Vortex creep heating in neutron star cooling with direct Urca processes in heavy neutron stars

TL;DR

This paper addresses the question of how vortex creep heating (VCH) interacts with rapid direct Urca (DUrca) cooling in massive neutron stars, potentially sustaining warm old objects. The authors implement VCH in a cooling code, derive the domain where the steady-state heating law $L_{ ext{h}} = J|\dot{\Omega}_{\infty}|$ is valid, and introduce a quantum-creep diagnostic $f_{ ext{Q}}(t)$ plus a 3D representation that includes magnetic field $B$ as an axis to break degeneracies seen in 2D projections. They show that DUrca+VCH can keep surface temperatures above $T_s^{\infty} \gtrsim 10^5$ K for $B \gtrsim 10^{11-12}$ G up to $P_0 \sim 10^2$ ms, with the transition timing depending more on $B$ and $P_0$ than mass. The results offer a physical framework to reinterpret warm old neutron stars as potentially more massive objects and emphasize treating $B$ as a co-equal axis in cooling analyses, supported by a practical $(B, P_0)$ validity map and 3D visualization.

Abstract

Old, thermally bright neutron stars imply internal heating at late times. Among candidate mechanisms, vortex creep heating (VCH) provides a robust link between spin-down and frictional dissipation in the pinned inner-crust superfluid, yet its interplay with fast DUrca cooling in massive stars remains insufficiently explored. We (i) implement VCH in our cooling code and validate it; (ii) identify the physically consistent domain where the steady-state form $L_{\text{h}}=J|\dotΩ_\infty|$ applies; (iii) quantify how $(B,P_0)$ regulate observable VCH signatures under DUrca cooling; and (iv) introduce a 3D representation that resolves degeneracies hidden in standard 2D projections. Cooling is computed with BSk24 and APR EoS, standard pairing gaps, and iron/carbon envelopes. VCH is modeled with $J\simeq10^{42.9\text{--}43.8}$ erg s, and a quantum-creep coverage fraction $f_{\text{Q}}(t)$ diagnoses when steady-state heating is valid. We survey $B=10^{10\text{--}13}$ G and $P_0=10$--$570$ ms for $1.4$ and $2.0\,M_\odot$, and compare with a curated set of ordinary pulsars with measured $(P,\dot P)$. Results: (1) Our implementation reproduces published VCH bands. (2) The $(B,P_0)$ validity boundary follows magnetic-dipole spin-down, confirming consistency with $|\dotΩ|$. (3) DUrca+VCH maintains $T_{\text{s}}^\infty\gtrsim10^5$ K for $B\gtrsim10^{11-12}$ G up to $P_0\sim10^2$ ms. (4) The 3D representation shows that sources appearing coincident in $(t,T_{\text{s}}^\infty)$ occupy distinct $B$-layers, removing degeneracies. VCH can substantially reshape late-time thermal states when spin-down power remains high; its observability depends chiefly on $(B,P_0)$ rather than on mass alone. We provide a practical $(B,P_0)$ validity map for $L_{\text{h}}=J|\dotΩ_\infty|$ and advocate treating $B$ as a co-equal axis in cooling analyses. (Shortened due to the arXiv words limit.)

Figures (14)

• Figure 1: Schematic illustration of the two-component model. (a) Meridional cross section and (b) three-dimensional view. The crust component consists of the solid outer crust, and the outer core with neutron ${}^3\mathrm{P}_2$ superfluid, which is expected to coexist with the proton superconductor and remain tightly coupled to the crustal motion Sauls_1982_core_is_crust_component_1Alpar_1984_core_is_crust_component_2. Quantized vortex lines (red) are aligned with the rotation axis and extend throughout the pinned superfluid region. Their outward creep in response to the external spin-down torque dissipates rotational energy, providing the physical basis for vortex creep heating.
• Figure 2: Schematic depiction of the forces acting on a pinned vortex line in the inner crust. A lag $\delta\Omega = \Omega_{\mathrm{s}} - \Omega_{\mathrm{c}}$ between the superfluid and the crustal rotation generates a relative azimuthal flow $\delta\mathbf{v}$ (blue), which produces an outward Magnus force $\mathbf{f}_{\rm Mag}$ (green) acting perpendicular to both $\delta\mathbf{v}$ and the vorticity vector $\boldsymbol{\kappa}$ (red). The balance between $\mathbf{f}_{\rm Mag}$ and the pinning force $f_{\rm pin}$ defines the critical lag $\delta\Omega_{\rm cr}$ that triggers vortex unpinning.
• Figure 3: Comparison between thermally activated (left) and quantum-tunneling (right) vortex creep. In the thermally activated regime, the creep rate $\mathcal{R}_{\rm VC}$ grows exponentially with increasing temperature, enabling a positive feedback loop between heating and thermal evolution that may trigger thermo-rotational instabilities. In contrast, quantum tunneling yields a temperature-independent $\mathcal{R}_{\rm VC}$ and stabilizes the system once the superfluid has cooled sufficiently below $T_{\mathrm{Q}}$.
• Figure 4: (a) Ordinary pulsars and (b) millisecond pulsars. Surface temperature evolution with (blue bands) and without (black dashed) vortex creep heating (VCH). The blue bands correspond to $J \simeq 10^{42.9\text{--}43.8}\,$erg s. Observational data from Fig. 5 of Ref. Fujiwara_2024 are shown with the same color scheme. Our results successfully reproduce the VCH cooling curves presented in Fig. 6 of Ref. Fujiwara_2024.
• Figure 5: Effect of including the fraction $\chi_{\mathrm{sf}}$ in the heating luminosity. (a) Red dashed (with $\chi_{\mathrm{sf}}$) and blue solid (without $\chi_{\mathrm{sf}}$) curves show the surface temperature evolution $T_{\mathrm{s}}^\infty(t)$; (b) Time evolution of $\chi_{\mathrm{sf}}$, representing the fraction of the moment of inertia occupied by superfluid neutrons in the inner crust; (c) Relative temperature difference ${|T_{\mathrm{w/}\,\chi_{\mathrm{sf}}}-T_{\mathrm{w/o}\,\chi_{\mathrm{sf}}}|}/{T_{\mathrm{w/o}\,\chi_{\mathrm{sf}}}}$ between the two cases. The maximum discrepancy is only $\sim0.65\%$, confirming that $\chi_{\mathrm{sf}}\rightarrow1$ early enough to justify the simplified implementation $L_{\mathrm{h}}=J|\dot{\Omega}_\infty|$ in practical applications.