We investigate three mechanisms that promote the loss of magnetic flux from an isolated neutron star. Ohmic decay produces a diffusion of the magnetic field with respect to the charged particles. It proceeds at a rate that is inversely proportional to the electric conductivity and independent of the magnetic field strength. Ohmic decay occurs in both the fluid core and solid crust of a neutron star, but it is too slow to directly affect magnetic fields of stellar scale. Ambipolar diffusion involves a drift of the magnetic field and charged particles relative to the neutrons. The drift speed is proportional to the second power of the magnetic field strength if the protons form a normal fluid. Variants of ambipolar diffusion include both the buoyant rise and the dragging by superfluid neutron vortices of magnetic flux tubes. Ambipolar diffusion operates in the outer part of the fluid core where the charged particle composition is homogeneous, protons and electrons being the only species. The charged particle flux associated with ambipolar diffusion decomposes into a solenoidal and an irrotational component. Both components are opposed by frictional drag. The irrotational component perturbs the chemical equilibrium between neutrons, protons, and electrons, thus generating pressure gradients that effectively choke it. The solenoidal component is capable of transporting magnetic flux from the outer core to the crust on a short time scale. Magnetic flux that threads the inner core, where the charged particle composition is inhomogeneous, would be permanently trapped unless particle interactions could rapidly smooth departures from chemical equilibrium. Magnetic fields undergo a Hall drift related to the Hall component of the electric field. The drift speed is proportional to the magnetic field strength. Hall drift occurs throughout a neutron star. Unlike ohmic decay and ambipolar diffusion which are dissipative, Hall drift conserves magnetic energy. Thus, it cannot by itself be responsible for magnetic field decay. However, it can enhance the rate of ohmic dissipation. In the solid crust, only the electrons are mobile and the tangent of the Hall angle is large. There, the evolution of the magnetic field resembles that of vorticity in an incompressible fluid at large Reynolds number. This leads us to speculate that the magnetic field undergoes a turbulent cascade terminated by ohmic dissipation at small scales. The small-scale components of the magnetic field are also transported by Hall drift waves from the inner crust where ohmic dissipation is slow to the outer crust where it is rapid. The diffusion of magnetic flux through the crust takes-5 x 10 8 /B 12 yr, where B 12 is the crustal magnetic field strength measured in units of 10 12 G.

Beca~se. a n~utron star is ~orn hot, its internal composition is close to chemical equilibrium. In the fluid core, thts tmphes_ that _the rat!o of the_ number densities of charged particles (protons and electrons) to neutrons, x ~.ncfnn, ts an mcreasmg function of the mass density. This composition gradient stably stratifies the m~tter glVlng ~ise to a Br~nt-Viiisiilii frequency N ~ (xgj2H) 112 ~ 500 s-1 , where g is the gravitational acceler-atto~, and H ts the denstty scale height. Consequently, a neutron star core provides a cavity that supports gra_vtty modes (g-modes). These g-modes are distinct from those previously identified with the thermal stratifi-catiOn of the surface layers and t~e chemical stratification of the crust. We compute the lowest-order, quadrupolar, g-modes for cold, Newtoman, neutron star models with M/M 0 = 0.581 and M/M 0 = 1.405 and show that t~e crustal and core g-modes have similar periods. We also discuss damping mechanisms and estimate ~ampmg ~at~s for the core g-modes. Particular attention is paid to damping due to the emission of gravitational radtatwn.

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.

The core of a neutron star contains several species of particles, whose relative equilibrium concentrations are determined by the local density. As the star spins down, its centrifugal force decreases continuously, and the star contracts. The density of any given uid element increases, changing its chemical equilibrium state. The relaxation towards the new equilibrium takes a nite time, so the matter is not quite in chemical equilibrium, and energy is stored that can be released by reactions. For a neutron star core composed of neutrons (n), protons (p), and electrons (e), the departure from chemical equilibrium is quanti ed by the chemical potential di erence p + e n . A nite increases the reaction rates and the neutrino emissivity. If large enough (j j > 5kT), it reduces the net cooling rate because some of the stored chemical energy is converted into thermal energy, and can even lead to net heating. A simple model (for non-super uid matter) shows the e ect of this heating mechanism on the thermal evolution of neutron stars. It is particularly noticeable for old, rapidly spinning stars with weak magnetic elds. If the timescale for variations of the rotation rate is much longer than the cooling time, a quasi-equilibrium state is reached in which heating and cooling balance each other and the temperature is completely determined by the current value of _ P=P 3 (or the spin-down power). If only modi ed Urca reactions are allowed, the predicted quasi-equilibrium x-ray luminosity of some millisecond pulsars approaches the upper limits obtained by Danner et al. (1994) from ROSAT data. The predicted x-ray luminosity is much lower if direct Urca or other fast reactions are allowed. In both cases, the luminosity is probably increased if the stellar interior is mostly super uid.

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.