The gravitational wave signal of the short rise fling of galactic run away pulsars
aa r X i v : . [ a s t r o - ph ] O c t Gravitational wave signal of the short rise fling of galactic run away pulsarsACCEPTED FOR PUBLICATION IN JCAP 17/10/2008
Herman J. Mosquera Cuesta , , Carlos A. Bonilla Quintero Instituto de Cosmologia, Relatividade e Astrof´ısica (ICRA-BR), Centro Brasileiro de Pesquisas F´ısicasRua Dr. Xavier Sigaud 150, CEP 22290-180, Urca Rio de Janeiro, RJ, Brazil ∗ (Dated: December 8, 2018)Determination of pulsar parallaxes and proper motions addresses fundamental astrophysical openissues. Here, after scrutinizing the ATNF Catalog searching for pulsar distances and proper motions,we verify that for an ATNF sample of 212 Galactic run away pulsars (RAPs), which currently runacross the Galaxy at very high speed and undergo large displacements, some gravitational-wave(GW) signals produced by such present accelerations appear to be detectable after calibrationagainst the Advanced LIGO (LIGO II). Motivated by this insight, we address the issue of the pulsarkick at birth, or short rise fling from a supernova explosion, by adapting the theory for emissionof GW by ultrarelativistic sources to this case in which Lorentz factor is γ ∼
1. We show thatduring the short rise fling each run away pulsar (RAP) generates a GW signal with characteristicamplitude and frequency that makes it detectable by current GW interferometers. For a realisticanalysis, an efficiency parameter is introduced to quantify the expenditure of the rise fling kineticenergy, which is estimated from the linear momentum conservation law applied to the supernovaexplosion that kicks out the pulsar. The remaining energy is supposed to be used to make the starto spin. Thus, a comparison with the spin of ATNF pulsars having velocity in the interval 400 − − is performed. The resulting difference suggests that other mechanisms (like differentialrotation, magnetic breaking or magneto-rotational instability) should dissipate part of that energyto produce the observed pulsar spin periods. Meanwhile, the kick phenomenon may also occur inglobular and open star clusters at the formation or disruption of very short period compact binarysystems wherein abrupt velocity and acceleration similar to those given to RAPs during the shortrise fling can be imparted to each orbital partner. To better analyzing these cases, pulsar astrometryfrom micro- to nano-arsec scales might be of much help. In case of a supernova, the RAP GW signalcould be a benchmark for the GW signal from the core collapse. PACS numbers: 04.30.Db, 04.80.Nn, 97.60.Gb
I. ASTROPHYSICAL MOTIVATION
The detection of gravitational waves (GW) is one ofthe greatest goals of today’s relativistic astrophysics.Because of the great success of Einstein’s generalrelativity (GR) in explaining the dynamical evolution ofthe binary pulsar PSR 1913+16; continuously observedover almost 30 years by Taylor and Hulse [14], mostresearchers in the field are confident that the very firstGW signal to be detected must come from coalescingbinary neutron star-neutron star or black hole-neutronstar (NS-NS, BH-NS) systems. Ongoing LIGO scienceruns [11] for the first time have set firm upper limits onthe rate of binary coalescences in our Galaxy: at the levelof a few events per year. Meanwhile the model-dependentestimate of such a rate at cosmic distance scales variesaround 10 − [12, 13, 21]. The expectation is enormousas well for an event like the supernova (SN) explosion.In this paper we focus on the radiation emitted whena pulsar is kicked out from the supernova remnantand is accelerated with very high speed, albeit nonultrarelativistic. Gravitational radiation is emittedwhenever the nascent neutron star or pulsar changes its ∗ Electronic address: hermanjc cbpf.br, herman icra.it velocity, a condition that is fulfilled by the large thrustgiven to the nascent pulsar during the SN explosion,and also due to the pulsar dragging against the SNejecta. (Recall that the gravitational field generated by amotion with constant velocity is nonradiative). We shallshow that each individual event of launching a nascentpulsar to drift or run across the Galaxy, after a SNexplosion, generates a gravitational wave signal that canbe detected by current and planned GW interferometerssuch LIGO I, LIGO II, VIRGO, GEO-600, or TAMA-300. Nonetheless, it is stressed from the very beginningthat the strain of the GW signal that we estimate belowis a lower bound . The reason is that we are not takinginto account the contribution to the GW amplitude, h ,stemming from the effective acceleration that the pulsarreceives at the kick at birth, which could be very large.Indeed, the distance over which the RAP is draggedagainst the supernova ejecta could also be astronomicallylarge.At first glance, it appears that the detection of theGW signal from the short rise must have an event rateon the order of that for SN events in our galaxy, i. e. ∼ •
1) In a seminal paper, Spruit and Phinney [5]demonstrated that regardless of the actual physicalengine driving the natal (instantaneous) pulsarkicks, that mechanism can only act upon the pulsarover a very restricted time scale, ∆ T kickmax , on theorder of ≃ .
32 s. This is a fundamental piece ofthe dynamics of observed pulsars that seems to beoverlooked by workers in the field. •
2) As shown in the next Section, pulsar surveyshave shown that the largest ( V max ⋆ ), smaller( V min ⋆ ) and average ( V ave ⋆ ) spatial (3-D) velocityof cataloged pulsars are 5000 km s − , 84 km s − ,and 450-500 km s − , respectively [2, 4, 10] (seeFig.1(c)). That is, V ⋆ changes by more than anorder of magnitude. •
3) Observations of NSs in binary systems haveallowed to estimate their masses with very highaccuracy. The current values of this physicalproperty gather around 1.4 M ⊙ [8]. •
4) According to GR the acceleration of a mass( M ⋆ ) to a relativistic velocity ( V ⋆ ); i. e. energy,should generate a GW pulse, the nature of whichhas a very distinctive imprint: the so-calledgravitational wave memory [6, 18, 19], which makesthe amplitude of this GW signal (the GW strain)not to fade away by the end of the traction phase(the kick), but rather leaves it variable in time asmuch as an energy source remains active in thesource. II. GALACTIC RAPS AND SAMPLESELECTION FROM ATNF CATALOG
Run away pulsars (RAPs) are observed to drift alongthe Galaxy with velocities ranging from 90 km s − to4000 km s − (Australian Telescope National Facility(ATNF) Catalog [1], [9]). Since most stars in ourgalaxy are observed to drift with average velocities of15 −
30 km s − , it has been suggested that such largevelocities should be given to the proto-neutron starat its birth. The kick velocity imparted to a just-born neutron star during the rise time is a fundamentalpiece for understanding the physics of the core collapsesupernovae, to have an insight on the sources ofasymmetries during the gravitational collapse and onthe emission of gravitational waves during this process.There seems to be a consensus that the core collapseasymmetry driving the kick velocity of a neutron star atbirth may have origin in a variety of mechanisms [25, 26].Although the mechanism responsible for the initialimpulses has not been properly identified yet, a verystrong constraint on the timescale over which it couldact on the nascent neutron star (NS) was put forwardin Ref.[5]. Whichever this thruster engine might beit cannot push the star over a timescale longer than∆ T kickmax ≃ .
32 s. Combining this with the NS mass,which piles-up around 1.4 M ⊙ [8], and the above averagevelocity, implies that a huge power should be expendedby the thruster during the pulsar early accelerationphase, and hereby a gravitational-wave (GW) burst with“memory” (understood as a steady time variation of themetric perturbation) must be released during this time,according to general relativity [6].The present velocity distribution (see Figs.1, 1(c)) isconstructed based on the analysis of the proper motionof radio pulsars in our Galaxy after estimating theirdistances. Pulsar distances are estimated by using annualparallax, H I absorption, or associations with globularclusters or supernova remnants, and from dispersionmeasures and galactic electron density models.This astronomy field has become a vivid researchenterprise since the pioneer works of Lyne and Lorimer(LL94) [2, 3]. However, the current view on thedistribution of kick velocities is far from the originalone they found through pulsar surveys fifteen years ago.LL94 showed that pulsars appear to run around theGalaxy (and its halo) with mean spatial (3-D) velocities ∼ −
500 km s − [2], but freeway (“cannon-ball”)pulsars with velocities as high as 5000 km s − andslow-down millisecond pulsars with speeds of 80 km s − wander also around the Galaxy [3]. Meanwhile, thesituation changed dramatically with the pulsar velocitysurvey performed by Arzoumian, Chernoff and Cordes(ACC02) [4], who found that such a speed distributionappeared to be bimodal indicating one distribution modearound 90 km s − and another near 500 km s − .However, this dynamic research field has shown, throughnew studies on pulsar proper motions, that the situation (a) Pulsar number (vertical axis) vs. distance (kpc) toEarth (horizontal axis) histogram (b) Pulsar number vs. Distance distribution of 212 pulsars Dist [kpc] V t r a n s [ k m s - ] (c) Velocity vs. Distance distribution of 212 ATNFpulsars having both parameters measured FIG. 1: Data taken from ATNF Pulsar Catalog is far from having been settled. For instance, Hobbs etal. [10] have recently challenged the ACC02 findings,and have shown that their studies lead to assert thatthere is no evidence of a bimodal distribution. Insteadthey suggest that mean 3-Dim pulsar birth velocity is400 ±
40 km s − , and that the distribution appears to beMaxwellian with 1-Dim rms σ = 265 km s − , showingalso a continuum distribution from low to high (present)run away speeds, being the last ones the tail of thevelocity distribution. Hence, looking back on the surveysby LL94, the study by Hobbs et al. [10] clearly agreeswith the mean velocity of 450 ±
90 km s − found by LL94[2, 3]. A. Sample selection from ATNF database
In the late times the number of pulsars has increasedconsiderably [10]. Although various researchers havekept update catalogues, since then; in general, thesehave been neither completed nor very accessible. TheATNF database is an effort among an australian scientificcommunity and some foreing partner institutions. Thepurpose is to compile the largest amount of informationabout pulsars, which is subsequently made available toevery researcher all over the world.The ATNF catalogue [10] registers all publishedrotation-powered pulsars, including those detected only G W S t r a i n h G W ( x − ) GW−PSR (10 kpc)GW−GRB (1 Mpc) (a)GW signals from pulsar acceleration and gamma-rayburst. Distribution (with the viewing angle) of the GWsignal produced during the early acceleration phase of apulsar (dashed–green line) with V ave ⋆ = 450 km s − , anddistance = 10 kpc. Comparison with the GW signal from agamma-ray burst (solid–red line), as a function of the jetangle to the line-of-sight, with parameters h E i = 10 erg, γ = 100, and distance = 1 Mpc. FIG. 2: Angular distribution of the GW signals from pulsaracceleration and gamma-ray burst at high energies. It also records Anomalous X-ray PulsarsAXPs and Soft Gamma-ray Repeaters (SGRs) for whichcoherent pulsations have been detected. However, itexcludes accretion-powered pulsars such as Her X-1 andthe recently discovered X-ray millisecond pulsars, forinstance SAX J1808.4-3658 [15].The ATNF catalogue [10] can be accessed in a numberof different ways. The simplest one is from a webinterface [28] allowing listing of the most commonlyused pulsar parameters, together with the uncertaintiesand information on the references. Several optionsfor tabular output format are provided. Currently, atotal of 67 “ predefined parameters ” are available. Atool is provided for plotting of parameter distributions,either as two-dimensional plots or as histograms. Zoomtools and interactive identification of plotted pointsare provided. Custom parameters can be defined bycombining parameters in expressions using mathematicaloperators and functions and these can be either listedor plotted. Finally, the sample of pulsars listedor plotted can be limited by logical conditions onparameters (see Table-I), pulsar name (including wild-card names) or distance from a nominated position.These operational tools are described in more detailbelow and links are provided within the web interfaceto relevant documentation [10].In this particular case, we select from ATNF, by clicking on the box to the left of the “parameter label” toselect a couple of pulsar parameters: DIST and VTRANS(see Fig.1(c)). The first parameter is the best estimateof the pulsar distance in kpc. The second parameter isthe transverse velocity, given in km s − , based on DISTestimates. By default the results will be sorted accordingto the pulsars’ J2000 names in ascending alphabeticalorder. However, sorting is possible on any parameter bytyping the parameter label in the “ sort on field text box ”and selecting whether the sort should be in ascending(default) or descending order. However, to obtain asmuch information as possible some careful proceduresmust be followed. Because of this, we selected thelisting fashion “ short format without errors ”. The shortformat is used to provide a condensed summary of thepulsars parameters (see Table-V). Finally, we selectedthe option “ header ”, that produces a table with headerinformation at the top and selected the output option“ Table ” (see Table-I). Besides, it is possible to displayfunctions of the pulsar parameters as a graph or as ahistogram. For a normal (x-y) graph, the values to plotare defined as regular expressions into the “ x, y-axis textbox ”, and the axes of the graph can be displayed linearlyor logarithmically. The expressions may also containcustom-defined variables or externally defined variables.
III. PRESENT GALACTIC PULSARDYNAMICS HINTS AT DETECTABILITY OFGW SIGNALS FROM THE SHORT RISE FLINGOF RAPS IN SUPERNOVAE
Next we discuss the prospective for the short rise flingof Galactic RAPs in a supernova to generate detectableGW signals. The analysis starts from the observedkinematic state of RAPs, and goes back to their kicksat birth. To such an approach we assume that before thekick the star velocity is V ⋆ = 0 [7].First, let us notice that one can combine the firstthree pieces discussed above together with the extremelyrestrictive character of the thrust timescale (∆ T kickmax ),the magnitude of the largest velocity ( V max ⋆ ), and theinstantaneous nature of the kick ( V ⋆ = 0). In so doing,one can then conclude that whichever the driving enginemight be, it should transfer to the star at the kick at birthan energy as large as h ∆ E i ∼ erg. (Recall thatthe NS masses are piled-up around 1.4 M ⊙ ). Besides,keep in mind that according to Newton’s second law: F ⋆ = M ⋆ a ⋆ , the star’s inertia depends directly on itsrest mass M ⋆ . Thus, putting together these ingredientsone can conclude that the effective acceleration, a ⋆ , atthe kick at birth of each RAP should be the same.[29]In other words, because pulsar velocities vary by morethan an order of magnitude, whilst the inertia (mass)of each neutron star is essentially the same (within anuncertainty no greater then a factor of 2), one concludesthat the acceleration at the kick at birth for each of thepulsars has definitely to be the same. In all our analysisbelow we will use this key feature.Further, as the kick is an “instantaneous action”,then the Newtonian kinematics states that the largestvelocity[30] should be attained after the longestacceleration phase. That is; over the longest tractiontime t kick ⋆ . Therefore, the RAP kinematics during theshort rise fling follows the law V fin ⋆ ( t ) = V ⋆ + a ⋆ · t kick ⋆ , (1)where V fin ⋆ ( t ) defines the final (at the end of the kick)velocity of a given RAP, which is reached after a totalimpulsion time t kick ⋆ . Consequently, if one assumes forthe sake of simplicity that the RAP transit through theinterstellar space does not modify its starting state ofdrifting, then one concludes that the largest (presentlyobserved velocity V max ⋆ ≃ − ) must havebeen reached after the elapsing of the longest timescalepermitted by the kick mechanism: ∆ T kickmax ≃ .
32 s [5].Thus, after collecting all the pieces on the RAPsphysics described above one can infer the characteristicacceleration received by a typical RAP. To this goalon can take the largest present velocity given by Refs.[2, 3, 4, 10] (see Fig.-1(c)), and the constraint on themaximum traction timescale for the kick at birth[5].That acceleration then reads a ⋆ = V max ⋆ ∆ T kickmax ∼ km s − . (2) A. Estimates of present h c , f GW GW characteristicshint at prospective detection during RAPs short risefling in supernovae
The above analysis suggests that the overall time scalefor acceleration of any given RAP to its present velocity(under the assumption stated above) depends exclusivelyon the total energy with which it was thrusted. Theactual amount of energy expended to thrust the starcertainly has something to do with the nature of thedriving mechanism. Hence, the GW dynamics to bediscussed below, combined with observations of GalacticRAPs, may help to enlighten the path to unravel theactual thruster. Next we provide order of magnitudeestimates for both GW amplitude and frequency basedon parameters of RAPs currently cataloged. (In the nextSection a rigurous derivation of these GW characteristicsfor the RAP kick at birth is presented).
1. Frequency estimate
From Eqs.(1)-(2) one can see that the traction timescale for any given pulsar is strictly directly proportionalto its present spatial velocity. This is a conclusion ofbasic importance. That is, the RAP current kinematics must retain information about what happened during itsshort rise fling in the supernova. Thus, the frequencyof the GW burst emitted during the early impulsion ofa pulsar in a supernova explosion can be assumed, in avery simplified picture, as being inversely related to thetotal acceleration time [18], that is, f kickGW ≃ t kick ⋆ . (3)As an example, let us take the values quoted abovefor V max ⋆ , V ave ⋆ , and V min ⋆ . In these cases, one obtainsthe GW frequencies f minGW = 3 . f aveGW = 34 . f maxGW = 186 . h and f GW for the 212 ATNF RAPs sample).
10 100 1000 f c [Hz] -26 -25 -24 -23 -22 -21 -20 -19 h c * [ H z ] - / LIGO AdvancedLIGO Adv. Cal. PSR SignalsLIGO ITAMAVIRGOGEONon-Calibrated PSR Signals
FIG. 3:
Color Online.
LIGO I, LIGO Advanced, VIRGO,GEO-600, TAMA-300 strain sensitivities and the h c and f c characteristics of the GW signal produced by each of the 212pulsars calibrated with respect to LIGO Advanced (greensquares), and the same sample without calibration (bluetriangles). The effect of the calibration procedure is evident.
2. Amplitude estimate
To have an order of magnitude estimate of theGW amplitude, h , from the present kinematic stateof a galactic RAP one can use the general relativisticquadrupole formula h ≃ Gc E k r , (4)where r is the pulsar distance as provided by the ATNFCatalog, and E k = M ⋆ V ⋆ is the RAP kinetic energy.Therefore, if one combines the theoretical predictionprovided by Eq.(3) with this approximate GWamplitude, one obtains the plot presented in Fig.(3). Theestimates were performed for a sample of 212 pulsaresselected from the Australian Telescope National Facility(ATNF) catalog [1].[31] Notice that Eq.(4) states thatwhenever the RAP velocity is time-varying the signals,whose representative points appear above the strainsensitivity limit of Advanced LIGO, would be detectableby this observatory. For a RAP with no time-varyingvelocity, the signal would be a direct-current-like (DC)signal that no detector might observe. (Regarding theabove estimates, keep in mind that at least the pulsarmotion around the Galactic center (the bulge) should be centripetally accelerated).In Fig.-3 blue triangles represent GW signalscomputed following this back-of-the-envelope technique.Green squares are the obtained after calibrating theapproximated signals against the strain sensitivity ofAdvanced LIGO. Clearly only a few RAPs would bedetectable if their GW signals were described in suchan approximate fashion. Moreover, this graph gives usa clear hint at the proper characteristics of the GWsignal that would be emitted by a run away pulsarduring its short rise fling in the supernova that makesit to drift across the galaxy. These order of magnitudeestimates suggest that a RAP that was kicked out fromthe supernova ejecta with higher speed and accelerationwould produce a clearly detectable GW signal. Thisperspective motivated us to study the problem using amore rigurous physical description: the theory for theemission of gravitational wave signals by ultrarelativisticsources, a GW signal which has memory. This is done inthe next Section. TABLE I: Full pulsar sample taken from ATNF. JName is the pulsarname based on J2000 coordinates, RAJ right ascension (J2000) inhh:mm:ss.s, DECJ declination (J2000) in +dd:mm:ss, PMRA propermotion in right ascension in mas yr − units, PMDEC proper motion indeclination in mas yr − units, DIST best estimate of the pulsar distancein kpc units, VTRANS transverse velocity - based on DIST in km s − TABLE I: Full sample taken from ATNF Catalog.
TABLE I: Full sample taken from ATNF Catalog.
TABLE I: Full sample taken from ATNF Catalog. TABLE I: Full sample taken from ATNF Catalog.
From the analysis of this Table one can realize thatmost of the SNR that were obtained gather about an
S/N ∼ O (1). However, an SNR of the order of unityis not enough for detection. The statistics which is theoutput of the matched filter must stand over the GWdetector noise. Thus, if one assumes Gaussian noise inthe detector and a Gaussian distributed statistics, thenan SNR of something like 5, what means a chance ofone in a million , is required to state that it is signal andnot noise . Notwithstanding, RAPs current kinematicssuggest that the dynamical situation could have beendramatically different during the short rise fling thatlaunches each pulsar to drift across the Galaxy. In virtueof the potentiality of this phenomenon for the emissionof powerful gravitational radiation signals, in the nextSection we analyze the pulsar kick at birth within theframework of a theory that appears more appropriate todescribe that process, and reestimate the characteristicsof the GW released.
IV. GRAVITATIONAL WAVES FROM PULSARACCELERATION
Observations of extragalactic astrophysical sourcessuggest that blobs of matter are ejected withultrarelativistic speeds (Lorentz factor γ >>
1) invarious powerful phenomena involving quasars, activegalactic nuclei, radio-galaxies, supernova explosions,and microquasars. For these sources the emission ofgravitational waves when such an ultrarelativistic blob isejected from the core of its host source have been studiedby Segalis and Ori[19] (see also [18] for the analysis ofsimilar phenomena in gamma-ray bursts).Supernova explosions are known to be the naturalplace of birth of neutron stars and pulsars, e. g.
TheCrab Nebula . In processes such these, the nascent pulsaris kicked out from the supernova remnant with speed thatis not so large as compared to that for the blobs in AGNor quasars, where a significant fraction of the speed oflight is typical for the ejecta. Nonetheless, the velocityat birth can be considered large if one bears in mindthat the typical mass of neutron stars is about one solarmass[8]. Although a galactic RAP cannot be consideredan ultrarelativistic source, the pulsar rise fling during thesupernova can kick it out with very high speed. That iswhy in the discussion below we analyze the gravitationalradiation emitted during the kick at birth of RAPs withinthe framework of Ref.[19]. Besides, it is also of worth to keep in mind that such an approach can be appliedto the early RAPs dynamics because in this case thelimit γ ∼ h provided by the well-known general relativisticquadrupole formula for the same dynamical situation [6].But bear in mind that this does not describe properlythe GW angular distribution.The pulsar is envisioned hereafter as a “particle” ofmass M ⋆ moving along the worldline r λ ( τ ) (with τ theproper time) and having an energy-stress tensor T µν ( x ) = M ⋆ Z V µ V ν δ (4) [ x − r ( τ )] dτ , (5)where V α = dr α /dτ is the particle 4-vectorvelocity. (Upper[sub])scripts are raised[lowered] with theMinkowski metric η µν ).Since we are using the linearized Einstein’s equationsto describe the emission of GW during the earlyacceleration phase of any pulsar, then the resultingmetric perturbation produced by the RAP canstraightforwardly be evaluated at the retarded time,which corresponds to the intersection time of r α ( τ ) withthe observer’s past light-cone, and then be transformedto the Lorentz gauge where it reads: h µν = ¯ h µν − η µν ¯ h αα .This strain can equivalently be written as h µν = 4 M ⋆ − V λ · [ x − r ( τ )] λ (cid:18) V µ ( τ ) V ν ( τ ) + 12 η µν (cid:19) . (6)Notice that is this factor − V λ · [ x − r ( τ )] λ ; whichdepends on the velocity in the denominator of Eq.(6),that is responsible for the nonvanishing amplitude (GW“memory” [6]) of the GW signal produced by thelaunching of the pulsar into its present trajectory.Eq.(6) must be finally rewritten in the tranverse-traceless ( T T ) gauge, i. e. h µν −→ h T Tµν ; which is thebest suited to discuss the GW detector’s response to thatsignal. This procedure leads to the result presented inEq.(7).A detailed analysis (see Refs.[6, 19]) shows that themaximum GW strain in the detector is obtained for awavevector, ~n , orthogonal to the detector’s arm, in the ~v - ~n plane. Here ~n is the unit spatial direction vectorfrom the (retarded) pulsar position to the observer, and ~v the unit spatial direction vector defining the 3-D pulsarvelocity ~V . In this case, the GW amplitude generated bythe pulsar kick becomes (c.g.s. units recovered)1 h max ( t ) = (cid:20) Gc (cid:21) Z θ V γ ( t ) M ⋆ β ( t ) D ⋆ (cid:18) sin θ sin 2∆ φ ∆Ω(1 − β cos θ ) (cid:19) dθ , (7)where θ is the angle between ~v and ~n , that is the anglebetween the unit spatial direction vector associated tovelocity of the pulsar, ~V , and the unit spatial directionvector linking the source point to the observer location,i. e. ~v · ~n = cos θ ). θ V is the particular (e. g. VelaPulsar ) viewing angle that a RAP velocity makes withthe direction ~n to the observer. ∆ θ is the angulardiameter of the pulsar as seen by the observer. (Forgalactic distances it can be assumed nearly constant foreach of the RAPs in the sample under analysis). Thus∆Ω ≃ π (∆ θ ) is the solid angle over which the RAP isviewed. Besides, β = | ~V | /c , being | ~V | the pulsar 3-Dvelocity V fin ⋆ ( t ) defined above, and γ the Lorentz factor;which can be taken here as ∼ V fin ⋆ ( t ) ≪ c . Itis also defined ∆ φ = cos − (cid:16) cos ∆ θ − cos θ V cos θ sin θ V sin θ (cid:17) . Finally,the distance to the pulsar is D ⋆ .This formula was obtained in the form presentedabove first by Sago et al.[18] in discussing the GWemission from gamma-ray bursts (GRBs). It is essentiallyequivalent to the formula obtained by Segalis and Ori[19] in an earlier paper in which they discussed theGW emission by ultrarelativistic sources. The maindifferent in between is related to the detailed descriptionof the angular distribution of the gravitational radiationemitted in each of the cases that those authors analyzed.The formula indicates that the GW strain depends on theLorentz factor, pulsar speed and distance to the observer,and also on the strong beaming effect in the case ofGRBs, as indicated by the factors (1 − cos θ ), and ∆Ω,appearing in the denominator of the equation. Theseeffects are no so much important in the case of GalacticRAPs, as is evident from the pulsar speed and angularsize.This result, shown in Fig.2, states that the GW space-time perturbation is not strongly beamed in the forwarddirection ~n , as opposed to the case of the electromagneticradiation in gamma-ray bursts (also shown in thesame Figure). Instead, the metric perturbation at theultra-relativistic limit (not applicable to RAPs) has adirectional dependence scaling as 1 + cos θ . In such acase, because of the strong beaming effect, as in gamma-ray bursts; for instance, the electromagnetic radiationemitted by the source over the same time interval isvisible only inside the very small solid angle ( θ ∼ γ − ) ,whereas the GW signal is observable within a widersolid angle; almost 2 π radians (see Fig.2). Besides, theobserved GW frequency is Doppler blueshifted in theforward direction, and therefore the energy flux carriedby the GWs is beamed in the forward direction, too.Quite noticeable, in the case of RAPs the GW signal willhave its maximum for viewing angles θ ∼ π/
2, that is; forpulsar motions purely in the plane of the sky, as shown inFig. 2. This key physical property of the emission process allows us to neglect the radial component of the pulsarvelocity because it does not contribute to the effectiveGW amplitude of the signal being emitted by the RAP,or in other words, to the signal to be detected on Earth.
TABLE II: Characteristic amplitude and frequency, h c , f c (Hz), and S/N = h c ( f c ) /h r ms ( f c ).Jname f c h c S/NˆA [Hz] ˆA ˆAJ0014+4746 37.1239 7.99071 × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − TABLE II: results of full calculation of h c , f c and S/N .Jname f c h c S/NˆA [Hz] ˆA ˆAJ0659+1414 54.243 9.24396 × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − h c , f c and S/N .Jname f c h c S/NˆA [Hz] ˆA ˆAJ1722-3207 41.6931 9.28191 × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − TABLE II: results of full calculation of h c , f c and S/N .Jname f c h c S/NˆA [Hz] ˆA ˆAJ1954+2923 45.8579 1.06684 × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − V. COMPUTING DETECTABILITY OF GWFROM RAPS
One can discuss the detectability of the GW (burst)signal of the rise fling of galactic RAPs by using the matching filter technique. This procedure is based on the definition of the signal-to-noise ratio (SNR), which isa quantity used as a criterium of detectability of a GWsignal. The SNR depends on the features in the GWwaveform of the process, the orientation of the sourcewith respect to the GW interferometer (observer) andalso on the source direction. It also depends on the totalenergy per unit frequency, dE/df ( f ) (the GW energyspectrum at the source), that is carried away from thesource by the GW, and on the distance to the source D ⋆ .Here f is the average GW frequency for a given signal.Because of this angular dependence, the SNR is usuallydefined as SN R = h ρ i = 25 π D ⋆ Z ∞ f S h ( f ) dEdf f df . (8)Symbol hi refers to an average of SNR square for allthe directions (angles) to the GW source. In other words,it is obtained from the average rms of the amplitudes ofthe GW signals for different orientation of the source andthe interferometer. In our computations above in SectionIII we applied this definition to estimate the S/N ratiogiven in Table II. As the attentive reader may notice, inwhat follows we shall use an equivalent definition of theSNR to quantify the detectability of RAPs GW signalsin the context of the formalism of Ref.[19]To complete this brief discussion on the SNR, wecollect below the power-laws that according to most ofthe experimental studies with GW interferometers arewhat better describe the interferometer noise in eachband of the spectrum. To this purpose, it is defined thedimensionless quantity: h rms ( f ) ≡ p f S h ( f ), wherein S h ( f ) is the interferometer noise spectral density. Thus,the strain sensitivity for each frequency band is given by h rms ( f ) = ∞ , f < f s ,h m ( αf /f m ) − / , f s f < f m /α,h m , f m /α f < αf m ,h m [ f / ( αf m )] / , αf m < f . (9)Understanding these laws is straightforward if onerecalls that the interferometer noise curves dependbasically on four parameters: • The threshold frequency f s : below of it the noisegrows almost asymptotically. For detectors onEarth the noise source is the seismic gradients • The frequency f m , centered in the almost flat partof the noise spectrum • A critical amplitude h m , which is the minimumvalue of h rms ( f ) • And the dimensionless parameter α , whichdetermines the width of the flat band of the noisecurve4For instance, for Advanced LIGO the following set ofparameters is estimated (see E. E. Flanagan and S. A.Hughes, [Phys. Rev. D 57, 8 (1998)])LIGO Advanced f s = 10 Hz f m = 68 Hz α = 1 . h m = 1 . × − . (10)Therefore, to know whether the RAPs GW signalscan be effectively detected by the currently operativeinterferometric detectors LIGO, VIRGO, GEO-600, etc.,one needs to compute the signal-to-noise ratio withrespect to the strain sensitivity of these instruments. (Werefer to this procedure as calibration ). In order to dothat we need first to compute the frequency at which thelargest part of the GW emission is expected to take place.This is the so-called characteristic frequency f c , which isdependent on the detector design properties.Should the detector have a power spectral density S h ( f ), have the characteristic frequency to be obtainedfrom the frequency first moment of the time-dependentGW amplitude h ( t ), i. e., f c ≡ Z ∞ h| ˜ h ( f ) | i S h ( f ) f df ! "Z ∞ h| ˜ h ( f ) | i S h ( f ) df − , (11)where h ˜ h ( f ) i is the Fourier transform of h ( t ) [32],and the symbol hi stands for an average over randomlydistributed angles of h ( f ), and it can be approximatedas h| ˜ h ( f ) |i = | ˜ h ( f ) | .The characteristic frequency f c is strictly related tothe characteristic amplitude h c , which is the physicalquantity associated to the GW pulse h ( t ) to be detectedby a given instrument. It can be computed as h c ( f c ) ≡ (cid:18) Z ∞ S h ( f c ) S h ( f ) h| ˜ h ( f ) | i f df (cid:19) / . (12)In this way we take ˜ h in the following form (seeRef.[18, 19]) | ˜ h | = (∆ h m )8 π f t m (1 − cos(2 πt m f )) . (13)Here t m defines the rise time of the memory of thesignal. This equation is obtained from the Eq.(7) givenabove. Thus, Eq.(13) is the Fourier transform of theGW waveform, where ∆ h is given by (after average overangles in Eq.(7)) ∆ h m = 4 Gβ γM ⋆ c D ⋆ (14) with G the universal constant of gravitation, γ Lorentzfactor, M ⋆ star mass, D ⋆ distance to Earth, and β = | ~V | /c .Therefore, the signal-to-noise ratio S/N is nowcomputed using the relation SN = h c ( f c ) h rms ( f c ) (15)where h rms ( f c ) = [ f c S h ( f c )] / is the averageamplitude of the interferometer spectral noise at thecharacteristic frequency.Guided by Table-IV, this allows us to compute thecharacteristic amplitude of the GW burst released duringthe short rise time of acceleration of each RAP for whichthose parameters are estimated. Since we are usingonly the transversal velocity of the pulsars, then we set θ ≃ in our calculations below. Our main resultsare presented in Figs.(4(a), 4(b), 5(a)). Those Figurescompare the GW signals from individual RAPs with theexpected sensitivities of LIGO I, its projected advancedconfiguration LIGO II, VIRGO, GEO-600, and TAMA-300 observatories.At this stage, a note of caution to the attentivereader is needed. It regards with the appearance ofthe noise curves of TAMA, GEO and VIRGO GWobservatories in figures from 4 through 9, in additionto the noise curves for LIGO I and LIGO Advanced,despite that the calibration of the expected signalsin those figures (symbol: triangles, circles, etc.) isperformed exclusively with respect to LIGO I andLIGO Advanced, independently. First, the reader mustkeep in mind that the position of those points in thediagram h c vs. f c will change when the calibrationof the expect GW strain is performed with respectto the noise curve of a different detector, say TAMA,VIRGO, etc.. For instance, compare Fig.4(a) andFig.5(a) which are calibrated for LIGO Advanced andLIGO I, respectively. Second, thinking on the benefitof the Scientific Community operating observatorieslike TAMA, GEO and VIRGO, those noise curves areincluded having in mind the expectation that suchdetectors might gain more sensitivity in both frequencyband and amplitude so as to be able to catch the GWsignals produced by Milky Way RAPs.Finally, but not to the end, a direct interpretation ofall the h c vs. f c diagrams in figures from 4 through 9is that, exception done for the TAMA-300 GW detector,the signals (points) plotted in those figures appear tobe not very much sensitive to the noise curves of theinstruments LIGO I, LIGO Advanced, VIRGO and GEO-600. This feature justifies to include those noise curvesin such figures.5
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMAVIRGOGEOLIGO Adv. (a) LIGO I, LIGO Advanced, TAMA, VIRGO andGEO600 strain sensitivities and the h c and f c characteristics of the GW signal produced by each of the212 pulsars calibrated with respect to LIGO Advanced, forparameters a = 10 km s − and V kick ∼ km s − .
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMAVIRGOGEOLIGO Advanced (b) Idem as Fig.(4(a)), but now for parameters a = 10 kms − and V kick ∼ km s − , calibrated with respect toAdvanced LIGO. It is possible to see that these signals havehigh enough chance to be detected. FIG. 4: Color Online. Pulsar GW signals for different parameters a and V kick . (a), (b)
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h* [ H z ] - / LIGO Adv.LIGO IGEOTAMAVIRGOPSR Signals (a) Idem as Fig.(4(a)), but now for parameters a = 10 kms − and V kick ∼ km s − , and calibrated against LIGOI. Clearly also these signals might be detected FIG. 5: Color Online. Pulsar GW signals for differentparameters a and V . (c) VI. MORE REALISTIC ANALYSIS OF GWEMISSION FROM RAPS
We can start with by recalling both the inertia(which is large for a neutron star progenitor) and linearmomentum conservation laws, which should dominatethe physics of the SN explosion that gives the kick tothe neutron star. Thus, one can write M SN V SN = M NS V NS − M RSN V RSN , (16)where M SN , M NS , and M RSN are the SN, neutronstar, and ejected material mass, respectively. Be awareof the role of the “ − ” signal in front of the term forthe ejected material. It is responsible for a correctdescription of the physics at the explosion. Without it,one would obtain immaginary values for the searchedparameter V NS . Now, by assuming parameters of atypical progenitor star M SN = 10 M ⊙ , V SN = 15 kms − , M NS = 1 . ⊙ for a canonical neutron star, andfor the material in the ejected envelope M RSN = 8 . M ⊙ ,and expansion velocity V RSN = 3000 km s − (see Table-IV), one obtains that the kick at birth velocity reads: V = V kick ∼ km s − (compare to velocities takenfrom ATNF in Fig.1(c)). The discrepancy between themwill be briefly discussed below.6
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 40% a = 10 km s -2 (a) Efficiency case ǫ = 0 .
40 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 10% a = 10 km s -2 (b) Efficiency case ǫ = 0 .
10 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 5% , a = 10 km s -2 (c) Efficiency case ǫ = 0 .
05 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced.
10 100 1000 f c [Hz] -26 -25 -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 1% , a = 10 km s -2 (d) Efficiency case ǫ = 0 .
01 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced. FIG. 6: Color Online. Pulsar GW signals for parameters: a = 10 km s − and V kick ∼ km s − A. Case a: Centered kick V = V kick and ǫ = 1 We first consider that the whole energy obtained fromthe conservation law discussed above goes into the pulsarkick, imparting to it no rotation at all. This is a veryidealistic situation. Nonetheless, it gives us an idea onthe potential detectability of those GW signals from theflinging of a pulsar out of the SN cocoon. In other words, in this first approach we assume that the efficiency, ǫ ,that relates the effective RAP velocity V (which entersthe calculation of h c ) and the kick speed V kick is ǫ = 1(In the other cases analyzed below these velocities willbe related through the expression: V = ǫV kick .7 TABLE III: ATNF sample of pulsars having 2-Dim speedgreater than 10 km s − . The current high velocities suggestthat a much higher speed was imparted at the kick at birth.Jname DIST VTrans[kpc] [km s − ]J0525+1115 7.68 1101.985J1509+5531 2.41 1104.013J1824-1945 5.20 2482.989J1829-1751 5.49 3945.954J2013+3845 13.07 2521.130J2149+6329 13.65 1113.385J2225+6535 2.00 1729.770B2011+38 ???? 1624.0B2224+65 ???? 1608.0B1830-1945 5.20 740.0 ⋆ RX J0822-4300 2.0 1600.0 ⋆ Ref. P. F. Winkler & R. Petre,
Direct measurement of neutronstar recoil in the Oxygen-rich supernova remnant Puppis A , Ap.J.670, 635 (2007)
1. Fling with a = 10 km s − , V kick = km s − Based on the dynamical analysis presented in theSection-III, we suppose here that the short rise time kickimparts to the pulsar an acceleration a = 10 km s − ,and a birth velocity V kick = 10 km s − . The GW signalobtained for these parameters is presented in Figs.(4(a),4(b)).
2. Fling with a = 10 km s − , V kick = km s − Now one can take a step further and consider that thebirth acceleration is actually more larger than the onecurrently inferred from the RAPs galactic parameters.Several physical mechanisms can act together to impartan acceleration as large as a = 10 km s − , see forinstance Refs.[23, 24, 26], and references therein. Sucha large thrust can be estimated as a ≃ V kick / ∆ T mech ,where (10 − . ∆ T mech . − ) s, depending on thespecific mechanism driving the fling. The GW signalobtained for these parameters is presented in Fig.(5(a)). B. Case b: Off-centered kick
Finally, we analyze the most astrophysically realistickicks at birth: a situation where the pulsar receivesboth a translational and a rotationl kick. In thiscase, the energy transferred to the translational kickcan be parameterized by an efficiency ǫ , which rangesfrom 0 to 1. Of course, this efficiency represents ourdifficulty in explaining how a specific kick mechanismbecomes the kick driver. The remaining energy is supposed to be expended in making the pulsar to spin.This parametrization can be translated into a power-law relationship between the GW amplitude, h , and theefficiency, ǫ , of energy conversion into pulsar kick, whichhas the form: h ∝ ǫ . (17)This relation stems from the quadratic dependence ofthe GW amplitude with the β parameter appearing inEq.(7) above. In this way, one can recover the wholespectrum of RAPs velocities as observed today and listedin ATNF catalog. To exemplify, we compute signalsfor values of ǫ = 0 . , . , .
10, and 0 .
40, which canreproduce the set of velocities V kick = 100 km s − , V kick =500 km s − , V kick = 10 km s − , and V kick = 4 × kms − , as shown in Fig.-(1(c)). For this case, the results arepresented in Figs.(6, 7, 8, 9). To the end of the papera brief discussion on the pulsar periods provided by theATNF catalog is given.
1. Efficiency ǫ = 0.05 The following figures illustrate the results obtainedfor this efficiency: Fig.(7(c)) with a = 10 km s − andFig.(6(c)) with a = 10 km s − for advanced. Fig.(8(c))with a = 10 km s − and Fig.(9(c)) with a = 10 km s − para LIGO I.
2. Efficiency ǫ = 0.10 The following figures illustrate the results obtained forthis efficiency: Fig.(7(b)) with a = 10 km s − andFig.(6(b)) with a = 10 km s − for Advanced LIGO.Fig.(9(b)) with a = 10 km s − and Fig.(8(b)) with a = 10 km s − for LIGO I.
3. Efficiency ǫ = 0.40 The following figures illustrate the results obtained forthis efficiency: Fig.(7(a)) with a = 10 km s − andFig.(6(a)) with a = 10 km s − for Advanced LIGO.Fig.(9(a)) with a = 10 km s − and Fig.(8(a)) with a = 10 km s − for LIGO I. VII. DISCUSSION AND CONCLUSIONSA. Computed rotation periods vs. Periods ofATNF sample of RAPs with V = 400 − km s − To the end, in order to make our analysis self-consistent we next discuss the relation between thetranslational speed and rotation frequency of ATNF8
TABLE IV: High expansion speeds of CaII and SiII lines of the ejected material in some supernovae. Data taken from A.Balastegui, P. Ruiz-Lapuente, J. M´endez, G. Altavilla, M. Irwin, K. Schahmaneche, C. Balland, R. Pain and N. Walton. arXiv:astro-ph/0502398v1 21 Feb 2005Event CaII Expansion Speed SiII Expansion Speed Days after maximum[km s − ] [km s − ] [days]SN 2002li 17200 ±
500 -7SN 2002lj 12300 ±
300 8900 ±
300 7SN 2002lp 13800 ±
900 10400 ±
400 3SN 2002lq 18900 ± ±
300 8600 ±
400 10SN 2002lk 14500 ±
500 -5 pulsars. To do this, we select, as an example, the 3-D average pulsar velocity as determined by Hobbs etal. [10] (see Table-V). On this basis we determinethe theoretical rotation periods that one could expectto find for that velocity sample if is factual that mostof the explosion energy, as inferred from the momentumconservation law in Eq.(16), goes into the pulsar rotationas discussed above. (Of course we compare with theactual periods measured by the ATNF surveys). Hencethe torque applied to the just-born neutron star reads τ ∼ V E k M NS R NS d l (18)where E k is the kinetic energy of each velocity chosenfor the present analysis, V pulsar velocity, M NS = 1 . ⊙ , R NS = 20 km and d l = 25 km are the neutronstar mass, radius and torque lever. Thus, for R NS = 20km and velocity V = 350 km s − we obtain p= 0 . V = 433 km s − one gets p= 0 . NS = 30 km and V = 350 km s − wehave p= 0 . V = 433 km s − one getsp= 0 . •
1) perhaps a large part of the initial rotationalenergy is dissipated by processes as differentialrotation or magnetic braking, especially if processesas the magneto-rotational instability[27] indeeddrives a large class of supernova explosions [22]. •
2) it may also happen that convection and neutronfingers also becomes highly effective dissipativeprocesses, thus reducing the pulsar initial spin to the level that we are currently finding in pulsarsurveys.In summary, from astronomical statistics the numberof observed Galactic RAPs is definitely much larger thanthe corresponding to relativistic NS-NS binaries. Thetypical lifetime of canonical pulsars is ∼
10 Myr, whichis defined as the duration of the radio emitting phasefor objects with B ∼ G, or the time needed forthe pulsar to emit less radiation or being entirely turnedoff. Hence, with a statistical Galactic rate of SNe, andthe likely fling out rate of just-born pulsars, of about 1per every 30-300 years, one expects that the GW signalof the kick at birth be observable as aftermath of theSN core collapse. In other words, the RAP GW signalmay become a benchmark for looking for GW signalsfrom the SN gravitational implosion. Besides, if the GWsignal from the kick to a RAP were detected, for instance,from a source inside a globular, or from either an openstar cluster [16], or a binary system harboring a compactstar, such an observation might turn these RAPs themost compelling evidence for the existence of Einstein’sgravitational waves.We conclude by stating that the detection withadvanced LIGO-type interferometers of individual GWsignals from the short rise fling of Galactic RAPs isindeed possible.
Acknowledgments
HJMC thanks Prof. Jos´e A. de Freitas Pacheco (OCA-Nice) for the hospitality during the initial developmentof this idea. Dr. Sebastien Peirani (OCA-Nice), MarceloPerantonio (CBPF-Rio de Janeiro), and folks as Dr. EricLagadec (OCA-Nice) are also thanked. HJMC is fellowof Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio deJaneiro (FAPERJ), Brazil. [1] R. N. Manchester, G. B. Hobbs, A. Teoh, M.Hobbs,
The ATNF pulsar catalogue (Australia, CSIRO, Epping), Astron. J 129, 1993-2006 (2005), e-Print:
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 40% , a = 10 km s -2 (a) Efficiency case ǫ = 0 .
40 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Advanced ε = 10% , a = 10 km s -2 (b) Efficiency case ǫ = 0 .
10 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 5%, a = 10 km s -2 (c) Efficiency case ǫ = 0 .
05 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 1% , a = 10 km s -2 (d) Efficiency case ǫ = 0 .
01 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGOAdvanced. FIG. 7: Color Online. Pulsar GW signals for parameters: a = 10 km s − and V kick ∼ km s −
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / Pulsars signalsLIGO ITAMA-300VIRGOGEO-600LIGO Advanced ε = 40% a = 10 km s -2 (a) Efficiency case ǫ = 0 .
40 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I. f c [Hz] -23 -22 -21 -20 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 10% a = 10 km s -2 (b) Efficiency case ǫ = 0 .
10 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 5% a = 10 km s -2 (c) Efficiency case ǫ = 0 .
05 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 1% , a = 10 km s -2 (d) Efficiency case ǫ = 0 .
01 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I. FIG. 8: Pulsar GW signals for parameters: a = 10 km s − and V kick ∼ km s − arXiv:0704.3368L[12] K. A. Postnov, L. Youngelson, Liv. Rev. Rel. 9, 6 (2006)[13] V. Kalogera et al. Phys. Rep. 442, 75 (2007)[14] Weisberg, J. M. Taylor, J. H., 2003. The RelativisticBinary Pulsar B1913+16. Conference Proceedings,Vol. 302. Held 26-29 August 2002 at MediterraneanAgronomic Institute of Chania, Crete, Greece. Edited byMatthew Bailes, David J. Nice and Stephen E. Thorsett.San Francisco: Astronomical Society of the Pacific, 2003.ISBN: 1-58381-151-6, p.93 [15] R. Wijnands and M. van der Klis, Nature 394, 344 (1998)[16] W.H.T. Vlemmings, S. Chatterjee, W.F. Brisken, T.J.W.Lazio, J.M. Cordes, S.E. Thorsett, W.M. Goss, E.B.Fomalont, M. Kramer, A.G. Lyne, S. Seagroves, J.M.Benson, M.M. McKinnon, D.C. Backer, R. Dewey, Pulsar Astrometry at the Microarcsecond Level . In theproceedings of the ”Stellar End Products” workshop, 13-15 April 2005, Granada, Spain (for publication in MmSAIvol.77). arXiv:astro-ph/0509025[17] Coryn A. L. Bailer-Jones
Microarcsecond astrometry
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / LIGO ITAMA-300VIRGOGEO-600LIGO AdvPSR Signals ε = 40% , a = 10 km s -2 (a) Efficiency case ǫ = 0 .
40 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 10% a = 10 km s -2 (b) Efficiency case ǫ = 0 .
10 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 5%a = 10 km s -2 (c) Efficiency case ǫ = 0 .
05 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I.
10 100 1000 f c [Hz] -24 -23 -22 -21 -20 -19 h c * [ H z ] - / PSR SignalsLIGO ITAMA-300VIRGOGEO-600LIGO Adv ε = 1% a = 10 km s -2 (d) Efficiency case ǫ = 0 .
01 for parameters a = 10 kms − and V kick ∼ km s − , calibrated against LIGO I. FIG. 9: Pulsar GW signals for parameters: a = 10 km s − and V kick ∼ km s − with Gaia: The Solar system, the Galaxy and beyond .Invited paper at IAU Colloquium 196: Transits of Venus:New Views of the Solar System and Galaxy, Lancashire,England, United Kingdom, 7-11 Jun (2004) Submitted toIAU Symp. e-Print: astro-ph/0409531. See also O. Forsin New observational techniques and analysis tools forwide field CCD surveys and high resolution astrometry (2006). e-Print: astro-ph/0604150. Also see E. Fomalont,M. Reid,
Microarcsecond Astrometry using the SKA ,New Astron. Rev. 48, 1473-1482 (2004) [18] N. Sago, et al., Phys. Rev. D 70, 104012-1,104012-8(2004).[19] B. Segalis, A. Ori, Phys. Rev. D 64, 064018 (2001)[20] H. J. Mosquera Cuesta and C. A. Bonilla Quintero,
Thegravitational-wave background produced by the kick atbirth of galactic run away pulsars , (accompanying paperin preparation) (2007)[21] Jose A. de Freitas Pacheco, Tania Regimbau, S. Vincent,A. Spallicci Int. J. Mod. Phys. D15:235-250 (2006).e-Print: astro-ph/0510727:
Expected coalescence rates of TABLE V: Selected sample from the ATNF catalog ofpulsars having speeds between 350-500 km s − , and theircorresponding periods. − ]1 J0255-5304 0.447708 381.6482 J0538+2817 0.143158 401.6823 J0543+2329 0.245975 377.1514 J0837-4135 0.751624 364.7725 J1300+1240 0.006219 350.6376 J1645-0317 0.387690 417.0227 J1803-2137 0.133617 351.2528 J1902+0615 0.673500 368.0869 J1941-2602 0.402858 351.02710 J2048-1616 1.961572 355.32811 J2113+2754 1.202852 386.79112 J2219+4754 0.538469 375.30413 J2305+3100 1.575886 373.54614 J2326+6113 0.233652 433.57615 J2354+6155 0.944784 357.846 ns-ns binaries for laser beam interferometers [22] A. Ud-Doula, J. Blondin, The Magneto-RotationalInstability in Core-Collapse Supernovae , AmericanPhysical Society, The 70th Annual Meeting of theSoutheastern Section, November 6-8, 2003, Wilmington,North Carolina, MEETING ID: SES03, abstract
Rotation and Magnetic Fields inSupernovae and Gamma-ray Bursts , American PhysicalSociety, 47th Annual DPP Meeting, October 24-28(2005), abstract
TheProto-neutron Star Phase of the Collapsar Model and theRoute to Long-soft Gamma-ray Bursts and Hypernovae ,arXiv: 0710.5789 (2007); M. Shibata, Y. T. Liu, S.L. Shapiro, C. B. Stephens, PRD 74, 104026 (2006),
Magnetorotational collapse of massive stellar cores toneutron stars: Simulations in full general relativity [23] Chris L. Fryer, Astrophys.J.601:L175-L178,2004. e-Print: astro-ph/0312265,
Neutron star kicks from asymmetriccollapse. . See also, Aristotle Socrates, Omer Blaes, AimeeL. Hungerford, Chris L. Fryer Astrophys. J. 632: 531-562 (2005). e-Print: astro-ph/0412144,
The Neutrinobubble instability: A Mechanism for generating pulsarkicks ; Chris L. Fryer, Alexander Kusenko, Astrophys. J.Suppl. 163:335 (2006), e-Print: astro-ph/0512033,
Effectsof neutrino-driven kicks on the supernova explosionmechanism [24] Leonhard Scheck, K. Kifonidis, H.-Th. Janka, E.Mueller,
Multidimensional supernova simulations withapproximative neutrino transport. 1. neutron star kicksand the anisotropy of neutrino-driven explosions in twospatial dimensions , Submitted to Astron.Astrophys. e-Print: astro-ph/0601302. See also Hans-Thomas Janka,L. Scheck, K. Kifonidis, E. Muller, T. Plewa
Supernovaasymmetries and pulsar kicks - Views on controversialissues . e-Print: astro-ph/0408439[25] D. Lai (2004). In cosmic explosions in three dimensions:asymmetries in supernovae and gamma-ray bursts , eds.P. Hoflich, P. Kumar, J. C. Wheeler (CambridgeUniverity Press, Cambridge), p. 276[26] Dong Lai,
Neutron star kicks andasymmetric supernovae . e-Print: astro-ph/0012049. Seealso Chen Wang, Dong Lai, JinLin Han, Astrophys.J.639,1007-1017 (2006). e-Print: astro-ph/0509484:
Neutronstar kicks in isolated and binary pulsars: observationalconstraints and implications for kick mechanisms [27] L. M. Ozernoy, B. V. Somov,Ap. S S 11, 264O(1971)
The Magnetic Field of a Rotating Cloud and Magneto-Rotational Explosions V ⋆ , distances, r ⋆ and orientations ( θ, φ )angles with respect to the line of sight.[32] The Fourier transform reads: ˜ h = R ∞−∞ e πift h ( t ))