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6.2 Approximate formulae

For astrophysically relevant neutron star masses M > M ⊙, the gravitational mass of the crust Mcr = M − m (rcc), where rcc is the radial coordinate of the crust-core interface, constitutes less than 3% of M. Moreover, for realistic EoSs and M > M ⊙, the difference ΔR = R − rcc does not exceed 15% of R. Clearly, Mcr ∕M ≪ 1 and an approximation in which the terms 𝒪 (Mcr∕M ) are neglected is usually sufficiently precise. Neglecting the terms 𝒪 (ΔR ∕R) gives a less accurate but still useful approximation. In what follows we will use the above “light and thin crust approximation” to obtain useful approximate expressions for the crustal parameters.

Let us first derive an approximate equation for hydrostatic equilibrium within the crust. Let z be the proper depth below the neutron star surface, Equation (96View Equation). Within the crust, one can approximate z by

R − r z ≈ ∘---------- , (97 ) 1 − rg∕R
where rg = 2GM ∕c2 is the Schwarzschild radius of mass M.

In Equation (92View Equation) one can use the approximation m ≃ M, and neglect P∕ρc2 and 4πP r3∕M c2 as compared to one. Then the equation for hydrostatic equilibrium can be rewritten in a Newtonian form,

dP- dz = gsρ, (98 )
where gs is the surface gravity,
g = ---∘-GM-------. (99 ) s R2 1 − rg∕R
Using P(z = 0) = 0, we obtain a formula relating the pressure at depth z to the mass ΔM (z) of the crust layer above z,
R2∘ ----r-- ΔM (z) = 4π --- 1 − -g P(z) . (100 ) gs R
In particular, putting z = zcc, and denoting the pressure at the crust-core interface by Pcc, we get
4πR2Pcc ( rg) Mcr ≈ --------- 1 − -- . (101 ) GM R
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Figure 37: Mass of the crust for the SLy EoS [125Jump To The Next Citation Point]. The neutron star mass is M = 1.4M ⊙. For this EoS, spherical nuclei persist down to the crust-core interface. Left panel: fractional mass of the crust shell, ΔM ∕M, vs. its bottom density ρ. Right panel: proper depth below the star surface, z vs. mass density ρ.
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Figure 38: Mass of the crust for the FPS EoS [274Jump To The Next Citation Point]. The neutron star mass is M = 1.4 M ⊙. Notice the presence of the pasta layer, which are absent for the SLy EoS. The pasta phases occupy a thin density layer, but contain about 48% of the crust mass. Left panel: fractional mass of the crust shell, ΔM ∕M, vs. its bottom density ρ. Right panel: proper depth below the star surface, z vs. mass density ρ.
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Figure 39: Mass of the crust shell for the ground-state crust and for the accreted crust. The total stellar mass is M = 1.4M ⊙. Accreted crust: EoS of Haensel & Zdunik [185Jump To The Next Citation Point]. Ground-state crust: same compressible liquid drop model of atomic nuclei of Mackie & Baym [277], but full thermodynamic equilibrium (cold catalyzed matter). The black dots indicate the neutron drip transition. Left panel: fractional mass of the crust shell, ΔM ∕M, vs. its bottom density ρ. Right panel: ΔM ∕M versus depth below the star surface, z.

Consider a slow rigid rotation, meaning that the angular frequency of rotation, Ω, as measured by an observer at infinity, is small compared to the mass shedding frequency, Ωms. It can be shown that for slow rigid rotation the moment of inertia, I, then involves only the structure of the nonrotating neutron star [190Jump To The Next Citation Point] (see also [166Jump To The Next Citation Point184Jump To The Next Citation Point]). Using our notation,

8π ∫ R ( P ) ¯ω I = --- drr4 ρ + 2- --e−λ−Φ , (102 ) 3 0 c Ω
where ¯ω is the local rotation frequency, as measured in the local inertial frame. It can be calculated from an ordinary differential equation derived by Hartle [190], with boundary conditions,
2GJ ¯ω(R ) = Ω − --3-2 , (103 ) R c
where J is the stellar angular momentum.

The contribution of the crust comes from the spherical outer shell, rcc < r < R,

8π ∫ R ¯ω Icr = --- dr r4ρ--e−λ−Φ , (104 ) 3 rcc Ω
where P∕ ρc2 was neglected compared to one. Approximating the integrand by its value at r = R, and using Φ(R) −λ(R) 1∕2 e = e = (1 − rg∕R ) and Equations (103View Equation) and (101View Equation), we get
( ) Icr = 2McrR2 1 − rg --I--- . (105 ) 3 R M R2

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