A strong candidate for a source of gravitational waves is a highly magnetized, rapidly rotating neutron star (magnetar) deformed by internal magnetic stresses. We calculate the mass quadrupole moment by perturbing a zeroth‐order hydrostatic equilibrium by an axisymmetric magnetic field with a linked poloidal–toroidal structure. In this work, we do not require the model star to obey a barotropic equation of state (as a realistic neutron star is not barotropic), allowing us to explore the hydromagnetic equilibria with fewer constraints. We derive the relation between the ratio of poloidal to total field energy Λ and ellipticity ε, and briefly compare our results to those obtained using the barotropic assumption. Then, we present some examples of how our results can be applied to astrophysical contexts. First, we show how our formulae, in conjunction with current gravitational wave (non‐)detections of the Crab pulsar and the Cassiopeia A central compact object (Cas A CCO), can be used to constrain the strength of the internal toroidal fields of those objects. We find that, for the Crab pulsar (whose canonical equatorial dipole field strength, inferred from spin‐down, is 4 × 108 T) to emit detectable gravitational radiation, the neutron star must have a strong toroidal field component, with maximum internal toroidal field strength Btm= 7 × 1012 T; for gravitational waves to be detected from the Cas A CCO at 300 Hz, Btm∼ 1013 T, whereas detection at 100 Hz would require Btm∼ 1014 T. Using our results, we also show how the gravitational wave signal emitted by a magnetar immediately after its birth (assuming it is born rapidly rotating, with Λ≲ 0.2) makes such a newborn magnetar a stronger candidate for gravitational wave detection than, for example, an SGR giant flare.

Magnetic fields in upper main-sequence stars, white dwarfs, and neutron stars are known to persist for timescales comparable to their lifetimes. From a theoretical perspective this is problematic, as it can be shown that simple magnetic field configurations are always unstable. In non-barotropic stars, stable stratification allows for a much wider range of magnetic field structures than in barotropic stars, and helps stabilize them by making it harder to induce radial displacements. Recent simulations by Braithwaite and collaborators have shown that, in stably stratified stars, random initial magnetic fields evolve into nearly axisymmetric configurations with both poloidal and toroidal components, which then remain stable for some time. It is desirable to provide an analytic study of the stability of such fields. We write an explicit expression for a plausible equilibrium structure of an axially symmetric magnetic field with both poloidal and toroidal components of adjustable strengths, in a non-barotropic, nonrotating, fluid star, and study its stability using the energy principle. We construct a displacement field that should be a reasonable approximation to the most unstable mode of a toroidal field, and confirm Braithwaite's result that a given toroidal field can be stabilized by a poloidal field containing much less energy than the former, as given through the condition E pol /E tor 2aE tor /E grav , where E pol and E tor are the energies of the poloidal and toroidal fields, respectively, and E grav is the gravitational binding energy of the star. We find that a ≈ 7.4 for main-sequence stars, and a ∼ 200 for neutron stars. Since E pol /E grav ≪ 1, we conclude that the energy of the toroidal field can be substantially larger than that of the poloidal field, which is consistent with the speculation that the toroidal field is the main reservoir powering magnetar activity. The deformation of a neutron star caused by the hidden toroidal field can also cause emission of gravitational waves.

Certain multi-wavelength observations of neutron stars, such as intermittent radio emissions from rotation-powered pulsars beyond the pair-cascade death line, the pulse profile of the magnetar SGR 1900+14 after its 1998 August 27 giant flare, and X-ray spectral features of PSR J0821−4300 and SGR 0418+5729, suggest that the magnetic fields of non-accreting neutron stars are not purely dipolar and may contain higherorder multipoles. Here, we calculate the ellipticity of a non-barotropic neutron star with (i) a quadrupole poloidal-toroidal field, and (ii) a purely poloidal field containing arbitrary multipoles, deriving the relation between the ellipticity and the multipole amplitudes. We present, as a worked example, a purely poloidal field comprising dipole, quadrupole, and octupole components. We show the correlation between field energy and ellipticity for each multipole, that the l = 4 multipole has the lowest energy, and that l = 5 has the lowest ellipticity. We show how a mixed multipolar field creates an observationally testable mismatch between the principal axes of inertia (to be inferred from gravitational wave data) and the magnetic inclination angle. Strong quadrupole and octupole components (with amplitudes ∼ 10 2 times higher than the dipole) in SGR 0418+5729 still yield ellipticity ∼ 10 −8 , consistent with current gravitational wave upper limits. The existence of higher multipoles in fast-rotating objects (e.g., newborn magnetars) has interesting implications for the braking law and hence phase tracking during coherent gravitational wave searches.

Wave dispersion in a pulsar plasma is discussed emphasizing the relevance of different inertial frames, notably the plasma rest frame K and the pulsar frame K ′ in which the plasma is streaming with speed β s . The effect of a Lorentz transformation on both subluminal, |z| < 1, and superluminal, |z| > 1, waves is discussed. It is argued that the preferred choice for a relativistically streaming distribution should be a Lorentz-transformed Jüttner distribution; such a distribution is compared with other choices including a relativistically streaming Gaussian distribution. A Lorentz transformation of the dielectric tensor is written down, and used to derive an explicit relation between the relativistic plasma dispersion functions in K and K ′ . It is shown that the dispersion equation can be written in an invariant form, implying a one-to-one correspondence between wave modes in any two inertial frames. Although there are only three modes in the plasma rest frame, it is possible for backward-propagating or negative-frequency solutions in K to transform into additional forward-propagating, positive-frequency solutions in K ′ that may be regarded as additional modes.

A recent laboratory experiment suggests that a Kelvin–Helmholtz (KH) instability at the interface between two superfluids – one rotating and anisotropic, the other stationary and isotropic – may trigger sudden spin‐up of the stationary superfluid. This result suggests that a KH instability at the crust–core (1 S0–3 P2–superfluid) boundary of a neutron star may provide a trigger mechanism for pulsar glitches. We calculate the dispersion relation of the KH instability involving two different superfluids including the normal fluid components and their effects on stability, particularly entropy transport. We show that an entropy difference between the core and crust superfluids reduces the threshold differential shear velocity and threshold crust–core density ratio. We evaluate the wavelength of maximum growth of the instability for neutron star parameters and find the resultant circulation transfer to be within the range observed in pulsar glitches.

We consider critically the three most widely favored pulsar radio emission mechanisms: coherent curvature emission (CCE), beam-driven relativistic plasma emission (RPE) and anomalous Doppler emission (ADE). We assume that the pulsar plasma is one dimensional (1D), streaming outward with a bulk Lorentz factor γs ≫ 〈γ〉 − 1 ≳ 1, where 〈γ〉 is the intrinsic spread in the rest frame of the plasma. We argue that the formation of beams in a multi-cloud model is ineffective in the intrinsically relativistic case for plausible parameters, because the overtaking takes too long. We argue that the default choice for the particle distribution in the rest frame is a Jüttner distribution and that relativistic streaming should be included by applying a Lorentz transformation to the rest-frame distribution, rather than the widely assumed relativistically streaming Gaussian distribution. We find that beam-driven wave growth is severely restricted by (a) the wave properties in pulsar plasma, (b) a separation condition between beam and background, and (c) the inhomogeneity of the plasma in the pulsar frame. The growth rate for the kinetic instability is much smaller and the bandwidth of the growing waves is much larger for a Jüttner distribution than for a relativistically streaming Gaussian distribution. No reactive instability occurs at all for a Jüttner distribution. We conclude that none of CCE, RPE and ADE in tenable as the generic pulsar radio emission mechanism for “plausible” assumptions about the pulsar plasma.

A recipe is presented to construct an analytic, self-consistent model of a non-barotropic neutron star with a poloidal-toroidal field of arbitrary multipole order, whose toroidal component is confined in a torus around the neutral curve inside the star, as in numerical simulations of twisted tori. The recipe takes advantage of magnetic-field-aligned coordinates to ensure continuity of the mass density at the surface of the torus. The density perturbation and ellipticity of such a star are calculated in general and for the special case of a mixed dipole-quadrupole field as a worked example. The calculation generalises previous work restricted to dipolar, poloidal-toroidal and multipolar, poloidal-only configurations. The results are applied, as an example, to magnetars whose observations (e.g., spectral features and pulse modulation) indicate that the internal magnetic fields may be at least one order of magnitude stronger than the external fields, as inferred from their spin downs, and are not purely dipolar.

Recent calculations of the hydromagnetic deformation of a stratified, non‐barotropic neutron star are generalized to describe objects with superconducting interiors, whose magnetic permeability μ is much smaller than the vacuum value μ0. It is found that the star remains oblate if the poloidal magnetic field energy is ≳40 per cent of the total magnetic field energy, that the toroidal field is confined to a torus which shrinks as μ decreases and that the deformation is much larger (by a factor of ∼μ0/μ) than that in a non‐superconducting object. The results are applied to the latest direct and indirect upper limits on the gravitational‐wave emission from the Laser Interferometer Gravitational Wave Observatory (LIGO) and radio pulse timing (spin‐down) observations of 81 millisecond pulsars, to show how one can use these observations to infer the internal field strength. It is found that the indirect spin‐down limits already imply astrophysically interesting constraints on the poloidal–toroidal field ratio and the diamagnetic shielding factor (by which accretion reduces the observable external magnetic field e.g. by burial). These constraints will improve following gravitational‐wave detections, with implications for accretion‐driven magnetic field evolution in recycled pulsars and the hydromagnetic stability of these objects’ interiors.