The effect of supernova natal kicks on compact object merger rate

11 Pages·0.89 MB·English

by  Krzysztof Belczynski

ASTRONOMY
A&A manuscript no.
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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
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9
Received , Accepted...
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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 compo-
nent. 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<q <0.95).
where q is the initial mass ratio.
When q > 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 thou-
sand 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⊙ <M′2 <5.5M⊙.The orbitalparam- naries of intermediate and large initial mass ratio.
eters of such system after an explosive mass loss (second In the case of non zero kick velocity the systems with
supernova explosion) depend only on the total mass loss, high q are very unlikely to survive. In fact there are only
see e.g. Pols and Marinus (1994). In our calculation the two such systems on the plot for σ = 200km s−1, and
v
newlyformedneutronstarhasamassof1.4M⊙,thusthe none for higher velocities. The escape velocity in long pe-
relative mass loss x=(1.4+M′ )/(M′ +M′ ) is in the riod systems is low, and if such systems existed after the
2 2 2
range 0.47<x<0.69.Systems that loose more than half first supernova event, they have are very likely disrupted
ofthemass(x<0.5)areunbound.Othersystemsbecome in the second supernova explosion.
eccentric and the eccentricity is given by e′ = (1−x)/x. As the kick velocity is increased the shape of the dis-
Hence the lowest eccentricity a system can have after the tribution in the P,e diagram also changes. In the case
secondsupernovaexplosioninoursimulationsise′ =0.44. σ = 200 km s−1 there is a small fraction of high ec-
v
10 Krzysztof Belczyn´ski and Tomasz Bulik: The effect of supernovanatal kicks...
centricity, low period systems. This region of the param-
eter space fills up as the kick velocity increases, see lower
panel in Figure 6 where the case σ = 400 km s−1 and
v
σ = 800 km s−1 are shown. This is due to the fact that
v
with the increasing kick velocity the long period systems
are easier to disrupt and the fraction of surviving short
period systems becomes larger. We note that the distri-
butions shown in Figure 6 are similar to those obtained
previously (Portegies Zwart and Spreeuw, 1996; Tutukov
and Yungelson, 1993).
In Figure 7 we present the cumulative distributions of
the lifetimes of compact object binaries for the same set
of kick velocities as in Figure 6. In the case of no kick
velocity σ = 0 km s−1 the distribution is bimodal: the
v
systems with small q merge typically within the Hubble
time (which we take to be 15Myr). However the systems
withhigherqremaininwideorbitsandtheirmergertimes
exceedtheHubble time.Withtheincreasingkickvelocity
only the tightly bound systems survive. The lifetime of
a system scales with the fourth power of the semi-major
Fig.8. Fraction of binary compact objects that merge
axis, and therefore the median lifetime decreases with in-
within the Hubble time (15Gyrs), as a function of the
creasing kick velocity. In the case of σ = 200 km s−1
v width of the kick velocity distribution. We also show the
the median lifetime is ≈ 4×108 years, and it decreases
fit discussed in the text. Each calculation produced 1000
by a factor of four when the kick velocity is doubled. For
merging binaries.
the highest kick velocity velocity σ = 800 km s−1 the
v
median lifetime is only ≈ 3×107 years. One should note
however, that the distribution of t is very skewed, rateas0.02f year−1,andthebinaryfractionas0.5f ,
merge SN bin
and we present it on a logarithmic axis. Thus there is a where f and f are factors of the order of unity.
SN bin
longtailextendingto aboutthe Hubble time.Mostofthe In the simplest case when we assume that the star
mergers take place in the Hubble time and only a small forming processhas been goingatthe samerate through-
fraction of the total population lasts longer. out the history of the Galaxy we obtain the compact ob-
Finally in Figure 8 we present f – the fraction ject merger rate
merge
of all binaries with the primary star more massive than
r =0.0001×f f f year−1, (14)
10M⊙ that produce a pair of compact objects in a binary SN bin merge
system. We calculate fmerge for a large range of the kick where fmerge is expressed in percents. Taking values of
velocity distribution widths. To calculate each point in f from our Figure 8, we may calculate number of
merge
Figure 8 we calculated one thousand of compact binaries. expected compact objects gravitational merging events,
Thisfractionfallsdownveryapproximatelyexponentially which for, let say, σ = 100 km s−1 will be 0.0002 per
v
withtheincreasingkickvelocity.Wehavefittedamodified year which yields approximately 1 event per 5000 years
exponential to this dependence per Galaxy.
f ∝exp(−4.21−8.51×10−3σ +2.6×10−6σ2).(13) Since the compact object merger rate is directly pro-
merge v v portional to f it also depends exponentially on the
merge
The fit is shown by the dashed line in Figure 8 and is kick velocity distribution width! The assumption about
accurate to about 6%. Note that equation (13) can be the constantstar formationthroughoutthe history of the
usedtodeterminethemergerfractionforanykickvelocity Galaxy is in fact not really crucial. As we have seen most
distributionthatcanbeexpressedasalinearcombination ofthemergerstakeplacewithinafewtimes108yearseven
ofthreedimensionalGaussians.Equation(13)canbeused for rather small velocity kicks, therefore in reality we are
together with Figure 5 to obtain the production rates of only concerned about the star forming history in the last
any type of compact object binaries as a function of the 109 years. The star forming rate in the Galaxy has most
width of the kick velocity distribution. probablybeen constantoverthe last5×109 years(Miller
The actual compact object merger rate in the Galaxy and Scalo, 1979).
can be calculated given the observed supernova rate, the The calculation of the detection rate in gravitational
fractionofstarsthatexistinbinaries,andassumingsome wave detectors (Abramovici et al., 1992) requires a num-
form of the star formation history. Assuming that there berofadditionalassumptions,likeforexamplethegalaxy
is one supernovaexplosionevery fifty yearsin the Galaxy density in our local and far neighborhood etc., for a
and that binary fraction is 50%,we denote the supernova discussion see e.g. Phinney (1991); Chernoff and Finn