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

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2. Radio emission of SNRs

2.1. Theoretically expected value

We 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 [FORMULA], [FORMULA], can then be calculated as:

[EQUATION]

Q(E) is the production rate of CREs and [FORMULA] 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:

[EQUATION]

where [FORMULA] is the SN-rate, [FORMULA] the number of CREs produced per energy interval per SN and [FORMULA] the energy injection spectral index. For CREs in SNRs [FORMULA] is given by the life-time of the SNR, [FORMULA], and for the diffuse CREs [FORMULA] can be expressed as:

[EQUATION]

where [FORMULA] is the escape time-scale from the galaxy and [FORMULA] is the energy loss time-scale. We take into account a possible energy dependence of [FORMULA] according to:

[EQUATION]

The exponent y can have different values depending on the characteristics of the propagation of CREs. It is [FORMULA] if [FORMULA] and synchrotron and inverse Compton losses are the dominant energy loss processes. If [FORMULA] and CREs propagate mainly by diffusion, the energy dependence of the diffusion coefficient, [FORMULA], gets reflected in [FORMULA]. If, on the other hand, CREs can escape freely ([FORMULA]) by convection or if adiabatic and bremsstrahlung losses are dominant, we expect [FORMULA].

In the case of a steady state ([FORMULA]) the solution of Eq. (1) is simply:

[EQUATION]

We make the simplification that an electron with energy E emits all the synchrotron radiation at the frequency

[EQUATION]

(Longair 1992, Vol. 1 p. 332) where [FORMULA] is the gyrofrequency. Then the total synchrotron emission is given by:

[EQUATION]

with the energy losses by synchrotron radiation

[EQUATION]

where [FORMULA] is the Thompson scattering cross section, B the magnetic field strength and [FORMULA] its energy density.

With the above equations, we can express the ratio of the diffuse synchrotron emission to the emission from SNRs as:

[EQUATION]

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 µG (SN 1006; Tanimori et al. 1998) to much higher values of 70 µG (Kepler's SNR; Matsui et al. 1984), [FORMULA] 80 µG (Cas A; Cowsik & Sarkar 1980) and even 300 µG (Crab Nebula; Kennel & Coroniti 1984). Taking these values as a guide, we assume for our present estimate [FORMULA]G and [FORMULA]G. Furthermore, we adopt for the injection spectral index [FORMULA] (Völk et al. 1988), for the life-time of a SNR its adiabatic phase [FORMULA] yr and for [FORMULA] the energy loss time scale due to synchrotron and inverse Compton losses. The latter yields for a magnetic field of 5 µG and a radiation field of energy density 1 eV cm-3 [FORMULA] yr at [FORMULA] GHz. With these values we get [FORMULA] at a frequency 1.5 GHz.

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, [FORMULA] 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 observations

In order to compare this theoretically derived ratio to observations we calculate the expected average radio emission of a SNR:

[EQUATION]

where we have inserted the relations from Eqs. (6) and (8). For [FORMULA] 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, [FORMULA], can be estimated from the total energy released by a SN which we take as [FORMULA] erg. Assuming that 10% of this energy goes into CR acceleration and of this 1% into the electron component, a total energy of [FORMULA]erg is available per SN to produce the relativistic electrons following a power-law spectrum:

[EQUATION]

With [FORMULA], assuming for the lower energy limit of the produced spectrum [FORMULA] GeV and adopting the scaling parameter [FORMULA] GeV this yields

[EQUATION]

In total, we obtain for the radio emission of a SNR, taking [FORMULA]G,

[EQUATION]

This can be compared to the average observed emission of a SNR which can be derived from the relation between the surface brightness [FORMULA] and the diameter D of SNRs. We adopt the relation found by Huang et al. (1994) who fitted SNRs from the Galaxy, M 82 and the Magellanic Clouds:

[EQUATION]

Following Condon & Yin (1990) one can connect the diameter and the age of the radio emitting SNR by:

[EQUATION]

where [FORMULA] is the energy of a SN going into particle acceleration in units of [FORMULA] erg and n is the gas density in particles per cm3. In the following we assume [FORMULA] and [FORMULA]. Using [FORMULA] we then calculate the average luminosity of a SNR by integrating the luminosity evolution from [FORMULA] to [FORMULA] yr and adopting the frequency dependence of Eq. (13), [FORMULA]:

[EQUATION]

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 [FORMULA] yr, [FORMULA]. This yields, together with Eq. (13), a total radio emission from SNRs of [FORMULA]. This can be compared to the total radio emission of our Galaxy which is at 408 MHz [FORMULA] (Beuermann et al. 1985). Extrapolating this value, with a radio spectral index of 0.8, to 1.5 GHz we get [FORMULA] W Hz-1. From this calculation we derive a contribution of about 9% from SNRs to the total radio emission, in satisfactory agreement with the value derived in the previous section in spite of the crudeness of both estimates. Therefore, we conclude that [FORMULA] 10% is a reasonable estimate for the contribution of SNRs to the total radio emission. This value is also in agreement with previous estimates by other authors, who compared the expected total synchrotron emission from SNRs with the observed radio emission of galaxies, and have found that SNRs alone can only explain about 10% of the total radio emission (Biermann 1976; Ulvestad 1982).

2.3. Influence on the spatially integrated radio spectral index

In Fig. 1 we show the total (i.e. spatially integrated) nonthermal spectral index, [FORMULA], as a function of the spectral index of the diffuse radio emission, [FORMULA], for different contributions of SNRs to the radio emission, expressed by [FORMULA]. We assume, as above, an (energy) injection spectral index for CREs of [FORMULA], yielding a (frequency) spectral index of SNRs of [FORMULA] (Eq. (10)). A lower source spectral index of [FORMULA] to 2.1 is indicated by recent calculations by Berezhko and Völk (1997). This would give [FORMULA] to 0.55 and would somewhat increase the difference between [FORMULA] and [FORMULA] for a given [FORMULA].

[FIGURE] Fig. 1. The nonthermal spectral index, [FORMULA], as a function of the diffuse radio spectral index, [FORMULA], between 1.5 and 5 GHz, for different contributions of the radio emission from SNRs to the radio emission: [FORMULA]5% (full line), 10% (dashed line), 20% (dotted line).

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 [FORMULA] where the spectral index is lowered by [FORMULA] for [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000
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