Theoretical Constraints on Neutron Star Superfluidity from Her X-1 Precession

Technical Summary: Theoretical Constraints on Neutron Star Superfluidity from Her X-1 Precession

Problem Statement
Recent observations by the Imaging X-ray Polarimetry Explorer (IXPE) of the X-ray binary Her X-1 reveal correlations between flux, polarization degree, and polarization angle across its 35-day superorbital cycle. These data have been interpreted as strong evidence that the 35-day period is driven by the nearly free precession of the neutron star crust. However, standard models of neutron star superfluidity suggest that quantized vortex lines in the inner crust are pinned to the nuclear lattice. If these vortices remain strongly pinned, they exert a dominant gyroscopic restoring force that forces the precessional frequency to be very high (on the order of minutes), inconsistent with the observed 35-day period. Conversely, if the precession is "nearly free," the superfluid must decouple from the crust, challenging conventional understandings of vortex pinning and mutual friction. This paper investigates the theoretical constraints imposed by the existence of long-lived free precession on the dynamics of the crustal superfluid and its implications for glitch phenomenology.

Methodology
The authors employ a two-component rotational model treating the neutron star as a system consisting of a coupled core and elastic crust (treated as a single deformed rigid body) and a crustal neutron superfluid. The analysis utilizes Euler's equations in the frame rotating with the normal crust to derive the evolution of the angular velocities of the crust ($\Omega_n$) and the superfluid ($\Omega_s$).

Key theoretical components include:

1. Torque Formulation: The internal torque exerted by the superfluid on the crust is derived from the mutual friction force density. This force is modeled using a phenomenological drag coefficient $R$, which characterizes the interaction between quantized vortices and the normal fluid. The torque is decomposed into a dissipative component ($N_{s1}$) and a non-dissipative component ($N_{s2}$).
2. Analytic and Numerical Solutions: The authors derive approximate analytic solutions for the evolution of the lag angle ($\delta\Omega = \Omega_s - \Omega_n$) and the precession parameters (wobbling angle $\theta$ and azimuthal angle $\phi$) under the assumption of weak coupling ($|\delta\Omega| \ll \Omega_n$) and small ellipticity ($|\epsilon| \ll 1$). These analytic results are validated against numerical solutions of the full Euler equations.
3. Application to Her X-1: The derived equations are applied to Her X-1, incorporating external torques arising from accretion, magnetospheric interactions, pulsar losses, and the back-reaction of a warped accretion disk. The authors simulate the rotational evolution over timescales of hundreds to thousands of years to determine if the observed 35-day cycle can be sustained under various drag regimes.

Key Contributions and Results

1. Constraint on Mutual Friction: The analysis demonstrates that for the 35-day precession to persist over the $\sim$50-year observational baseline of Her X-1, the superfluid vortices must remain effectively unpinned for centuries. This requires an extremely weak mutual friction, characterized by a dimensionless drag coefficient $R \lesssim 0.01$.
2. Damping Timescales: The paper shows that the damping timescale of the slow precession mode is $\tau_p \approx (1 + \kappa_s^2 R^2) / (\epsilon \kappa_s R \Omega_n)$. For $R \gtrsim 0.01$, the precession damps on timescales significantly shorter than the observational baseline, ruling out standard strong-pinning scenarios. Only in the regime of very weak drag ($R \lesssim 0.01$) does the precession survive for hundreds of years.
3. Incompatibility with Glitch Dynamics: A critical finding is that the differential rotation (lag) developed during the quasi-steady precession phase is orders of magnitude smaller than the critical lag required to trigger vortex unpinning in standard glitch models. Specifically, the fractional lag during precession is $|\delta\Omega|/\Omega_n \lesssim 10^{-9}$, whereas glitch-triggering lags are typically $\sim 10^{-6}$ to $10^{-1}$.
4. External Torque Compensation: The authors evaluate whether external torques (from accretion and magnetic interactions) could compensate for the dissipation caused by the superfluid. They find that for $R \gtrsim 0.002$, the internal dissipative torque dominates the external torque, meaning the precession evolution is controlled by the superfluid coupling rather than external driving.

Significance and Claims
The paper claims that the interpretation of Her X-1's 35-day cycle as nearly free precession carries "far-reaching implications" for the dynamics of the crustal superfluid. The primary conclusion is that long-lived precession is incompatible with the buildup of the large differential rotation required to trigger pulsar glitches.

The authors posit two potential resolutions to this tension:

1. The superfluid regions responsible for storing angular momentum in glitching pulsars are largely decoupled from the precessing component in Her X-1.
2. The effective vortex drag and pinning conditions depend sensitively on the dynamical state of the star (i.e., the presence of precession alters the pinning landscape).

Alternatively, the paper notes that if the superfluid physics in Her X-1 is standard, the 35-day cycle may not be caused by neutron star precession, suggesting the need for alternative explanations (such as a precessing accretion disk) that can account for the IXPE polarization data. The authors maintain a modest stance, acknowledging that while their results constrain the superfluid parameters under the precession hypothesis, further observational and theoretical work is required to definitively distinguish between precession and alternative mechanisms.