ASTRONOMY A&A manuscript no. AND (will be inserted by hand later) ASTROPHYSICS Your thesaurus codes are: 02.07.2, 08.02.03, 08.05.03 February 1, 2008 The effect of supernova natal kicks on compact object merger rate Krzysztof Belczyn´ski and Tomasz Bulik Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa,Poland 9 9 Received , Accepted... 9 1 n Abstract. Mergersofcompactobjectsmayleadtodiffer- thus obtain the expected merger rate in the Galaxy. This a entastrophysicalphenomena:theymayprovidesourcesof approach suffers from several weaknesses: it is based on J observable gravitational radiation, and also may be con- observationsofonlythreeobjects,sosmallnumberstatis- 4 1 nected with gamma-ray bursts. Estimate of the rate with tics strongly affects the results. Moreover,the estimate of which such mergers take place are based on assumptions therateisbasedontheoreticalassumptionsregardingthe 1 aboutvariousparametersdescribingthebinaryevolution. selection effects. These assumptions may lead to system- v The distribution of one of these parameters - the kick ve- atic errors of uncertain value. 3 9 locity a neutron star receives at birth will strongly influ- The other approach that we can use, (let us call it a 1 ence the number and orbital parameters of compact ob- theoretical one), is based on population synthesis of bi- 1 jects binaries. We calculate these effect using population narysystems.Allcompactobjectsthatwillfinallymerge, 0 synthesisofbinarystarsandshowthattheexpectedcom- musthavetheiroriginasordinarystarsinbinarysystems. 9 pact object merger rate changes by a factor of 30 when FirststudiesinthatfieldwereperformedforMonteCarlo 9 / the kick velocity varies within its current observational simulationsofradiopulsarsbyDewey andCordes(1987). h bounds. Thus it seems that by modeling the evolution of stellar p binaries and the statistical properties of large ensembles - o of binaries one can calculate the expected population of r Key words: general relativity: gravitational waves — compact object binaries. Analyzing this population and t s stars: binaries, evolution thenconsideringthesystemsthatwillhavemergedwithin a : the Hubble time one can also estimate the present com- v pact object merger rate in the Galaxy. This calculation i 1. Introduction X requires the supernova rate in the Galaxy, and also the r The gravitational wave detectors LIGO and VIRGO will fraction of all stars that are in binaries. This ”theoreti- a soonbeoperational.Theircompletionbringsuptheques- cal approach” also suffers from the fact that it is based tionofthepossiblesourcesofgravitationalwavesandalso onseveralassumptions.Inorderto createapopulationof their brightness and observable rate. The most often con- binaries one requires distributions of initial parameters, sideredsourcesofgravitationalwavesaremergersofcom- like the mass of the primary star or the initial mass ratio pactobjects;neutronstarsorblackholes.Anumberofau- in the binary. Some stages in the stellar evolution require thors have already considered these phenomena, and sev- parameterization as well, e.g. mass loss and angular mo- eral estimates have been published (Narayan et al., 1991; mentum loss through stellar wind, mass exchange in the Phinney, 1991; Tutukov and Yungelson, 1993; Portegies commonenvelopephaseetc.Anotherparameterthatmay Zwart and Spreeuw, 1996). strongly influence the number of compact object binaries Therearetwoapproachesleadingtothe calculationof as well as their lifetime is the value of the velocity kick the merger rate. The first method, let us call it the ex- that a neutron star receives as a result of supernova ex- perimental approach, is based on the observational fact plosion.Theimportanceofthisparameterhasbeenshown that we do see three binary systems of neutron stars: by Portegies Zwart and Spreeuw (1996). PSR1913+16 (Taylor and Weisberg, 1989), PSR1534+12 The measurement of the distribution of kicks in su- (Wolszczan,1991),andPSR2303+46(Taylorand Dewey, pernova explosions is not easy. One approachthat can be 1988). Based on this number, and considering various taken is to measure velocities of pulsars and use this dis- observational selection effects, like e.g. the pulsar beam tribution as the distribution of supernova kicks. Here one width,onecanestimatethenumberofsuchsystemsinour has to take into account selectioneffects, like for example Galaxy. Given the observed orbital parameters for these the fact the fast neutron stars may leave the Galaxy and systemsonecanthencalculatethelifetimeofeachoneand not be visible as radio pulsars. A comprehensive study of 2 Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... thisandotherselectioneffects(Arzoumanianetal.,1997) Werestricttherangeofthemassestotheintervalbetween shows that a large fraction of neutron stars has veloci- 10M⊙ to 50M⊙ aswewantatleastthe moremassivestar ties above 500 km s−1. Iben and Tutukov (1996) argue to undergo a supernova explosion. that the velocities of radio pulsars can be explained by We adopt the following distribution of the mass ratio just taking into account the recoil velocity due to mass q loss in supernova explosions in binaries, and neglecting a possible ”natal kick”! An independent study of veloci- Φ(q)∝const (2) ties of young pulsar through measuring offsets from the centers of supernova remnants (Frail et al., 1994; Lyne There is some uncertainty regarding Φ. Portegies Zwart and Lorimer, 1994) confirms that the distribution initial andVerbunt(1996)useΦ∝(1+q)−2,whileTutukovand velocities of neutron stars may have a large high velocity Yungelson (1994) use Φ ∝ q. Yet another distribution is tail.OntheotherhandBlaauwandRamachandran(1998) used in the simulation by Pols and Marinus (1994), who argue that the observed properties of pulsars can be ex- useadistributionslightlypeakedforsmallvaluesofqand plained in a model with a single unique value of the kick falling down when q goes to unity. v =200kms−1.Asimilarresulthasbeenobtainedby kick Initial binary eccentricity e is assumed to have value Lipunov et al. (1997) who show that the observed pop- between 0 and 1 and its initial distribution is taken from ulation of neutron stars is consistent with the kick ve- (Duquennoy and Mayor, 1991) : locities in the range 150 − 200 km s−1. The distribution of the kick velocities has been parameterized by differ- Ξ(e)=2e. (3) ent authors: Portegies Zwart and Spreeuw (1996) used a Gaussianwiththe widthof450kms−1,while Cordesand The distribution of the initial semi-major axis a used Chernoff (1997) proposed a weighted sum two Gaussians: in population synthesis codes is flat in the logarithm, i.e. 80percentwiththewidth175kms−1 and20percentwith thewidth700kms−1.Hereweattempttoinvestigatesys- 1 tematically the effect of the kick velocity distribution on Γ(a)∝ . (4) a the compactobjects merger rate.Insection 2 we describe themodelofbinaryevolutionusedforsimulatingthepop- Both stars are initially massivemain sequence stars,with ulation of binaries,in section3 we show the results of the the radiiofat least4–5R⊙, so we takeminimum value of simulation and calculate the compact object merger rate. atobe10R⊙.Howeverifthe sumofradiioftwozeroage Finally we discuss our results in section 4. mainsequencestarsexceeds10R⊙,wechosethe minimal separation to be twice the radius of the primary component. This should give the stars the space to evolve with- 2. Population and evolution of binary systems out merging before they go through their main sequence While the evolution of a single star is only a function of lifetime. Nevertheless some systems might be born with itsmassandmetallicitytheevolutionofabinarypresents high eccentricity, and thus be in contactand merge form- a more complicated problem.It depends on the mass of a ing a single very massive star. At this point we stop our more massive component, the mass ratio of the smaller calculationaswearenotinterestedinthistypeofmergers to the larger star q, and on the initial parameters of in our study. their orbit: e - the eccentricity and a the semi-major axis Although very wide binaries with periods up to 1010 of the orbit. Following the general approach in the field days are observed (Duquennoy and Mayor, 1991) and we assume that these parameters are independent, i.e. models with initial separation up to 106 R⊙ are used the probability density can be expressed as a product: (Portegies Zwart and Verbunt, 1996), we have set the p(M,q,e,a) = Ψ(M)Φ(q)Ξ(e)Γ(a). Our population syn- maximal separation to 105 R⊙ as wider binaries are very thesis code is mainly based on Bethe and Brown (1998). unlikely to survive two supernova explosions and form a The evolution of a single star is described by Eggleton compact object system. Our models showed that setting et al. (1989). In the following all masses are in the units the upper limit to 106 R⊙ does not change the results of solar mass M⊙. noticeably. We assume that the distribution of the kick velocity a 2.1. Distribution of the initial parameters newlybornneutronstarreceivesinasupernovaexplosion isathreedimensionalGaussianandparameterizeitbyits Beyond 10M⊙ the distribution of the masses of the pri- width σ .We varyσ to determine the dependence of the v v mary stars that we use follows Bethe and Brown (1998), mergerrateandpropertiesofthe compactobjectbinaries Ψ(M)∝M−1.5 (1) population on this parameter. Alternatively, we draw the natal kicks from a weighted sum two Gaussians (Dewey Other authors have used similar distributions, e.g. Porte- and Cordes,1987):80 percentwith the width 175km s−1 gies Zwart and Verbunt (1996) use the exponent of −1.7. and 20 percent with the width 700 km s−1. Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... 3 2.2. Evolution of binaries where a is binary separation, M is the mass of the giant 1 losing material,M is the mass ofits main sequence com- 2 In our calculations we neglect the effects of stellar wind panionandindices i,f correspondrespectivelyto the val- on evolution of binaries. uesbeforeandaftertheRochelobeoverflow.Theparame- Tidal circularization takes place in binaries for which terβ describesthespecificangularmomentumofmaterial the size of any component (or both components) is large leavingbinary(PolsandMarinus,1994).Theaboveequa- in relation to their separation(Zahn,1978). This happens tion assumes that β is constantwith time. The value of β when the stars evolve to the giant stage, and/or if the is uncertain; typically the values such as β ≥ 6 or β = 3 initial binary separation is small. We follow the prescrip- areused,fordiscussionseePolsandMarinus(1994).Here tionproposedbyPortegiesZwartandVerbunt(1996);the all calculations are performed for β = 6. The initial pri- circularizationtakesplaceifthestellarradiusofonecom- mary, now either a helium star (after the mass transfer) ponent is larger then 0.2 of the periastron binary separa- or a giant goes supernova, leaving behind a neutron star. tion.The orbitalelements (a,e)changewithconservation The initial secondary evolves, becomes a giant and may of angular momentum until the new binary separation is begin to transfer material to the newly formed neutron 5stellarradiiofthe componentinwhichtidaleffects take starincreasingitsmass,andpossiblyturningittoablack place, or the binary is totally circularized (e=0). hole. We assume that the maximum mass of a neutron Binary stars may undergo mass transfer and common star is 2.2M⊙. envelope phases in different evolutionary stages. Depend- In describing accretion on a neutron star we follow ingonthephysicalconditionsthiswillresultinthechange BetheandBrown(1998),andwewillcallthisregimetype of mass of each star, mass loss from the system, and also II mass transfer. A neutron star accretes matter while in change of the size and shape of the orbit. We describe moving in orbit through the extended envelope of the gi- below three schematic evolutionary paths and accompa- ant. We use the Bondi-Hoyle accretion rate nying mass transfer types used in the code. M˙ =πρvR2 (7) ac 2.2.1. Evolution for small mass ratios, (q <0.8897) whereρisthedensityinthevicinityofthecompactobject The more massive star (primary) evolves faster in the bi- of mass M1, v is its velocity, and Rac = 2GM1/v2 is the nary. We use the following approximate formula for the accretion radius. Consideration of the energy loss equa- main sequence lifetime of a star with mass M: T = tions lead to the final mass of the compact object (Bethe MS 20×106 (M/10M⊙)−2 yrs. After this time the primary and Brown, 1998) evolves to the giant stage which lasts about 20 per cent Mi 1/6 of its main sequence lifetime, and increases its size. For Mf =2.4× 2 Mi. (8) q <0.8897 the secondary is still on main sequence at the 1 (cid:18)M1i(cid:19) 1 endofprimarygiantstage.Ifthegiantprimaryoverfillsits We assume that the giant loses its envelope so its final Rochelobemasstransfertothemainsequencecompanion mass is just that of the helium core Mf = 0.3Mi. The (star 2, with mass Mi) and mass loss takes place. We de- 2 2 2 orbital separation follows from the energy integration i.e. notethisregimebytypeImasstransfer.Themasstransfer equations 5.18 through 5.23 in (Bethe and Brown, 1998): continues until the giant is stripped of its envelope, and thus the final mass of the giant will be just that of the Mi its helium core. We use the approximation M1f = 0.3M1i af =0.145×ai(cid:18)M12i(cid:19) . (9) (Bethe and Brown, 1998). A part of the envelope mass 0.7Mi islostfromthesystemwhileanotherpartistrans- Finally the initial secondary undergoes a supernova ex- 1 plosion,andprovidedthat the systemsurvives,weobtain ferredto the companion.The fractionof masstransferred a compact object binary, consisting of either two neutron tothecompanionisproportionaltothesquareofthemass stars or of a black hole and a neutron star. ratio (Vrancken et al., 1991; Bethe and Brown, 1998), so the mass of the companion after the mass transfer is 2.2.2. Evolution of two stars for intermediate mass ratio, Mf =Mi(q+0.7q2), (5) 2 1 (0.8897 0.8897 the secondary is already a giant when Theorbitwasalreadycircularizedbytidalforces,when the primary explodes as a supernova. The upper limit is the giant was filling its Roche lobe, and now its size explained in the next subsection. changes due to mass transfer and mass loss. We describe For the case of small orbital separations the evolution thechangeinsemi-majoraxisby(PolsandMarinus,1994) begins in a similar way as for small q described above. −2 2β+1 The evolution begins in a similar way as in the case of af = M1f M2f M1f +M2f , (6) small mass ratio, and the primary may transfer mass to ai M1i M2i! M1i+M2i ! the secondary as described by type I above. When the 4 Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... secondarybecomesagianttheprimaryisalreadyahelium star,strippedofits envelope.The secondarymaytransfer masstotheprimaryandweapproximatethisprocessalso bytype I.Theprimaryexplodesasasupernovaandforms a neutron star, which may accrete from the secondary, provided that it is still a giant. The mass transfer in this phase is described by type II. The secondary undergoes a supernova explosion and if the system survives we obtain a compact object binary. In the case of large orbital separations the first stage of mass transfer from the giant to the main sequence star (type I)does notoccur.The starsmightbeginto interact only when both of them are already in the giant stage. In this case they loose the common envelope. To describe thisstageofevolutionweassumethatthegiantsloosethe entireenvelopesandbecomeheliumstarsof30percentof the initial giantmasses.We then approximatethe change in the size of the orbit by: −1 a (Mi+Mi)Mi(1−q2) f ≈ 1+ 1 2 1 . (10) ai " αCEM1fM2f # As above a is binary separation, M ,M are the masses 1 2 of primary and secondary. The indices i,f correspond to the values before and after common envelope phase, re- spectively, and α describes the efficiency of the orbital CE energy expenditure in the dispersal of common envelope. Equation (10) was adopted from Yungelson and Tutukov Fig.1. An example evolutionary path leading to forma- (1993) and describes the change of the orbital separation tionofablackholeneutronstarbinary.Units ofmassare a following decrease of the orbital energy which was used solarmasses,andunitsofdistanceareastronomicalunits: to eject a common envelope. They have also showed that (a)initialbinary;(b)aftertidalcircularizationwhenstar1 0.6 ≤ α ≤ 1, and we have performed our calculations expandedtogiantsize;(c)systemafterfirstmasstransfer, CE for different values: α = 0.4,0.6,0.8,1.0. Our calcula- star 1 explodes as supernova;(d) systemafter first super- CE tions show that the results (the production rate of the nova; (e) after tidal circularization when star 2 expanded compact object binaries, and their properties) are almost to giantsize;(f) system after secondmass transfer,star2 indistinguishable for these three values, so for simplicity explodesassupernova;(g)systemaftersecondsupernova: we present our results for one value of α =0.8. black hole neutron star binary. CE Similarly to the caseofsmallorbitalseparationsifthe secondary is still a giant it may transfer mass to the neu- After each process which changes the orbital param- tronstarformedinthesupernovaexplosionoftheprimary. eters and sizes of the components we check if the com- This mass transfer is described as type II. ponents sizes are not larger than the periastron binary separation. 2.2.3. Evolution of two stars of almost equal mass, (q >0.95) 2.3. Supernova explosion Two stars almost simultaneously (within 10% of their Thesupernovaexplosionsareverylikelytohaveanimpact main sequence lifetime) leave main sequence and become onthe binarysystems.We assumethatineachsupernova giants. If their radii are large enough in relation to the explosion a neutron star with mass of 1.4M⊙ is formed periastron orbital separation, the orbit is tidally circular- andweneglectthe interactionofthecompanionstarwith ized. If the binary separation is small enough the giants the envelope ejected in the supernova explosion. may finally overfill their Roche lobe and enter a common We draw a random time for a supernova explosion in envelope phase, which we describe by equation (10). The the orbital motion of the binary. We find the relative po- twostarsgosupernovaoneafteranother,andifthesystem sition and velocity of the two stars for this moment. The survives,we obtain a compact object binary consisting of exploding star is then replaced by a neutron star and we two neutron stars. add the kick velocity to its orbital velocity, while the ex- Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... 5 Fig.3.Possibleresultsoftheevolutionofabinarywiththeinitialorbitalseparationa=20R⊙ andeccentricitye=0.5 for different values of the initial primary mass M and the initial mass ratio q. The top left panel shows the systems that were disrupted in the first supernova explosion. The solid line corresponds to the initial secondary mass 10M⊙. The top right panel shows the systems in which the neutron star, born in the first supernova explosion, merged with the companion.The bottom left panel shows the systems disrupted in the second supernova explosion. The crosses in thebottomrightpanelshowtheneutronstarbinariesandthefilledcirclesaretheblackholeneutronstarbinaries.We present one thousand of each type of binaries. In this calculation we used the Cordes & Chernoff (1997) kick velocity distribution. panding supernova shell carries away a part of the total sudden mass loss and a random kick velocity on binary momentum of the binary. If the energy of such a system parameters see Hills (1983). is less than zero the system is bound and we calculate the parameters of a new orbit of the binary and its new Afterthefirstsupernovaexplosionweverifyiftheneu- spatial velocity. For a detailed discussion of the effect of tron star does not collide and merge with the companion, 6 Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... Fig.4. Possible results of the evolution of a binary with the initial primary mass 20M⊙ and the mass ratio q = 0.5. The topleft panelshowsthe systemsdisruptedinthe firstsupernovaexplosion,the toprightpanelshowsthe systems in which the neutron star, born in the first supernova explosion, merged with the companion. The bottom left panel shows the systems disrupted in the secondsupernovaexplosion.The bottom rightpanelshows the black hole neutron star binaries, and there are no neutron star binaries formed for this mass ratio (q =0.5). We present one thousand of each type of binaries. In this calculation we used the Cordes & Chernoff (1997) kick velocity distribution. i.e. if the radius of the companion is larger than the peri- bital energy loss through radiation of gravitationalwaves astron binary separation. becomes important once a compact object binary is in a tight and/or highly eccentric orbit. The evolution of the semi-major axis a and eccentricity of the orbit e, in a bi- 2.4. Energy loss through gravitational radiation Once a compact object binary is formed its orbit will evolve because of the gravitational wave energy loss. Or- Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... 7 Fig.5. Population of the compact object binaries in the Fig.2. An example evolutionary path leading to forma- four categories: double neutron star binaries that merge tion of a neutron star neutron star binary formation: (a) within the Hubble time (empty squares), and these that initial binary; (b) after tidal circularization when both do not (empty circles), neutron star black hole binaries stars expanded to giant sizes, system ejects common en- that merge within the Hubble time (filled squares), and velope; (c) system after common envelope phase, star 1 these that do not (filled circles). The error bars represent explodes as supernova; (d) system after first supernova, the counting statistics of the simulation. nowstar2 explodesas supernova;(e)systemaftersecond supernova: neutron star neutron star binary. effects we fix two of them and present the types of sys- naryemittinggravitationalwaveshavebeencalculatedby tems obtained in the course of the binary evolution. In Peters (1964). The lifetime of a compact object binary is Figure 3 we fix the initial orbital separation a = 20R⊙ and eccentricity e = 0.5. The graphs are empty in the 5c5a4(1−e2)7/2 73 37 −1 lower left part for which M ×q < 10M⊙. This is the re- t = 1+ e2+ e4 (11) merg 256G3M M (M +M ) 24 96 gion for which the secondary is not massive enough to 1 2 1 2 (cid:18) (cid:19) undergoasupernovaexplosion.However,insomesystems where a is the semi-major axis of the orbit, e is its eccen- for which the primary mass is M > 20M⊙, the mass of tricity,andM1, M2arethemassesofthecompactobjects. the secondary is increasedby accretionwhen the primary goes through the giant phase. Most of the systems end upeitherby disruptioninthe firstsupernovaexplosionor 3. Results by a spiral in of the neutron star to the secondary (top The code described above allows to generate populations panels) The remaining systems may be disrupted in the ofcompactobjectbinariesandtracetheirstatisticalprop- second supernova explosion. The compact object binary erties. In this paper we mainly concentrate on the depen- population (bottom right panel) is bimodal. The neutron denceofthesepropertiesasafunctionofthekickvelocity star neutron star binaries are formed from the systems distribution. In Figures 1 and 2 we present two example with q very close to unity, while the black hole neutron evolutionary paths leading to formation of compact ob- star systems are primarily formed when q is intermedi- ject binaries: black hole neutron star binary and double ate. Thus the initial distribution of the mass ratio q has neutron star systems. a strong influence on the production rates of these two Letusfirstconsiderdifferentevolutionarypathsofthe types of compact object binaries. The number of systems binaries depending on their initial parameters. We first shownineachpaneldoesnotcorrespondtotheactualpro- consider a model with the Cordes and Chernoff (1997) duction rates. We present one thousand systems in each kick velocity distribution. Tracing the evolution now de- panel. The actual calculation produces 51% systems with pends on four parameters: M (primary initial mass), q thesecondarynotmassiveenoughtoundergoasupernova (initialmassratio),a(initialsemi-majoraxis),ande(ini- explosion,40%systemstornafterthefirstexplosion,6.2% tial eccentricity). In order to visualize the evolutionary of systems merged after the first explosion, 2.7% systems 8 Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 P [days] P [days] 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 P [days] P [days] Fig.6.DistributionoftheinitialorbitalcompactbinarysystemsparametersinthespacespannedbytheperiodP and eccentricity e, for four different Gaussiandistributions of the kick velocity σ =0 km s−1 - top left panel, 200 km s−1 v - top right panel, 400 km s−1 - bottom left panel, and 800 km s−1 - bottom right panel. Each panel contains 1000 objects. torn after the second explosion, and 0.35% of compact tems.Thepopulationofsystemsthatendupmergingafter object binaries. the first explosion, are disrupted in the second explosion, or form a compact object binary originate from the same In Figure 4 we present the results of the evolution of region of the parameter space. One should note that be- systems for which we fix the initial primary mass and the cause of the circularization of orbits already in the initial initial mass ratio, while varying the the initial orbital pa- stages of the binary evolution the final population is not rameters a and e. The systems with small orbital separa- verysensitivetotheinitialeccentricity.Ontheotherhand tionsandhigheccentricitymergeintheearlyphaseofthe the distribution of the initial orbital separation is impor- evolutionandarenotconsideredinthispaper.Disruption tant, as the compact object binaries originate in systems after the first supernova explosion may occur to all sys- Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks... 9 withrelativelysmallorbitalseparations(seebottomright panel in Figure 4). One should note that all the compact object binaries shown in Figure 4 are black hole neutron starbinaries.Doubleneutronstarsystemsareformedonly when q is nearly unity in our simulations. As before the number of systems shown in each panel does not corre- spondto the actualproduction rates,andshow one thousand systems in each panel. The actual calculation pro- duces12%systemsbornincontact,43%systemswiththe secondarynotmassiveenoughtoundergoasupernovaex- plosion, 34% systems torn after the 1st explosion, 0.7% systems merged after the first explosion, 0.50% systems torn in the second explosion and only 0.15% mergers. We present the dependence of the production rates of different types of compactobject binaries onthe width of the kick velocity distribution σ in Figure 5. The num- v ber of double neutron star systems that merge within the Hubble time increases with the kick velocity, while the production rate of black hole neutron star systems be- Fig.7. Cumulative distributions of the t - the life- mrg comes smaller.These two ratesare nearly equalwhenthe times of compact object binaries for four different kick kickvelocityisroughlythatgivenbyCordesandChernoff velocities: the solid line for σ = 0 km s−1, the short v (1997). dashed line for σ = 200 km s−1, the long dashed line v for σ = 400 km s−1, and the dash dotted line for v σ =800kms−1.Eachlinewasplottedfromadistrbution v In Figure 6 we present the distributions of the orbital of 1000 objects. parametersof compactobject binaries for a few represen- tative values of the width ofthe kick velocity distribution σ . Each panel contains one thousand compact object bi- The relation between the new orbital period P′ and the v naries. To understand these plots let us first consider the new eccentricity e′ for different relative mass loss x is propertiesofthepopulationofobjectsjustbeforethesec- (1+e′)1/2 ond supernova explosion in the case σv =0kms−1. Some P =P0(1−e′)3/2 , (12) ofthesystemsarewide,withtheeccentricityvaryingfrom zeroto unity,however,mostofthe systemspopulate are- where P is the orbital period before the explosion. This 0 gionwith eccentricitiesabove≈0.45.These systemsorig- relationdefinesthecurvedshapeofthedistributioninthe inate in binaries with small initial mass ratios. A char- P,e diagram. The objects populating the right hand side acteristic property of this group is a correlation between of the P,e plot, originate from systems with intermediate eccentricityandperiod.Objectsinthis grouporiginatein or large q. The intermediate q objects went through ac- systems with small initial value of q. The masses of the cretionontotheneutronstar(regimeIIofamasstransfer primary compact objects in these systems have a narrow describedabove)andthereforesimilarreasoningasabove distribution,themassofthesecondaryaftertheaccretion applies to them. However systems with high initial value phaseweaklydependsonthe initialmassbeforeaccretion ofq resultsinwidebinaries(periodslargerthan100days) -seeequation8,andistypicallyM′2 ≈2.3M⊙.Themass with different eccentricities. The compact object binaries of the secondary just before the second supernova explo- in Figure 6 (top left panel) with eccentricities below 0.45 sion,isthemassoftheheliumcoreofsecondarystarandis and orbital periods smaller then 103 days originate in bi- intherange3.0M⊙

The effect of supernova natal kicks on compact object merger rate

Similar The effect of supernova natal kicks on compact object merger rate