Equation (6.7) is called the Tolman-Oppenheimer-Volkoff equation of hydrostatic equilibrium (Tolman 1939; Oppenheimer & Volkoff 1939; also see § 1.2). Equation (6.8) describes mass balance; its apparently Newtonian form is illusive because the proper volume of a spherical shell, given by Eq. (6.4), is not simply 4nr2 dr. Finally, Eq. (6.9) is a relativistic equation for the metric function $(r). These equations should be supplemented by an EOS, P = P(p). The above equations constitute a closed system of equations to be solved for obtaining P(r), p(r), m(r), and $(r). Actually, Eqs. (6.7) and (6.8) do not contain $(r) and can be solved separately to determine P(r), p(r) and m(r). The function $(r) can be found then from Eq. (6.9).
In the stellar layers where P ^ pc2 and Pr3 ^ mc2, the Tolman-Oppenheimer-Volkoff equation can be rewritten in the quasi-Newtonian form:
where g can be called the local gravitational acceleration and dl is given by Eq. (6.4). This form will be used in § 6.9 for studying the structure of the outer neutron star envelope.
In the stellar interior, P > 0 and dP/dr < 0. The stellar radius is determined from the condition P (R) = 0. Outside the star (for r > R), we have P = 0 and p = 0. Then Eq. (6.8) gives m(r > R) = M = const. The latter quantity is called the total gravitational mass of the star; the total energy content of the star is E = Mc2. Combining Eqs. (6.7) and (6.9) in the limit of P —> 0, we obtain that e2^ = 1 — rg/r outside the star. Therefore, for r > R, Eqs. (6.7)-(6.9) yield the well-known Schwarzschild metric, ds2 = c2 dt2 (l — ^ — (l — y)dr2 — r2 (d92 + sin2 9d^2), (6.11)
which describes a curved space-time around any massive, spherically symmetric body (not necessarily static). Thus, the gravitational redshift of signals emitted from the neutron star surface (r = R) is
= (1 — rg/R)1/2 ^(R), w = (1 — rg/R)"1/2 — 1 . (6.12)
At large distances from the star (r » rg) the Schwarzschild space-time (6.11) becomes asymptotically flat. Therefore, Schwarzschild time t is a proper time for a distant observer.
Finally, a non-relativistic star with P ^ pc2, Pr3 ^ mc2, and rg ^ R creates a weak space-time curvature. In this case Eqs. (6.7)-(6.9) reduce to the familiar Newtonian equations of stellar equilibrium, dP Gmp dm 2 d$ Gm a 2 ' ^ -4nr p, — = ^^. (6.13)
dr r2 dr dr c2r2
We see that in the non-relativistic limit $(r) c2 becomes the Newtonian gravitational potential.
6.2. Baryon number, mass and chemical potential. Binding energy of neutron stars
Let us consider an element of matter of a proper volume dV. The baryon number in this volume is nb dV, where nb is the baryon number density measured in a local reference frame. An isolated star has a fixed baryon number, which will be denoted by Ab. Although some theories of fundamental interactions predict the breaking of baryon number conservation, the timescales involved are much longer than the Universe age ~ 1010 years.1
In a static, spherically symmetric star the number of baryons confined within a sphere of radius r is r r ab(r) = 4W nb (r') eA(r'> r'2 dr' . (6.14)
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