Explaining the thermal emission of old neutron stars with rotochemical heating and magnetized superconducting protons

TL;DR

This work investigates rotochemical heating in neutron stars with magnetized superconducting protons, introducing a magnetized fraction f that creates normal-proton regions and two chemical-thresholds for reactions. By simulating MSPs and CPs with a fixed proton gap Δp^∞ ≈ 1.5 MeV and varying the neutron gap Δn^∞ and f, the authors compare to HST ultraviolet observations of old NSs. They find that old NS thermal emission can be explained if Δp^∞ is large and Δn^∞ is small or vanishing, with MSPs requiring an extremely small f (weak internal fields) and CPs allowing a broader range of f depending on Δn^∞. The results imply cores with large proton pairing gaps and small neutron gaps, and suggest millisecond pulsars possess very weak internal magnetic fields; rotochemical heating alone can account for observed emission under these conditions, though vortex creep could relax some requirements.

Abstract

The detection of likely thermal ultraviolet emission from a few old neutron stars suggests that at least one internal heating mechanism is present in these stars. One proposed mechanism is rotochemical heating, in which the continuous contraction of the neutron star due to its spin-down produces chemical imbalances that induce Urca reactions, and the latter deposit heat in the neutron star core. If the protons in the star are superconducting, their energy gap suppresses the reactions, except in microscopic magnetized regions (such as quantized flux tubes) in which the protons act as if they were normal. Therefore, the strength of the internal magnetic field controls the rate at which reactions proceed and thus affects the thermal evolution of the neutron star. Here, we present the first comprehensive study of the effect of an internal magnetic field in the superconducting interior on rotochemical heating. We simulate the evolution of neutron stars for different internal magnetic field strengths and neutron energy gaps, comparing the results to Hubble Space Telescope observations of old neutron stars. All the observational data can be accounted for if the proton energy gap is large ($\sim 1.5\,\mathrm{MeV}$) and the neutron energy gap is small ($\lesssim 0.1\,\mathrm{MeV}$) or vanishing, while the millisecond pulsar PSR~J0437$-$4715 needs to have a very weak internal magnetic field. Our results suggest that neutron-star cores are characterized by a large proton pairing gap and a small or vanishing neutron gap, and that millisecond pulsars have very weak internal magnetic fields. Under these conditions, rotochemical heating alone can account for the observed thermal emission of old neutron stars.

Figures (6)

• Figure 1: Schematic representation of the net reaction rate $\Delta\tilde{\Gamma}_{np\ell}^\infty$ as a function of the chemical imbalance $\eta_\ell^\infty$, for four different cases: (a) Normal neutrons and protons; (b) $\Delta_{\rm thr}^{\rm in,\infty}<\eta_\ell^\infty\approx\Delta_{\rm thr}^{\rm out,\infty}$ and $f\lesssim f^*$; (c) $\Delta_{\rm thr}^{\rm in,\infty}\lesssim\eta_\ell^\infty\lesssim\Delta_{\rm thr}^{\rm out,\infty}$ and $f\approx f^*$; (d) $\Delta_{\rm thr}^{\rm in,\infty}\approx\eta_\ell^\infty<\Delta_{\rm thr}^{\rm out,\infty}$ and $f\gtrsim f^*$.
• Figure 2: Evolution of the redshifted surface temperature ($T_{\rm s}^\infty$) and electron ($\eta_e^\infty$) and muon ($\eta_\mu^\infty$) chemical imbalances for rotochemical heating due to spin-down with the inferred magnetic dipole moment of PSR J0437-4715, assuming only Murca reactions are allowed. The temperature measurement for this pulsar is given by the error bar, placed at the time when the spin parameters $P$ and $\dot P$ reach the measured values. The solid black lines correspond to normal neutrons and protons. The solid lines in cyan, blue, purple, brown, red, chocolate, and orange consider normal neutrons and superconducting protons with a uniform Cooper pairing gap, $\Delta_p=1.5$ MeV, core magnetic flux densities $B_{\rm int}=10^{13}, 10^{11}, 10^{9}, 10^7, 10^5, 10^3$, and 0 G, respectively, assuming a critical magnetic field $H_{\rm crit}=10^{15}$ G. For each colour, from top to bottom, there are three curves representing $\eta_\mu^\infty/k_B$, $\eta_e^\infty/k_B$, and $T_{\rm s}^\infty$, where $k_B$ is the Boltzmann constant. All curves were computed considering a NS with an initial rotation period $P_0=1$ ms, initial core temperature $T_0^\infty=10^9$ K, equation of state A18+$\delta v$+UIX$^*$Akmal1998, mass $M=1.44\,M_{\odot}$, and coordinate radius $R=11.45\,\mathrm{km}$.
• Figure 3: The same as Fig. \ref{['Bint']}, now for superconducting protons and superfluid neutrons with $\Delta_p=\Delta_n=1.5$ MeV, for different internal magnetic fields $B_{\rm int}=10^{12}$ G (cyan), $10^{8}$ G (purple) and $10^{4}$ G (blue), assuming $H_{\rm crit}=10^{15}\,\mathrm{G}$
• Figure 4: Evolution of the redshifted surface temperature ($T_{\rm s}^\infty$; lower curves) and electron chemical imbalance ($\eta_e^\infty$; upper curves) for rotochemical heating with normal neutrons and superconducting protons with a uniform proton Cooper pairing gap $\Delta_p=1.5$ MeV, assuming only Murca reactions are allowed. The spin-down evolution was calculated with the magnetic dipole moment inferred for PSR B0950+08, which is almost identical to that of PSR J0108-1431; the temperature measurement for the former and the upper limit for the latter are shown. The blue, green, cyan, purple, brown, and red curves consider core magnetic flux densities $B_{\rm int}=10^{15}, 10^{13}, 10^{11}, 10^9, 10^7$, and $10^5$ G, respectively, and a critical magnetic field $H_{\rm crit}=10^{15}$ G. (Strong $B_{\rm int}$ are shown with continuous lines, weak $B_{\rm int}$ with segmented lines, in order to emphasize the two different regimes discussed in the text.) For clarity, we do not show the curves for $\eta_\mu^\infty/k_B$, which are very similar to those for $\eta_e^\infty/k_B$. All curves were computed considering an initial rotation period $P_0=5$ ms, initial core temperature $T_0^\infty=10^{11}$ K, equation of state A18+$\delta v$+UIX$^*$Akmal1998, mass $M=1.44\,M_{\odot}$, and coordinate radius $R=11.45\,\mathrm{km}$.
• Figure 5: The same as Fig. \ref{['BintCL']}, but considering superfluid neutrons and superconducting protons. For reference, the black line corresponds to rotochemical heating with normal protons and normal neutrons. For all other curves, $\Delta_p=1.5$ MeV is used for the superconducting protons, while two different gaps are considered for the superfluid neutrons: $\Delta_n = 0.01$ MeV (segmented lines) and $\Delta_n=0.1$ MeV (solid lines). The purple lines represent a weak internal magnetic field regime with $B_{\rm int}=10^5$ G, and the green lines represent a strong internal magnetic field regime with $B_{\rm int}=10^{14}$ G, assuming $H_{\rm crit}=10^{15}\,\mathrm{G}$