Figure 6.9. Apparent radius of neutron stars versus gravitational mass for selected EOSs of dense matter (labeled as in Table 6.1). The dashed curve is for hybrid neutron stars with a mixed baryon-quark phase (EOS from Table 9.1 of Glendenning 2000). The dashed-and-dot curve is for an EOS with first-order phase transition to a pure kaon-condensed phase (Kubis, 2001). The dash-and-dot straight line is the minimum value R™ln = 7.66 (M/Mq) km.

(except for a tiny region close to Mmax). In contrast, R decreases in the same mass interval.

A strict lower bound on R^ (M) results from the definition of Rœ (Lat-timer & Prakash, 2001; Haensel, 2001). Specifically, Eq. (6.51) can be rewritten as R^/rg = x-R(1 — xGR)-1/2, which means that R^/rg is a function of one parameter, xGR = rg/R. This function diverges at xGR = 0 and xGR = 1, and has a single minimum at xGR = 2/3. Therefore, the minimum value of R^(M) is R™in(M) = 7.66 (M/M©) km. This value is only 0.6% smaller than the apparent radius 7.71 (M/M©) km for the "true" maximum compactness xGR = 0.7081 consistent with vs < c (§ 6.6.4).

While the subluminal upper bound on xGR at a given M is slightly larger than 2/3, actual maximum values xGR(Mmax) for various EOSs are lower than 2/3. However, if we restrict ourselves to medium stiff and stiff EOSs with

Mmax > 1.8 Mq, then xGR(Mmax) ~ 0.6, which is only by 0.07 lower than 2/3. In such cases R^(Mmax) is close to R™n(Mmax) (Fig. 6.9).

According to Lattimer & Prakash (2001) and Haensel (2001) one can expect R^ > 12 km for any baryonic EOS, independently of neutron star mass. Our Fig. 6.9 confirms this "practical lower bound" on R^.

6.7. Binding energy

A configuration with M = Mmax has maximum baryon number Ab (and baryon mass Mb), whereas a configuration with M = Mmin has minimum baryon number (and baryon mass) (Zeldovich, 1962). To prove this statement let us consider two infinitesimally close configurations built of cold catalyzed matter. Let the first one consist of Ab baryons, and the second consist of Ab + dAb baryons. The first configuration can be transformed into the second by adding dAb baryons on the stellar surface (and keeping the system in full equilibrium). The change of the total gravitational mass accompanying this transformation is given by the small-increment theorem which relates small increments dM and dAb (Zeldovich, 1962; Zeldovich & Novikov, 1971)

Therefore, the extremum condition dM/dpc = 0 implies dAb/dpc = 0 (and dMb/dp c — 0), and vice versa. This means that the extremum of M is reached simultaneously with the extremum of Ab (and Mb).

An example of the Mb(pc) dependence is displayed in Fig. 6.10. The configuration with M = Mmax corresponds also to the maximum mass defect, which is about 15%. As it turns out, the maximum relative mass defect depends rather weakly on the EOS of baryon matter. Instead of the mass defect, it is convenient to use its energy equivalent, the binding energy Ebind, defined in § 6.2. Figure 6.11 shows Ebind as a function of M for several EOSs. For M = 1.4 Mq, the binding energy relative to the dispersion of the star into the gas of 56Fe ions is about 2 x 1053 erg. The maximum binding energy increases with the growth of Mmax, and ranges from & 3 x 1053 erg for the softest to & 1054 erg for the stiffest EOS of Table 6.1.

The behavior of Ebind at M & Mmax deserves a comment. The spike at M = Mmax (Fig. 6.11) becomes nonsingular, if considered as a function of pc (rather than M). To show this, let us use the equation dEbind/dAb = io {1 — V1 — Xgr } > 0 , (6.53)

where fi0 = m0 c2. It is valid for configurations built of cold catalyzed matter, as a consequence of the relation (6.52). Therefore, dEbind/dpc = (dEbind/dAb) (dAb/dpc) is continuous at pc & pc,max and vanishes at pc = pc,max. We have dEWnd/dpc > 0 for pc < Pc,max and dEWnd/dpc < 0 for pc > pc .max-

Was this article helpful?

0 0

Post a comment