2. The Fourier power spectrum

Quasi-periodic oscillations (QPO) were discovered with EXOSAT observations of Cen X-3 (Tennant (1988)). They are also present in Ginga (Takeshima et al. (1991)) and BBXRT data. Takeshima et al. argue that the Ginga data are consistent with the beat frequency mass accretion model of QPOs (Alpar and Shaham (1985); Shibazaki and Lamb (1987)). This model assumes the presence of a Keplerian disk in which there are density and magnetic field fluctuations and is thus appropriate since Cen X-3 is believed to be a magnetized neutron star with disk accretion. Interactions between the magnetosphere of the neutron star and fluctuations in the inner disk give rise to modulation of the mass accretion rate, and hence the X-ray luminosity. This modulation is at a harmonic of the beat frequency between the rotation frequency of the neutron star and that of the magnetospheric boundary layer. Thus the QPOs in Cen X-3 are consistent with the existence of an accretion disk.

The QPO detected with BBXRT has a centroid frequency of 40.7$\pm$1.1 mHz while the value obtained by Takeshima et al. was 35$\pm$2 mHz. If the beat frequency model is correct, the angular frequency of the QPO will be $\omega$ = $\Omega_{K}^{}$(r_A) - $\Omega_{0}^{}$ if the pulsar is spinning up or $\omega$ = $\Omega_{0}^{}$ - $\Omega_{K}^{}$(r_A) if it is spinning down, assuming that $\omega$ is the fundamental beat frequency. Here r_A is the Alfvén radius which represents the inner edge of the accretion disk, $\Omega_{K}^{}$(r_A) = (GM)^1$\over$2 r_A^{- ${3\over2}$} is the Keplerian angular frequency at that radius, and $\Omega_{0}^{}$ is the angular frequency of the magnetosphere which co-rotates with the neutron star. Long term monitoring of Cen X-3 with the BATSE instrument on the Compton Gamma Ray Observatory (Finger et al. (1994)) has revealed that the spin up of the pulsar is accomplished by alternating episodes of increase and decrease in the pulse period at rates greater than the average spin up rate. This indicates that the accretion rate is such that the neutron star is close to equilibrium between magnetic braking and angular momentum accretion. The low frequency of the QPO also suggests that the Keplerian rotation frequency at the magnetic boundary layer is close to the rotation frequency of the magnetosphere, as would be expected if the system is close to equilibrium between spinning up and spinning down. At the time of the BBXRT observation the pulsar could be either spinning up or down so that $\Omega$(r_A) = $\Omega_{0}^{}$$\pm$$\omega$ which implies $$r_A=5.1\times10^8\,(\Omega_0\pm\omega)^{-{3\over2}}\biggl({M\over M_\odot}\biggr)^{1\over3}\ \rm cm.$$

The Alfvén radius may also be estimated from the magnetic dipole moment and mass accretion rate (Elsner and Lamb (1977)): $$r_A=3.2\times10^8\,\dot M_{17}^{-{2\over7}}\mu_{30}^{4\over7}\biggl({M\over M_\odot}\biggr)^{-{1\over7}}\ \rm cm.$$
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. Thus the magnetic dipole moment is $$\mu_{30}=2.4\,(\Omega_0\pm\omega)^{-{7\over6}} L_{37}^{1\over2}... ...psilon_{0.1}^{-{1\over2}}\biggl({M\over M_\odot}\biggr)^{5\over6}$$
where $\varepsilon_{0.1}^{}$ is the conversion efficiency in units of 0.1. With $\Omega_{0}^{}$ = 1.3040 rad s^-1, $\omega$ = 0.2557 rad s^-1, L₃₇ = 3.29, and assuming that the pulsar is spinning up this yields the value $$\mu=2.6\times10^{30}\,\varepsilon_{0.1}^{-{1\over2}}\biggl({M\over M_\odot}\biggr)^{5\over6}\ \rm G\ cm^{3}.$$
This corresponds to a surface magnetic field $$B = {{2.6\times10^{12}\, \varepsilon_{0.1}}^{-{1\over2}}\biggl({M\over M_\odot}\biggr)^{5\over6} \over{{R_6}^3}}\ \rm G$$
where R₆ is the neutron star radius in units of 10⁶ cm.

The value obtained from the BBXRT data for the surface magnetic field of Cen X-3 may be used to estimate the energy of the fundamental cyclotron resonance. The observed fundamental energy of a cyclotron resonance feature is $$E_{obs} = {{E_0}\over{1+z}}$$
where $$E_0 = 11.6\times \biggl({B\over{10^{12}\ \rm G}}\biggr)\ \ \rm keV$$
and $$z = \biggl(1 - {{2GM}\over{Rc^2}}\biggr)^{-{1\over 2}} - 1$$
is the gravitational redshift. Thus the observed energy of the fundamental cyclotron resonance is expected to be $$E_{obs} = {{29\, \varepsilon_{0.1}}^{-{1\over2}}\biggl({M\over ... ...over6}\over{{R_6}^3}}\, \sqrt{1 - {{0.41}\over{R_6}}}\ \ \rm keV.$$
If on the other hand the pulsar is spinning down $$B = {{4.2\times10^{12}\, \varepsilon_{0.1}}^{-{1\over2}}\biggl({M\over M_\odot}\biggr)^{5\over6} \over{{R_6}^3}}\ \rm G$$
and $$E_{obs} = {{48\, \varepsilon_{0.1}}^{-{1\over2}}\biggl({M\over ... ...over6}\over{{R_6}^3}}\, \sqrt{1 - {{0.41}\over{R_6}}}\ \ \rm keV.$$
However this situation is unlikely as in this case the incoming matter would be repelled by a centrifugal barrier and the source would switch off.

If the emission region extends far enough above the neutron star surface to sample a smaller B field the observed energy of the cyclotron feature will be less than this. Nagase et al. (1992) report that the shape of the spectrum of Cen X-3 is consistent with a high energy roll off of the form exp(- L(E)) where $$L(E) = {\tau(WE/E_0)^2\over(E-E_0)^2+W^2}$$
The form of this roll off is that of a cyclotron scattering resonance feature where E₀ is the energy of the resonance and W is its width. The fit obtained to the Ginga data by Nagase et al. suggested that E₀ = 30$\pm$2 keV. However this result is inconclusive as the inferred resonance energy was at the upper limit of the LAC's bandpass. Makishima et al. (1992) described the cyclotron resonance features detected with Ginga in several X-ray pulsars. The observed resonance energies ranging from 7 to 40 keV imply surface magnetic fields of (0.6 - 3.5) x 10¹² G. The lack of definite evidence for a cyclotron feature below 30 keV suggests that Cen X-3 may have a relatively larger surface magnetic field. The values estimated from the QPO are consistent with this.