Crust, Superfluidity, and Glitches

A pulsar glitch is a sudden increase in the observed rotation frequency, but the timing step does not by itself reveal which internal component supplied the angular momentum. The standard picture combines an elastic ionic crust, a neutron superfluid threaded by quantized vortices, and imperfect coupling to the charged component. Pinning can store differential rotation and vortex motion can release it. Entrainment, core coupling, and the trigger determine how large the usable reservoir really is, so the often-repeated phrase “the crust causes glitches” is a hypothesis with testable qualifications rather than a direct observation.

Required background. The Einstein Equation supplies the stellar structure that maps local crust microphysics into a global moment of inertia.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies stress and momentum conventions; Relativistic Matter and Compact Objects supplies the multifluid setting; Computational Relativity supplies sequence and uncertainty checks.

An elastic lattice and a quantum fluid

Below neutron drip, the crust is a Coulomb lattice of neutron-rich ions in a degenerate electron gas. Above neutron drip, unbound neutrons coexist with clusters and can form Cooper pairs. The lattice supports shear stress, while the condensate permits nearly dissipationless flow relative to it. Near the crust–core transition, nonspherical clusters and uncertain nuclear symmetry energy complicate both the shear modulus and the superfluid density. The microphysical regimes and their characteristic scales are developed in Chamel and Haensel 2008, §§3.2, 4.2, and 8.

In a local nonrelativistic description, single-valuedness of the paired neutron condensate phase quantizes the circulation of canonical momentum per neutron. When entrainment is neglected this is $m_n\boldsymbol v_s$, giving

$$\oint \boldsymbol v_s\mathbin{\cdot}d\boldsymbol\ell =N\kappa, \qquad \kappa=\frac{\pi}{m_n},$$

where $\hbar=1$. With entrainment, it is the canonical momentum circulation that remains quantized; the transport-velocity circulation need not obey this simple formula. On scales large compared with the vortex spacing, the same unentrained local model gives an areal vortex density $n_v\kappa=2\Omega_s$. Spinning the superfluid down therefore requires vortices to move outward. Pinning to the nuclear lattice or to magnetic structures can impede that motion, allowing a lag $\Delta\Omega_{\rm lag}=\Omega_s-\Omega_c$ to grow as the electromagnetically coupled component spins down.

The angular-momentum account

With constant moments of inertia and negligible entrainment, a minimal two-component model writes

$$I_c\dot\Omega_c=N_{\rm ext}+N_{\rm int}, \qquad I_s\dot\Omega_s=-N_{\rm int}.$$

Here $c$ denotes the component coupled to the observed crust on the timescale of interest, not merely the solid lattice, and $s$ denotes the reservoir. During a sufficiently brief glitch the external torque is negligible, giving

$$I_c\,\Delta\Omega_c+I_s\,\Delta\Omega_s\simeq0.$$

Thus a measured positive $\Delta\Omega_c$ requires a compensating decrease in the reservoir’s angular velocity or an explicitly different source of angular momentum. Repeated glitches constrain the time-integrated reservoir, while post-glitch exponentials constrain coupling timescales. Haskell and Melatos derive these relations and compare trigger and recovery models in Haskell and Melatos 2015, §I.C, Eqs. (13)–(20).

A useful observable is the glitch activity over an interval $\mathcal T$,

$$A_g=\frac{1}{\mathcal T}\sum_i\frac{\Delta\Omega_{c,i}}{\Omega_c}.$$

For a nearly stationary statistical reservoir, complete pinning between glitches, and approximately constant long-term spin-down rate, the reservoir can supply at most $I_s|\dot\Omega_c|\mathcal T$ in that interval. The observed transfers require approximately $I_c\Omega_c A_g\mathcal T$. Hence

$$\frac{I_s}{I_c}\gtrsim \frac{\Omega_c A_g}{|\dot\Omega_c|} =2\tau_c A_g, \qquad \tau_c=\frac{\Omega_c}{2|\dot\Omega_c|}.$$

Here $\dot\Omega_c$ is the long-term secular rate, averaged over glitches. Using the slightly larger interglitch spin-down rate instead gives the corresponding bound on $I_s/(I_c+I_s)$. The distinction is small only when the reservoir fraction is small. This is an activity bound within the two-component assumptions; it is not a direct measurement of crustal thickness. Leakage between glitches reduces the angular momentum available to glitches. Incomplete lag release affects how a single event constrains the reservoir; the activity bound uses the cumulative transfer over many events.

Entrainment changes the interpretation. In a periodic lattice, the neutron momentum need not be parallel to the transport velocity, so the density that can flow independently is not simply the density of unbound neutrons. A large effective neutron mass reduces the crustal reservoir inferred from a naive geometric moment of inertia. The resulting “reservoir problem” was made explicit by comparing band-structure entrainment with glitch activity in Andersson et al. 2012, Eqs. (1)–(7) and Fig. 1. Pairing, disorder, nuclear dynamics, and participation of core superfluid all alter the quantitative conclusion.

Competing trigger and coupling pictures

Evidence cutoff: 2026-08-18. The evidence axes are the pinning landscape, entrainment calculation, mutual-friction law, stellar EOS and crust thickness, fraction of the core coupled during the rise, trigger mechanism, dimensionality and resolution of a simulation, and which timing observables it predicts. Timing establishes the step, recovery, and long-term activity. It does not directly image vortices or locate the reservoir.

In the vortex-unpinning picture, a growing Magnus force initiates local motion; knock-on interactions or stress redistribution may create an avalanche. This picture explains angular-momentum transfer naturally and can generate broad event-size distributions. Competing triggers include crust failure, thermal feedback, hydrodynamic instabilities, and magnetospheric state changes. A starquake can change the moment of inertia or perturb pinning, while a magnetospheric change can alter the measured torque; neither alternative automatically supplies the observed angular momentum for every large glitch.

The strongest contrary evidence to a crust-only model is an inferred activity that exceeds its effective entrained reservoir under a stated EOS and coupling assumption. Evidence against that conclusion comes from calculations with weaker entrainment, thicker crusts, or partial core participation. These are different physical formulations, not inconsistent arithmetic. The unresolved questions are which region participates on the unresolved rise time, how vortices cross the crust–core interface, and whether one trigger explains both giant and small glitches.

A re-review should be triggered by a new controlled entrainment calculation, a simulation suite that closes its resolution and boundary-condition tests, a resolved glitch-rise or recovery data release, or a systematic timing reanalysis that changes activity or coupling constraints. A successful model must distinguish storing a lag, initiating a failure, transferring angular momentum, and reproducing recovery; demonstrating only one stage is not a complete glitch theory.

Scope checks and limiting cases

If pinning vanishes and mutual friction couples all components rapidly, $\Omega_s\simeq\Omega_c$ and no long-lived reservoir accumulates. If coupling is extremely weak, a large lag may accumulate, but the model must still explain a rapid transfer. If entrainment locks nearly all dripped neutrons to the lattice, a geometric crust fraction greatly overestimates available angular momentum. These limits expose which assumption is doing the work.

Relativistic stellar structure enters through the moment-of-inertia weighting: a local mass fraction is not the same as a global $I_s/I$. Magnetic stresses, temperature-dependent pairing, and composition can evolve between events. Consequently, a two-component model is a controlled diagnostic, not a claim that a real star has exactly two rigid rotors.

Check your understanding

During a glitch, take $I_s/I_c=0.02$ and neglect the external torque. If the observed component jumps by $\Delta\Omega_c=10^{-4}\,\mathrm{rad\,s^{-1}}$, what change in the reservoir angular velocity is required?

Solution

Angular-momentum conservation gives

$$\Delta\Omega_s=-\frac{I_c}{I_s}\Delta\Omega_c =-50\left(10^{-4}\,\mathrm{rad\,s^{-1}}\right) =-5\times10^{-3}\,\mathrm{rad\,s^{-1}}.$$

The negative sign shows that the reservoir spins down while the observed component spins up. The calculation is conditional on which material belongs to $I_c$ and $I_s$ during the unresolved rise; entrainment and rapid core coupling change that partition.

References

• Andersson, Nils, Kostas Glampedakis, Wynn C. G. Ho, and Cristóbal M. Espinoza. “Pulsar Glitches: The Crust Is Not Enough.” Physical Review Letters 109 (2012): 241103. DOI.
• Chamel, Nicolas, and Pawel Haensel. “Physics of Neutron Star Crusts.” Living Reviews in Relativity 11 (2008): 10. DOI.
• Haskell, Brynmor, and Andrew Melatos. “Models of Pulsar Glitches.” International Journal of Modern Physics D 24 (2015): 1530008. arXiv.