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High Neutron Star Birth Velocities and Gravitational Radiation During Supernova Explosions

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High Neutron Star Birth Velocities and Gravitational Radiation During Supernova Explosions A&A manuscript no. (will be inserted by hand later) ASTRONOMY AND Your thesaurus codes are: ASTROPHYSICS 01 (02.07.2; 08.14.1; 08.19.4) 19.5.2018 High Neutron Star Birth Velocities and Gravitational Radiation during Supernova Explosions S.N.Nazin1 and K.A. Postnov1,2 1 Sternberg Astronomical Institute, Moscow University, 119899 Moscow, Russia 2 Faculty of Physics, Moscow University, 117234 Moscow, Russia Received ... 1996, accepted ..., 1996 Abstract. Assuming the observed pulsar velocities to precessing binary pulsar orbit in PSR J0045-7319 in the originate during asymmetric collapse of stellar cores, we SMC (Kaspi et al. 1996). The collapse asymmetry result- compute the amplitude of gravitational waves emitted ing in the kick velocity of a neutron star at birth may during type II and Ib supernova explosions and their de- be due to different reasons. For example, recent calcula- tection rate from within a distance of 30 Mpc. At the tions of Burrows & Hayes (1996) has shown the ability rms-level of advanced laser interferometers h 10−22 at of neutrino anisotropic emission to produce kick velocities frequencies 300 1000 Hz the expected rate is about≈ 1 per of 400-500 km/s, as observed (see also Imshennik 1992; year. − Bisnovatyi-Kogan 1993). One of the consequences of Lyne and Lorimer’s result Key words: Gravitational waves — Stars: neutron — was a recognition that the kick velocity imparted to a neu- Supernovae: general tron star at birth has a power-law asymptotic form at high velocities (Lipunov, Postnov & Prokhorov 1996a,b). Using direct Monte-Carlo calculations of binary star evolution (the so-called “Scenario Machine”), they found that the Supernova explosions are among the most violent Lyne-Lorimer pulsar transverse velocities are best repro- events known in nature. Supernovae type II and Ib are duced assuming the space (3-D) kick velocity distributed triggered by the gravitational collapse of evolved degen- as erate cores of massive (> 10 M⊙) stars and leave be- x0.19 hind neutron stars or, possibly, black holes as a rem- f(x) 6.72 1/2 (1) nant. An enormous amount of energy is released during ∝ (1 + x ) the collapse (roughly, the binding energy of a neutron where x = v/v0, v0 = 400 km/s. An important thing 2 star 0.15M⊙c ). Although most of this energy is car- about having this distribution is that at high velocities ∼ −3 17 ried away by neutrinos, a part of it can escape in the (v > 500 km/s) it goes practically as v . , that is much form of gravitational radiation provided that the collapse slower than the maxwellian tail exp( v2), which some- arXiv:astro-ph/9701073v2 15 Jan 1997 is not purely symmetrical. This collapse anisotropy (ex- times is assumed for the kick velocity∝ − distribution (e.g. pressed conveniently in terms of the fraction of the energy Portegies Zwart & Spreeuw 1996). 2 released, ǫM⊙c ) was first introduced by Ozernoy (1965) The anisotropic stellar collapse may be a source of and Shklovskii (1970). gravitational waves (GW). To be detectable from the dis- It required more than 20 years for the idea of the tance of Virgo cluster ( 18 Mpc) by future ground- collapse anisotropy to receive serious observational sup- based laser interferometric∼ and bar detectors, the GW port. First, the revision of the pulsar distance scale led energy emitted during a SN explosion should be about −3 2 to increasing the observed pulsar velocities by a factor of ∆E = 10 M⊙c (Thorne 1995; Schutz 1996). However, two up to values of about 400-500 km/s (Lyne & Lorimer in numerical calculations this energy has been found to 1994). Observations of pulsars associated with young su- be very tiny, typically 10−9 10−7 of the total energy pernova remnants revealed even higher pulsar velocities output (see M¨uller 1996 for a− review). Unfortunately, all up to 900 km/s (Frail et al. 1994). Such high velocities these calculations are still far from being realistic con- of young pulsars could hardly be obtained without addi- sidering enormous numerical difficulties and uncertainties tional recoil (“kick velocity”) during the supernova explo- involved. sion. Next, a direct evidence for a kick velocity of at least Nevertheless, the recognition of high additional veloc- 100 km/s has recently been obtained from observations of ities neutron stars acquire at birth suggests a way to esti- 2 S.N.Nazin et al.: High Neutron Star Birth Velocities and Gravitational Radiation during Supernova Explosions mate ǫ from observations, regardless of the unknown col- lapse anisotropy mechanism(s). Indeed, the kinetic energy 2 of the neutron star motion, ∆E = MNSv /2, where MNS is the neutron star mass, may be considered as a lower limit to the energy emitted in GW. Assuming MNS =1.4 2 M⊙, we obtain ǫ =0.7 (v/c) . With the distribution law (1) the fraction of high× velocity pulsars is ∞ P (> x)= f(ξ)dξ 0.38 x−2.17 (2) Zx ≃ (the power-law asymptotics is valid for x > 2). For ex- ample, the fraction of pulsars with v > 1000 km/s is 0.05. Of course, there should exist a maximum cut- off≈ velocity; it must be higher than the maximum pulsar velocities observed, 2000 km/s (x > 5), but its exact value only slightly changes∼ the normalization coefficient. For ǫ = 10−4 (v 3200 km/s) we obtain P (> 8) 0.004, i.e. every 250-th≈ neutron star may be born with the≈ high velocity required. The neutron star galactic birth-rate is determined by that of massive (> 10 M⊙) stars, which is 1/25 years ≈ −2 35 assuming Salpeter mass function f(M)dM M . dM Fig. 1. The log N(> hc) − log hc curve of the number of ∝ GW-bursts caused by core collapse supernovae as seen by a and the mean galactic star formation rate of 1 M⊙ per h year. Using the relations ǫ =0.7(v/c)2, x = v/(400km/s) GW-detector with an rms-sensitivity c at the unitary sig- nal-to-noise ratio in 1-year integration time. The supernova and Eq. (2) we may write galactic birth rate of high- rate are calculated for the baryonic matter distribution within velocity pulsars as the region of 30 Mpc from Sun according to Tully’s Nearby −2.17 −1 −5 −1.085 −1 Galaxies Catalog (Tully 1988) with account for the SN-rate R =0.38/25 x yr 1.3 10 ǫ−3 yr (3) ≈ × dependence on the morphological type of the galaxies. The (here and below subscripts indicate the quantities mea- distribution of SN-generated GW-amplitudes is taken assum- sured in units of the corresponding power of ten, e.g. ing Lyne-Lorimer kick velocity distribution (see the text). The −3 hatched region encompasses the assumed GW frequencies from ǫ−3 ǫ/10 ) The≡ characteristic amplitude of the GW burst from a 1 kHz (lower boundary) to 300 Hz (upper boundary). The bro- ken curve represents the SN rate assuming constant GW energy SN located at a distance r is (Thorne 1987; Schutz 1996) −4 fraction ǫ = 10 (from Lipunov et al. 1995) with contributions 1/2 of local group of galaxies indicated. −22 1/2 1kHz 18Mpc hc =5 10 ǫ−3 (4) × fc r where fc is the characteristic frequency of the burst. This the realistic baryon matter distribution within 30 Mpc level should be detectable by the advanced LIGO inter- ≈ ferometer. The event rate from the volume V of the Uni- according to Tully’s Nearby Galaxies Catalog (Tully 1988) verse is (e.g. Phinney 1991; see also Lipunov, Postnov & (see Lipunov et al. 1995 for more detail). The result is pre- 3 Prokhorov 1996b) 0.01 (galactic rate) (V/Mpc ). sented in Fig. 1. The hatched region corresponds to GW- Using Eqs. (3) andR≈ (4) we find× × frequencies of the signal from 300 Hz (upper boundary) to 1 kHz (lower boundary). For comparison, we reproduce the 4 −1 0.42 −3/2 −3 log N log h calculated for constant ǫ = 10− (Lipunov et (0.08yr ) ǫ−4 f1 h−22 R≈ kHz (5) al. 1995).− Clearly, the more realistic ǫ-distribution yields −1 0.42 −3/2 −3 (1yr ) ǫ−3 f h−22 . an order of magnitude smaller event rate because the rate ≈ 300Hz of SN explosions with higher ǫ strongly decreases. The This means that we have chance to observe 1 GW-burst shape of the curve is smoothed by the f(v) distribution. per year caused by SN II and Ib explosions at a level of −22 The mean ǫ for Lyne-Lorimer velocity distribution (1) is h = 10 provided that the energy is carried out at 2 6 rms ǫ v f(v)dv 4.4 10− . the frequency 300 Hz. h i∝ ≈ × R Since hc √ǫ v, the distribution of the GW-bursts It is seen from the figure that a few events per year are caused by the∝ SN∼ explosions from a given distance r will expected at the noise level of the advanced-LIGO rms- have the form (1). This enable us to compute the cumula- sensitivity 10−22. At the initial laser interferometers ∼ −21 tive distribution of the number of events with the ampli- sensitivity level hrms 10 the expected SN rate is tude higher than a given one (log N(>h ) log h ) using very low, 0.01 per year.≈ c − c ∼ S.N.Nazin et al.: High Neutron Star Birth Velocities and Gravitational Radiation during Supernova Explosions 3 In our analysis, we have not used any particular mech- Phinney, E.S.
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