We develop the formalism for computing the magnetic field within an axisymmetric neutron star with a strong Type II superconductor core surrounded by a normal conductor .
The formalism takes full account of the constraints imposed by hydrostatic equilibrium with a barotropic equation of state .
A characteristic of this problem is that the currents and fields need to be determined simultaneously and self-consistently .
Within the core , the strong Type II limit B \ll H allows us to compute the shapes of individual field lines .
We specialize to purely poloidal magnetic fields that are perpendicular to the equator , and develop the “ most dipolar case ” in which field lines are vertical at the outer radius of the core , which leads to a magnetic field at the stellar surface that is as close to a dipole as possible .
We demonstrate that although field lines from the core may only penetrate a short distance into the normal shell , boundary conditions at the inner radius of the normal shell control the field strength on the surface .
Remarkably , we find that for a Newtonian N = 1 polytrope , the surface dipole field strength is B _ { surf } \simeq H _ { b } \epsilon _ { b } / 3 where H _ { b } is the magnetic field strength at the outer boundary of the Type II core and \epsilon _ { b } R is the thickness of the normal shell .
For reasonable models , H _ { b } \approx 10 ^ { 14 } G and \epsilon _ { b } \approx 0.1 so the surface field strength is B _ { surf } \simeq 3 \times 10 ^ { 12 } G , comparable to the field strengths of many radiopulsars .
In general , H _ { b } and \epsilon _ { b } are both determined by the equation of state of nuclear matter and by the mass of the neutron star , but B _ { surf } \sim 10 ^ { 12 } G is probably a robust result for the “ most dipolar ” case .
We speculate on how the wide range of neutron star surface fields might arise in situations with less restrictions on the internal field configuration .
We show that quadrupolar distortions are \sim - 10 ^ { -9 } ( H _ { b } / 10 ^ { 14 } { G } ) ^ { 2 } and arise primarily in the normal shell for B \ll H _ { b } .