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Context. When a neutron star's rotation slows down, its internal density increases,
causing deviations from beta equilibrium that induce reactions, heating
the stellar interior. This mechanism, named rotochemical heating,
has previously been studied for non-superfluid neutron stars.
However, the likely presence of superfluid nucleons will
affect the thermal evolution of the star by suppressing the specific heat
and the usual neutrino-emitting reactions, while at the same time
opening new Cooper pairing reactions.
Aims. We describe the thermal effects of Cooper pairing
with spatially uniform and isotropic energy gaps of neutrons Δn
and protons Δp, on the rotochemical heating in millisecond pulsars (MSPs) when only
modified Urca reactions are allowed. In this way, we are able to determine the amplitude
of the superfluid energy gaps for the neutron and protons needed to
produce different thermal evolution of MSPs.
Methods. We integrate numerically, and analytically in some approximate cases,
the neutrino reactions for the modified Urca processes with superfluid nucleons
to include them in the numerical simulation of rotochemical heating.
Results. We find that the chemical imbalances in the star grow up to the
threshold value Δthr = min (Δn + 3Δp, 3Δn + Δp),
which is higher than the quasi-steady
state achieved in the absence of superfluidity. Therefore, the superfluid MSPs
will take longer to reach the quasi-steady state than their nonsuperfluid
counterparts, and they will have a higher a luminosity in this state, given
by Lγ∞,qs = (1–4) × 1032(Δthr/MeV)
(P˙−20/Pms3) erg s-1, where
P˙−20 is the period derivative in units
of 10-20 and Pms is the period in milliseconds.
We are able to explain the UV emission of the PSR J0437-4715 for
0.05 [MeV] ≲ Δthr ≲ 0.45 [MeV].
These results are valid if the energy gaps are uniform and isotropic
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