New probability distributions in astrophysics: IV. The relativistic Maxwell-Boltzmann distribution

Two relativistic distributions which generalizes the Maxwell Boltzman (MB) distribution are analyzed: the relativistic MB and the Maxwell-J{\"u}ttner (MJ) distribution. For the two distributions we derived in terms of special functions the constant of normalization, the average value, the second moment about the origin, the variance, the mode, the asymptotic behavior, approximate expressions for the average value as function of the temperature and the connected inverted expressions for the temperature as function of the average value. Two astrophysical applications to the synchrotron emission in presence of the magnetic field and the relativistic electrons are presented.


Introduction
The equivalent in special relativity (SR) of the Maxwell-Boltzmann (MB) distribution, see [1,2], is the so called Maxwell-Jüttner distribution (MJ), see [3,4]. The MJ distribution has been recently revisited, we select some approaches among others: a model for the anisotropic MJ distribution [6], an astrophysical application of the MJ distribution to the energy distribution in radio jets [7], a new family of MJ distributions characterized by the parameter η [5] and an application to counter-streaming beams of charged particles [8]. The above approaches does not cover the determination of the statistical quantities of the MJ distribution. In this paper the statistical parameters of the relativistic MB distribution are derived in Section 2 and those of the MJ distribution are derived in Section 3. Section 4 derives the spectral synchrotron emissivity in the framework of the two relativistic distributions here analyzed. The mode is the real solution of the following cubic equation in γ which has the real solution At the moment of writing a closed form for the distribution function (DF) which is does not exists and we therefore present a numerical integration, see Figure 2.
The integration of the above approximate PDF gives an approximate DF which has a maximum percentage error of 7% in the interval 1.1 < γ < 4 when T = 1. The random numbers belonging to the relativistic MB can be generated through a numerical computation of the inverse function following the algorithm outlined in Sec. 4.9.1 of [14]. The above PDF has only one parameter which can be derived approximating the average value with a Pade approximant [2,2] µ(T ) ≈ −0.061723842 + 1.542917977 T + 0.3269078746 (T − 1) The above approximation in the interval 0.1 ≤ T < 10 has a percent error less than 1%. The inverse function allows to derive T as .
Herex is the sample mean defined asx formula which is useful to derive the variance of the sample where x i are the n-data, see [15]. An example of random generation of points is reported in Figure 3 where we imposed T = 1 and we found T = 1.0397 from the generated random sample.

Variable velocity
We now return to the variable velocity, the PDF is where v is expressed in c = 1 units. The mode is a solution of a sextic equation, see [16], in v which has the following real solution The position of the mode for the PDF in v is different from that one in γ, see Figure 4. At the moment of writing the other statistical parameters cannot be presented in a closed form.

The Maxwell Jüttner distribution
The PDF for the Maxwell Jüttner (MJ) distribution is where Θ = kT M B m c 2 , m is the mass of the gas molecules, k is the Boltzmann constant, T M B is the usual thermodynamic temperature and K 2 (x) is the Bessel function of second kind, see [3,4,6,7]. Figure 5 reports the above PDF for three different values of Θ and Figure 6 displays the PDF as a 2-D contour.
The average value is and the variance is  The mode can be found by solving the following cubic equation The real solution is mode = 1 The asymptotic expansion of order 10 for the PDF is The DF is evaluated with the following integral which cannot be expressed in terms of special functions.
We now present some approximations for the distribution function A first approximation is given by a series expansion when, ad example , Θ = 1 which has a percent error less < 0.6% in interval 1.1 < γ < 10 when T = 1. A second approximation is given by an asymptotic expansion of order 50 for the PDF followed by the integration, see Figure 7. The An example of random generation of points is reported in Figure 8 where we imposed T = 10 and we found T = 9.97 from formula (30) and T = 9.98 from formula (31).

Variable β
We now change the variable of integration γ in β = v c , the PDF of the MJ is where 0 ≤ β ≤ 1, see Figure 9. We have only one analytical result, the mode, which is found solving the following equation in β As an example when Θ = 0.1 the mode is at β = 0.4866 and Figure 10 reports the mode as function of Θ.     The mean and the variance of the MJ distribution does not have an analytical expression and they are reported in a numerical way, see Figures 11 and 12. The DF of the MJ is given by the following integral with β in [0,1] which does not have an analytical expression. An approximation is given by the Riemann sums, see [17], when Θ = 1 see Figure 13. The above DF has a maximum percentage error of ≈ 10% at β = 1.

The astrophysical applications
This section reviews the synchrotron emissivity for a single relativistic electron, derives the spectral synchrotron emissivity for the two relativistic distributions here analyzed and models the observed synchrotron emission in some astrophysical sources.

Synchrotron emissivity
The synchrotron emissivity of a single electron is where, according to eqn.(8.58) in [18], e is the electron charge, B is the magnetic field, α is the pitch angle, 0 is the permittivity of free space, c is the light velocity, m e is the electron mass, x = is the ratio of the angular frequency (ω) to the critical angular frequency (ω c ) and where K 5/3 (z) is the modified Bessel function of second kind with order 5/3 [19,13]. The modified Bessel function is also known as Basset function, modified Bessel function of the third kind or Macdonald function see pag. 527 in [20]. The above function has the following analytical expression where 2 F 1 (a, b; c; v) is a regularized hypergeometric function [19,21,22,13]. Figure 14 displays F (x) as function of x.

The synchrotron relativistic MB distribution
We start from the PDF for the relativistic MB distribution as represented by equation (7) and we perform the following first change of variable where E is the relativistic energy. The resulting PDF in relativistic energy is A second change of variable is produces where We know that ν g = 2.799249 10 12 B where B is the magnetic field expressed in gauss and therefore the above PDF in frequency becomes

The synchrotron Maxwell Jüttner distribution
We start from the PDF for the Maxwell Jüttner distribution as given by equation (22) and we perform two changes in variable as in the previous section. The resulting PDF in relativistic energy is The second PDF in ν is The astrophysical PDF in frequency for the Maxwell Jüttner distribution is The mismatch between measured flux in Jy and theoretical flux, S theo , can be obtained introducing a multiplicative constant C S theo = C × f M J (ν; Θ, B) .

The spectrum of the radio-sources
As a first example we analyze the spectrum of an extended region around M87, see as example Figure 1 in [23] where the flux in Jy as function of the frequency is reported in the range 9×10 9 Hz < ν < 2×10 18 Hz. Figure 15 reports the measured and theoretical flux in the range 9 × 10 9 Hz < ν < 2 × 10 12 Hz for the quiet core of M87. A second example is given by the radio sources with ultra steep spectra (USS) which are characterized by a spectral index, α, lower than -1.30 when the radio flux, S, is proportional to S α , see [24]. As a practical example we select the cluster Abell 1914 where the measured total flux densities at 150 MHz and 1.4 GHz are S 150 = 4.68 Jy and S 1.4 = 34.8 mJy which means α = −2.17. We now evaluate the theoretical spectral index of synchrotron emission for the relativistic MB distribution between 150 MHz and 1.4 GHz when B is fixed and T variable, see Figure 16 and Figure 17 when T and B are both variables. The two Figures above show that the theoretical spectral index is always smaller than -2 which can be considered as an asymptotic limit for high values of relativistic temperature. As an example when B = 1.0 × 10 −5 gauss the spectral index is -2.17 when T = 10.