4. Inclusion of additional loss processes
In itself the "box" model would be of little interest beyond providing a simple "derivation" of the acceleration time scale. Its main interest is as a potential tool for investigating the effect of additional loss processes on shock acceleration spectra. One of the first such studies was that of Webb et al. (1984) where the important question of the effect of synchrotron losses was investigated (see also Bregman et al. 1981). An interesting question is whether or not a "pile-up" occurs in the accelerated particle spectrum at the energy where the synchrotron losses balance the acceleration. Webb et al. (1984) found that pile-ups only occured if the spectrum in the absence of synchrotron losses (or equivalently at low energies where the synchrotron losses are insignificant) was harder than . However Protheroe and Stanev obtain pile-ups for spectra as soft as .
It is relatively straightforward to include losses of the synchrotron or inverse Compton type (Thomson regime) in the model. These generate a downward flux in momentum space, but one which is distributed throughout the acceleration region. Combined with the fact that the size of the "box" or region normally increases with energy this also gives an additional loss process because particles can now fall through the back of the "box" as well as being advected out of it (see Fig. 1). Note that particles which fall through the front of the box are advected back into the acceleration region and thus this process does not work upstream.
If the loss rate is the basic equation becomes
This equation is easily generalised to the case of different loss rates upstream and downstream. Simplifying Eq. (12) gives
Note that for convenience we have dropped the explicit vector (and tensor) notation; all non-scalar quantities are to be interpreted as normal components, that is means etc. Note also that our model differs from that of Protheroe and Stanev in that they do not allow for the extra loss process resulting from the energy dependence of the "box" size.
In the steady state and away from the source region this gives immediately the remarkably simple result for the logarithmic slope of the spectrum,
Note that at small values of p we recover the standard result, that the power-law exponent is .
Under normal circumstances both and are monotonically increasing functions of p. Thus both the numerator and denominator of the above expression, regarded as functions of p, have single zeroes at which they change sign. The denominator goes to zero at the critical momentum
where the losses exactly balance the acceleration. If the numerator at this point is negative, the slope goes to and there is no pile-up. However the slope goes to and a pile-up occurs if
In the early analytic work of Webb et al. the diffusion coefficient was taken to be constant, so that and this condition reduces to in agreement with their results. However if, as in the work of Protheroe and Stanev, the diffusion coefficient is an increasing function of energy or momentum, the condition becomes less restrictive. For a power-law dependence of the form the condition for a pile-up to occur reduces to
(The equivalent criterion for the model used by Protheroe and Stanev is slightly different, namely
because of their neglect of the additional loss process.)
For the case where and with this condition predicts that shocks with compression ratios greater than about will produce pile-ups while weaker shocks will not. In Figs. 1 and 2 we plot the particle spectra up to for a range of values of and with and respectively.
Thus there is no contradiction between the (exact) results of Webb et al. and those of Protheroe and Stanev; the apparent differences can be attributed to the energy dependence of the diffusion coefficient. Indeed, looking at the results presented by Protheroe and Stanev, it is clear that the pile-ups they obtain are less pronounced for those cases with a weaker energy dependence.
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999