Rotation of stars containing strange quark matter

Volume 243, number 3 PHYSICS LETTERS B 28 June 1990

Rotation of stars containing strange quark matter -A-

M a n j u Prakash , E. Ba ron a n d M a d a p p a P rakash Physics Department, State University of New York at Stony Brook, Stony Brook, NY 11794-3800, USA

Received 5 March 1990; revised manuscript received 6 April 1990

We calculate the maximum keplerian frequencies of stars containing strange matter that are in uniform rotation. We consider both self-bound stars and stars containing quark matter cores but normal surfaces. For both cases studied, use of perturbative QCD results for the equation of state of quark matter implies a limit on the maximum keplerian frequency of ~ 1 X 104 s- 1.

1. Introduction

Stars which have a mass in the range 1-2 Mo within a radius of about 10 km contain cores that are at several times the equi l ibr ium nuclear density, no = 0.16 fm -3. The study of the composit ion and the interaction laws of such dense matter enables us to test ex- pectations of perturbative QCD [ 1 ]. The rotational frequency of a pulsar left in the aftermath of a super- nova explosion is particularly useful in this regard. For uniform rotation including the effects of general relativity, the maximum rotational frequency of a star (i.e., approximately the keplerian frequency, coK) is proportional toM~ZxR ma3~/2 , where Mmax and Rmax are the mass and the radius of the max imum mass star in the non-rotating configuration, respectively [ 2-4 ]. If independent constraints on Mma x can be found, then OK provides an addit ional constraint ( involving Rmax) which may severely limit the properties of dense matter. The more massive object in the binary pulsar, PSR 1913 + 16, has a gravitational mass of 1.444 Mo [ 5 ]. thus, Mr~ax is constrained to be at least this value. We thus would have two constraints on Mma× and Rmax which in turn strongly constrain the equation of state (EOS) of dense matter [ 2-4 ]. The report of a 0.5 ms pulsar [6] in SN 1987A provided a strong impetus for examining such constraints. Even though the observation turned out to be erroneous [ 7 ] it led to the conclusion that it is very difficult for

Supported in part by the Department of Energy under Grant No. DE-FG02-88ER40388.

realistic EOSs to yield such rapid rotation [ 2-4 ]. There have been conflicting claims in the literature

as to whether stars containing strange matter self- bound or otherwise could exist at all [ 1,8-17 ]. In an attempt to clarify this situation we examine closely the mass-radius relation obtained for stars containing strangeness-rich matter using perturbative QCD results for the EOS of matter containing u, d and s quarks. We calculate the max imum keplerian frequency, at which mass is shed from the equator, for both self-bound stars and stars which have matter with strangeness to baryon ratio of the order of uni ty at their cores but normal surfaces.

2. Self-bound strange stars

Witten [12] has conjectured that strange matter may be the true ground state of matter. For this to be true the energy per particle would have to be less than the nucleon mass of 939 MeV at the baryon density, nb, where the pressure is equal to zero. If it were larger than this then strange matter would decay into nucleons on a strong interaction timescale. In the con- text of the bag model and considering gluon exchange corrections to first order in the coupling constant #~ ac (=g~2/4Jr), attributes of stars containing strange-

#~ In this work we use the convention ofFJ [ 13 ] for the value of ac =gZc/4n. The work of Freedman and McLerran [ 11 ], whose results we will use later, employs ac =g~/16n. Earlier bag model fits to hadron spectra have used the notation as which is half the o~c used in this work.

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 17 5


ness-rich mat ter were calculated in refs. [ 13-16 ]. We will begin our discussion employing the EOS of Fahri and Jaffe (FJ) [13].

First, consider the case of massless u, d and s quarks. For density independent C~c, the bag model relation between the pressure P and energy density is

P = ~ ( g - 4 B ) , (1)

170 ' I ' I ' I

'3° r t2o[ ' I , I , I

{. 2 . 4 0 .9

2 . 2 2

1.8

1.8 ate =

where B is a constant positive energy density. The existence of an energy ceiling of 939 MeV for zero pressure matter implies a m a x i m u m value for the bag constant

Bma x m4 ( 1 - ~ ) - - 1 0 8 7 r 2

=94.92 ( 1 _ 2~c~ MeV sr ,/ fm 3 '

(2a)

I ' I ' I ' I ' .

f ,,,oop-

0 , 0 0 .5 I .O 1.5 2 . 0

m,/B II~

1.4,0 , , I ' I ' I '

, I , I , J L 0 .0 0 .5 1.0 1.5 2 . 0

m, /B t/4

o r

( --maxRt/4 ---- 1 6 4 . 3 4 1 - M e V , (2b)

which decreases with increasing c~c. Inclusion of a finite mass for the s quark (details about the EOS and the renormalization scheme employed may be found in refs. [ 13-15 ] ) brings about a further reduc- tion in the allowed values of Bmax. Fig. 1 shows this decreasing trend of BJm/ax with ms corresponding to e / n b ( P = O ) of 939 MeV at zero pressure for c~c=0, 0.3, 0.6 and 0.9.

Since the To lman-Oppenhe imer -Vo lkov (TOV) equations of hydrostatic equilibrium [ 18,19 ] with the EOS in eq. ( 1 ) are invariant under the t ransforma- tions m--, m ' x ~ o o , r-~ r ' x / ffoo , ~ ~ ~' /17o and P - . P' / Bo, the max imum mass configurations scale with B as [121

-- --1/2

M m a x = 2 . 0 3 3 ( ~ ~ 0 ) M o ,

/ B .,% 1/2

R m a x = 1 1 . 0 9 t 7 ) k m , ( 3 )

where Bo=56 MeV fm -3 is a fiducial value B. The surface density of the star is given by

Fig. 1. The upper bound on the bag constant B~m/4,, the maximum mass, radius, and maximum keplerian frequency versus the strange quark mass for several values of the interaction strength ~xc. Results are for the O(ac) quark matter EOS of FJ.

( 4 B x 3 / 4 ( ~ ) i / 4

r i b ( P = 0 ) = \5~57~) \ 1 -

- \ 1 / 4

=0.7(B1/4) 3 ( I -- ~ ) (4)

while the energy per particle at zero pressure is given by

( 1087C2 B ~1/4

t~b ( P = 0 ) = \1-2e~c/TrJ

B1/4 =5.714 ( l _ 2 a j n ) , / 4 . (5)

Turning now to the max imum keplerian frequencies, we first note that general relativistic calculations assuming uniform rotation using several EOSs are re- produced to within 4% by the relation [2-4]

// . , , I / 2 , , -- , , - - 3 /2 • M m a x \ [ /~max \ s-,.

Here, the subscripts " m a x " denote the max imum mass non-rotating star of a given EOS, and Mmax is the gravitational mass. Combining the results in eqs.

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(3) and (6) the maximum keplerian frequency for the EOS in eq. ( 1 ) reads

B ~1/2 1 OK(B) = 9 . 4 × 103 \BoJ s- , (7)

with Bo=56 MeV f m - 3 (B~/4 ~- 144 MeV). This result implies that larger values of the bag constant lead to larger values of OK, and hence the maximum possible value of OK is obtained for the n m a x given in eq. ( 2 ). Setting ac = 0 in eq. (2) the limit ~2K = 1.20 × 104 s- ~ is obtained. This result for massless non-interact- ing quarks has been obtained earlier in refs. [3,4,20,21 ]. In fig. 1, we also show M . . . . R . . . . and OK, the maximum mass, radius and rotation velocity as a function ofms/B ~/4 for ac=0, 0.3, 0.6 and 0.9. For ms in the range 150-200 MeV required by fits [22] to data such as the kaon mass, OK is limited to ~l×104S -I"

We turn now to examine the claim [23 ] that self- bound strange quark stars can rotate with OK sub- stantially larger than 1 × l 0 4 S-1 . Ref. [23 ] makes use of the scaling of OK with B in eq. (7) which is valid only for massless quarks with ac = 0, abandoning at the same time the constraint in eq. (2) in that values of BW4> 170 MeV (B> 100 MeV fm -3) are used. The arguments adduced in support of ignoring the constraint stemming from energetic considerations are (i) for an opposite comparison of energies in the baryon and the quark phases, the energies must be accurately calculate, (ii) perturbative QCD cannot guarantee the required accuracy for the quark phase energy at zero pressure; and (iii) relevant lattice calculations are, at present, non-existent. Since the TOV equations require only the relation between P and but not their specific behaviour with rib, ref. [23 ] as- serts that the functional form in eq. ( 1 ), which is a result ofperturbative QCD, can nevertheless be used to describe strangeness-rich quark matter with no constraints on B.

This procedure suffers from the drawback that the constituents and the laws of interaction are not spec- ified and therefore we cannot attest to the fact that the star is indeed made up of strangeness-rich quark matter. In fact, many other schematic EOSs can be written down [3,4] which yield O K > I × 1 0 4 s -~. Consider, for example, the relation P=s(e-eo), where eo is a constant energy density and v/~ is the

speed of sound of matter independent of density [eq. ( 1 ) corresponds to s= ] and eo=4B]. Since P and e are linearly dependent, the scaling relations eq. (3) still apply, albeit with different coefficients [3,4 ]. Subject only to the constraint that Mmax be at least equal to 1.44 Mo, but abandoning any physical constraints on eo, the largest possible value of OK from eq. (6) are OK= 1.33, 1.77, 1.85, and 1.98× 104 s -~ fo rs= ~ 2 4 3, 3, ~ and 1, respectively. While all these EOSs give fast rotators, none can be considered realistic in that connections to identifiable physics have not been established. Furthermore, such EOSs imply that the surface density of the star is at much a higher density than no, with e/no > 939 MeV [ see eqs. (4) and (5) ]. For example, for s= ~ and eo=4× 111.6 MeV fm -3, nb(P=O)=2.85no and e/no_~978 MeV, if the bag model physics is retained. Such stars will be unstable with respect to decay into nucleons on a strong interaction time scale. Similar considerations apply for EOSs with s> ].

3. Stars with strangeness-rich quark matter cores

If the constraint that strange matter be the absolute ground state of matter is relaxed, then a phase transition from ordinary nucleonic matter to strangeness- rich quark matter can still occur at some density greater than no. For a given EOS on the quark side, the magnitude of nt depends sensitively on whether the hadronic EOS is soft or stiff at the relevant densities. For stiff hadronic EOSs nt is often larger than the central density of the maximum mass star, no Such stars cannot contain quark cores [8,9,17]. In most such cases, we have found that effects due to beta equilibrium were not considered in the hadronic phase. In general, effects of beta equilibrium tend to soften the EOS and lead to larger values of nt than pure neutron matter; however, a soft EOS also allows for matter at much higher densities to exist in the core. In contrast to the significant effects of beta equilibrium on the hadronic side, quark matter is not greatly affected by the presence of leptons, since there are very few of them in the quark phase.

In the nuclear phase and for densities below no, the dominant contribution to the energy is from leptons and nuclei, for the description of which we use the EOS of ref. [24] for nb<0.001 fm -3, and for

177


0.001 < 1% < 0.08 f m - 3, we use the results of ref. [ 25 ].

Above no we use the recent EOS of Wiringa, Fiks and Fabrocini [26 ] (WFF) whose calculations are based on a hamil tonian that fits n - n scattering data and the properties of deuteron and includes explicit forms of three-nucleon interaction to obtain a fit to nuclear matter equil ibrium properties. This work also includes effects of beta equilibrium in the nuclear phase. We consider two of the four EOSs calculated by WFF and refer to them as WFF1 (interaction terms A V 1 4 + U V I I ) and WFF2 (interaction terms U V 1 4 + U V I I ) .

For the EOS in the quark phase we first consider the perturbative QCD results of FJ calculated to O (a t ) . Choosing ac = 0.45, ms = 200 MeV and B = 60 MeVfm -3, one obtains e/nb(P=O)=970 MeV at nb ( P = 0 ) = 0.282 fm-3. This situation represents the

case of unbound strangeness-rich quark matter [ 14 ]. Our results are shown in table 1. Note that these EOSs permit strangeness-rich quark matter to exist in the cores of stars (since n ,<nc ) with Mmax> 1.44 Mo. The rotational frequencies are smaller than that of the self-bound star obtained using the EOS of FJ. This result is only to be expected as the radius is signifi- cantly influenced by the EOS near the surface and surfaces containing leptons and nuclei are at densities orders of magnitude below those occurring for self-bound stars.

Next, we consider the quark matter EOS calculated to O ( a 2) by Freedman and McLerran [ 11 ] (FM)

who compare their results to those of Bethe and Johnson [27 ] (model BJVH) for the hadronic phase.

Their findings for n, are shown in table 1, where we also present results of maximum mass configurations calculated by us for the two EOSs considered by FM. The values of B used are 56 MeV fm -3 (referred to as FM 1 ) and 0 (referred to as FM2 ), respectively. In both cases, ms= 280 MeV. With FM's choice of renormalization scheme, C~c decreases with density from ~0.6 for n b ~ 0 to ~0 .2 at rib=2 fm -3. Their EOSs also permit quark cores with strangeness to baryon ratio of the order of unity and result in Mmax> 1.44 Mo. The rotational frequencies fall below 1 × 104 s - i

again due to the presence of very low density matter in the surface.

We therefore conclude that perturbative QCD calculations of the EOS of unbound matter do permit strangeness-rich quark matter to exist within the cores of stars but their rotational velocities are limited to ~ 1 × 104 s - i. This conclusion seems to be valid for a

wide variety of nuclear EOSs as verified in refs. [2,3 ]. The EOSs of W F F used here do not include effects of strange baryons at supra-nuclear densities. The influ- ence of hyperons on the nuclear EOS has been recently investigated [ 28 ]. Uncertainties in the strength of interactions of hyperons among themselves and with nucleons lead to an uncertainty in the maximum allowed neutron star mass of nearly a factor of two. For Mma x > 1.44 Mo, one again finds OK ~< 1 X 104 s - ~ for stars with quark matter cores. Our emphasis here has been to investigate the dependence on the quark EOSs calculated to O(~xc) and O(~x 2) and the

effects of beta equil ibrium on the hadronic EOS. To account for pulsar glitches, Benvenuto et al.

Table 1 Maximum mass neutron stars with cores containing u, d and s quarks, n~ and n2 are the minimum and maximum densities across the phase transition and are quoted in units of no = 0.16 fm-3,/2 and P are the chemical potential and pressure at the transition density rt t. The central density, mass and radius of the non-rotating configuration are listed as no Mmax and R .... OK is the maximum keplerian frequency, eq. (6).

EOS reference FJ + WFF 1 FJ + WFF2 FM 1 FM2 BBC + WFF 1 BBC + WFF2

nl/no 5.32 4.37 - - 4.23 3.01 nt/no 5.86 4.61 2.13 1.75 4.55 3.29 n2/no 7.11 5.12 - - 5.09 3.63 ,u (MeV) 1494.3 1345.5 1031.3 1003.8 1235.1 1109 P (MeV fm -3) 340.5 195.74 18.45 9.86 141.35 53.93 nc/no 7.33 6.88 7.35 6.67 8.52 9.32 Mmax (Mo) 1.89 1.79 1.68 1.78 1.56 1.51 Rma x (km) 10.16 10.54 10.23 10.67 9.43 9.28 g2~ ( 104 s- ~ ) 1.03 0.95 0.97 0.93 1.05 1.06

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[ 29 ] have recently explored the structure of strange stars containing stable quark complexes (quark al- phas, Q~). They find stable configurations of a strange matter core surrounded by shells of solid and fluid Q~ matter with a thin envelope of normal matter for which P-~0 for nb~0. For the two 1.4 M e stars pre- sented, the values of R are 11.45 and 14.2 km, and the corresponding ~2K are 0.74 and 0.54 X 104 s - i. We do not expect Mmax to be much larger than 1.4 M o since the high density EOS is that of three-flavor quarks and rather soft.

responding central density, implying that it is possible for stars to contain strangeness-rich matter. The earlier conclusions of BBC hold, for the larger values o f c~c used in their calculations. As stated earlier, the effects of beta equilibrium for the baryonic phase are significant in that it leads to large values of the central densities. While Mmax> 1.44 M e the rotational frequencies are smaller than ~ 1 X 104 s - 1.

5. Conclusions

4. Reliability of perturbative QCD around nuclear densities

In ref. [ 17] (BBC) (see also ref. [30] ) it was ar- gued that perturbative QCD cannot be trusted at densities near no where matter must be confined, and, thus, any strange quarks would appear in the form of A's. To infer whether a transition to strangeness-rich quark matter can occur at higher densities, BBC calculate the energy of massless u, d and s quarks to O ( a c ) taking its density dependence as a c = 2.2 (no/n)1/3. For this choice the pressure turns out to be independent of ac once thermodynamic self- consistency is imposed. For several choices of the pure neutron matter EOS they find that the transition to quark matter occurs at densities larger than those found in neutron stars. The interaction strengths in this calculation are larger than considered by FJ who argue that values of C~c inferred for single baryon properties may not necessarily apply for calculating star structure.

As opposed to an upper bound on B implied by Witten's hypothesis, there now exists a lower bound on B since near no baryonic matter is the preferred phase. To meet this criterion, the values of c~¢ and B are correlated. The value ofc~c used by BBC is so large that only EOSs with an uncomfortably small value o f B can give rise to a transition to quark matter. Smaller values Ofac, however, permit values of B more in ac- cordance with those used in FJ. Table 1 contains our results using the EOS of BBC with c~=0.45 and B = 100 MeV fm -3 and the EOSs of WFF for baryonic matter which include the effects of beta equilibrium. In both cases considered, the transition to quark matter occurs at densities lower than the cor-

We find that the EOS calculated using perturbative QCD gives self-bound stars with Mmax > 1.44 M e but their rotational frequencies are limited to ~ 1 × 104 s- l . We also find that the existence of stars with strangeness-rich quark matter cores but normal surfaces cannot be ruled out at the present level of calculation. Allowing ac near no to be lower than given by bag model fits to hadron spectroscopy, we find that neutron stars can have sufficiently high densities due to effects of beta equilibrium such that strange matter becomes energetically favored. Observationally, it is difficult to distinguish normal neutron stars from those containing strange matter. Stars containing strange matter cool more rapidly than normal neutron stars due to enhanced neutrino emission from URCA-like processes; however, there exist other types of exotic matter, e.g. meson condensates, which also lead to enhanced cooling (cf. ref. [31 ] ). Thus pro- viding an unambiguous case for strange stars remains a challenging task.

While completing this work, we received a preprint by Zdunik and Haensel who reach conclusions similar to ours for the case of self-bound strange stars.

Acknowledgement

We thank Gerry Brown and James M. Lattimer for a careful reading of the paper.

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Rotation of stars containing strange quark matter