Superfluid spin up and pulsar glitch recovery

Abstract

When a massive star comes to the end of its life, it explodes in a supernova, leaving behind a compact remnant known as a neutron star. The core density of a neutron star exceeds that of terrestrial nuclei, making these cosmic objects the only known means to probe the properties of bulk nuclear matter at extreme densities. Radio waves are beamed along a neutron stars' magnetic axis, which is misaligned with its rotation axis, creating a lighthouse effect which has earned these stars the name `pulsars'. The pulse arrival times are measured with ultra-high precision, rivalling that of the best terrestrial clocks. Some pulsars undergo timing irregularities, known as glitches, during which the spin frequency of the star suddenly increases and then recovers quasi-exponentially over a period of days to weeks. In this thesis, glitch recovery is used to extract information about the pulsar interior in two ways: high-resolution timing data, and gravitational radiation.

The interior of a pulsar has long been thought to consist of superfluid neutrons coexisting with a proton-electron plasma. In Chapter 2, a hydrodynamic model of the global flow induced during glitch recovery is constructed using the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) two-component equations. The impulsive spin up of the two-component superfluid and its container is solved analytically in arbitrary geometry, generalising the extensively studied case of single-fluid spin up. The spin-up time depends on the geometry, mutual friction coefficients, $B$, $B'$, the Ekman number $E$, and the superfluid density fraction $\rho_n$. For $B\sim O(1)$, the inviscid component undergoes Ekman pumping due to strong coupling to the viscous component, and the azimuthal velocities are “locked together” during the spin-up. For $B\lesssim E^{1/2}$, there is no Ekman pumping in the inviscid component, and the inviscid azimuthal velocity spins up via mutual friction on a combination of the mutual friction and Ekman time-scales. The spin-up process is studied in spheres, cylinders (with co- and counter-rotating lids), and cones, and occurs faster in spheres and cones which become shorter at larger radius.

In Chapter 3, the coupled, dynamic response of a rigid container filled with a two-component superfluid undergoing Ekman pumping is calculated self-consistently. The container responds to the back-reaction torque exerted by the viscous component of the superfluid and an arbitrary external torque. The resulting motion is described by a pair of coupled integral equations for which solutions are easily obtained numerically. If the container is initially accelerated impulsively then set free, it relaxes quasi-exponentially to a steady state over multiple time-scales, which are a complex mix of $B$, $B'$, $E$, $\rho_n$ and the varying hydrodynamic torque at different latitudes. The spin down of light containers (compared with the contained fluid) depends weakly on $B$, $B'$, $E$, $\rho_n$ and occurs faster than the Ekman time. When the fluid components are initially differentially rotating, the container can “overshoot” its asymptotic value before increasing again. When a constant external torque is applied, the superfluid components rotate differentially and non-uniformly in the long term. For an oscillating external torque, the amplitude and phase of the container response are most pronounced when the container is light compared with the contained fluid.

The coupled spin-up of a two-component superfluid and its container are fitted to radio pulsar timing data in Chapter 4. All glitches recorded to date in the Crab and Vela pulsars are considered, with specific attention given to the 1985 and 1988 Vela glitches recorded at Mount Pleasant observatory in Australia and the 1975 Crab glitch recorded at Jodrell Bank in England. The model successfully accounts for the quasi-exponential recovery observed in pulsars like Vela and the “overshoot” observed in pulsars like the Crab. By fitting the model to high-resolution timing data, three constitutive coefficients in bulk nuclear matter can be extracted: the shear viscosity, the mutual friction parameters, and the charged fluid fraction. The fitted coefficients for the Crab and Vela are compared with theoretical predictions for several equations of state, including the colour-flavour locked and two-flavour colour superconductor phases of quark matter. Good agreement is found between the bulk-averaged, effective parameters extracted from observations and the theory of condensed protons and neutrons, giving support to the hydrodynamic model.

The spin-down recovery of impulsively spun-up containers filled with superfluid helium has been studied in the laboratory by Tsakadze & Tsakadze (1980). In Chapter 5, theory derived in Chapters 2 and 3 is applied to interpret the Tsakadze data. The dependence of the spin-down time on temperature and the mass fraction of the viscous component are investigated. Excellent agreement at the 0.5% level is obtained for experiments at $1.4<T<1.6\,{\rm K}$, after correcting for the non-uniform rotation in the initial state, confirming that vortex tension and pinning (which are omitted from the theory) play a minimal role for small impulsive accelerations and smooth-walled containers. For $T\gtrsim1.6\,{\rm K}$ turbulence is expected and experimental agreement diverges.

A new, practical method for measuring the kinematic viscosity and Hall-Vinen mutual friction coefficients of a two-component superfluid is presented in Chapter 6. The proposed experiment is modelled on that of Tsakadze & Tsakadze (1980). A step-by-step recipe is presented for extracting the transport coefficients using the theory derived in Chapters 2 and 3. The experiment is straight-forward to conduct and is independent of alternative techniques involving tuning forks, vibrating wires, and Kapton diaphragms.

The gravitational wave signal emitted by a neutron star following a rotational glitch is calculated in Chapter 7. The signal is generated by nonaxisymmetric Ekman pumping excited during the glitch. The fluid is assumed to be stratified and compressible. For the largest glitches, the gravitational wave strain produced by the hydrodynamic mass quadrupole moment approaches the sensitivity range of advanced long-baseline interferometers, e.g., the Laser Interferometer Gravitational Wave Observatory (LIGO) for signals persisting for a sufficient length of time. It is shown that the viscosity, compressibility, and orientation of the star can be inferred in principle from the width and amplitude ratios of the Fourier peaks (at the spin frequency and its first harmonic) observed in the gravitational wave spectrum in the + and x polarisations. In principle, the viscosity, stratification, compressibility and inclination angle of the pulsar can be extracted. The transport coefficients constrain the equation of state of bulk nuclear matter, because they depend sensitively on the degree of superfluidity.

There are many applications and extensions to the studies in Chapters 2-7. In Chapter 8, preliminary results are reported for some examples. It is shown that the viscous stress exerted by Ekman pumping is not sufficient to break the neutron star crust. The magnetic field induced by Ekman pumping is calculated and is found to reach $10^{12}\,{\rm G}$ over the lifetime of the star. Glitch recovery including an electromagnetic braking torque is considered, and can result in recovery times significantly longer than the purely hydrodynamic case. The crust and fluid components tend towards co-rotation in the long term, with braking index $n=3$. Using multiple glitches in a single pulsar, the theory in Chapter 4 is used to investigate the statistics of pulsar glitches. An extension of the theory in Chapter 3 to test the hypothesis of a crystalline core in pulsars is discussed.