## 2. Radio emission of SNRs## 2.1. Theoretically expected valueWe estimate the fraction of the nonthermal radio emission of a galaxy that is expected to come from SNRs in the framework of a simple leaky-box model. The number of CREs in a certain energy range , , can then be calculated as: Q(E) is the production rate of CREs and the life-time of CREs. The main sources of CREs are gravitational supernovae (SN) (SN II and Ib/c) (e.g. Xu et al. 1994a and discussion in Xu et al 1994b). The energy spectrum of the CREs produced can be described by a power law: where is the SN-rate, the number of CREs produced per energy interval per SN and the energy injection spectral index. For CREs in SNRs is given by the life-time of the SNR, , and for the diffuse CREs can be expressed as: where is the escape time-scale from the galaxy and is the energy loss time-scale. We take into account a possible energy dependence of according to: The exponent In the case of a steady state () the solution of Eq. (1) is simply: We make the simplification that an electron with energy (Longair 1992, Vol. 1 p. 332) where is the gyrofrequency. Then the total synchrotron emission is given by: with the energy losses by synchrotron radiation where is the Thompson scattering
cross section, With the above equations, we can express the ratio of the diffuse synchrotron emission to the emission from SNRs as: Thus, the ratio of the diffuse synchrotron emission to the emission
from SNRs is mainly determined by the ratio of the life-times of CREs
in the two zones and by the ratio of the magnetic field strengths. The
magnetic field strength in SNRs is expected to be higher than in the
diffuse ISM because the gas in which the magnetic field is frozen-in
gets compressed by the shock. In general, the magnetic field strength
found in SNRs exceeds the value expected by this compression which
would be a factor of 4 in a strong shock, indicating that the magnetic
field is not only compressed, but enhanced by the shock. Especially in
young SNRs, the inferred values span a wide range from a few
A further contribution to the radio emission of a galaxy are radio supernovae (RSNe, e.g. Weiler et al. 1986). Their contribution is difficult to estimate due to uncertainties in their luminosity and life-time. Pérez-Olea & Colina (1995) have included RSNe in a model for the radio emission of starburst galaxies by generalizing and extrapolating the luminosities of 9 observed RSNe. Adopting their numbers, we calculate that for a galaxy in a steady state the total radio emission of RSNe is, for a life-time of a RSNe of 100 yr, the total radio emission of SNRs. For a shorter life-time of 10 yr, their total radio emission would be a factor of 2 lower. Thus, in spite of the uncertainties, it seems plausible that RSNe contribute at a level of at least a few percent to the total radio emission of a galaxy. ## 2.2. Comparison to observationsIn order to compare this theoretically derived ratio to observations we calculate the expected average radio emission of a SNR: where we have inserted the relations from Eqs. (6) and (8). For this predicts a radio spectral index of SNRs of 0.6, in reasonable agreement with the observations. The total number of CREs per energy interval produced by a SN, , can be estimated from the total energy released by a SN which we take as erg. Assuming that 10% of this energy goes into CR acceleration and of this 1% into the electron component, a total energy of erg is available per SN to produce the relativistic electrons following a power-law spectrum: With , assuming for the lower energy limit of the produced spectrum GeV and adopting the scaling parameter GeV this yields In total, we obtain for the radio emission of a SNR, taking G, This can be compared to the average observed emission of a SNR
which can be derived from the relation between the surface brightness
and the diameter Following Condon & Yin (1990) one can connect the diameter and the age of the radio emitting SNR by: where is the energy of a SN going
into particle acceleration in units of
erg and which is very close to the above theoretical value (Eq. 13). We can now estimate the total emission of SNRs expected for our
Galaxy: Adopting a SN rate of 1 SN per 30 yr (Berkhuijsen 1984), the
total number of the SNRs present in our Galaxy is, with
yr,
. This yields, together with Eq.
(13), a total radio emission from SNRs of
. This can be compared to the total
radio emission of our Galaxy which is at 408 MHz
(Beuermann et al. 1985).
Extrapolating this value, with a radio spectral index of 0.8, to 1.5
GHz we get W Hz ## 2.3. Influence on the spatially integrated radio spectral indexIn Fig. 1 we show the total (i.e. spatially integrated) nonthermal spectral index, , as a function of the spectral index of the diffuse radio emission, , for different contributions of SNRs to the radio emission, expressed by . We assume, as above, an (energy) injection spectral index for CREs of , yielding a (frequency) spectral index of SNRs of (Eq. (10)). A lower source spectral index of to 2.1 is indicated by recent calculations by Berezhko and Völk (1997). This would give to 0.55 and would somewhat increase the difference between and for a given .
The radio emission of SNRs has a noticeable effect on the nonthermal radio spectral index. Due to the contribution of SNRs it gets flatter, especially for large values of where the spectral index is lowered by for . © European Southern Observatory (ESO) 2000 Online publication: February 9, 2000 |