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6 Crust in Global Neutron Star Structure

The equilibrium structure of the crust of a neutron star results from the balance of pressure, electromagnetic and elastic stresses, and gravitational pull exerted by the whole star. The global structure of the crust can be calculated by solving Einstein’s equations

1- 8-πG- R μν − 2Rg μν = c4 Tμν , (83 )
where G is the gravitational constant and c the speed of light. The Ricci scalar Rμν and the scalar curvature R are determined from the spacetime metric gμν, which represents gravity. Pressure, electromagnetic and elastic stresses are taken into account in the energy-momentum tensor T μν.

As a first approximation, treating the star as an ideal fluid, the energy-momentum tensor is given by (see, e.g., Landau & Lifshitz [248Jump To The Next Citation Point])

μ(liq) 2 μ μ T ν = (ρc + P )u uν + P δν , (84 )
where ρ is the mass-energy density and u μ is the 4-velocity of the fluid. Equation (84View Equation) can be written in an equivalent but more general form [70Jump To The Next Citation Point]
μ(liq) μ μ T ν = n πν + Pδν , (85 )
where nμ = nu μ is the total 4-current, n is the total particle number density in the fluid rest frame and π ν is the momentum per particle of the fluid given by
πν = μu ν . (86 )
The quantity μ is a dynamic effective mass defined by the relation
2 P = (n μ − ρ)c , (87 )
where c is the speed of light. As shown by Carter & Langlois [80Jump To The Next Citation Point], Equation (85View Equation) can be easily transposed to fluid mixtures (in order to account for superfluidity inside the star) as follows
μ (mix) ∑ μ X μ T ν = n Xπν + Ψ δν , (88 )
where X labels matter constituents and Ψ is a generalized pressure, which is not simply given by the sum of the partial pressures of the various constituents (see Section 10).

The electromagnetic field can be taken into account by including the following contribution to the stress-energy tensor

μ(em) -1- μρ μ 1-- σρ T ν = 4π F F νρ − δν 8πF Fσρ , (89 )
where Fμν is the electromagnetic 2-form. Likewise elastic strains in the solid crust contribute through an additional term μ(elast) T ν. While the components of μ(elast) T ν are small compared to P (remember that the shear modulus is ∼ 10−2 of P, Section 7), it can produce nonaxial deformations in rotating neutron stars, and nonsphericity in nonrotating stars, as discussed in Section 12.5. For the time being, we will use the ideal-fluid approximation, μ μ (liq) μ(em) T ν ≈ T ν + T ν, and consider the effect of μ(elast) T ν on neutron star structure in Section 12.5. We will, therefore, consider purely-hydrostatic equilibrium of the crust, instead of a more general hydro-elastic equilibrium.

 6.1 Spherical nonrotating neutron stars
 6.2 Approximate formulae
 6.3 Crust in rotating neutron stars
 6.4 Effects of magnetic fields on the crust structure

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