4. Interaction of the accretion disk with the magnetosphere

One property which HMXB do not share with the rotation-powered pulsars is the presence of accreting matter that can spin the star up or down. How this occurs depends on how the magnetic field of the neutron star is coupled to the accretion flow.

In the case of a HMXB with a large ($\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}10^{11}\rm G$
) magnetic field the accretion disk cannot extend all the way down to the neutron star's surface. This is because movement of the plasma perpendicular to field lines is inhibited due to its electrical conductivity. The region surrounding the neutron star in which the flow of matter is dominated by the magnetic field is the magnetosphere. Figure 5.4 shows how the motion of the infalling matter is controlled by the magnetic field. We can make a very crude estimate of its size by equating the magnetic pressure with the ram pressure of the infalling material for spherical accretion. The radius at which these two pressures balance is the Alfvén radius r_A.

$$\dot{M}_{17}^{-{2\over7}} \mu_{30}^{4\over7} \biggl( {M\over M_\odot} \biggr)^{-{1\over7}}_{}$$ (52)

Here $\mu_{30}^{}$ is the magnetic dipole moment of the neutron star in units of 10³⁰ G cm³ and $\dot{M}_{17}^{}$ is the mass accretion rate in units of 10¹⁷ g s^-1. In terms of the luminosity $\dot{M}$ = L/c²$\varepsilon$ where $\varepsilon$ is the energy conversion efficiency for 2-10 keV X-rays. This means that $\dot{M}_{17}^{}$ = 1.1 L₃₇/$\varepsilon_{0.1}^{}$ where $\varepsilon_{0.1}^{}$ = $\varepsilon$/10 $\sim$ 1. Thus $$r_{\rm A}=3.1\times10^8\,L_{37}^{-{2\over7}}\varepsilon_{0.1}^{... ...0}^{4\over7}\biggl({M\over M_\odot}\biggr)^{-{1\over7}}\ \rm cm.$$
The magnetosphere co-rotates with the neutron star. If we make the simplistic assumption that the inner edge of the accretion disk is at the magnetospheric radius r_m, there are three possible cases for spin-period change depending on the angular velocity of the inner accretion disk. We define the co-rotation radius R_co as the radius at which the Keplerian orbital angular velocity $\Omega_{\rm K}^{}$ = $\sqrt{GM_{\rm x}\over r^3}$ equals the neutron star's spin angular velocity $\Omega_{0}^{}$. If r_m < r_co, the inner disk will be rotating faster than the magnetosphere and it will exert a positive torque which will spin up the pulsar. If r_m < r_co, the inner disk will be rotating more slowly than the magnetosphere and it will exert a negative torque which will cause the pulsar to spin down. In this simple picture, for r_m = r_co there will be no net torque. These results hold for a prograde disk. Clearly, for a retrograde disk, the net torque on the neutron star will always be negative. The Alfven radius r_A depends on the mass accretion rate so it is possible for the neutron star to have episodes of spinning up and spinning down.

Figure 14: An accreting X-ray pulsar. Inside the magnetosphere the the infalling matter flows along the magnetic field lines. The dimensions of this system have been chosen to be similar to those of Cen X-3. The radius of the neutron star ( $\sim$ 10 km) has been exaggerated by a factor of 10 for clarity.

The disk-torque model predicts that the rate of change of the spin period will depend on luminosity. Cen X-3 and Her X-1 were found to have spin-up rates an order of magnitude smaller than those predicted. Treating the neutron star and magnetosphere as a unit, Lamb et al. (1973) considered the transport of angular momentum across a surface just outside r₀ and found

$${\dot P\over P} {\dot M\over M_{\rm x}} \left(\vphantom{{M_{\rm x}\over I_{\rm x}} {dI_{\rm x}\over dM_{\rm x}}-{l(r_0)\over l_x}}\right. {M_{\rm x}\over I_{\rm x}} {dI_{\rm x}\over dM_{\rm x}} {l(r_0)\over l_x} \left.\vphantom{{M_{\rm x}\over I_{\rm x}} {dI_{\rm x}\over dM_{\rm x}}-{l(r_0)\over l_x}}\right) {N^\prime\over I_{\rm x}\Omega}$$ (53)

where I_x is the moment of inertia of the neutron star, l (r₀) is the component parallel to $\Omega$ of the specific angular momentum of matter just outside the surface, l_x = I_x$\Omega$/M_x is the specific angular momentum of the neutron star, and N^$\prime$ is the sum of the external magnetic and viscous torques acting just outside the surface. For a Keplerian disk l (r₀) = $\sqrt{GM_{\rm x}r_0}$ and for typical neutron star equations of state (1983)

$${M_{\rm x}\over I_{\rm x}} {dI_{\rm x}\over dM_{\rm x}} \approx$$ (54)

Then, assuming N^$\prime$ = 0 (1983),

$$\dot{P} \approx \left(\vphantom{\mu_{30}^{2\over7}R_6^{6\over7}\left({M_{\rm x}\over M_\odot}\right)^{-{3\over7}}I_{45}^{-1}}\right. \mu_{30}^{2\over7} \over \left(\vphantom{{M_{\rm x}\over M_\odot}}\right. {M_{\rm x}\over M_\odot} \left.\vphantom{{M_{\rm x}\over M_\odot}}\right)^{-{3\over7}}_{} \left.\vphantom{\mu_{30}^{2\over7}R_6^{6\over7}\left({M_{\rm x}\over M_\odot}\right)^{-{3\over7}}I_{45}^{-1}}\right) \left(\vphantom{PL_{37}^{3\over7}}\right. \over \left.\vphantom{PL_{37}^{3\over7}}\right)^{2}_{}$$ (55)

where I₄₅ is the neutron star's moment of inertia in units of 10⁴⁵ g cm².

Ghosh et al. (1977) and Ghosh and Lamb (1979a,b) assumed that the neutron star's magnetic field threaded a thin disk. In a broad outer zone the disk is Keplerian and exerts a negative torque on the neutron star through the magnetic field. The disk makes the transition from Keplerian rotation to co-rotation with the magnetosphere in a narrow inner zone. The torque exerted by this region of the disk on the neutron star depends on whether the magnetosphere radius r_m is larger or smaller than the corotation radius r_co. The radius at which the disk deviates from Keplerian rotation is $r_0\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}r_{\rm m}$
. Then if r_co < r₀ the disk in the inner transition region is rotating more slowly than the magnetosphere and will exert a negative torque on it. If r_co > r₀ the disk in this region rotates faster than the magnetosphere and will exert a positive torque. Ghosh and Lamb (1979a) found that $\delta$ $\approx$ 0.03r₀ and

$$\xi \approx$$ (56)

The fastness parameter $\omega_{s}^{}$ is defined by (Elsner and Lamb (1977))

$$\omega_{s}^{} \equiv {\Omega_s\over \Omega_{\rm K}(r_0)} \left(\vphantom{{r_{\rm m}\over r_{\rm co}}}\right. {r_{\rm m}\over r_{\rm co}} \left.\vphantom{{r_{\rm m}\over r_{\rm co}}}\right)^{3\over2}_{}$$ (57)

Ghosh and Lamb (1979b) parameterized the total torque N in terms of a dimensionless torque n($\omega_{s}^{}$) and the material torque N₀ $\equiv$ $\dot{M}$$\sqrt{G M_{\rm x}r_{\rm m}}$ (Pringle and Rees (1972)) exerted by a Keplerian accretion disk so that

$$\approx \omega_{s}^{}$$ (58)

They found that n($\omega_{s}^{}$) could be fitted by the approximate expression

$$\omega_{s}^{} \approx \left(\vphantom{{1-{\omega_s\over\omega_c}\over1-\omega_s}}\right. {1-{\omega_s\over\omega_c}\over1-\omega_s} \left.\vphantom{{1-{\omega_s\over\omega_c}\over1-\omega_s}}\right)$$ (59)

There is a critical value of the fastness parameter, $\omega_{c}^{}$ $\approx$ 0.8-0.9 above which the neutron star will spin down and below which the neutron star will spin up. For $\Omega_{s}^{}$ $\gg$ $\omega_{c}^{}$, accretion to the neutron star will cease because of the the propeller effect (Davidson and Ostriker (1973)). Because n($\omega_{c}^{}$) = 0 there will be an equilibrium pulsar spin period P_eq for which $\dot{P}$ $\approx$ 0 on average. Fluctuations in $\dot{M}$ can still cause episodes of spin-up and spin-down. With the external magnetic torques in N^$\prime$ absorbed into N, Equation 1.65 becomes (Ghosh and Lamb (1979b))

$$\dot{P} \approx \left(\vphantom{\mu_{30}^{2\over7}n(\omega_s)R_6^{6\over7}\left({M_{\rm x}\over M_\odot}\right)^{-{3\over7}}I_{45}^{-1}}\right. \mu_{30}^{2\over7} \omega_{s}^{} \over \left(\vphantom{{M_{\rm x}\over M_\odot}}\right. {M_{\rm x}\over M_\odot} \left.\vphantom{{M_{\rm x}\over M_\odot}}\right)^{-{3\over7}}_{} \left.\vphantom{\mu_{30}^{2\over7}n(\omega_s)R_6^{6\over7}\left({M_{\rm x}\over M_\odot}\right)^{-{3\over7}}I_{45}^{-1}}\right) \left(\vphantom{PL_{37}^{3\over7}}\right. \over \left.\vphantom{PL_{37}^{3\over7}}\right)^{2}_{}$$ (60)

The relationship between average spin-up rate and luminosity for several different accreting pulsars is shown in Figure 5.4. All of the systems plotted agree with the theoretical prediction except for Vela X-1. This confirms the presence of an accretion disk in these systems. Vela X-1 is believed to be a wind accretor, although Anzer and Börner (1995) have suggested that a Keplerian disk may form just outside the magnetosphere and that this may account for the magnitude of the random variations in the system's pulse period.

Figure 15: Relationship between spin-up rate and luminosity for accreting pulsars (Shapiro and Teukolsky (1983) after Ghosh and Lamb (1979b)). The shaded region represents a range of neutron star masses between 0.5 and 1.9M_$\odot$. The magnetic moment is assumed to be $\mu_{30}^{}$ = 0.48 for all of the systems shown.

Wang (1987) assumed a different threading parameter for the magnetic field in the inner accretion disk. Wang found that $\xi$ $\approx$ 1 and thus the magnetospheric radius is r_m = $\xi$r_A $\approx$ r_A.