The shear-Hall instability in newborn neutron stars

Abstract

Aims. In the first few minutes of a newborn neutron star's life the Hall effect and differential rotation may both be important. We demonstrate that these two ingredients are sufficient for generating a "shear-Hall instability" and for studying its excitation conditions, growth rates, and characteristic magnetic field patterns. Methods. We numerically solve the induction equation in a spherical shell, with a kinematically prescribed differential rotation profile Ω(s), where s is the cylindrical radius. The Hall term is linearized about an imposed uniform axial field. The linear stability of individual azimuthal modes, both axisymmetric and non-axisymmetric, is then investigated. Results. For the shear-Hall instability to occur, the axial field must be parallel to the rotation axis if Ω(s) decreases outward, whereas if Ω(s) increases outward it must be anti-parallel. The instability draws its energy from the differential rotation, and occurs on the short rotational timescale rather than on the much longer Hall timescale. It operates most efficiently if the Hall time is comparable to the diffusion time. Depending on the precise field strengths B 0 , either axisymmetric or non-axisymmetric modes may be the most unstable. Conclusions. Even if the differential rotation in newborn neutron stars is quenched within minutes, the shear-Hall instability may nevertheless amplify any seed magnetic fields by many orders of magnitude.

Linear stability analysis of the Hall magnetorotational instability in a spherical domain

“…Only for very strong anti-parallel magnetic fields, the growth rates exceed the parallel case, but never exceeds the growth rate of the pure MRI for S 1. The maximal growth rate was of the order of the inverse rotation period, the smallest time-scale of the system, similar to what was found by Kondić et al (2011) for the SHI above. parallel here.…”

“…The shear feeds the energy into the toroidal field, and the Hall effect channels it back to poloidal creating a positive feedback. This is the same as in Kondić et al (2011). The consequence is that, due to the shear-Hall mechanism, there is an instability even in the range S = 0 to S = 25.…”

“…9). If, however, the flow becomes turbulent, the turbulent diffusivity is much larger (Naso et al 2008), leading to much lower Rm (∼10 4 , as noted by Kondić et al 2011, and which are doable numerically), but still above the threshold for the onset of the Hall-MRI. An interesting feature of turbulent flows is that S is close to unity for densities larger than or equal to 10 13 g cm −3 .…”

“…This is even more obvious in models with pure shear without rotation (Bejarano et al 2011;Kunz 2008). Kondić et al (2011) examined the instability driven by the Hall effect itself, the shear-Hall instability (SHI), in the context of protoneutron stars. Like the MRI, it feeds off the shear energy and grows on the time-scale of the rotation period.…”

“…Unlike Kondić et al (2011), this works includes the back-reaction leading to the interplay between the Hall effect and MRI. Stability maps and growth rates of the global Hall-MRI instability follow from numerical calculations of the Hall-MHD equations in a spherical shell.…”