  A superconductor of paired protons is thought to form in the core of neutron
stars soon after their birth. Minimum energy conditions suggest magnetic flux
is expelled from the superconducting region due to the Meissner effect, such
that the neutron star core is largely devoid of magnetic fields for some
nuclear equation of state and proton pairing models. We show via neutron star
cooling simulations that the superconducting region expands faster than flux is
expected to be expelled because cooling timescales are much shorter than
timescales of magnetic field diffusion. Thus magnetic fields remain in the bulk
of the neutron star core for at least 10^6-10^7 yr. We estimate the size of
flux free regions at 10^7 yr to be <~ 100 m for a magnetic field of 10^11 G and
possibly smaller for stronger field strengths. For proton pairing models that
are narrow, magnetic flux may be completely expelled from a thin shell of
approximately the above size after 10^5 yr. This shell may insulate lower
conductivity outer layers, where magnetic fields can diffuse and decay faster,
from fields maintained in the highly conducting deep core.
