6.2 Approximate formulae
For astrophysically relevant neutron star masses , the gravitational mass of the crust
, where is the radial coordinate of the crust-core interface, constitutes less than
3% of . Moreover, for realistic EoSs and , the difference does not exceed
15% of . Clearly, and an approximation in which the terms are neglected is
usually sufficiently precise. Neglecting the terms 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 be the
proper depth below the neutron star surface, Equation (96). Within the crust, one can approximate by
where is the Schwarzschild radius of mass .
In Equation (92) one can use the approximation , and neglect and as
compared to one. Then the equation for hydrostatic equilibrium can be rewritten in a Newtonian form,
where is the surface gravity,
Using , we obtain a formula relating the pressure at depth to the mass of the
crust layer above ,
In particular, putting , and denoting the pressure at the crust-core interface by , we get
||Mass of the crust for the SLy EoS . The neutron star mass is . For
this EoS, spherical nuclei persist down to the crust-core interface. Left panel: fractional mass of the
crust shell, , vs. its bottom density . Right panel: proper depth below the star surface,
z vs. mass density .
||Mass of the crust for the FPS EoS . The neutron star mass is . 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,
, vs. its bottom density . Right panel: proper depth below the star surface, z vs. mass
||Mass of the crust shell for the ground-state crust and for the accreted crust. The
total stellar mass is . Accreted crust: EoS of Haensel & Zdunik . Ground-state
crust: same compressible liquid drop model of atomic nuclei of Mackie & Baym , but full
thermodynamic equilibrium (cold catalyzed matter). The black dots indicate the neutron drip
transition. Left panel: fractional mass of the crust shell, , vs. its bottom density . Right
panel: versus depth below the star surface, z.
Consider a slow rigid rotation, meaning that the angular frequency of rotation, , as
measured by an observer at infinity, is small compared to the mass shedding frequency, .
It can be shown that for slow rigid rotation the moment of inertia, , then involves only
the structure of the nonrotating neutron star  (see also [166, 184]). Using our notation,
where is the local rotation frequency, as measured in the local inertial frame. It can be
calculated from an ordinary differential equation derived by Hartle , with boundary conditions,
where is the stellar angular momentum.
The contribution of the crust comes from the spherical outer shell, ,
where was neglected compared to one. Approximating the integrand by its value at
, and using and Equations (103) and (101), we get