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Figure 8.8. Dimensionless ratio I/MR2 versus M/R for several EOSs of SQM. The best-fit curve, Eq. (8.24), for strange stars (SS) is plotted by the upper thick line. For comparison, we show also the best-fit curve (NS) for neutron stars. From Bejger & Haensel (2002).

dependence of the fractions of all matter constituents in the interior of a 1.4 Mq bare strange star is displayed in Fig. 8.9.

8.12. The nonexistence of quark stars with heavy quarks

Apart from the light u, d quarks and the moderately massive s quark, the complete set of known quarks includes also heavy c, b, and t quarks. Their running masses are estimated as mcc2 ~ 1.2 GeV, mbc2 = 4 GeV, and mtc2 ~ 170 GeV (Yao et al.,2006).

A c-quark could be produced in the uds matter via the weak interaction process

Physical constraints on the initial and final quark energies (e and e') result from the degeneracy of the SQM matter. The initial states of d and u quarks should be occupied, ed < ¡id and eu < ¡i,u, while the final d quark state should be empty, e'd > ¡j.d. The energy needed to create a single c quark is e'c > mcc2. The necessary condition resulting from the above constraints is d + u —> c + d

Figure 8.9. Number densities of u, d, s quarks and electrons versus radial coordinate r for a model of a bare strange star of M = 1.4 Mq, calculated using the MIT Bag Model EOS with msc2 = 150 MeV, as = 0.17, B = 60 MeV fm~3.

Figure 8.9. Number densities of u, d, s quarks and electrons versus radial coordinate r for a model of a bare strange star of M = 1.4 Mq, calculated using the MIT Bag Model EOS with msc2 = 150 MeV, as = 0.17, B = 60 MeV fm~3.

It yields the lower limit on the number density of u quarks, nu > ^ (^f) = 22.8 fm-3 . (8.27)

At such a high density even s quarks are ultrarelativistic, so that

P > Pcrit,c = m f) 3 = L1 x 1016 g cm-3 . (8.28)

This critical density is much higher than the maximum central density of strange stars, pc,max (§ 8.11). A detailed analysis of quark star models with central densities pc higher than pc,max was performed by Kettner et al. (1995). Let us consider the M(pc) curve in Fig. 8.10. For pc > pc,max, the stellar mass decreases and reaches a minimum at pc ~ 1017 g cm-3. In the pc > pc,crit part of the S1C1 segment of the curve c quarks are present in stars, but stellar configurations are unstable with respect to the fundamental mode of radial pulsations because dM/ dpc < 0. As we know from § 6.5.2 at the minimum at C1 the stability of one of the radial modes has to change. The character of this change is determined by the static stability criterion in the M — R plane (§ 6.5.1). The condition dM/ dp c > 0 is only necessary for stability. As we

15 16 17 18 log Pc [g cm"3]

Figure 8.10. Left: Gravitational mass versus central density for stars built of SQM. The solid branch of the line ending at Si corresponds to equilibrium configurations of non-rotating strange stars, containing only u, d, and s quarks. The dotted segment SiCi refers to equilibrium configurations unstable with respect to the fundamental mode of radial pulsations. The solid segment CiC2 exhibits equilibrium configurations of charmed stars containing u, d, s, and c quarks. They are unstable against first two modes of radial pulsations. The dotted segment to the right of C2 describes equilibrium configurations of charmed stars which suffer additional instability against the next mode of radial pulsations. Right: The same configurations in the M - R plane.

see in the right panel of Fig. 8.10, the radius increases with increasing pc on the C1Ci segment in the vicinity of C1. Therefore, according to the static stability criterion, passing through Ci leads to the instability of the first overtone of radial pulsations. Charmed stars are unstable with respect to the two lowest modes of radial pulsations and cannot exist in the Universe. This can be confirmed by the calculation of the spectrum of radial pulsations which shows that < 0 and wi < 0 in the C1C2-branch (Kettner et al., 1995). In the dotted segment to the right of C2 in the left panel of Fig. 8.10 the second overtone of radial pulsations becomes unstable. Superdense branches of quark stars containing b and t quarks are susceptible to more unstable radial modes.

8.13. Scaling properties

As we have shown in § 8.8, the approximate and reasonably precise linear representation of the EOS of the SQM is determined by two parameters, a and ps. Using this representation, one can rewrite the equations of hydrostatic equilibrium of strange stars in a dimensionless form, derived in Appendix E.

Consider first two EOSs with the same a but with different values of the second parameter, ps and p's. The equilibrium configurations for these two EOSs form two different families, parameterized by the central density. There fore, we can construct curves M(R), I(M), ..., parameterized by pc. As we show in Appendix E, points of a curve obtained for a given ps transform into points of a curve obtained for pS by a scaling transformation. For example, the transformation of M (R) curves reads R — R' = (ps/pS)1/2R, M — M' = (ps/pS)1/2 M. An extremum of the unprimed curve transforms into an extremum of the primed curve. Particularly interesting is the scaling of parameters of the maximum-mass configuration,

pC,max/pc,max = p's/ps , I'(M'^)/I(Mmax) = (psM)^ ■ (8.29b)

According to the scaling of Mmax and RMmax, the maximum surface redshift (reached at M = Mmax) does not depend on ps. The maximum value of the moment of inertia, Imax, is reached at a mass slightly lower than Mmax, but of course it scales via the same factor (ps/p'sj3/2 as I(Mmax). At a fixed a, the ratio Imax/(MmaxRMmax) does not depend on ps and is just a number. This number depends weakly on a (Appendix E) and is slightly lower than 1 for realistic EOSs of the SQM (Bejger & Haensel 2002 got the value 0.97).

There exists a maximum circumferential radius Rmax of bare strange stars (Fig. 8.5). Its value scales according to the same relation as for RMmax.

Scaling properties are particularly simple (and actually exact) for the MIT Bag Model EOS with non-interacting massless quarks (a = 1/3, ps = 4B/c2, model SQM0 of § 8.5). The scaling formulae for Mmax, RMmax and pc,max were derived by Witten (1984), while those for Mb,max and Imax were obtained by Haensel et al. (1986a),

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