2 The spin-down of PSR B1509-58

It is known that the electromagnetic torque acting on a rotating, magnetized body (e.g. a neutron star) immersed in a plasma is enhanced as compared with the torque in vacuum (Ginzburg 1971, see also Istomin 1994). It was mentioned by Istomin (1994) that for the increase of the slow-down torque of a pulsar it is suffient to have a dense plasma in the vicinity of the light cylinder, since just in this region the pulsar loses its rotational energy due to the acceleration of particles. The particles of the ambient medium penetrating into the region of the light cylinder are accelerated there to velocities comparable with the speed of light and then leave this region (Istomin 1994). This presumably equatorial outflow (cf. Brinkmann et al. 1985; King & Cominsky 1994) carries away the pulsar's angular momentum and is responsible for the enhanced braking of the pulsar. We suggest (see also Gvaramadze 1999a; cf. Yusifov et al. 1995; Istomin & Komberg 2000) that just this effect is responsible for the present high spin-down rate of PSR B1509-58, i.e. that the pulsar loses its rotational energy mainly due to the acceleration of protons of the ambient medium arriving at the light surface at the rate $\dot{M}$:

$$\vert\dot{E}\vert = \vert I\Omega \dot{\Omega}\vert \, = \, \gamma _{\rm p} \dot{M} c^2 ,$$ (1)

where $I \simeq 10^{45} \, {\rm g}\,{\rm cm}^2$ is the moment of inertia of the pulsar, c is the speed of light, and $\gamma _{\rm p}$ is the Lorentz factor of accelerated protons. For $\Omega = 41.89 \, {\rm s}^{-1}$ and $\dot{\Omega} = -4.25\times 10^{-10} \, {\rm s}^{-2}$(e.g. Kaspi et al. 1994), and $\gamma _{\rm p} \simeq 1$(Istomin 1994), one has $\dot{M} \simeq 2.2\times 10^{16} \, {\rm g}\,{\rm s}^{-1}$. We also suggest that the pulsar moves through the inhomogeneous ambient medium and episodically plunges into clumps of dense matter. If the density of clumps is sufficiently high (see Sect. 3), one can expect a temporary increase of $\dot{\Omega}$ every time the pulsar travels through a clump (we believe that just this situation takes place now). The corresponding spin-down age will be less than estimated when the pulsar moves through the low-density interclump medium.

It is clear that the presence of radio emission of the pulsar means that the ambient medium does not penetrate far beyond the light surface. Assuming that the ram pressure of the accreting medium is equal to the magnetic pressure at the light surface (cf. King & Cominsky 1994), one has an estimate of the surface magnetic field of the pulsar:

$$B_{\ast} = \left[2(2GM_{\ast})^{1/2} \dot{M} r_{\rm L}^{7/2} ... ...st} ^6 \right]^{1/2} \, \simeq 2.9 \times 10^{12} \, {\rm G} ,$$

where G is the gravitational constant, $M_{\ast}=1.4 \, M_{\odot}$ and $r_{\ast} =10^6$ cm are the mass and the radius of the pulsar, and $r_{\rm L} = c/\Omega \simeq 7.1\times 10^8$ cm is the radius of the light cylinder; for simplicity we assumed that the pulsar magnetic field is dipolar. Given this value of $B_{\ast}$, one can estimate the "vacuum" values of $\dot{\Omega}$ and $\tau$: $\dot{\Omega} _0 = -2\mu ^2 \Omega ^3 /3c^3 I \simeq -1.36\times 10^{-11} \, {\rm s}^{-2}$, where $\mu = B_{\ast} r_{\ast}^3$ is the magnetic momentum of the pulsar, and $\tau _0 \leq - \Omega/2\dot{\Omega} _0 \simeq 4.9\times 10^4 \, {\rm yr} \simeq 30\tau$ (here we assumed that the "vacuum" braking index is equal to 3). This means that the "true" age of the pulsar could be as large as $\tau _0$ (provided that $P_{\rm i} \ll P$) and that the SNR MSH15-52 could be a middle-aged remnant similar to the Vela SNR (G263.9-3.3).