**Physica A: Statistical Mechanics and its Applications**

Volume 387, Issue 14, 1 June 2008, Pages 3480-3494

### Abstract

Fermionic power-law distributions are derived by the second
quantization of classical power-law ensembles, and applied to
ultra-relativistic electron populations in the Galactic center. The *γ*-ray
flux from the direction of the compact central source Sagittarius A* is
fitted with a superluminal cascade spectrum. In this way, estimates of
the radiating electron plasma in the Galactic center region are
obtained, such as the power-law index, temperature, particle number,
and internal energy. The spectral averaging of the tachyonic radiation
densities with Fermi power-laws is explained. Fugacity expansions of
the thermodynamic variables (thermal equation of state, entropy,
isochoric heat capacity, and isothermal compressibility) are obtained
in the quasiclassical high-temperature/low-density regime, where the
spectral fit is carried out. The leading quantum correction to these
variables is calculated, and its dependence on the electronic power-law
index and the thermal wavelength is discussed. Excess counts of cosmic
rays from the Galactic center region are related to the plasma
temperature inferred from the cascade fit.

**Keywords: **Superluminal
radiation; Tachyonic cascade spectra; Fermi power-law ensembles;
Ultra-relativistic electron plasma; Quasiclassical fugacity expansion;
Spectral averaging

**PACS classification codes: **05.30.Fk;
05.70.Ce; 52.25.Kn; 95.30.Tg

### Article Outline

- 1. Introduction
- 2. Thermodynamic variables of Fermi power-law ensembles
- 3. Equations of state, entropy, and heat capacity of fermionic power-law densities in the quasiclassical regime
- 3.1. Low-temperature asymptotics
- 3.2. Fugacity expansion and thermal wavelength in the high-temperature limit
- 4. Superluminal
*γ*-rays from the Galactic center: Tachyonic spectral maps and electronic source distributions - 5. Conclusion
- Acknowledgements
- References

### 1. Introduction

Electronic power-law distributions are commonly used in electromagnetic spectral averages to model the synchrotron emission of astrophysical sources, such as the X-ray spectra of supernova remnants [1]. In this article, we quantize Boltzmann power-law densities, exponentially cut power-law distributions , where*H*=

*m*

*c*

^{2}

*γ*is the free electronic Hamiltonian. In Fermi–Dirac statistics, we arrive at densities

where the momentum integration has been parametrized with the electronic Lorentz factor

*γ*.

*δ*is the power-law index, and we use the shortcuts and

*α*=−

*β*

*μ*/(

*m*

*c*

^{2}), where

*μ*is the chemical potential. Electronic and protonic power-law densities have been invoked to perform electromagnetic and hadronic spectral fits to various TeV

*γ*-ray sources recently discovered with imaging air Cherenkov detectors [2]. Tachyonic spectral fits to the cascade spectra of

*γ*-ray pulsars and microquasars are based on electronic power-law averages as well [3] and [4]. Power-law indices inferred from these spectral fits or, for that matter, from the magnetospheric radio emission of planets [5] and [6], usually range in the interval 0

*δ*4.

Here, we give evidence for superluminal radiation from the
Galactic center by fitting recently obtained spectral maps [7], [8] and [9] with
tachyonic cascade spectra. The tachyonic *γ*-ray
wideband consists of two cascades generated by ultra-relativistic
electron populations, and the spectral fit allows us to infer the
thermodynamic parameters. The observed spectra are clearly
distinguishable from electromagnetic synchrotron and inverse-Compton
fits, due to the emergence of extended spectral plateaus. We show that
the TeV spectral map of the Galactic center admits a tachyonic
extension into the GeV region, providing an excellent fit to the
spectrum of the unidentified *γ*-ray
source 3EG J1746 − 2851 [10] associated
with the Galactic central source Sagittarius A*. The superluminal
cascades are generated by an ultra-relativistic electron gas at high
temperature and low density, so that we can use distribution (1.1) in the
quasiclassical regime to calculate the thermodynamic variables and the
spectral averages.

In Section 2, we set up the thermodynamic formalism of fermionic power-law densities (1.1). Starting with the grand partition function, we derive the quasiclassical fugacity expansion of the thermodynamic variables, such as the caloric and thermal equations of state, entropy, specific heat, and compressibility. In Section 3, we discuss the leading quantum correction to the classical thermodynamic functions. In the high-temperature regime, the quantum corrections become more pronounced with increasing electronic power-law index, exhibiting the same temperature scaling as the classical limit usually dominant in the high-temperature/low-density regime. This is illustrated by calculating the mentioned variables for an increasing sequence of electron indices. We study the thermal wavelength of fermionic power-law ensembles in the low- and high-temperature regime, and discuss the range of applicability of the quasiclassical fugacity expansion.

In Section 4, we average
the quantized superluminal radiation densities with the power-law
distributions (1.1), and
derive the fugacity expansion of the spectral averages. We perform a
tachyonic cascade fit to the *γ*-ray
spectrum of the Galactic center, and obtain estimates of the
thermodynamic parameters of the electron plasma generating the
superluminal radiation. The cutoff energy of the high-energy cascade
fitting the TeV spectrum can be related to anisotropies in the cosmic
ray spectrum detected by the AGASA and SUGAR air shower arrays [11] and [12]. If the TeV
cascade is generated by a protonic source population, this requires a
thermal proton density at .
This cutoff temperature very closely matches the upper energy edge of
both the AGASA and SUGAR excess counts from the Galactic center region,
suggesting that the compact radio source Sagittarius A* is capable of
accelerating protons into the 10^{18} eV
region. In Section 5, we present
our conclusions.

### 2. Thermodynamic variables of Fermi power-law ensembles

We start with the fermionic partition function,

*ρ*(

*H*) stands for density

and

*H*=

*m*

*γ*is the free Hamiltonian. The electronic Lorentz factors

*γ*=(1−

*υ*

^{2})

^{−1/2}range in an interval

*γ*

_{1}

*γ*<

*∞*.

*H*

_{1}=

*m*

*γ*

_{1}is the lower threshold energy, and

*γ*

_{1}≥1 the lower edge of Lorentz factors of the electron distribution. The momentum parametrization is , so that we can substitute , where the angular integration has already been performed, as there is no angular dependence in the integrand. The Heaviside step function

*θ*restricts the integration in (2.1) to

*γ*≥

*γ*

_{1}. The exponent

*α*defines the fugacity , and is related to the chemical potential by

*μ*=−

*m*

*α*/

*β*, cf. after (2.26).

*δ*is the electronic power-law exponent, and the cutoff parameter in the Boltzmann factor, so that the relativistic Fermi–Dirac distribution is recovered with

*δ*=0 and

*γ*

_{1}=1. Here, we study power-law ensembles of arbitrary real power-law index

*δ*. We use

*c*=ħ=1 for most of this article; the units can easily be restored, e.g., , where the ratio of electron rest energy and Boltzmann constant is .

The grand partition function (2.1) is
obtained via a standard trace calculation in fermionic occupation
number representation,

*ω*exceeding the lower threshold

*ω*

_{1}

*H*

_{1}. We briefly sketch the derivation of partition function (2.1) from the trace (2.3). A basis |

*n*for the occupation number representation of the fermionic creation/annihilation operators (labeled by the wave vector ) is explicitly defined in Eq. (4.11) in Ref. [13]. In (2.3), we substitute the particle number operator , where , as well as the energy operator , where . In occupation number representation, the Hermitian number operators are diagonal, , and the fermionic occupation numbers attached to a wave vector can only take the values zero and one. We also note the dispersion relation , relating electron energy and wave number . Finally, is the wave number corresponding to the lower threshold energy

*ω*

_{1}=

*m*

*γ*

_{1}.

We employ box quantization, discretizing the wave vector as .
The following summations are taken over integer lattice points
in *R*^{3},
corresponding to periodic boundary conditions on a box of size *L*^{3}.
By making use of ,
we may write trace (2.3) over the
basis states |*n*
as

The continuum limit

*L*→

*∞*amounts to replacing the summation over the lattice wave vectors by the integration [14]; the factor of two accounts for the spin degeneracy. As we have put ħ=

*c*=1, we can identify , so that . Performing these substitutions in (2.5), we arrive at integral representation (2.1) of the partition function, since the Heaviside function in the integrand (2.2) is equivalent to the restriction

*p*≥

*k*

_{1}of the momentum integration (2.1).

The internal energy,

are obtained from the integral representation (2.1),

Here we have reparametrized the momentum integration in (2.1) with the Lorentz factor as explained after (2.2). The integral representations (2.6), (2.7) and (2.8) are the key ingredients of the Legendre formalism employed below; in particular, log

*Z*(

*δ*,

*β*,

*α*,

*V*) in (2.8) is the starting point for the quasiclassical fugacity expansion (2.10) (at high temperature and low density) of the thermodynamic functions. The opposite asymptotic limit, the nearly degenerate quantum regime (low temperature, high density), is based on the representation

where the logarithm in the integrand of (2.8) has been removed by a partial integration. That is, we write the integrand in (2.8) as

*f*(

*γ*)

*g*

^{′}(

*γ*)/3, with and

*g*=(

*γ*

^{2}−1)

^{3/2}, and integrate by parts to arrive at (2.9).

For the remainder of this section, we derive the
quasiclassical fugacity expansions of the thermodynamic variables. In
Section 3, we will
discuss the low- and high-temperature limits of these expansions and
conditions for their applicability. We start by expanding partition
function (2.8) in
ascending powers of ,

so that

*K*

_{1}=

*K*and

*K*

_{n}=

*K*

_{n,0}. Differentiation with respect to

*β*is denoted by a prime, . The fugacity expansions of the partition function, internal energy, and particle number thus read

We eliminate the fugacity in the partition function and the internal energy by inverting in (2.15):

with expansion parameter

The following expansions are ascending series in . On substituting (2.16) into (2.13) and (2.14), we find the fugacity expansion of the partition function in Helmholtz parametrization

as well as the internal energy

*U*

_{cl}is the classical limit, the internal energy of a Boltzmann power-law density [4], and the first two series coefficients in (2.19) read

The fugacity expansion of the entropy function

is obtained by substituting the ascending -series (2.18), (2.19), and, cf. (2.16),

We find

where the coefficients

*u*

_{1,2}are defined in (2.20), and

is the entropy of a classical Boltzmann power-law density, cf. Ref. [4] and (4.13), with a term added owing to the multiplicity factor in (2.1).

The thermal equation of state is derived from the Helmholtz
free energy,

*P*=−

*∂*

*F*/

*∂*

*V*. By making use of the -series (2.18) and (2.22), we obtain

The fugacity expansion of the chemical potential

*μ*=

*∂*

*F*/

*∂*

*N*=−

*m*

*α*/

*β*is found by substituting the series expansion (2.22) of

*α*.

The isochoric specific heat and the isothermal compressibility
are

*V*(

*δ*,

*β*,

*P*,

*N*) is obtained by solving the thermal equation (2.26). As for the heat capacity, we find the fugacity expansion by substitution of (2.23) and (2.24)

where

*C*

_{V,cl}denotes the classical limit

To obtain the quantum correction of the isothermal compressibility, we have to iteratively solve (2.26) for

*V*(since , cf. (2.17)),

where

*κ*

_{T,cl}=1/

*P*is the classical limit. Thermodynamic stability requires

*C*

_{V}≥0 and

*κ*

_{T}≥0. The leading order of the fugacity expansion of

*C*

_{V}in (2.28) and

*κ*

_{T}in (2.30) is in either case positive, as it coincides with the classical limit based on a Boltzmann power-law density [4].

### 3. Equations of state, entropy, and heat capacity of fermionic power-law densities in the quasiclassical regime

#### 3.1. Low-temperature asymptotics

In the quasiclassical regime covered by the fugacity
expansions in Section 2, the
low-temperature limit of the thermodynamic variables is determined by
the *β*1 asymptotics of integral *K*(*δ*,*β*,*γ*_{1})
in (2.11). At *γ*_{1}=1,
the asymptotic 1/*β*-series
reads

*γ*

_{1}=1 in partition function (2.8). (An ultra-relativistic lower threshold energy can be treated in like manner; the low-temperature expansion of

*K*(

*δ*,

*β*,

*γ*

_{1}1) is given in Ref. [15].) The series coefficients in the fugacity expansions are composed of certain

*K*-ratios. Using the notation (2.12), we find

and the same for

*K*

_{2,1}/(

*K*

_{1,1}

*K*

_{1}), with 45/16 replaced by 57/16. As for coefficient

*u*

_{1}in (2.20), we note , up to terms of

*O*(1/

*β*). Finally,

*K*

_{1,1}/

*K*

_{1}1 in leading order. Since , cf. (2.17), these ratios already suffice to obtain the temperature and density scaling of the leading quantum correction to the thermodynamic variables. The low-temperature limit of the internal energy (2.19) is found as

where the ellipsis indicates terms of , that is, higher orders in

*N*/

*V*. The leading factor

*U*

_{cl}is the low-temperature expansion of the internal energy of a classical power-law density [4]. The thermal equation of state (2.26) reads

where the quantum correction ∝

*N*/

*V*stems from the linear -term in (2.26). We may replace

*β*by the thermal wavelength, , to find condition for the quasiclassical fugacity expansion to apply; at low temperatures, the gas has to be sufficiently dilute. The mean kinetic energy per particle in the low-temperature regime is

*E*

_{av}

*U*/

*N*−

*m*, so that

*β*∝

*m*/

*E*

_{av}in leading order. We thus recover the nonrelativistic wave-mechanical scaling .

The leading quantum correction to entropy, heat capacity, and
compressibility scales linearly with density *N*/*V*.
As for the entropy, cf. (2.23) and (2.24), we find

The classical isothermal compressibility is

*κ*

_{T,cl}=1/

*P*, and the quantum correction is found as, cf. (2.30),

The series expansions (3.3)–(3.7) are in ascending powers of

*N*/

*V*or

*P*. We have only stated the leading quantum correction, that is, the term linear in

*N*/

*V*or

*P*. The indicated temperature scaling of this term is meant in leading order as well, the next-to-leading order being smaller by a factor of

*O*(1/

*β*), cf. (3.3).

#### 3.2. Fugacity expansion and thermal wavelength in the high-temperature limit

Like in Section 3.1, the
quasiclassical high-temperature asymptotics of the thermodynamic
variables are based on the ascending -series
derived in Section 2, namely *U*
in (2.19), *S*
in (2.23), the
thermal equation of state (2.26), and *C*_{V}
in (2.28). As for
the compressibility *κ*_{T},
we will use the ascending *P*-series
in (2.30). The
leading quantum correction to these variables is the term linear in
or *P*,
which is in all cases composed of the *K*-ratios,
cf. (2.11) and (2.12),

*β*1) of these ratios is calculated from the ascending

*β*-series of integral

*K*(

*δ*,

*β*,1) in (2.11), cf. Ref. [15]. The structure of the high-temperature expansion of

*K*(

*δ*,

*β*,1) depends on the power-law index

*δ*. In the following, we list the thermodynamic functions at integer power-law indices

*δ*=0,1,…,4. These indices exhaust all qualitatively different cases; non-integer power-law indices can be dealt with in the same way, based on the

*β*-expansions of

*K*(

*δ*,

*β*,

*γ*

_{1}) in Ref. [15], which cover real

*δ*as well as ultra-relativistic threshold Lorentz factors

*γ*

_{1}1. In this subsection, we study

*γ*

_{1}=1 and integer power-law indices 0

*δ*4.

At *δ*=0,
a thermal Fermi–Dirac distribution is recovered, admitting the
quasiclassical high-temperature expansions [14]

*κ*

_{T}

*P*for

*κ*

_{T}/

*κ*

_{T,cl}, cf. (2.30). The quantum corrections are the terms linear in

*N*/

*V*or

*P*; the logarithms in the expansion of

*S*stem from the classical term

*S*

_{cl}in (2.24). At index

*δ*=1, we find

There is no qualitative difference to the Fermi–Dirac variables (3.9) as yet. The entropy has the usual logarithmic divergence, the heat capacity approaches a finite limit, and the internal energy has a 1/

*β*divergence. All quantum corrections vanish for

*β*→0. At

*δ*=2, we obtain

In the thermal equation of state and the compressibility, the quantum correction is augmented by a factor of 1/

*β*as compared to the previous cases

*δ*=0,1, which has implications for the thermal wavelength, cf. after (3.16). The quantum correction to the compressibility is of the same order as in the thermal equation, since

*P*

*m*

*N*/(

*β*

*V*).

For index *δ*=3,
there emerges a logarithmic temperature dependence already in the
classical internal energy and heat capacity [4]. Introducing
the shortcut

*γ*

_{E}≈0.5772 is Euler’s constant, we find

The entropy diverges very slowly for

*β*→0, owing to a double-logarithmic divergence. The heat capacity approaches zero logarithmically, but stays positive at finite temperature. The quantum corrections vanish logarithmically, except for

*κ*

_{T}. At

*δ*=4, the expansions read

with

*l*

_{E}(

*β*) defined in (3.12). In this case, the internal energy diverges logarithmically, and the entropy approaches a finite limit at

*β*=0. (For power-law indices

*δ*>4, the internal energy reaches saturation, attaining a finite limit at

*β*=0 like the entropy, since integral (2.6) stays finite without exponential cutoff.) In (3.14), the quantum correction to the thermal equation is in leading order independent of

*β*, approaching the indicated finite limit linear in

*N*/

*V*. The quantum correction to the specific heat has the same linear

*β*-dependence as the classical term, in leading order that is, and both terms are positive. (At

*δ*=5, the classical term as well as the quantum correction scale ∝

*β*

^{2}

*l*

_{E}(

*β*) in leading order, with positive proportionality constants.)

In the fugacity expansions (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14), the
quantum correction is the term linear in *N*/*V*
(or *P*
in the case of *κ*_{T}*P*).
Only the leading order in *β*
is indicated, in the classical term as well as the quantum correction,
except for the entropy function in (3.14), where
the next-to-leading order in *β*
is included in the quantum correction, so that the heat capacity can be
recovered by differentiation, cf. (2.27).
Otherwise, the omitted *β*-terms
are by at least a factor of *O*(*β*log*β*)
smaller than the indicated ones. (In the expansion procedure, the logs
are treated as constants.) The ellipses in expansions (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14) indicate
higher-order quantum corrections in powers of *N*/*V*.
The units are restored by replacing *m*
by *m**c*^{2}
on the left-hand side of the above equations as well as in *β*,
cf. after (2.2); on the
right-hand side, *m*
is replaced by *m**c*/ħ.

As mentioned after (3.8),
high-temperature expansions at a non-integer power-law exponent *δ*
are calculated analogously, by making use of ratios (3.8) and the
expansions of *K*(*δ*,*β*,1)
in Ref. [15].
High-temperature expansions of thermodynamic variables are differently
structured in different *δ*-intervals,
*n*−1<*δ*<*n*,
joining the expansions (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14) at
integer *δ*.
Ultra-relativistic high-temperature expansions of *K*(*δ*,*β*,*γ*_{1}1) have been obtained in Ref. [15] as well, to
be substituted into the ratios (3.8) in the
case of power-law distributions with Lorentz factors exceeding a
high-energy threshold, cf. (2.3).

Returning to the thermal equation of state (2.26), we
define the thermal wavelength *λ*_{T}
by writing the leading quantum correction in (2.26) as

*λ*

_{T}is chosen in a way to recover the usual definition, , in the low-temperature limit: for

*β*1. The numerical proportionality factors in and

*λ*

_{T}are a mere convention;

*λ*

_{T}provides a second length scale to be compared to (

*V*/

*N*)

^{1/3}, which sets the scale for the quantum correction, the quasiclassical regime being defined by , cf. after (3.4). The high-temperature scaling of

*λ*

_{T}at integer power-law indices

*δ*=0,1,…,4 is

where

*l*

_{E}(

*β*) denotes the logarithmic temperature dependence (3.12). If

*δ*3, we invert the caloric equation of state to find in leading order, cf. (3.9), (3.10), (3.11), (3.12) and (3.13),

where

*E*

_{av}stands for the mean particle energy

*U*/

*N*, and

*β*1 is implied. On combining (3.16) and (3.17), we find the high-temperature dispersion relation

*λ*

_{T}(

*E*

_{av}). At

*δ*=0,1, we recover the ultra-relativistic wave-mechanical scaling

*λ*

_{T}∝1/

*E*

_{av}. For

*δ*≥4, the thermal wavelength is proportional to the Compton wavelength,

*λ*

_{T}∝1/

*m*. For intermediate power-law indices, the thermal wavelength is a hybrid of ultra-relativistic and Compton wavelength, e.g., .

### 4. Superluminal *γ*-rays
from the Galactic center: Tachyonic spectral maps and electronic source
distributions

In this section, we average tachyonic radiation
densities [15] with
fermionic power-law distributions, and use the spectral averages to
perform a cascade fit to the *γ*-ray
broadband of the Galactic central source Sagittarius A*, cf. Fig. 1
and Fig. 2.
The thermodynamic parameters of the electron plasma in the Galactic
center generating the superluminal cascades are extracted from the
spectral fit, cf. Table 1.
First we briefly summarize the tachyonic radiation densities, cf. (4.1) and (4.2), then we
explain the spectral averaging, in particular the fugacity expansion of
the spectral functions and their low- and high-temperature asymptotics,
cf. (4.3), (4.4), (4.5), (4.6), (4.7), (4.8), (4.9), (4.10), (4.11), (4.12), (4.13), (4.14), (4.15), (4.16), (4.17), (4.18) and (4.19). The
spectral fit in Fig. 1
and Fig. 2
is discussed after (4.20).

Fig. 1. *γ*-ray
wideband of the Galactic center. EGRET data points from Ref. [2], HESS points
from Refs. [7] and [8], MAGIC
points from Ref. [9]. EGRET
points refer to the source 3EG J1746 − 2851, HESS and
MAGIC points to HESS J1745 − 290. The solid line T+L
depicts the unpolarized differential tachyon flux ,
obtained by adding the flux densities *ρ*_{1,2}
of two electron populations and rescaled with *E*^{2}
for better visibility of the spectral curvature, cf. (4.20). The
transversal (T, dot-dashed) and longitudinal (L, double-dot-dashed)
flux densities
add up to the total flux T+L. The *χ*^{2}-fit
is done with the unpolarized tachyon flux T+L, and subsequently split
into transversal and longitudinal components. The exponential decay of
the cascades *ρ*_{1,2}
sets in at about ,
implying cutoffs at 5.5 TeV for the *ρ*_{1}
cascade and at 38 GeV for *ρ*_{2},
which terminate the spectral plateaus. The unpolarized flux T+L is the
actual spectral fit, the parameters of the electron densities are
recorded in Table 1.

Electronic source densities *ρ*_{1,2}
generating the *γ*-ray
broadband of the TeV source HESS J1745 − 290 and the
associated EGRET source 3EG J1746 − 2851 in the
Galactic center region

Each *ρ*_{i}
stands for a Maxwell–Boltzmann density ,
cf. (4.13). *β*
is the cutoff parameter in the Boltzmann factor.
determines the amplitude of the tachyon flux generated by *ρ*_{i},
from which the electron count
is inferred at the indicated distance of 8 kpc, cf. after (4.21). (The
subscript 1 in
and
has been dropped.)
is the electron temperature, and *U*
(erg) the internal energy of the thermal densities *ρ*_{i}.
The cascades labeled *ρ*_{1,2}
in Fig. 1
and Fig. 2
are obtained by averaging the tachyonic radiation densities (4.1) with the
electron densities *ρ*_{i},
cf. (4.14), (4.15) and (4.16). The
parameters *β*
and
are extracted from the least-squares fit T+L in Fig. 1.

Fig. 2. Close-up of the HESS spectrum in Fig. 1.
The TeV spectral map coincides with the *ρ*_{1}
cascade, since the *ρ*_{2}
flux is exponentially cut at 38 GeV. T and L stand for the transversal
and longitudinal flux components, and T+L labels the unpolarized flux.
The HESS points define a spectral plateau in the high GeV range typical
for tachyonic cascade spectra [15] and [16]. The
spectral curvature is intrinsic, being generated by the Boltzmann
factor in the electron densities.

The quantized tachyonic radiation densities of a uniformly
moving spinning charge read [16]

*γ*is the electronic Lorentz factor,

*α*

_{q}the tachyonic fine structure constant, and

*m*

_{t}the tachyon mass. A spectral cutoff occurs at

Only frequencies in the range 0

*ω*

*ω*

_{max}(

*γ*) can be radiated by a uniformly moving charge, the tachyonic spectral densities being cut off at the break frequency

*ω*

_{max}. The units ħ=

*c*=1 can easily be restored. We use the Heaviside–Lorentz system, so that

*α*

_{q}=

*q*

^{2}/(4

*π*ħ

*c*)≈1.0×10

^{−13}. The tachyon mass is , and the tachyon–electron mass ratio

*m*

_{t}/

*m*≈1/238. These estimates are obtained from hydrogenic Lamb shifts [17]. A positive

*ω*

_{max}(

*γ*) requires Lorentz factors exceeding the threshold

*μ*

_{t}in (4.2), since

*ω*

_{max}(

*μ*

_{t})=0. The lower threshold on the speed of the electron for radiation to occur is thus

*υ*

_{min}

*m*

_{t}/(2

*m*

*μ*

_{t}). The tachyon–electron mass ratio gives

*υ*

_{min}/

*c*≈2.1×10

^{−3}, cf. Ref. [18].

The radiation densities (4.1) refer to a
single charge with Lorentz factor *γ*.
We average them with a Fermi power-law distribution,

*A*

_{F}

*m*

^{3}

*V*/

*π*

^{2}, cf. (2.6), (2.7) and (2.8). The particle number is , where

*γ*

_{1}is the lower edge of Lorentz factors in the source population. The exponential cutoff in (4.3) is related to the electron temperature by . In this section, we write the exponent of the fugacity with a hat, cf. after (2.2), to distinguish it from the electron index customarily defined as

*α*=

*δ*−2, cf. (4.13). The normalization factor

*A*

_{F}is dimensionless via

*m*→

*m*

*c*/ħ, cf. (1.1); the volume factor in the quantum corrections discussed in Section 3 is thus found as , where is the reduced electronic Compton wavelength.

The spectral average of the radiation densities (4.1) is carried
out as

*θ*is the Heaviside step function. The spectral range of densities (4.1) is bounded by

*ω*

_{max}, so that the solution of defines the minimal electronic Lorentz factor for radiation at this frequency,

The average (4.4) can be reduced to the fermionic spectral functions

with lower integration boundary

*γ*

_{1}≥

*μ*

_{t}, cf. after (4.2). The threshold Lorentz factor

*γ*

_{1}defines the break frequency [19]

which separates the spectrum into a low- and high-frequency band. In particular, , and if

*ω*>

*ω*

_{1}. By making use of the spectral functions (4.6), we can write the averaged radiation densities (4.4) as

with in (4.5) and

*ω*

_{1}in (4.7). The superscripts T and L denote the transversal and longitudinal radiation components, cf. (4.1). The spectral functions in (4.8) are assembled by substituting the radiation densities (4.1) into the integral representation (4.6),

where the coefficients

*f*

_{k}read

with density in (4.3).

The quasiclassical fugacity expansion of the spectral
functions (4.9) is found
by expanding density (4.3) in
ascending powers of ,
cf. (2.10),

*f*

_{k}(

*γ*

_{1}) in (4.10) reads

The classical limit of the fermionic density is defined by the leading term in series (4.11), the Boltzmann power-law density

with normalization . (As mentioned after (4.3), exponent in the normalization factor defines the fugacity, and is not to be confused with the electron index

*α*=

*δ*−2.) The electron count based on the classical density (4.13) is , to be identified with the renormalized electron count obtained from the spectral fit, cf. after (4.21).

The classical limit of the fermionic spectral functions
in (4.6) is the
Boltzmann average [15]

*m*

_{t}/

*m*factors in (4.1) and (4.2). In particular, and , cf. (4.1). Carrying out the integration in (4.14), we find the classical spectral functions

which can also be obtained from the Fermi functions in (4.9) by dropping the

*m*

_{t}/

*m*terms and substituting the leading order of the fugacity expansion of

*f*

_{k}(

*γ*

_{1}), cf. (4.12). The classical limit of the fermionic spectral average in (4.8) thus reads

where is the classical limit of the break frequency (4.7).

In the low-temperature limit, *β**γ*_{1}1, the incomplete -functions
occurring in the fugacity expansion of *f*_{k}(*γ*_{1})
in (4.12) can be
replaced by the asymptotic series

*β*

*γ*

_{1}1. In the fugacity expansion (4.12) of the reduced spectral functions, we can therefore substitute the ascending series

If

*k*−

*δ*

*n*is zero or a negative integer,

*k*−

*δ*

*n*=−

*m*,

*m*≥0, we use instead of (4.18) the ascending series of the exponential integral

*E*

_{m+1}(

*β*

*γ*

_{1}),

The spectral fit in Fig. 1 and Fig. 2 is based on the

*E*

^{2}-rescaled flux densities [20]

where

*d*is the distance to the source and the spectral average (4.16) (with

*ω*=

*E*/ħ). The fit is done with the unpolarized flux density of two electron populations

*ρ*

_{i=1,2}, thermal Maxwell–Boltzmann distributions (4.13) with

*α*=−2 and

*γ*

_{1}=1, cf. Table 1. Each electron density generates a cascade

*ρ*

_{i}, and the wideband fit is obtained by adding two cascade spectra, labeled

*ρ*

_{1}and

*ρ*

_{2}in Fig. 1. As for the electron count, , we use a rescaled parameter for the fit,

which is independent of the distance estimate in (4.20). Here, implies the tachyon mass in keV units, that is, we put

*m*

_{t}≈2.15 in the spectral density (4.1). At

*γ*-ray energies, only a tiny -fraction (the ratio of tachyonic and electric fine structure constants) of the tachyon flux is absorbed by the detector, which requires a rescaling of the electron count

*n*

_{1}, so that the actual number of radiating electrons is . We thus find the electron count to be , where defines the tachyonic flux amplitude extracted from the spectral fit [20]. This renormalized count is to be identified with the particle number

*N*in the thermodynamic variables of the electron plasma. The electron temperature and cutoff parameter in the Boltzmann factor are related by , and the energy estimates in Table 1 are based on , cf. (3.9).

Fig. 1
and Fig. 2
show the spectral map of the TeV *γ*-ray
source HESS J1745 − 290 and the GeV source 3EG
J1746 − 2851. Both sources are located in the
vicinity of the unidentified compact radio source Sagittarius (Sgr) A*
at the core of the Galactic center region, which comprises the
supernova remnant Sgr A East and the pulsar wind nebula
G359.95 − 0.04, as well as molecular cloud complexes
such as Sgr B and Sgr C [21] and [22].

In Table 1,
we list the thermodynamic parameters of the electron populations
generating the tachyonic cascade spectra *ρ*_{1,2}
depicted in the figures. In the case of protonic source densities (4.13), the
cutoff energy
in the Boltzmann factor has to be multiplied by 1.84×10^{3},
the proton/electron mass ratio, resulting in a cutoff energy of 10^{16.2} eV
for a protonic *ρ*_{2}
population. The protonic high-energy population *ρ*_{1}
is cut at 10^{18.4} eV, which is to be
compared to the excess fluxes from the Galactic center region observed
by AGASA in the 10^{18}–10^{18.4} eV
range [11], and the
SUGAR array in the 10^{17.9}–10^{18.5} eV
interval [12].

The tachyonic spectral maps are further explained in the
figure captions, and can be compared to electromagnetic inverse-Compton
fits [23] or hadronic
fits based on proton–proton scattering and pion decay [2] and [24]. The
electronic source count for the Crab Nebula at
is 1.9×10^{50}, cf. Ref. [16], as
compared to 3.4×10^{49} for HESS
J1745 − 290 and 3EG J1746 − 2851,
at a distance of 8 kpc, cf. Table 1.
The internal energy of the electron gas is 7.2×10^{51} erg,
to be compared to hadronic model estimates predicting 5×10^{45} erg [24], 10^{49}–10^{50} erg [25], and 5.1×10^{50} erg [23] for a
protonic source population.

### 5. Conclusion

We have investigated superluminal radiation from electron
populations in the Galactic center region, and found the thermodynamic
parameters of the source densities. We demonstrated that the *γ*-ray
wideband of the Galactic center can be fitted with a tachyonic cascade
spectrum. In particular, the extended spectral plateau in the MeV–GeV
range as well as the spectral curvature apparent in double-logarithmic
plots are reproduced by the cascade fit in Fig. 1
and Fig. 2.
Estimates of the temperature, the source count, and the internal energy
of the electron plasma generating the superluminal cascades are given
in Table 1.

In Section 4, we averaged
the superluminal radiation densities with fermionic power-law
distributions, and derived the quantized spectral functions. The
power-law densities in Sections Sections 2 and 3 and the
spectral averages were studied mostly in the quasiclassical limit, as *γ*-ray
spectral fits are done in the high-temperature regime. The opposite
asymptotic limit of nearly degenerate fermionic power-law ensembles at
low temperature and high density will be discussed elsewhere.

We quantized Boltzmann power-law densities in Fermi–Dirac
statistics, derived the fermionic partition function, developed the
Legendre formalism of fermionic power-law distributions, and calculated
the fugacity expansion of the thermodynamic variables, cf.
Section 2. The
qualitative dependence of the variables on the electronic power-law
index, the temperature scaling of the quantum corrections, and the
thermal wavelength of power-law ensembles were investigated in
Section 3. We have
focused on *γ*-ray
spectra, which can be fitted with ultra-relativistic electron densities
at high temperature, cf. Section 4. As for
tachyonic X-ray spectra obtained from diffraction gratings [21], the
tachyon mass of 2 keV has to be included in the dispersion relation
when parametrizing the glancing angle in the Bragg condition with
energy, which affects the shape of the spectral maps in the X-ray
bands; this will be discussed elsewhere. The spectral fit in Fig. 1
and Fig. 2
is performed with the unpolarized tachyon flux. At *γ*-ray
energies, the speed of tachyons is close to the speed of light, the
basic difference to electromagnetic radiation being the longitudinal
flux component [26]. The
polarization of tachyons can be determined from transversal and
longitudinal ionization cross-sections of Rydberg atoms, which peak at
different scattering angles [27].

### Acknowledgements

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.