one could also study the strong interactions in strange matter in terms of the timescale of the giant glitch.

What we discuss above is that a neutron star behaving as a pulsar (of which typical temperature is about 108K) could convert into a strange star. When the age of a pulsar exceeds 107yr, it will die based on the current radio-pulsar the- ory (Ruderman and Sutherland, 1975). Because a great number of neutrinos and photons have radiated previously, the neutron star becomes cold (Shapiro and Teukolsky, 1983). According to the results shown above, if this star converts into a strange star, the conversion timescale is much smaller than ls. Owing to rapid changing of the neutron-star structure, a new-born strange star possibly has great radial vibrations, of which the energies might be about 1047 ergs, many people (Wang and Lu, 1984; Sawyer, 1989; Madsen, 1992; Goyal et al., 1994; Dai and Lu, 1995) have shown that these radial vibrations are efficiently damped by the bulk viscosity of strange matter, and the timescale for this damping is smaller than ls. The surface vibrations of a strange star should result in variations of its surface magnetic field, which produce electric fields parallel to the background magnetic field. These electric fields will accelerate charged particles such as elec- trons and positrons. If the surface magnetic field is very high, the vibrations with large amplitudes lead to the charged particles accelerated to high energies along the magnetic field. Because the field lines are curved, the relativistic electrons emit increasing amounts of curvature radiation as their energies increase. Recently Smith and Epstain (1993) showed that a maximum energy is reached where cur- vature radiation losses balance the energy gains from the electric field, and the curvature photons can produce electron-positron-synchrotron cascades. Further- more, they showed that an observed gamma-ray burst spectrum can be explained

13 8 ZIGAO DAI ET AL.

by the addition of the synchrotron radiation spectrum from the last generation of electron-positron pairs and the inverse-Compton spectrum from X-rays scattered by the pairs. Therefore, the conversion of cold neutron stars to strange stars are likely to produce a population of gamma-ray bursts, which may be simple bursts with single peaks and the duration timescales smaller than ls. Future work will make a statistical study of such a kind of gamma-ray burst.

5. Acknowledgements

This work was supported in part by the National Natural Science Foundation, the National Climbing Programme on Fundamental Researches and the Foundation of the Committee for education of China. Z.G. Dai would like to thank Prof. Fan Wang for his encouragement.

References

Alcock, C.: 1991, Nucl. Phys. B (Proc. Suppl.) B24, 93. Alcock, C., Farhi, E. and Olinto, A.: 1986, Astrophys. J. 310, 261. Baym, G., Bethe, H.A. and Pethick, J.C.: 1971a, Nucl. Phys. A175, 225. Baym, G., Pethick, J.C. and Sutherland, P.: 1971b, Astrophys. J. 170, 299. Baym, G., etal.: 1985, Phys. Lett. B160, 181. Colpi, M. and Miller, LC.: 1992, Astrophys. J. 388, 513. Dai, Z.G. and Lu, T.: 1995, Phys. Rev. D, submitted. Dai, Z.G., Peng, Q.H. and Lu, T.: 1995, Astrophys. J. 440, 815. Farhi, E. and Jaffe, R.L.: 1984, Phys. Rev. D30, 2379. Gentile, N.A., et al.: 1993, Astrophys. J. 414, 701. Glendenning, N.K. and Weber, E: 1992, Astrophys. J. 400, 647. Goyal, A., et aI.: 1994, Z. Phys. A349, 93. Haensel, P., Zdunik, J.L. and Schaeffer, R.: 1986, Astron. Astrophys. 160, 121. Heiselberg, H.: 1992, Phys. Scr. 46, 485. Kutschera, M. and Kotlorz, A.: 1993, Astrophys. J. 419, 752. Lattimer, J.M., et al.: 1991, Phys. Rev. Letters 66, 2701. Madsen, J.: 1992, Phys. Rev. D46, 3290. Madsen, J.: 1993, Phys. Rev. D47, 325. Olesen, M.L. and Madsen, J.: 1991, Nucl. Phys. B (Proc. Suppl.) B24, 170. Olinto, A.: 1987, Phys. Lett. B197, 71. Pizzochero, P.M.: 1991, Phys. Rev. Letters 66, 2425. Ruderman, M.A. and Sutherland, RG.: 1975, Astrophys. J. 196, 51. Sawyer, R.E: 1989, Phys. Lett. B233, 412. Shapiro, S.L. and Teukolsky, S.A.: 1983, Black holes, white dwarfs and neutron stars (Wiley-

Interscience, New York). Smith, I.A. and Epstein, R.I.: 1993, Astrophys. J. 410, 315. Tsuruta, S. and Cameron, A.: 1966, Can. J. Phys. 44, 1895. Waldauser, B., etal.: 1987, Phys. Rev. C36, 1019. Wang, Q.D. and Lu, T.: 1984, Phys. Lett. B148, 211. Witten, E.: 1984, Phys. Rev. D30, 272. Wu, X.J.: 1992, Ph.D. Thesis (Eberhard-Karls-Universitfit). Wu, X.J., et al.: 1991, Astron. Astrophys. 246, 411.