2. Strange stars at finite temperatures
The interesting point made in D98 is that starting from an empirical form for the density dependent masses of the up (u), down (d) and strange (s) q-s given below one can constrain the parametric form of this mass from recent astronomical data 1:
Restoration of chiral quark masses at high density is incorporated in this model. Using the model parameter () for this restoration one can calculate the density dependence of the strong coupling constant (Ray et al. 2000b). In other words the masses of stars in units of solar mass, (), found as a function of the star radius R, calculated using the above Eq. 1, - produces constraints which enable us to restrict the parameter . At high the q- mass falls from its constituent value to its current one . The parameter is taken to be 310 MeV to match up with constituent q- masses assuming the known fact that the hadrons have very little potential energy. The results are not very sensitive in so far as changing to 320 MeV changes the maximum mass of the star from 1.43735 to 1.43738 and the corresponding radius changes from 7.0553 kms to 7.0558 kms.
It is interesting to plot the up (u), down (d) and strange (s) q- masses at various radii in a star. This is done with strong coupling constant , chiral symmetry restoration parameter and the QCD scale parameter already discussed in D98 & Li et al. (1999a,b). Fig. 1 shows that the quarks do not have the constituent masses as in zero density hadrons nor do they have the current masses of the bag model. Upto a radius about 2 kms the quarks have their chiral current mass but in the major portion of the star their masses are substantially higher. At the surface the strange q- mass is about 278 MeV and the u, d q- masses MeV.
We use the large (colour) approximation of 't Hooft (1974) for quarks, where quark loops are suppressed by and the calculation can be performed at the tree level with a mean field derived from a qq interaction (Witten 1979). This was done for baryons (Dey et al. 1986; Dey et al. 1991; Ray et al. 2000a) and extended to dense systems like stars (D98). Following is the Hamiltonian, with a two-body potential :
The vector potential in Eq. 2 between quarks originate from gluon exchanges, and the -s are the color SU(3) matrices for the two interacting quarks. In the absence of an accurate evaluation of the potential (e.g. from large planar diagrams) we borrow it from meson phenomenology, namely the Richardson potential (Richardson 1979). The potential reproduces heavy as well as light meson spectra (Crater et al. 1984). It has been well tested for baryons in Fock calculations (Dey et al. 1986; Dey et al. 1991). Recently, using the Vlasov approach with the Richardson potential, Bonasera (1999) finds a transition from nuclear to quark matter at densities 5 times .
We had to calculate the potential energy (PE) contribution in two steps: the single particle potential, , for momentum k, is first calculated and this is subsequently integrated to get the PE. The is needed for doing finite T calculation.
The is parametrized as sum of exponentials in k (i.e., for a given flavour), where the parameters for a given set, reported in this paper, with , MeV, and a typical density are given in Table 1.
Table 1. Parameters of
Finite temperature can be incorporated in the system through the Fermi function:
Now we evaluate
For we get the number density and for , the energy density. is the spin-colour degeneracy.
Even at very high T which is around 70 MeV, the chemical potential is very large, of the order several hundred MeV. The entropy is calculated as follows:
The pressure (P) is calculated from the free energy
We find P = 0 points only upto T = 70 MeV on plotting P as a function of at different T (Fig. 2).
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001