THE EFFECT OF SUPERNOVA ASYMMETRY ON COALESCENCE RATES
OF BINARY NEUTRON STARS
K.A. POSTNOV, A.G. KURANOV
Sternberg Astronomical Institute, Universitetskij pr. 13,
119992 Moscow, Russia
We study the effect of the kick velocity – neutron star spin alignment observed in young
radio pulsars on the coalescence rate of binary neutron stars. The effect is shown to be
especially strong for large kick amplitudes and tight alignments, reducing the galactic rate
of binary neutron star coalescences up to an order of magnitude with respect to the rates
calculated for random kicks. The spin-kick correlation also leads to much narrower NS spin-
orbit misalignments compared to random kicks.
1 Introduction
Kick velocity imparted to a newborn neutron star is known to be an important phenomenological
parameter of core collapse supernovae. The origin of the kicks remains unclear (see, for example,
1 and references therein). Recently, new important observational results appeared suggesting
possible NS spin-kick alignment. Tight spin-kick alignment follows from measurements of radio
pulsar polarization 2, as well as from X-ray observations of pulsar wind nebulae around young
pulsars3,4. Implications of these observations to the formation of double pulsars were discussed
by Wang et al. 5. Here we explore the effect of NS spin – kick correlation on the formation and
galactic coalescence rate of double neutron stars (DNS) which are primary targets for modern
gravitational wave detectors. We show that the tighter alignement, the smaller is the DNS
merging rate with respect to models with random kick orientation. The effect is especially
important for large kick amplitudes (∼ 400 km/s).
2 Effect on the binary neutron star coalescence rates
The effect of NS kick velocity on merging rates of compact binaries was studied earlier (e.g. 6).
The observed tight NS spin – kick alignment may have important implications to the formation
and evolution of binary compact stars (see especially earlier paper by Kalogera 7). Let us
consider the standard evolutionary scenario leading to formation of binary NS from a massive
binary system (see8 for discussion and references) focusing on the effect of the NS kick velocity.
We shall assume that the kick velocity vector is confined within a cone which is coaxial with the
progenitor’s rotation axis and characterized by angle θ < π/2. We shall consider only central
kicks thus ignoring theoretically feasible off-center kicks affecting the NS spin 9,10. The value
of the kick velocity is assumed to obey the Maxellian distribution f(v) ∼ v2 exp(−(v/v0)
2), as
suggested by pulsar proper motion measurements11. The velocity v0 varied from 0 to 400 km/s.
http://arxiv.org/abs/0704.1204v2
0 50 100 150 200 250 300 350 400 450
Kick velocity (km/s)
Double NS coalescence rate
0,0 0,2 0,4 0,6 0,8 1,0
 (in units /2)
 50 km/s
 75 km/s
100 km/s
125 km/s
150 km/s
200 km/s
400 km/s
Figure 1: Galactic coalescence rate of DNS vs. kick
parameter v0 (random kicks). An almost exponential
decay with v0 is seen for v0 > 100 km/s
Figure 2: Relative change of DNS merging rate for
NS spin-kick correlation
The rotational axes of both components are assumed to be aligned with the orbital angular
momentum before the primary collapses to form first NS. The SN explosion is treated in a
standard way as instantaneous loss of mass of the exploding star. The effect of the kick on the
post-explosion binary orbital parameters is treated from the point of view of energy-momentum
conservation in two point-mass body problem (see e.g. in 7,12). The first SN explosion most
likely occurs when the binary orbit is circular (unless the initial binary is very wide so that tidal
circularization is ineffective), while the second explosion can happen before the orbit has been
tidally circularized; in the latter case we choose the position of the star in the orbit distributed
according to Kepler’s 2d law.
We use the population synthesis method to calculate the expected coalescence rate of DNS
(see 6,8 and references therein). The standard assumptions about binary evolution have been
made: Salpeter’s mass function for the primary’s mass, f(M1) ∼ M
−2.35, flat initial mass ratio
(q = M2/M1 < 1) distribution f(q) = const, initial semi-major axes distribution in the form
d log a = const. The common envelope phase is treated in the standard way8 with the efficiency
αCE = 0.5. The calculations were normalized to the galactic star formation rate 3M⊙ per year,
with binary fraction 50%. We also have carefully taken into account rotational evolution of
magnetized compact stars, as described in13,14, assuming no magnetic field decay. The galactic
DNS merging rate is shown in Fig. 1 as a function of the kick parameter v0 and assuming random
central kicks. Note an almost exponential decay of the rate with v0 for v0 > 100 km/s. Fig. 2
shows the relative change in the DNS merging rate when we allow for NS spin-kick alignment
with different values of the confinement angle θ. It is seen that tight alignment generally reduces
the DNS merging rate, with the effect being especially strong for large kick velocity amplitudes.
Such a reducing relative to calculations with random kicks is in fact expected, because the NS
spin – kick correlation excludes kicks in the orbital plane which, if directed opposite to the
orbital velocity, can additionally bind the post-explosion binary system.
3 Neutron star spin – orbit misalignment
There is another observational consequence of the kick in DNS systems: NS spin – orbit mis-
alignment, which can be tested by geodetic precession measurements in binary pulsars 15. Such
a misalignment is potentially very interesting for GW studies 16. After SN explosion in a bi-
nary system, additional kick imparted to newborn NS results, in general, in a misalignment
between the new orbital angular momentum and the NS (and the secondary component’s) spin
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=100 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=100 km/s
Cos (
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=100 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=100 km/s
Cos (
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=400 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=400 km/s
Cos (
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=400 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=400 km/s
Cos (
Figure 3: NS spin-orbit misalignment cosΨ in coalescing DNS for v0 = 100 km/s (upper row) and 400 km/s
(bottom row) and different NS spin-kick alignment angles θ0. NS1: spin aligned with orbit before SN explosion;
NS2: spin aligned with original binary’s orbital angular momentum
vector characterized by angle Ψ1. After the second SN explosion in the system, there are several
possibilities for the NS spin-orbit misalignment.
1) In sufficiently wide binaries, when tidal interactions between the components are inef-
ficient, the orientation of the NS1 spin and the secondary’s spin vector may remain unchaged
until the second SN explosion, after which the orbital angular momentum vector changes again
due to NS2 kick. So in this case we would expect two coaxial NS with spins misaligned by angle
Ψ2 with orbital angular momentum. However, such binaries, unless very eccentric, may be too
wide to coalesce over the Hubble time.
2) In close binaries, tidal interactions tend to rapidly align angular momentum vector of the
normal star with the orbital angular momentum. To spin-up the NS rotation up to observed ms
periods (in binary ms pulsars), a modest amount of matter (∼ 0.1M⊙) should be accreted by
NS. This amount is sufficient to align the NS rotation with the orbital angular momentum. So if
NS1 accretes matter before the second SN explosion, both NS1 and the secondary component’s
spins are most likely aligned with orbital angular momentum (see discussion in5). Note that the
NS1 spin tends to align with the orbital angular momentum even if NS1 does not accrete matter
but spins-down by the propeller mechanism before the second SN explosion, since in that case
very strong currents must flow through its polar cap and the alignment torques is as strong as
during accretion. So the NS1 remains misaligned prior to the second SN explosion only in rare
cases where the secondary collapses shortly after the first SN in the binary. If both NS1 and
secondary were aligned with orbital angular momentum prior to the second SN explosion, both
neutron stars will be equally misaligned with orbital angular momentum, Ψ1 = Ψ2.
In our population synthesis simulations we take into account the discussed spin alignment
effects. In Fig. 3 we show the calculated distribution between the NS1 and NS2 spins and orbital
angular momentum in coalescing DNS systems (angles Ψ1 and Ψ2, respectively) assuming spin-
orbit alignment (angle Ψ1) and conservation of the secondary’s angular momentum (angle Ψ2).
Clearly, the real distribution must be intermediate between the two, depending on the degree
of misalignment of the secondary’s angular momentum prior to the collapse. It is seen that the
Table 1: Mean NS spin-orbit misalignment Ψ (in units π/2)
Kick v0 NS1 (Ψ1) NS2 (Ψ2)
(km/s) Kick confinement angle θ (in units π/2)
0.01 0.1 0.2 0.5 1.0 0.01 0.1 0.2 0.5 1.0
50 0.061 0.060 0.059 0.073 0.212 0.307 0.310 0.316 0.308 0.345
100 0.110 0.110 0.109 0.186 0.444 0.378 0.381 0.381 0.423 0.633
200 0.192 0.195 0.207 0.337 0.535 0.417 0.419 0.442 0.575 0.813
400 0.257 0.262 0.291 0.451 0.670 0.442 0.447 0.481 0.670 0.909
misalignment angles can be very different (and even with negative cosines) for random or loosely
constrained (θ ∼ π/2) kicks, while tight spin-kick alignment (θ ≪ π/2) results in much narrow
distributions (see also 7). The mean misalignement angles Ψ are presented in Table 1.
4 Conclusions
We have shown that the spin-velocity correlation observed in radio pulsars, suggesting NS spin-
kick velocity alignment, may have very important implications to GW studies. First, the tight
alignment reduces the galactic rate of double neutron star coalescences (especially for large kicks
300-400 km/s – up to ten times) relative to models with random kicks. Second, the spin-kick
correlation results in specific distribution of NS spin – orbit misalignments, which can be tested
by analysing GW signals from DNS mergings.
Acknowledgments
KAP acknowledges the financial support from the Meeting Organizers and RFBR grant 07-02-
08065z.
References
1. D. Lai in Cosmic Explosions in Three Dimensions: Asymmetries in Supernovae and
Gamma-ray Bursts, ed. P. Hoflich, P. Kumar, and J. C. Wheeler (Cambridge Univ. Press,
Campridge, 2004), p. 276.
2. S. Johnston et al., MNRAS 364, 1397 (2005).
3. D.J. Helfand et al, Astrophys. J. 556, 380 (2001).
4. O. Kargaltsev et al, arXiv:astro-ph/0611599.
5. C. Wang et al, Astrophys. J. 639, 1007 (2006).
6. V.M. Lipunov et al, MNRAS 288, 245 (1997).
7. V. Kalogera, Astrophys. J. 541, 319 (2000).
8. K.A. Postnov and L.R. Yungelson, LRR 9, 6 (2006).
9. H.C. Spruit and E.S. Phinney, Nature 393, 139 (1998).
10. C. Wang et al, Astrophys. J. 656, 399 (2007).
11. G. Hobbs et al, MNRAS 360, 974 (2005).
12. L.P. Grishchuk et al, Physics-Uspekhi 44, 1 (2001); arXiv:astro-ph/0008481.
13. V.M. Lipunov, Astrophysics of Neutron Stars (Springer Verlag, Berlin, 1992)
14. V.M. Lipunov, K.A. Postnov and M.E. Prokhorov, in The scenario machine: Binary star
population synthesis, ed. R.A. Sunyaev (Harwood Academic Publ., Amsterdam, 1996).
15. M. Bailes, Astron. Astrophys. 202, 109 (1988).
16. T. Apostolatos et al, Phys. Rev. D 49, 6274 (1994).
http://arxiv.org/abs/astro-ph/0611599
http://arxiv.org/abs/astro-ph/0008481
	Introduction
	Effect on the binary neutron star coalescence rates
	Neutron star spin – orbit misalignment
	Conclusions

  We study the effect of the kick velocity -- neutron star spin alignment
observed in young radio pulsars on the coalescence rate of binary neutron
stars. The effect is shown to be especially strong for large kick amplitudes
and tight alignments, reducing the galactic rate of binary neutron star
coalescences up to an order of magnitude with respect to the rates calculated
for random kicks. The spin-kick correlation also leads to much narrower NS
spin-orbit misalignments compared to random kicks.

Introduction
Kick velocity imparted to a newborn neutron star is known to be an important phenomenological
parameter of core collapse supernovae. The origin of the kicks remains unclear (see, for example,
1 and references therein). Recently, new important observational results appeared suggesting
possible NS spin-kick alignment. Tight spin-kick alignment follows from measurements of radio
pulsar polarization 2, as well as from X-ray observations of pulsar wind nebulae around young
pulsars3,4. Implications of these observations to the formation of double pulsars were discussed
by Wang et al. 5. Here we explore the effect of NS spin – kick correlation on the formation and
galactic coalescence rate of double neutron stars (DNS) which are primary targets for modern
gravitational wave detectors. We show that the tighter alignement, the smaller is the DNS
merging rate with respect to models with random kick orientation. The effect is especially
important for large kick amplitudes (∼ 400 km/s).
2 Effect on the binary neutron star coalescence rates
The effect of NS kick velocity on merging rates of compact binaries was studied earlier (e.g. 6).
The observed tight NS spin – kick alignment may have important implications to the formation
and evolution of binary compact stars (see especially earlier paper by Kalogera 7). Let us
consider the standard evolutionary scenario leading to formation of binary NS from a massive
binary system (see8 for discussion and references) focusing on the effect of the NS kick velocity.
We shall assume that the kick velocity vector is confined within a cone which is coaxial with the
progenitor’s rotation axis and characterized by angle θ < π/2. We shall consider only central
kicks thus ignoring theoretically feasible off-center kicks affecting the NS spin 9,10. The value
of the kick velocity is assumed to obey the Maxellian distribution f(v) ∼ v2 exp(−(v/v0)
2), as
suggested by pulsar proper motion measurements11. The velocity v0 varied from 0 to 400 km/s.
http://arxiv.org/abs/0704.1204v2
0 50 100 150 200 250 300 350 400 450
Kick velocity (km/s)
Double NS coalescence rate
0,0 0,2 0,4 0,6 0,8 1,0
 (in units /2)
 50 km/s
 75 km/s
100 km/s
125 km/s
150 km/s
200 km/s
400 km/s
Figure 1: Galactic coalescence rate of DNS vs. kick
parameter v0 (random kicks). An almost exponential
decay with v0 is seen for v0 > 100 km/s
Figure 2: Relative change of DNS merging rate for
NS spin-kick correlation
The rotational axes of both components are assumed to be aligned with the orbital angular
momentum before the primary collapses to form first NS. The SN explosion is treated in a
standard way as instantaneous loss of mass of the exploding star. The effect of the kick on the
post-explosion binary orbital parameters is treated from the point of view of energy-momentum
conservation in two point-mass body problem (see e.g. in 7,12). The first SN explosion most
likely occurs when the binary orbit is circular (unless the initial binary is very wide so that tidal
circularization is ineffective), while the second explosion can happen before the orbit has been
tidally circularized; in the latter case we choose the position of the star in the orbit distributed
according to Kepler’s 2d law.
We use the population synthesis method to calculate the expected coalescence rate of DNS
(see 6,8 and references therein). The standard assumptions about binary evolution have been
made: Salpeter’s mass function for the primary’s mass, f(M1) ∼ M
−2.35, flat initial mass ratio
(q = M2/M1 < 1) distribution f(q) = const, initial semi-major axes distribution in the form
d log a = const. The common envelope phase is treated in the standard way8 with the efficiency
αCE = 0.5. The calculations were normalized to the galactic star formation rate 3M⊙ per year,
with binary fraction 50%. We also have carefully taken into account rotational evolution of
magnetized compact stars, as described in13,14, assuming no magnetic field decay. The galactic
DNS merging rate is shown in Fig. 1 as a function of the kick parameter v0 and assuming random
central kicks. Note an almost exponential decay of the rate with v0 for v0 > 100 km/s. Fig. 2
shows the relative change in the DNS merging rate when we allow for NS spin-kick alignment
with different values of the confinement angle θ. It is seen that tight alignment generally reduces
the DNS merging rate, with the effect being especially strong for large kick velocity amplitudes.
Such a reducing relative to calculations with random kicks is in fact expected, because the NS
spin – kick correlation excludes kicks in the orbital plane which, if directed opposite to the
orbital velocity, can additionally bind the post-explosion binary system.
3 Neutron star spin – orbit misalignment
There is another observational consequence of the kick in DNS systems: NS spin – orbit mis-
alignment, which can be tested by geodetic precession measurements in binary pulsars 15. Such
a misalignment is potentially very interesting for GW studies 16. After SN explosion in a bi-
nary system, additional kick imparted to newborn NS results, in general, in a misalignment
between the new orbital angular momentum and the NS (and the secondary component’s) spin
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=100 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=100 km/s
Cos (
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=100 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=100 km/s
Cos (
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=400 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=400 km/s
Cos (
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Cos (
v=400 km/s
0,0 0,2 0,4 0,6 0,8 1,0
v=400 km/s
Cos (
Figure 3: NS spin-orbit misalignment cosΨ in coalescing DNS for v0 = 100 km/s (upper row) and 400 km/s
(bottom row) and different NS spin-kick alignment angles θ0. NS1: spin aligned with orbit before SN explosion;
NS2: spin aligned with original binary’s orbital angular momentum
vector characterized by angle Ψ1. After the second SN explosion in the system, there are several
possibilities for the NS spin-orbit misalignment.
1) In sufficiently wide binaries, when tidal interactions between the components are inef-
ficient, the orientation of the NS1 spin and the secondary’s spin vector may remain unchaged
until the second SN explosion, after which the orbital angular momentum vector changes again
due to NS2 kick. So in this case we would expect two coaxial NS with spins misaligned by angle
Ψ2 with orbital angular momentum. However, such binaries, unless very eccentric, may be too
wide to coalesce over the Hubble time.
2) In close binaries, tidal interactions tend to rapidly align angular momentum vector of the
normal star with the orbital angular momentum. To spin-up the NS rotation up to observed ms
periods (in binary ms pulsars), a modest amount of matter (∼ 0.1M⊙) should be accreted by
NS. This amount is sufficient to align the NS rotation with the orbital angular momentum. So if
NS1 accretes matter before the second SN explosion, both NS1 and the secondary component’s
spins are most likely aligned with orbital angular momentum (see discussion in5). Note that the
NS1 spin tends to align with the orbital angular momentum even if NS1 does not accrete matter
but spins-down by the propeller mechanism before the second SN explosion, since in that case
very strong currents must flow through its polar cap and the alignment torques is as strong as
during accretion. So the NS1 remains misaligned prior to the second SN explosion only in rare
cases where the secondary collapses shortly after the first SN in the binary. If both NS1 and
secondary were aligned with orbital angular momentum prior to the second SN explosion, both
neutron stars will be equally misaligned with orbital angular momentum, Ψ1 = Ψ2.
In our population synthesis simulations we take into account the discussed spin alignment
effects. In Fig. 3 we show the calculated distribution between the NS1 and NS2 spins and orbital
angular momentum in coalescing DNS systems (angles Ψ1 and Ψ2, respectively) assuming spin-
orbit alignment (angle Ψ1) and conservation of the secondary’s angular momentum (angle Ψ2).
Clearly, the real distribution must be intermediate between the two, depending on the degree
of misalignment of the secondary’s angular momentum prior to the collapse. It is seen that the
Table 1: Mean NS spin-orbit misalignment Ψ (in units π/2)
Kick v0 NS1 (Ψ1) NS2 (Ψ2)
(km/s) Kick confinement angle θ (in units π/2)
0.01 0.1 0.2 0.5 1.0 0.01 0.1 0.2 0.5 1.0
50 0.061 0.060 0.059 0.073 0.212 0.307 0.310 0.316 0.308 0.345
100 0.110 0.110 0.109 0.186 0.444 0.378 0.381 0.381 0.423 0.633
200 0.192 0.195 0.207 0.337 0.535 0.417 0.419 0.442 0.575 0.813
400 0.257 0.262 0.291 0.451 0.670 0.442 0.447 0.481 0.670 0.909
misalignment angles can be very different (and even with negative cosines) for random or loosely
constrained (θ ∼ π/2) kicks, while tight spin-kick alignment (θ ≪ π/2) results in much narrow
distributions (see also 7). The mean misalignement angles Ψ are presented in Table 1.
4 Conclusions
We have shown that the spin-velocity correlation observed in radio pulsars, suggesting NS spin-
kick velocity alignment, may have very important implications to GW studies. First, the tight
alignment reduces the galactic rate of double neutron star coalescences (especially for large kicks
300-400 km/s – up to ten times) relative to models with random kicks. Second, the spin-kick
correlation results in specific distribution of NS spin – orbit misalignments, which can be tested
by analysing GW signals from DNS mergings.
Acknowledgments
KAP acknowledges the financial support from the Meeting Organizers and RFBR grant 07-02-
08065z.
References
1. D. Lai in Cosmic Explosions in Three Dimensions: Asymmetries in Supernovae and
Gamma-ray Bursts, ed. P. Hoflich, P. Kumar, and J. C. Wheeler (Cambridge Univ. Press,
Campridge, 2004), p. 276.
2. S. Johnston et al., MNRAS 364, 1397 (2005).
3. D.J. Helfand et al, Astrophys. J. 556, 380 (2001).
4. O. Kargaltsev et al, arXiv:astro-ph/0611599.
5. C. Wang et al, Astrophys. J. 639, 1007 (2006).
6. V.M. Lipunov et al, MNRAS 288, 245 (1997).
7. V. Kalogera, Astrophys. J. 541, 319 (2000).
8. K.A. Postnov and L.R. Yungelson, LRR 9, 6 (2006).
9. H.C. Spruit and E.S. Phinney, Nature 393, 139 (1998).
10. C. Wang et al, Astrophys. J. 656, 399 (2007).
11. G. Hobbs et al, MNRAS 360, 974 (2005).
12. L.P. Grishchuk et al, Physics-Uspekhi 44, 1 (2001); arXiv:astro-ph/0008481.
13. V.M. Lipunov, Astrophysics of Neutron Stars (Springer Verlag, Berlin, 1992)
14. V.M. Lipunov, K.A. Postnov and M.E. Prokhorov, in The scenario machine: Binary star
population synthesis, ed. R.A. Sunyaev (Harwood Academic Publ., Amsterdam, 1996).
15. M. Bailes, Astron. Astrophys. 202, 109 (1988).
16. T. Apostolatos et al, Phys. Rev. D 49, 6274 (1994).
http://arxiv.org/abs/astro-ph/0611599
http://arxiv.org/abs/astro-ph/0008481
	Introduction
	Effect on the binary neutron star coalescence rates
	Neutron star spin – orbit misalignment
	Conclusions
