## 5. Energy lossesAn 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 lossesTo find the allowed regimes for particle acceleration we solve the equation for various cases of and initial magnetic fields. The synchrotron and Coulomb losses for electrons are given by (e.g. Leach & Petrosian 1981) where is the classical electron radius,
the Coulomb logarithm, and
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.
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., 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 At a radius of m for example, the electron
thermal velocity in the blob is m
s So in regions of density of particles
m ## 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
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 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
where is the time
taken to diffuse along the loop, and where 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 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 . Equation (33) together with (31) and the requirement that the trapping time exceeds the acceleration time then lead to the (conservative) condition which for the determined values of and 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 , where is the radial distance from the stellar centre. Let us apply this to particles of . The maximum field strength (Fig. 5) is 0.005 T. The acceleration time is s for and s for . Particles initially attaching to the field at radii greater than, respectively, ( T, ) and ( T, ) will not precipitate in less than the acceleration time. © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |