Abstract

We study tachyonic synchrotron densities of ultra-relativistic electrons in helical motion. There is a longitudinally polarized spectral component due to the negative mass square of the superluminal quanta. The helical pitch-angle scaling of the transversal and longitudinal spectral densities is investigated, in particular the transition from circular to rectilinear motion. The magnetic field induces oscillations along the power-law slopes of the superluminal radiation densities, whose amplitude depends on the pitch angle. At moderate field strength and ultra-relativistic orbital speed, the modulations are tiny, but they become quite pronounced in the surface fields of γ-ray pulsars, resulting in cockscomb distributions. A tachyonic spectral fit to the γ-ray flux of the binary pulsar PSR B1259–63 is performed. The γ-ray wideband of this millisecond pulsar is reproduced by a tachyonic cascade spectrum, capable of generating the spectral curvature in double-logarithmic plots as well as the extended spectral plateau defined by ISGRI, OSSE, COMPTEL, and EGRET flux points.

1. Introduction

We investigate tachyonic synchrotron emission from electrons in pulsar magnetospheres. The goal is to provide quantitative evidence for this radiation by working out a specific example, a tachyonic spectral fit to the γ-ray spectrum of the binary pulsar PSR B1259–63 orbiting around the massive Be star SS 2883. When tachyonic spectral densities are averaged with thermal or power-law electron densities, this leads to cascade spectra [1] and [2], extended spectral plateaus followed by power-law decay terminating in exponential decay. A glance at the spectral map of PSR B1259–63 suggests that it can readily be fitted by a cascade spectrum.

In Section 2, the tachyonic spectral densities of electrons in helical orbits and their pitch-angle scaling are derived. In Section 3, we discuss the pitch-angle averaging and its effect on modulations along the spectral slopes induced by the negative mass-square of the radiation field. In particular, we illustrate the helical cross-over from circular orbits to rectilinear motion in the surface magnetic field of a γ-ray pulsar. In Section 4, we average the tachyonic spectral densities with electronic source populations and perform a wideband fit to the PSR B1259–63/SS 2883 binary system, ranging from high-energy X-rays to TeV γ-rays. In Section 5, we present our conclusions.

We consider electrons in a constant magnetic field B=(0,0,B), B>0. The equations of motion read d(γmv)/dt=ev×B. The helical orbits can be parametrized as

where α is the pitch angle between B and v, 0απ. The speed υ along the orbit is constant. A pitch angle of π/2 gives circular orbits in the (x,y)-plane, and α=0 or π means straight uniform motion. We put =c=1, and use the Heaviside–Lorentz system. The electric charge e is defined negative for electrons, γ is the Lorentz factor, and m the electron mass. We also admit sign(e)=0 in (1), that is, planar transversal oscillations, realizable by undulators in storage rings [3], [4], [5], [6], [7] and [8].

2. Superluminal radiation densities

2.1. High-frequency regime

The tachyonic radiation theory pertinent to a charge in arbitrary motion has been given in Ref. [1]. The multipole expansion of the superluminal Poynting vectors is summarized in Eqs. (2), (3) and (4). We introduce polar coordinates with B as polar axis and polar angle θ, and consider the tachyonic wave vector in the (y,z)-plane, so that k:=k(ω)n, n=(0,sinθ,cosθ). Wave number and frequency are related by the superluminal dispersion relation, , where m_t is the mass of the tachyonic quanta. The sign convention is , so that the plus-sign in the dispersion relation implies a negative mass square. We define two transversal polarization vectors, ε:=(0,−cosθ,sinθ), and ε:=ε×n, so that n, ε and ε constitute an orthonormal triad, ε=−ε×n. We study the ultra-relativistic limit, γ1.

The angular-integrated power radiated in the nth harmonic can readily be assembled from Eqs. (2.27), (2.28) and (2.31) of Ref. [1],

Here, α_q=q²/(4πc)≈1.0×10⁻¹³ is the tachyonic fine structure constant. The superluminal radiation field, a Proca field with negative mass square, is coupled by minimal substitution to the subluminal current, and q is the tachyonic charge of the source particle. The tachyon mass is m_t≈2.15 keV/c². These estimates are obtained from hydrogenic Lamb shifts [9]. The tachyonic frequencies and wave numbers in (2) read

The squared multipole coefficients can be calculated in closed form [1],

We refrain from writing sign²(e)=1, since in this way the contributions of the two linear transversal polarizations (defined by ε_,) to the radiated power can easily be distinguished; the term stemming from the ε-polarized component of the radiation field has the sign²(e)-factor attached. Moreover, by putting sign²(e)=0 in (4), we find the transversal power radiated by oscillating charges in undulator fields. The electromagnetic counterpart of (2), (3) and (4) is recovered in the limit of vanishing tachyon mass [10].

The integrals in (2) give the power transversally or longitudinally radiated in the harmonic ω_n. We are interested in the large-n asymptotics,

the expansion parameter is . The 1/n²-expansion of the wave numbers reads

and the argument z_n of the Bessel functions in (4) is expanded as

In (2), we expand the integrands in θ−θ_c, at the stationary phase, dz_n(θ_c)/dθ=0, which reads

Hence,

We introduce a new integration variable ψ in (2),

where γ=(1−υ²)^−1/21, and approximate

In the vicinity of θ_c, the -factors in (4) can be replaced by

and the Bessel functions in (12) by their Nicholson approximation. The latter applies for large positive n and z, so that z/n≈1 and z<n,

In the preceding notation, this means

The integration boundaries in (2) can be extended to infinity, given the decay of the integrands. Assembling the enumerated approximations, we arrive at the transversal power coefficients,

and the longitudinal coefficients read

Here, we defined

with ε in (10). The transversal power radiated in ε-polarization is given by the sign²(e) term, and the ε-polarized transversal radiation by the -integration in (15). The integrals in (15) and (16) can be simplified by the identities [10] and [11]

where the spectral functions on the right-hand side read

The easiest way to derive these identities is to write K_1/3 and K_2/3 in terms of Ai(x) and its derivative, and then to use the standard integral representation for Ai(x), cf. Section 2.2.

The total power transversally and longitudinally radiated is obtained by summing over the individual modes, . We pass to continuous frequencies by identifying ω=ω_n≈ω_Bn/sin²α, so that the continuous spectral densities read , with n=(ω/ω_B)sin²α. We find ξ in (17) as

This is ξ(ω) as in Ref. [1], rescaled by1/sinα.

The restriction z<n on the Nicholson asymptotics (13) translates into ω>ω_b. In this frequency band, we find the transversal density, cf. (15) and (18),

where ξ and ω_b are defined in (20). The lower plus-sign in (22) refers to the -polarization. p^T(ω) in (21) stands for the total transversal radiation; we have sign²(e)=1. The longitudinal density is assembled with (16) and (18),

The spectral functions F_∞(ξ), G_∞(ξ), and L_∞(ξ) are defined in (19). The pitch angle enters in p^T,L(ω) via the rescaled gyrofrequency, ω_B→ω_Bsinα, cf. (20).

2.2. Analytic continuation into the low-frequency band

Densities (21), (22) and (23) can be continued into the low-frequency regime, ωω_b, by means of Airy functions. The Airy representation of the spectral functions (19) reads

where z=(3ξ/2)^2/3. We define the shortcut

so that ξ=(2/3)|η|^3/2, cf. (20), and find

These densities coincide with (22) and (23) for ωω_b, and they also constitute the analytic continuation into the lower frequency band ωω_b, where η(ω,α) is negative. The lower plus-sign in (26) refers to the -polarization. The pitch angle enters only via η(ω,α), cf. (25).

3. Pitch-angle averaging

We perform a pitch-angle average with density [11]

where λ>−1. This density is normalized to one over the interval [0,π/2]. The averaging is done by replacing in (26) and (27) with

and Ai^′(η)/η in (26) is replaced by

Here, η₀(ω):=η(ω,α=π/2), cf. (25), so that η=η₀/sin^2/3α. The averages (29) and (30) remain unaltered, if we replace the integration over by . This also holds for the normalization of density (28), . More generally, we may average over the full pitch-angle range [0,π] with density

where λ_1,2>−1 and 0A1. dρ_λ is defined in (28). The distribution (31) is normalized to one, . Thus the averaging of the spectral densities (26) and (27) is effected by the substitutions

We denote the averaged densities by p^T,L(ω)_σ.

The averages ∫Ai_λ and Ai^′_λ in (29) and (30) depend on ω only via η₀(ω), cf. after (30). We briefly discuss their |η₀|1 asymptotics, starting with the low-frequency band, ωω_b, cf. after (27), where η₀ is negative. In leading order,

where β0. The systematic asymptotic expansion of the integral (35) can be obtained by substituting the asymptotic series of the Airy function and applying term-by-term integration, which results in a series of incomplete Γ-functions. The latter are then replaced by their asymptotic series. The leading order stated in (35) is independent of the exponent β, which enters linearly in the next-to-leading order; if β is large, we have to require β/|η₀|^3/21 for the O-term to be small. In (29), we also need

Substituting the leading-order asymptotics (34), (35) and (36) into the averages (29) and (30), we find

Thus the averaged spectral densities (as defined by (26) and (27) with substitutions (32) and (33)) converge for large |η₀| to the densities generated by a charge in uniform motion, cf. Section 4, irrespectively of the choice of parameters in the pitch-angle distribution (31).

We turn to the upper frequency range, ωω_b, where η₀(ω) is positive. The asymptotics of ∫Ai_λ and Ai^′_λ is obtained from

The next-to-leading orders are by a factor of smaller, like in (34), (35) and (36). We thus find, above the break frequency,

The averaged spectral densities vanish for large (and even moderate) positive η₀(ω), reflecting the fact that the spectral densities of a charge in uniform motion vanish identically for ωω_b, cf. Section 4.

The pitch-angle distribution dσ_A,λ1,λ2(α) is peaked in the interval [0,π/2], at , provided that λ₁>0. If −1<λ₁0, it decreases monotonically to zero. The peak in the interval [π/2,π] occurs at . If −1<λ₂<0, the density increases in this interval from zero to infinity. For instance, λ_1,2=1 gives α_p1=π/4; λ_1,2=1/3 gives α_p1=π/6, and λ_1,2=3 amounts to a peak at α_p1=π/3. The respective second peak in [π/2,π] is obtained as α_p2=π−α_p1. If A=1/2 and λ₁=λ₂, the density is symmetric around α=π/2. If λ_1,2>0, it vanishes at the interval boundaries and π/2. These three pitch angles correspond to the limit cases of rectilinear motion parallel and antiparallel to the magnetic field, and to circular orbits orthogonal to B. If λ_1,2 is negative or close to zero, the average is dominated by pitch angles close to zero and π, so that the radiation density generated in uniform motion is prevalent. The latter is obtained by performing the zero-pitch-angle limit sinα→0 in (26) and (27), cf. Ref. [12] and Fig. 1 and Fig. 2. If λ_1,21, the distribution (31) has two peaks close to π/2, even though it vanishes at π/2, so that the synchrotron densities (26) and (27) at sinα≈1 dominate the average.

The units =c=1 can easily be restored. Gyroradius (magnetic bending radius, helical curvature radius) and gyrofrequency are connected by R=c/ω_B in the ultra-relativistic regime. The length of one helical turn is 2πR, and Rsinα is the radius of the helical base circle. Electronic energy and Lorentz factor are related as γ≈1957E [GeV], so that and , and the tachyonic break frequency scales as . Fig. 1, Fig. 2 and Fig. 3 illustrate the pitch-angle scaling and averaging of the superluminal radiation densities of ultra-relativistic electrons orbiting in the polar field of a γ-ray pulsar.

4. Tachyonic spectral fit to the γ-ray broadband of the binary pulsar PSR B1259–63

To compare to observational spectral maps, we have to perform a further average over the electron density [2]. We demonstrated that a pitch-angle average attenuates the modulations in the spectral slope, cf. Fig. 3, so that the synchrotron densities approach the spectral densities of a uniformly moving charge,

where Δ^T=1 and Δ^L=0 for the transversal and longitudinal radiation component, respectively. The spectral cutoff occurs at , to be identified with the break frequency ω_b in the ultra-relativistic limit, cf. (20). Only frequencies in the range 0<ω<ω_max(γ) can be radiated by a uniformly moving charge [16].

We average with electronic power laws exponentially cut with a Boltzmann density, . Parametrized with the Lorentz factor, this reads

The normalization A_α,β is determined by , where n₁ is the electron count and γ₁ the smallest Lorentz factor of the source population. β=mc²/(kT), and α is the electronic power-law index (not to be confused with the pitch angle). Here, we consider thermal distributions, defined by α=−2 and γ₁=1. The average is carried out as [2]

The spectral fit is based on the E²-rescaled flux density dN^T,L/dE, which is related to the averaged energy density by

where d is the distance to the pulsar. In Fig. 4, Fig. 5 and Fig. 6, we plot the flux density of two thermal electron populations ρ_1,2 specified in Table 1. Each electron density generates a cascade ρ_i, the wideband spectral map is obtained by adding the cascade spectra.

As for the electron count n₁, it is convenient to use a rescaled parameter for the fit,

which is independent of the distance estimate. Here, [keV s] implies the tachyon mass in keV units, that is, we put m_t≈2.15 in the spectral density (40). 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 actually absorbed by the detector [17], 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 [kpc], where is determined by the tachyonic flux amplitude obtained from the spectral fit. Finally, electron temperature and cutoff parameter in the electronic Boltzmann factor are related as .

Fig. 4, Fig. 5 and Fig. 6 show the least-squares fit to the γ-ray wideband of the binary system PSR B1259–63/SS 2883, cf. Refs. [18], [19], [20], [21], [22], [23] and [24]. The extended spectral plateau as well as the spectral curvature over the entire frequency range can readily be reproduced by a tachyonic cascade spectrum. (As for the high-magnetic-field pulsar PSR B1509–58, cf. Fig. 1, Fig. 2 and Fig. 3, there are currently not enough data points available for a spectral fit [14]. A tachyonic spectral fit to the pulsed Crab flux is given in Ref. [25], where the spectral asymptotics of the averaged densities and the technical details of the nonlinear χ²-fit are discussed as well.) We assume d≈1.5 kpc as distance to the pulsar, based on optical and scintillation observations [22], although it could be as high as 4.5 kpc if inferred from the dispersion measure [23]. In the latter case, the electronic source count has to be rescaled accordingly, ; the spectral maps are not affected by the distance estimate. The total flux above 380 GeV is 5% of the unpulsed Crab flux [21]. The electronic source count for the Crab pulsar at 2 kpc is 2.6×10⁴⁹, cf. Ref. [25], as compared to 3.2×10⁴⁷ for PSR B1259–63, cf. Table 1.

Several electromagnetic radiation mechanisms have been invoked to model the spectra of γ-ray pulsars, most notably curvature radiation due to electric fields followed by pair creation, accompanied by synchrotron radiation and inverse Compton scattering [26]. An electromagnetic spectral fit to the γ-ray wideband of the PSR B1259–63/SS 2883 binary was performed in Ref. [20], in the context of a synchro–inverse-Compton model; the tachyonic wideband fit in Fig. 4 is directly comparable to Fig. 16 of this reference. The tachyonic TeV fit in Fig. 6 is directly comparable to the inverse-Compton fit in Fig. 15 of Ref. [24].

5. Conclusion

We have studied superluminal radiation from ultrarelativistic electrons helically 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 (40) for uniform motion. Orbital curvature induces modulations along the spectral slopes. If integrated over the pitch angles, these oscillations are averaged out, cf. Fig. 1, Fig. 2 and Fig. 3.

We demonstrated that the γ-broadband of the binary pulsar PSR B1259–63 can be reproduced by a tachyonic cascade spectrum, cf. Fig. 4, Fig. 5 and Fig. 6. The cascades are obtained by averaging the radiation densities with thermal electron distributions. At γ-ray energies, the speed of tachyons is close to the speed of light; the basic difference to electromagnetic radiation is the longitudinally polarized flux component. The polarization of tachyons can be determined from transversal and longitudinal ionization cross-sections of Rydberg atoms, which peak at different scattering angles [27].

We have focused on the γ-ray spectrum, including the high end of the X-ray tail. At X-ray energies, if a grating spectrometer is used [28], the interference peaks are determined by wavelength. In this case, the data points have to be reparametrized by wavelength, and then the tachyonic dispersion relation is substituted (which substantially differs from the photonic counterpart due to the 2 keV tachyon mass located in the X-ray regime) to obtain the tachyonic energy-flux relation. Tachyonic spectral fits in the X-ray band 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.