Are gravitational waves from giant magnetar flares observable?

Burkhard Zink, Paul D. Lasky, and Kostas D. Kokkotas

Theoretical Astrophysics, Institute for Astronomy and Astrophysics, University of Tubingen,Auf der Morgenstelle 10, Tubingen 72076, Germany(Received 15 July 2011; published 18 January 2012)

Are giant flares or storms in soft-gamma repeaters viable sources of gravitational radiation? Few

theoretical studies have been concerned with this problem, with the small number using either highly

idealized models or assuming a magnetic field orders of magnitude beyond what is supported by

observations. We perform nonlinear general-relativistic magnetohydrodynamic simulations of large-scale

hydromagnetic instabilities in magnetar models, which mimic the magnetic field reconfiguration expected

in magnetar flares. We utilize these models to find gravitational-wave emission over a wide range of

energies, from 1040 to 1047 erg. This allows us to derive a systematic relationship between the surface field

strength and the gravitational-wave strain. In particular, for typical magnetar fields of a few times 1015 G,

we conclude that a direct observation of f-modes excited by global magnetic field reconfigurations is

unlikely with present or near-future gravitational-wave observatories, though we also discuss the

possibility that modes in a low-frequency band up to 100 Hz could be sufficiently excited to be relevant

for observation.

DOI: 10.1103/PhysRevD.85.024030 PACS numbers: 04.30.Db, 04.40.Dg, 95.30.Sf

I. INTRODUCTION

Bursts of soft gamma-rays, storms, and occasionallygiant flares in soft-gamma repeaters (SGRs) and anoma-lous x-ray pulsars are commonly understood as consequen-ces of magnetic field activity in magnetars, neutron starsendowed with strong magnetic fields. These violent eventshave been considered in the literature as possible efficientsources of gravitational radiation, since they are very com-pact and the high electromagnetic luminosity may repre-sent only a small part of the overall energy contained in themechanism. It is thus not surprising that operationalgravitational-wave observatories, in particular, LaserInterferometric Space Antenna (LIGO) and VIRGO havebeen employed to establish upper limits on SGR bursts andstorms. Presently, the best empirically derived upper limitsare 1:4� 1049 erg for the f-mode and 3:5� 1044 erg for awhite noise band around 100–200 Hz [1]. For the f-mode,gravitational-wave energies of this order would still implya very substantial excitation of the star. Could luminositiesof this order actually be realized in a nearby giant flare?

There has been little theoretical work addressing thisquestion, for a large part, because the system is verycomplex and the giant flare mechanism is not yet wellunderstood. Possible flare motors are discussed in [2–5]and, as described by Levin and van Hoven [6], fall into twobroad categories: internal mechanisms based on a large-scale rearrangement of the interior magnetic field, andexternal mechanisms likely following magnetic reconnec-tion in the magnetosphere. (Of course, these two processesmay also occur concurrently.)

If we accept that magnetic field dynamics are closelycorrelated with the electromagnetic signal, and the recentobservations of quasiperiodic oscillations in flare tails

seem to support this point of view (e.g. [7–9]), then anoptimistic scenario for gravitational-wave emission is alarge-scale dynamical rearrangement of the core magneticfield inside the star. Ioka [10] has investigated the maxi-mum gravitational-wave energy released by the change inmoment of inertia induced by such a mechanism andplaced an upper limit of about 1049 erg under ideal con-ditions, including optimistic values of the internal mag-netic field. More recently, Corsi and Owen [11] foundsimilar values to be possible under more generic conditionsstill tapping into the full energy reservoir associated withan instantaneous change in the magnetic potential energyof the star. In contrast, Levin and van Hoven [6] do not findf-mode detection to be very likely in the near future. Theirmodel is based on the aforementioned external mechanismtriggering f-modes in the star. In the present paper, wefocus on the other side of the problem—attempting totrigger large-scale mass motions that could generate gravi-tational radiation through a global rearrangement of theinternal magnetic field.Our recent publication on dynamical instabilities [12]

focused on the dynamics and evolution of magnetic fieldsinside relativistic stellar models, and we followed thedevelopment of these instabilities until saturation andring-down to establish the nature of the quasistationarystates, which result from these processes. Most recently,Ciolfi et al. [13] performed numerical simulations of thesame hydromagnetic instabilities and concluded that giantflares could give rise to observable gravitational radiation,employing a stellar model with a surface field strength of6:7� 1016 G to reach this conclusion. Actual SGR fieldstrengths are closer to 1=60 of this value, and the luminos-ity depends on the field strength in a highly nonlinearfashion, as we shall see below.

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In this paper, we will address the following questions:(i) Assuming a large-scale rearrangement of the core mag-netic field to be involved in a giant flare, how does thegravitational-wave luminosity scale with the field strengths?(ii) As a consequence of this relation, can we expect toobserve magnetar giant flares in gravitational-wavedetectors?

II. RESULTS

In order to investigate gravitational-wave luminosities,we consider a large-scale restructuring of an initially purelypoloidal magnetic field. This provides us with a toy-modelthat mimics the catastrophic reconfiguring of the internalmagnetic field during a magnetar flare. An actual magnetarmay well contain a substantial toroidal field component butfor purposes of obtaining an order of magnitude estimate ofthe gravitational-wave emission, a highly unstable initialstate should provide a good first approximation.

As with our recent publication [12], we are using thegraphics processing unit (GPU)-accelerated HORIZON code[14] for general-relativistic magnetohydrodynamics in theCowling approximation, in conjunction with the LORENE

library for the construction of magnetized equilibriumneutron stars [15]. Horizon is based on THOR code[16,17], but employs GPUs for high-throughput parallelprocessing. Gravitational waves are extracted using thefirst-moment form of the quadrupole formula [18]. Formore details on the numerical method and stellar modelssee Lasky et al.[12], Zink [14].

Our initial models are relativistic equilibrium n ¼ 1polytropes (mass 1:3M�, radius 15 km) with purelypoloidal magnetic fields of varying strength. In particular,we have investigated a full sequence of models withpolar surface magnetic field amplitudes between Bpole ¼3:1� 1015 G and 5:5� 1016 G, corresponding to mag-netic field strengths of 1:6� 1016 G to 2:7� 1017 G inthe center of the star. While typical magnetar fieldstrengths reside near the low end of this sequence, wehave decided on a full spectrum of models both for nu-merical reasons (since low field strengths give rise to weakgravitational-wave signals which compete with numericalnoise), as well as to gain fundamental insights between therelation of initial field configuration and the gravitational-wave amplitude.

Purely poloidal fields are known to be dynamically un-stable [12,19], and we follow the evolution throughoutdevelopment, saturation and, ring-down of the instability.This is particularly important for lower field strengths, sincecouplings between magnetic motions and fluid modes re-quires us to follow the system for many Alfven times.

All our models exhibit the poloidal field instability asexpected. In Fig. 1, we show the measured gravitational-wave strain h� at 10 kpc in a particular model with Bpole ¼1:8� 1016 G. Other components and other models in thesequence show a similar structure, although the growth

time scale of the instability and the strain amplitude aredifferent. An analysis of the signal spectrum shows that thefundamental quadrupole f-mode is excited by the mag-netic field instability. Intriguingly, we also see some evi-dence that low-frequency modes, likely either Alfven or/and g-modes (due to stratification induced by gradientsin the magnetic pressure), in the range of 100 Hz maycontribute to the gravitational-wave signal. This evidenceis not yet conclusive and warrants further investigation, inparticular, very long-term simulations in the order of asecond or more to gain a sufficiently high resolution inthis part of the spectrum.Figure 2 collects the gravitational-wave amplitudes

as a function of the magnetic field strength. We find an

0 50 100 150 200 250Time [ms]

-2 10-23

-1 10-23

0

1 10-23

2 10-23

h x (at

10

kpc)

FIG. 1. Gravitational wave signal obtained from one of oursimulations. This particular stellar model has an initial surfacemagnetic field strength of 1:8� 1016 G and develops a poloidalfield instability during the first 50 ms (see also [12]), whichrestructures the global magnetic field inside the star. Thegravitational-wave signal, h� for an assumed source distanceof 10 kpc, has the expected 1.8 kHz oscillations associated withthe f-mode, and also exhibits low-frequency components whichare further discussed in the text.

1015

1016

1017

Surface magnetic field strength [G]

10-26

10-25

10-24

10-23

10-22

10-21

Stra

in

FIG. 2 (color online). Gravitational-wave strain in relation tosurface magnetic field strength at the stellar pole. This plotrepresent results from our entire sequence of simulations, withsurface magnetic fields ranging from 3:1� 1015 G to 5:5�1016 G. The signal amplitude is found to depend strongly onthe magnetic field strength, ranging from almost 10�21 for ourmost extreme model down to less than 10�25 for models closer torealistic magnetar strengths. The dashed line is a power-law fit tothe data (see text).

BURKHARD ZINK, PAUL D. LASKY, AND KOSTAS D. KOKKOTAS PHYSICAL REVIEW D 85, 024030 (2012)

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approximate power-law relation between hmax and Bpole

given by

hmax � 7:6� 10�27 ��10 kpc

d

��

�Bpole

1015 G

�3:3: (1)

Most of the energy in the signal is in the f-mode.Assuming a gravitational-wave damping time of approxi-mately 100 ms (e.g. [20,21]), we find a correspondingpower-law relation for the energy emitted in gravitationalradiation [6]

Egw � 5:1� 1036 ��10 kpc

d

�2 �

�Bpole

1015 G

�6:5

erg: (2)

From this result, which can be directly compared withobservations [1], we make two notes: (i) the gravitational-wave amplitude is a function of the surface magnetic fieldstrength given by a power-law. From linear analysis, wewould expect a quadratic dependence of the strain on themagnetic field strength if direct emission of Alfven wavesare responsible. We observe a more complex scaling,probably because nonlinear saturation effects are involved.(ii) typical magnetar field strengths of 1015 G give rise tostrains below 10�25 and energies lower than 1040 erg for asource at 10 kpc, even if we assume a catastrophic globalrestructuring of the field to be associated with a giant flare.

We have indicated the signal-to-noise ratios we derivefrom our data for different detector sensitivity curves in

Fig. 3. Here we have plotted the signal amplitudeffiffiffiffiT

p j~hðfÞj,

where T is the damping time of the oscillation and ~hðfÞ isthe Fourier transform of h�, as a function of the frequencyfor various magnetic field strength models. At approxi-mately 1.8 kHz we show the signal amplitude for thef-mode as excited by the hydromagnetic instability andassuming a damping time of 50 ms � T � 200 ms. In thelower part of the spectrum we plot the other maximal modeseen in the Fourier transform, assuming a damping timebetween of 10 ms � T � 1 s. Over this, we plot theentire spectrum (assuming T ¼ 100 ms) for two modelswith Bpole ¼ 8:8� 1015 G and Bpole ¼ 1:8� 1016 G, re-

spectively. Finally, we also plot the root of the noise power

spectral density,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijShðfÞj

p, as a function of frequency for

the LIGO, Advanced LIGO, and Einstein Telescope (ET)detectors [22].The amplitude signal-to-noise ratio, defined byffiffiffiffiT

p j~hðfÞj=jShðfÞj1=2, can be read off the graph as the ratiobetween the signal and the noise curve for the respectivedetector. The main conclusion from this figure is the fol-lowing: Assuming that a giant flare is associated with acatastrophic large-scale rearrangement of the core mag-netic field, the gravitational-wave signal associatedwithf-modes are not observable with present or near-futuregravitational-wave observatories.This statement is in line with the conclusion of [6]

concerning the efficiency of the internal mechanism inexciting f-mode radiation. Moreover, it is in contrastwith the scenarios presented in [10,11], which are bothbased on equilibrium models as opposed to full dynamicalsimulations. While we cannot exclude that different initialmodel configurations could produce higher levels off-mode radiation, we consider it, at present, unlikely thatthe result will change by many orders of magnitude.Our results can also be compared with the numerical

study of [13], which obtains a much more optimisticprediction than we do, although they investigate the samekind of instability. However, the magnetar model consid-ered by these authors has a very strong surface magneticfield of 6:7� 1016 G, which is even stronger than the mostextreme case we have considered here. Clearly, the ob-served soft-gamma repeaters do not fall into this category.

III. DISCUSSION

We have performed three-dimensional general relativis-tic magnetohydrodynamics (GRMHD) simulations of neu-tron stars, inducing large-scale reconfigurations of theinternal magnetic field to model gravitational-wave emis-sions from magnetar flares. We have found power-lawrelations governing the surface, polar magnetic fieldstrength as a function of the maximal gravitational-wavestrain [Eq. (1)], and also the energy emitted in gravitationalwaves [Eq. (2)]. We find that the gravitational-wave emis-sions due to f-mode excitations are unlikely to be observedin current or near-future gravitational-wave observatories(Fig. 3).

0 500 1000 1500 200010–27

10–26

10–25

10–24

10–23

10–22

Frequency [Hz]

Sign

al A

mpl

itude

[Hz-1

/2]

ET

AdvLIGO

LIGOBpole = 18 B15Bpole = 13 B15Bpole = 9 B15Bpole = 3 B15

FIG. 3 (color). Signal amplitudeffiffiffiffiT

p j~hðfÞj against the oscilla-tion frequency. The colored boxes on the right represent themaximum for the f-mode assuming a constant periodic sourcelasting between 50 and 200 ms. The colored boxes on the leftrepresent themaximummode seen in the Fourier transform for anygiven frequency below the f-mode frequency, assuming a constantperiodic source lasting between 10 ms and 1 s. These scale withffiffiffiffiT

p, implying one can easily extrapolate to alternative values of the

damping time.We have further plotted the entire spectrum (assum-ing a damping time of T ¼ 100 ms) for both the Bpole ¼ 1:8�1016 G model (blue line) and the Bpole ¼ 8:8� 1015 G model

(black line). The thick curves represent the root of the noise powerspectral density, k ShðfÞ k , for ET, Advanced LIGO and LIGO,respectively. These curves have been taken from the review articleof Sathyaprakash and Schutz [22].

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There are two factors which could modify our aboveconclusion. The first is that other equations of state andinitial magnetic field topologies could be investigated, e.g.those with a dominating toroidal field component. Mixturesof poloidal and toroidal fields in neutron stars may be stablefor a range of energy ratios [23] and could indeed representtypical field equilibria. However, there is a possibility thatconfigurations with a strong toroidal component (and com-paratively weak surface dipole field) could be unstable andlead to higher coupling into the f-modes and, consequently,stronger gravitational-wave luminosities, while still beingconsistent with SGR spindown rates. Recent studies ofhydromagnetic equilibria in stratified neutron stars [24,25],however, cast a shadow of doubt over this possibility. Themagnetar also contains a crust which we do not model here.Recent studies of the coupled crust-core magnetic systemindicate that core Alfven modes are only weakly affected bythe crust [8,9,26].

The other factor is the possible low-frequency compo-nents of the signal (see also [6,27]). As discussed above,we have found indications (but no solid confirmation) of

spectral components in the band typically associated withg-modes or Alfven modes, i.e. below 200 Hz. This isparticularly interesting because the detectors are mostsensitive in this regime. Moreover, these modes havemuch longer damping times and are assumed to be partof the post-flare signal. We leave these possibilities forfuture work.

ACKNOWLEDGMENTS

We would like to thank K. Glampedakis and S. Landerfor helpful discussions. This work was supported bythe Sonderforschungsbereich/Transregio 7 on‘‘Gravitational Wave Astronomy’’ by the DeutscheForschungsgemeinschaft. Some computations were per-formed on the Multi-modal Australian ScienceS Imagingand Visualisation Environment (MASSIVE) (www.massive.org.au) and the graphics processing unit (GPU)nodes on the nehalem cluster at the High PerformanceComputing Center Stuttgart (HLRS). P. Lasky. is supportedby the Alexander von Humboldt Foundation.

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