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
where
is the gravitational constant and
the speed of light. The Ricci scalar
and the scalar
curvature
are determined from the spacetime metric
, which represents gravity. Pressure,
electromagnetic and elastic stresses are taken into account in the energy-momentum tensor
.
As a first approximation, treating the star as an ideal fluid, the energy-momentum tensor is given by
(see, e.g., Landau & Lifshitz [248
])
where
is the mass-energy density and
is the 4-velocity of the fluid. Equation (84) can be written
in an equivalent but more general form [70
]
where
is the total 4-current,
is the total particle number density in the fluid rest frame and
is the momentum per particle of the fluid given by
The quantity
is a dynamic effective mass defined by the relation
where
is the speed of light. As shown by Carter & Langlois [80
], Equation (85) can be easily
transposed to fluid mixtures (in order to account for superfluidity inside the star) as follows
where X labels matter constituents and
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
where
is the electromagnetic 2-form. Likewise elastic strains in the solid crust contribute through an
additional term
. While the components of
are small compared to
(remember that
the shear modulus is
of
, 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,
, and
consider the effect of
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.