MNRAS 000,1–10 (0000) Preprint 18 January 2021 Compiled using MNRAS LATEX style file v3.0 Modeling Magnetohydrodynamic Equilibrium in Magnetars with Applications to Continuous Gravitational Wave Production S. G. Frederick1,¢ K. L. Thompson1, M. P. Kuchera1 1Department of Physics, Davidson College, 405 N Main St, Davidson, NC 28035, USA 18 January 2021 ABSTRACT Possessing the strongest magnetic fields in the Universe, magnetars mark an extremum of physical phenomena. The strength of their magnetic fields is sufficient to deform the shape of the stellar body, and when the rotational and magnetic axes are not aligned, these deformations lead to the production of gravitational waves (GWs) via a time-varying quadrupole moment. Such gravitational radiation differs from signals presently detectable by the Laser Interferometer Gravitational-Wave Observatory. These signals are continuous rather than the momentary ‘chirp’ waveforms produced by binary systems during the phases of inspiral, merger, and ringdown. Here, we construct a computational model for magnetar stellar structure with strong internal magnetic fields. We implement an = = 1 polytropic equation of state (EOS) and adopt a mixed poloidal and toroidal magnetic field model constrained by the choice of EOS. We utilize fiducial values for magnetar magnetic field strength and various stellar physical attributes. Via computational simulation, we measure the deformation of magnetar stellar structure to determine upper bounds on the strength of continuous GWs formed as a result of these deformations inducing non-axisymmetric rotation. We compute predictions of upper limit GW strain values for sources in the McGill Magnetar Catalog, an index of all detected magnetars. Key words: stars: magnetars – gravitational waves – MHD 1 INTRODUCTION consistent periodicity of these GW signals, they are referred to as ‘continuous’ GWs. Under extended survey, stellar spin down due to Magnetars are an exceptional classification of pulsars, characterized loss in rotational kinetic energy through magnetic braking or energy by surface magnetic field strengths in excess of 1014 G and dipolar loss in the form of gravitational radiation will increase the rotational magnetic energies exceeding the star’s rotational energy (Thomp- period and GWs emitted will drift to lower frequencies (Creighton son & Duncan 1995). Olausen & Kaspi(2014) provide a catalog & Anderson 2011). However, under shorter observation, continuous of 23 confirmed and 6 candidate sources, and document consider- GWs appear as constant frequency sinusoidal waveforms. able progress in magnetar detection via W-ray burst events in recent years following the launch of the Swift and Fermi space telescopes. Continuous GWs are expected to be detected following improve- Given the rapid growth in confirmed magnetar sources, these stars ments in GW detector sensitivity, as their signals are often far fainter present a wealth of opportunity for improving current understanding than GWs produced by binary inspiral events. Evaluation of their regarding the influence of strong magnetic fields in extreme stellar signal strength, or wave strain, can be made by estimating the mag- environments. nitude of stellar deformations responsible for producing such signals Chandrasekhar & Fermi(1953) first showed for an incompressible (Zimmermann & Szedenits 1979). stellar model that a strong internal magnetic field will deform a star Recent work places upper limits on the GW strain of pulsar sources arXiv:2002.02619v3 [astro-ph.SR] 15 Jan 2021 away from spherical symmetry. For deformations induced along a capable of producing GWs within the operating range of the Laser In- magnetic field axis which is misaligned with the stellar rotational terferometer Gravitational-Wave Observatory (LIGO) (Abbott et al. axis, a time-varying gravitational quadrupole will result in the pro- 2017). The authors compute the spin-down limit; the GW strain sen- duction of gravitational waves (GWs). Thus, magnetars are com- sitivity produced by attributing the loss in rotational kinetic energy pelling candidates for the detection of GWs from deformed stellar completely to gravitational radiation. For a rigidly rotating triaxial sources. star, the frequency of gravitational waves produced by the source Such GWs differ from former event detections, as unlike the ‘chirp’ will be twice the rotational frequency. As magnetars are slowly rotat- waveform of binary inspiral mergers, GWs produced by a rapidly ro- ing stars (with rotational periods ∼ 5–8 s) (Olausen & Kaspi 2014), tating stellar source are nearly constant-frequency, sinusoidal signals GWs produced by these sources fall outside the sensitivity range of due to the source returning to the same spatial configuration in the LIGO and corresponding wave strain estimates were not addressed span of a complete revolution about its rotational axis. Due to the by Abbott et al.(2017). The principal goal of this paper is to provide estimates for upper- limit calculations of the GW strain for all confirmed magnetar sources ¢ E-mail: [email protected] in the McGill Magnetar Catalog (Olausen & Kaspi 2014) by con- © 0000 The Authors 2 S.G Frederick et al. structing a computational model for magnetar stellar structure and 2.2 Polytropic Equations of State magnetic field configuration. We determine the degree of structural The time-independent equations of hydrostatic equilibrium and mass deformation introduced by a strong internal magnetic field as the conservation provide an initial description of stable Newtonian stars. stellar structure reaches magnetohydrodynamic (MHD) equilibrium. In order to fully specify stellar structure, an equation of state (EOS) These results for stellar deformation subsequently inform wave strain is required to relate pressure to a number of state variables describ- estimates. ing stellar structure. We adopt a barotropic EOS, which defines the To compute upper-limit estimates of the GW strain for magnetars, relationship between pressure and density as %¹dº. While an EOS we adopt a simple barotropic EOS for an n=1 polytrope and seek parameterized by numerous state variables such as %¹d, G , ), ...º is dipolar solutions to the mixed poloidal-toroidal magnetic field model ? more physically representative of the interior of a neutron star, sub- derived by Haskell et al.(2008). We then compute the ellipticity that sequent discussion will show that our particular choice of barotropic arises from this solution and calculate the GW strain for the magnetars EOS allows analytic equations for stellar structure and magnetic field. in the McGill catalog. We use a polytropic EOS of the form For numerically stable computation, we use the fiducial values for stellar attributes: stellar mass, "¢ = 1.4 M ; stellar radius, '¢ = 10 W 15 −3 %¹dº = d , (4) km; and central density, d2 = 2.2 × 10 g·cm . where is the polytropic constant and the real, positive constant W is defined via the polytropic index = as 2 STRUCTURAL MODEL = ¸ 1 W = . (5) Prior authors (Owen 2005, and references therein) note that cumula- = tive errors introduced by excluding relativistic gravity and rotational effects largely cancel; while relativistic gravity results in a more com- Polytropic equations of state are often categorized by the com- pact model of stellar structure than the Newtonian framework, stellar pressibility of stellar matter, whereby a lowering of the constant W rotation has an opposing effect. Thus, in constructing a stellar model, corresponds to lower compression (Haensel et al. 2007). Thus, the we adopt the Newtonian gravitational theory and neglect rotational structural composition of the stellar interior sets a constraint on rep- effects. resentative equations of state. This gives us the following set of non-relativistic MHD equations Prior work has established neutron star structure as well approxi- to describe the time evolution of the system. mated by the choice of polytropic EOS corresponding to 0 < = . 1 md (Cho & Lee(2010), Woosley(2014)). In addition, an = = 1 polytrope ¸ v · rd ¸ dr · v = 0 (1a) has the property that the stellar radius is unaffected by mass nor cen- mC tral density, which reflects the insensitivity of radius to mass within mv 1 1 ¸ v · rv ¸ B × ¹r × Bº ¸ rd = −rΦ ¸ g (1b) normal neutron stars. Under these considerations, we implement an mC d d = = 1 polytropic EOS in modeling neutron star structure. An equation mB ¸ B¹r · vº − ¹B · rºv ¸ ¹v · rºB = v¹r · Bº (1c) for density as a function of radius can be determined via solutions mC to the Lane-Emden equation for a specified polytropic index, =. The m? 2 = = 1 polytrope possesses the following analytic solutions for d¹Aº ¸ v · r? ¸ d2Br · v = 0 (1d) mC and %¹Aº from which stellar structure can be fully determined: where v is the velocity vector, B is the magnetic field, d is density, ? is momentum, g is the gravitational acceleration vector, Φ is the time- sin ¹cA/'¢º'¢ d¹Aº = d2 for A < '¢ (6) independent gravitational potential, and 2B is the adiabatic speed of Ac sound as described in Mignone et al.(2018). All of the computations and in this paper evolve these sets of equations numerically using the third-order Runge Kutta algorithm for time evolution. 2 %¹Aº = d¹Aº for A < '¢, (7) where the polytropic constant = 4.25 × 104 cm5 · g−1 · s−2 for 2.1 Hydrostatic Equilibrium Conditions a neutron star with radius '¢ = 10 km, mass "¢ = 1.4 " , and 15 −3 Our choice of stellar model is constrained to configurations which are central density d2 = 2.2 × 10 g·cm . in equilibrium. Thus, the construction of this stellar model requires a crucial balance between the force of gravity and stellar structure. The equilibrium condition 3% " d 2.3 Gravitational Potential Model = −퐺 A , (2) 3A A2 Hydrostatic equilibrium requires the balance of an inward gravita- where % and d are the stellar pressure and density, respectively, 퐺 is tional force with the radial change in pressure. We determine so- the gravitational constant, and "A is the mass interior to the radius lutions to the spherically symmetric form of Poisson’s equation for for A < '¢, provides the basis for balancing the gravitational force gravitational potential per unit mass, with structural variation throughout the stellar interior.