Chin. Astron. Astrophys. (1995)19/4,513-521 A translation of Acta Astron. Sin. (1995)36/2,165-172

Copyright @ 1995 Ekvier Science Ltd Printed in Great Britain. All rights resewed

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02751062(%)00@68-2

The effects of strong interaction on the observational discrimination between neutron

and strange stars t

DA1 Zi-gad2 LU Tan’ 1 Deparlmenl of Astronomy, Nanjing University, Nanjing 210008

2Department of Physics, Nanjing University, Nanjing 210008

Abstract Strange stars are compact objects similar to neutron stars composed

of strange matter. This paper investigates the observational effects of the strong

interaction between quarks. We believe: 1) that the conversion of a neutron

star to a strange star is a large “period glitch” which is determined by the

strong interaction; 2) that the strong interaction results in effective damping

of oscillation of hot strange stars, which could be a new mechanism of driving

supernova explosions; 3) that the strong interaction increases the difference in

rotation between strange and neutron stars under high temperatures, making

the minimum period for strange stars lower than that for neutron stars.

Key words: neutron stars-strong interaction-vibration-rotation

1. INTRODUCTION

Witten[‘l surmised that strange quark matter (briefly, strange matter) is more stable than

hadron matter. Strange matter consists of almost equal amounts of up (u), down (d) and

strange (s) quarks, and a small amount of electrons to maintain electrical neutrality. Com-

pact stars made of such matter are called strange stars. A detailed studyi has shown that,

within the uncertainties of strong interaction calculation, the existence of stable, strange

matter is reasonable. In astronomy, strange matter may be produced in two way&l: by

hadron-quark phase changes in the early universe and by conversion of neutron stars into

strange stars. The second way can be further divided into two paths: ordinary conversion

of neutron stars[3141 and conversion of the proto neutron star formed during the collapse of a

supernova core L51. The first may produce neutrino bursts and period glitch and the second

may drive the supernova explosion. In physics, strange matter may be produced in collisions

of heavy relativistic ion@], for which formal thermodynamic conditions have recently been

studied[q .

t Received 1994-0613; revised version 1994-0822

514 DA1 Zi-gao & LU Tan

Observational discrimination between strange and neutron stars appears difficult. Firstly,

at a mass of 1.4 MO, both have the same radius, hence there will be no difference in such ob-

servables as surface gravitational redshiftl ‘~‘1 Secondly, both have the same magnetosphere .

at the pulsar stage, hence will give the same radio pulsar propertieslsl. Thirdly, The neutrino

emission rate of strange matter is of the same order of magnitude as that of neutron stars

having the direct URCA processI’Ol, so that it is difficult to distinguish them from X-ray

observations. Since strange and neutron stars are different in composition and structure,

they may be distinguished by the phenomenon of period glitch; but up to now, it is not clear

whether the scenario of strange star can reasonably explain the glitch phenomenonI’ll.

Theoretical ways for distinguishing the two are being explored. These relate to vibration

and rotation of the star. Vibrational damping and instability of gravitational radiation

of strange stars are being investigated. The latter gives an upper limit to the rotational

velocity. Both are closely related to the bulk viscosity of strange matter while the bulk

viscosity is determined by the reaction rate of the following non-leptonic process:

~(1) +&2)-u(3) + s(4). (I)

WANG Qing-de and LU Tanl’2~‘31 were the first to point out that for strange stars and

neutrons stars that contain quark matter, process (1) is the most effect mechanism of vibra-

tional damping, because the reaction rate of this process is comparable to the vibrational

frequency of the star. On this basis, Sawyerl’4l and Madsenl151 went on to calculate the

bulk viscosity of strange matter and found it to be far greater than that of neutron matter.

Therefore, the vibrational damping of a strange star is faster than that of a neutron star,

and the maximum rotational velocity of a strange star is greater than that of a neutron star.

Whether when investigating the conversion of a neutron star into a strange star13-51,

or when calculating the bulk viscosity of strange matter 114~151, the rate of reaction (1) is

calculated. But calculations so far have always neglected the strong interaction between

the quarks. Since the strong interaction directly bears on the neutrino emission ratel’6-‘g1,

we can expect that it will markedly change the rate of (1) and hence the bulk viscosity of

strange matter. This is the motivation of the present paper. We shall also be discussing the

observational implications of the modifications.

2. THE REACTION RATE

The strong interaction between the quarks of strange matter can be described by the strong

coupling constant o,l21. Its value has quite a wide range. As in Ref. [20], we shall take

oe = 0.0-1.5. Under the condition that the temperature is far below the quark chemical

potential, the relation between the Fermi momentum and chemical potential is

PFj - spir i - u, d, (2)

[ 3m3 ln pLI + (rT - mf>I”]}} 113

l I%- (pt - mf>‘t2

,

PR

(3)

Strange Stars 515

where a E (1 - 2c~/?r)‘/~,pR = 313MeV, and m, is the mass of the s quark. Here we have

neglected the masses of the u and d quarks. We take units in which the Planck constant

h and the velocity of light c are equal to 1. These two expressions were deduced from the

quark thermodynamic potential given in Ref. [2].

We now derive the reaction rate of process (1). Using the Weinberg-Salaam theory[*l),

we obtain the positive reaction rate of (1) (p er unit time and unit volume) to be

j-+ i= 72 - G:sin2e,c0s2e, (2flY

~(fJd3+ -u2cose~2)(l -0bc0se3,)

x ss(E,+ E,- E3 - E+)6’3’(~, + ~2 - ~3 - P,), (4)

where

5 - n(E,)n(E,)[l - n(E,)][ 1 - n(E,)], (5)

n(Ei) - [ 1 + exp(~)]-’ , (6)

a%, b=-

aI4 ’ (7)

where Gp = 1.435 X 1O-4g erg cmB3 is the Fermi constant and fIc is the Cabbido angle

(co52 ec x 0.948). Because of conservation of particle momentum before and after the

reaction, the momenta of particles l-4 must form a quadrilateral. Let &,6& be the an-

gles between the momenta of particles 1 and 2, and 3 and 4, and Ei, pi, the energy and

momentum of particle i and T, the temperature of the matter. Expression (4) contains the

integral,

11 s [( fi dEi) $a( El + Er - E3 - E,), . i=i

When m, < pa, this becomes

I, _ J_ b4Apz +(2xU')21 6 l--c -Ap/kT '

where Ap = fid - ps. Expression (4) contains also the integral,

12 z I(fi dQi) (I - o’cose,,)(l - abcos8,,)

x ~‘3’(P, + P2 - P3 - P,>,

(9)

Because AT << pi, reaction (1) takes place on the separate Fermi surface of each quark and

so this becomes

516 DA1 Zi-gao & LU Tan

E

sin (PF,x) l cos (P&X) - ~-- 1 x sin ( pF,x) sin (pF,x)

+ ab sin (pF,x) sin (p~,x) sin (pF,x)

cos (PF,~) - ~- PF,x I

sir1 (PF,X)

PF,x I

(11)

(12)

This defines the function f(a,). From (9) and (12), we have

(13)

The inverse reaction rate of (1) is obtained by substituting -Ap for A,x in the above. Hence,

the net rate of process (1) is

r = 5 C:sin2B,c0s28,

78 AL0c9 r~rrr3W~3f)~d~~2 + 47r2(k7-)21. (14)

For (Y, = 0, (14) reduces to the result of Madsenlz21 and Heiselberglz31. Note, in our deriva-

tion of (14), we made two assumptions, kT < pi and m, << p,. Both these assumptions are

satisfied in strange stars or in the quark cores of neutron stars.

Because of the strong interaction between quarks, the net reaction rate of process (1)

falls, this is because the strong interaction in quark matter manifests itself like some kind

of repulsion, which reduces the probability of collision between the quarks. Let us define

attenuation factor

cr - o%‘J. (15)

For given P,, pd, pa, ma and % we can calculate Cr. Fig. 1 shows the calculated Cr as a

function of a,. We see that the larger the strong interaction, the greater is the decrease in

the reaction rate; and for a, > 0.8, the decrease is more than one order of magnitude. Fig. 2

shows the function f(o,) defined at (12). We see that, although f decreases with increasing

ocr the variation is not very strong.

3. BULK VISCOSITY COEFFICIENT

As the strange star vibrates, because the mass of the s quark is far larger than that of the d quark, the chemical potential of the s quark during the expansion and contracting phases is

Strange Stars 517

‘J, 0.5 I i.E ff c

Fig. 1 Fig. 2

Figs. 1 and 2 The attenuation factor Cr (Fig. 1) and the function f (Fig. 2) as functions of a=. cc, = pd = 260, ,us = 250 MeV. Solid, dotted and dashed lines correspond

to m, = 0,100,150 MeV, respectively

not equal to the chemical potential of the d quark, equilibrium of process (1) is destroyed, leading to dissipation of vibrational energy. WANG Qing-de and LU Tan112~‘3J investigated the evolution of the vibrational energy of the a neutron star with a quark core of 0.2Mo, and found the time scale of vibrational damping to be of the order of millisecond. Thus, the rate of (1) is comparable to the vibrational frequency. Too slow or too fast reactions will not effectively dissipate the vibrational energy. Sawyerli4J went on to express this mechanism by

the physical quantity of bulk viscosity and further assumed 2?rT B A\c1. Recently, Madsenl’5J extended Sawyer’s calcutation to 2rT < Ap. In order to give an analytical expression for the bulk viscosity, we shall continue with Sawyer’s assumption. Also, we shall assume that

Pp, EJ 44 - mt)“‘,

which is obviously valid when ma << ps.

Let the volume of unit mass be

(16)

v(t) - us + Avsin (UN), (17)

where ue is the equilibrium volume, Av is the perturbed amplitude and w = 2zr/r is the vibrational frequency. Usually, r = 10m3s is taken.

In unit mass of quark matter, the number of d quarks, nd, varies with time according to

d% - - - f Gtsin28,cos2t3,p~Ap( ZntT)'( a’f)vo, at (18)

where

518 DA1 Zi-gao & LU Tan

f * 1: $- [ sin’x + uz ( ~0s x - %)2]‘.

Correspondingly, the time variation of Ap is

1

dAP -s- 2~: - -cm: dn

at wcos (WZ) +

3P.4% --c-.

at

w-8

(20)

Let Ap = A sin wt + l? cos wt. Substituting in (18) and (20) then gives an analytic expression for Ap. Another substitution in (18) then gives an analytic expression for dmdldt.

Using Swayer’s definition, we found the bulk viscosity coefficient of strange matter to be

which further simplifies into

5- u(kT I 1 MeV)’

u? + ,9(kT/lMeV)”

where

a I 47r2 6G+ &2e cos2e -- 9 72 c d&1( a'f >

w 7.04 X 102’ 3 mf(df)g l cm-’ l se3, ( ) *o

(21)

(22)

(23)

sin28 cos2e ’ ’

> - mflrib’f I’( z)’

m 4.0 x IO’O 3 ( >

2 no (~6f2)J-*> (24)

where nb is the baryon number density of the strange matter, no(- 0.16fmm3) is the nuclear matter density and m,, pd are in units of MeV.

Analyze now the effect of the strong interaction on the bulk viscosity coefficient. Define a critical temperature,

- 2.9 x 10+-Ly”2(p)-1’2 (O”f’)-‘/+K.

T, increases with increasing a,. At a, = 1.5, Tc is about 5 times its value at a, = 0. According to (22)-( 24)) we find that when T < Te, the strong interaction has a marked effect on the bulk viscosity, the greater oc is, the smaller C will be. When T > T,, although C increases with increasing oe, because of the function f, the effect is not large. We shall see that these conclusions have important astrophysical implications.

Strange Stars 519

4. DISCUSSION

4.1 ‘Ihnsformation of Neutron star into Strange Star At present, there are two mechanisms for a neutron star to change into a strange star.

One is diffusion of strange matter. 01into[31 assumes that at the center of the neutron

star, a strange matter seed is generated (due to instantaneous density fluctuation etc.).

Because neutrons are not charged, they can freely diffuse into the seed of strange matter,

and becomes strange matter after deconfinement: this implies strange matter diffusing into

its surroundings. After a certain time, the neutron star becomes a strange star. The

speed of diffusion is proportional to the square root of the reaction rate of process (1). For

Q, = 1.0 and 1.5, the reaction rate will be down from its cy, = 0.0 value by a factor of 20

and lo*, respectively, and the diffusion speed therefore down by a factor of about 5 and

102, and the time scale of the neutron star changing into strange star is up by the same

factor. Because this transformation will lead to changes in the rotational inertia of the star,

the period will be changed, and the time scale of period changes is more easily observed.

Therefore, we may witness transformation of neutrons stars with longer time scales into

strange stars, somewhat in the form of giant glitches. Since the features of giant glitches

depend on the strong coupling constant a,, we can from observations of giant glitches infer

the strong interaction in quark matter. The second mechanism is transformation through

a two-flavor quark matter stage. Recently, we[*t5) assumed that in the interior of neutron

stars or in supernova cores deconfinement of neutron matter on a large scale takes place,

generating two-flavor quark matter and so investigated the phase transition from two-flavor

quark matter to three-flavor quark matter and found the time scale of the phase transition

was as short as lo-‘s. The effect of this phase transition should not be greatly altered even

when strong interaction is taken into consideration.

4.2 Vibrational Damping of Strange Stars

Assuming a uniform interior density, Sawyer estimated the time scale of vibrational

damping of a strange star to be

= l(2) L&)X Io’&ls-’ >-’ $3 (26)

where pnuc - 2.8x10’*g/cm3 is the mass density of nuclear matter, and R is the radius.

For the case of low temperatures (T << T,), the strong interaction reduces < and increases

TD. For the case of high temperatures (2’ >> T,), the effect of the strong interaction on rD

is not large, although the larger o, is, the smaller rg will be.

A supernova during its collapse may well produce a core of strange quark matter[5~241,

and this core usually vibrates[2J]. Will decay of the vibration be of any significance for the

supernova explosion ? As the vibration is damped, vibrational energy is converted into heat,

the temperature rises, raising the neutrino emission rate. For p - pnuc, T - lOMeV, R - 5x 105cm, m, - 100 MeV, we have zeta N 1025gcm-‘s-1 and TD - 0.25s. This implies

that, on time scales as short as this, the neutrino luminosity can increase markedly. And

a high neutrino luminosity is necessary for supernova explosion, because for the scenario

DA1 Zi-gao & LU Tan

of delayed supernova explosion 126,2rl (time scale of delay about 0.5s), many researchers are

searching for mechanisms that will increase neutrino luminosity12sl. The effect discussed

in this paper may possibly drive supernova explosion. If, however, as in the case of low

temperatures, the effect of the strong interaction in the case of high temperatures is again

a marked decrease in the bulk viscosity, then the above effect will not take place.

4.3 Maximum Rotational Velocity of Pulsars We know that for a rotating neutron star, gravitational radiation instability fixes an

upper limit to the angular velocity, and this limit is always less than the Kepler value. The

effect of viscosity is to damp down the gravitational radiation instability, making the angular

velocity closer to the Kepler value. The greater the viscosity, the closer is the upper limit

to the Kepler value12gl.

The viscosity coefficient has two measures, one is the bulk viscosity coefficient calculated

above, the other is the tangential viscosity coefficient. Usually, in young pulsars, bulk

viscosity is dominant1141. For the lower temperatures (T << T,), the strong interaction

reduces the bulk viscosity, hence reduces the maximum rotational angular velocity. And

this reduces the difference between strange stars and neutrons stars in respect of rotation.

When T >> Tcr the strong interaction increases slightly the maximum angular velocity, and

so accentuates the difference between the two.

In the above, we have assumed 2aT >> Ap and this implies the vibration of the strange

star is a small vibration. Let < be the maximum variation in the volume of the strange star

in the course of the vibration. For 2?rT, = A/J(&), we have

5, = 2.7 X 10-L (20~~ev-2 (g)-"'(y" (o"fYl',

therefore, the small vibration here is in fact vibration with c << CC.

5. CONCLUSIONS

It is very probable that there exists a core of quark matter in the interior of a neutron star.

It is also possible that some neutron stars are in fact composed of strange quark matter.

Such stars are called strange stars. Are there observational differences between neutron stars

and strange stars ? This is obviously a question of great current interest. In this paper,

we first investigated the effect of strong interaction on the reaction rate of the non-leptonic

processes and bulk viscosity in strange matter and we found: 1) that the strong interaction

markedly reduces the reaction rate; 2) that the strong interaction markedly reduces the bulk

viscosity when T << T, and slightly increases it when T >> T,. These results can lead to

certain observational effects.

First, in regard to transformation of neutron star into strange star, we should observe

giant glitch phenomenon on quite long time scales. At present, period glitches have been

observed in a small number of pulsars, but during the glitch stage, the relative variation of

period is far smaller than 10s2. If giant glitches ate seen in a pulsar, then it will mean that

this pulsar is a strange star, and the phenomenon will provide information on the strong

interaction in quark matter.

Strange Stars 521

Next, in regard to vibrational damping of strange stars, we hope to overcome the difficulty

of supernova explosion-the lack of a high neutrino luminosity. Vibrational damping of the

strange star generated during the supernova collapse can, within a short time scale, produce

a high neutrino luminosity.

Lastly, in regard to rotation of strange stars, the strong interaction increases the differ-

ence in the maximum rotational velocity between young strange stars and young neutron

stars. If we observe a pulsar with a very short period, then the pulsar may well be a strange

star, because the rotational velocity of a neutron star cannot exceed a certain limit.

111 121 [31 141

151 [‘31 [71

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ml P21 1131 I141

1151 [=I P71 k31 WI

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