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Astron. Astrophys. 322, 242-255 (1997)

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6. Emission from AE Aquarii

Having outlined the acceleration mechanism and found the regimes in which it may operate, we now proceeed to see whether it can explain the radio emission which we observe in AE Aqr.

6.1. Quiescent radio emission

Can acceleration by magnetic compressions power the quiescent radio emission from AE Aqr? Since the radio spectrum is observed to increase with frequency up to at least [FORMULA] 394 GHz (Abada-Simon et al. 1995a) we shall assume that the majority of quiescent emission is output around this frequency. Assuming that the emission is synchrotron radiation, the characteristic individual particle energy is (Pacholczyk 1970)


or [FORMULA] (91 in a [FORMULA] T field). The particles would have to be accelerated to this energy and then kept there by pumping, balancing out with synchrotron and Coulomb energy losses in a steady-state situation. The loss of energy in synchrotron radiation of each particle is given by Eq. (27), and inserting the value for the Lorentz factors of emitting particles gives the total synchrotron power for a volume V of particles at density [FORMULA],


(integrating over solid angle, assuming an isotropic distribution and an optically thin cloud). From Fig. 5 we see that the magnetic field must be [FORMULA] T for a net acceleration to appropriate energy to take place (determined by [FORMULA]). Therefore we need [FORMULA] emitting electrons to explain the quiescent emission at [FORMULA] W. Acceleration can occur (giving a sustainable emission of synchrotron radiation) in cold plasma densities [FORMULA]. Also, for a steady source the particle energy density should remain smaller than the energy density of the magnetic field, [FORMULA] which leads to an upper limit on the fast particle density of [FORMULA] m-3 (for a Lorentz factor [FORMULA] and a field strength [FORMULA] T). Then the source volume is at least [FORMULA] m3. If the white dwarf surface field at the poles is 100 T, this corresponds to a spherical shell of minimum thickness [FORMULA] m at the distance at which the field equals [FORMULA] T ([FORMULA] m). As detailed in the previous section precipitation losses increase this distance further.

It therefore seems possible that the quiescent radio emission can be explained by continuous energisation of the particles by the large-scale standing field oscillations.

6.2. Flare radio emission

To produce flares, we would envisage a situation in which enough particles are trapped on the field lines of the white dwarf and subsequently take part in the acceleration process. As the energy density in fast particles becomes too large with respect to that in the field, a MHD instability sets in and a magnetic cloud containing accelerated particles is expelled.

The number of energetic particles that may be stored in the white dwarf magnetosphere depends both on the magnetic field geometry and on where the energetic particles are produced. If they are deposited within a narrow flux tube a ballooning instability is likely, leading to an outward expansion of the flux tube releasing a cloud of energetic particles. To find the conditions for this it is necessary to specify the size and location of the region where the particles are deposited.

If the energy of stored particles [FORMULA], deposited at a radius [FORMULA], is comparable to the magnetic energy [FORMULA] required to open the field beyond [FORMULA], the closed field lines will open up releasing them. Approximating the open field [FORMULA] by a uniform radial field which reverses direction at a current sheet in the equatorial plane, with a flux given by a dipole magnetic flux at [FORMULA], we have [FORMULA]. Neglecting the change in magnetic energy for [FORMULA], the condition for releasing the stored particles is, [FORMULA]. If we take [FORMULA] T, [FORMULA] m as canonical values for the white dwarf magnetic field and radius this maximum stored energy is [FORMULA] J. For example if the minimum radius at which energisation dominates over losses is [FORMULA], we expect the stored particles to be released if their energy exceeds [FORMULA] J.

Once the cloud of trapped particles has been released from the magnetosphere much of the particle energy will go towards expanding the plasmoid. The individual particle energy will decrease as [FORMULA] as the source radius [FORMULA] increases. For an observed luminosity of [FORMULA] W, a source size [FORMULA], and a bandwidth [FORMULA] the brightness temperature at 400 GHz is given by


and the source is initially optically thin for [FORMULA]. Therefore at this frequency the brightness temperature decreases as the plasmoid expands. However at lower frequencies, taking into account the observed average positive slope of the spectrum (index [FORMULA]), the brightness temperature increases [FORMULA] and the source is initially optically thick. In fact the observed positive frequency slope of the quiescent average radio flux and the transition from optically thick to thin emission in a flare (Bastian et al. 1988) can be understood if a constant number of electrons are gradually accelerated to larger and larger energies within the same loop structure until it bursts open.

Finally, note that the emission at the time of the outbreak of the plasmoid is rather efficient as the synchrotron loss-time equals the acceleration time.

6.3. Hard X-ray emission by trapped and precipitating particles

To determine if trapped particles would emit observable quantities of X-rays we calculate the hard X-ray luminosity from:


where L is the hard X-ray luminosity, T is the temperature of the emitting plasma, [FORMULA] and [FORMULA] are the electron and ion densities with electric charge Z, and V is the volume of the emitting plasma. The term in brackets is the so-called emission measure (EM) which can be simplified as [FORMULA] V. The above equation can therefore be re-written as:


From § 5.1 and § 6.1 we take [FORMULA] m-3, V [FORMULA] [FORMULA] m3, [FORMULA] (the thermal electron velocity) = [FORMULA] m s-1 (which implies an electron temperature of [FORMULA] K) and find [FORMULA] W. This figure is many orders of magnitude less than the observed hard X-ray flux, [FORMULA] W (Eracleous et al. 1991). We conclude that the particles energised by magnetic pumping would not contribute a significant proportion to the observed hard X-ray flux.

If we assume that a small fraction of these energised particles (say 1%) can escape from the magnetosphere (see x4) and generate thick target emission near the stellar surface, the hard X-ray emission from these particles would be [FORMULA] W for [FORMULA] [FORMULA] and [FORMULA] hour. If the hard X-rays are generated from tall accretion curtains of the sort described by Eracleous et al (1995), it is clear that the emission from the accelerated particles would not contribute a significant amount to the overall X-ray flux and are not the origin of the X-rays seen in AE Aqr (Clayton & Osborne 1995).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998