Vortex Creep Heating in Neutron Stars

• Vortex creep heating is frictional dissipation in neutron stars where quantized superfluid vortices overcome pinning forces, converting spin energy into heat.
• The mechanism is modeled by integrating pinning strengths over the inner crust and correlating heating luminosity with spin-down rate via the universal parameter J.
• This heating process sustains higher surface temperatures in old neutron stars, providing a robust explanation for observed thermal emissions in pulsars.

Vortex creep heating (VCH) describes frictional dissipation arising from the outward motion of quantized neutron superfluid vortices across pinning sites in the inner crust of neutron stars (NSs). This mechanism, fundamentally linked to the spin-down of the star, converts rotational energy into heat and plays an essential role in maintaining surface temperatures of old neutron stars above predictions from standard cooling scenarios. VCH is governed by the properties of the superfluid, pinning strength, crust composition, and macroscopic parameters such as the magnetic field and spin period. Both theory and late-time pulsar temperature measurements consistently support a nearly universal relationship between heating luminosity and spin-down rate, parameterized by a coefficient $J$, which encapsulates microphysical and structural dependencies.

1. Physical Framework and Mechanism

In the inner crust ($\rho \sim 10^{11}$–$10^{14}$ g cm$^{-3}$), neutrons form a $^1S_0$ superfluid characterized by an array of quantized vortex lines, each carrying circulation $\kappa=h/(2m_n)$ (Fujiwara et al., 2023, Nam et al., 17 Nov 2025, Gonzalez et al., 2010). The global rotation of the superfluid is maintained by a vortex area density $n_v=2\Omega_s/\kappa$, with $\Omega_s$ denoting the angular velocity. As the neutron star undergoes electromagnetic spin-down, the rigid crust decelerates at a rate $\dot\Omega_c<0$ while the superfluid lags, as its vortices are pinned to the nuclear lattice.

The differential angular velocity $\delta\Omega=\Omega_s-\Omega_c$ builds up, generating a Magnus force $f_{\rm Mag}\sim\rho_s\kappa r\,\delta\Omega$ that acts transversely on vortices. When $f_{\rm Mag}$ exceeds the local pinning force $f_{\rm pin}$, vortices thermally activate or quantum tunnel ("creep") from one pinning site to the next, dissipating rotational energy into heat during each transition.

At late epochs ($t\gtrsim10^5$ yr), steady-state conditions prevail, wherein the superfluid and crust decelerate at the same rate $\dot\Omega_s=\dot\Omega_c\equiv\dot\Omega_\infty$, and the relative lag saturates at a critical value $\delta\Omega_{\rm cr}$ determined by the force balance $f_{\rm Mag}=f_{\rm pin}$ (Fujiwara et al., 2023, Nam et al., 17 Nov 2025).

2. Mathematical Formulation of Heating Luminosity

The VCH luminosity in the steady creep regime is given by the integral over the pinned region:

$L_{\rm heat} = J |\dot\Omega_\infty|$

where

$J = \int_{\rm pin} dI_p\,\delta\Omega_\infty$

and $dI_p=\rho r^2 dV$ is the moment of inertia element of the pinning region (Fujiwara et al., 2023, Nam et al., 17 Nov 2025). Microscopically,

$$\delta\Omega_{\rm cr} \simeq \frac{f_{\rm pin}}{\rho \kappa r}$$

leading to

$$J \simeq \int_{R_{\rm in}}^{R_{\rm out}} dR \int_0^\pi d\theta \int_0^{2\pi} d\phi\, R^3 \sin^2\theta\, \frac{f_{\rm pin}(R)}{\kappa}$$

$J$ thus depends on the pinning force $f_{\rm pin}$, local superfluid density $\rho(R)$, and the geometric distribution of pinning sites throughout the inner crust.

3. Universality and Parameter Determination

Despite variations in the nuclear equation of state, pairing models, and pinning microphysics, state-of-the-art mesoscopic calculations—averaging vortex-nucleus forces across realistic orientations—yield pinning strengths $f_{\rm pin}\sim10^{-7}$–$10^{-4}$ MeV fm$^{-2}$, spanning the inner crust (Fujiwara et al., 2023, Fujiwara et al., 2023). Integrating these over a crustal thickness $\Delta R\sim1$ km and density $\rho\sim10^{13}$–$10^{14}$ g cm$^{-3}$ yields predicted $J_{\rm theory}\sim10^{40}$–$10^{43}$ erg s.

Surface temperature observations of old pulsars, in which photon cooling dominates, allow extraction of $J$ via the balance

$J|\dot\Omega| = 4\pi R^2 \sigma_{\rm SB} T_s^4$

yielding $$J_{\rm obs} = (4\pi R^2 \sigma_{\rm SB} T_s^4)/|\dot\Omega|$$ (Fujiwara et al., 2023, Fujiwara et al., 2023). Empirically, for neutron stars with $R\sim 11$ km, $J_{\rm obs}\sim 10^{42.9}$–$10^{43.8}$ erg s is consistently found. This narrow range corroborates theoretical predictions and affirms VCH as the dominant internal heating channel in observed old neutron stars.

4. Thermal Evolution: Theory and Observational Signatures

VCH modifies classical cooling trajectories. In canonical neutron stars ($M=1.4\,M_\odot$), standard neutrino–photon cooling predicts $T_s < 10^4$ K after $t > 10^7$ yr (Gonzalez et al., 2010). Including VCH with $J \sim 10^{43}$ erg s maintains $T_s \sim 10^5$ K for millisecond pulsars (MSPs, $P \sim 1$–10 ms) out to $t \sim 10^9$–$10^{10}$ yr, matching observed UV temperatures of sources such as PSR J0437–4715. Classical pulsars ($B \sim 10^{11}$ G, $P \sim 0.1$–1 s) can sustain $T_s \sim 3 \times 10^4$–$10^5$ K over $t \sim 10^7$–$10^9$ yr (Gonzalez et al., 2010, Gonzalez et al., 2010, Nam et al., 28 Oct 2025). In massive stars where direct Urca (DUrca) cooling is active, VCH can partially offset the rapid cooling, as demonstrated numerically for $M=2.0\,M_\odot$ (Nam et al., 28 Oct 2025, Nam et al., 17 Nov 2025).

At late times, steady-state thermal balance requires photon emission to match VCH heating:

$$J|\dot\Omega_\infty| = 4\pi R^2 \sigma_{\rm SB} T_s^4$$

directly linking the observed $T_s$ to the current spin-down rate $|\dot\Omega|$ and universal parameter $J$ (Fujiwara et al., 2023, Nam et al., 17 Nov 2025).

5. Dependencies and Domain of Validity

VCH is regulated by several macroscopic and microscopic parameters:

• Magnetic field ($B$) and birth spin ($P_0$): Higher $B$ or lower $P_0$ increase spin-down power, amplifying VCH. For $B \lesssim 10^{11}$ G, even $P_0 \sim 10$ ms yields insufficient heating for $T_s\gtrsim 10^5$ K; for $B \gtrsim 10^{12}$ G and $P_0$ (10–100 ms), VCH is dominant (Nam et al., 17 Nov 2025).
• Equation of state and pairing gaps: Influence the volume, density profile, and moment of inertia of the pinned region, affecting $J$ and neutrino-cooling rates.
• Envelope composition: Light-element envelopes (carbon) elevate $T_s$ for the same interior temperature, making VCH signatures more prominent.
• Quantum-creep regime: VCH becomes temperature-independent when inner crust temperatures fall below the quantum cutoff ($T_Q$). The quantum-creep fraction $f_Q(t)$ measures whether the whole crust has entered this regime; steady-state heating ($L_h=J|\dot\Omega|$) only holds when $f_Q \approx 1$ (Nam et al., 17 Nov 2025).
• Steady-state boundary: Precise validity maps in the $(B, P_0)$ plane (with $P_0 \propto B^{2/3}$), define where the heating law applies (Nam et al., 17 Nov 2025).

6. Comparison with Alternative Heating Mechanisms

Competing mechanisms include magnetic field decay, crust cracking, dark matter heating, and rotochemical heating. Magnetic field decay, crust cracking, and dark matter accretion typically yield sub-detectable heating for old pulsars (Gonzalez et al., 2010, Fujiwara et al., 2023). Dark-matter heating would only dominate if $J \ll 10^{39}$ erg s, several orders below observed values. Rotochemical heating can be important, especially in classical pulsars, but is sensitive to initial spin period, field strength, and superfluid suppression (Gonzalez et al., 2010, Gonzalez et al., 2010). VCH is exceptionally robust, requires only current spin parameters, and is largely insensitive to initial conditions once the quantum-creep regime is established (Gonzalez et al., 2010).

7. Observational Implications and Modeling

VCH is central to explaining old, unexpectedly warm neutron stars observed in X-ray/UV surveys. Current models (e.g., Nam & Sekizawa (Nam et al., 17 Nov 2025, Nam et al., 28 Oct 2025)) incorporating both VCH and DUrca processes recover the observed clustering of $T_s \sim 10^5$–$3\times10^5$ K with $J \sim 10^{42.9}$–$10^{43.8}$ erg s across ordinary and millisecond pulsars. Three-dimensional mappings in $(t, T_s, B)$ space resolve degeneracies inherent in two-dimensional cooling tracks and highlight the necessity of accounting for magnetic field in cooling analyses.

Parameter constraints derived from pulsar temperature measurements feed back on nuclear EoS and crustal microphysics, tightening the acceptable range of $f_{\rm pin}$ and informing superfluidity models.

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A plausible implication is that vortex creep heating, parameterized by a nearly universal $J$, constitutes a cornerstone mechanism in the late-time thermal evolution of neutron stars. Its theoretically predicted and observationally inferred magnitude provides a strong feedback loop connecting crustal superfluid dynamics, dense-matter nuclear microphysics, and pulsar phenomenology.