Spheroidal magnetic stars rotating in vacuum

Abstract

Context. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton's theory. However, several mechanisms break this isotropy like for instance their rotation generating a centrifugal force, magnetic pressure or anisotropic equations of state. The stellar surface therefore deviates slightly or significantly from a sphere depending on the strength of these anisotropic perturbations. Aims. In this paper, we compute analytical and numerical solutions of the electromagnetic field produced by a rotating spheroidal star of oblate or prolate nature. This study is particularly relevant for millisecond pulsars for which strong deformations are produced by the rotation or a strong magnetic field, leading to indirect observational signatures of the polar cap thermal X-ray emission. Methods. First we solve the time harmonic Maxwell equations in vacuum by using oblate and prolate spheroidal coordinates adapted to the stellar boundary conditions. The solutions are expanded in series of radial and angular spheroidal wave functions. Particular emphasize is put on the magnetic dipole radiation. Second, we compute approximate solutions by integrating numerically the timedependent Maxwell equations in spheroidal coordinates. Results. We show that the spin down luminosity corrections compared to a perfect sphere are to leading order given by terms involving (a/r L ) 2 and (a/R) 2 where a is the stellar oblateness or prolateness, R the smallest star radius and r L the light-cylinder radius. The corresponding perturbations in the electromagnetic field are only perceptible close to the surface, deforming the polar cap rims. At large distances r a, the solution tends asymptotically to the perfect spherical case of a rotating dipole.

Spheroidal force-free neutron star magnetospheres

“…The present study relies on our previous work (Pétri 2021), adding an electric charge and current density in the force-free approximation. We succinctly remember the model before diving into the salient features of a force-free spheroidal magnetosphere.…”

“…ε 0 and µ 0 are the electric permittivity and magnetic permeability respectively, ρ e is the charge density and j the current density. For an extensive and detailed discussion of the spheroidal coordinate systems, we refer to Pétri (2021) in order to avoid reproducing lengthy formulae here.…”

“…The numerical setup for performing simulations is similar to the one used in Pétri (2021). The gravitating fluid outer boundary is denoted by ρ in = R and should not to be confused with the spherical radius because we use spheroidal coordinates.…”

“…The spin down depends on the surface shape but also on the normalisation convention used to compute the luminosity. This important issue is discussed in depth by Pétri (2021). In order to get rid of effects not directly related to the change in shape, we used a normalisation where the magnetic dipole tends asymptotically to the same magnetic moment value at infinity, irrespective of the oblateness or prolateness parameter a/R.…”

“…Magnetars are also expected to show strong magnetic constrictions leading to oblate or even prolate surfaces not directly connected to a geometry imposed by their rotation. A detailed analysis of such systems evolving in vacuum was recently performed by Pétri (2021). If vacuum is replaced by an ideal plasma like in electron/positron pairs, a spheroidal force-free neutron star magnetosphere can be constructed, generalizing the extensive literature about the force-free magnetosphere of spherical stars (Spitkovsky 2006;Komissarov 2006;McKinney 2006;Pétri 2012;Cao et al 2016).…”