2. Magnetic powering
The light cylinder at a radius m reaches out past the inner Lagrangian point but does not contain the secondary star entirely. The figure also shows a number of unperturbed dipolar field lines, including the field lines through the inner Lagrangian point. The cones of 'open' - reaching outside the light cylinder - field lines on opposite hemispheres encompass practically the entire set of (unperturbed) field lines going through the inner Lagrangian point during a rotation period of the white dwarf. These polar caps of open field lines at the white dwarf surface are small: for a dipole with obliquity the half-width in the azimuthal direction is while in the meridional plane it extends and away from the magnetic pole for a white dwarf radius m.
Since the rotation of the white dwarf is fast compared to the orbital period ( hours) the white dwarf magnetosphere is strongly perturbed by the relative motion of the companion. Moreover, any gas streaming from the inner Lagrangian point onto the white dwarf will similarly perturb the fast moving magnetosphere, until the gas reaches its Alfvén radius (defined loosely by equating the ram pressure , v is the relative speed between gas and magnetosphere, to the ambient stellar magnetic pressure ). Thereafter, depending on the location of the Alfvén radius with respect to the radius of corotation
the gas will either be propelled away (when ) or, when the reverse inequality is satisfied, be caught by the field and fall onto the white dwarf.
2.1. Power radiated in MHD waves
The power radiated in MHD (mainly Alfvén) waves by a good conductor of spherical shape (cross-section A) moving at subalfvénic speed v through an ambient magnetic field of strength B can be obtained from Barnett & Olbert (1986) under a number of assumptions
where is the Alfvén speed in the ambient medium. It is assumed that the radius of the sphere is much larger than an effective gyroradius ( is the proton Larmor frequency), the cold plasma approximation is used (valid if ), the ambient plasma is dense enough (, where is the electron plasma density, is the electron Larmor frequency), and finally, the Alfvén crossing time over the dimension of the sphere is much less than the Ohmic diffusion time. In our case all but one of these assumptions are correct: the ratio is probably much less than unity for the magnetosphere of AE Aqr. However we do not expect that this changes the above result significantly as long as one uses the correct relativistic Alfvén speed given by .
The radiated power by an antenna is known to be very sensitive to matching of impedances between antenna and the external medium. The detailed shape of the emitter (Lüttgen & Neubauer 1994) and the far-field boundary conditions are therefore crucial in these calculations. For instance, if the magnetosphere acts as a cavity, standing waves can be set up and the power radiated by an antenna then depends on the dissipation of the waves in the cavity. If the antenna were placed in a homogeneous medium the radiated power would not necessarily depend on the wave dissipation. Further, if the plasma is not cold and if moreover the conductor is injecting plasma into the magnetosphere, such as is the case for Io moving in Jupiter's magnetosphere, the slow mode can also be excited and magnetic field perturbations arise not only along so-called Alfvén wings but also in the wake of the satellite (Kopp 1996). This case may be particularly relevant for AE Aqr as the infalling gas blobs are expected to become ablated by the rotating magnetosphere.
Finally we mention another process for radiation of MHD waves by infalling gas. Scheurwater & Kuijpers (1988) calculated the efficiency of MHD wave excitation from a localized pressure pulse, generated by a plasma sphere falling along the stellar magnetic field and hitting the stellar surface at subalfvénic speed. Of the total infall energy a fraction is radiated into Alfvén waves and a fraction into fast magnetosonic waves, where is the particle density inside the cloud and n the ambient density above the surface. This demonstrates that the matter which finally accretes (with an accretion energy up to W) onto the surface of the white dwarf is an efficient source of MHD waves.
Let us now compare the power radiated in MHD waves by the companion with that radiated by infalling gas, applying the simple estimate (2) with a coefficient one.
2.2. Power radiated by the companion
Since the companion is near the light cylinder, its relative motion through the magnetosphere of the white dwarf is probably superalfvénic and it will possess a bow shock. We can then estimate the MHD power generated in the stellar wake by putting in the previous equations and writing :
where we have taken a radius for the companion m, a distance from the white dwarf m and a field strength T at the surface pole of the white dwarf with radius m.
2.3. Power radiated by infalling gas
To estimate the power radiated in MHD waves by infalling gas we use an accretion rate kg/s based on the quiescent X-ray luminosity. For simplicity we assume that a steady flow of gas escapes at a rate from the inner Lagrangian point at a sound speed km/s ( K) through a nozzle with cross-section and km (much smaller than the scale height near the inner Lagrangian point, see below). Further, we assume that the gas stream remains isothermal and keeps the same cross-section and that its velocity quickly reaches the free-fall speed in the gravitational field of the white dwarf km/s (). We take the cross-section to be constant until the stream reaches the pressure balance radius, defined by the distance at which the gas pressure becomes equal to the ambient stellar field pressure. For a dipolar magnetic field dependence we find a pressure balance radius,
Inside the pressure balance radius we assume that infall is again steady at the same rate, at the same temperature, and at the free-fall speed, now however with the cross-section determined by balance between the gas pressure and the ambient magnetic field pressure. Assuming the relative transverse speed between gas stream and magnetosphere to be supra-Alfvénic everywhere we arrive at the following estimate for the power radiated in MHD waves by a steady stream with a transverse extent extending from the Lagrangian point at a distance m down to the Alfvén radius (or the white dwarf radius at m, whichever is larger)
Under our assumptions the stream becomes compressed inside m and extends down to the white dwarf (because of the assumptions of isothermality and pressure equilibrium, the stream is very dense and never reaches its Alfvén radius (), which in this case simply reduces to m ). Note that the spin-down luminosity in MHD waves (estimate (5)) can be comparable to the quiescent gravitational X-ray luminosity which we used to estimate the mass loss. Of course for consistency it is required that the MHD power is not ultimately converted into X-rays.
Note that we have assumed a very small thickness of the gas stream, km, in comparison to the scale height at the inner Lagrangian point, km (Frank et al. 1992). As the efficiency of MHD wave excitation increases with the transverse extent of the gas stream, a small value of the thickness leads to a conservative estimate. In reality, the gas blobs will first fall freely towards the white dwarf. In the comoving frame the gas will expand along the equatorial plane at the speed of sound (which will of course decrease quickly due to adiabatic expansion). Transverse to the orbital plane the gas also expands as long as the gravitational acceleration is large compared to the centrifugal acceleration. When the latter increases, the gas tries to reach pressure equilibrium with the remaining effective field of gravity but will not succeed completely as it needs at least a free-fall or Keplerian time to do this. As a rough approximation for the transverse thickness we could therefore take km, as long as the ambient magnetic field pressure remains much smaller than the gas pressure (that is outside the pressure balance distance). As can be seen from (5) an increased transverse extent H causes an increase of radiated MHD power. Because of the unknown amount of equatorial spreading we shall use the conservative estimate (5). In principle the MHD power could increase by over two orders of magnitude, up to the observed spin-down luminosity W.
We conclude that the fast spinning white dwarf emits a spin-down luminosity in the form of MHD waves with a lower limit of W determined by the presence of the non-corotating companion, and a value of W if mass transfer takes place at a rate kg/s provided the local rotation speed is (super)Alfvénic. In a realistic magnetosphere the radiated waves will not be purely Alfvénic but have a compressive component because of the inhomogeneity and line-tying of the fields.
Is this sufficient in principle to power the observed radio emission? The quiescent radio power is W (Abada-Simon et al. 1995b), and an order of magnitude larger during flaring. The net conversion efficiency of MHD waves into radio emission should then on average be up to . In the next section we examine the properties of particle orbits in the present system and how violation of particular adiabatic invariants can cause stochastic acceleration.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998