3.2. Equipartition magnetic fields derived from the synchrotron emission
From the synchrotron emissivity it is not possible to derive unambiguously the magnetic field value. The usual way to estimate the magnetic field strength in a radio source is to minimize its total energy content U_{tot} [10]. The total energy of a synchrotron source is due to the energy in relativistic particles (U_{el} in electrons and U_{pr} in protons) plus the energy in magnetic fields (U_{B}):
(12) |
The magnetic field energy contained in the source volume V is given by
(13) |
where is the fraction of the source volume occupied by the magnetic field (filling factor). The electron total energy in the range _{1} - _{2}:
(14) |
can be expressed as a function of the synchrotron luminosity L_{syn}:
(15) |
by eliminating VN_{0} and by writing _{1} and _{2} in terms of _{1} and _{2} (Eq. 1):
(16) |
where sin has been taken equal to 1 and
(17) |
The energy contained in the heavy particles, U_{pr}, can be related to U_{el} assuming:
(18) |
Finally the total energy is obtained as a function of the magnetic field:
(19) |
In order to obtain an estimate for the magnetic fields, it is necessary to make some assumptions about how the energy is distributed between the fields and particles. A convenient estimate for the total energy is represented by its minimum value (see Fig. 2). The condition of minimum energy is obtained when the contributions of the magnetic field and the relativistic particles are approximately equal:
(20) |
For this reason the minimum energy is known as equipartition value:
(21) |
The magnetic field for which the total energy content is minimum is:
(22) |
The total minimum energy is:
(23) |
and the total minimum energy density is:
(24) |
where c_{13} = 0.921 c_{12}^{4/7}. The constants c_{12} and c_{13}, depending on the spectral index and on the frequency range, are tabulated [10] for cgs units.
By including the K-correction, assuming = 1, and expressing the parameters in commonly used units, we can write the minimum energy density of a radio source in terms of observed quantities:
(25) |
where z is the source redshift, I_{0} is the source brightness at the frequency _{0}, d is the source depth, and the constant (, _{1}, _{2}) is tabulated in Table 1 for the frequency ranges: 10 MHz - 10 GHz and 10 MHz - 100 GHz. I_{0} can be measured directly by the contour levels of a radio image (for significantly extended sources), or can be obtained by dividing the source total flux by the source solid angle.
(, 10 MHz, 10 GHz) | (, 10 MHz, 100 GHz) | |||
0.0 | 1.43 × 10^{-11} | 2.79 × 10^{-11} | ||
0.1 | 9.40 × 10^{-12} | 1.63 × 10^{-11} | ||
0.2 | 6.29 × 10^{-12} | 9.72 × 10^{-12} | ||
0.3 | 4.29 × 10^{-12} | 5.97 × 10^{-12} | ||
0.4 | 2.99 × 10^{-12} | 3.79 × 10^{-12} | ||
0.5^{*} | 2.13 × 10^{-12} | 2.50 × 10^{-12} | ||
0.6 | 1.55 × 10^{-12} | 1.72 × 10^{-12} | ||
0.7 | 1.15 × 10^{-12} | 1.23 × 10^{-12} | ||
0.8 | 8.75 × 10^{-13} | 9.10 × 10^{-13} | ||
0.9 | 6.77 × 10^{-13} | 6.92 × 10^{-13} | ||
1.0^{*} | 5.32 × 10^{-13} | 5.39 × 10^{-13} | ||
1.1 | 4.24 × 10^{-13} | 4.27 × 10^{-13} | ||
1.2 | 3.42 × 10^{-13} | 3.43 × 10^{-13} | ||
1.3 | 2.79 × 10^{-13} | 2.79 × 10^{-13} | ||
1.4 | 2.29 × 10^{-13} | 2.29 × 10^{-13} | ||
1.5 | 1.89 × 10^{-13} | 1.89 × 10^{-13} | ||
1.6 | 1.57 × 10^{-13} | 1.57 × 10^{-13} | ||
1.7 | 1.31 × 10^{-13} | 1.31 × 10^{-13} | ||
1.8 | 1.10 × 10^{-13} | 1.10 × 10^{-13} | ||
1.9 | 9.21 × 10^{-14} | 9.21 × 10^{-14} | ||
2.0 | 7.76 × 10^{-14} | 7.76 × 10^{-14} | ||
^{*} for these values of the constant defined in Eq. 17 diverges, thus the corresponding parameters have been computed by solving directly the integrals in Eqs. 14 and 15. |
The equipartition magnetic field is then obtained as:
(26) |
One must be aware of the uncertainties inherent to this determination of the magnetic field strength. The value of k, ratio of the energy in relativistic protons to that in electrons, depends on the mechanism of generation of relativistic electrons, which is so far poorly known. Uncertainties are also related to the volume filling factor . Values usually assumed in literature for clusters are k = 1 (or k = 0) and = 1. Another parameter difficult to infer is the extent of the source along the line of sight d.
In the standard approach presented above, the computation of equipartition parameters is based on the integration of the synchrotron radio luminosity between the two fixed frequencies _{1} and _{2} (Eq. 16 and followings). The electron energies corresponding to these frequencies depend on the magnetic field value (see Eq. 1), thus the integration limits are variable in terms of the energy of the radiating electrons. The lower limit is particularly relevant, owing to the power-law shape of the electron energy distribution and to the fact that electrons of very low energy are expected to be present. If a low-energy cutoff in the particle energy distribution is imposed, rather than a low-frequency cut-off in the emitted synchrotron spectrum, the exponent 2/7 in Eq. 22 should be replaced by 1 / (3 + ), as pointed out by Beck & Krause [11]. The equipartition quantities obtained following this approach are presented by Brunetti et al. [12]. Indicating the electron energy by its Lorentz factor , assuming that _{min} < < _{max}, the new expression for the equipartition magnetic field B'_{eq} in Gauss is (for > 0.5):
(27) |
where B_{eq} is the value of the equipartition magnetic field obtained with the standard formulae by integrating the radio spectrum between 10 MHz and 100 GHz. If the equipartition magnetic field obtained with the standard formulae is ~ µG, the magnetic field derived considering electrons of _{min} ~ 100 is 2 to 5 times larger, using in the range 0.75 - 1.