J. Astrophys. Astr. (2020) 41:14 Ó Indian Academy of Sciences

https://doi.org/10.1007/s12036-020-09631-0Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

Bursts of gravitational waves due to crustquake from

BISWANATH LAYEK* and PRADEEPKUMAR YADAV

Birla Institute of Technology and Science, Pilani Campus, Pilani, Jhunjhunu 333 031, India. E-mail: [email protected]

MS received 13 November 2019; accepted 29 March 2020

Abstract. We revisit here a possibility of generation of (GW) bursts due to a very quick change in the quadrupole moment (QM) of a deformed spheroidal as a result of crustquake. Since it was originally proposed as a possible explanation for sudden spin-up () of pulsars, the occurrence of crustquake and its various consequences have been studied and discussed quite often in the literature. Encouraged by recent development in gravitational wave (GW) astronomy, we re-investigate the role of crustquake in the emission of GWs. Assuming exponential decay of excitation caused by crustquake, we have performed a Fourier analysis to estimate the GW strain amplitude h(t), characteristic signal amplitude hcðf Þ and signal-to-noise ratio (SNR) of the burst for the Crab pulsar. For exotic , multifold enhancement of these quantities are expected, which might make quark a potential source of gravita- tional waves. The absence of such bursts may put several constraints on pulsars and such hypothetical stars.

Keywords. star—pulsar—crustquake—gravitational wave.

1. Introduction based on suggestions made by several authors (see Abadie et al. 2011 and the references therein) through The remarkable first-ever direct detection of GWs in their works in this area. As per those suggestions, 2015 by LIGO (Abbott et al. 2016) opened a new era several sources namely the flaring activity, the for- in gravitational astronomy. Observations supple- mation of hypermassive NS following coalescence of mented with numerical simulations identified a black binary neutron stars, etc. are capable of exciting hole merger as the source for ripples’ in the space- quasinormal modes of a pulsar and hence may emit time. Since then there have been quite a few signifi- GWs. The timing glitch can be one of these sources, cant detections of GWs. The peak strain amplitude (h) which has the potential to excite quasinormal modes for all these detections have been in the range 10À21– in the parent pulsar. Although the searches for GWs 10À22. In view of these detections, we should be during August 2006 Vela pulsar (Abadie et al. 2011) hopeful that the gravitational waves produced by timing glitch produced no detectable GWs, with the compact isolated astrophysical objects like isolated improving sensitivity of advanced detectors, continu- pulsars can be measurable in the near future by more ous attempts in this direction may produce more sensitive upcoming ground-based advanced detectors, conclusive results in near future. namely, aLIGO, VIRGO, the third generation Einstein In the literature, there have been a few other theo- Telescope (ET), etc. Prior to the GWs detections as retical works that advocate gravitational wave mentioned above, there have been a few attempts for astronomy in the context of isolated neutron stars. GWs searches from isolated pulsars as well. The There have been discussions on the emission of GWs attempt for GWs search associated with the timing from deformed neutron stars (Zimmermann & Sze- glitch in the Vela pulsar in August 2006 (Abadie et al. denits 1979), crustal mountains in radio pulsars or in 2011) was one among those which are worth men- low mass X-ray binaries (Haskell et al. 2015), con- tioning. The motivation of the above searches were tinuous emission of GWs from a triaxially symmetric rotating due to permanent ellipticity 14 Page 2 of 8 J. Astrophys. Astr. (2020) 41:14

(Jones 2002), etc. Further details on the above-men- damped oscillator (as we propose in this paper). This tioned sources of GWs from isolated pulsars can be causes a change in the pulsar’s oblateness and hence it found in a detailed review by Lasky (2015). In the should emit gravitational radiation. Note that the context of bursts of GWs from an isolated NS, there detailed mechanism of excitation or de-excitations are has also been an interesting suggestion by Bagchi still unknown, neither the consequences of these are et al. (2015), where the authors have proposed a established experimentally. In this work, we assume the t unified model for glitches, antiglitches, and generation evolution of ellipticity behaves like ðtÞ/expðÀ sÞ. of GW from isolated pulsars due to phase transitions where s characterizes the relaxation time scale during inside the core of a pulsar. which major changes of D or equivalently, DQ of QM In view of these theoretical proposals, we explore occurs following crustquake. Such an assumption can here another possible source of GWs from isolated be justified from the fact that the oscillations caused by pulsars, which may arise due to a very fast-changing excitation is not expected to have a well-defined single elastic deformation () and hence the change in (angular) frequency (x). Rather, it is more natural to quadrupole moment (Q) of a pulsar as a result of expect that it consists of an infinitely large number of crustquake. Nearby Crab pulsar exhibiting small size normal mode frequencies. The study of de-excitation glitches of order 10À8 that can be explained success- can then be achieved through Fourier analysis by fully using the crustquake model will be the most modeling the evolution of ellipticity like critical/over- likely candidate to test our proposal. In this context, critical damped oscillators. Also, as the strain amplitude 2 we should mention the work by Keer and Jones (2015) h(t) depends on Q€ ðtÞ, generation of GWs with a sig- on crustquake initiated excitation of various oscilla- nificant strain amplitude is expected only if Q changes tion modes in a pulsar and hence the emission of in a very short duration (i.e., smaller s) which can be possible GWs from such oscillations. In their work, achieved only if the star behaves like critical/overcrit- the authors had assumed that excitations decay after ical damped oscillators. It will be interesting to study executing several oscillations and they made an order the properties of matter in detail to check if at all a star of estimate for the strain amplitude of GWs behaves like such a damped oscillator. arising due to the excitation of f-mode. According to The motivation for theoretical studies on possible their picture, the estimated strain amplitude depends emission of GWs from isolated pulsars can be manifold. on various factors such as amplitude of oscillations, Firstly, the event rate of (number of events which can rotational frequency of the star, etc. in contrast to our produce significant GWs in a per year) GWs picture where h turns out to be independent of these emission from pulsars is expected to be quite high in quantities. comparison to the event rate from various other sources In this work, we assume that the decay of crustquake of gravitational radiation, namely mergers of neutron initiated excitation behaves like a critically damped stars (NS) and/or black holes (BH). For example, it has oscillator and estimate the characteristics of GW bursts. been mentioned in Riles (2013) that event rates lie in the We will take pulsar of spheroidal (oblate) shape, the range 2  10À4–0.2 per year for initial LIGO detection ellipticity/oblateness () of which can be defined of a NS–NS coalescence, and 7  10À5–0.1 per year for IzzÀIxx through, Izz 6¼ Ixx ¼ Iyy and  ¼ . Here, Izz ¼ I0 a NS–BH coalescence, etc. In contrast, there is a huge 2 2 1 2 2 5 Mc ; Ixx ¼ 5 Mða þ c Þ are the MI of the star about catalogue of pulsars and more than 450 glitch events the (symmetric) z-axis and x-axis, respectively (c\a). (crustquake can account for small size 10À8–10À12 1 I0 is the MI of the spherical star. The quadrupole glitches) have been recorded and analyzed (Ezpinoza moment of the spheroidal star can be taken as Qzz ¼ et al. 2011) quite extensively. Thus crustquake can be 2 2 2 quite a frequent phenomenon when one considers a very 5 ðc À a ÞQ which is related to the ellipticity large number of pulsars in our galaxy. The other pur- through, Q ¼À2I0. The negative value of Q arises solely due to the oblate shape of the star. Obviously, a pose of the GWs study from pulsar might be to under- spheroidal star does not radiate continuous gravitational stand a few features of the pulsar itself. For example, in waves due to its spherical symmetry. However, the this proposal, the strain amplitude of GWs crucially crustquake can initiate excitations to the star and the star depends on the relaxation time scale (s) towards the new de-excites itself to achieve a new equilibrium (more equilibrium value of oblateness (hence, QM). Thus oblate to less oblate) through several oscillations (like gravitational wave studies from pulsars can shed light an underdamped oscillator) as discussed by Keer and Jones (2015) or it can decay like a critical/overcritical 1http://www.jb.man.ac.uk/pulsar/glitches/gTable.html. J. Astrophys. Astr. (2020) 41:14 Page 3 of 8 14 on crustquake, possible mechanism of strain relaxation, about 105 cm. The deformation parameter of the crust etc. To the best of our knowledge, so far no serious can be characterized by its ellipticity,  ¼ IzzÀIxx as I0 attempt has been made to determine strain relaxation defined earlier in Section 1. Here, we briefly discuss time scale. This is because of the fact that in the context the basic features of crustquake model (Baym & Pines of crustquake model for glitches, s is not relevant in 1971). At an early stage of formation, the crust determining glitch size and other features of glitches. solidified with initial oblateness o (unstrained value However, as we show below, s is crucial to have a or reference value) at a much higher rotational fre- significant value of h in our model. Similarly, if a few quency. As the star slows down, the ellipticity ðtÞ small size glitches are caused by crustquake, then the decreases, leading to the development of strain in the glitches will be followed by little bursts of GWs which crust due to the inherent crustal rigidity. Finally, once can be a falsifiable prediction of a crustquake. the breaking stress is reached, the crust cracks and The paper is organized in the following manner. In releases its (partial) energy. The total energy of the Section 2, we briefly review the basic relevant fea- pulsar can be written as (Baym & Pines 1971) tures of crustquake model as pictured by Baym and 2 Pines (1971). We shall also discuss here the maximum L 2 2 E ¼ E0 þ þ A þ Bð À 0Þ ; ð1Þ possible values of ellipticity parameter () of a neutron 2I star crust which can sustain the crustal stress. The where A and B are two coefficients, the values of methodology is presented in Section 3. Here, the which are available (Baym & Pines 1971). The equi- expressions of GW strain amplitude h(t), characteristic librium value of  can be obtained by minimizing E time scale s, characteristic signal amplitude hcðf Þ and while keeping angular momentum (L ¼ IX) fixed. The the signal-to-noise ratio (SNR) are presented. The value thus obtained is given by results are presented in Section 4, followed by the X2 dI B B concluding remarks in Section 5.  ¼ þ  ’  : ð2Þ 4ðA þ BÞ d A þ B 0 A þ B 0

2. Crustquake: The basic relevant features The crustquake causes the shift in the reference value of 0 by D0, which in turns changes the equilibrium value  by D, where Crustquake was originally proposed by Ruderman (1968) as a possible explanation for the sudden spin- B D ¼ D : ð3Þ up (glitches) of otherwise highly periodic pulsar PSR A þ B 0 0833-45 (Vela). Currently, the superfluid vortex For normal neutron star of 10 km radius and 1 km model (Anderson & Itoh 1975) is the leading model to crust thickness, we have A ’ 6  1052 erg and B ’ explain glitches. However, in the context of glitches or 47 otherwise, crustquake model has been revisited 6  10 erg (Baym & Pines 1971) which in turn repeatedly. There have been discussions in the litera- provides the value of  as À5 ture suggesting the involvement of crustquake in NS  ¼ 10 0: ð4Þ physics, such as an explanation for the giant magnetic flare activities observed in (Thompson & The upper limit of ellipticity can be constrained by noting Duncan 1995; Lander et al. 2015), a trigger mecha- that the crust has a critical strain, say, Dcr such that nism for angular momentum transfer by vortices A j À  j¼ \D : ð5Þ (Eichler & Shaisultanov 2010; Melatos et al. 2008; 0 B cr Warszawski & Melatos 2008) or to explain the change À2 À4 in spin down rate that continues to exist after a glitch The theoretical value of Dcr (10 –10 ) had been event (Alpar et al. 1994). In view of these discussions, estimated earlier by Jones (2002). However, in a crustquake seems to be an inevitable event for a large recent work (Horowitz & Kadau 2009) on crustal number of NS. This work will assume the occurrence breaking strain of NS, the authors have done detailed of crustquake in a NS and study one of its many molecular dynamics simulations by modeling the crust possible consequences, namely, the generation of by a single pure crystal and obtained the value, short duration (bursts) GWs due to crustquake. Dcr ¼ 0:1. As per their work, the value of Dcr remains The crustquake in a neutron star is caused due to the around 0.1 even in the presence of impurities, defects, existence of a solid elastic (Ruderman 1969; Smolu- etc. Note that this value is at least an order of mag- chowski & Welch 1970) deformed crust of thickness nitude higher than the maximum value quoted in Jones 14 Page 4 of 8 J. Astrophys. Astr. (2020) 41:14 Z (2002). Here, for the estimate of maximum GWs 2I  1 a þ ix QðtÞ¼À2I ðtÞ¼À 0 i eÀixtdx: strain amplitude, we take Dcr ¼ 0:1 and we obtain the 0 2 2 p 0 a þ x upper limit of  as ð10Þ B \ D ¼ 10À6: ð6Þ A cr Expansion of ðtÞ and/or Q(t) in a Fourier series in frequency domain allows us to apply linear pertur- At the onset of crustquake, the above value will be bation theory for the estimate of strain amplitude À6 taken as an initial oblateness, i ¼ 10 for the cal- from a source oscillating with infinitely large number culation of GW strain amplitudes. We will assume the of normal modes. The resultant strain amplitudes D  change of ellipticity ¼ g (where 0\g\1isa hijðtÞ can be obtained by summing up the contribu- numerical factor characterizing the fraction of strain tion from each frequency. The strain amplitude (i.e., released due to crustquake), to estimate strain relax- one of the components say, hzzðtÞ¼hðtÞ) in a par- ation time s. The above estimate has been provided for ticular frequency interval) can then be written as normal NS. It has been proposed that more exotic (Riles 2013) (see Keer & Jones (2015) and references Z therein) may exist having a very large solid core 2G 4GI xmax hðtÞ¼ Q€ ¼ 0 dxðxÞx2eÀixt; ð11Þ which can sustain a large ellipticity. For such a star, 4 4 c d pc d xmin 52 50 A ’ 8  10 erg and B ’ 8  10 erg and hence i À3 where d is the distance of the pulsar from the Earth can be as large as 10 (for the same value of Dcr ¼ 0:1 as above) as obtained from Equation (6). Thus, the and we have taken a frequency interval between xmin quark star can also be a potential source of gravita- and xmax which are the minimum and maximum tional wave burst as a result of crustquake. angular frequencies respectively. The upper cut-off frequency is a requirement to validate the slow motion approximation. In such an approximation, the wave- 3. Gravitational wave burst caused by crustquake: length k of gravitational radiation must be much larger Methodology than the size of the source. Thus a frequency of about 1 kHz implies the wavelength of radiation as about As per the discussion in Section 1, the strain ampli- k ¼ 300 km, which is quite large compared to the size tude due to change of ellipticity following starquake of the source (  10 km). We will set 1 kHz frequency will be determined by assuming that the star behaves as an upper limit and slow motion approximation will like a critically damped oscillator (from now onwards, be assumed up to this frequency. Error arising due to we will do the analysis for critically damped oscilla- small deviation from this approximation for a one kHz tor; however, the methodology will be equally appli- frequency will be neglected. There is no such cable for overcritical damped oscillator). We will take requirement on minimum limit of frequency, however, the evolution of ellipticity as we will estimate h in the frequency range 100 Hz–1 kHz in accordance with the sensitivity of existing Àt=s Àat ðtÞ¼ie ¼ ie ; ð1=s  aÞ: ð7Þ interferometers. Thus the strain amplitude (the real part of Equation (11)) can be written as Here, i is the initial value of ellipticity of the neutron star at the onset of crustquake. As we are interested to Z " # study the evolution of h(t) within characteristic time s, 4GI  xmax acosðxtÞþxsinðxtÞ hðtÞ¼ 0 i x2dx: we neglect the eÀatt term in the above expression. 4 2 2 pc d xmin a þ x Expanding ðtÞ in a Fourier series, Z ð12Þ 1 1 ðtÞ¼pffiffiffi dxðxÞeÀixt: ð8Þ p Expressing in terms of dimensionless quantity 0 0 x x ¼ a  xs, the above expression can be re-written as Here, ðxÞ is the corresponding (complex) amplitude Z " ÀÁ ÀÁ# x0 x0t x0t 4GI  max cos þ x0sin in frequency domain and is given by hðtÞ¼ 0 i s s x02dx0 4 2 02 ! pc ds x0 1 þ x Z min 1 ! ! ! 1 ixt i a þ ix 2 ðxÞ¼pffiffiffi dtðtÞe ¼ pffiffiffi : ð9Þ 1 kpc  10À4 s p p a2 þ x2 ¼3:5  10À24 i KðtÞ: ð13Þ 0 d 10À6 s Hence, the quadrupole moment can be written as J. Astrophys. Astr. (2020) 41:14 Page 5 of 8 14

The numerical prefactor is calculated by taking MI of with the strain energy released by the star using 45 2 the star I0 as 10 gm-cm . The function K(t) (for a Equation (1)as 0 0 fixed set of values of xmin and xmax) is given by 3G I22 Z "#ÀÁ ÀÁ DE ¼ BD ’ 0 i ; ð18Þ x0 x0t x0t 5 5 max cos þ x0sin 5c s KðtÞ¼ s s x02dx0: ð14Þ 02 x0 1 þ x i.e., min ! ! 2=5 1=5 The frequency-dependent information in the burst can  1039erg s ¼ 5  10À5 i s: ð19Þ also be inferred from the frequency distribution, 10À6 BD dhðtÞ hx0 ðtÞ¼ dx0 and is given by (using Equation (13)) " ÀÁ ÀÁ# We will use Equations (13), (14) and (19) to deter- x0t 0 x0t cos s þ x sin s 02 mine the GW strain amplitude h by taking the hx0 ðtÞ¼N x : ð15Þ 02 1 þ x appropriate values of i and D. Here, we should mention that while calculating rate of energy loss Note that hx0 ðtÞ is a dimensionless quantity, to be using Equation (16), we have essentially taken con- multiplied by s to get the proper frequency distribu- tribution of all frequencies. However, for consistency, tion hxðtÞ. The prefactor (N) in the above equation is one should apply Equation (10) and carry out the 24 of order 10À (to be discussed in the next section). integration over frequency up to a maximum limit Now, as the strain amplitude depends on the relax- (say, 1 kHz) to validate slow motion approximation. ation time s, we will provide a rough estimate for s as 1 However, the rate of energy loss being s5 dependent, follows. Here we should mention that, so far no seri- the order of magnitude of s will not be affected sig- ous attempt has been made to determine the precise nificantly due to the above factor. value of s (which is a typical glitch time). It is understood, since in the context of crustquake model for glitches, this is not necessary as the size of glitches 4. Results and other relevant features of glitches are not sensitive to the value of s. If crustquake causes glitches, then The strain amplitude in Equation (13) of gravitational uncertainty of such time scale may be resolved waves in a given frequency interval in our scenario through pulsar timing by narrowing down to the now depends on  and on the time scale s of strain characteristic time-scale for spin up events/glitches. i relaxation. Following the arguments provided in At this stage, we may take it as a parameter which can Section 2, for a normal NS the maximum possible be fixed by observation in future by studying the À6 impact of s on various consequences. However, we value of i is in the order of 10 and the value of D will provide a rough estimate of s by assuming that (required to determine s) will be determined by the the strain energy goes into gravitational radiation and prefactor g (0\g\1). The value of pre-factor g and the emission is dominated within the time interval s. hence, D can be constrained through the crustquake Thus the estimate provided below may set a lower model for glitches. For small size glitches in Crab À8 limit on s. With this assumption, we can now write the pulsar, D ¼ 10 (Ezpinoza et al. 2011), which cor- rate of energy loss due to GW radiation as responds to g ¼ 0:01 (i.e., 1% release of strain). For consistency, we should mention that for a Crab pulsar, ... dE 3G 6G 2 2 6 À2at the above value of D corresponds to only few years ¼À Q 2 ¼À I0 i a e : ð16Þ dt 10c5 5c5 waiting time for the next glitch to occur, which is 3 Here we have used Qzz ¼À2Qxx ¼À2Qyy relevant much less than the spin-down time scale (10 years) for spheroidal pulsar and quadrupole moment tensor of Crab pulsar and this fact is fairly consistent with the observations. For a quark star, the value of  can be of Qij which is traceless, i.e., Qzz þ Qxx þ Qyy ¼ 0. The i À3 3 net energy loss DE in a typical time s now can be the order 10 , an enhancement of order 10 times obtained from Equation (16)as compared to a normal neutron star. Z We will take the value of s as obtained from 1=a 2 2 6G 3G I  47 À6 2 2 6 À2at 0 i Equation (19). Assuming B ¼ 6  10 erg, i ¼ 10 DE ¼À 5 I0 i a dte ’ 5 5 : ð17Þ 5c 0 5c s and D ¼ 10À8, we obtain the value of s ’ 10À4 s. For 2 i 1=5 We can estimate the value of the relaxation time scale exotic quark stars, s being proportional to ðBDÞ is of s by (approximately) equating the above energy loss the same order as for a normal NS. The strain 14 Page 6 of 8 J. Astrophys. Astr. (2020) 41:14 amplitude hðsÞ for Crab as obtained from Equa- tions (13)–(14) is provided in Table 1. The time evolution of h(t) is shown in Figure 1(a). Here, h(t)is the frequency integrated (over the frequency range 100 Hz–1 kHz) strain amplitude. It is obvious that shorter the time scale s for spin up events, larger the strain amplitude one expects. As we see from the table, the value of strain amplitude for Crab is expected to be of the order 10À25. For a same distance quark star h (=10-22) is enhanced by order 103 com- pared to Crab. The frequency density distribution of dh the strain amplitude dx0 =hðsÞ (dividing by hðsÞ is just for convenience) evaluated at t ¼ s as obtained from Equation (15) is shown in Figure 1(b). The plot shows dh that the proper frequency density dx =hðsÞ (using dh dh À3 dx ¼ s dx0) is of order 10 . Thus, the strain amplitude hðt; xÞ is almost frequency-independent in the fre- quency range 100 Hz–1 kHz. Now whether such bursts of GWs have the potential to be detectable against signal noises depend on var- ious factors such as sensitivity of interferometer, proper ‘template’ to analyze burst, etc. There are several ways to characterize the detectability of signal strength from a source against the background noise. One such commonly used method (Sathyaprakash & Schutz 2009; Moore et al. 2014) is the comparison of square rootp offfiffiffiffiffiffiffiffiffiffiffi power spectral density (PSD) for the hcðfffiffiÞ source, Shðf Þ ¼ p and for the noise, Figure 1. (a) Plot shows the time evolution of strain f pffiffiffiffiffiffiffiffiffiffiffi amplitude h(t) within characteristic time, s ¼ 10À4 s. Here, hnðfffiffiÞ Snðf Þ ¼ p . Here, hcðf Þ is the characteristic signal h(t) is the frequency integrated (over the frequency range f 100 Hz–1 kHz) amplitude. (b) Plot shows the frequency amplitude in the frequency domain and hnðf Þ is the dhðx0Þ effective noise that characterizes the sensitivity of a density distribution of the strain amplitude as a dx0 detector. Note that Shðf Þ and Snðf Þ both have a 0 0 dhðx Þ À1=2 function of x (¼ xs). has been calculated at time dimension of Hz (see Sathyaprakash & Schutz dx0 2009; Moore et al. 2014 for details), whereas hcðf Þ t ¼ s. Note, the above distribution is dimensionless and it and hnðf Þ are dimensionless. The characteristic signal has to be multiplied by s to get the proper frequency density dhðxÞ amplitude hcðf Þ is defined as hcðf Þ¼f jh~cðf Þj, where distribution dx .

Table 1. Typical order of magnitude for strain amplitude, h and characteristic signal amplitude, hcðf Þ for the bursts. h~cðf Þ is the Fourier transform of the signal amplitude The values of h are provided for Crab and a near-by ~ À4 h(t). In our case, hcðf Þ can be estimated (Sathyapra- hypothetical quark star at t ¼ s for s ¼ 10 s. The values kash & Schutz 2009)as of i are chosen as per the argument provided in the text. Z ! Characteristic signal amplitude h ðf Þ is quoted for a fre- 1 c ~ i2pft a þ i2pf quency of 1 kHz. hcðf Þ/ dthðtÞe ’ h 2 2 2 : ð20Þ 0 a þ 4p f

Star i hðt ¼ sÞ hcðf ¼ 1 kHzÞ In the above equation, we have used the fact that the Crab (d ¼ 2 kpc) 10À6 10À25 10À26 strain amplitude is approximately constant during the Quark star (d ¼ 2 kpc) 10À3 10À22 10À23 Dh time interval Dt ¼ s ( h ’ 0:1 as shown in Figure 1(a)). The characteristic signal amplitude is then given by J. Astrophys. Astr. (2020) 41:14 Page 7 of 8 14

result in emission of GW bursts. Our estimated strain ~ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif hcðf Þ¼jf hcðf Þj ¼ h : ð21Þ amplitude turns out to be smaller than the values as a2 þ 4p2f 2 obtained in Bagchi et al. (2015). This is mainly due to For s ¼ 10À4 s and for a frequency range 100 Hz–1 smaller value of characteristic time scale s (about one kHz, the characteristic signal strain lies in the range microsecond, typical value for a QCD phase transition hcðf Þ¼(0.01–0.1) h. Hence, the signal-to-noise ratio, time). We have also presented our estimate of h for a SNR ¼ hcðf Þ can be estimated using the data provided typical quark star (assumed to be at a same distance as hnðf Þ Crab pulsar). The strain amplitude for such for the noise amplitude h ðf Þ. For h ðf Þ, we utilize the n n (about 10À22) is increased by about 103 order of data available in the literature (Sathyaprakash & magnitude compared to a normal star. This happens Schutz 2009; Hild & Abernathy 2011) for the Einstein due to high ellipticity  at the onset of crustquake. Telescope2 and the values lie between 10À23–10À22 i for the frequency range 100 Hz–1 kHz. For these values of hnðf Þ, SNR (at f ¼ 1 kHz) in our case turns out be of order 10À4 for Crab and 0.1 for a same 5. Conclusion distance quark star. We conclude this section by comparing strain We investigated a possibility of generation of gravita- amplitudes as obtained from isolated pulsars (through tional wave burst caused by crustquake due to a very different mechanism) which were suggested by sev- fast change of QM of the star. Modeling the decay of eral authors earlier. For example, as we have already crustquake initiated excitation behaves like a critically pointed out in Section 1 that Keer and Jones (2015) damped system, we estimated the possible values of had estimated the value of strain amplitude resulting strain amplitudes of the burst. As we have mentioned, from neutron star oscillations initiated due to star- such modeling (i.e., exponential decay) can be justified quake (through somewhat different model). Their as the excitation expected to have a very large number suggested values for a kHz frequency are larger by an of frequencies instead of a well-defined single fre- order of magnitude, however, there was an assumption quency. With this model, GW strain amplitude depends that energy transfer to oscillations is instantaneous on ellipticity i at the onset of crustquake and relaxation -6 (i.e., causality is not taken into consideration). time scale s. We fixed the value of i (¼10 ) as sug- Similarly, GW emission from a triaxial pulsar has gested in Horowitz and Kadau (2009) based on detailed been discussed in the literatures (Jones 2002). molecular dynamics simulations on NS crust. For s,we Although the triaxiality is not a criterion in our sce- have provided an order of magnitude estimate (’10-4) nario, however, even a triaxial star may undergo by assuming that the strain energy goes into emission of crustquake. In this case, the continuous emission of GWs. For this set of values, hðsÞ (integrated over fre- GWs will be followed by sudden emission of GW quency range 100 Hz–1 kHz) turns out to be of order burst. The strain amplitudes from a triaxial Crab 10À25 for Crab and the value is almost constant within À26 pulsar turns out to be of order ðhÞtr  10 for the time s. We also estimated the characteristic signal À6 values of ellipticity, tr ¼ 10 . The above quoted amplitude hcðf Þ and the signal-to-noise ratio (SNR) for values of ðhÞtr is provided for Crab (time period T ¼ the burst. We noticed that there is a multifold increase in 33 ms). As ðhÞtr depends on time period of rotation, the values of these quantities for an exotic quark star. the magnitude can be even smaller for slowly rotating We again point out that the value of h(t) depends on s, NS. Our estimate of strain amplitude for the burst of the precise value of which is still unknown. It is there- GW as a result of crustquake (refer Table 1), turns out fore important to fix this characteristic time scale to be of an order of magnitude larger compared to the (which is a typical spin-up time of glitch) through corresponding value for triaxial stars. We should observation to reduce the uncertainty in estimating emphasize that GW bursts in our case is expected strain amplitude in this model. So far, the best resolved even for a spheroidal star for which continuous time for spin-up has been reported (Dodson et al. 2002) emission will be completely absent. to be  40 s for Vela pulsar, which is far away from the Also, there is a proposal by Bagchi et al. (2015) requirement. We are hopeful that next-generation that several phase transitions inside the core of a telescopes such as MeerKAT, Giant Magellan, Square pulsar can generate quadrupole moment which can Kilometer Array, etc. will enhance the chances of direct glitch (sort of as it happens) detections and constrain the 2refer http://www.et-gw.eu/index.php/etsensitivities#references. characteristic spin-up time for pulsar glitches. 14 Page 8 of 8 J. Astrophys. Astr. (2020) 41:14

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