Volume 385, Issue 2, 15 November 2007, Pages 558-572
Roman Tomaschitz^{}^{, }^{a}^{, }^{}
Abstract
Tachyonic spectral densities of ultra-relativistic electron populations are fitted to the γ-ray spectra of two microquasars, LS 5039 and LSI +61°303. The superluminal spectral maps are obtained from BATSE, COMPTEL, EGRET, HESS, and MAGIC data sets. The spectral averaging is done with exponentially cut power-law densities. Estimates of the electron distributions generating the tachyon flux are obtained from the spectral fits, such as power-law indices, electron temperature and source counts. The internal energy and heat capacities of the source populations are calculated. An extensive entropy functional is defined for Boltzmann power-law densities and its stability is checked. The high-temperature limit of the thermodynamic variables is determined by the power-law index of the electron plasma, which enters in the scaling exponents as well as the amplitudes.
Keywords: Superluminal radiation; Tachyonic γ-rays; Boltzmann power-law ensembles; Spectral averaging; γ-Ray binaries
PACS classification codes: 05.70.Ce; 05.20.Gg; 52.25.Kn; 95.30.Tg
Article Outline
- 1. Introduction
- 2. Tachyonic γ-ray spectra of microquasars
- 3. Caloric equation of state and heat capacities of electronic power-law distributions
- 4. Ultra-relativistic Boltzmann power laws
- 5. Conclusion
- Acknowledgements
- Appendix A. High-temperature asymptotics of Gibbs free energy and entropy
- References
1. Introduction
We study the tachyonic spectral maps of two microquasars, LS 5039, cf. Refs. [1] and [2], and LSI +61°303, cf. Refs. [3] and [4], and reconstruct the electron distributions emitting the superluminal radiation. The spectral maps of these γ-ray binaries suggest cascade spectra which are generated by electronic power-law densities exponentially cut with the Boltzmann factor. Classical statistics applies, as the high-temperature regime is invoked at γ-ray energies. We average the tachyonic spectral densities with electronic power-law distributions, and infer the power-law index, the electron count, and the temperature from the spectral fits. Based on these parameters, we calculate the internal energy and the specific heat of the ultra-relativistic electron plasma of the microquasars. To this end, we set up the thermodynamic formalism for Boltzmann power-law ensembles. Starting with the grand partition function, we find an entropy functional for exponentially cut power-law densities that is extensive and stable. We derive the Helmholtz and Gibbs free energies as well as the low- and high-temperature expansions of the caloric equation of state and the heat capacities, in particular their high-temperature scaling depending on the electronic power-law index.In Section 2, we briefly sketch the transversal and longitudinal tachyonic spectral densities and their averaging with Boltzmann power laws, which results in cascade spectra. We perform the spectral fits to the γ-ray wideband of the microquasars, and extract the parameters of the electronic source populations, such as power-law index, temperature and source number. In Section 3, we develop the thermodynamic formalism for exponentially cut power-law densities and derive the low- and high-temperature expansions of the thermodynamic variables, focusing on entropy, equations of state, and the specific heats. In Section 4, we investigate ultra-relativistic Boltzmann ensembles, power-law densities with Lorentz factors exceeding a high-energy threshold, and we calculate the internal energy and heat capacities of the electron gas in the two microquasars. In Section 5, we present our conclusions. In Appendix A, we list the asymptotic expansions of the partition function, entropy, and the free energies at integer electronic spectral index, which requires special treatment due to logarithmic temperature dependence in the high-temperature regime.
2. Tachyonic γ-ray spectra of microquasars
The tachyonic radiation densities of uniformly moving
electrons were derived in Ref. [5],
The radiation densities (2.1) refer to a
single charge with Lorentz factor γ. We average
them with ultra-relativistic electron distributions, power-law
densities exponentially cut with the Boltzmann factor:
and the spectral fit is based on the E^{2}-rescaled flux densities:
where d is the distance to the source. The spectral maps of the microquasars in Fig. 1, Fig. 2 and Fig. 3 are fitted with the unpolarized flux density dN^{T+L}=dN^{T}+dN^{L} of electron populations ρ_{i}_{=1,2} specified in 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 the figures.
Fig. 1. γ-Ray wideband of the microquasar LS 5039. BATSE data points from Ref. [8], EGRET points from Refs. [9] and [10], and HESS points from Ref. [11]. BATSE and EGRET data refer to the associated EGRET source , HESS points to at the inferior conjunction as defined in Ref. [11]. The solid line T+L depicts the unpolarized differential tachyon flux dN^{T+L}/dE, 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. (2.4). The transversal (T, dot-dashed) and longitudinal (L, double-dot-dashed) flux densities dN^{T,L}/dE add up to the total flux T+L. The exponential decay of the cascades ρ_{1,2} sets in at about E_{cut}≈(m_{t}/m)kT, implying cutoffs at 5 TeV for the ρ_{1} cascade and at 2 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.
Fig. 2. Close-up of the HESS spectrum of LS 5039 in Fig. 1. The TeV spectral map coincides with the ρ_{1} cascade, since the ρ_{2} flux is exponentially cut at 2 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 cascade spectra, followed by exponential decay. The spectral curvature is generated by the Boltzmann factor in the electron densities. If rescaled, this spectral map is quite similar to the cascade spectra of the Markarian galaxies Mkn 501 and Mkn 421 in Figs. 5 and 6 of Ref. [6].
Fig. 3. Spectral map of the microquasar LSI +61°303, associated with the EGRET source 3EG J0241+6103. COMPTEL points from Ref. [12], EGRET data from Refs. [13] and [14], and MAGIC points from Ref. [15]. The plots are labeled as in Fig. 1. The χ^{2}-fit is done with the total unpolarized tachyon flux T+L and subsequently split into transversal (dot-dashed) and longitudinal (double-dot-dashed) components. A spectral break at is visible as edge in the spectral map; exponential decay sets in at the cutoff frequency . In the extended crossover regime between the spectral break and the spectral cutoff, the decay is gradual but not power law, with electron index α=1. (A power-law crossover, , occurs for electron indices α>1, cf. Ref. [6]).
Electronic source densities ρ_{1,2} generating the γ-ray broadband of the microquasars LS 5039 and LSI +61°303
Each ρ_{i} stands for a Boltzmann power-law density defined by parameters , cf. (2.2). α is the electronic power-law index, β=mc^{2}/(kT) the cutoff parameter in the Boltzmann factor, and γ_{1} the lower edge of Lorentz factors in the electronic source population ρ_{i}. The amplitude of the tachyon flux generated by ρ_{i} is determined by , from which the electron count n^{e} is inferred at the indicated distance, cf. (2.5). (The subscript 1 in and n^{e} has been dropped). kT is the electron temperature. In the thermodynamic variables, we identify δ=α+2 and put N=n^{e}, cf. Sections 3 and 4. The cascades labeled ρ_{i} in Fig. 1, Fig. 2 and Fig. 3 are obtained by averaging the tachyonic radiation densities (2.1) with the electron distributions ρ_{i}, cf. (2.3) and (2.4). The parameters α, β, γ_{1}, and are extracted from the least-squares fit T+L in the figures. As for the thermal high-energy population ρ_{1} of LSI +61°303, it is not yet possible to determine the spectral cutoff and the electron temperature from the presently available TeV data.
As for the electron count n_{1},
we use a rescaled parameter
for the fit:
In Fig. 1 and Fig. 2, we depict the spectral map of the microquasar LS 5039, located at a distance of 2.5 kpc, a compact object orbiting a massive O-star [8], [9], [10] and [11]. Mass estimates and orbital parameters are given in Refs. [1] and [2]; we do not list them here, as they are not required to infer the thermodynamic parameters of the electron plasma. LS 5039 is associated with the unidentified EGRET source and the TeV source , cf. Refs. [11] and [18]. The tachyonic spectral maps are further explained in the figure captions. The GeV spectrum in Fig. 1 can be compared to the spectral map in Fig. 2 of Ref. [10], based on inverse Compton scattering.
Fig. 3 shows the tachyonic spectral map of the microquasar LSI +61°303, at d≈2 kpc, associated with the unidentified EGRET source 3EG J0241+6103, cf. Refs. [12], [13], [14], [15], [19] and [20]. Estimates of the orbital parameters of this binary system, a massive Be star with a compact companion, are given in Refs. [3] and [4]. Inverse-Compton fits of the EGRET spectrum can be found in Figs. 2–4 of Ref. [14]. The thermodynamic variables of the electron populations are discussed in Section 3, and estimates of the internal energy and the specific heats are derived in Section 4. The spectral fits in Fig. 1, Fig. 2 and Fig. 3 are 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. The polarization of tachyons can be determined from transversal and longitudinal ionization cross-sections [21] and [22].
3. Caloric equation of state and heat capacities of electronic power-law distributions
The exponentially cut power-law distribution (2.2) is obtained from the momentum-space density dρ∝H^{-δ}e^{-H/(kT)}d^{3}p, where H=mγ is the free Hamiltonian, and the electronic Lorentz factors range in the interval γ_{1}γ<∞. The power-law exponent δ is related to the electron index in Eq. (2.2) by δ=α+2. The thermodynamic formalism employed is standard, and no derivations are given, although we will consider stability conditions such as the positivity of the specific heat and compressibility. We will discuss the low- and high-temperature expansions of the caloric equation of state and the isochoric heat capacity, in particular their qualitative dependence on the power-law index δ. The typical range of δ in spectral averages is narrow, very rarely outside the interval 0δ4, cf. Refs. [23], [24], [25], [26], [27], [28], [29], [30] and [31]. The relativistic thermal Maxwell–Boltzmann distribution in 3D is recovered with δ=0 and γ_{1}=1, cf. Ref. [32]. Here, we will study power-law ensembles with arbitrary real index δ.
We start with the classical grand partition function
where H/m=γ and β=m/(kT). The thermal wavelength reads , and is required for classical statistics to apply. We parametrize the momentum integration in Eq. (3.1) with the Lorentz factor, , and perform the angular integration, so that . In the ultra-relativistic limit, p mγ, the factor γ^{−δ} in density (3.2) can be generated by analytic continuation in the space dimension. The fact that the exponent δ varies only in a narrow range also suggests treating it like a dimension rather than an intensive variable when setting up the thermodynamic formalism for power-law ensembles. We find
where
The momentum integration (3.3) is over R^{3}, as we at first consider the full range of Lorentz factors, 1γ<∞. In Section 4, we will discuss a truncated electron density, restricting the integration range in Eq. (3.3) to HH_{1}=mγ_{1}, γ_{1}>1, which amounts to replacing the lower integration boundary in Eq. (3.4) by γ_{1}; in particular, we will study ultra-relativistic ensembles with high-energy threshold γ_{1}1. In this section, we assume γ_{1}=1.
The summation in Eq. (3.1) gives
Here, m^{3}V is dimensionless, so that m→mc/ when restoring units, and β=mc^{2}/(kT). On eliminating the fugacity e^{-α}, we obtain the caloric equation of state:
The chemical potential reads
and the entropy is defined as
By making use of N=log Z, we find
The homogeneity relation, S/N=S(δ,β,V/N,1), is satisfied, and we may substitute β(U/N) by solving (3.8).
We note the Helmholtz free energy U – TS,
This is the energy required to shift the exponential cutoff of the power-law slope, that is, to vary the electron temperature without changing volume and power-law index. The isothermal compressibility is
Thermodynamic stability, d^{2}S0 at dS=0 (with δ kept fixed) requires C_{V}0 and κ_{T}0, the latter is obviously satisfied. As for the first, we note ∂K(δ,β)/∂β=-K(δ-1,β) and the Schwarz inequality K^{2}(δ-1,β)<K(δ,β)K(δ-2,β), which implies C_{V}>0. As for energy and number fluctuations, we note the variances
which result in decay of the fluctuations ΔU/U and ΔN/N. The isobaric expansivity is
where , and the isobaric heat capacity C_{P} is calculated by means of the identity
The adiabatic compressibility and the adiabatic expansion coefficient read
and are calculated via the identities [33]
where γ_{C} denotes the heat capacity ratio C_{P}/C_{V}. The positivity conditions C_{V}>0 and κ_{T}>0 thus imply C_{P}>C_{V}, γ_{C}>1, and κ_{T}>κ_{S}>0; the adiabatic expansivity α_{S} is negative.
The low-temperature expansion of the equation of state (3.8) reads
The leading order is just the non-relativistic result; at low temperature, the power-law index enters only into the relativistic corrections. Here and in the following, the asymptotic expansions are obtained by term-by-term differentiation of the log K series in Appendix A.
The high-temperature expansions of the internal energy and the
isochoric specific heat differ qualitatively in different δ
ranges. We define the coefficients
In the range 1<δ<3, the next-to-leading order is modified as
In the interval 3<δ<4, we obtain
and for 4<δ<5
Finally, for δ>5
The leading-order coefficients are all positive, which is easy to see except for C_{V} in Eq. (3.27), where positivity follows from the Schwarz inequality, cf. after (3.14). Regarding units, we note and .
In the case of integer δ, there appear
singularities in the expansion coefficients, which cancel if ε
expanded, cf. Appendix A and
Ref. [7]. We list the
equation of state and the specific heat for integer power-law index 0δ5.
Outside this range, the indicated orders of expansions (3.23) and (3.27) are
singularity free and can be used even at integer δ.
As for the caloric equation, we find
The case δ=0 is the high-temperature limit of a Maxwell–Boltzmann distribution [32]. The specific heat is safely positive, but approaches zero in the high-temperature limit for power-law indices δ3.
4. Ultra-relativistic Boltzmann power laws
We proceed as in Section 3, but
restrict the momentum integration in the partition function (3.1) to
ultra-relativistic Lorentz factors, so that
with γ ranging above a high-energy edge γ_{1}1.
In the partition function (3.1), we
substitute the truncated density
with density ρ in Eq. (3.2), so that
Eqs. (3.5), (3.6), (3.7), (3.8), (3.9), (3.10), (3.11), (3.12), (3.13), (3.14), (3.15), (3.16), (3.17), (3.18) and (3.19) remain valid with K(δ,β) replaced by K(δ,β,γ_{1}); internal energy and heat capacity read
Schwarz's inequality as stated after (3.14) also applies to K(δ,β,γ_{1}), so that the maximum condition d^{2}S0 at fixed δ and γ_{1} is met. The low-temperature expansion (3.20) of the internal energy is replaced by [7]
where . The asymptotic parameter is , so that this expansion remains valid for small β and large βγ_{1}. The low-temperature limit of the heat capacity (4.5) reads
If γ_{1}≈1, these expansions break down; the low-temperature expansion of K(δ,β,γ_{1}) in the case of small but non-vanishing υ_{1} is discussed in Ref. [7]. At γ_{1}=1, the low-temperature expansions (3.20) and (3.21) apply. In this section, we consider ultra-relativistic power-law ensembles, with Lorentz factors exceeding a high-energy threshold γ_{1}1.
The high-temperature expansion of the internal energy and the
specific heat as stated in Eqs. (3.23), (3.24), (3.25), (3.26) and (3.27) can also
be used for γ_{1}>1,
if we replace the coefficients c_{k}(δ)
in Eq. (3.22) by
The corresponding high-temperature expansions of the ultra-relativistic heat capacity at integer power-law index 0δ5 are
The expansions in Eqs. (4.9) and (4.10) are ultra-relativistic, terms of order are systematically dropped, cf. (A.22). The high-temperature expansions valid for γ_{1}=1 are stated in Eqs. (3.28) and (3.29). In Table 2, we list the internal energy and heat capacities of the electron populations in the microquasars LS 5039 and LSI +61°303. The input parameters are inferred from the spectral fits in Fig. 1, Fig. 2 and Fig. 3, as explained in the table captions.
Internal energy and heat capacities of the electronic source densities ρ_{1,2} in the microquasars
U/(n^{e}mc^{2}) | U (erg) | C_{V} (erg/K) | C_{P}/C_{V} | |
---|---|---|---|---|
LS 5039 | ||||
ρ_{1} | 6.83×10^{9} | 5.4×10^{50} | 4.0×10^{31} | 1.33 |
ρ_{2} | 2.79×10^{6} | 1.7×10^{48} | 3.1×10^{32} | 1.33 |
LSI +61°303 | ||||
ρ_{1} | — | — | 1.7×10^{31} | 1.33 |
ρ_{2} | 2.57×10^{7} | 5.9×10^{49} | 3.4×10^{31} | 12.4 |
U/(n^{e}mc^{2}) denotes the internal energy normalized with the rest mass of the respective electron population ρ_{i}. The internal energy U (erg) of the thermal densities ρ_{1,2} in LS 5039 and ρ_{1} in LSI +61°303 is calculated via the high-temperature expansion U(δ=0) in Eq. (3.28). The series U(δ=3,γ_{1}) in Eq. (4.9) is used for the ultra-relativistic power-law density ρ_{2} of LSI +61°303. The input parameters δ=α+2, β, γ_{1}, and N=n^{e} are listed in Table 1. The isochoric heat capacity is calculated by means of the series expansions C_{V}(δ=0) in Eq. (3.29) and C_{V}(δ=3,γ_{1}) in Eq. (4.10). The adiabatic index C_{P}/C_{V} is obtained from identity (3.17). The internal energy of the thermal population ρ_{1} in LSI +61°303 remains undetermined, as the cutoff parameter β of this distribution could not be extracted from the spectral fit in Fig. 3. Regarding the ultra-relativistic power-law density ρ_{2} of LSI +61°303, we note βγ_{1}≈1.9×10^{-5}, cf. Table 1, so that the quoted high-temperature expansions of internal energy and specific heat are applicable, obtained from the ultra-relativistic expansion K(δ=3,β,γ_{1}) listed in Eq. (A.22).
5. Conclusion
We have investigated superluminal radiation from ultra-relativistic electron populations in γ-ray binaries and explained how to obtain the thermodynamic variables of the source densities. We demonstrated that the γ-ray spectra of the microquasars can be fitted with tachyonic cascade spectra. The spectral curvature is intrinsic and reproduced by the tachyonic spectral densities (2.1) averaged with thermal and power-law electron distributions. In Table 2, we have given estimates of the internal energy and the heat capacities of the electron plasmas in the microquasars LS 5039 and LSI +61°303. More generally, we have shown that Boltzmann power-law densities admit a stable and extensive entropy function (3.11). On that basis, we have derived the caloric equation of state and studied the temperature scaling of the internal energy and the heat capacities in the high-temperature regime, in particular the dependence of the scaling exponents on the electronic power-law index.
The tachyonic radiation densities (2.1) are generated by electrons in uniform motion. In Ref. [17], we studied superluminal radiation from ultra-relativistic electrons orbiting in strong magnetic fields and derived the pitch-angle scaling of the radiation densities. In the zero-magnetic-field limit, the tachyonic synchrotron densities converge to the spectral densities for uniform motion [34]. Orbital curvature induces modulations in the spectral slopes of densities (2.1), but these ripples are attenuated when performing a pitch-angle average, cf. Figs. 1–3 in Ref. [17]. Thus we can use uniform radiation densities in the spectral fits even in the presence of magnetic fields, for instance, if the compact companion is a magnetized neutron star [1] and [4]. The orbital parameters of the binary system do not directly enter into the spectral fit either. If one considers microquasars as scaled-down Galactic counterparts of active galactic nuclei [2] and [10], one would rather expect the spectra to be determined by the global thermodynamic parameters of the electron populations in these objects, which are in turn inferred from the spectral maps, cf. Table 1.
We have focused on γ-ray spectra, which can be fitted with ultra-relativistic electron populations in the high-temperature regime. Otherwise we have to consider the quantized tachyonic spectral densities [6] instead of the classical limit (2.1). In the low-temperature regime, we also have to replace the Boltzmann power-law density in the spectral average (2.3) with its quantized fermionic counterpart. As for tachyonic X-ray spectra obtained from diffraction gratings, 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.
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.
References
Appendix A. High-temperature asymptotics of Gibbs free energy and entropy
The low- and high-temperature expansions of the partition
function
The first few orders of the low-temperature expansion read [7]
The high-temperature expansion is composed of two series:
where
The general term of the series indicated in Eq. (A.5) is (-)^{k}(1/2)_{k}Γ(3-δ-2k)β^{2k}/k!, where (1/2)_{k} denotes the falling factorial, cf. Ref. [7]. The second series in Eq. (A.4) reads
with coefficients c_{k}(δ) defined in Eq. (3.22). To get an overview, we summarize the leading orders of expansion (A.4), with power-law index δ ranging in the indicated intervals:
At integer δ, pole singularities emerge in the coefficients of series (A.5) and (A.6), which cancel in Eq. (A.4) if ε expanded [7]:
γ_{E} is Euler's constant. At integer δ<0 and δ>5, expansions K(δ<1,β) and K(δ>5,β) in Eq. (A.7) can be used, as the indicated orders are singularity free. This settles the high-temperature expansion of the partition function (A.1) for arbitrary power-law index δ.
The high-temperature expansion of the chemical potential (3.9), the
Helmholtz free energy (3.12) and the
Gibbs potential F+PV,
Internal energy and specific heat are found by term-by-term differentiation of the log K series. For integers 0δ5, singularities occur in the series coefficients in Eq. (A.10). In this case, we use (A.8) to find
The high-temperature expansion of the entropy function (3.11) is obtained by substituting the asymptotic series of the caloric equation of state and the chemical potential into
Alternatively, we may use the log K series in Eqs. (A.10) and (A.11) and term-by-term differentiation to find the β→0 asymptotics of S, starting with
We note the leading-order asymptotics
and S_{0} approaches a finite limit, , for δ>3, cf. (A.10). At δ=3, S_{0} has a double-logarithmic divergence in the high-temperature limit, cf. (A.11).
We finally discuss the low- and high-temperature asymptotics
of the ultra-relativistic partition function, cf. (4.1):
so that
Here, N is to be identified with the renormalized electron count as defined after (2.5), and the power-law exponent δ is related to the electron index α defined in Eq. (2.2) by δ=α+2. The low-temperature expansions of the chemical potential (3.9) and the free energies (3.12) and (A.9) are obtained by substituting [7]
where .
It remains to settle the high-temperature limit, the β→0
asymptotics of K(δ,β,γ_{1})
in Eq. (A.16). As in
Eq. (A.4), the
expansion is composed of two series [7]:
with coefficients c_{k}(δ,γ_{1}) defined in Eq. (4.8). The high-temperature expansion of reads as stated in Eq. (A.10) for , with coefficients c_{k}(δ) replaced by c_{k}(δ,γ_{1}). In the ultra-relativistic limit, we can use the hypergeometric series in Eq. (4.8):
so that the expansion parameter of series in Eq. (A.20) is βγ_{1}. If βγ_{1}1, despite of β being small, the low-temperature expansion (A.18) applies.
At integer δ, the coefficients c_{k}(δ,γ_{1})
in Eq. (A.20) become
singular owing to poles in the coefficients of the hypergeometric
series. They can be dealt with by ε expansion and
are canceled by corresponding poles of
in Eq. (A.19), which
arise in the Γ-functions in Eq. (A.5). In the
ultra-relativistic limit, γ_{1}1
(but still βγ_{1}1), we find, for integers 0δ5,
There are two expansion parameters in these series, β in and βγ_{1} in , cf. (A.19). It is possible to perform a systematic high-temperature expansion in β, keeping γ_{1} fixed and assuming β→0, just by reordering the terms in Eq. (A.22):
Here, we have dropped the estimates indicated in Eq. (A.22). These expansions are valid in the ultra-relativistic regime, γ_{1}1, provided that β is sufficiently small to ensure βγ_{1}1.
The ultra-relativistic high-temperature asymptotics of entropy is obtained by means of (A.13), with K(δ,β) replaced by K(δ,β,γ_{1}) in the reduced entropy function S_{0}. For power-law indices δ3, the leading order of S_{0} is independent of the ultra-relativistic threshold γ_{1}, cf. (A.23), so that the high-temperature limit of S_{0} in Eq. (A.14) remains unchanged. For power-law indices δ>3, the reduced entropy admits a finite limit, , cf. (A.10) and (A.21).