# Distortion of Neutron Stars

With a Toroidal Magnetic Field

###### Abstract

Models of rotating relativistic stars with a toroidal magnetic field have been computed for a sample of eight equations of state of cold dense matter. Non-rotating models admit important levels of magnetization and quadrupole distortion accompanied by a seemingly unlimited growth in size. Rotating models reach the mass-shedding limit at smaller angular velocities than in the non-magnetized case according to the larger circumferential equatorial radius induced by the magnetic field. Moreover, they can be classified as prolate–prolate, oblate–prolate, or oblate–oblate with respect to surface deformation and quadrupole distortion. Simple expressions for surface and quadrupole deformation are provided that are valid up to magnetar field strengths and rapid rotation.

## 1 Introduction

Neutron stars with a strong toroidal magnetic field have attracted increasing interest as the magnetically induced distortion of their matter distribution may lead to the quasi-periodic emission of gravitational waves [Cutler2002, Bonazzola1996], for example, in the case of low-mass X-ray binaries (LMXBs). Moreover, strong magnetic fields are believed to power the electromagnetic activity of magnetars, which subsume both anomalous X-ray pulsars (AXPs) and soft-gamma repeaters (SGRs) [Duncan1992, Thompson1996]. Models of relativistic stars with a toroidal magnetic field can be obtained within the standard formalism for stationary and axisymmetric relativistic stars [Bonazzola1993], since the electromagnetic stress–energy tensor then satisfies the same compatibility condition [Oron2002] as the stress–energy tensor of an unmagnetized perfect fluid in purely rotational motion. Based on this finding, numerical models of relativistic stars with a toroidal magnetic field have emerged [Kiuchi2008, Frieben2012] whereas the poloidal case was already studied a long time ago [Bocquet1995].

## 2 Method and results

The neutron star matter is modeled as a perfectly-conducting perfect fluid at zero temperature, described by a one-parameter equation of state (EOS). For stationary and axisymmetric models in rigid rotation as considered hereafter, the general-relativistic line element in spherical coordinates can be chosen as

(1) |

with gravitational potentials , , , and that are functions of alone. The toroidal magnetic field must then ensure that the Lorentz force is the gradient of a scalar potential, which is the case for , where is the proper energy density of the fluid, is the fluid pressure, and is the magnetization parameter. The field and matter equations are derived from the perfect-fluid case [Bonazzola1993] by taking into account additional magnetic source terms, expressed in terms of , and the magnetic potential , supplemented by the above relation for and the EOS.

The numerical models have been computed by means of a multidomain and
surface-adaptive pseudo-spectral code for stationary and axisymmetric
relativistic stars from the
lorene^{1}^{1}1http://www.lorene.obspm.fr package,
extended to the case of the toroidal magnetic field specified
above, and employing its standard sample of nuclear matter EOSs.

All models built with a certain EOS have the same rest mass corresponding to a gravitational mass of in the non-rotating and non-magnetized case. For the polytropic Pol2 EOS, defined by with the polytropic exponent and the rest-mass density , the adopted polytropic constant (in units in which ) implies a circumferential radius of .

Non-rotating models have been obtained up to large values of (limited only by computational resources) for all EOSs, and the surface deformation , computed from the equatorial coordinate radius and the polar coordinate radius , as well as the quadrupole distortion , obtained from Thorne’s quadrupole moment and the moment of inertia , attain considerable negative values as the magnetization is increased. The dimensions of the star even appear to grow without bounds. In turn, the volume-averaged magnetic field strength always falls off after attaining a maximum value of several .

The solution space of magnetized and rotating models, parametrized by and , has been determined for the Pol2 EOS. Its lower part up to the maximum field strength limit, beyond which decreases, is schematically shown in \freffig1 (a). Since the curves of vanishing surface deformation, , and of vanishing quadrupole distortion, , are different, the models can be divided into three classes for which surface deformation and quadrupole distortion are (1) both prolate, (2) oblate and prolate, or (3) both oblate, depending on the relative strength of magnetic and centrifugal forces. In the rotating case, the mass-shedding limit of a magnetized star is reduced with increasing magnetization in agreement with the condition of geodesic motion at the stellar equator since the circumferential equatorial radius is enlarged by the toroidal magnetic field.

Magnetic field strengths and angular velocities of all known magnetars are small enough that can be well approximated by a linear function of and , , with the distortion coefficients and shown in \freffig1 (b), adopting normalized variables and . An estimate for the type II superconducting case[Lander2012] is then given by below the second critical magnetic field strength . Likewise, can be computed by using and instead of and . The Newtonian model Pol2N10 with and its relativistic counterpart Pol2R10 demonstrate that relativistic effects strongly attenuate both the surface deformation induced by the toroidal magnetic field and the quadrupole deformation in general. In contrast, the rotational surface deformation is only slightly reduced since the centrifugal force is more effective at larger distances from the rotation axis where relativistic effects have already weakened.

## Acknowledgments

This work was supported in part by the DFG grant SFB/Transregio 7. JF gratefully acknowledges financial support from the Daimler und Benz Stiftung.