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

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7. Discussion

To put our results into perspective we briefly discuss the injection of the particles into the accelerator and other acceleration mechanisms expected to occur in AE Aqr.

The maximum number of radiating electrons is [FORMULA]. Assuming that they have to be refreshed every hour (the flare timescale) an equivalent hydrogen mass accretion rate of [FORMULA] kg/s [FORMULA] [FORMULA] yr has to be supplied for, which is a fraction [FORMULA] of the nominal accretion rate. Two sources of gas injection are the wind from the secondary and Roche lobe overflow from the secondary in the form of clumps of matter.

Considering Roche lobe overflow first, gas clumps falling towards the white dwarf quickly reach supersonic speeds. Assuming that the motion is supersonic (or super-alfvénic) with respect to both the magnetosphere and the internal velocity of sound in the blob, a system of two shocks forms, separated by a contact discontinuity. At the outer shock the incoming magnetospheric plasma is compressed and heated. The inner shock pressurises the gas blob and vanishes when there has been enough time for the shock to reach the blob center. The details depend on the blob density and dimensions as they determine the cooling rate. In any case Kelvin-Helmholtz and Rayleigh-Taylor instabilities are expected to develop along the contact discontinuity toward the wake, gradually stripping the gas blob. Note that the short-periodic, dramatic increase of ambient magnetic field pressure (with a factor 4 for a perpendicular rotator) shocks the gas blobs continuously. Ablation turns the stream of gas blobs into a spray (cf. Wynn et al. 1995). Ultimately the ambient field penetrates the fragments or part of them (Arons & Lea 1980) and magnetic pumping starts acting on the plasma. This process is enhanced by the so-called pickup process (see below) of initially neutral gas atoms escaping from the blob surface into the magnetosphere.

The second source of plasma is the stellar wind of the companion. If the stellar mass loss rate is comparable to the, extremely weak, solar one ([FORMULA] [FORMULA] /yr) a fraction [FORMULA] of the particles have to get injected into the accelerator. Of course strong magnetic activity (see below) in the corona of the secondary enhances the mass loss rate in the stellar wind. Moreover erupting stellar prominences, in particular unstable prominences in the reversed gravity field above the inner Lagrangian point could enhance the mass transfer. Such 'prominences' with estimated densities of [FORMULA] m-3 can act as long-term injectors of particles in dwarf novae (Steeghs et al. 1996).

Finally, in the fast rotating magnetosphere cool matter can be held-up against gravity at preferred locations (Steeghs 1996).

We now briefly turn to acceleration processes other than magnetic pumping.

7.1. Pickup acceleration

Neutral gas atoms evaporating from the infalling blobs can penetrate the white dwarf magnetosphere. As soon as the atoms get ionized their charged constituents become trapped onto the white dwarf magnetic field with particle speeds at least equal to the differential speed of the blob and the magnetosphere [FORMULA]. For instance at a hundred white dwarf radii the kinetic pickup energy of an electron would be 150 keV, and 300 MeV for a proton. The pickup process can be enhanced by microinstabilities driven by the relative motion between the new ions and the background plasma. This occurs in the critical ionisation velocity mechanism. The theory (Raadu 1978) invokes the modified two stream instability which accounts for the observed electron energisation parallel to the magnetic field (Danielsson & Brenning 1975) with energies comparable to the kinetic pickup energy of the ions. Thus the pickup process can produce electrons with energies much greater than their pickup energies. Clearly, the pickup process can be a very important injection process for magnetic pumping.

7.2. Transit-time damping

For small-amplitude wave-like perturbations transit-time magnetic pumping requires a Cerenkov resonance to be fulfilled


where [FORMULA] are the wave frequency and the wave vector, respectively, and the relevant components are taken parallel to the ambient magnetic field. Further only compressional magnetic waves -the fast and the slow mode - contribute (Achterberg 1981). Equation (40) is familiar from Landau damping when the wave has a longitudinal electric field component. As has been shown by Stix (1992) transit-time pumping is the magnetic analogue of Landau damping. Actually both effects interfere and should be taken into account to determine the rate of transfer of energy from waves to particles. For the nearly perpendicular fast magnetosonic wave, however, the electric field component along the magnetic field is much smaller than the electric field component transverse to both the magnetic field and the wave vector and Landau damping can be neglected compared to transit-time damping. For an isothermal plasma primarily electrons rather than ions are heated by collisionless transit-time damping of fast magnetosonic waves. Transit-time damping could play an important role in AE Aqr, if only to preheat electrons (and ions). In particular a non-linear version of transit-time damping is expected to operate on those field lines where the bounce time of a particle equals the oscillation time which in our case is the rotation period of the white dwarf. From Eq. (15) it follows that loops with shell parameter


will be heated by this process. We conclude that a future investigation of transit-time damping in AE Aqr for large amplitudes is needed.

7.3. Resonant heating

Another process of interest, which has been investigated in the context of heating the solar corona (Kuperus et al. 1981, Goedbloed & Halberstadt 1994, Poedts & Boynton 1996), is resonant heating by Alfvén waves. The mechanism is based on the resonance between the Alfvén propagation time along a particular field line of length l, which is anchored at its footpoints in the star, and the frequency of an MHD wave [FORMULA] impinging on the inhomogeneous magnetic loop structure


with integer Z. In the layer of field lines satisfying this equation for given [FORMULA], a standing kinetic Alfvén wave can be excited to a large amplitude. Radial inhomogeneity then leads to phase mixing and the development of large gradients. The incoming Poynting flux of the MHD wave is then efficiently tapped by strong dissipation either by viscous damping and Ohmic heating, or collisionless damping. Taking the speed of light as an upper limit to the Alfvén parameter resonant heating by MHD waves at the spin period is found to be important for dipolar loops with shell parameter


Again this is an interesting process requiring further study.

7.4. Magnetic flares and reconnection

As we have seen, magnetic pumping can lead to eruptions. These can be called magnetic flares as they originate from a large-scale instability of an MHD structure. It is important, however, to realize that our flares are pressure-driven by the high energy particles stored in the magnetic fields. This is at variance with solar flares which are considered to be driven by the evolving magnetic field of a low-beta structure (Kuijpers 1992, [FORMULA] is the ratio of gas pressure to magnetic field pressure). We merely point out here that such 'classical' flares could occur as well in AE Aqr and lead to particle acceleration. Two obvious cases present themselves: firstly, reconnection between coronal magnetic fields of the companion and of the white dwarf may lead to coronal current systems (Lamb et al. 1983) and subsequent explosions. The formation of such flaring structures would primarily occur where the fields of the companion and white dwarf are comparable. For the above dipolar field of the white dwarf and a surface field of the companion at the inner Lagrangian point ([FORMULA] m from the white dwarf) of 0.3 T with a dipolar scale height of [FORMULA] m the magnetic fields are comparable ([FORMULA] T) at the midpoint between the stars (about [FORMULA] from the white dwarf). The total energy which could be stored in such a huge weak field structure is of the order of the potential energy, [FORMULA] J. Of course this estimate implies reconnection at the speed of light, assumes that the surface of the companion is covered with strong fields, and moreover neglects the complicating shock structure which engulfs the companion. Therefore, it seems unlikely that such large-scale reconnections between the fields of the companion and white dwarf are sufficiently powerful to cause the observed radio outbursts. Secondly and more promising, the ambient white dwarf field can be distorted (and reconnect) by infalling blobs in a manner such as described by Aly & Kuijpers (1990).

Finally other acceleration processes can be important such as shock acceleration by bullets (Jones et al. 1994), magnetic field aligned electric fields in AC or DC circuits (Raadu 1989), and modulational instabilities of magnetosonic waves (Bingham 1995 private communication), and it would be important to have observations which discriminate between these (such as TeV [FORMULA] -rays, De Jager et al. 1994, Meintjes et al. 1994).

7.5. Flares on other magnetised binaries?

If our proposal that magnetic pumping creates plasmoids of cosmic rays in AE Aqr is correct, what are the predictions for other magnetic CVs and for X-ray binaries with neutron stars? Qualitatively, we expect that magnetic pumping is at work if the compact object is spinning fast, if its magnetic field is strong, and if the mass transfer rate is, on one hand large enough to tap the spin energy, and, on the other hand sufficiently small that low densities can be maintained in the magnetosphere required for magnetic pumping. In the case of white dwarfs, the magnetic field should not be too large because otherwise, as the observations show, the white dwarf becomes locked to the companion and the spin rate adjusts to the orbital revolution rate. In the case of neutron stars the latter effect is not important because their magnetic moments are less than for white dwarfs. To get some idea of the permitted initial density of non-relativistic particles we put [FORMULA] to obtain (see Eqs. (19) and (28))


for a star of period [FORMULA] in seconds, [FORMULA], and [FORMULA]. Transforming this number into an effective mass accretion rate we find [FORMULA] with [FORMULA] m, and [FORMULA]. This number remains many orders of magnitude below the critical mass accretion rate even for the fastest spinning neutron star. We expect this effective mass accretion rate to depend monotonically on the real mass accretion rate onto the compact star, and to set an upper limit to it if magnetic pumping operates. Further the pumping process and the loop eruption require the particles to be trapped and therefore located inside the light cylinder [FORMULA]. This sets a lower limit to the spin period. Also note in this respect that a trapped cloud of fast electrons near a neutron star is only observable as a synchrotron source at radio frequencies if it is large enough ([FORMULA] m, Kuijpers 1989). Finally from [FORMULA] we obtain an estimate of the characteristic equilibrium Lorentz factor (see Eqs. (19) and (27))


for the same field oscillation amplitude as before. Note that B is the field strength in the source. We intend to elaborate these points in a future paper.

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Online publication: June 30, 1998