3 Diffusion-driven accretion

Although the interchange instabilities of the magnetospheric boundary during the subsonic propeller state are suppressed, the "magnetic gates'' are not closed completely: the atmospheric plasma is able to penetrate into the stellar magnetic field due to diffusion. The diffusion rate is limited to (see e.g. Ikhsanov 2001b)

$$% \dot{M}_{\rm B} \la 10^{11}~{\rm g~s^{-1}}\ \zeta_{0.1}^{1/... ...{\dot{M}_{\rm c}}{10^{15}~{\rm g~s^{-1}}}\right)^{11/14}\cdot$$ (10)

Here $\zeta_{0.1}=\zeta/0.1$ is the diffusion efficiency, which is normalized following the results of experiments on the nuclear fusion (e.g. Hamasaki et al.1974) and the measurements of the solar wind penetrating the magnetosphere of the Earth (Gosling et al. 1991).

This means that the hot atmosphere surrounding the magnetosphere of the star in the subsonic propeller state cannot be purely stationary. For the atmosphere to remain in an equilibrium state, the amount of material flowing from its base into the magnetosphere must be compensated for by the same amount of material coming into the atmosphere through its outer boundary. Thus the radial drift of plasma through the atmosphere with the rate $\dot{M}_{\rm B}$ and the velocity

$$% V_{\rm dr} = (\dot{M}_{\rm B}/\dot{M}_{\rm c})\ V_{\rm ff}$$ (11)

towards the neutron star is expected.

The structure of the atmosphere with the radial plasma drift can be explain in terms of the quasi-stationary model of Davies & Pringle (1981) as long as the characteristic time of the accretion process,

$$% t_{\rm dr} = R/V_{\rm dr},$$ (12)

is larger than the characteristic time of turbulent motions $t_{\rm t} = R/V_{\rm t}$. According to Davies & Pringle (1981), the velocity of turbulence motions can be approximated as $V_{\rm t} \simeq (R_{\rm m}/R)^{1/6} \omega R_{\rm m}$. Therefore, the condition $t_{\rm t} \ll t_{\rm dr}$proves to be satisfied if the spin period of the star is $P_{\rm s} \ll P_{\rm qs}$, where

$$% P_{\rm qs, B} \simeq 1.7 \times 10^5\ \zeta_{0.1}^{-1/2}\ \dot{M}_{15}^{-3/14}\ m^{-6/7}\ \mu_{30}^{13/14}\ {\rm s}.$$ (13)

Thus, as long as the spin period of the star is $P_{\rm cd} \la P_{\rm s} \ll P_{\rm qs, B}$, the magnetosphere of the star is surrounded by a hot ( $T \simeq T_{\rm ff}$) atmosphere in which the plasma pressure is $p \propto R^{-5/2}$, the sound speed is $c_{\rm s} \propto R^{-1/2}$ and the number density is $n \propto R^{-3/2}$ (for discussion see Davies & Pringle 1981).

The radial drift of plasma through the atmosphere towards the neutron star leads to the release of the accretion (potential) energy, which is mainly spent in heating the atmospheric plasma. The heating rate due to this process is $L_{\rm dr}(R)\simeq \dot{M}_{\rm B} GM_{\rm ns}/R$. This value is small in comparison with that of the spin-down power (see Davies & Pringle 1981, Eq. (3.2.4)),

$$% L_{\rm ssp} = 1.2 \times 10^{36}\ \mu_{30}^2\ m^{-1} P_{\rm s}^{-3}\ {\rm erg~s^{-1}},$$ (14)

as long as the spin period of the star is $P_{\rm s} \la P_{\rm acc, B}$, where

$$% P_{\rm acc, B} \simeq\ 450\ \zeta_{0.1}^{1/6}\ \mu_{30}^{37/42}\ \dot{M}_{15}^{-5/14}\ m^{-16/21}\ {\rm s}.$$ (15)

However, under the condition $P_{\rm s} > P_{\rm acc}$ the heating of the atmospheric plasma is governed by the accretion power.

The characteristic time of the heating due to accretion process is $t_{\rm dr}$. On the other hand, the cooling time of the atmospheric plasma due to the bremsstrahlung emission is

$$% t_{\rm cool} \approx t_{\rm br} \simeq 633~{\rm s}\ \left(\... ...ght)^{1/2} \left(\frac{n}{10^{13}~{\rm cm^{-3}}}\right)^{-1},$$ (16)

where n is the number density of the atmospheric plasma, which can be expressed as $n(R)=n(R_{\rm m})(R_{\rm m}/R)^{3/2}$, where

$$% n(R_{\rm m}) = \frac{\mu^2}{4 \pi R_{\rm m}^6 k T(R_{\rm m})}\cdot$$ (17)

Hence, for the temperature of the atmospheric plasma to be smaller than the critical value, $T_{\rm cr}$, the following condition should be satisfied

$$% t_{\rm cool}(R_{\alpha}) < t_{\rm dr}(R_{\alpha}).$$ (18)

I require this condition to be satisfied at the outer boundary of the atmosphere since $t_{\rm c} \propto R$ and $t_{\rm dr} \propto R^{3/2}$. Combining Eqs. (11), (12), (16), and (17), one finds that the bremsstrahlung cooling dominates the heating due to accretion power only if the strength of the stellar wind is $\dot{M}_{\rm c} \ga \dot{M}_0$, where

$$% \dot{M}_0 \simeq 3 \times 10^{14}\ \zeta_{0.1}^{7/17}\ \mu_{30}^{-1/17}\ m^{16/17}\ V_{8}^{14/17}\ {\rm g~s^{-1}}.$$ (19)

This means that a magnetized isolated neutron star is able to switch its state from subsonic propeller to steady accretor only if the mass capture rate by this star from the interstellar medium is $\dot{M}_{\rm c} \ga \dot{M}_0$. Otherwise, the corresponding state transition does not occur and the star remains surrounded by the hot atmosphere. In this case the mass accretion rate onto the stellar surface is limited to $\dot{M} \la \dot{M}_{\rm B}$ (see Eq. (10)) and, correspondingly, the accretion luminosity is $L_{\rm x} \la L_{\rm max}$, where

$$% L_{\rm max} \simeq 10^{30}\ \zeta_{0.1}^{1/2}\ \mu_{30}^{-1/14}\ m^{8/7}\ R_6^{-1}\ \dot{M}_{14}^{11/14}\ {\rm erg~s^{-1}}.$$ (20)

Here R₆ is the radius of the neutron star expressed in units of 10⁶ cm, and $\dot{M}_{14}=\dot{M}_{\rm c}/10^{14}~{\rm g~s^{-1}}$.