Magnetic fields generated by r-modes in accreting millisecond pulsars

📝 Abstract

In millisecond pulsars the existence of the Coriolis force allows the development of the so-called Rossby oscillations (r-modes) which are know to be unstable to emission of gravitational waves. These instabilities are mainly damped by the viscosity of the star or by the existence of a strong magnetic field. A fraction of the observed millisecond pulsars are known to be inside Low Mass X-ray Binaries (LMXBs), systems in which a neutron star (or a black hole) is accreting from a donor whose mass is smaller than 1 $M_\odot $. Here we show that the r-mode instabilities can generate strong toroidal magnetic fields by inducing differential rotation. In this way we also provide an alternative scenario for the origin of the magnetars.

💡 Analysis

In millisecond pulsars the existence of the Coriolis force allows the development of the so-called Rossby oscillations (r-modes) which are know to be unstable to emission of gravitational waves. These instabilities are mainly damped by the viscosity of the star or by the existence of a strong magnetic field. A fraction of the observed millisecond pulsars are known to be inside Low Mass X-ray Binaries (LMXBs), systems in which a neutron star (or a black hole) is accreting from a donor whose mass is smaller than 1 $M_\odot $. Here we show that the r-mode instabilities can generate strong toroidal magnetic fields by inducing differential rotation. In this way we also provide an alternative scenario for the origin of the magnetars.

📄 Content

arXiv:0903.0349v2 [astro-ph.HE] 13 Mar 2009 Magnetic fields generated by r-modes in accreting millisecond pulsars Carmine Cuofano and Alessandro Drago Dipartimento di Fisica, Universit´a di Ferrara and INFN sez. Ferrara, 44100 Ferrara, Italy E-mail: cuofano@fe.infn.it Abstract. In millisecond pulsars the existence of the Coriolis force allows the development of the so-called Rossby oscillations (r-modes) which are know to be unstable to emission of gravitational waves. These instabilities are mainly damped by the viscosity of the star or by the existence of a strong magnetic field. A fraction of the observed millisecond pulsars are known to be inside Low Mass X-ray Binaries (LMXBs), systems in which a neutron star (or a black hole) is accreting from a donor whose mass is smaller than 1 M⊙. Here we show that the r-mode instabilities can generate strong toroidal magnetic fields by inducing differential rotation. In this way we also provide an alternative scenario for the origin of the magnetars.

1. Introduction The r-mode oscillations in all rotating stars are unstable for emission of gravitational waves [1]. These modes play therefore a very important role in the astrophysics of compact stars and in the search for gravitational waves. On the other hand the existence of millisecond pulsars implies the presence of damping mechanisms of the r-modes. Damping mechanisms are associated with bulk and shear viscosity and with the possible existence of the so called Ekman layer. The latter is located at the interface between the solid crust and the fluid of the inner core and in this region friction is significantely enhanced respect to friction in a purely fluid component. All these mechanisms are strongly temperature dependent. An important class of rapidly rotating neutron stars are the accreting millisecond pulsars associated with Low Mass X-ray Binaries (LMXBs). For these objects the internal temperature is estimated to be in the range 108-108.5 K [2, 3] and their frequencies can be as large as ∼650 Hz. In this range of temperatures and in the case of a purely nucleonic star, bulk and shear viscosities alone cannot stabilize stars whose frequency exceeds ∼100 Hz. A possible explanation of the stability of stars rotating at higher frequencies is based on the Ekman layer, but recent calculations show that this explanation holds only for rather extreme values of the parameters [4]. In this contribution we propose a new damping mechanism based on the generation inside the star of strong magnetic fields produced by r-mode instabilities. This same mechanism has been proposed in the case of rapidly rotating, isolated and newly born neutron stars in [5, 6]. In that paper the mechanism which generates the magnetic field is investigated only during the relatively short period in which the star remains always in the instability region. In our work we consider accreting stars and we investigate the interplay between r-modes and magnetic field on an extremely long period and we show that in this scenario a very strong magnetic field can be produced.
2. R-mode equations in the presence of magnetic field R-mode instabilities are associated to kinematical secular effects wich generate differential rotation in the star and large scale mass drifts, particularly in the azimuthal direction. Differential rotation in turn can produce very strong toroidal magnetic fields in the nucleus and these fields damp the instabilities extracting angular momentum from the modes. In order to derive the equations regulating the evolution of r-modes in the presence of a pre-existent poloidal magnetic field we have modified the equations derived in [7], taking into account also the magnetic damping. We use the estimate given in [1] for the gravitational radiation reaction rate due to the l = m = 2 current multipole Fg = 1 47M1.4R4 10P −6 −3 s (1) as well as the bulk and shear viscosity damping rates Fb = 1 2.7 × 1011 M−1 1.4 R10P −2 −3 T 6 9 s Fs = 1 6.7 × 107 M5/4 1.4 R−23/4 10 T −2 9 s (2) where we have used the notation M1.4 = M/1.4M⊙, R10 = R/10 Km, P−3 = P/1 ms and T9 = T/109 K. The total angular momentum J of a star can be decomposed into a equilibrium angular momentum J∗and a perturbation proportional to the canonical angular momentum of a r-mode Jc: J = J∗(M, Ω) + (1 −Kj)Jc, Jc = −Kcα2J∗ (3) where K(j,c) are dimensionless constants and J∗∼= I∗Ω. Following Ref.[8] the canonical angular momentum obeys the following equation: dJc/dt = 2Jc{Fg(M, Ω) −[Fv(M, Ω, Tv) + Fmi(M, Ω, Bp)]} (4) where Fv = Fs + Fb is the viscous damping rate and we have introduced the magnetic damping rate Fmi that we discuss in the next section. The total angular momentum satisfies instead the equation: dJ/dt = 2JcFg + ˙Ja(t) −I∗ΩFme (5) where ˙Ja is the rate of accretion of angular momentum and we have assumed it to be ˙Ja = ˙M(GMR)1/2, and Fme is the magnetic braking rate associated to the poloidal magnetic field. Combining the equations (4) and (5) than we give the dynamical evolution re

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