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6.1 Spherical nonrotating neutron stars

A static (nonrotating) neutron star, built of a perfect fluid, has spherical symmetry. The spacetime is also spherically symmetric, with the metric (see, for instance, Landau & Lifshitz [248Jump To The Next Citation Point]; here we use the notation of Haensel, Potekhin and Yakovlev [184Jump To The Next Citation Point])
ds2 = c2dt2e2Φ − e2λdr2 − r2(dθ2 + sin2 θdφ2 ), (90 )
where t is a time coordinate, r is a radial coordinate called “circumferential radius” (see below), and θ and φ are polar and azimuthal angular coordinates. The dimensionless metric function Φ = Φ(r) and λ = λ(r) have to be determined from Einstein’s equations. For a flat (Minkowski) spacetime we would have Φ = λ = 0. One can show that λ(r) is determined by the mass-energy contained within radius r, divided by 2 c, to be denoted by m (r) [248],
( )1 ∕2 −λ(r) 2Gm--(r) e = 1 − rc2 . (91 )

Let us limit ourselves to the static case t = const. Fixing r, θ = π∕2, and then integrating ds over φ from zero to 2π, we find that the proper length of the equator of the star, i.e., its circumference, as measured by a local observer, is equal to 2πr. This is why r is called the circumferential radius. Notice that Equation (90View Equation) implies that the infinitesimal proper radial distance (corresponding to the infinitesimal difference of radial coordinates dr) is given by λ dℓ = e dr.

From Einstein’s equations, we get the (relativistic) equations of hydrostatic equilibrium for a static spherically-symmetric star

( ) ( 3) ( ) −1 dP- = − Gρm-- 1 + P--- 1 + 4πP-r-- 1 − 2Gm-- , (92 ) dr r2 ρc2 mc2 c2r
dm--= 4πr2ρ , (93 ) dr
d Φ 1 dP ( P )− 1 --- = − --2--- 1 + ---2 . (94 ) dr ρc dr ρc
Equation (92View Equation) is the famous Tolman–Oppenheimer–Volkoff equation of hydrostatic equilibrium [407313]. Equation (93View Equation) enables one to calculate m (r) within a radius r. Finally, Equation (94View Equation) determines the metric function Φ (r ). Equations (92View Equation)–(94View Equation) have to be supplemented with an equation of state (EoS) P = P (ρ).

Let us consider the differential Equations (92View Equation) and (93View Equation), which determine the global structure of a neutron star. They are integrated from the star center, r = 0, with the boundary conditions ρ(0) = ρ c [P (0) = P(ρ ) c] and m (0) = 0. It is clear from Equation (92View Equation), that pressure is strictly decreasing with increasing r. The integration is continued until P = 0, which corresponds to the surface of the star, with radial coordinate r = R, usually called the star radius.

The gravitational mass of the star is defined by M = m (R ). The mass M is the source of the gravitational field outside the star (r > R), and creates an outer spacetime described by the Schwarzschild metric,

( ) ( ) 2 2 2 2GM--- 2 2GM--- − 1∕2 2 2 2 2 r > R : ds = c dt 1 − rc2 − dr 1 − rc2 − r (dθ + sin θdφ ) . (95 )

The crust corresponds to the layer rcc < r < R, where rcc determines the crust-core interface. The depth below the stellar surface, z, is defined as the proper radial distance between the star surface and a given surface of radius r. It is given by

∫ R λ z(r) = e dr . (96 ) r
The structure of the crust depends on its EoS, stellar mass, and the EoS of its liquid core. In Figures 37View Image and 38View Image we present the structure of the crust of a 1.4M ⊙ star, for two EoSs of the neutron star interior.

An accreted crust has a different composition and thus a different EoS (stiffer) than the ground-state crust (see Sections 4 and 5). For the comparison to be meaningful, however, these two EoSs have to be calculated from the same nuclear Hamiltonian. We satisfied this by using in both cases the same compressible liquid drop model of Mackie & Baym [277Jump To The Next Citation Point]. The plots of ΔM (z ) for the ground state and accreted crust of a 1.4 M ⊙ neutron star are shown in Figure 39View Image.

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