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Astron. Astrophys. 328, 107-120 (1997)

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3. EPs energy spectrum

Apart from the EP composition, the energy spectrum of the EPs must be specified in order to calculate the gamma-ray line shapes and ratios from EP interactions in the cloud material. In the case of Orion, however, no decisive information can be drawn from the gamma-ray line shapes. This is due to the poor energy resolution of the instruments and to the ambiguity on the EP composition itself. On the other hand, the gamma-ray emission at higher energy ([FORMULA] MeV), due to [FORMULA] decay principally, has been measured first by COS-B (Bloemen et al. 1984), and most recently by EGRET (Digel et al. 1995) and was found compatible with the GCR flux observed at Earth. This strongly suggests that the energy spectrum of the extra EP component breaks at an energy lower than the [FORMULA] production threshold of a few hundreds MeV/n.

It is quite remarkable that independently of these phenomenological considerations, and actually before the observations by COMPTEL of the gamma-ray line emission in Orion, Bykov & Toptygin (1990) and Bykov & Fleishman (1992) presented a model of `non-thermal particle generation in superbubbles' leading to hard spectra of energetic nuclei up to tens or hundreds of MeV. Their calculated spectra (see also Bykov 1995) are rather similar to that adopted by Ramaty and co-workers on the basis of shock acceleration calculations (e.g. Ellison & Ramaty 1985) with a finite acceleration time or a finite shock size:


As emphasized e.g. in RKL96, this analytical form allows one to explore a variety of phenomenologically possible spectra by varying one single parameter, namely the `break energy' [FORMULA]. Therefore we adopt this formalism, varying [FORMULA] from 2 to 100 MeV. The normalisation coefficients [FORMULA], are proportional to the abundances by number of each isotope i, according to the compositions discussed in Sect.  2. Finally, we normalise the EP spectra so as to reproduce the detected flux from Orion in the range 3-7 MeV, i.e. [FORMULA] (Bloemen et al. 1997).

Knowing the injection function [FORMULA], one can derive the `propagated' energy spectrum [FORMULA] of EP species i from the usual transport equation:


where [FORMULA] is the total lifetime of nuclei i against catastrophic losses (escape, nuclear destruction and/or decay). In the steady-state approximation, we have:


In Fig. 1, we show the energy distribution of different nuclei, [FORMULA], representing the amount of energy between E and [FORMULA]. For clarity, all the nuclear species have been given the same abundance at injection, so that the differences between nuclei are only due to their different energy losses, as a consequence of their different [FORMULA] ratio. We also plot the same quantity for the injection spectrum, i.e. [FORMULA].

[FIGURE] Fig. 1. Energy content of the propagated EP spectrum for different nuclei and for [FORMULA] MeV/n. We also show the same quantity for the injected spectrum (dashed line), with arbitrary relative scale. All the nuclear species have the same abundance at injection.

Although the injected energy is a monotonically decreasing function of energy, it appears that the energy of the propagated EPs is concentrated around the break energy of the injection spectrum. This indicates that the gamma-ray emission is mainly governed by the relative behaviour of the different cross sections of interest at energies around the break energy.

Because of their interactions in the surrounding medium, the EPs suffer ionisation energy losses, at a rate:


The discrete sum is over all EP species, [FORMULA] is the mean density of the interaction region, [FORMULA] is the velocity of the particles, and [FORMULA] is the ionisation energy loss per g.cm-2.

The energy loss rate of the EPs and the gamma-ray production rates are both proportional to the ISM density (see Eq.  7). Therefore, the normalisation to the Orion fluxes allows us to obtain absolute (i.e. density independent) energy loss rates, which are shown in Fig. 2 as a function of the break energy [FORMULA] of the injection spectrum. For the sake of clarity, we only plot the results for a few representative mean-wind compositions, including the solar system (SS) and late-WC compositions, for comparison. The gamma-ray production efficiency is higher for larger values of [FORMULA], since: i) more particles are above the excitation thresholds and ii) the ionisation energy losses are lower at higher energies. However, this effect is compensated by the decrease of the excitation cross sections above [FORMULA] MeV/n, which explains the flatening of the curves in Fig. 2.

[FIGURE] Fig. 2. Ionisation energy losses (in units of [FORMULA] as a function of the break energy [FORMULA], for different EP compositions. For example, 60/E refers to the mean-wind of a [FORMULA] star with model E. The target composition is solar.

In a general way, all the compositions that we consider provide gamma-ray emission efficiencies intermediate between the solar composition (the worse) and the late-WC composition (the best). This results from the higher proton and helium abundances in our mean-wind and mean-OB compositions as compared to the late-WC case, as discussed in Sect.  2. Indeed, since we normalise the EP energy spectrum to the gamma-ray flux emitted in the 3-7 MeV band, i.e. in first approximation to the 12 C and 16 O line fluxes (see Sect.  6), the highest efficiency is obtained for compositions having the highest C and O abundances.

It should be realised from Fig.2 that the gamma-ray fluxes detected from Orion actually imply a very high energy consumption within the cloud complex, namely [FORMULA] in the best case. This represents an energy of [FORMULA] erg over a period of [FORMULA] years. However, if the gamma-ray emission in Orion is to be attributed to the excitation of energetic C and O nuclei, one has to admit that nature actually manages to supply such a mechanism. Taking this as a fact, we shall focus on the implications of the detected Orion flux. The energetics will be discussed in a separate paper. Nevertheless, the energetic considerations favour values of [FORMULA] greater than about 20 MeV/n. Moreover, they tend to exclude hydrogen and helium rich EP compositions, such as SS and probably also CRS compositions, as shown by Ramaty et al.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998