2 Break period

According to the picture presented by Davies & Pringle the magnetosphere of the neutron star in the state of subsonic propeller is surrounded by the adiabatic ( $p\propto R^{-5/2}$) spherically symmetrical plasma envelope. Until the energy input to the envelope dominates the energy losses the temperature of the envelope plasma is of the order of the free-fall temperature, $T\simeq T_{\rm ff}$ and, correspondingly, the sound speed is of the order of the free-fall velocity, $V_{\rm s} \simeq V_{\rm ff}$. Under this condition the height of the homogeneous atmosphere through out the envelope is comparable to R and thus, the envelope is extended from the magnetospheric radius,

 

$$R_{\rm m} \simeq r_{\rm m} \equiv \left(\frac{\mu^{2}}{\dot{M}_0 \sqrt{2GM_{\rm ns}}}\right)^{2/7}$$

$$1.2 ~ 10^9\,{\rm cm}\ \mu_{30}^{4/7}\ \dot{M}_{15}^{-2/7}\ m^{-1/7},$$ = (3)

up to the accretion radius of the neutron star,

$$R_{\alpha} \equiv \frac{2GM_{\rm ns}}{V_{\rm rel}^2} = 2.7 ~ 10^{10}\,{\rm cm}\ m\ V_8^{-2}.$$

Here, V₈ is the relative velocity between the neutron star and the stellar wind plasma, $V_{\rm rel}$, expressed in units of $10^8\,{\rm cm\,s^{-1}}$ and $\dot{M}_0$ is strength of the stellar wind which is expressed following DP81 in terms of the maximum possible accretion rate:

$$\dot{M}_0 = \pi R_{\alpha}^2 \rho_{\infty} V_{\rm rel},$$

where $\rho_{\infty}$ is the density of the stellar wind plasma. Within the considered picture the envelope is quasi-static, i.e. the mass flux through the envelope is almost zero. In this situation the physical meaning of the parameter $\dot{M}_0$ is the rate by which the stellar wind plasma overflows the outer edge of the envelope compressing the envelope plasma.

The interaction between the fast rotating magnetosphere and the base of the envelope leads to the turbulization of the envelope plasma. The velocity of the convection motions at the magnetospheric boundary is obviously limited as

$$V_{\rm t} \mathrel{\mathchoice {\vcenter{\offinterlineskip\ha... ...scriptscriptstyle ...$$ (4)

Under the condition $R_{\rm m} < R_{\rm cor}$ the maximum velocity of the convective motions is smaller than $V_{\rm s}$ and hence, the Mach number at the base of the envelope is $M_{\rm Mach}=V_{\rm t}/V_{\rm s} < 1$.

The rate of energy loss by the neutron star and, correspondingly, the energy input to the envelope in this case can be expressed as (see Eqs. (3.2.1) and (3.2.2) in DP81)

$$L_{\rm d} = 2 \pi R_{\rm m}^2 \rho V_{\rm t}^3.$$

As it has been argued by Davies & Pringle the cooling of the envelope plasma is determined by the combination of convective motions and bremsstralung radiation. In order to evaluate the energy balance between the heating and cooling processes they introduced the convective efficiency parameter (see also Cox & Guili 1968):

$$\Gamma = \frac{\rm Excess\ heat\ content\ of\ convective\ blob}{\rm Energy\ radiated\ in\ the\ lifetime\ of\ a\ blob}\cdot$$

Under the conditions of interest this parameter can be expressed as (for discussion see DP81, p. 221)

$$\Gamma = M_{\rm Mach}^2 \left[\frac{V_{\rm t} t_{\rm br}}{R}\right] = \frac{V_{\rm t}^3 t_{\rm br}}{V_{\rm s}^2 R},$$ (5)

where $t_{\rm br}$ is the bremsstralung cooling time:

$$t_{\rm br} = 6.3 ~ 10^4 \left[\frac{T}{10^9\,{\rm K}}\right]^... ...} \left[\frac{n}{10^{11}\,{\rm cm^{-3}}}\right]^{-1} {\rm s}.$$ (6)

Here, n is the number density of the envelope plasma which at the base of the envelope can be evaluated as

$$n(R_{\rm m}) = \frac{\mu^2}{4 \pi k T_{\rm ff}(R_{\rm m}) R_{\rm m}^6}\cdot$$ (7)

As it has been shown in DP81 the cooling of the envelope during the subsonic propeller state occurs first at its inner radius. Thus, the energy input to the envelope due to the neutron star propeller action dominates the radiative losses if .

Combining Eqs. (1-7) I find the condition to be satisfied if the spin period of the neutron star is , where the break period is

$$P_{\rm br} \simeq\ 450\ \mu_{30}^{16/21}\ \dot{M}_{15}^{-5/7}\ m^{-4/21}\ {\rm s}.$$ (8)

This value of the break period exceeds the value of $P_{\rm br}$ presented in DP81 by a factor of 7.5 (see Eq. (4.8) in DP81). Hence the natural question about the reason of this inconsistency arises. One of the most possible reasons is that Davies & Pringle have mistakenly used the value of the magnetospheric radius: $4.4~ 10^8 \mu_{30}^{4/7} \dot{M}_{15}^{-2/7} m^{-1/7}$cm (see Eq. (3.2.3) of their paper) instead of the correct value which is expressed in their paper by Eq. (2.5), i.e. $10^9 \mu_{30}^{4/7} \dot{M}_{15}^{-2/7} m^{-1/7}$cm. Taking into account that $P_{\rm br}(R_{\rm m})\propto R_{\rm m}^{5/2}$ one finds that the correct value of $P_{\rm br}$ should be larger than that derived in DP81 by a factor of 7.7, i.e. very close to the value of the break period obtained in this letter.