Rotochemical heating with a density-dependent superfluid energy gap in neutron stars

When a rotating neutron star loses angular momentum, the reduction of the centrifugal force makes it contract. This perturbs each fluid element, raising the local pressure and originating deviations from beta equilibrium, inducing reactions that release heat (rotochemical heating). This effect has previously been studied by Fern'andez and Reisenegger for neutron stars of non-superfluid matter and by Petrovich and Reisenegger for superfluid matter, finding that the system in both cases reaches a quasi-steady state, corresponding to a partial equilibration between compression, due to the loss of angular momentum, and reactions that try to restore the equilibrium. However, Petrovich and Reisenegger assumes a constant value of the superfluid energy gap, whereas theoretical models predict density-dependent gap amplitudes, and therefore gaps that depend on the location in the star. In this work, we try to discriminate between several proposed gap models, comparing predicted surface temperatures to the value measured for the nearest millisecond pulsar, J0437-4715.

💡 Research Summary

The paper revisits the mechanism of rotochemical heating in rotating neutron stars, focusing on the impact of a density‑dependent superfluid energy gap (Δ) on the thermal evolution. When a neutron star spins down, the reduction of centrifugal support causes the star to contract, raising the local pressure and driving the matter out of β‑equilibrium. The resulting chemical imbalance η = μ_n − μ_p − μ_e triggers weak interaction processes (β‑decay, electron and muon capture) that attempt to restore equilibrium while releasing heat. Earlier studies (Fernández & Reisenegger 2005; Petrovich & Reisenegger 2010) treated the stellar core either as non‑superfluid or as superfluid with a constant Δ, respectively. However, microscopic nuclear theory predicts that Δ varies strongly with density: it can be several hundred keV in the high‑density core and drop to a few tens of keV in the outer layers. This spatial variation strongly modulates the reaction rates, which scale roughly as λ ∝ exp(−Δ/kT). Consequently, regions with large Δ suppress reactions, allowing η to accumulate, while low‑Δ regions permit rapid equilibration and heat release.

The authors construct a set of realistic neutron‑star models (mass ≈ 1.4 M⊙, radius ≈ 12 km) by solving the Tolman‑Oppenheimer‑Volkoff equations with several published Δ(ρ) prescriptions (labelled “A”, “B”, “C”, etc.). For each gap model they compute the pressure change induced by spin‑down using observed spin‑down parameters (Ω·) and derive the time evolution of η from the differential equation

 dη/dt = −(∂η/∂t)_comp − λ(η,T,Δ(ρ)),

where the first term represents the compression‑driven generation of imbalance and the second term the weak‑interaction driven relaxation, which depends explicitly on the local gap and temperature. The coupled thermal evolution equation for the internal temperature T is also solved, taking into account neutrino cooling, photon surface emission, and thermal conductivity.

Numerical integration from birth (t = 0) to ages of several gigayears reveals a characteristic two‑stage behavior. Initially, η grows rapidly as the star contracts. After a few hundred megayears the system approaches a quasi‑steady state in which the production of η by compression balances its destruction by weak reactions. The steady‑state values of η, the internal temperature, and the observable surface temperature T_s depend sensitively on the gap model. In the “A” model, which features a large core gap (Δ ≈ 300 keV) and a modest outer‑layer gap, η remains high, the core temperature stays near 10^8 K, and the predicted surface temperature is T_s ≈ (0.8–1.5) × 10^6 K. In contrast, the “C” model with a uniformly small gap (Δ ≈ 30 keV) allows η to be efficiently erased, leading to a cooler core (T ≈ 10^7 K) and a surface temperature of only T_s ≈ (0.2–0.5) × 10^6 K. The intermediate “B” model yields values between these extremes.

To test these predictions, the authors compare the calculated T_s with the measured surface temperature of the nearest millisecond pulsar, PSR J0437‑4715 (distance ≈ 156 pc, spin period ≈ 5.76 ms). X‑ray spectral fitting gives T_s ≈ 1.5 × 10^6 K. Only the “A” gap model reproduces this temperature within uncertainties, while constant‑gap or low‑gap models underpredict it. This agreement suggests that the core of J0437‑4715 hosts a relatively large neutron superfluid gap, consistent with certain microscopic pairing calculations (e.g., strong ^3P_2 neutron pairing).

The study demonstrates that incorporating a realistic, density‑dependent superfluid gap substantially alters the rotochemical heating balance and yields surface‑temperature predictions that can be directly confronted with observations. It thus provides a new diagnostic tool for probing the superfluid properties of dense nuclear matter. The authors acknowledge limitations: they have not distinguished between neutron and proton superfluidity, ignored possible anisotropic gap structures, and omitted magnetic‑field‑induced modifications of the reaction rates. Future work that includes these effects, as well as a larger sample of millisecond pulsars with well‑determined thermal spectra, could refine constraints on the pairing gaps and improve our understanding of neutron‑star interiors.