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Astron. Astrophys. 354, 423-430 (2000)

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3. CR propagation and energy losses

The types of propagation of CREs in a galaxy are diffusion, i.e. the CREs are scattered randomly by magnetic field irregularities, and convection, a systematic movement of the scattering centres outwardly. While propagating through a galaxy, CREs suffer energy losses. Eventually, CREs might escape from the galaxy.

In order to discuss the radio spectral index in the various cases, we calculate two simple models, one that considers diffusion and one in which we study convection. In both models we take into account energy losses by synchrotron and inverse Compton losses. Here, we ignore adiabatic and bremsstrahlung losses which might also be important, especially in starburst galaxies, but since they do not change the spectral index, they are in the framework of the present study equivalent to no energy losses at all.

3.1. Diffusion model

The diffusion model can be described by the following equation:

[EQUATION]

where z is the height above the galactic midplane. We assume that the sources of CRs are situated in the midplane at [FORMULA] and that their energy dependence is given by a power-law according to Eq. (2). The synchrotron and inverse Compton losses, [FORMULA], are assumed to be spatially homogeneous and are given by:

[EQUATION]

[FORMULA] is the energy density of the radiation field below the Klein-Nishina limit (e.g. Longair 1992). We adopt a free escape boundary at [FORMULA], the outer boundary of the halo, so that [FORMULA]. An approximate solution is given by (for further details see Lisenfeld et al. 1996a):

[EQUATION]

with

[EQUATION]

[FORMULA] denoting the confluent hypergeometric function (or Kummer function), and

[EQUATION]

and

[EQUATION]

where

[EQUATION]

3.2. Convection model

The convection model is described by the following equation:

[EQUATION]

where V is the convection speed, assumed here to be constant. The solution of this equation can be obtained with the method of characteristics:

[EQUATION]

3.3. Asymptotic spectral indices

The diffuse synchrotron emission is calculated from [FORMULA] in the same way as for SNRs (Eqs. (7) and (8)). The total diffuse synchrotron emission is obtained by integrating [FORMULA] over the extent of the halo, [FORMULA], which in the case of the diffusion model coincides with the free-escape boundary, [FORMULA].

We define the escape probability, [FORMULA], as the ratio of the measured total nonthermal radio emission (including the diffuse synchrotron emission and the radio emission from SNRs), [FORMULA], to the maximum possible nonthermal radio emission, [FORMULA], that would be measured if the halo were infinitely large.

The expected diffuse and nonthermal spectral indices for different asymptotic cases are:

Case (1): Synchrotron and inverse Compton losses are very strong so that no escape is possible. Then, independent of whether diffusion or convection is the principle mode of propagation, we have [FORMULA]. Including the contribution of SNRs, the spectral index becomes (see Fig. 1) [FORMULA]. For a lower injection spectral index, [FORMULA], we would get [FORMULA].

Case (2): CREs can escape freely from the galaxy by diffusion without suffering synchrotron and inverse Compton losses. The diffusion coefficient can be energy dependent, [FORMULA]. For our Galaxy, µ has been determined from the energy-dependence of the fraction of secondary to primary CR nuclei. For CRs with energies above a few GeV, the energy range relevant for radio observations in the GHz range, [FORMULA] has been found (García-Muñoz et al. 1987). The expected diffuse spectral index is then, adopting [FORMULA], [FORMULA], and including the contribution of SNRs it becomes [FORMULA], with corresponding reductions for [FORMULA].

Case (3): CREs propagate by convection and do not suffer considerable synchrotron or inverse Compton losses. In this case we expect [FORMULA] for [FORMULA]; the spectral index remains unchanged when including the emission from SNRs.

3.4. The spatially integrated spectral index

The results of the models are shown in Fig. 2 where the nonthermal spectral index (assuming a contribution of SNRs of 10% at 1.5 GHz) is plotted as a function of the escape probability for the case of diffusion and of convection. The values for [FORMULA] predicted by the two models are compared to the observed mean nonthermal spectral index which is about [FORMULA]. This observational value is obtained as the average of the results from results by Klein (1988) ([FORMULA]) and Niklas et al. (1997) ([FORMULA]) who determined the average [FORMULA] of samples of galaxies by subtracting the thermal radio emission from the observed total radio emission.

[FIGURE] Fig. 2. The nonthermal spectral index between 1.5 and 5 GHz, assuming [FORMULA] and [FORMULA], is shown as a function of the escape probability [FORMULA]. The full line is calculated for a convection model, the dashed line for a diffusion model with [FORMULA] and the dotted line for a diffusion model with [FORMULA]. The full and dashed-dotted horizontal lines give the observational value for [FORMULA] and its dispersion, calculated as the average of the results by Niklas et al. (1997) and Klein (1988). For [FORMULA] all model curves would be lower by about 0.1.

Within the pure diffusion model with [FORMULA] the observed [FORMULA] is explained by a high escape probability ([FORMULA]). For a weaker energy dependence of the diffusion coefficient, [FORMULA], however, the predicted escape probability is lower, [FORMULA]. The conclusion from the convection model is very different: Here the observed [FORMULA] is explained by [FORMULA]. The different conclusions with respect to the escape probability given by the diffusion and the convection model are due to the fact that (i) convection is an energy independent process and (ii) in the case of convection the relation between the spectral index and the escape probability is not linear, in the sense that low escape probabilities are predicted for almost the whole range of spectral indices.

We conclude that from the observed average radio spectral index of galaxies it can neither be decided to which degree CREs can escape from galaxies nor whether diffusion or convection is the dominant mode of propagation. Only for asymptotic cases partial conclusions can be drawn: If the spectral index of a galaxy is very steep ([FORMULA]) energy losses are important and the escape rate is low. Yet, it cannot be determined whether diffusion or convection is the dominant type of propagation. A very flat overall spectral index ([FORMULA]), on the other hand, indicates strong convection or almost energy independent diffusion. Since measurements in our Galaxy indicate that diffusion does depend significantly on energy ([FORMULA], García-Muñoz et al. 1987), convection is the more likely explanation. In this case, however, we cannot draw any conclusions with respect to the confinement of CREs.

3.5. Spatially resolved spectral index

The question of how important energy losses are for the propagation of CREs in a galaxy can be decided if the spectral index distribution in the halo is known. If the spectrum steepens considerably away from the disk, it is a clear sign that CREs suffer considerable synchrotron and inverse Compton losses as they propagate outwardly. This is illustrated in Figs. 3 and 4, where we show the diffuse radio spectral index, [FORMULA], calculated from the convection and diffusion models of Sect. 3.1 and 3.2, as a function of the distance from the galactic plane, z. In Fig. 3 we show the results for a large halo ([FORMULA] kpc) in which the energy losses affect the CREs considerably before they manage to escape. The escape probabilities are here 9% for convection and 6% ([FORMULA]), respectively 10% ([FORMULA]) for diffusion. In Fig. 4 the opposite case is shown: A small halo ([FORMULA] kpc) where the CREs can escape without suffering significant energy losses, so that the escape probabilities lie between 78% for convection and 85% ([FORMULA]), respectively 87% ([FORMULA]) for diffusion. We find that important energy losses are reflected clearly in a steepening of the spectral index with increasing distance from the plane. On the contrary, if the CREs can escape almost freely, or if only adiabatic or bremsstrahlung losses occur, no such steepening is seen.

[FIGURE] Fig. 3. The diffuse radio spectral index as a function of the height over the galactic plane, z, in the case of important energy losses. The full line is calculated for a convection model, the dashed line for a diffusion model with [FORMULA] and the dotted line for a diffusion model with [FORMULA]. We have assumed a magnetic field [FORMULA]G, a radiation field of energy density 1 eV cm-3, convection velocity [FORMULA] km s-1 and diffusion coefficient [FORMULA] cm2 s-1.

[FIGURE] Fig. 4. The diffuse radio spectral index as a function of the height over the galactic plane, z, in the case of almost free escape. The line coding and parameters are as in Fig. 3.

3.6. Comparison to observations: Are energy losses important?

The distribution of the spectral index in the halo is a definite diagnostic of whether energy losses are important. The observational study of radio halos in external galaxies is difficult, because the halos are extended and intrinsically faint (see Dahlem 1997). Therefore, not many galaxies with multi-frequency radio data for the halo are known.

One of the best studied radio halos is that of the spiral galaxy NGC 891 (Hummel et al. 1991). Its spatially integrated nonthermal radio spectral index is 0.78 (Niklas et al. 1997), only slightly below the average spectral index of galaxies (see Fig 2). In Fig. 5 we show its radio spectral index distribution in the halo. The data are at a resolution of 40", corresponding to 1.8 kpc at the distance of 9.5 kpc of NGC 891. The spectrum steepens very quickly within the first 2-3 kpc. It continues to steepen more slowly outside this range, especially in the eastern side of the halo. Here, the steepening becomes more pronounced at [FORMULA] kpc.

[FIGURE] Fig. 5. A comparison of our model results to the data of NGC 891 (Hummel et al. 1991). The triangle refer to the spectral index in the western part of the halo, and the stars to the eastern side of the halo. The full line corresponds to the convection model and the other lines to the diffusion model with [FORMULA] and different values for the diffusion coefficient and thus escape probability (see text).

In Fig. 5 we also show the results of our model in which we have included, in order to correspond to the data, apart from the diffuse radio emission, the contribution from SNRs and the thermal radio emission. For the latter two components we have assumed, [FORMULA], as before, and a contribution of the thermal radio emission of 10% at 1.5 GHz, a typical value for normal galaxies (Condon 1992). We furthermore assume that the thermal radio emission and the radio emission from SNRs is restricted within 100 pc above the disk (the exact value of this spatial extent is not important because of the low resolution of the data). Finally, we convolve the model results with a Gaussian beam of HPBW 1.8 kpc, corresponding to the resolution of the data.

We have adjusted the diffusion coefficient and convection velocity such that we achieved the best possible fit to the data. Because of the simplicity of the models described here, we cannot expect to describe the data in all detail, we are aiming rather at a qualitative comparison with the main features. For a quantitative comparison a more complex model would be necessary, combining diffusion and convection, taking into account spatial variations of the parameters ([FORMULA], etc.) and describing more realistically the distribution of SNRs and the contribution of the thermal radio emission. Nevertheless, the diffusion model is able to describe a large range of the data reasonably well. The fast steepening of the spectral index within the first 2-3 kpc above the disk can to a large extent be attributed to the contribution of SNRs and the thermal radio emission close to the disk. The steepening outside this range requires the presence of dominant energy losses: The diffusion models that describe the data well have escape probabilities of 11% (dotted line) and 0% (dashed line). For comparison a model with a high escape rate is shown (78% - dashed-dotted line) which is not able to describe the data. In order to reproduce the fast steepening of the spectral index at [FORMULA] kpc in the western part of the halo, we would possibly have combine the diffusion model with the convection model.

Radio halos have been observed in several other galaxies, although mostly not in such detail. In all of these galaxies a steepening of the spectral index at least up to [FORMULA] is found with increasing distance from the disk. This is the case for both starburst galaxies, as M 82 (Seaquist & Odegard 1991; Reuter et al. 1992); NGC 253 (Carilli et al. 1992); NGC 2146 (Lisenfeld et al. 1996b); NGC 4666 (Sukumar et al. 1988; Dahlem et al. 1997) and non-starburst spiral galaxies like NGC 4631 (Hummel & Dettmar 1990); NGC 5775 (Duric et al. 1998); NGC 5055 and NGC 7331 (Hummel & Bosma 1982). We conclude therefore that for all galaxies for which a multi-frequency radio halo has been observed, energy losses are important and it seems likely that this is generally the case in galactic halos.

Niklas & Beck (1997) have found a trend that actively star-forming galaxies tend to have lower spectral indices and more quiescent galaxies steeper ones. They interpret this as an indication that in galaxies with a high star-formation efficiency (SFE) CREs escape more easily due to a galactic wind. On the basis of the above consideration, however, it seems more likely that in these galaxies convection causes the flat spatially integrated spectrum. Energy losses play an important role also in galaxies with a high SFE which is shown by the steepening of the spectral index in the halo. A good example is the starburst galaxy M 82. For this galaxy the observed overall synchrotron spectral index [FORMULA] (Niklas et al. 1997) indicates strong convection (see Fig. 2) whereas at the same time the steepening of the spectral index in the halo shows the presence of important energy losses (Seaquist & Odegard 1991; Reuter et al. 1992).

3.7. Consequences for the FIR/radio correlation

The question whether CREs can escape freely from galaxies or whether synchrotron and inverse Compton losses determine their spectra has been asked in the framework of the FIR/radio correlation. Völk (1989) has proposed a calorimeter model in which the tightness of the correlation can be explained by the fact that CREs loose their energy by synchrotron and inverse Compton losses below the energy level corresponding to the observing frequency before they are able to escape from a galaxy. Lisenfeld et al. (1996a) have generalized this model, allowing for moderate escape and a spatially inhomogeneous magnetic field. Other authors have claimed the opposite, i.e. that CREs can escape more or less freely from the galaxy (Chi & Wolfendale 1990; Helou & Bicay 1993; Niklas & Beck 1997).

It has been argued (Niklas & Beck 1997) that the observed spectral index of galaxies, [FORMULA], is in contradiction with the calorimeter model which predicts in its asymptotic version [FORMULA], corresponding to case (1) of Sect. 3.3. The difference between the prediction of the calorimeter model and observations of the radio spectral index can be decreased by allowing for moderate escape and a spatially inhomogeneous magnetic field (Lisenfeld et al. 1996a) and even more by including the contribution of SNRs (this work), so that such a modified calorimeter model would predict [FORMULA], corresponding to the observed average value. However, even in this case this model is not able to account for the flatter spectra that some of the galaxies (e.g. M 82) show because it is based on diffusion. In this paper, we have shown, that for galaxies with such a low spectral index, convection is likely to be important. With respect to the question of whether confinement or escape is taking place the overall spectral index is not a good diagnostic. Spatially resolved observations are necessary and these indicate that energy losses indeed do play a dominant role. Thus, even for galaxies with a low overall spectral index, like M 82, escape seems to be negligible as predicted by the calorimeter model.

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© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000
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