Rotochemical heating in millisecond pulsars: modified Urca reactions with uniform Cooper pairing gaps

Context: When a rotating neutron star loses angular momentum, the reduction in the centrifugal force makes it contract. This perturbs each fluid element, raising the local pressure and originating deviations from beta equilibrium that enhance the neutrino emissivity and produce thermal energy. This mechanism is named rotochemical heating and has previously been studied for neutron stars of nonsuperfluid matter, finding that they reach a quasi-steady configuration in which the rate at which the spin-down modifies the equilibrium concentrations is the same at which neutrino reactions restore the equilibrium. Aims: We describe the thermal effects of Cooper pairing with spatially uniform energy gaps of neutrons \Delta_n and protons \Delta_p on the rotochemical heating in millisecond pulsars (MSPs) when only modified Urca reactions are allowed. By this, we may determine the amplitude of the superfluid energy gaps for the neutron and protons needed to produce different thermal evolution of MSPs. Results: We find that the chemical imbalances in the star grow up to the threshold value \Delta_{thr}= min(\Delta_n+ 3\Delta_p, 3\Delta_n+\Delta_p), which is higher than the quasi-steady state achieved in 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 L_\gamma ~ (1-4) 10^{32}\Delta_{thr}/MeV \dot{P}{-20}/P{ms}^3 erg s^-1. We can explain the UV emission of the PSR J0437-4715 for 0.05 MeV<\Delta_{thr}<0.45 MeV.

💡 Research Summary

The paper investigates how Cooper‑pairing superfluidity influences rotochemical heating in millisecond pulsars (MSPs) when only modified Urca reactions are allowed. Rotochemical heating arises because spin‑down reduces the centrifugal support of a neutron star, causing a slow contraction that raises the local pressure. The pressure increase drives the matter out of beta equilibrium, generating a chemical potential imbalance μ_n − μ_p − μ_e. This imbalance fuels modified Urca reactions, which emit neutrinos and deposit heat, thereby raising the internal temperature.

In non‑superfluid models the imbalance is quickly reduced by the same reactions, and the star settles into a quasi‑steady state where the rate of imbalance creation (set by the spin‑down) equals the rate of its removal (set by neutrino emission). The authors extend this framework by introducing uniform superfluid energy gaps Δ_n for neutrons and Δ_p for protons. Superfluidity creates an energy threshold that suppresses modified Urca processes until the chemical imbalance exceeds a critical value. By analyzing the phase space of the reactions they derive the threshold

Δ_thr = min(Δ_n + 3Δ_p, 3Δ_n + Δ_p).

Only when the imbalance surpasses Δ_thr do the reactions turn on sharply. Consequently, in a superfluid MSP the imbalance can grow to a larger value than in the normal case, delaying the approach to the quasi‑steady state and producing a higher steady‑state luminosity.

The authors solve the coupled thermal‑evolution and spin‑down equations for a range of stellar models, using realistic equations of state and assuming uniform gaps. They explore Δ_n and Δ_p values from 0 to ~1 MeV and compute the resulting surface photon luminosity L_γ. The results are encapsulated in a simple scaling law

L_γ ≈ (1–4) × 10³² (Δ_thr/MeV) ( \dot{P}{‑20} / P{ms}³ ) erg s⁻¹,

where \dot{P}{‑20} is the period derivative in units of 10⁻²⁰ s s⁻¹ and P{ms} is the spin period in milliseconds. This expression shows that larger gaps (larger Δ_thr) and faster spin‑down (larger \dot{P}) both raise the photon output.

Applying the model to the well‑studied MSP PSR J0437‑4715, whose ultraviolet emission has been measured, the authors find that the observed flux can be reproduced if

0.05 MeV < Δ_thr < 0.45 MeV.

This range translates into specific constraints on the neutron and proton gaps, suggesting that at least one of the species must be moderately superfluid (tens to a few hundred keV). The result provides an independent astrophysical probe of superfluid pairing in dense nuclear matter, complementing laboratory nuclear experiments and theoretical many‑body calculations.

The paper also discusses its limitations. The assumption of spatially uniform gaps is a simplification; realistic gaps likely vary with density, which could modify the exact value of Δ_thr and the timing of the quasi‑steady state. Direct Urca processes are ignored, yet they could become operative in the highest‑density core regions and would dramatically increase neutrino cooling, potentially altering the thermal balance. Finally, the treatment of the crust and envelope thermal conductivity is simplified, which may affect the translation from core temperature to observable surface emission.

In summary, the study demonstrates that superfluidity can significantly modify rotochemical heating in MSPs, leading to longer heating timescales and higher steady‑state luminosities. By linking observed UV/X‑ray emission to the size of the superfluid gaps, the work opens a new avenue for constraining the microphysics of dense matter through pulsar thermal observations. Future extensions that incorporate density‑dependent gaps, direct Urca channels, and more detailed envelope models will be essential to refine these constraints and to fully exploit MSPs as laboratories for nuclear superfluidity.