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

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.)