5. A limit-cycle model for RX J0513
The X-ray and optical light curves of RX J0513 suggest that the system follows a kind of a limit cycle behaviour with four typical time-scales: the 140 days of the optical high/ X-ray off state, the rapid transition ( 4 days) to the X-ray on/ optical low state, the 30 days duration of this state, and, again, the rapid transition ( 2 days) to the optical high/ X-ray off state.
We propose here that this behaviour results from expansion of the white dwarf photosphere in response to enhanced accretion onto the white dwarf, together with the reaction of the disk to increased irradiation by this expanded photosphere while the mass transfer from the companion star remains constant. At the (arbitrary) start of the cycle (Fig. 3 (d)), let us assume we have an accretion disk supplying matter to a white dwarf with its non-expanded radius cm. Because the mass supply rate is close to the Eddington critical accretion rate (Fujimoto 1982; Kato 1985), the white dwarf radius begins to expand (Fig. 3 (d)-(e)), as explained in Sect. 2 above. This will, in turn, influence the disk temperature. An extended central source with radius , where H is the scale height of the disk, produces a surface temperature at disk radius R in an optically thick disk (e.g. Adams et al. 1988) given by
Here , is the temperature of the white dwarf photosphere, limb-darkening has been neglected, and is the reprocessing efficiency of the disk surface. Increasing has two effects: (i) at given radius R, the disk temperature rises approximately as , and (ii) the inner disk disappears in the hot envelope of the star (see also Sect. 4 above).
The increase in disk temperature raises the mass-flow rate in the disk, since the disk viscosity coefficient is increased. Here is the sound speed, and is the scale height of the disk. The disk is now no longer in a steady state, since the mass-flow rate within it exceeds the mass supply rate from the companion star at its outer edge. Its mass is therefore gradually drained onto the white dwarf on a viscous timescale
where cm is a characteristic disk radius (the radius of the Roche lobe is about cm), , cm s-1 for K, and is a typical value of the viscosity parameter. With the disk being drained, the accretion rate onto the white dwarf eventually drops below , and the white dwarf reverts to its unexpanded state, the disk becomes cooler, and the system enters an optical low and X-ray on state (Fig. 3 (a)-(b)). The collapse of the expanded stellar envelope leaves the accretion disk with an inner hole of approximately the envelope radius which is gradually refilled by accretion from the outer disk. The disk temperature at a given radius R decreases as , but with cm in the optical low state the temperature at the edge of the hole ( cm) becomes K, similar to the outer disk temperature in the optical high state. The viscous time scale for refilling the hole, assuming again and cm s-1 then is
This picture predicts a long X-ray off state and a shorter X-ray on state, with rapid (thermal-timescale) transitions between them, in quantitative agreement with what is observed.
© European Southern Observatory (ESO) 2000
Online publication: January 31, 2000