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.

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],

where Δ^T=1 refers to the transversal superluminal radiation, and Δ^L=0 holds for the longitudinal component. We use the classical limit of the radiation densities, neglecting terms depending on the electron–tachyon mass ratio, as the spectral fits are done in the ultra-relativistic high-temperature regime [6]; γ denotes the electronic Lorentz factor, α_q the tachyonic fine structure constant, and m_t the tachyon mass. The 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 p^T,L(ω) are cut off at the break frequency ω_max. We use the Heaviside–Lorentz system, so that α_q=q²/(4πc)≈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 [7].

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:

The electronic Lorentz factors range in an interval γ₁γ<∞. The normalization is determined by , where n₁ is the electron count and γ₁1 the lower edge of Lorentz factors in the source population. The exponential cutoff is related to the electron temperature by β=mc²/(kT), and α is the electronic power-law index. When discussing the thermodynamics of Boltzmann power-law ensembles, we will redefine the power-law index as δ=α+2, so that a thermal Maxwell–Boltzmann distribution corresponds to δ=0 and γ₁=1. In spectral averages it is customary to define the electron index by α as done in Eq. (2.2). The average is carried out as [6]

and the spectral fit is based on the E²-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 ρ₁ and ρ₂ in the figures.

As for the electron count n₁, we use a rescaled parameter for the fit:

which is independent of the distance estimate. Here, implies the tachyon mass in keV units, that is, we put m_t≈2.15 in the spectral density (2.1). At γ-ray energies, only a tiny α_q/α_e-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 n₁, so that the actual number of radiating electrons is . We thus find the electron count as , where defines the tachyonic flux amplitude extracted from the spectral fit [6] and [7]. As for electron temperature and cutoff parameter in the Boltzmann factor, we note kT[TeV]≈5.11×10^-7/β. The electronic source count for the Crab pulsar at d≈2 kpc is 2.6×10⁴⁹, cf. Ref. [16], as compared to 8.6×10⁴⁷ for LS 5039 and 2.8×10⁴⁸ for LSI +61°303; these estimates are obtained by adding the source counts of the high- and low-energy populations ρ_1,2 in Table 1. A similar source number of 3.2×10⁴⁷ was derived in Ref. [17] for the binary pulsar at 1.5 kpc, which suggests a neutron star as compact companion in both microquasars.

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^-δe^-H/(kT)d³p, where H=mγ is the free Hamiltonian, and the electronic Lorentz factors range in the interval γ₁γ<∞. 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 γ₁=1, cf. Ref. [32]. Here, we will study power-law ensembles with arbitrary real index δ.

We start with the classical grand partition function

with density

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 R³, 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 HH₁=mγ₁, γ₁>1, which amounts to replacing the lower integration boundary in Eq. (3.4) by γ₁; in particular, we will study ultra-relativistic ensembles with high-energy threshold γ₁1. In this section, we assume γ₁=1.

The summation in Eq. (3.1) gives

and we find the internal energy and the particle number as



Here, m³V is dimensionless, so that m→mc/ when restoring units, and β=mc²/(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,

and the thermal equation of state, P=-∂F/∂V or PV=mN/β. The isochoric specific heat reads

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, d²S0 at dS=0 (with δ kept fixed) requires C_V0 and κ_T0, the latter is obviously satisfied. As for the first, we note ∂K(δ,β)/∂β=-K(δ-1,β) and the Schwarz inequality K²(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

As for the heat capacities at constant volume/pressure, we use (3.13) to find

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

and find, for δ<1,

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

where γ_E≈0.5772. The internal energy admits a finite high-temperature limit for power-law indices δ>4. The enthalpy is U+mN/β. The high-temperature expansion of the corresponding isochoric heat capacity reads

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 γ₁1. In the partition function (3.1), we substitute the truncated density

where H₁=mγ₁ is the threshold energy. The step function restricts the d³pd³q integration in Eq. (3.1) to H(p,q)H₁:

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,γ₁); internal energy and heat capacity read



Schwarz's inequality as stated after (3.14) also applies to K(δ,β,γ₁), so that the maximum condition d²S0 at fixed δ and γ₁ 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 βγ₁. The low-temperature limit of the heat capacity (4.5) reads

If γ₁≈1, these expansions break down; the low-temperature expansion of K(δ,β,γ₁) in the case of small but non-vanishing υ₁ is discussed in Ref. [7]. At γ₁=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 γ₁1.

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 γ₁>1, if we replace the coefficients c_k(δ) in Eq. (3.22) by

We note c_k(δ,1)=c_k(δ). In the ultra-relativistic limit, γ₁1, the hypergeometric function in c_k(δ,γ₁) can be dropped, cf. (A.21). However, care must be taken if the power-law exponent δ is integer, as singularities emerge in the coefficients of the hypergeometric series, cf. (A.19), (A.20), (A.21), (A.22) and (A.23). For integers 0δ5 and ultra-relativistic γ₁1, the high-temperature expansions of the internal energy read

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 γ₁=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.

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.