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

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5. Energy losses

  An energy increase can only occur and be sustained if acceleration by magnetic pumping energises the system faster than energy is lost. Whilst they are accelerating, electrons can also lose energy by synchrotron losses, Coulomb losses or plasma instabilities of the loss-cone type. This will occur only in restricted regimes of background particle density, field strength and injected particle energy. In addition, the accelerated particles must be prevented in some way from leaving the magnetosphere so that there are sufficient left to account for a flare. We shall assume that plasma instabilities scatter the particles without absorbing much of their energy.

5.1. Coulomb and synchrotron losses

To find the allowed regimes for particle acceleration we solve the equation [FORMULA] for various cases of [FORMULA] and initial magnetic fields. The synchrotron and Coulomb losses for electrons are given by (e.g. Leach & Petrosian 1981)

[EQUATION]

and

[EQUATION]

where [FORMULA] is the classical electron radius, [FORMULA] the Coulomb logarithm, and n the background electron density. We arrive at a series of curves in [FORMULA] space plotted here in Fig. 5. In this figure, acceleration can take place in the regions above and to the left of the curves.

[FIGURE] Fig. 5. Allowed acceleration regimes for [FORMULA] = 1 and various field strengths ([FORMULA] T). A net acceleration will occur in the regions inside the curves.

A critical region is around the energy with which the particles are injected onto the field lines. It must be possible for the particles to begin accelerating right away via the magnetic pumping effect, otherwise we are forced to invoke a first step acceleration mechanism to take them up to some critical threshold energy, at which magnetic pumping can proceed. To examine this question we look at the low energy end of Fig. 5.

[FIGURE] Fig. 6. The low energy end of Fig. 5.  

If we assume that the accreting gas consists of fully ionised hydrogen with components in thermal equilibrium, then the thermal energy of electron and proton components is equal, i.e.,

[EQUATION]

If we further assume that the thermal energy of the gas derives from the infall energy, which is dominated by the more massive proton component, then

[EQUATION]

At a radius of [FORMULA] m for example, the electron thermal velocity in the blob is [FORMULA] m s-1, so the minimum energy it will have as it attaches to the field is then [FORMULA]. It is a minimum energy since by virtue of the rapid rotation of the white dwarf, the velocity of a 'pickup' electron in the frame of the moving magnetic field may be higher, depending on the radius at which it attaches.

So in regions of density of [FORMULA] particles m-3 betatron acceleration (assuming [FORMULA]) of free-falling material may take place. Of course if they have a higher energy, particles may continue to accelerate in higher densities.

5.2. Particle losses and trapping.

 

If particles are to be accelerated by magnetic pumping, they must be maintained in the magnetosphere. Depending on factors such as the particle energy, pitch-angle and position in the magnetosphere, a particle will either precipitate onto the poles (due to synchrotron losses and Coulomb collisions) or be kept in the atmosphere by the magnetic mirror force (due to the converging magnetic field). The competition between energy loss and gain processes (as a function of density, field etc) determines the region where particles lose more energy than they gain, as described in the above section, whilst the magnetic field convergence and the particle energy and pitch-angle determine which particles will enter this region. Let the position [FORMULA] be the position on a given field line at which losses dominate energy gains and particles precipitate out. Only a fraction of the particles will ever reach this critical position, and the loss-cone - the region in pitch-angle space encompassing all particles which will ultimately precipitate - at position r has a half-angle [FORMULA] defined by

[EQUATION]

For magnetic pumping to work efficiently we must be in a regime of efficient scattering, which puts us in the strong diffusion limit. In this limit the loss-cone is always approximately full, with particles being scattered in and out continuously. The escape timescale is then [FORMULA] where [FORMULA] is the time taken to diffuse along the loop, and F is the fraction of solid angle contained in the loss-cone. In a scattering medium this is the time taken to random walk across the structure,

[EQUATION]

where L is the length of the structure which the electrons have to cross, and [FORMULA] is the spatial diffusion coefficient. For small loss-cones, [FORMULA]. Taking an optimum value for [FORMULA] and using [FORMULA] we get

[EQUATION]

We see from Fig. 5 that synchrotron losses limit the maximum Lorentz factor. To derive the conditions on the loops in which particles can be accelerated up to a chosen Lorentz factor we determine the maximum field [FORMULA] allowed from Fig. 5 for that value of the Lorentz factor, and the corresponding acceleration time. This field value is then identified with that at the mirror point [FORMULA]. Equation (33) together with (31) and the requirement that the trapping time exceeds the acceleration time then lead to the (conservative) condition

[EQUATION]

which for the determined values of [FORMULA] and [FORMULA] identifies the smallest loop in the given field structure where acceleration up to the required energy can take place. For a dipolar structure the loop half-length is [FORMULA], where [FORMULA] is the radial distance from the stellar centre.

Let us apply this to particles of [FORMULA]. The maximum field strength (Fig. 5) is 0.005 T. The acceleration time is [FORMULA] s for [FORMULA] and [FORMULA] s for [FORMULA]. Particles initially attaching to the field at radii greater than, respectively, [FORMULA] ([FORMULA] T, [FORMULA]) and [FORMULA] ([FORMULA] T, [FORMULA]) will not precipitate in less than the acceleration time.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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