4 Discussion

Application of our findings to the case of INSs leads to the following conclusions. First, comparing Eqs. (4) and (19) one finds that under the conditions of interest the maximum possible strength of the stellar wind of an isolated neutron star, $\dot{M}_{\rm max}$, is smaller (at least by a factor of 3) than $\dot{M}_0$. Therefore, the interchange instabilities of the magnetospheric boundary of these stars are suppressed and the plasma penetration from the base of the hot atmosphere into the stellar magnetic field is governed by the Bohm diffusion. In this case the accretion luminosity of INSs, whose age is smaller than the characteristic time of the magnetic field decay, $t_{\rm mfd}$, is limited to

$$% L_{\rm x}(t<t_{\rm mfd})= \dot{M}_{\rm B} \frac{GM_{\rm ns}}{R_... ... R_6^{-1}\ \left(\frac{\dot{M}_{\rm c}}{10^{12} {\rm g~s^{-1}}}\right)^{11/14}.$$ (21)

This radiation is to be emitted from local regions situated at the magnetic poles of the star. That is why the star could be observed as a low luminous pulsing X-ray source. The minimum period of this pulsar is (see Eq. (6))

$$% P_{\rm min}\ \simeq\ 450\ \mu_{30}^{6/7}\ m^{-5/7}\ \dot{M}_{12}^{-3/7}~{\rm s}.$$ (22)

Second, as the age of an INS becomes comparable with $t_{\rm mfd}$, the magnetic field is almost unable to control the accretion flow in the stellar vicinity and the direct accretion onto the stellar surface occurs. The accretion luminosity of the star in this case is $L_{\rm x} = \dot{M}_{\rm c} GM_{\rm ns}/R_{\rm ns}$, i.e. by a factor of $\dot{M}_{\rm c}/\dot{M}_{\rm B}$ larger than the accretion luminosity of a neutron star in the state of the subsonic propeller. This radiation is to be emitted from the whole surface of the star and, therefore, the pulsations are not expected to be observed.

Finally, for $\dot{M}_{\rm c} \la 10^{12}~{\rm g~s^{-1}}$, the spin-down time scale of a neutron star in the subsonic propeller state is (see Davies & Pringle 1981, Eq. (3.2.5))

$$% t_{\rm acc, B}\ \ga\ 10^9\ \zeta_{0.1}^{-1/2}\ m^{1/7}\ \mu_{30}^{-15/14}\ \dot{M}_{12}^{-3/14}\ P_6\ {\rm yr},$$ (23)

where $P_6= P_{\rm acc, B}/10^6$ s (see Eq. (15)). This time scale is comparable with the characteristic time scale of the magnetic field decay (see e.g. Urpin et al.1996). Hence, the hot atmosphere surrounding the neutron star in the propeller state can be treated within the quasi-static approximation suggested by Davies & Pringle (1981).

Acknowledgements

I would like to thank the referee, Dr. Marina Romanova, for useful comments and suggested improvements. I acknowledge the support of the Alexander von Humboldt Foundation within the Long-term Cooperation Program.