DUrca Cooling & Vortex Creep Heating

• DUrca cooling is a rapid neutrino emission process in neutron stars that lowers core temperatures while necessitating complementary heating mechanisms.
• The integrated model leverages vortex creep heating—frictional dissipation from moving superfluid vortices—to explain persistent thermal luminosity in aged neutron stars.
• Quantitative studies show that key parameters like the lag reservoir coefficient (J), magnetic field, and birth spin critically determine the steady-state thermal balance.

Vortex creep heating (VCH) is a robust, microphysically motivated internal heat source in neutron stars, arising from frictional dissipation as quantized neutron superfluid vortices migrate through the pinned nuclear lattice in the inner crust during stellar spin-down. VCH explains the persistent thermal luminosity observed in aged, isolated neutron stars, notably those whose temperatures exceed predictions from minimal cooling models. The quantitative implementation and empirical validation of VCH have positioned it as a cornerstone mechanism in modern models of neutron-star thermal evolution (Fujiwara et al., 2023, Nam et al., 28 Oct 2025, Nam et al., 17 Nov 2025, Fujiwara et al., 2023, Gonzalez et al., 2010, Gonzalez et al., 2010).

1. Physical Basis: Superfluid Dynamics and Vortex Pinning

Neutron star inner crusts (densities $10^{11}$–$10^{14}$ g cm$^{-3}$) host a $^1S_0$ neutron superfluid threaded by an array of quantized vortex lines, each carrying circulation $\kappa = h/(2m_n)$. The areal density of vortices is $n_v = 2\Omega_s/\kappa$, with $\Omega_s$ denoting the superfluid angular velocity. The overlying crust and coupled charged fluids decelerate due to external electromagnetic spin-down torques, but pinned vortices impede the superfluid’s immediate response, causing a lag $\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$ between superfluid and crustal rotation.

This lag generates a Magnus force per unit length, $f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$, acting on each vortex. When $f_{\text{Mag}}$ equals the microscopic pinning force $10^{14}$0, vortices overcome the pinning barrier via thermal activation or quantum tunneling—“creep”—and move outward. Each vortex hop dissipates rotational energy as frictional work against pinning sites, producing local heating in the inner crust (Fujiwara et al., 2023, Nam et al., 17 Nov 2025).

2. Mathematical Formulation: Heating Luminosity and the J Parameter

The global heating power from vortex creep is encapsulated in the steady-state, two-component dynamical model:

$10^{14}$1

where $10^{14}$2 is the late-time spin-down rate and $10^{14}$3—Editor’s term: “lag reservoir coefficient”—is a nearly universal parameter for neutron stars with similar mass and structure. $10^{14}$4 integrates the critical rotational lag over the moment of inertia of the pinned inner-crust superfluid:

$10^{14}$5

At late times, the system settles into a steady regime where creep rate matches crustal deceleration, yielding a constant lag $10^{14}$6 determined by Magnus–pinning force equilibrium. Microscopically, $10^{14}$7 depends on the spatial profile of $10^{14}$8, superfluid density $10^{14}$9, and the crust geometry. Empirical and many-body theoretical calculations consistently constrain $^{-3}$0 to $^{-3}$1–$^{-3}$2 erg s for $^{-3}$3 neutron stars (Fujiwara et al., 2023, Nam et al., 17 Nov 2025, Fujiwara et al., 2023, Nam et al., 28 Oct 2025, Gonzalez et al., 2010).

3. Implementation in Thermal Evolution Equations

VCH enters the neutron-star global energy-balance equation as an internal heat source:

$^{-3}$4

Here, $^{-3}$5 is the total heat capacity, $^{-3}$6 is the neutrino luminosity (modified or direct Urca, depending on EoS and mass), $^{-3}$7 is the redshifted photon surface luminosity, and $^{-3}$8 the core temperature. In steady-state, old neutron stars with $^{-3}$9 yr achieve $^1S_0$0, so the observed surface temperature $^1S_0$1 is directly set by the VCH luminosity:

$^1S_0$2

Empirically, this relation is validated across diverse neutron star populations (Fujiwara et al., 2023, Nam et al., 28 Oct 2025).

4. Microphysical Inputs and Parameter Dependencies

The value of $^1S_0$3 and, consequently, the heating rate are regulated by:

• Pinning Force ($^1S_0$4): Derived from many-body calculations of vortex–nucleus interactions; sensitive to the nuclear EoS, pairing gaps, and lattice geometry. State-of-the-art quantum HFB and semiclassical approaches yield $^1S_0$5 in the empirically supported range, with uncertainties up to two orders of magnitude (Fujiwara et al., 2023, Fujiwara et al., 2023).
• Magnetic Field ($^1S_0$6) and Birth Spin ($^1S_0$7): The instantaneous spin-down $^1S_0$8 for magnetic dipole braking. Observability of VCH-induced thermal signatures requires $^1S_0$9 above a threshold, ensuring $\kappa = h/(2m_n)$0 at late times (Nam et al., 17 Nov 2025).
• Envelope Composition: Light-element envelopes can elevate $\kappa = h/(2m_n)$1 at a given $\kappa = h/(2m_n)$2, broadening the parameter space for observable VCH signatures (Nam et al., 28 Oct 2025, Nam et al., 17 Nov 2025).
• Mass and EoS: Crust thickness, moment of inertia, and DUrca threshold all modify the degree and longevity of VCH heating (Nam et al., 28 Oct 2025, Nam et al., 17 Nov 2025).

5. Domain of Validity: Steady-State and Quantum-Creep Regimes

Application of the canonical VCH law ($\kappa = h/(2m_n)$3) requires the quantum-creep regime, where thermal activation is suppressed and vortex motion is dominated by quantum tunneling. This is quantified by the quantum-creep coverage fraction $\kappa = h/(2m_n)$4, measuring the fraction of the pinned region at temperature $\kappa = h/(2m_n)$5 ($\kappa = h/(2m_n)$6 few $\kappa = h/(2m_n)$7 K). The prescription holds when $\kappa = h/(2m_n)$8.

The (B, $\kappa = h/(2m_n)$9) parameter space must ensure that VCH becomes relevant only after quantum creep dominates (i.e., $n_v = 2\Omega_s/\kappa$0), else the use of $n_v = 2\Omega_s/\kappa$1 is invalid (Nam et al., 17 Nov 2025). Results indicate that, for ordinary field strengths $n_v = 2\Omega_s/\kappa$2 G, $n_v = 2\Omega_s/\kappa$3–$n_v = 2\Omega_s/\kappa$4 ms are needed to guarantee steady-state applicability.

6. Observational Signatures and Comparative Heating Processes

VCH naturally explains the observed clustering of old pulsar surface temperatures $n_v = 2\Omega_s/\kappa$5 K for ages $n_v = 2\Omega_s/\kappa$6–$n_v = 2\Omega_s/\kappa$7 yr, including the well-measured PSR J0437–4715. Observed $n_v = 2\Omega_s/\kappa$8 values extracted from fits to surface temperatures and spin-down rates are tightly constrained, in excellent accord with microphysical estimates (Fujiwara et al., 2023, Fujiwara et al., 2023, Gonzalez et al., 2010). Competing mechanisms—rotochemical heating, crust cracking, magnetic field decay—are subdominant in this late-time, parameter-independent regime, except possibly in sources with unusual initial conditions or strong field decay (Gonzalez et al., 2010, Gonzalez et al., 2010, Nam et al., 28 Oct 2025).

A comparison with dark matter (DM) heating models demonstrates that realistic $n_v = 2\Omega_s/\kappa$9 values ensure VCH dominates over expected DM-induced luminosity for any plausible Galactic DM properties (Fujiwara et al., 2023).

7. Impact on Cooling Models and Future Applications

Integrating VCH into neutron-star cooling codes, particularly in conjunction with rapid DUrca neutrino emission in massive stars (M $\Omega_s$0 1.6$\Omega_s$1), reveals that VCH can maintain $\Omega_s$2 K out to $\Omega_s$3–$\Omega_s$4 yr—regimes otherwise unreachable by standard cooling alone. Modern approaches utilize 3D representations (log $\Omega_s$5, log $\Omega_s$6, log $\Omega_s$7) to resolve degeneracies in observed populations and to constrain combinations of $\Omega_s$8, $\Omega_s$9, and microphysical inputs (Nam et al., 17 Nov 2025, Nam et al., 28 Oct 2025). These studies emphasize the necessity of including $\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$0 and its interplay with $\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$1 in any quantitative model.

Table: Key Quantitative Ranges in VCH Models

Parameter	Typical Range	Reference
$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$2 (erg s)	$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$3–$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$4	(Fujiwara et al., 2023, Nam et al., 17 Nov 2025)
$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$5 (G)	$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$6–$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$7	(Nam et al., 17 Nov 2025)
$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$8 (ms)	$\delta\Omega(r,t)=\Omega_s(r,t)-\Omega_c(t)$9–$f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$0	(Nam et al., 17 Nov 2025)
$f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$1 (K, old NSs)	$f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$2	(Fujiwara et al., 2023, Gonzalez et al., 2010)
$f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$3 (MeV fm$f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$4)	$f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$5–$f_{\text{Mag}} = \rho_s \kappa r \delta\Omega$6	(Fujiwara et al., 2023)

These quantitative results and validations across different neutron star subpopulations cement VCH as a physically inevitable, empirically confirmed mechanism in the late-time evolution of neutron star temperatures (Fujiwara et al., 2023, Nam et al., 28 Oct 2025, Nam et al., 17 Nov 2025, Fujiwara et al., 2023).