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Appendix A: Minimum energy calculation for an isotropic electron population with a power law spectrum in a disordered magnetic field

The total non-thermal energy density of the radio lobes is written as the sum of the ions, electrons, and magnetic field contributions: (A.1)The energy density in non-radiating particles is set simply proportional to that of electrons (A.2)The electron energy spectrum (number density of electrons with a Lorentz factor between γ and γ + dγ) is taken to be a power law (A.3)where the electron’s Lorentz factor γ is in the range from γ_min to γ_max and we assumed an isotropic distribution of the pith angle θ between the local direction of the magnetic field and the electron velocity. Thus, the electron energy density is (A.4)The radio emissivity as a function of the pitch angle in a uniform magnetic field of strength B is given by (A.5)where F_syn(ν/ν_c) is the synchrotron kernel (see e.g. Blumenthal & Gould 1970; Rybicki & Lightman 1979), while (A.6)The constants C_f = 2.3444 × 10^-22 (erg Gauss^-1) and C_ν = 4.1989 × 10⁶ (Gauss^-1 s^-1) depend only on fundamental physical constants.

We now suppose that the magnetic field is completely tangled in an infinitesimally small scale compared to the source’s size. With this assumption, the synchrotron luminosity, after averaging Eq. (A.5) over all possible magnetic field directions with respect to the line-of-sight (LOS), is (A.7)\newpage\noindentwhere V is the source’s volume, see also the discussion in Murgia et al. (2010). In the power law regime, ν_min ≪ ν ≪ ν_max, we obtain the well-known formula for the synchrotron monochromatic power (A.8)where α = (p − 1)/2 while (A.9)In the literature an idealized situation is often assumed in which the source’s magnetic field lies in the plane of the sky. In this case, due to the high beaming of the synchrotron radiation, only electrons with pith angle θ = 90° can be observed and the constant in Eq. (A.8) has to be replaced by (A.10)The synchrotron radiation is then amplified by 60 − 70% with respect to the more realistic situation in which the field direction is disordered in the source.

For α in the range between 0.5 and 1.0, Eq. (A.9) can be approximated by the polynomial expansion: (A.11)Substituting Eq. (A.8) in (A.4), we found that the source’s non-thermal energy is minimum for (A.12)For this value of the magnetic field strength the energy is nearly equipartitioned between particles and field: (A.13)so that the total minimum energy density is (A.14)

© ESO, 2012