In this paper we investigate the space and velocity distributions of old neutron stars (aged 10 9 to 10 10 yr) in our Galaxy. Galactic old Neutron Stars (NSs) population fills a torus-like area extending to a few tens kiloparsecs above the galactic plane. The initial velocity distribution of NSs is not well known, in this work we adopt a three component initial distribution, as given by the contribution of kick velocities, circular velocities and Maxwellian velocities. For the spatial initial distribution we use a Γ function. We then use Monte Carlo simulations to follow the evolution of the NSs under the influence of the Paczyński Galactic gravitational potential. Our calculations show that NS orbits have a very large Galactic radial expansion and that their radial distribution peak is quite close to their progenitors' one. We also study the NS vertical distribution and find that it can well be described by a double exponential low. Finally, we investigate the correlation of the vertical and radial distribution and study the radial dependence of scale-heights.
Aims. During the first 40 s after their birth, proto-neutron stars are expected to be subject to at least two types of instability. The first one, the convective instability, is excited in the inner regions, where the entropy gradient produces a Rayleigh-type convection. The second one, the neutron-finger instability, is instead excited in the outer layers where the lepton gradients are large. Both instabilities involve convective motions and hence can trigger dynamo actions that may be responsible for the large magnetic fields in neutron stars and magnetars. However, because they have rather different mean turbulent velocities, they are also likely to give rise to different types of dynamo. Methods. We have solved the mean-field induction equation in a simplified one-dimensional model of both the convective and the neutron-finger instability zones. Although very idealized, the model includes the nonlinearities introduced by the feedback processes that tend to saturate the growth of the magnetic field (α-quenching) and suppress its turbulent diffusion (η-quenching). The possibility of a dynamo action is studied within a dynamical model of turbulent diffusivity where the boundary of the unstable zone is allowed to move. A large number of numerical simulations have been performed in which the relevant parameters, such as the spin-period, the strength of the differential rotation, the intensity of the initial magnetic field, and the extent of the neutron finger instability zone, have been suitably varied. Results. We show that the dynamo action can also be operative within a dynamical model of turbulent diffusivity and that the amplification of the magnetic field can still be very effective. Furthermore, we confirm the existence of a critical spin-period, below which the dynamo is always excited independently of the degree of differential rotation, and whose value is related to the size of the neutron-finger instability zone. We provide a relation for the intensity of the final field as a function of the spin of the star and of its differential rotation. Conclusions. Although they were obtained by using a toy model, we expect that our results are able to capture the qualitative and asymptotic behaviour of a mean-field dynamo action developing in the neutron-finger instability zone. Overall, we find that such a dynamo is very efficient in producing magnetic fields well above equipartition, and thus that it could represent a possible explanation for the large surface magnetic fields observed in neutron stars.
We report results from calculations investigating stationary magnetic field configurations in accretion discs around magnetised neutron stars. Our strategy is to start with a very simple model and then progressively improve it, providing complementary insight into results obtained with large numerical simulations. In our first model, presented here, we work in the kinematic approximation and consider the stellar magnetic field as being a dipole aligned with the stellar rotation axis and perpendicular to the disc plane, while the flow in the disc is taken to be steady and axisymmetric. The behaviour in the radial direction is then independent of that in the azimuthal direction. We investigate the distortion of the field caused by interaction with the disc matter, solving the induction equation numerically in full 2D. The influence of turbulent diffusivity and fluid velocity on the poloidal field configuration is analysed, including discussion of outflows from the top and bottom of the disc. We find that the distortions increase with increasing magnetic Reynolds number R m (calculated using the radial velocity). However, a single global parameter does not give an adequate description in different parts of the disc and we use instead a "magnetic distortion function" D m (r, θ) (a magnetic Reynolds number defined locally). Where D m 1 (near to the inner edge of the disc) there is little distortion, but where D m > 1 (most of the rest of the disc), there is considerable distortion and the field becomes weaker than the dipole would have been. Between these two regions, there is a transition zone where the field is amplified and can have a local minimum and maximum. The location of this zone depends sensitively on the diffusivity. The results depend very little on the boundary conditions at the top of the disc.
Millisecond pulsars are rather weakly-magnetized neutron stars which are thought to have been spun up by disc accretion, with magnetic linkage between the star and the disc playing a key role. Their spin history depends sensitively on details of the magnetic field structure, but idealized models from the 1980s and 1990s are still commonly used for calculating the magnetic field components. This paper is the third in a series presenting results from a stepby-step analysis which we are making of the problem, starting with very simple models and then progressively including additional features one at a time, with the aim of gaining new insights into the mechanisms involved. In our first two papers, the magnetic field structure in the disc was calculated for a standard Shakura/Sunyaev model, by solving the magnetic induction equation numerically in the stationary limit within the kinematic approximation; here we consider a more general velocity field in the disc, including backflow. We find that the profiles of the poloidal and toroidal components of the magnetic field are fairly similar in the two cases but that they can be very different from those in the models mentioned above, giving important consequences for the torque exerted on the central object. In particular we find that, contrary to what is usually thought, some regions of the disc outward of the co-rotation point (rotating more slowly than the neutron star) may nevertheless contribute to spinning up the neutron star on account of the detailed structure of the magnetic field in those parts of disc.
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