4 Discussion

We now discuss some consequences of our proposal that the braking of PSR B1509-58 is mostly due to the interaction of the pulsar's magnetosphere with the dense matter of a circumstellar clump.

First, we consider the contribution of the circumstellar matter to the neutral hydrogen absorption toward the pulsar. The low covering factor of clumps (see Sect. 3) implies that this contribution is

$$N_{\rm cl} = \int_{r_{\rm L}}^{r_{\rm acc}} \limits n_{\rm ff} (r){\rm d}r + n_{\rm cl} (r_{\rm cl} -r_{\rm acc}) ,$$ (3)

where $n_{\rm ff} (r)=\dot{M}/4\pi m_{\rm p} r^2 v_{\rm ff}$ is the number density of the infalling gas captured inside the accretion radius, $v_{\rm ff} =(2GM_{\ast}/r)^{1/2}$ is the free-fall velocity, $r_{\rm cl}$ is the line of sight thickness of the dense neutral gas located between the pulsar and the observer ( $r_{\rm cl} \leq l_{\rm cl}$).

It was pointed out by Strom (1994) that a ROSAT observation of the SNR MSH15-52 indicates a larger absorption towards PSR B1509-58 than seen from the bright northwest part of the SNR's shell (known as RCW89). The comparison of neutral hydrogen absorption data (see Greiveldinger et al. 1995; Trussoni et al. 1996; Tamura et al. 1996; Marsden et al. 1997; Rots et al. 1998) shows that the excess of absorption towards the pulsar could be as large as $\simeq$(1-5) $\times\ 10^{22} \, {\rm cm}^{-2}$. This discrepancy could be interpreted as a sign that the pulsar is more distant than the SNR, and therefore that these two objects are not physically associated with each other (Strom 1994). It is also possible that "the spectral analysis is not detailed enough to provide the correct parameters" (Trussoni et al. 1996). Another possibility is that the HI column density distribution is really inhomogeneous across the SNR (cf. Trussoni et al. 1996). We favour the last possibility and suggest that the excess of absorption towards PSR B1509-58 is due to the dense neutral gas around the pulsar. One can use Eq. (3) to set an upper limit on $r_{\rm cl}$. For the parameters adopted above, and assuming that $N_{\rm cl} \leq$ (1-5)  $\times\ 10^{22} \, {\rm cm}^{-2}$, one has $r_{\rm cl} \leq$ (0.2-1.2) $\times\ 10^{16} v_{\ast, 50} ^{-3}$ cm. This estimate shows that if our explanation of the age discrepancy is correct, then one might expect that in the near future (i.e. after a lapse of $\simeq$ $r_{\rm cl}/v_{\ast, \parallel}$, where $v_{\ast, \parallel}$ is the line of sight component of $v_{\ast}$) the first derivative of the pulsar's spin period will suffer a significant decrease.

Second, let us discuss the candidate optical counterpart for PSR B1509-58 proposed by Caraveo et al. (1994). Caraveo et al. pointed out that the luminosity of the optical counterpart ( $L_{\rm V} \sim 6.5\times 10^{32} d_5 ^2 \, {\rm ergs} \, {\rm s}^{-1}$) exceeds by a few orders of magnitude the value derived for magnetospheric optical emission of young pulsars (Pacini 1971). This fact together with the negative result of searching of optical pulsations at the radio period led to the conclusion that the proposed identification could be erroneous (Mignami et al. 1998; see also Shearer et al. 1998; Chakrabarti & Kaspi 1998). We suggest, however, that the observed optical emission should rather be attributed to the bow shock around the pulsar than to the pulsar itself. This suggestion is supported by the estimate of the total luminosity of the bow shock, $L_{\rm T} \simeq Sn_{\rm cl} v_\ast (m_{\rm p} v_\ast ^2 /2)$, where $S\simeq \pi r_{\rm b} ^2$ and $r_{\rm b} \simeq 3^{1/2} r_{\rm s}$ are the area and the characteristic radius of the bow shock, respectively. For the adopted parameters this gives $L_{\rm T} \simeq \alpha \times 10^{33} \, {\rm ergs} \, {\rm s}^{-1}$, i.e. $L_{\rm V} \simeq L_{\rm T}$ for $\alpha \simeq 1$. It is obvious that if our suggestion is correct, there can be no correspondence between the observed luminosity and the luminosity expected from the results of Pacini (1971). The optical pulsations should be absent as well.

To conclude, we point out a curious coincidence of the accretion rate derived in Sect. 2 with accretion rates required in accretion-based models to explain high spin-down rates of anomalous X-ray pulsars and soft gamma-ray repeaters (e.g. Mereghetti & Stella 1995; Ghosh et al. 1997; Chatterjee et al. 2000; Alpar 2000). This coincidence allows us to believe (Gvaramadze 1999a, 2000b) that these objects could lose a significant part of their rotational energy due to the process discussed in this paper, and that their "true" ages could be much larger than the respective characteristic spin-down ones.

Acknowledgements

I am grateful to N. D'Amico and A. D'Ercole for discussions, to D. Page (the referee) for comments, and to J. K. Katgert-Merkelijn (the Deputy Editor) for carefully reading the manuscript. This work was partially supported by NPS.