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Astron. Astrophys. 353, 729-740 (2000)

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4. Bremsstrahlung radiation

4.1. Efficiency

We define bremsstrahlung efficiency of a beam as the bremsstrahlung power emitted per unit beam power input. As discussed further below the answer depends somewhat on the photon energy ([FORMULA]) range considered, on the spectrum emitted, and on the bremsstrahlung cross-section approximation adopted. Here, however, we are mainly interested in comparing the thick target neutral beam bremsstrahlung efficiency with that of a thick target electron beam. Therefore, so long as we consider essentially the same energy ranges and use the same cross-section in both cases, these issues should be secondary. For simplicity in this comparison we will therefore use the Kramer's bremsstrahlung cross-section:

[EQUATION]

At first sight the neutral beam model might be expected to be `efficient' in that the electrons are at least quasi-thermal, and pure thermal bremsstrahlung sources can be very efficient. This is, however, only the case for the ideal isolated (iso)thermal source for which radiation is the only energy loss. In real non-isothermal sources there are power losses by conduction and escape. In the present case the situation is that the hot electrons only remain so because they are continuously heated by extracting energy from the protons to offset the collisional deceleration they would otherwise undergo in collision with the cold background. One can in fact think of the beam electrons as producing their bremsstrahlung in a cold thick target background but with continuous driving by the protons. As we have seen, due to the warm target effect, however, this transfer of energy from the protons to the electrons is rather inefficient and we may consequently expect a rather high beam power requirement to get enough into the beam electrons for thick target HXR production - i.e. a rather low efficiency.

Calculation of the radiation efficiency is complicated by the fact that the actual distribution of the beam random electron energies is unknown. In Sect. 3 we estimated only a mean value [FORMULA] and its evolution, though we also observed on the basis of our simulations that the distribution possibly has a quasi-Maxwellian form. We will therefore proceed by considering the efficiency in the two limiting cases where the local electron distribution is treated as monoenergetic (delta-function at [FORMULA]) and as Maxwellian with [FORMULA].

4.1.1. Mono-energetic [FORMULA]

Assuming that radiation of beam electrons is due to collisions with background plasma protons, the power in the bremsstrahlung emission, per unit photon energy [FORMULA], from the entire length of the beam, for neutral injection flux [FORMULA], can be expressed as

[EQUATION]

where the time interval is governed by the proton deceleration, spanning the range of proton energies [FORMULA] from 0 to that at which [FORMULA] first exceeds [FORMULA], namely by (5)

[EQUATION]

(Note, in this monoenergetic case [FORMULA].) Thus, writing [FORMULA] we get, using (4)-(7), for the HXR spectral luminosity

[EQUATION]

where [FORMULA] is the beam power and [FORMULA].

The most natural definition of neutral beam bremsstrahlung efficiency [FORMULA] in this case is the ratio of the total bremsstrahlung power (all [FORMULA]) to [FORMULA], namely

[EQUATION]

where B is the beta function.

We want to compare this with the efficiency of a cold thick target electron beam. In particular, we will compare it with the efficiency of a monoenergetic electron beam of energy [FORMULA] at injection since this produces a photon spectrum, like (13) restricted to [FORMULA]. The result is

[EQUATION]

Thus the relative efficiency of the neutral beam is by (13) and (14)

[EQUATION]

independent of [FORMULA]. This figure bears out our earlier expectation that [FORMULA] but in fact the ratio is not very small and makes the neutral beam model of considerable interest.

Result (15) says that we need 5 times more power in a monoenergetic neutral beam to produce a prescribed bremsstrahlung flux than we need in an electron beam. On the one hand this suggests that neutral beams are not a serious candidate to explain HXR burst intensities from HXR-rich flares since to explain these with an electron beam already requires a beam power of the order of the total flare power. That is the neutral beam compounds the beam power problem for such events, though it does reduce the beam number flux required by a factor of [FORMULA]. On the other hand, result (15) also means that a [FORMULA]MeV neutral beam carrying a power of order the flare power (i.e. able to act as a flare energy transport mechanism) will produce an easily detectable HXR burst in the [FORMULA] keV range.

Result (15) must be treated with some caution, however, in that we are not really comparing like with like. All we have done is to compare the photon energy integrated bremsstrahlung from monoenergetic electron and neutral beams. But, apart from having the same upper cut off ([FORMULA]), the bremsstrahlung spectra from those two models are not the same - that from a monoenergetic electron beam is [FORMULA] [FORMULA] which is not the same as (10) from a monoenergetic neutral beam. To discuss the relative efficiencies of the two models in a reliable way, we really want them to produce the same photon spectrum as well as total luminosity. This is made difficult by two facts - that we do not really know the electron energy distribution about [FORMULA] for the case of a monoenergetic neutral beam. Second, it is not clear how to deal with more general neutral beam injection spectra which may be required to match HXR spectral observations - cf. Sect. 4.2. To deal with the first problem, we examine the effects of replacing a monoenergetic [FORMULA] distribution at [FORMULA] with a local Maxwellian at [FORMULA].

4.1.2. Maxwellian [FORMULA]

The bremsstrahlung spectrum from a thermal source in the Kramer's approximation can be written

[EQUATION]

where [FORMULA] is the differential emission measure function which in our 1D case is

[EQUATION]

where [FORMULA] is the beam electron density given by [FORMULA], and [FORMULA] is the background proton density, since in our problem the bremsstrahlung is from collisions of hot beam electrons with background ions (assuming the latter to be denser than the beam). Writing [FORMULA] and using (1)-(3) we get

[EQUATION]

where [FORMULA] and the total emission measure is

[EQUATION]

Fig. 7 shows [FORMULA] which emphasises that the differential emission measure is concentrated near [FORMULA].

[FIGURE] Fig. 7. Emission measure of the beam electrons differential in temperature as a fraction of the total emission measure ([FORMULA]) vs. beam electron `temperature' as a fraction of the maximum (final) temperature ([FORMULA]).

Inserting (18) in (16) and integrating over [FORMULA] we find the neutral beam radiation efficiency in this case to be

[EQUATION]

so that now, using (14), the relative efficiency is

[EQUATION]

which is essentially the same as when [FORMULA] was regarded as monoenergetic at [FORMULA].

4.2. Spectrum

Inserting (18) in (16) allows us to compute the spectrum [FORMULA] expected from a monoenergetic neutral beam when [FORMULA] is taken to have a local Maxwellian distribution of [FORMULA], namely:

[EQUATION]

The result is shown in Fig. 8 along with the spectra for the monoenergetic case (13) and also of an isothermal source, with the same total emission measure, all at [FORMULA]. The thermal and isothermal spectra are very similar since [FORMULA] peaks sharply at [FORMULA] while the monoenergetic spectra cuts off at [FORMULA] because no Maxwellian tail exists in this case. The overall behaviour of the thermal case is that [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA].

[FIGURE] Fig. 8. Spectrum of hard X-rays under various assumed initial beam distributions: thermal (solid), iso-thermal (dashed), power law (dotted), and monoenergetic (dashed-dotted).

Except over a limited range, none of these spectra resemble observed HXR spectra which are well approximated as power laws. In order to compare accurately the efficiency of the neutral beam with an electron beam we should really do so for sources which yield the same HXR spectrum, and for spectra similar to those observed. The natural way to think of getting a `broader' spectrum from the neutral beam is to have a range of [FORMULA] (such as a power law), rather than a single [FORMULA] at injection. This poses a physical problem if the distributed [FORMULA] occurs across the area A for then, along each beam path, there is a range of proton speeds and it is not clear how this modifies the electrostatic/charge separation arguments on which our neutral beam model has been based. Electrodynamics does not demand that each proton be accompanied by an electron of the same longitudinal speed but only that the mean longitudinal speed of the beam electrons (or electron current) equals the mean proton speed (current). If in fact the electrons do follow the protons on a one-to-one basis then the resulting HXR spectrum, and the radiation efficiency, can be found simply by integrating our previous [FORMULA] results over a distribution [FORMULA], but establishing whether or not this is the case physically is beyond the aims of our present study. However, one situation where this approach is valid is where the neutral beam comprises monoenergetic proton beams along each path in A but with a distribution of [FORMULA] across A. We therefore here evaluate the spectrum and efficiency in this situation for the particular case where the total beam power injected is distributed in [FORMULA] over A with a power law form - that is, with [FORMULA], the beam power per unit [FORMULA] given by

[EQUATION]

where [FORMULA] is the total power at [FORMULA]. If we then replace P in (13) by [FORMULA] from (23) and integrate over [FORMULA] we obtain (treating [FORMULA] as locally mono-energetic at [FORMULA])

[EQUATION]

[EQUATION]

where [FORMULA]. Thus a power law flux in [FORMULA] of index [FORMULA] yields a power law flux [FORMULA] of index [FORMULA] just as for a thick target electron beam for which the result is explicitly

[EQUATION]

where [FORMULA] is the electron beam flux above [FORMULA]. To yield exactly the same flux and spectrum, i.e. [FORMULA] for all [FORMULA] this implies a relative efficiency

[EQUATION]

which is shown in Fig. 9 (solid curve) as a function of [FORMULA].

[FIGURE] Fig. 9. Relative efficiency of neutral beam hard X-ray production (compared to an electron beam) vs. power law index [FORMULA] of initial beam distribution.

We see that the relative efficiency is of the same order as found for the monoenergetic case for small [FORMULA] but increases somewhat for larger [FORMULA]. We note, however, that we are again not quite comparing like with like here. The electron power law cut-off below [FORMULA] yields a photon spectrum [FORMULA] with [FORMULA] at [FORMULA] and [FORMULA] at [FORMULA]. The neutral beam power law cutoff at [FORMULA] yields exactly the same spectrum (with [FORMULA]) for [FORMULA] but for [FORMULA] produces a photon spectrum of the same form (13) as a monoenergetic neutral beam which is not [FORMULA] but rather falls below the spectrum emitted by the electron beam case.

We can also compare the electron beam with the neutral beam where allowance is made for the local spread in the neutral beam [FORMULA] by assuming this distribution to be Maxwellian. That is, we insert (23) in (16) and (17) and obtain

[EQUATION]

so that here, comparing with the electron beam (25) we get

[EQUATION]

which increases with [FORMULA] faster than (26) by the factor [FORMULA] and [FORMULA] from (15) becomes [FORMULA] for large [FORMULA]. Here, however, the problem of comparing like with like becomes very serious because (27) is only a power law above [FORMULA] if [FORMULA] is a power law down to [FORMULA] for which the total beam power is infinite. This is also true of the electron beam if one requires [FORMULA] for all [FORMULA] and we are then comparing two infinite quantities.

Since [FORMULA] is measured only over a finite interval and with finite accuracy, we can usefully ask how much [FORMULA] differs (e.g. at [FORMULA]) between power laws [FORMULA] with and without a cut-off - i.e. cut-off at [FORMULA] and at [FORMULA]. The relevant expression is

[EQUATION]

which is shown in Fig. 10 as a function of [FORMULA] for [FORMULA]. We see in fact the spectra differ enormously for [FORMULA], i.e. for photon energies below the final electron energies [FORMULA] corresponding to the cutoff in proton energy [FORMULA]. The difference is more pronounced for larger [FORMULA], and is only small when [FORMULA]. Thus the efficiency comparison (28) is not really meaningful in terms of comparing electron and neutral beams producing similar photon spectra, other than in the unphysical limit of [FORMULA].

[FIGURE] Fig. 10. Comparisons of the HXR spectra resulting from a power-law neutral beam injection spectrum, with a locally Maxwellian beam electron distribution. The curves are plots of [FORMULA] from (29) as functions of [FORMULA] for spectral indices [FORMULA].

In summary, what these results show is that the exact bremsstrahlung spectrum produced by a neutral beam is rather sensitive to the spread in the local electron energies [FORMULA] along the beam path and that the actual efficiency will need to be computed using more elaborate treatments of the distribution functions (including electrodynamics). However, from the cases where we have managed to compare the efficiencies of the neutral and electron beams in a convincing manner in our approximations to the [FORMULA] distribution, the efficiency of neutral beams seems to be of order 20% of the electron beam efficiency.

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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