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

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4. Acceleration by magnetic pumping

  Magnetic pumping is one of a number of stochastic particle acceleration mechanisms (known also as Fermi-type mechanisms) whose essential property is the combination of a particle interaction with a spatial or temporal change in the magnetic field - which causes the particle energy to change - with a randomising process (Swann 1933, Fermi 1954, Alfvén 1954, Spitzer 1962). Usually a distinction is made between two kinds of magnetic pumping: transit-time and 'collisonal' magnetic pumping (Stix 1992, Berger et al. 1958, Schlüter 1957). In both cases a confining magnetic field is being modulated at frequency f (assumed to be small in comparison with the particle cyclotron frequencies) over a distance d. Transit-time pumping occurs when the bounce time of the particles is approximately equal to the modulation frequency: [FORMULA]. In 'collisional' pumping particles are scattered in pitch angle by an independent process, sometimes but not necessarily by particle collisions.

Here we will investigate the latter process of 'collisional' magnetic pumping for arbitrary large field modulations, now however invoking unspecified collisionless wave-particle scattering instead of collisional pitch angle scattering, so that acceleration instead of heating occurs. This complements Kulsrud & Ferrari (1971) who calculated acceleration by small-amplitude slowly-varying large-scale hydromagnetic turbulence in a uniform background field in the presence of pitch-angle scattering. It also complements the 'gyro-relaxation' by Schlüter (1957) and the collisonal magnetic pumping by Berger et al. (1958), both for a one-temperature gas and small amplitude variations of a uniform background field. In the Discussion at the end we briefly discuss the effects of transit-time pumping.

Although particles may gain or lose energy in the interaction, randomisation ensures that on average an increase occurs. In the case of magnetic pumping, an increase in the magnitude of the magnetic field causes the particle momentum perpendicular to the magnetic field to increase, by conservation of the first adiabatic invariant [FORMULA] (11). If there is no other process to change the particle momentum, a decrease of the field to the old value results in the particle losing exactly the amount of energy it gained. However if it can be arranged that the particle scatters, moving some of its newly gained perpendicular energy into the parallel direction, decompression of the field (which only affects the perpendicular component) leaves the particle with slightly more energy than it had to begin with. On average the increase in energy ([FORMULA] in units of the rest-mass energy [FORMULA]) by pumping of angular frequency [FORMULA] is given by

[EQUATION]

where [FORMULA] is a small coefficient and the field varies periodically between the values B and [FORMULA]. The value [FORMULA] is the average increase in the particle population energy with time due to the magnetic pumping process; [FORMULA] depends primarily on the ratio of field oscillation time to pitch-angle scattering timescale.

In a few illustrative cases (e.g. Alfvén & Fälthammar 1963, Melrose 1980), [FORMULA] can be estimated from heuristic considerations, but we shall approach it numerically, deriving it from the evolution for the distribution function [FORMULA] of particles undergoing pumping. From (11) the equation for individual particle trajectories in a time-varying magnetic field is

[EQUATION]

and in the absence of other processes the continuity equation for the distribution function [FORMULA] (averaged over gyration phase) of many such particles is

[EQUATION]

If we rewrite this equation in terms of [FORMULA], where [FORMULA], [FORMULA] is the pitch-angle, now including a term representing particle scattering, the distribution evolution equation becomes (correcting Kirk 1995)

[EQUATION]

This represents an intrinsically reversible energisation process, made irreversible by the isotropisation of the energised particles in the period between compression and re-expansion of the field. If we solve for the particle distribution function, we may fit the results to the averaged form for energy increase (19), thus identifying the coefficient [FORMULA] for different cases.

We solve equation (22) with a scattering term

[EQUATION]

where [FORMULA], the diffusion coefficient, is the inverse of the scattering timescale. At present we have neither energy nor pitch-angle dependence of the diffusion coefficient.

The solution of equation (22) is accomplished by means of the method of stochastic simulations (MacKinnon & Craig 1991) which has been used recently in the solution of transport equations for the case of electrons propagating in the solar corona and chromosphere (Fletcher 1995, Fletcher & Brown 1995) and of cosmic rays accelerated at shocks (Achterberg & Krülls 1992, Krülls & Achterberg 1994). The method has been described in some detail in the above references, and here we state only a few important points. It is based on the formal mathematical equivalence of a Fokker-Planck type of equation (such as (22)) with a set of stochastic differential equations describing the motion of individual particles and is similar to a Monte Carlo formulation in implementation, in that the evolution of the particle distribution function is calculated by generating many realisations of individual particle orbits, and binning the results. The difference with a conventional Monte Carlo simulation lies in the interpretation of diffusion terms. Whilst a Monte Carlo calculation simulates individual collisions, the stochastic method scales the mean free path and collision frequency in a manner which is entirely determined by the physics which appear in the transport equation (cf Achterberg & Krülls 1992), making it a very 'physical' method, as well as extremely easy to implement computationally. As the scattering is a Gaussian process, the error on each bin of the histogram is [FORMULA], for a bin containing N particles. In the calculations we shall impose an oscillating magnetic field on a population of test electrons in a background of electrons and ions. The test electrons represent electrons that have been injected into, or have attached themselves onto, the magnetic field of the white dwarf.

We can consider two categories of acceleration. The first is standing compressional oscillations, and the second travelling compressional oscillations. In general both standing and travelling oscillations can be excited by the companion as well as by the infalling gas. Large oscillations are generated by the field lines opening and closing around the companion and can be expected to cause a standing wave pattern. However, the passage of a stream of smaller blobs of material through the field can also lead to standing oscillations in the inner magnetosphere if a resonance condition exists, and the spin period and the Alfvén travel time along the field lines have a rational relation. As has been demonstrated, the power in the latter can easily be 100 times that in the former. We investigate the possibility that the acceleration occurs by oscillations at the spin frequency, either as standing or as travelling magnetic compressions.

4.1. Acceleration by standing magnetic compressions.

Standing magnetic compressions are caused by the large-scale oscillations of the field as it sweeps by the companion. The field oscillation is purely transverse - there is no component of the wave vector along the field. If we restrict our considerations to the non-relativistic regime, Eq. (19) becomes

[EQUATION]

which is linear in a plot of [FORMULA] versus t, with gradient [FORMULA], to which we can easily fit numerical results.

In Fig. 2 we plot the mean energy increase with time for 5,000 numerical particles in a few runs, demonstrating clearly the linear relationship found by the stochastic simulation.

[FIGURE] Fig. 2. The increase of energy with time by homogeneous magnetic pumping in the non-relativistic regime, for 3 values of the pitch-angle scattering to field oscillation time. The straight lines are fits to the numerical data, given by the points. [FORMULA] is calculated given the field parameters indicated.

Repeating a number of times we generate Fig. 3, showing [FORMULA] against [FORMULA].

[FIGURE] Fig. 3. The parameter [FORMULA] as a function of the ratio of pitch-angle scattering time to field oscillation time.

Note that, although we have chosen specific values of [FORMULA] and [FORMULA] for this calculation, the values of [FORMULA] found are robust to changes in these parameters.

In this type of acceleration, similar to second-order Fermi acceleration, we find that a particle initially in the non-relativistic energy regime spends much of its time being accelerated to relativistic energies, thereafter the increase to ultra-relativistic energies happens very quickly.

It is evident that the efficiency of particle acceleration reaches a maximum at [FORMULA] for [FORMULA]. Alfvén & Fälthammar (1963) find a value of [FORMULA], half as large, in an idealised situation in which scatterings only occur at the precise minima and maxima of the field. This type of wave can, for [FORMULA] accelerate particles to relativistic energies within a few thousand seconds starting from their characteristic free-fall energy down the potential well of the white dwarf. However, in a high field environment the energy loss of the particles in synchrotron emission should be included. This may result in an equilibrium between energy gain and loss by electrons, and is investigated in §   5.

4.2. Acceleration by travelling magnetic disturbances.

Here we look at travelling waves, which are typically those caused by the infall of blobs of material. We envisage that a train of Alfvén and compressional waves is set up in the ambient field by the opening and closing of the field around a blob. As one might imagine, the fact that the blobs are moving through a field of varying magnitude means that there will be a whole spectrum of waves set up in the atmosphere, dependent on position and time. The frequency of the oscillation is now determined by the blob size l and the local Alfvén speed (if we neglect the speed of sound in the magnetosphere and take [FORMULA])

[EQUATION]

However to keep things simple we shall give here values of the efficiency of acceleration in the atmosphere at the spin frequency [FORMULA] but at a number of values of the parallel wave number [FORMULA] corresponding effectively to a number of values of Alfvén speed, which can be associated with regions of particular density at a particular position.

We model the field oscillation as a travelling wave of form

[EQUATION]

where [FORMULA] determines the amplitude of the field oscillation. Since the wave is travelling the effective oscillation time (the time which elapses between a particle meeting field compressions) is Doppler shifted and depends not only on the oscillation frequency but also on the particle velocity along the field. In this case we do not obtain a linear relationship at low energies. As particle velocity increases towards the speed of light, the relationship tends to a linear one. By comparing Figs. 2 and   4 it can be seen that at low energies acceleration by travelling waves can be more efficient than that by standing waves, but that the efficiency decreases as the particles become relativistic. Long wavelength oscillations are more efficient accelerators than short wavelength oscillations. The efficiency eventually tends to that of standing (infinite wavelength) oscillations.

We now turn our attention to the energy losses of the particles which have to be overcome by the acceleration.

[FIGURE] Fig. 4. The increase of energy with time of particles accelerated by travelling magnetic compressions.
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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