Roman Tomaschitz, a,
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.
Keywords: Tachyonic synchrotron radiation; Superluminal curvature radiation; Longitudinal polarization; Cascade spectra; Pulsar magnetospheres
PACS classification codes: 41.60.Ap; 03.50.Kk; 95.30.Gv; 97.60.Gb
1. IntroductionWe 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  and , 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.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 , , , ,  and .
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. . 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 mt 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. ,
The squared multipole coefficients can be calculated in closed
zn of the Bessel functions in (4) is expanded as
γ=(1−υ2)−1/21, and approximate
θc, the -factors in (4) can be replaced by
(12) by their Nicholson approximation. The latter applies for large positive n and z, so that z/n≈1 and z<n,
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
(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  and 
The total power transversally and longitudinally radiated is
obtained by summing over the individual modes, .
We pass to continuous frequencies by identifying ω=ωn≈ωBn/sin2α,
so that the continuous spectral densities read ,
We find ξ in (17) as
ωb are defined in (20). The lower plus-sign in (22) refers to the -polarization. pT(ω) in (21) stands for the total transversal radiation; we have sign2(e)=1. The longitudinal density is assembled with (16) and (18),
2.2. Analytic continuation into the low-frequency bandz=(3ξ/2)2/3. We define the shortcut
ξ=(2/3)|η|3/2, cf. (20), and find
(22) and (23) for ωωb,
and they also constitute the analytic continuation into the lower
frequency band ωωb,
is negative. The lower plus-sign in (26) refers to
The pitch angle enters only via η(ω,α),
3. Pitch-angle averaging
We perform a pitch-angle average with density 
Ai′(η)/η in (26) is replaced by
η0(ω):=η(ω,α=π/2), cf. (25), so that η=η0/sin2/3α. 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
λ1,2>−1 and 0A1. dρλ 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 pT,L(ω)σ.
The averages ∫Aiλ
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, ωωb,
cf. after (27), where η0
is negative. In leading order,
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
(34), (35) and (36) into the averages (29) and (30), we find
(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). smaller, like in (34), (35) and (36). We thus find, above the break frequency,
η0(ω), reflecting the fact that the spectral densities of a charge in uniform motion vanish identically for ωωb, cf. Section 4.
The pitch-angle distribution dσA,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, λ1,2=1 gives αp1=π/4; λ1,2=1/3 gives αp1=π/6, and λ1,2=3 amounts to a peak at αp1=π/3. The respective second peak in [π/2,π] is obtained as αp2=π−αp1. If A=1/2 and λ1=λ2, the density is symmetric around α=π/2. If λ1,2>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 λ1,2 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.  and Fig. 1 and Fig. 2. If λ1,21, 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.
Fig. 1. Transversal tachyonic spectral density pT(ω), cf. (26), in the polar surface field, B≈3.1×1013 G, of the γ-ray pulsar PSR B1509–58, cf. Refs. ,  and . The electronic orbital energy is 10 GeV. The density is shown for three pitch angles, α=π/2 (circular orbit), α=0.1, and α=0. The latter corresponds to a charge in straight uniform motion, cf. (40). The radiation densities are rescaled with the tachyonic fine structure constant αq, cf. after (2). The gyrofrequency is ωB≈18 eV, the helical curvature radius R≈1.1×10−6 cm. The break frequency is ωb≈42 MeV. The spectral peak is located at about the tachyon mass, 2 keV, followed by an extended X-ray tail with power-law decay, in contrast to the exponential cutoff in the upper frequency band, cf. (19).
Fig. 2. Longitudinal spectral density pL(ω), cf. (27). Magnetic field, electron energy and pitch angles as in Fig. 1. The oscillations are superimposed by the magnetic field and vanish at zero pitch angle, for rectilinear motion along the magnetic field lines. The transversal and longitudinal spectral densities of a uniformly moving ultra-relativistic electron (dotted curve in Fig. 1 and Fig. 2) coalesce in the depicted frequency range, ωωb.
The units =c=1 can easily be restored. Gyroradius (magnetic bending radius, helical curvature radius) and gyrofrequency are connected by R=c/ωB 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.
Fig. 3. Transversal and longitudinal spectral densities pT,L(ω)σ, averaged with the symmetric pitch-angle distribution dσ(A=1/2,λ1=λ2=1), cf. (31). This distribution is peaked at pitch angles of 45° and 135°, in the cross-over region between the synchrotron regime and rectilinear motion. Magnetic field and orbital energy as in Fig. 1 and Fig. 2. For comparison, we also indicate the unaveraged transversal distribution pT(ω) at these pitch angles, ; the modulations are wiped out by the averaging.
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 . 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,
E2-rescaled flux density dNT,L/dE, which is related to the averaged energy density by
Fig. 4, Fig. 5 and Fig. 6, we plot the flux density of two thermal electron populations ρ1,2 specified in Table 1. Each electron density generates a cascade ρi, the wideband spectral map is obtained by adding the cascade spectra.
Fig. 4. γ-ray wideband of the binary pulsar PSR B1259–63. RXTE and ISGRI data points from , OSSE points from , COMPTEL and EGRET upper limits from , HESS points from . The solid line T+L is the unpolarized differential tachyon flux dNT+L/dE, obtained by adding the flux densities ρ1,2 of two thermal electron populations. The χ2-fit is based on the first three ISGRI points, the first four OSSE points, and the HESS points. The fourth ISGRI point, the remaining five OSSE points, as well as the COMPTEL and EGRET points are upper limits only. The four depicted RXTE points are at the upper edge of the hard X-ray spectrum detectable with the RXTE proportional counter , they suggest a seamless transition into the X-ray band, but have not been included in the least-squares fit.
Fig. 5. EGRET plateau and HESS slope of PSR B1259–63. The solid line T+L is the unpolarized tachyon flux dNT+L/dE rescaled with E2 to render the spectral curvature visible. The latter is caused by the Boltzmann factor in the averaged spectral densities (42); the exponential decay of the cascades ρ1,2 sets in at about Ecut≈(mt/m)kT, implying cutoffs at 0.8 TeV for the ρ1-cascade and 80 GeV for ρ2. The unpolarized flux T+L can be split into transversal (dot-dashed) and longitudinal (dot-dot-dashed) components dNT,L/dE, cf. (43). The spectral plateau extends from the ISGRI and OSSE points in the keV range to the first spectral cut at 80 GeV, cf. Fig. 4.
Fig. 6. Close-up of the HESS spectrum of PSR B1259–63 in Fig. 5. The tachyon flux T+L is obtained by adding the cascades ρ1,2. The TeV spectral map coincides with the ρ1-cascade, since the ρ2-flux is exponentially cut at 80 GeV, cf. Fig. 5. Temperature and source number of the electron populations generating the cascades are listed in Table 1.
Thermal electronic source densities ρ1,2 generating the γ-ray broadband of the binary pulsar PSR B1259–63
The cascades ρ1,2 in Fig. 4, Fig. 5 and Fig. 6 are obtained by averaging the superluminal radiation densities with electron populations, cf. (42), which are in turn inferred from the spectral fit. Each distribution is defined by electron temperature kT and source count ne, the latter based on a distance estimate of 1.5 kpc. β is the cutoff parameter in the electronic Boltzmann factor. is the normalized amplitude of the tachyonic flux density defined in (44). The parameters and β are extracted from the χ2-fit.
As for the electron count n1,
it is convenient to use a rescaled parameter
for the fit,
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. , , , , ,  and . 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 . A tachyonic spectral fit to the pulsed Crab flux is given in Ref. , 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 , although it could be as high as 4.5 kpc if inferred from the dispersion measure . 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 . The electronic source count for the Crab pulsar at 2 kpc is 2.6×1049, cf. Ref. , 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 . An electromagnetic spectral fit to the γ-ray wideband of the PSR B1259–63/SS 2883 binary was performed in Ref. , 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. .
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 .
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 , 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.
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.
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