arXiv:1102.4830v3 [astro-ph.HE] 28 Mar 2011 uniso h aeom n ol open could eigenfre- and asteroseismology from (NS) waveform, wave interior star the NS neutron of the the quencies infer One is can We targets signals. important oscillations. GW) most and the ray of cosmic (electro- bring multimessenger neutrino, will among magnetic, interferometers synergy GW greater the a These in about [4] band. ET like Hz-kHz ones generation 10 3rd VIRGO the advanced and LIGO, LCGT, de- advanced and GW as generation such 2nd [1–3], the tectors by started be will astronomy eopei msinvamgei ed [18–20]. mag- fields the magnetic to inter- via propagating emission be oscillations netospheric can axial the X-QPOs by [16]. preted energy electromagnetic the mtrpae ussadsmtmsgatflrs(GFs) flares giant energy sometimes huge and SGRs with energy. bursts magnetic repeated of to thought dissipation emit is the emission by whose powered LMC, (AXPs) be the pulsars and galaxy x-ray our anomalous in gamma-ray GW and soft the (SGRs) as for observed repeaters are candidates They promising asteroseismology. are [7–9] magnetars eomto n ei oa siltosaon new a around oscillations the polar equilibrium. least stellar begin the At change and would [17]. deformation stress GWs magnetic tap emit the will of and vary reduction energy that inertia GF of modes the moment polar the of the especially in part oscillations, stress A stellar magnetic the region. accumulated inner the produced or be of crust to release thought are a GFs by The [14–16]. GWs internalwith the of not GW-scopy field, the indispensable. magnetic is [13], the field accretion is mass source energy the the that prove rmmgea F.TeG nrywti h X-QPO the (X- within GWs energy GFs of GW duration of searches The tails the GFs. activated magnetar x-ray from recently the and in [18] discovered QPOs) been have Hz ihymgeie Sswith NS’s magnetized Highly nteucmn er,tegaiainlwv (GW) wave gravitational the years, upcoming the In culy h usproi siltoswith oscillations quasiperiodic the Actually, oscillations seismic the excite would GFs magnetar The ∼ 2 antcfil,wihi rca o h Wdtcin ihth s With field magnetic detection. interior GW the the probe for can search crucial we becau current is time-derivatives, time than which with to evolve field, also compared magnetic would months, frequency to oscillation day the (X-QPO a oscillations lasting quasiperiodic yses x-ray th that observed suggest We the than (GWs). waves gravitational of sources ing ASnmes 95.85.Sz,97.60.Jd numbers: PACS in generation next ET. the and with LCGT asteroseismology VIRGO, GW advanced new a open to 0 e a enlmtdbelow limited been has sec 200 hoyCne,KKHg nryAclrtrRsac Organ Research Accelerator Energy KEK(High Center, Theory antrAtrsimlg ihLn-emGaiainlW Gravitational Long-Term with Asteroseismology Magnetar antcflrsadidcdoclain fmgeas(supe magnetars of oscillations induced and flares Magnetic ∼ 10 5 6]. [5, 44 − 1 eateto hsc,KooUiest,Koo6680,J 606-8502, Kyoto University, Kyoto Physics, of Department 46 r 1–2.I re to order In [10–12]. erg ∼ auiKashiyama Kazumi 10 14 - 16 gravitational so-called G ∼ 0 of 10% ∼ 100 1 n uiioIoka Kunihito and ainisd h S eas h oefrequency mode the reconfigu- because or NS, decay the field inside magnetic ration the measure directly [17]. frequencies discrete have Alfv´en modes h nenlmgei edfo h bevdfrequency observed the from X-QPOs. f measure field directly axial-type magnetic to internal the possibility the interesting oscilla- Alfv´en than an polar longer opens This the last by which produced tions, largely the are believe any, to reasons the [8]. GFs of of one origin is magnetic which frequency Alfv´en crossing Eq.(4)], internal the [see frequency. about this also have is X-QPOs This Actually had resonant. GF most of is spike time initial rise the because a GFs magnetar the in However, GWs. for ployed modes. than longer hs fteGW, the of phase on pends than budgets, nuh hsi eas h Wlmnst spropor- is luminosity GW the (frequency) to because tional is this enough; fteG nryi oprbet h electromagnetic of even the searches GW, to current the comparable The is detect energy. energy to GW necessary the from are analyses if months GW to long-term day [see the a frequencies that X-QPO the show the if We X-QPOs to Eq.(6)]. the close than are longer frequencies much GW last would GFs tar ietepla iigsac.TeF em ffc the affect terms FD The on) so search. and timing also (2FD pulsar and terms the term derivative like (1FD) frequency derivative higher frequency the first the has a h ogtr Weiso lomksi osbeto possible it makes also emission GW long-term The o epooeta W rmmgea F,if GFs, magnetar from GWs that propose we So, em- were modes fluid the studies, previous most In nti etr esgetta h W rmmagne- from GWs the that suggest we Letter, this In ∝ Ψ( B ∼ t = ) ) aln o h ogrtr Wanal- GW longer-term the for calling s), eo h ea rrcngrto fthe of reconfiguration or decay the of se Weiso,i n,wudls longer last would any, if emission, GW e ic h etrn oc smgei n h polar the and magnetic is force restoring the since rnt of trength H udmdswl eectdmr effectively more excited be will modes fluid kHz mgeie eto tr)aepromis- are stars) neutron rmagnetized B ∼ ψ slsig ietepla timing, pulsar the Like lasting. es zto) skb 0-81 Japan 305-0801, Tsukuba ization), ∼ efrmtr ieavne LIGO, advanced like terferometers 0 hteovswt ie h alrexpanded Taylor The time. with evolves that bevdG rqec n its and frequency GW observed e 0H W ol ess pt 10 to up persist could GWs 100Hz ∼ 2 + 0 H W ftpclfli oe iep- like modes fluid typical of GWs kHz . 1sc[01]adisinverse its and [10–12] sec 01 π ∼ 2  f 10 6 ( hs yfiigteG energy GW the fixing by Thus, . t 16 − n t evolution its and G apan t 0 ∼ ) − 0 zoclain rather oscillations Hz 100 2 1 . f ˙ ( t 0 e r o long not are sec 200 aves − t 0 ) 2 + ... ∼  6 0 Hz 100 , times f de- (1) 2 signal to noise ratio (S/N) even if we assume the mini- where d10 = d/10kpc is the distance from the source. mum magnetic field decay required to supply energy for Therefore, if the GW energy is & 1044erg (only a fraction the observed emission from magnetars [see Eq.(9)]. of the GF energy 1044−46 erg) and the data taking In our picture, GWs are produced by the polar oscil- lasts for a day to months,∼ the GWs from magnetar GFs lations while X-QPOs are caused by the axial ones. This within 10 kpc are detectable with enough S/N by the next is implied by the amplitude of X-QPOs 0.1 [18], which generation detectors. The event rate will be 1/10yr, is much larger than the expected polar∼ oscillation am- and more for weaker flares. We should note that∼ the GW plitude . 10−4 [see Eq. (2)]. Thus, the GW searches energy or the deformation ǫ has a large uncertainty. The beyond the X-QPO duration are meaningful. minimum deformation would be ǫ ǫmax/100 in Eq.(2) To clarify the above statement, we first make an since 1% of magnetic energy is released∼ in a GF. We ∼ order-of-magnitude estimate before the detailed analy- can rescale the following results with S/N hc ǫ. sis. We take the fiducial minimum magnetic field in- Meanwhile, Eqs.(4) and (6) do not depend on ∝ǫ, and∝ thus side the NS as B = 1016G because the total magnetic are relatively robust. energy (B2/8π) (4πR3/3) 1049 erg should sup- Since the magnetic fields of magnetars are considered ∼46 4· 2 ∼ 48 4 ply 10 erg 10 yr/10 yr 10 erg for the GF ac- to decay significantly in the lifetime τ 10 yr, the ∼ × ∼ mag tivities once per century (with three GFs out of 10 magnetic field are decaying at least at the rate∼ sources within 30 years) during the lifetime of magnetars,∼ 4 5 τ 10 yr, and the ionospheric observations suggest B˙ B/τmag 10 B16 [G/sec]. (8) mag ≈− ∼− that ∼ 9 times energy is released outside the X-γ ray band∼ [21]. This leads to the evolution of the Alfv´en frequency When the interior magnetic fields are deformed by the ˙ GFs, the induced difference in the moment of inertia of 1 B −9 f˙a 10 B16 [Hz/sec]. (9) the star ǫ is given by ≈ R · √4πρ ∼− 2 3 ∆I (B /8π) (4πR /3) −5 2 During the GW emission, the oscillation frequency ǫ . ǫmax · 3 10 B16 , I ≡ ≡ GM 2/R ∼ × changes by (2) 2 −3 −5 where I 2MR /5 is the moment of inertia of the ∆fa f˙a τGW 4 10 B16 Hz. (10) ≈ 16 | | ∼ | × |∼ × magnetar, B16 (B/10 G) is the interior magnetic field, and M, R≡are the mass and the radius, respec- In principle, the frequency resolution of GW is given by 6 tively [14, 22]. Hereafter, we fix R = 10 cm and M = the inverse of the observational time Tobs,

1.4M⊙. The oscillation energy or the GW energy is about −1 −1 −7 6 ∆f T τ 3 10 B16 Hz (11) the gravitational energy shift caused by the stellar defor- obs ∼ obs ∼ GW ∼ × mation, for long enough data taking. Therefore, we could resolve 2 2 43 4 EGW (ǫ /5) (GM /R) . 8 10 B16 [erg]. (3) the GW frequency evolution to prove the magnetic field ≈ · × decay rate inside the magnetar for B16 . 3.0. In other The oscillation frequencies are determined by the Alfv´en words, the 1FD term could be crucial for keeping S/N in sound crossing, the GW detection. If the magnetic field does not decay va 1 B constantly in the GF phase, higher FD terms may be im- fa = = 100B16 [Hz], (4) R R · √4πρ ∼ portant, where the long-term frequency-time analysis [24] may be useful. where v is the Alfv´en velocity, and ρ M/(4πR3/3) is a Next, we shall move on to more details with the the mass density. If these oscillations are≈ polar mode [17], matched filtering Fisher analyses. We only consider the in which the moment of inertia changes, GWs are emit- l = 2,m = 0 mode as the magnetar oscillation because ted. The luminosity of the GWs are estimated to be one-sided outflow has been confirmed by the radio obser- 5 ...... 5 3 2 LGW (G/5c ) I ij I ij (G/5c ) (ǫI(2πfa) ) vation of the GF from SGR 1806-20 [25]. This suggests ≈ 37 · h−10− i≈ · . 2 10 B16 [erg/sec]. (5) that a GF is a pointlike energy release from a NS, which × would mainly trigger the l =1,m = 0 plus l =2,m = 0 Then the GW duration after the GF becomes mode oscillations. We take m = 0 since there is no spe- 6 −6 cific direction along the azimuthal angle and the rotation τGW EGW /LGW 4 10 B16 [sec]. (6) ≈ ∼ × is slow ( 0.1 Hz) compared with the Alfv´en oscillations. ∼ The characteristic GW amplitude hc [23] is given by The l = 1,m = 0 mode is the dipole oscillation which 1 2 does not contribute to the GW emission. GL / h GW f τ Taking a misalignment between the polar oscillation c π2c3d2f 2  a GW ≈ a · p and the rotation axes as in Fig.1, the waveform of the −23 3/2 −1 . 5 10 B16 d10 , (7) observed plus (+) and cross ( ) GWs are obtained from × × 3 the quadrupole formula, matrix of the parameter estimation error ∆γi is given by the inverse of the Fisher information matrix Γij as ǫI t ∆γ ∆γ = (Γ−1) . The Fisher matrix becomes h+ + ih× = ( + i )cosΨ(t)exp , (12) h i j i ij d A B −τGW  ˜∗ ˜ df ∂hα(f) ∂hα(f) where Γij = 4Re , (15) Z S (f) ∂γ ∂γ α=+X,× n i j 2 2 dΨ 2 2 = (cos θ + 1) sin α cos2(2πνt + φ) where S (f) is the noise spectrum. The accuracies of the A √3  dt  n 2 2  interested parameters are described as ∆B/B = ∆fa/fa, + sin θ(3 cos α 1) + sin 2θ sin 2α cos(2πνt + φ) ,(13) ˙ ˙ ˙ ˙ − ∆B/B = ∆fa/fa and ∆EGW /EGW = 2∆ǫ/ǫ. The reso- lution for the position of the magnetar is 2 2 dΨ 2 2 2 = cos θ sin α sin 2(2πνt + φ) ∆Ω=2π ∆µ ∆φ ∆µ∆φ , (16) B √3  dt  { ph ih i − h i + sin θ sin 2α sin(2πνt + φ) . (14) where µ = cos θ. The S/N is given by } 2 df 2 The stellar oscillation phase Ψ(t) can be Taylor expanded (S/N) =4 h˜α(f) . (17) Z S (f)| | as Eq.(1). In this Letter, we stop the expansion at the α=+X,× n 1FD term, and add ψ0, fa and f˙a as the parameters for the GW waveform. We neglect the proper motion of the We integrate the waveform in Eqs.(15) and (17) for the magnetar. As a whole, the parameters are the strength duration Tobs. We refer advanced LIGO [1] (2nd gener- of deformation ǫ, the mean moment of inertia I, the dis- ation) and ET [4] (3rd generation) for a detector noise tance to the magnetar d, the rotational frequency of the spectrum Sn(f). magnetar ν, the angle between the magnetar rotation and the oscillation axes α, the frequency of the oscillation f , a TABLE I: S/N and parameter determination accuracy its evolution rate f˙ , the initial phase of the oscillation a for the optimal case (ǫ = ǫmax) ψ0, the duration of the GW emission τGW , and the obser- . vational polar and azimuthal angle θ and φ. The position (θ, φ) can be determined independently by the x-ray or detector 2nd generation 3rd generation radio observations. Here, I, d and ǫ are completely de- B16 1.0 2.0 1.0 2.0 generated. We fix I = 1045erg sec2, d = 10kpc and Tobs(& τGW ) 4 months 2 days 4 months 2 days leave ǫ as a free parameter. According· to Eqs.(2),(4),(6) −9 S/N 11 23 170 350 and (9), we set fa = 100B16Hz, f˙a = 10 B16Hz/sec, 6 −5 −−5 2 −10 −8 −11 −10 τGW =4 10 B16 sec and ǫ = 3 10 B16 (i.e., op- ∆B/B 6.4 × 10 1.0 × 10 4.0 × 10 6.5 × 10 × × − − − − timal case). We also set α = π/4, ν = 0.1Hz, θ = π/2, ∆B/˙ B˙ 8.9 × 10 6 9.0 × 10 3 5.6 × 10 7 5.8 × 10 4 φ = π/4 and ψ0 = π/4. − − − − ∆ν/ν 4.2 × 10 8 1.3 × 10 6 2.7 × 10 9 8.4 × 10 8 −2 −3 ∆EGW /EGW 0.27 0.14 1.7 × 10 8.8 × 10 rotation axis magnetic axis −2 −3 ∆τGW /τGW 0.18 0.090 1.2 × 10 5.8 × 10 − − α ∆α/α 0.030 0.015 1.9 × 10 3 9.8 × 10 4 giant flare detectors −4 −5 ∆Ω 0.044 0.010 1.7 × 10 4.2 × 10 ν θ φ),( Table.I shows the S/N and the parameter determina- magnetar tion accuracy assuming enough data taking, Tobs & τGW . l = 2, m = 0 oscillation For the optimal case (ǫ = ǫmax), even 2nd generation de- tectors can detect the GWs with enough S/N. The de- f a , f& a … tection of the GW is possible when ǫ & 0.1ǫmax and ǫ & 0.01ǫmax for 2nd and 3rd generation detectors, respec- 8 FIG. 1: The configuration of the magnetar and detectors. tively. Thanks to the large Q value f τ 10 , ∼ a × GW ∼ parameters related to the phase of the GW (i.e., fa, f˙a and ν) could be determined accurately. When fixing B, In the matched filtering analysis, we can compute the errors scales as (S/N)−2 for ∆Ω, and (S/N)−1 for other determination accuracy of these parameters using the parameters. Fisher matrix formalism with a waveform template in the Figs.2 and 3 show the dependence of the S/N and the frequency domain h˜(f) [26, 27]. The variance-covariance accuracies on the data taking time Tobs. We can clearly 4

5⋅1016 the GF energy are initially injected to the polar Alfv´en 45 mode, and the dependence of Alfv´en mode frequencies 4⋅1016 25 on the magnetic field may become different from Eq.(4). 3⋅1016 10 The effects of the crust and superfluidity are also to be investigated in the future. 2⋅1016 S/N = 1 (2nd) The change in the rotational frequencyν ˙ could be- come important as the 1FD. Although the magnetar −12 B [G] spindown rate is usually small 10 Hz/sec com- pared with Eq.(9), the sudden increase∼ − in the GF phase 1⋅1016 ∆ν/ν 10−4 has been reported in the August 27, 1998 9⋅1015 ∼− 8⋅1015 S/N = 10 (3rd) event from SGR 1900+14 [28]. Since ν = fa, we could 15 6 7⋅10 also measureν ˙ independently. ⋅ 15 6 10 Finally, we should mention the strategic change in the 5⋅1015 102 103 104 105 106 107 analysis of the GW waveform. When discussing GWs Tobs [sec] from a single NS or black hole, the waveform usually has a short damping time . 10sec [29, 30], hence the evolution of the waveform due to the daily and yearly FIG. 2: The contour lines of S/N in the plane of the motion of the earth could be neglected. However in the magnetic field of magnetars and the GW observation time. case of the long-term oscillations with & a day, we have Solid (dashed) lines show S/N = 1, 10, 25, 45 (S/N = 10) using 16 to take into account these effects as in the analysis of a 2nd (3rd) generation detector. B = 10 G is the minimum interior magnetic field required for magnetar activities. GW from a pulsar [31, 32]. We thank N. Seto, M. Ando, K. Yagi and T. Naka- mura for much useful advice. This work was supported by the Grant-in-Aid for the Global COE Program ”The 1 Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and 0.1 KAKENHI, 19047004, 21684014, 22244019, 22244030.

0.01

accuracy · | B | 0.001 EGW [1] Harry, G.M., (for the LIGO Scientific Collabora- ∆ Ω tion) 2010, Class. Quantum. Grav. 27, 084006. 0.0001 [2] wwwcascina.virgo.infn.it/advirgo/ B = 1.5 × 1016 G [3] Kuroda, K.,(on behalf of the LCGT Collabora- tion) 2010, Class. Quantum Grav, 27, 084004. 102 103 104 105 106 107 [4] Hild, S. et al, 2010, arXiv:1012.0908. [5] Andersson, N. and Kokkotas, K. D., 1998, MN- Tobs [sec] RAS, 299, 1059. [6] K.D. Kokkotas and B.G. Schmidt, Living Rev. Relativity FIG. 3: The parameter determination accuracies with respect 2, 2 (1999), http://www.livingreviews.org/lrr-1999-2. to the GW data taking time. The solid, dotted and dotted- [7] Thompson, C. and Duncan, R. C., 1992, ApJ, 392,L9. dash lines show |B˙ |, EGW and ∆Ω, respectively. Thin (thick) [8] Thompson, C. and Duncan, R. C., 1995, MN- lines shows the results using 2nd (3rd) generation detectors. RAS, 275, 255. We set B = 1.5 × 1016G. [9] Duncan, R. C., 1998, ApJ, 498, L45. [10] Mazets, E. P. et al., 1979, Nature, 282, 587. [11] Hurley, K. et al., 1999, Nature, 397, 41. 5 [12] Hurley, K. et al., 2005, Nature, 434, 1098. find that the long-term data taking for a day ( 10 sec) [13] Chatterjee, P., Hernquist, L. and Narayan, R. 2000, ApJ, 7 ∼ to months ( 10 sec) is necessary for high S/N and pa- 534, 373. ∼ rameter accuracies, especially B˙ . [14] Ioka, K., 2001, MNRAS, 327, 639. Since a NS has various oscillation modes, it is hard, at [15] Corsi, A. and Owen, B. J., arXiv:1102.3421. this stage, to answer which oscillation mode is mainly ex- [16] Abadie, B. P. et al., (The LIGO Science Collabora- tion), arXiv:1011.4079. cited at the GF. Also, we should consider the mode cou- [17] Sotani, H. & Kokkotas, K. D., 2009, MNRAS, 395, 1163. pling between the polar Alfv´en mode and other modes. [18] Israel, G. L. et al., 2005, ApJ, 628, L53. If the coupling is strong, the polar Alfv´en mode oscilla- [19] Sotani, H., Kokkotas, K. D. & Stergioulas, N., 2008, MN- tion cannot last so long as a day to months even if RAS, 385, 2161. ∼ 5

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