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Astron. Astrophys. 322, 533-544 (1997)

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4. Discussion

4.1. Evolution of accretion stars

The massive star may have previously accreted matter from the progenitor of the neutron star. In this paper, the main-sequence star evolves as a single star until it first fills its Roche lobe, according to the works of Hellings (1983) and Vanbeveren (1990) who showed that the further evolution of accretion stars is almost identical to the evolution of a corresponding single star. However, while considering the effect of molecular weight gradients on convection, Braun & Langer (1995) found that this rejuvenation process of the mass gaining star does not always take place, but appears to depend mainly on the rate of mixing in the semiconvective layer which forms above the convective core upon accretion. The main result of a non-rejuvenation is an internal chemical structure which is quite unlike that occurring in any single star, leading to a different track in the HR diagram. According to Braun & Langer, a non-rejuvenated star spends its whole post main-sequence evolution as a blue supergiant whereas the corresponding massive single star evolves into a red supergiant.

This feature may have important consequences on the outcome of high-mass binary evolution. Taking the characteristic values of a blue supergiant (e.g. [FORMULA]), one finds that evolutionary expansion of a non-rejuvenated star will lead to a spiral-in of the neutron star companion if the orbital separation is less than [FORMULA] 150 [FORMULA]. Such a configuration implies that very few systems can survive the mass transfer phase. Furthermore, binary pulsars with large orbital periods like PSR J1518+49 cannot be formed in this way. Among other basic parameters, Braun & Langer showed that rejuvenation is more likely for accreting stars of initial moderate mass. For instance, according to their model #5, a 12 [FORMULA] which accretes 8 [FORMULA] adopts a chemical structure comparable to a 20 [FORMULA] single star, whereas a 20 [FORMULA] (model #12) does not perform rejuvenation for the same amount of accreted matter. Therefore, we expect little changes at least for the left part of the evolutionary diagrams computed in this paper (i.e., M [FORMULA] [FORMULA]). However, Braun & Langer pointed out that none of their models lead to rejuvenation if a low value is adopted for the efficiency parameter of semiconvection. In such a case, the merger rate of NS-NS systems would be considerably smaller than expected.

4.2. The formation of PSR 2303+46 and PSR J1518+49

In Sect. 3.2, it was shown that the formation of binary pulsars with large orbital periods requires a moderate orbital shrinkage during the preceding mass transfer phase. This can be accomodated if one adopts a full CE-evolution with [FORMULA]. However, using the same formalism for CE-evolution, Van den Heuvel et al. (1994) argue that a progenitor cannot be found for PSR 2303+46. This result was established in the case of a symmetric supernova explosion, for which the smallest possible pre-SN separation is obtained, namely [FORMULA]. If a kick is imparted to the neutron star, the pre-SN separation is larger, which seems a less favourable configuration. Actually, we do not find any progenitors for PSR 2303+46 nor for PSR J1518+49 in the spherically symmetric case. For instance, from the parameters of PSR 2303+46, we derive a helium star of 3.13 [FORMULA] at the time of explosion, which corresponds to a 11.5 [FORMULA] initial companion mass. Equation (3) then yields a pre-spiral-in separation 165 times larger than that of the post-spiral-in system, whereas Van den Heuvel et al. found a value of 235. This departure comes from the different assumptions we have used for the neutron star mass, the normalized Roche lobe radius [FORMULA] (Eq. 1) and the relationship between the helium core and the hydrogen star masses (Eq. 10). Nevertheless, both values lead to a pre-spiral-in separation much too large to allow a mass transfer event.

On the other hand, a binary can survive an asymmetric supernova explosion even if the amount of mass ejected is high enough to disrupt the system in the symmetric case. So, let us consider a binary containing a 20 [FORMULA] initial hydrogen star with an orbital separation of 1500 [FORMULA] (cf. system [FORMULA] 5 in Table 1). Due to the large separation, the massive star overflows its Roche lobe in the late burning phases. As a result of the time spent on the RSG branch, the stellar mass has dropped to 16.6 [FORMULA] while the separation has slowly increased up to 1790 [FORMULA] at the onset of mass transfer. The CE-evolution then gives a 7.68 [FORMULA] helium star with a separation of 18.3 [FORMULA], i.e. an orbital shrinkage factor of 98. Therefore, a stage of heavy mass loss by stellar wind prior to spiral-in leads to a moderate orbital shrinkage and hence, favors the formation of binary pulsars with large orbital periods.

From their results, Van den Heuvel et al. (1994) propose an alternative evolutionary picture for the formation of PSR 2303+46. In this model, a very massive star ([FORMULA] 40-45 [FORMULA]) spontaneously sheds its hydrogen-rich envelope during a LBV phase, without transferring mass to the neutron star companion (see also Kaper et al. 1995). Then, a decrease of the orbital separation may result from the frictional torque acting on the neutron star immersed in the dense wind. Nice et al. (1996) point out that the same scenario may also explain the wide orbit of the newly discovered binary pulsar PSR J1518+49.

The restricted region of potential progenitors (Fig. 4b) may suggest that PSR J1518+49 is unlikely to be formed through standard evolutionary models. However, the high sensitivity of the outcome of the mass transfer phase to the spiral-in prescriptions implies that the inverse problem (i.e. the finding of progenitors) is not robust. In addition, the efficiency parameter [FORMULA] may be formally larger than unity if additional energy sources other than orbital energy contribute to mass ejection, such as accretion energy or recombination energy in ionization zones (see e.g. the review of Iben & Livio 1993 for other possible processes).

Furthermore, we have considered so far the formalism for CE-evolution according to Eq.(3), which is the most commonly used in the Monte-Carlo simulations. In principle, these simulations can limit the possible range of [FORMULA] by trying to reproduce the parameters of well-studied binaries (e.g. De Kool 1990). Another approach in literature consists of performing hydrodynamical calculations that can follow the dynamical phase of CE-evolution (Livio & Soker 1988; Taam & Bodenheimer 1989, 1991; Terman et al. 1995 and references therein). These numerical computations start from an initial configuration consisting of the two cores surrounded by an envelope of diameter [FORMULA] and lead to small values of [FORMULA] in the range [FORMULA] 0.3-0.6. These works attempt to quantify the CE-phase according to the following expression:


It is important to realize that differences between these two formalisms (i.e. Eqs. 2, 21), and hence between the values of [FORMULA], reflect the difficulty in estimating the binding energy of the envelope at early and late times at the CE-phase (Rasio & Livio 1996).

As a last model, we perform evolutionary calculations according to this formalism. When the massive star begins overflowing its Roche lobe, we assume that some drag force brings the two stars into contact during the early dynamical phase, so that the new separation [FORMULA] corresponds to the stellar radius of the primary (cf. Terman et al. 1994). Then, we apply Eq.(21) for the phase of strong dynamical interaction with an efficiency parameter [FORMULA] of 0.5 (cf. Fig. 5). Such an initial configuration for CE-evolution may also appear after a substage of fast mass transfer in the form of jets (see Sect. 2.1.3) or as a consequence of the Darwin instability. In this model, the configuration just prior the spiralling-in process for the 20 [FORMULA] +1.35 [FORMULA] system#5 consists of a 16.6 [FORMULA] +1.35 [FORMULA] binary with an orbital separation of 1060 [FORMULA]. Then, the system achieves a post-spiral-in separation of 33.2 [FORMULA]. We emphasize that the assumption of a pre-spiral-in separation [FORMULA] equal to the separation at the onset of mass transfer (e.g. 1790 [FORMULA] in system #5) would lead to an even more favourable situation for the formation of binary pulsars (cf. Tutukov & Yungelson 1993, 1994).

[FIGURE] Fig. 5a and b. Regions of the progenitors systems of the four known binary pulsars in the ([FORMULA]) plane. In this model, the first mass transfer phase involves the formation of a CE as soon as the neutron star comes into contact with the evolved star. The efficiency parameter is [FORMULA], see Sect. 4.2 for the formalism used for CE-evolution. During the second stage of mass transfer, the excess overflowing matter is thought to be ejected in the form of jets from the neutron star. The gray-scale code is the same as in Fig. 4.

Therefore, from Figs. 4-5, we conclude that it is well conceivable that binary pulsars with a wide orbit may have been formed through standard evolutionary models involving a CE-event. In particular, this picture holds for PSR J1518+49 provided that CE-evolution leads to a moderate orbital shrinkage.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998