Indeed, the distance to a pulsar is usually derived from the dispersion measure evaluation and crucially depends on the assumed model of the electron density distribution in the Galaxy. In the middle of the 1990s, Lyne and Lorimer  derived a very high mean space velocity of pulsars with known proper motion of about 450 km s–1. This value was difficult to adopt without invoking an additional natal kick velocity of NSs. It was suggested  that the observed 2D pulsar velocity distribution found by Lyne and Lorimer  is reproduced if the absolute value of the (assumed to be isotropic) NS kick has a power-law shape, give similar results, and later we shall not distinguish these kicks. The high mean space velocity of pulsars, consistent with earlier results by Lyne and Lorimer, was confirmed by the analysis of a larger sample of pulsars . The recovered distribution of 3D velocities is well fit by a Maxwellian distribution with the mean value and a 1D rms .
Possible physical reasons for natal NS kicks due to hydrodynamic effects in core-collapse supernovae are summarized in [209, 208]. Neutrino effects in the strong magnetic field of a young NS may be also essential in explaining kicks up to 100 km s–1 [57, 84, 207]. Astrophysical arguments favouring a kick velocity are also summarized in . To get around the theoretical difficulty of insufficient rotation of pre-supernova cores in single stars to produce rapidly spinning young pulsars, Spruit and Phinney  proposed that random off-center kicks can lead to a net spin-up of proto-NSs. In this model, correlations between pulsar space velocity and rotation are possible and can be tested in further observations.
Here we should note that the existence of some kick follows not only from the measurements of radio pulsar space velocities, but also from the analysis of binary systems with NSs. The impact of a kick velocity 100 km s–1explains the precessing binary pulsar orbit in PSR J0045–7319 . The evidence of the kick velocity is seen in the inclined, with respect to the orbital plane, circumstellar disk around the Be star SS 2883 – an optical component of binary PSR B1259–63 .
Long-term pulse profile changes interpreted as geodetic precession are observed in the relativistic binary pulsars PSR 1913+16 , PSR B1534+12 , PSR J1141–6545 , and PSR J0737–3039B . These observations indicate that in order to produce the misalignment between the orbital angular momentum and the neutron star spin, a component of the kick velocity perpendicular to the orbital plane is required [450, 453, 454]. This idea seems to gain observational support from recent thorough polarization measurements  suggesting alignment of the rotational axes with pulsar’s space velocity. Such an alignment acquired at birth may indicate the kick velocity directed preferably along the rotation of the proto-NS. For the first SN explosion in a close binary system this would imply the kick to be perpendicular to the orbital plane.
It is worth noticing that the analysis of the formation of the double relativistic pulsar PSR J0737–3039  may suggest, from the observed low eccentricity of the system , that a small (if any) kick velocity may be acquired if the formation of the second NS in the system is associated with the collapse of an ONeMg WD due to electron-captures. The symmetric nature of electron-capture supernovae was discussed in  and seems to be an interesting issue requiring further studies (see, e.g., [311, 206] for the analysis of the formation of NSs in globular clusters in the frame of this hypothesis). Note that electron-capture SNe are expected to be weak events, irrespective of whether they are associated with the core-collapse of a star which retained some original envelope or with the AIC of a WD [350, 194, 78].
We also note the hypothesis of Pfahl et al. , based on observations of high-mass X-ray binaries with long orbital periods ( 30 d) and low eccentricities (), that rapidly rotating precollapse cores may produce neutron stars with relatively small kicks, and vice versa for slowly rotating cores. Then, large kicks would be a feature of stars that retained deep convective envelopes long enough to allow a strong magnetic torque, generated by differential rotation between the core and the envelope, to spin the core down to the very slow rotation rate of the envelope. A low kick velocity imparted to the second (younger) neutron star ( 50 km s–1) was inferred from the analysis of large-eccentricity binary pulsar PSR J1811–1736 . The large orbital period of this binary pulsar (18.8 d) then may suggest an evolutionary scenario with inefficient (if any) common envelope stage , i.e. the absence of deep convective shell in the supernova progenitor (a He-star). This conclusion can be regarded as supportive to ideas put forward in .
In principle, it is possible to assume some kick velocity during BH formation as well [229, 113, 329, 331, 286, 23, 469]. The similarity of NS and BH distribution in the Galaxy suggesting BH kicks was noted in . A recent analysis of the space velocity of some BH binary systems  put an upper limit on the BH kick velocity of less than 200 km s–1. However, no kick seems to be required to explain the formation of other BH candidates (Cyg X-1, X-Nova Sco, etc.) .
To summarize, the kick velocity remains one of the important unknown parameters of binary evolution with NSs and BHs. Further constraining this parameter from various observations and theoretical understanding of possible asymmetry of core-collapse supernovae seem to be of paramount importance for the formation and evolution of close compact binaries.
The collapse of a star to a BH, or its explosion leading to the formation of a NS, are normally considered as instantaneous. This assumption is well justified in binary systems, since typical orbital velocities before the explosion do not exceed a few hundred km/s, while most of the mass is expelled with velocities about several thousand km/s. The exploding star leaves the remnant , and the binary loses a portion of its mass: . The relative velocity of stars before the event is
Let us start from the limiting case when the mass loss is practically zero (, ), while a non-zero kick velocity can still be present. This situation can be relevant to BH formation. It follows from Equation (49) that, for relatively small kicks, , the system always (independently of the direction of ) remains bound, while for the system always unbinds. By averaging over equally probable orientations of with a fixed amplitude , one can show that in the particular case the system disrupts or survives with equal probabilities. If , the semimajor axis of the system becomes smaller than the original binary separation, (see Equation (40)). This means that the system becomes more hard than before, i.e. it has a greater negative total energy than the original binary. If , the system remains bound, but . For small and moderate kicks , the probabilities for the system to become more or less bound are approximately equal.
In general, the binary system loses some fraction of its mass . In the absence of the kick, the system remains bound if and gets disrupted if (see Section 3.3). Clearly, a properly oriented kick velocity (directed against the vector ) can keep the system bound, even if it would have been disrupted without the kick. And, on the other hand, an unfortunate direction of can disrupt the system, which otherwise would stay bound.
Consider, first, the case . The parameter varies in the interval from 1 to 2, and the escape velocity varies in the interval from to . It follows from Equation (44) that the binary always remains bound if , and always unbinds if . This is a generalization of the formulae derived above for the limiting case . Obviously, for a given , the probability for the system to disrupt or become softer increases when becomes larger. Now turn to the case . The escape velocity of the compact star becomes . The binary is always disrupted if the kick velocity is too large or too small: or . However, for all intermediate values of , the system can remain bound, and sometimes even more bound than before, if the direction of happened to be approximately opposite to . A detailed calculation of the probabilities for the binary survival or disruption requires integration over the kick velocity distribution function (see, e.g., ).
© Max Planck Society and the author(s)