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Two relativistic distributions which generalize the Maxwell Boltzman (MB) distribution are analyzed: the relativistic MB and the Maxwell-Jüttner (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.

The equivalent in special relativity (SR) of the Maxwell-Boltzmann (MB) distribution, see [

The usual MB distribution, f ( v ; m , k , T M B ) , for an ideal gas is

f ( v ; m , k , T M B ) = 2 v 2 e − 1 2 v 2 m k T M B π ( k T M B m ) 3 2 , (1)

where m is the mass of the gas molecules, k is the Boltzmann constant and T M B is the usual thermodynamic temperature. In SR, the total energy of a particle is

E = m γ c 2 , (2)

where m is the rest mass, c is the light velocity, γ is the Lorentz factor 1 1 − β 2 , β = v c and v is the velocity. The relativistic kinetic energy, E k , is

E k = m c 2 ( γ − 1 ) , (3)

where the rest energy has been subtracted from the total energy, see formula (23.1) in [

f r ( v ; T ) = v 2 e 1 T ( 1 − 1 1 − v 2 c 2 ) ∫ 0 c w 2 e 1 T ( 1 − 1 1 − w 2 c 2 ) d w , (4)

where the relativistic temperature; T, is expressed in m c 2 / k units; up to now the treatment is the same of [

· has the velocity of the light as maximum velocity,

· becomes the usual MB distribution in the limit of low velocities,

· is not invariant for relativistic transformations.

We now change the variable of integration

v = γ 2 − 1 γ . (5)

The differential of the velocity, d v ,

d v = 1 γ 2 − 1 γ 2 d γ , (6)

and therefore the relativistic MB distribution in the variable γ is

f r ( γ ; T ) = 32 γ 2 − 1 e 1 − γ T T 3 γ 4 e T − 1 G 1,3 3,0 ( 1 4 T 2 | − 1 / 2 , − 1, − 3 / 2 1 ) , (7)

where G is the Mejier G-function [

The average value or mean, μ , is

μ ( T ) = 2 T G 1,3 3,0 ( 1 4 T 2 | − 1 / 2 , − 1 / 2 , − 1 1 ) G 1,3 3,0 ( 1 4 T 2 | − 1 / 2 , − 1, − 3 / 2 1 ) , (8)

the second moment about the origin is

μ ( T ) 2 = 4 T 2 G 1,3 3,0 ( 1 / 4 T − 2 | 0, − 1 / 2 , − 1 / 2 1 ) G 1,3 3,0 ( 1 / 4 T − 2 | − 1 / 2 , − 1, − 3 / 2 1 ) , (9)

the variance, σ 2 is

σ 2 ( T ) = 4 T 2 ( G 1,3 3,0 ( 1 4 T 2 | − 1 / 2 , − 1, − 3 / 2 1 ) G 1,3 3,0 ( 1 4 T 2 | 0, − 1 / 2 , − 1 / 2 1 ) − ( G 1,3 3,0 ( 1 4 T 2 | − 1 / 2 , − 1 / 2 , − 1 1 ) ) 2 ) ( G 1,3 3,0 ( 1 4 T 2 | − 1 / 2 , − 1, − 3 / 2 1 ) ) 2 . (10)

The mode is the real solution of the following cubic equation in γ

γ 3 + 3 T γ 2 − γ − 4 T = 0 , (11)

which has the real solution

m o d e = 1 6 324 T − 216 T 3 + 12 − 1296 T 4 + 621 T 2 − 12 3 − 6 − 1 / 3 − T 2 324 T − 216 T 3 + 12 − 1296 T 4 + 621 T 2 − 12 3 − T . (12)

At the moment of writing a closed form for the distribution function (DF) which is

F r ( γ ; T ) = ∫ 1 y f r ( γ ; T ) d γ , (13)

does not exists and we therefore present a numerical integration, see

The asymptotic behavior of the PDF, f a , is

f a ( γ ; T ) = 1 G 1,3 3,0 ( 1 / 4 T − 2 | − 1 / 2 , − 1, − 3 / 2 1 ) 2048 γ 21 T 3 ( 65536 γ 18 − 32768 γ 16 − 8192 γ 14 − 4096 γ 12 − 2560 γ 10 − 1792 γ 8 − 1344 γ 6 − 1056 γ 4 − 858 γ 2 − 715 ) e − γ T . (14)

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 [

μ ( T ) ≈ − 0.061723842 + 1.542917977 T + 0.3269078746 ( T − 1 ) 2 0.1069596119 + 0.8930403881 T + 0.1511024609 ( T − 1 ) 2 . (15)

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

T = − 1 2 5.908 × 10 9 x ¯ − 8.89 × 10 9 + 1.931 × 10 19 x ¯ 2 − 5.528 × 10 19 x ¯ + 4.437 × 10 19 1.511 × 10 9 x ¯ − 3.269 × 10 9 . (16)

Here x ¯ is the sample mean defined as

x ¯ = 1 n ∑ i = 1 n x i , (17)

formula which is useful to derive the variance of the sample

V a r = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2 , (18)

where x i are the n-data, see [

We now return to the variable velocity, the PDF is

f r ( v ; T ) = 32 − v 2 v 2 − 1 e − v 2 + 1 − 1 − v 2 + 1 T − v 2 + 1 T 3 v e T − 1 G 1,3 3,0 ( 1 / 4 T − 2 | − 1 / 2 , − 1, − 3 / 2 1 ) , (19)

where v is expressed in c = 1 units. The mode is a solution of a sextic equation, see [

− 4 T 2 v 6 + 12 T 2 v 4 − v 4 − 12 T 2 v 2 + 4 T 2 = 0, (20)

which has the following real solution

m o d e = 1 6 ( 3 24 3 27 T 2 − 1 T 3 − 216 T 4 + 36 T 2 − 1 3 T 2 − 3 24 T 2 − 1 T 2 24 3 27 T 2 − 1 T 3 − 216 T 4 + 36 T 2 − 1 3 + 3 12 T 2 − 1 T 2 ) 1 / 2 . (21)

The position of the mode for the PDF in v is different from that one in γ , see

At the moment of writing the other statistical parameters cannot be presented in a closed form.

The PDF for the Maxwell Jüttner (MJ) distribution is

f M J ( γ ; Θ ) = γ γ 2 − 1 e − γ Θ Θ K 2 ( 1 Θ ) , (22)

where Θ = k T 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 [

The average value is

μ ( Θ ) = − 2 Θ 2 G 1,3 2,1 ( 1 4 Θ 2 | 3 / 2 , − 1 / 2 , − 2 1 ) K 2 ( 1 Θ ) (23)

and the variance is

σ 2 ( Θ ) = 1 Θ 2 ( K 2 ( 1 Θ ) ) 2 ( − 4 Θ 5 ( 2 K 1 ( Θ − 1 ) G 1,3 2,1 ( 1 / 4 Θ − 2 | 5 / 2 , − 1 / 2 , − 2 1 ) Θ + ( G 1,3 2,1 ( 1 / 4 Θ − 2 | 3 / 2 , − 1 / 2 ,2 1 ) ) 2 Θ + K 0 ( Θ − 1 ) G 1,3 2,1 ( 1 / 4 Θ − 2 | 5 / 2 , − 1 / 2 ,2 1 ) ) ) . (24)

The mode can be found by solving the following cubic equation

d d γ f M J ( γ ; Θ ) ∝ − γ 3 + 2 Θ γ 2 + γ − Θ = 0. (25)

The real solution is

m o d e = 1 6 − 36 Θ + 64 Θ 3 + 12 − 96 Θ 4 − 39 Θ 2 − 12 3 × ( ( − 36 Θ + 64 Θ 3 + 12 − 96 Θ 4 − 39 Θ 2 − 12 ) 2 3 + 4 Θ − 36 Θ + 64 Θ 3 + 12 − 96 Θ 4 − 39 Θ 2 − 12 3 + 16 Θ 2 + 12 ) . (26)

The asymptotic expansion of order 10 for the PDF is

f M J ( γ ; Θ ) ∼ 1 Θ K 2 ( 1 Θ ) ( 128 γ 8 − 64 γ 6 − 16 γ 4 − 8 γ 2 − 5 ) e − γ Θ 128 γ 6 . (27)

The DF is evaluated with the following integral

F M J ( γ ; Θ ) = ∫ 1 γ f M J ( γ ; Θ ) d γ , (28)

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

F M J ( γ ; 1 ) = 1 K 2 ( 1 ) ( K 2 ( 1 ) + π ∑ m = 0 ∞ ( − 1 ) 1 + m Γ ( 3 − 2 m , γ ) Γ ( 1 + m ) Γ ( 3 2 − m ) ) , (29)

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

Θ = 0.1661 x ¯ − 0.3085 + 1.36051 × 10 − 10 × 1.4908 × 10 18 x ¯ 2 + 5.913 × 10 18 x ¯ − 6.5835 × 10 18 . (30)

An analogous formula allows to derive Θ from the variance V a r of the sample

Θ = 1 4 × 1.818 × 10 10 V a r + 5.972 × 10 11 + 5 2.277 × 10 20 V a r 2 + 7.814 × 10 23 V a r − 3.597 × 10 22 5.436 × 10 8 V a r + 1.978 × 10 12 . (31)

An example of random generation of points is reported in

We now change the variable of integration γ in β = v c , the PDF of the MJ is

f M J ( β ; Θ ) = ( 1 − β 2 ) − 1 − 1 e − 1 Θ 1 1 − β 2 β ( 1 − β 2 ) 2 Θ K 2 ( 1 Θ ) , (32)

where 0 ≤ β ≤ 1 , see

− 3 ( β − 1 ) 3 ( β + 1 ) 3 ( Θ ( β 2 + 2 / 3 ) − β 2 + 1 − 1 / 3 β 2 ) e − 1 − β 2 + 1 Θ β 2 = 0. (33)

As an example when Θ = 0.1 the mode is at β = 0.4866 and

The mean and the variance of the MJ distribution does not have an analytical expression and they are reported in a numerical way, see

The DF of the MJ is given by the following integral

F M J ( β ; Θ ) = ∫ 0 β f M J ( β ; Θ ) d β , (34)

with β in [0, 1] which does not have an analytical expression. An approximation is given by the Riemann sums, see [

F M J ( β ; Θ ) = β ∑ i = 0 9 β 10 K 2 ( 1 ) ( − β 2 100 ( i + 1 2 ) 2 + 1 ) − 1 − 1 e − 1 − β 2 100 ( i + 1 2 ) 2 + 1 ( i + 1 2 ) ( − β 2 100 ( i + 1 2 ) 2 + 1 ) − 2 10 , (35)

see

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.

The synchrotron emissivity of a single electron is

3 e 3 B sin ( α ) 8 π 2 ϵ 0 c m e F ( x ) , (36)

where, according to eqn.(8.58) in [

F ( x ) = x ∫ x ∞ K 5 / 3 ( z ) d z (37)

where K 5 / 3 ( z ) is the modified Bessel function of second kind with order 5/3 [

F ( x ) = − 9 3 2 3 π 320 Γ ( 2 / 3 ) x 11 3 F 1 2 ( 4 3 ; 7 3 , 8 3 ; x 2 4 ) − x 3 π 3 + x 3 2 2 3 Γ ( 2 3 ) F 1 2 ( − 1 3 ; − 2 3 , 2 3 ; x 2 4 ) , (38)

where 1 ( a , b ; c ; v ) is a regularized hypergeometric function [

We start from the PDF for the relativistic MB distribution as represented by Equation (7) and we perform the following first change of variable

γ = E m e c 2 , (39)

where E is the relativistic energy. The resulting PDF in relativistic energy is

f r ( E ; T ) = 32 E 2 m e 2 c 4 − 1 e 1 T ( 1 − E m e c 2 ) T 3 m e 3 c 6 E 4 e T − 1 G 1,3 3,0 ( 1 4 T 2 | − 1 2 , − 1, − 3 2 1 ) . (40)

A second change of variable is

E = ν ν g m e c 2 , (41)

produces

f r ( ν ; T , ν g ) = 16 ν ν g − 1 e 1 T ( 1 − ν ν g ) T 3 ν g ν 2 e T − 1 G 1,3 3,0 ( 1 4 T 2 | − 1 2 , − 1, − 3 2 1 ) ν ν g , (42)

where

ν g = e B 2 π m e . (43)

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

f r ( ν ; T , B ) = 7.49345 × 10 19 3.57238 × 10 − 13 ν B − 1 e 1 T ( 1 − 5.97694 × 10 − 7 ν B ) T 3 B ν 2 e T − 1 G 1,3 3,0 ( 1 4 T 2 | − 1 2 , − 1, − 3 2 1 ) ν B . (44)

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

f M J ( E ; Θ ) = E E 2 m e 2 c 4 − 1 e − E m e c 2 Θ m e 2 c 4 Θ K 2 ( 1 Θ ) . (45)

The second PDF in ν is

f M J ( ν ; Θ , ν g ) = ν ν g − 1 e − 1 Θ ν ν g 2 Θ K 2 ( 1 Θ ) ν g . (46)

The astrophysical PDF in frequency for the Maxwell Jüttner distribution is

f M J ( ν ; Θ , B ) = 1.78619 × 10 − 13 3.57238 × 10 − 13 ν B − 1 e − 5.97694 × 10 − 7 1 Θ ν B Θ K 2 ( 1 Θ ) B . (47)

The mismatch between measured flux in Jy and theoretical flux, S t h e o , can be obtained introducing a multiplicative constant C

S t h e o = C × f M J ( ν ; Θ , B ) . (48)

As a first example we analyze the spectrum of an extended region around M87, see as example

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 [

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 .

The relativistic MB distribution has been derived in [

We derived the following statistical parameters of the MJ distribution when γ is the main variable: average value, variance, mode, asymptotic expansion, two approximate expressions for the distribution function, a first evaluation of Θ from the average value and a second evaluation of Θ from the variance.

Following the usual argument which suggests a power law behavior for the spectral distribution of the synchrotron emission in presence of a power law distribution for the energy of the electrons, we derived the spectral distribution for the relativistic MB and MJ distributions which are now function of the selected generalized temperature and the magnetic field. Two astrophysical applications are given: the spectral distribution of emission in the core of M87 in the framework of the synchrotron emissivity and an explanation for the steep spectra sources in the framework of the synchrotron emissivity for the relativistic MJ distribution.

The author declares no conflicts of interest regarding the publication of this paper.

Zaninetti, L. (2020) New Probability Distributions in Astrophysics: IV. The Relativistic Maxwell-Boltzmann Distribution. International Journal of Astronomy and Astrophysics, 10, 302-318. https://doi.org/10.4236/ijaa.2020.104016