3. Comparison with observations
The hypothesis of natal kicks imparted to neutron stars during asymmetrical supernova explosion (see e.g. Dewey & Cordes 1987) is supported by recent observations of pulsar's space velocities. In particular, Lyne & Lorimer (1994) found a mean pulsar birth velocity of km/s. In order to find the potential progenitors of the observed binary pulsars (Table 2), the next step will then consist of formulating the post-SN state of the binary relative to the kick velocity and the pre-SN orbital parameters.
Table 2. Parameters for the binary pulsars. Errors in the last quoted digits are given in parentheses. Limits on the pre-SN orbital separation are derived according to (i.e. Eq. 20). References: (1) Taylor & Weisberg 1989; (2) Wolszczan 1991; (3) Lyne & Bailes 1990; (4) Nice et al. 1996.
3.1. Effects of the kick velocity
Several authors have considered the effect of a random kick velocity on the post-SN orbital characteristics (see e.g. Flannery & Van den Heuvel 1975, Hills 1983, Yamaoka et al. 1993, Brandt & Podsiadlowski 1995, Kalogera 1996). In this section, we will summarize the relationships of interest with regards to this work. Effects of the expanding supernova shell on the companion are ignored since these are thought to be small (Fryxell & Arnett 1981, Yamaoka et al. 1993).
Let us consider a pre-SN binary consisting of a massive star and a neutron star companion with masses and m, moving in a circular orbit with orbital separation . The reference frame is centered on just before the explosion. In this frame, the pre-SN orbital velocity of the massive star relative to its companion chosen to be at rest is given by . The x-axis corresponds to the line pointing from m to , the y-axis lies along the direction of and the z-axis is perpendicular to the orbital plane. After the explosion, energy conservation allows to express the relative velocity at a distance r as:
where the subscript f refers to the post-SN state of the binary. Assuming that the instantaneous position r is not changed by the explosion, we can write and just after the supernova, where is the kick-velocity vector. From energy and angular momentum conservation, this yields:
Since evolutionary models provide the pre-SN parameters , the potential progenitors of a peculiar binary pulsar specified by can be found according to Eq.(20), independently of the kick velocity. Among these potential progenitors, Eq.(19) with the condition allows one to find, for a restricted range of the kick, the pre-SN binaries that can produce the required parameters of a specific binary pulsar.
3.2. The formation of binary pulsars
Fig. 4 illustrates the regions of parameter space () which allow the formation of the four known binary pulsars. By increasing the kick range velocity, these regions tend to cover all the part of the parameter space corresponding to the potential progenitors as defined above. This is the case for PSR 2303+46, 1534+12 and J1518+49 with kick velocities ranging from 0 to 300 km/s. For PSR 1913+16, the region of potential progenitors is entirely covered for a kick range of 0-500 km/s. Indeed, kick velocities above these limits are allowed but they cannot extend the regions of parameter space any more. Therefore, the more severe constraints on evolutionary models to form a specific binary pulsar come from two necessary conditions independent of the kick: the first one concerning the pre-SN separation is expressed in Eq.(20), i.e. , and the second one requires that the initial orbital separation is not too large so that binary components can interact.
As an illustration, let us consider the formation of NS-NS systems with large orbital periods like PSR 2303+46 and PSR J1518+49. From Fig. 4, one can easily see that the extent of the progenitors region is extremely sensitive to the spiral-in prescriptions. Actually, the case of PSR J1518+49 is the most interesting because of its moderate eccentricity. As a consequence, the minimum pre-SN orbital separation is 18.5 , whereas the corresponding value is 10.7 for PSR 2303+46 (cf. Table 2). The huge orbital shrinkage in models A2 and B can never lead to such a large value of . Therefore, under the assumption that the same evolutionary picture holds for the four binary pulsars, the newly discovered system PSR J1518+49 favors a full CE-evolution with close to unity according to the formalism used for the mass transfer phase.
In all models, the pre-SN binaries of these two binary pulsars consist of He-NS systems which are formed through a case C mass transfer. On the contrary, PSR 1913+16 and PSR 1534+12 are likely to have experienced a case B mass transfer. In model A1, evolutionary expansion of the He star progenitor may lead to a second mass exchange phase. Since ejection of matter in the form of jets induces a small decrease in the separation, PSR 1913+16 and PSR 1534+12 may be the product of a type Ic supernovae in a CO-NS system. This feature seems very probable in the case of low kicks for PSR 1534+12 (i.e., km/s) or moderate kicks for PSR 1913+16 (i.e., 300 km/s). On the other hand, the high mass transfer rate involved in the second mass exchange phase may lead to the formation of a CE (model A3). In this case, the orbit of remnant CO-NS systems is too close to produce the required parameters. Following this scenario, PSR 1913+16 and PSR 1534+12 are the product of a type Ib supernovae in a He-NS system.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998