Physica A: Statistical and Theoretical Physics
Volume 335, Issues 3-4, 15 April 2004, Pages 577-610

Tachyonic synchrotron radiation

Roman TomaschitzCorresponding Author Contact Information, E-mail The Corresponding Author

Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan

Received 17 July 2003. 
Available online 12 December 2003.


The emission of superluminal quanta (tachyons) by ultra-relativistic electrons gyrating in magnetic fields is investigated. The tachyonic Liénard–Wiechert potentials of helically orbiting charges are derived, as well as the superluminal energy flux and the transversal and longitudinal spectral densities. We calculate the tachyonic synchrotron power, its angular dependence, the mean energy of the superluminal quanta radiated, tachyonic emission rates, the spectral maxima, critical and break frequencies, and we identify the Stokes parameter of the longitudinal radiation. The tachyonic energy densities are averaged with electronic power-law distributions, and the spectral indices are determined. Quantitative estimates are given for superluminal synchrotron radiation generated in storage rings, the Jovian magnetosphere, and supernova remnants. The spectral density of Jupiter's tachyonic X-ray emission is inferred from radio fluxes obtained from SL9 pre-impact observations and the Cassini fly-by, and we identify the tachyonic spectral peak at 2 keV in the ROSAT and Einstein spectral maps. We scrutinize multiwavelength observations of galactic supernova remnants, pointing out evidence in their wideband spectra for the TeV γ-radiation to be tachyonic rather than a consequence of inverse Compton scattering or pion decay. In the Crab Nebula, the electronic source population generating this radiation extends beyond the ‘knee’ of the cosmic ray spectrum.

Author Keywords: Author Keywords: Superluminal radiation; Curvature radiation; Tachyons; Proca equation; Negative mass-square; Storage rings; Planetary magnetospheres; Van Allen belts; Jupiter's aurora; Supernova remnants; Synchrotron nebulae; Crab Nebula; TeV γ-rays

41.60.Ap; 03.50.Kk; 29.20.Dh; 96.35.Kx; 98.38.Mz

Article Outline

1. Introduction
2. Superluminal radiation by helically moving charges
3. Tachyonic spectral densities, curvature radiation, and the longitudinal Stokes parameter
4. Radiant power, tachyonic number counts, and spectral indices
5. Tachyonic X-rays from Jupiter's magnetosphere
6. Tachyonic gamma-rays from the shock-heated plasmas of supernova remnants
Appendix A. Superluminal spectral and power asymptotics

1. Introduction

We will study the spectral densities of superluminal radiation fields generated by ultra-relativistic particles in helical and circular orbits (tachyonic synchrotron radiation). We will discuss examples with orbital radii varying over some 15 orders of magnitude, from storage rings via planetary radiation belts to shell supernova remnants and plerions. Superluminal quanta are a sort of photons with negative mass-square, a quantized Proca field minimally coupled to subluminal matter [1, 2, 3 and 4]. The coupling strength of tachyons to matter is determined by the tachyonic fine structure constant; the ratio of electric and tachyonic fine structure constants is estimated from Lamb shifts in hydrogenic ions, αqe≈1.4×10−11, like the tachyon-electron mass ratio, mt/me≈1/238, cf. [5].

Although the Proca equation with negative mass-square is formally close to electromagnetic theory, there are some marked differences such as the third degree of freedom, longitudinally polarized quanta. In the ultra-relativistic limit studied here, the longitudinal radiation is even more pronounced than its transversal counterpart. If tachyons are radiated in the MeV range or above, the tachyon mass is negligible in the dispersion relation, so that they propagate very nearly at the speed of light, and then the longitudinal polarization gets crucial in distinguishing them from photons. At low energies, from soft X-rays down to radio frequencies, the tachyon mass dominates, so that tachyons cannot exceed a wavelength of roughly 6 Å. Low-energy tachyons can thus be discerned from photons by their wavelength, polarization, and speed. A basic difference to Maxwell's theory is the residual radiation in the limit of infinite curvature radius. Particles in linear uniform motion with regard to the cosmic microwave background can radiate superluminal quanta. In the ultra-relativistic limit, the spectrum of this radiation is not sharply peaked, extending over a broad frequency range, from the spectral maximum determined by the tachyon mass up to a break frequency depending on the Lorentz factor of the inertial source [6]. We will consider a finite gyroradius, so that the frequencies radiated extend beyond this break frequency by virtue of curvature radiation, which generates a smooth exponential cutoff in the tachyonic spectral densities.

The existence of superluminal quanta has not been established so far; Lamb shifts in hydrogen-like ions, hyperfine splittings, radiative transitions in Rydberg atoms, and ionization cross-sections provide circumstantial evidence [5]. Here, we will search the Jovian magnetosphere and galactic synchrotron nebulae. As for Jupiter's aurora, thanks to the SL9 comet impact and the Cassini spacecraft encounter, the synchrotron radio flux has been measured at various λ, from 90 to 2 cm, and can be used to piece together the electron density. This in turn allows us to infer the tachyon flux emitted in hard X-rays, as well as the slope and the break frequencies of the high-energy tail of the tachyonic spectral density. The ROSAT and Einstein spectral maps indicate line spectra of highly stripped heavy ions in soft X-rays, but also a third radiation peak at 2 keV, well separated from the line spectra, which will be shown to coincide with the peak of the tachyonic spectral density.

In search of high-energy tachyon radiation, we will study galactic supernova remnants with TeV γ-ray spectra, that is, the Crab, Cas A, the remnant of SN 1006, and RX J1713.7–3946, the latter in connection with the nearby GeV γ-ray source 3EG J1714–3857. We will analyze their broadband spectra, from soft X-rays upwards, the spectral breaks and slopes. We will conclude that the EGRET flux data and, in the case of the Crab, the COMPTEL data in the low MeV region, support the identification of their MeV to TeV γ-ray spectra as tachyonic. We will determine the range and index of the power-law electron density generating the superluminal γ-radiation, and conclude that it is distinct from the electron population emitting the electromagnetic synchrotron radiation in the radio-to-X-ray bands.

In Section 2, we derive the tachyonic flux vectors and the integral energy flux emitted by helically moving charges, the formalism of tachyonic synchrotron and cyclotron radiation, that is. In Section 3, we specialize to charges in circular, ultra-relativistic motion (subluminal, with high Lorentz factors). We perform the asymptotic summation of the multipole expansion of the superluminal radiation field and calculate the transversal and longitudinal spectral densities. In Section 4, we integrate these densities to obtain the power radiated, the tachyonic number counts (emission rates), mean energies, and polarization ratios. We average the superluminal spectral densities with electronic power-law distributions, and relate the power-law index to the tachyonic spectral index. In Section 5, we first give numerical estimates for tachyon radiation in storage rings and compare with electromagnetic synchrotron radiation. We then turn to the Jovian magnetosphere, and infer the slopes of the tachyonic spectral density from the radio electrons. Jupiter's tachyon spectrum extends over the hard and high-energy X-ray bands, and we identify the tachyonic spectral peak in the ROSAT and Einstein spectral maps. In Section 6, we scrutinize galactic supernova remnants with known TeV γ-ray fluxes, and identify tachyonic spectral slopes in their multi-band spectra. The electron density generating the superluminal γ-rays is inferred from the tachyonic break energies and spectral slopes. The conclusions are stated at the end of 5 and 6. In Appendix A, we sketch the stationary phase asymptotics of the tachyonic spectral densities and the power asymptotics.

2. Superluminal radiation by helically moving charges

The superluminal radiation field solves the Proca equation with negative mass-square, (□+mt2)Aα=−c−1jα, subjected to the Lorentz condition Aμ=0. The sign conventions for metric and d'Alembertian are ημν=diag(−1,1,1,1) and □colon, equalsημνμν, respectively. The tachyon mass has the dimension of an inverse length, a shortcut for mtc/planck constant over two pi, estimated as mt/me≈1/238 from Lamb shifts in hydrogenic systems [5]. We will mainly work in Fourier space, representing the spatial component of the vector potential as

and analogous relations hold for the time component and the current density. Fourier transforms will be defined as in (2.1) and denoted by hats.

In (2.2), (2.3), (2.4), (2.5), (2.6), (2.7), (2.8), (2.9) and (2.10), we summarize the classical radiation theory, tachyon radiation by arbitrarily moving charges [4 and 6]. The Fourier transform of the current is split into a transversal and longitudinal component,

where n=x/r, so that the respective components of the asymptotic radiation field read,


We consider the trajectory of a subluminal particle x0(t), Image, carrying tachyonic charge q, so that the current density admits the Fourier transform

The transversal and longitudinal components of the velocity are vT(x,t)colon, equalsvn(n·v) and vL(x,t)colon, equalsn(n·v), respectively, cf. (2.2), so that transform (2.4) of the current can be assembled as

The asymptotic Liénard–Wiechert potentials of the tachyonic charge are thus given by (2.3) with this Image inserted. Once these potentials are known, the time-averaged energy flux can be derived by a standard procedure [6]. We find the transversal flux,

and the averaged longitudinal Poynting vector,

with (2.6) substituted. Here, we use a standard limit definition of the Dirac function,

so that δ(1)(ω;T→∞)=δ(ω), and the same for δ(2). The purpose of these limit representations is to avoid ill-defined squares of δ-functions. The radiant power is obtained by integrating the flux through a sphere of radius r→∞,

P=PT+PL, PT,Lcolon, equalsr2n·left angle bracketST,Lright-pointing angle bracket dΩ ,
with the solid angle element dΩ=sin θ dθ dphi.

We specialize to helical motion; the subluminal source moves with constant speed, otherwise there are no restrictions on the velocity. The ultra-relativistic limit, tachyonic synchrotron radiation from particles in circular motion, will be studied in greater detail in 3, 4, 5 and 6. Tachyonic cyclotron radiation will be dealt with elsewhere; in fact, tachyon radiation by non-relativistic particles always requires quantization, cf. the beginning of Section 5. We consider a constant magnetic field B=(0,0,B), B>0, and a particle with constant speed υ, electric charge e and tachyonic charge q. The equations of motion read d(γmv)/dt=(e/c)v×B, where γcolon, equals(1−υ2/c2)−1/2. Heaviside–Lorentz units are used throughout, and e is defined negative for electrons. We so find the helix


ωBcolon, equals|e|B/(mcγ), υperpendicularcolon, equalsυ sin α, υ||colon, equalsυ cos α ,
where α is the pitch angle between B and v, and 0<α<π. The gyrofrequency relates to the Larmor frequency as ωLBγ/2. For circular motion in the (x,y)-plane, we have of course α=π/2. We will also admit sign(e)=0 in (2.11), that is, planar transversal oscillations, realizable by undulators in storage rings [7], though we will not discuss this here. The notation is kept close to electromagnetic synchrotron radiation [8, 9, 10 and 11].

We introduce polar coordinates with B as polar axis and polar angle θ, and consider the wave vector in the (y,z)-plane, without loss of generality, so that kcolon, equalsk(ω)n, n=(0,sin θ,cos θ), with k(ω) defined in (2.3). There is no necessity to specify k(ω) for ω=0, as there is no wave propagation in this case. The ei are unit vectors along the coordinate axes, and we define two further unit vectors,

so that n, var epsilon|| and var epsilonperpendicular constitute an orthonormal triad. We may thus write the transversal and longitudinal components of the velocity as

vT=var epsilon|||| sin θ−υperpendicular cos θ cos(ωBt))−var epsilonperpendicular υperpendicular sign(e)sin(ωBt) ,

vL=nperpendicular sin θ cos(ωBt)+υ|| cos θ)
and we also note

n·x(t)=(υperpendicularB)sin(ωBt)sin θ+υ||t cos θ .

When studying electromagnetic synchrotron radiation, it is customary to choose a rest frame where the helical orbit appears circular. A noticeable exception is [10], and we will follow the reasoning there. The introduction of a frame comoving along the helical axis is not useful when studying superluminal radiation, as in such frames longitudinal and transversal modes can appear tangled or even advanced. We will study superluminal radiation in the comoving galaxy frame, locally realized by a Minkowskian frame in which the microwave background is isotropic, that is, Planckian with an isotropic temperature. Both the causality and the energy concept for tachyons are based on the universal cosmic time order, on the absolute space–time defined by the galaxy grid, locally manifested by the background radiations and the cosmic ether [4, 12, 13 and 14]. The helix (2.11) is meant in this frame, in a locally geodesic neighborhood in which the microwave radiation is isotropic.

We start by truncating the integral representation (2.6) of the current,

which can be evaluated via ((2.13) and (2.14)) and Jacobi's expansion,

To this end, we use the smooth limit representation (2.9) of the δ-function and the truncated Fourier transforms


as well as the identities Jn+1Jn−1=−2Jn′(z) and Jn+1+Jn−1=2(n/z)Jn(z). Expansion (2.16) amounts to a reordered multipole expansion, frequently used in radiation problems with periodically moving sources [11]. In this way, we arrive at




z(ω)colon, equalsk(ω)(υperpendicularB)sin θ, αn(ω)colon, equalsnωB−ω+k(ω)υ|| cos θ .
Collecting terms, we may write


λnT(z)colon, equalsvar epsilon||||sin θ−(n/zperpendicular cos θ)Jn(z)−ivar epsilonperpendicularυperpendicular sign(e)Jn′(z) ,

λnL(z)colon, equalsn((n/zperpendicular sin θ+υ||cos θ)Jn(z) ,
As for the flux vectors ((2.7) and (2.8)), we need to know the zeros ωn of αn(ω)=0, cf. (2.19). For every n, there are at most two solutions,

βngreater-or-equal, slanted0, and we define

Apparently, kn+>0 and kn<0, and thus only positive ωn+ and negative ωn can be solutions, because only then kn+=kn+) and kn=kn), respectively. Hence, for sufficiently large |n|, there is exactly one solution, namely ωn+ for positive n, and ωn for negative integers. If the mass term is dominant in (2.22), there may be two solutions of αn(ω)=0 or none. Solutions ωn±=0 are discarded, as they do not correspond to wave modes. Hence,

We note Jn(−z)=Jn(z) and Jn′(−z)=−Jn′(z), define

zn±colon, equalszn±)=kn±)(υperpendicularB)sin θ ,
and find the symmetries

With these preparations, we can readily compile the Poynting vectors ((2.7) and (2.8)),



|λnT(zn±)|2=(υ|| sin θ−(n/zn±perpendicular cos θ)2Jn2(zn±)+sign2(eperpendicular2Jn2(zn±) ,

|λnL(zn±)|2=((n/zn±perpendicularsin θ+υ||cos θ)2Jn2(zn±) ,
where |λn|2colon, equalsλn·λn*. In ((2.27) and (2.28)), we may replace ∑nset membership, variantZ by ∑n=0,±+2 ∑ngreater-or-equal, slanted1,± without further changes, due to symmetries (2.26). In (2.29) we refrain from writing sign2(e)=1, since in this way the contributions of the two linear transversal polarizations (defined by var epsilon||,perpendicular in ((2.12) and (2.21))) to the radiated power can easily be distinguished; the terms stemming from the var epsilonperpendicular-polarized component of the radiation field are proportional to sign2(e). Moreover, by putting sign2(e)=0, we find the radiation generated by oscillating charges in undulator fields, mentioned after (2.11). The power transversally and longitudinally radiated is thus, cf. (2.10),

where the summation signs stand for the series in the flux vectors ((2.27) and (2.28)), respectively. In the next section, we will evaluate these vectors for ultra-relativistic circular motion, and derive explicit formulas for the tachyonic spectral distributions. In Section 4, we will have a closer look at the radiant powers (2.31).

3. Tachyonic spectral densities, curvature radiation, and the longitudinal Stokes parameter

We derive the transversal and longitudinal spectral densities for tachyon radiation emitted by ultra-relativistic (subluminal) particles, υ/c≈1, in circular motion. The context is given in Section 2, specialized to circular orbits, υ||=0, υperpendicular=υ. The spectral modes are ωn=nωB, cf. (2.22), so that in the notation of ((2.19) and (2.23)),

Here, ωn means ωn>0+ or ωn<0, cf. after (2.23). The multipole expansion of the time averaged Poynting vectors, cf. ((2.27), (2.28), (2.29) and (2.30)), reads left angle bracketST,Lright-pointing angle bracket=∑n=1 left angle bracketSnT,Lright-pointing angle bracket, where



Carrying out the angular integration, we find the transversal and longitudinal components of the total power radiated as

More explicitly,


where the power radiated in the transversal linear var epsilon|| and var epsilonperpendicular polarizations is determined by


respectively, and the longitudinally radiated power is defined by

cf. Appendix A for further comments on these identities. Radiation generated by transversal planar oscillations corresponds to sign2(e)=0; we will always indicate sign2(e) to distinguish the polarization components of the transversal radiation, cf. after (2.30).

The preceding calculations are exact, apart from the r→∞ asymptotics, which is sufficient to calculate the radiated energy. We will evaluate PnT,L for large n and υ/c≈1. This can be done by means of the Nicolson asymptotics sketched in Appendix A. As n is large, we may pass to continuous frequencies via ω=nωB, so that PnT,L dnpT,L(ω) dω, with the spectral densities

pT,L(ω)colon, equalsωB−1Pn=ω/ωBT,L .
There are two asymptotic regimes, depending on whether xn/n>1 or xn/n<1, cf. (3.1). To study that, it is almost indispensable to introduce, cf. (A.9),

First we consider the case xn/ngreater-or-equal, slanted1, which means frequencies satisfying ωless-than-or-equals, slantωb. In this regime, the spectral densities follow from ((3.5) and (3.6)), with substitutions ((A.7) and (A.8)). The transversal density is assembled as

pT(ω)=p||T(ω)+sign2(e)pperpendicularT(ω) ,
where, for ωless-than-or-equals, slantωb,



with ξ and ωb in (3.11). The lower plus-sign in (3.13) refers to the perpendicular-polarization. The longitudinal spectral density reads


The second case, xn/nless-than-or-equals, slant1, refers to the upper frequency range, ωgreater-or-equal, slantedωb. Explicit expressions for the spectral densities again follow from ((3.5) and (3.6)), but now with substitutions ((A.17) and (A.18)). The transversal density is composed as in (3.12), with



where ξ and ωb are defined in (3.11). Finally, the longitudinal density in this frequency range,

The asymptotic expansions of the spectral functions F0,∞, G0,∞ and L0,∞ are listed in ((A.10), (A.11), (A.19) and (A.20)).

The massless limit is easily recovered, electromagnetic radiation that is, where ξ reduces to ξmt=0=ω/ωc, with the critical frequency ωccolon, equals(3/2)ωBγ3. As ωbmtc→0, we find from (3.16),

with q replaced by the electric charge, of course.

Even though densities ((3.13) and (3.16)) have a very different shape in the upper and lower spectral range, they smoothly join at ωb, in fact analytically, and the same holds for the longitudinal densities ((3.15) and (3.18)). To see this, we define

so that ξ=(2/3)|η|3/2, cf. (3.11), and we find, via (A.22),


valid throughout the spectral range. The lower plus-sign again refers to the perpendicular-polarization. In the next section, we will integrate these densities to obtain the radiant power. This will be done asymptotically, with κ as expansion parameter, cf. (3.11).

We return to the lower spectral range, ωless-than-or-equals, slantωb, cf. ((3.13) and (3.15)). In the limit κ→∞, i.e., for ξ→∞ (at a fixed frequency), the spectral functions F0, G0 and L0 all vanish, which happens in the limit of infinite gyroradius, since κ∝R, as will be shown in (4.2). This suggests to split the spectral densities into

p||,perpendicularTless-than-or-equals, slantωb)=pT,lin/2−p||,perpendicularT,curv, pLless-than-or-equals, slantωb)=pL,linpL,curv ,
where the densities pT,lin and pL,lin stand for the tachyon radiation generated by a charge in linear uniform motion (in the ultra-relativistic limit, that is) [6],

The curvature radiation subtracted in (3.22) reads

p||,perpendicularT,curv(ω):=−(pT,lin/2)(F0minus-or-plus signG0), pL,curv(ω)colon, equalspL,linL0 ,
where the upper minus-sign in (3.24) refers to p||T,curv. The Lorentz factor of the charge enters in the transversal linear density pT,lin via ωb. It also enters in pL,lin, again by ωb, which is the cutoff frequency for uniform motion. A uniformly moving ultra-relativistic charge can only radiate frequencies ωless-than-or-equals, slantωb. The radiation in the upper frequency range, ωgreater-or-equal, slantedωb, is pure curvature radiation, the spectral functions F, G and L vanish in the limit of zero orbital curvature, κ→∞. The densities p||,perpendicularT,L in (3.22) are positive definite, but not so the curvature terms pT,curv and pL,curv in (3.24), which oscillate for large ξ, cf. (A.11). Densities (3.21) are positive throughout the spectral range, of course. In the next section, we will show that the curvature terms in densities (3.22) give negative contributions to the radiated power, nearly equal in magnitude to the power radiated as curvature radiation in the upper frequency range.

Once the spectral densities are known, the transversal and longitudinal polarization functions, ΠωT,Lcolon, equalspT,L/(pT+pL), are readily assembled. We find in the lower spectral range,

and in the high-frequency regime,

The argument in the spectral functions is always ξ(ω). We will content ourselves with the integrated versions of these ratios, and replace the densities in ΠωT,L by the powers radiated in the respective polarizations, cf. (4.17).

Tachyon radiation can be longitudinally polarized, and the longitudinal component may even overpower the transversal radiation. I therefore conclude this section with the Stokes parameter for longitudinal radiation. Tachyonic E and B-fields are related to the vector potential in the usual way, by E=(1/c)(backward differenceA0−∂A/t) and B=rot A, cf. [15]. We consider a Fourier mode

the var epsilonk,1 and var epsilonk,2 are real unit vectors (linear polarization vectors) orthogonal to var epsilonk,3colon, equalsk0=k/|k|, so that the var epsilonk,n constitute an orthonormal triad. The ak,ncolon, equalsrk,n(t)exp(iphik,n(t)) are complex amplitudes, slowly varying in time as compared to the phase factor in (3.27). Defining Ek,mcolon, equalsvar epsilonk,m·Ek, the polarization can be determined from the time averages left angle bracketEk,mEk,n*right-pointing angle bracketleft angle bracketak,mak,n*right-pointing angle bracket taken over the period 2π/ω. In the following we will suppress the subscript k. The transversal degrees, m,n=1,2, are settled by the Stokes parametrization


sTcolon, equalsleft angle bracket|a1|2right-pointing angle bracket+left angle bracket|a2|2right-pointing angle bracket, s1colon, equals2left angle bracket|a1|2right-pointing angle bracketsT=left angle bracketr12right-pointing angle bracketleft angle bracketr22right-pointing angle bracket ,

s2+is3colon, equals2left angle bracketa2a1*right-pointing angle bracket=2left angle bracketr1r2 exp(i(phi2phi1))right-pointing angle bracket .
Hence, Trleft angle bracketEmEn*right-pointing angle bracket=sT, and

detleft angle bracketEmEn*right-pointing angle bracket=(s2T/4)(1−Πtrans2), Πtrans2colon, equals(s12+s22+s32)/sT2 ,
0less-than-or-equals, slantΠtransless-than-or-equals, slant1. Complete polarization, Πtrans=1, in the transversal plane is achieved if the ak,n are time independent, and totally unpolarized radiation, Πtrans=0, is indicated by vanishing s1,2,3, so that no direction is preferred in the polarization tensor (3.28). The parameters sT,1,2,3 account for the transversal component only, a fifth parameter is needed for the longitudinal radiation,

The real tachyonic field strength defined by (3.27) is 2 Re Ek, cf. (2.1); the time averaged (over the period) transversal and longitudinal energy densities of this nearly plane wave are left angle bracketρETright-pointing angle bracketnot, vert, similar2sT and left angle bracketρELright-pointing angle bracketnot, vert, similar2sL, respectively [4 and 13]. The transversal and longitudinal degrees of polarization read accordingly ΠT,L=sT,L/(sT+sL), which explains the definition of sL. For instance, we may assume complete polarization in the transversal plane, Πtrans=1. If two of the amplitudes rn vanish, we have linear polarization. If one of them vanishes and the other two are equal in magnitude with a phase difference of ±π/2, the wave is circularly polarized. The longitudinal component does not show in the four transversal parameters; a field strength rotating orthogonally to the transversal plane can still be interpreted as linear transversal on the basis of sT,1,2,3 only, erroneously, as one ignores the longitudinal energy density. If the superluminal velocity is close to the speed of light, the tachyon mass can be neglected in the dispersion relation, cf. (4.22), but high-energy tachyons can still be discerned from photons by their longitudinal polarization. In the examples discussed in 5 and 6, the longitudinal radiation is always more pronounced than the transversal counterpart. Differential cross sections are perhaps the most practical means to discriminate longitudinal radiation from transversal tachyons and photons. Ionization cross sections have been scrutinized to that effect in [13]. The polarization of the ionizing radiation affects the angular maxima, the peaks in the transversal and longitudinal cross sections occur at different scattering angles.

4. Radiant power, tachyonic number counts, and spectral indices

We will study the tachyonic power emitted in the three polarizations, based on the ultra-relativistic spectral densities derived in the previous section, that is, the transversal densities p||,perpendicularT(ω) in ((3.13) and (3.16)), and the longitudinal pL(ω) in ((3.15) and (3.18)). We will discuss the integral number counts (tachyons per unit time) in the respective polarizations, and compare with electromagnetic synchrotron radiation. Finally, we will discuss tachyonic spectral slopes generated by averages over electronic source populations.

We start with some estimates concerning the helix (2.11); the notation summarized or introduced here will also be used in the tables of Section 5. Gyroradius and gyrofrequency relate as R=υ/ωB, where ωB=eB/(γmc), e>0, so that

We write the Lorentz factor as γ=E/mc2, where E and m denote energy and mass of the gyrating subluminal particle, usually electrons or positrons, so that E and m stand for electron energy and mass, and eB=(υ/c)E/R. In the ultra-relativistic limit, we approximate υ≈c whenever possible, e.g., ωBc/R. The following discussion also applies to protonic source populations, after some obvious rescaling with the electron–proton mass ratio. We use the Heaviside–Lorentz system; e2/(4πplanck constant over two pic)=:αe≈1/137 and q2/(4πplanck constant over two pic)=:αq≈1.0×10−13 are the electric and tachyonic fine structure constants. The tachyon mass mtm/238≈2.15 keV/c2 gives a reduced Compton wave length of planck constant over two pi/(mtc)≈0.92 Å; the quotient of tachyonic and electric fine structure constant reads αqe≈1.4×10−11, all inferred from Lamb shifts in hydrogenic systems [5]. We restore the natural units, mtmtc/planck constant over two pi≈1.09×108 cm−1, cf. the beginning of Section 2, so that the break frequency reads ωbmtc2/planck constant over two pi, cf. (3.11). In the subsequent asymptotics, we will use κ in (3.11) as expansion parameter, which is the quotient of break frequency and critical photon frequency, ωc=3ωBγ3/2, cf. (3.19), relating to the bending radius as

In the examples discussed in the following sections, κ will always be large, and the asymptotic expansions will be in κ−2/3, cf. (A.25). The opposite limit, κ→0, can be realized in the surface magnetic fields of pulsars, but we won't consider this here. The peak of the electromagnetic spectral distribution (3.19) is located approximately at 0.286ωc, cf. [16], but the analytically defined ωc is the customary reference value for the location of the bulk of the photon distribution. The tachyonic curvature radiation is peaked at the break frequency ωb, where the linear densities (3.23) terminate, cf. the discussions following ((3.24) and (4.14)).

We turn to the transversal power radiated in the low-frequency regime, ωless-than-or-equals, slantωbmtc, cf. ((3.12), (3.13), (3.14) and (3.15)). This power can be split into polarization components like the spectral densities,

Alternatively, we may decompose Pless-than-or-equals, slantωbT into a linear and a curvature component according to ((3.22), (3.23) and (3.24)),

Pless-than-or-equals, slantωbT=PT,linPT,curv, PT,curvcolon, equalsP||T,curv+PperpendicularT,curv ,

so that P||,perpendicularT=PT,lin/2−P||,perpendicularT,curv. The respective number counts such as Nless-than-or-equals, slantωbT are defined in the same way, with the spectral densities divided by planck constant over two piω. The power stemming from the linear transversal density reads

which is the leading order in the ultra-relativistic 1/γ-expansion of the first integral in (4.4); the second integral gives the transversal curvature radiation in this frequency range,

up to terms of O(κ−2/3), cf. ((A.23), (A.24), (A.25), (A.26) and (A.27)). The lower plus-sign in (4.6) refers to the perpendicular-polarization, and PT,curv is the total transversally polarized curvature radiation as defined in (4.4). The linear power PT,lin in (4.5) is the residual radiation in the limit of infinite gyroradius; this is the radiation emitted by an ultra-relativistic charge in uniform motion [6]. The curvature correction, P||,perpendicularT,curv, is generated by the F0(ξ) and G0(ξ)-terms in (3.13), and tends to reduce the radiation in the low-frequency regime, cf. (4.4) and the discussion following (3.24).

The longitudinal radiant power can likewise be decomposed into a linear and a curvature term,


An elementary integration gives the ultra-relativistic linear power,

up to O(γ−1) like in (4.5). The κ−2/3-expansion of the longitudinal curvature radiation can be assembled from ((A.23), (A.24), (A.25), (A.26) and (A.27)). In leading order,

Here and in (4.6), we use 22/3Γ(2/3)≈2.15 and 21/3Γ(1/3)≈3.375.

The transversal low-frequency number counts are defined like the power components in ((4.3) and (4.4)), with the integrands divided by planck constant over two piω. The transversal count is assembled as Nless-than-or-equals, slantωbT=NT,linNT,curv, where NT,curv=N||T,curv+NperpendicularT,curv, and we find


The longitudinal low-frequency count reads Nless-than-or-equals, slantωbL=NL,linNL,curv, with NL,linnot, vert, similarNT,lin, cf. (4.10), and

The κ−2/3-expansion of the powers and count rates beyond the leading order is sketched in ((A.23), (A.24), (A.25), (A.26) and (A.27)).

The high-frequency regime, ωgreater-or-equal, slantedωb, is determined by the spectral densities ((3.16), (3.17) and (3.18)) defining the radiant powers

This is all curvature radiation, there is no linear contribution in contrast to the low-frequency regime. The integral number counts Ngreater-or-equal, slantedωbT,L are defined as in (4.13), with the densities divided by planck constant over two piω. The κ−2/3-expansion of these integrals is outlined in ((A.28), (A.29), (A.30) and (A.31)), and we find in leading order,

Pgreater-or-equal, slantedωbTnot, vert, similarPT,curv, Pgreater-or-equal, slantedωbLnot, vert, similarPL,curv, Ngreater-or-equal, slantedωbTnot, vert, similarNT,curv, Ngreater-or-equal, slantedωbLnot, vert, similarNL,curv ,
where PT,L,curv is the low-frequency curvature radiation calculated in ((4.6) and (4.9)), and NT,L,curv is the corresponding low-frequency count in ((4.11) and (4.12)). This equivalence of high- and low-frequency curvature radiation also persists for the individual transversal polarizations, P||,perpendicularTnot, vert, similarP||,perpendicularT,curv, with P||,perpendicularT in (4.13) and P||,perpendicularT,curv in (4.6). Relations (4.14) also hold in next to leading order in the κ−2/3-expansion, but beyond that the asymptotic series start to differ, as can be seen by comparing the expansions in ((A.23), (A.24), (A.25), (A.26), (A.27), (A.28), (A.29), (A.30) and (A.31)). Nevertheless, in leading order this symmetry holds, and it means that the power radiated in the upper frequency range is more or less drained from the linear component of the radiation at low frequencies, cf. ((4.4) and (4.7)). The curvature radiation vanishes for κ→∞ in both regimes; for large κ, the bulk of the curvature radiation is restricted to a small frequency range centered at the break frequency ωb, roughly defined by ξ(ω)less-than-or-equals, slant1, cf. ((3.11), (A.11) and (A.20)). In the low-frequency regime, the curvature term in the spectral densities is not exponentially damped but oscillating, so that it averages itself out when integrated over a frequency range where ξ(ω)much greater-than1; the asymptotic expansion in ((A.28), (A.29), (A.30) and (A.31)) is based on that. For large κ, the peak frequency of the spectral distributions (3.22) is ωpeaknot, vert, similarmtc2/planck constant over two pi≈3.27×1018 Hz, determined by the linear densities only, and so are the tachyonic mean frequencies, left angle bracketωT,Lright-pointing angle bracketcolon, equalsPT,L/NT,L,

In the tables of Section 5, we will compare tachyonic with electromagnetic synchrotron radiation. To this end, we quote the photonic synchrotron power, Pph=P||ph+Pperpendicularph, the photon count in the usual linear polarizations, and the photonic mean energy [11],


with ωBc/R, cf. (4.1). The photon density is defined in (3.19) and integrated in ((A.30) and (A.31)).

Degrees of polarization, ΠT,Lcolon, equalsPT,L/(PT+PL), are defined analogously to the polarization functions in ((3.25) and (3.26)), with the spectral densities replaced by the radiant powers Pless-than-or-equals, slantωbT,L or Pgreater-or-equal, slantedωbT,L,


In the low-frequency regime, the linear radiation is much more intense than the curvature radiation, cf. ((4.5), (4.6), (4.7), (4.8) and (4.9)), and the polarization Πless-than-or-equals, slantωbT,L is thus determined by the linear powers ((4.5) and (4.8)), for large κ, that is. In the high-frequency range, the longitudinal curvature radiation overpowers the transversal one by a factor ∝κ2/3, cf. ((4.6) and (4.9)), which shows in Πgreater-or-equal, slantedωbT. In the transversal plane, we find in both regimes PperpendicularT,curv:P||T,curvnot, vert, similar5:1, cf. ((4.6) and (4.14)), as compared with the photonic ratio Pperpendicularph:P||ph=7:1.

The tables in Section 5 are compiled with the foregoing formulas; we introduce dimensionless quantities, E0colon, equalsE[GeV] (electron energy), B0colon, equalsB[kG], and R0colon, equalsR[m], so that

R0≈33.36E0B0−1, γ≈1957E0, ωB[GHz]≈0.2998R0−1 ,
where we used the relations following (4.1) as well as 1 kG·e≈2.998×10−4 GeV cm−1. The expansion parameter (4.2) scales as

κ≈1.89×103E0−2R0≈6.32×104E0−1B0−1 ,
the tachyonic powers and number counts in ((4.5), (4.6), (4.7), (4.8), (4.9), (4.10), (4.11) and (4.12)) depend on κ and γ only. The critical frequency, the break and peak frequencies, and the corresponding energies scale as

νc[1018 Hz]≈0.536E03R0−1≈1.61×10−2E02B0 ,

planck constant over two piωc[keV]≈2.22E03R0−1≈6.65×10−2E02B0 .

νb[1018 Hz]≈1.02×103E0, planck constant over two piωb[keV]≈4.21×103E0 ,

νpeak[1018 Hz]≈0.520, planck constant over two piωpeak[keV]≈2.15 .
In the tables of Section 5, we will list ν rather than the circular frequencies used throughout the paper, and the corresponding energies planck constant over two piω will be denoted by var epsilonb, var epsilonc, and var epsilonpeak. The electromagnetic power and the photonic number count, cf. (4.16), scale as

Pph[GeV s−1]≈4.22×103E04R0−2≈3.79E02B02 ,

Nph[s−1]≈6.18×109E0R0−1≈1.85×108B0 .

Wave length, energy and speed (group velocity) of a tachyon relate as [17],

where hc≈12.40 keV Å. For instance, λbtachnot, vert, similarhc/var epsilonb, cf. (4.20), to be compared with λComptontach≈5.8 Å (unreduced) and the photonic λcph[Å]≈12.4/var epsilonc[keV]. As for the tachyonic velocity, we find υtach/c−1not, vert, similarmt2c4/(2var epsilonb2) at the break energy, and a modest υtach/c≈1.41 at the spectral peak. Numerical examples will be given in the tables.

In 5 and 6, we will need to know how a non-singular electron distribution affects the spectral densities (3.22). To this end, we average them with an electronic power-law distribution, dn(γ)∝γs dγ, of index s>1; the electronic Lorentz factors range in a finite interval, γ1less-than-or-equals, slantγless-than-or-equals, slantγ2, the source count reads Image . In the subsequent sections, we will piece together multi-band spectra with broken power-laws. A synchrotron model of γ-ray burst spectra with an analytic cross-over between power-laws is studied in [18 and 19]. A cross-over from power-law to exponentially damped power-law seems to apply to the pervading electron density in the Coma Cluster, derived on the basis of synchrotron and inverse Compton models [20]. Here, we will content ourselves with broken power-laws; the introduction of curvature always involves some arbitrariness in the analytic shape of n′(γ), and tends to make integrations rather clumsy. Like the cosmic ray spectrum, the wideband spectra of supernova remnants can be assembled quite convincingly with broken power-laws, cf. Section 6; there is very little evidence for curvature. The TeV flare spectra of the BL Lac object Mrk 501 are genuinely curved [21], suggesting an exponential cutoff factor in dn(γ), possibly with a further power-law in the exponential [22]. The averaged linear density (4.23) including an exponential cutoff is still analytically tractable with incomplete Γ-functions, but we would not consider blazars here, and broken power-laws will do for Jupiter's aurora and synchrotron nebulae. We also restrict ourselves to the linear densities (3.23); the leading order of the curvature radiation drops out in the averaging procedure, cf. after (4.24). These densities are generated by ultra-relativistic particles, so that γ1much greater-than1. The averaging is carried out via

and analogously for left angle bracketpL,lin(ω)right-pointing angle brackets. The densities pT,lin and pL,lin are cut off at ωb=mtcγ, which is accounted for by the step function Θ in (4.23), cf. the discussion after (3.24). The longitudinal density pL,lin depends on the Lorentz factor only via the cutoff frequency ωb. We thus find the averages,


ω1(ω)colon, equalsmax(ω,ωb1), ω2(ω)colon, equalsmax(ω,ωb2), ωb1,2colon, equalsmtcγ1,2 ,

Clearly, these densities vanish for ω>ωb2, where ω12=1. In the high-frequency band, ωb1much less-thanωmuch less-thanωb2, we find left angle bracketpT,L,linright-pointing angle brackets∝ωs, so that the tachyonic spectral index coincides with the electronic power-law index, in strong contrast to the photon index, cf. (4.26). In the low-frequency regime, we recover left angle bracketpT,L,linright-pointing angle brackets∝ω for ωmuch less-thanmtc, the linear frequency scaling of the distributions (3.23), and left angle bracketpT,L,linright-pointing angle brackets∝ω−1 holds for mtcmuch less-thanωmuch less-thanωb1. The curvature radiation generated by the densities ((3.16), (3.18) and (3.24)) can be averaged like in (4.23), the procedure outlined in ((A.23), (A.24), (A.25), (A.26), (A.27), (A.28), (A.29), (A.30) and (A.31)) also applies here, in particular the same variable transformations can be used, with ω replaced by ωb in (A.25). In leading order (large κ), the high- and low-frequency components cancel each other when integrated with ωbsb, and the higher orders are overpowered by the averaged linear densities (4.24) and do not affect their slopes. The same happens with the integral curvature radiation stemming from a singular electron distribution, cf. ((4.4), (4.7) and (4.14)).

We integrate the averaged densities (4.24) by making use of γ1much greater-than1, to find the averaged transversal power,

and the same for left angle bracketPLright-pointing angle brackets, with −1/2 in the parenthesis dropped. (We have again restored the units, mtmtc/planck constant over two pi). This can also be derived by directly averaging the powers ((4.5) and (4.8)) with dn(γ). The linear count rates do not depend in leading order on the Lorentz factor, cf. (4.10), and are therefore not affected by the averaging, apart from the factor N1,2, of course. The tachyonic mean frequencies are thus left angle bracketωT,Lright-pointing angle bracketsnot, vert, similar(2/π)ωpeak(log γ1+cdots, three dots, centered), with the parenthesis completed as in (4.25). This is to be compared with the mean frequencies (4.15) stemming from a singular electron distribution.

For comparison, we quote the electromagnetic average [11],

with p||,perpendicularph(ω,γ) defined in (3.19) and the critical photon frequencies ωc1,2 taken at γ1,2. (The critical frequency scales as ωc∝γ2, since ωB∝γ−1, cf. ((4.1) and (4.2)).) This is derived by extending the integration boundaries in (4.26) to zero and infinity and applying (A.30). The photon distribution is strongly peaked, more or less at the mean frequency, cf. (4.16) and after (4.2). The tachyon densities pT,L,lin are much more extended, with a mean frequency largely deviating from the spectral peak at 2.15 keV, cf. (4.15) and Table 2 in Section 5. The extension of the integration boundaries in (4.26) is only permissible if ω lies well within the band ωc1much less-thanωmuch less-thanωc2. (If the integration boundary in the integrals (A.30) is cut, they admit antiderivatives in terms of Lommel functions with simple asymptotic expansions.) It is only in this band that the averaged photon density has a power-law decay ∝ω−α, with αcolon, equals(s−1)/2. As κ=ωbc is large throughout the averaging interval, this band noticeably differs from the tachyonic high-frequency band ωb1much less-thanωmuch less-thanωb2, where the tachyon densities (4.24) admit a power-law slope of index s, even though these bands can overlap.

5. Tachyonic X-rays from Jupiter's magnetosphere

A theory of tachyons should provide clues as to where to find them. In Table 1, Table 2, Table 3 and Table 4,we give some estimates to that effect by comparing tachyonic with electromagnetic synchrotron radiation. The examples cover a wide range of electron energies and gyroradii, and a smaller range of magnetic fields. Extremely high magnetic field strengths require different asymptotics, cf. after (4.2), let alone quantum corrections which can completely change the shape of the classical spectral densities by generating a cutoff before the classical spectral peak is reached [11 and 23]. Even for moderate magnetic field strengths, there is a pronounced tachyonic quantum effect emerging in the low-energy cyclotron limit, unparalleled in electromagnetic theory. There is a threshold velocity, a lower bound on the speed of the (always subluminal) source, below which tachyons can only be radiated as curvature radiation [6]. This threshold velocity is without counterpart in the classical radiation theory, and it coincides, most remarkably, with the speed of the Galaxy in the microwave background, υLG/cmt/(2m)≈2.10×10−3. Clearly, the ultra-relativistic synchrotron radiation discussed here is not affected by this.

Table 1. Entries as defined in Section 4: electron energy E (input), electronic Lorentz factor γ, magnetic field B (input), gyroradius R, critical photon energy var epsilonc, tachyonic break energy var epsilonb, tachyonic spectral maximum var epsilonpeak≈2.15 keV. Magnetic field strengths: 100 and 300 μG for the Crab, 1 and 3 mG for Cas A, 10 and 30 μG for SN 1006. The parameters of SN 1006 apply to RX J1713.7–3946 as well, and the Crab entries E, γ and var epsilonb also hold for the other remnants. References for the broadband spectra, from soft X-rays to TeV γ-rays, are cited in the text. The wave lengths at 1.4 Jovian radii refer to the photonic spectral peak at λcph/0.286, cf. Table 2; the respective radio fluxes [28, 29, 30, 31 and 32] are quoted in the text. The magnetic field strengths are estimates derived in [34, 35, 36, 42, 44, 51, 55 and 56]. The URLs (a–d) point to the technical data sheets of the storage rings


Table 2. Entries as defined in Section 4: tachyonic mean energy left angle bracketvar epsilonT,Lright-pointing angle bracket, tachyonic mean frequency left angle bracketνT,Lright-pointing angle bracket (transversally and longitudinally radiated), critical photon frequency νc, tachyonic break frequency νb. Crab entries apply unless listed otherwise. The continuous parameter k labels the electron energy, cf. Table 1, one may envisage 0less-than-or-equals, slantkless-than-or-equals, slant7 as typical range, cf. Section 6. The tachyonic peak frequency, νpeak(1018 Hz)≈0.52, is independent of the electron energy


Table 3. The power radiated, cf. Section 4: unpolarized electromagnetic radiation Pph, transversal and longitudinal tachyon radiation PT,L,lin (residual, in the limit of infinite curvature radius), tachyonic curvature radiation PT,L,curv. Input parameters as in Table 1


Table 4. Count rates, wave lengths, group velocity, cf. Section 4: unpolarized photon count Nph, polarized tachyon count NT,L,curv (curvature radiation). The ultra-relativistic count rate for the residual radiation (infinite gyroradius) is NT,L,lin≈5.1×105 s−1. Critical photonic wave length λcph, tachyonic wave length at break energy λbtach, group velocity at break energy υtach. To be compared with λpeaktach≈4.1 Å and υtach/c≈1.4 at the spectral maximum at 2.15 keV


A first orientation as to what one can expect is provided by the storage rings listed in the tables. The frequency peak of the photonic energy density (3.19) is located at 0.286var epsilonc, very close to the photonic mean energy at 0.308var epsilonc, cf. (4.17). This is in contrast to tachyon radiation, where the peak energy of the densities (3.23) at 2.15 keV is clearly separated from the mean frequencies left angle bracketvar epsilonT,Lright-pointing angle bracket in Table 2. The photon density is sharply peaked with exponential decay toward higher frequencies, whereas the linear tachyon densities decay as pT,L,lin∝ω−1 between 2.15 keV and the break energy in the low MeV region. The tachyonic power radiated as curvature radiation is negligible compared to the linear power PT,L,lin, cf. Table 3; this applies to all examples given in the tables, since the expansion parameter ωbc (the ratio of break and critical frequency) is always very large, cf. (4.2). In pulsar magnetospheres, however, the opposite limit is realized, and then the curvature radiation dominates; this will be studied elsewhere. In the ultra-relativistic limit, the power of the linear tachyon radiation varies only weakly with the electronic Lorentz factor, and it can easily surpass the photonic power radiated, in Jupiter's case by a staggering factor of 109. The tachyonic break energies of the four storage rings range between 0.4 and 50 MeV, as compared with the tachyonic spectral peak at 2.15 keV; the photonic spectral peaks of SURF III lie in the near IR and UV, Aladdin's peak frequency in the extreme UV, and SPring-8 and Petra II radiate hard X-rays. The tachyonic wavelength at 2.15 keV is 4.1 Å, cf. (4.22), to be compared with the photonic counterpart λcph/0.286 and the tachyonic wavelengths λbtach at the break energies in the soft γ-ray band, cf. Table 4. The photon count Nph of the storage rings is by three to four orders higher than the tachyonic emission rate, NT,L,lin≈5.1×105 s−1, cf. Table 4. This does not seem insurmountable, but it comes on top of the very weak coupling of tachyons to matter, by a factor of αqe≈1.4×10−11 smaller than the electromagnetic interaction, cf. the beginning of Section 4. This factor shows in ratios of tachyonic and photonic ionization cross sections and induced transition amplitudes [13 and 15]; in the case of Petra II, only one in 1014 ionizations is tachyonic. The chances to detect tachyon radiation in storage rings are therefore rather slim, even though the longitudinal polarization and the tachyon speed at the peak frequency can be used as sieves.

We turn to Jupiter's radiation belts [24, 25, 26 and 27]. The non-thermal flux at 90 cm (0.33 GHz or 0.286νc, cf. Table 1) is 6.2 Jy [28]. At 21 cm (1.4 GHz), a non-thermal flux of 5.3 Jy is quoted in [29]. At 11 cm (2.7 GHz) a flux of 4.3 Jy and at 6 cm (5.0 GHz) a non-thermal flux of 3.5 Jy is quoted in [30] and revised in [31] as stated. All these fluxes stem from SL9 pre-impact observations. At 2.2 cm (13.8 GHz), a non-thermal flux of 0.42 Jy was observed with Cassini [32]. The photonic spectral density scales with ν−α when averaged with an electronic power-law density of index s=2α+1, cf. after (4.26). In the 90–21 cm range, we thus find the photon index α≈0.1 (as quotient of log(6.2/5.3) and log(1.4/0.33)), a virtually flat photon spectrum. The averaged tachyonic spectral densities inherit the electron index as pointed out after (4.24), and thus scale with ν−1.2 between the break energies of 33 and 67 keV, cf. Table 1. The tachyonic mean energies (averaged with the electronic power-law and Lorentz factors ranging between 15.3 and 31.3, cf. Table 1) are left angle bracketωTright-pointing angle brackets=1.2≈3.5 keV and left angle bracketωLright-pointing angle brackets=1.2≈4.2 keV, as indicated after (4.25). The electron energies range in the 8–16 MeV interval, with power-law index 1.2. Between the spectral maximum at 2.15 keV and the spectral break at 33 keV, the tachyonic energy density scales with ν−1, cf. the discussion after (4.24).

In the 21–11 cm range, we find α≈0.32, and in the 11–6 cm interval α≈0.33, so that the tachyonic frequency scaling ∝ν−1.6 applies between the break energies of 67 and 130 keV. The 6–2.2 cm interval admits the slopes α≈2 and s≈5, so that the tachyonic spectral densities rapidly decay in the 130–210 keV range, with a steep power-law tantamount to an exponential cutoff. The spectral peak is always located at the tachyon mass, at 2.15 keV, in the ultra-relativistic limit, that is. Hence, tachyon emission from Jupiter's radio electrons should be detectable in hard and high-energy X-rays, throughout the 2–130 keV range. Further SL9 pre-impact fluxes at other wavelengths can be found in [33 and 34], to the same effect.

Soft X-rays from Jupiter's aurora have been detected on two occasions. The ROSAT spectral map [37, 38 and 39] shows three peaks in the number count; the peaks at 0.2 and 0.8 keV are probably due to line emissions from precipitating oxygen and sulfur ions, but there is a third tiny peak located at 2 keV, that has not gained attention so far, coinciding with the peak of the tachyonic spectral densities. This peak also shows in the Einstein map [40]. The ROSAT count rate gets sparse above 0.4 keV, and the Einstein map has only a bandpass of 0.2–3 keV, but the three radiation peaks are clearly discernible in the spectral maps. However, the third peak is too tiny to check for the slope of the differential number count above 2 keV, which should scale with ν−2 in hard X-rays up to the spectral break at 30 keV. It is tempting to identify this peak with tachyon radiation from the radio electrons. Detection of longitudinally polarized X-rays could be the crucial test as to whether the third radiation peak is tachyonic; transversal and longitudinal tachyons are radiated in equal rates and with nearly equal power in Jupiter's magnetosphere, cf. Table 3.

6. Tachyonic gamma-rays from the shock-heated plasmas of supernova remnants

We will identify tachyonic spectral slopes in the broadband spectra (soft X-ray to TeV γ-ray) of four galactic remnants. We start by approximating the averaged longitudinal density left angle bracketpL,lin(ω)right-pointing angle brackets in (4.24) by

The proportionality constant is N1,2αqmt2c4/planck constant over two pi in both regimes, Eb1,2 denotes the break energies planck constant over two piωb1,2, and dleft angle bracketPLright-pointing angle bracketspL(E) dE, cf. (4.25). This holds for E/Eb2much less-than1, implying γ12much less-than1. In the same approximation, we write the transversal density left angle bracketpT,lin(ω)right-pointing angle brackets in (4.24) as

We will consider unpolarized radiation, that is, the density p(E)=pT+pL. The differential energy flux is thus S′(E)=p(E)/(4πd2), where d is the distance to the remnant. This distance and the proportionality factor in (6.1) will drop out in the subsequent ratios. The differential number count, N′(E), relates to the energy flux via dS(E)=E dN(E), of course. The following discussion is based on the rescaled flux density f(E)colon, equalsE2N′(E). We find, by assembling ((6.1) and (6.2)),

provided mtc2less-than-or-equals, slantEmuch less-thanEb1. We will consider energies much larger than the tachyon mass, so that this ratio is constant for Emuch less-thanEb1 and scales as f(E)/f(Eb1)≈(E/Eb1)1−s for Eb1less-than-or-equals, slantEmuch less-thanEb2. The tachyonic energy density p(E) attains its maximum at the tachyon mass, and f(E) stays nearly constant between break energy and maximum. The tachyonic spectral index s coincides with the electron index and typically ranges in 1<s<2, in the X- and γ-ray bands.

TeV γ-rays from the Crab have been detected in the 0.5–50 TeV range. A differential flux of N′(E)≈2.79×10−11(E/1 TeV)−2.59 cm−2 s−1 TeV−1 in the 1–20 TeV interval is quoted by the HEGRA Collab. [41]. A recent power-law of the Whipple Group with amplitude 3.12×10−11 and exponent 2.57 applies in the 0.5–8 TeV range [42]. The CANGAROO count [43] extends from 7 to 50 TeV, with amplitude 2.76×10−11 and exponent 2.53. The subsequent calculations are based on NTeV′(E)≈2.9×10−11(E/1 TeV)−2.6 and Eb1≈0.6 TeV. Above this break energy, the tachyonic spectral index s≈1.6 applies, as defined after (4.24). We write fTeV(E)=E2NTeV′(E), so that fTeV(Eb1)≈44.5 eV cm−2 s−1, and infer from (6.3) that fTeV(E) cuts as a plateau horizontally through the error bars of four EGRET points and one COS-B point located in the 1–30 GeV range [42 and 44]. This fTeV-plateau also hits a CELESTE point at 60 GeV [45]. The remaining EGRET flux points below 1 GeV define a power-law slope [46], NMeV′(E)≈9.1×10−9(E/118 MeV)−2.9 cm−2 s−1 MeV−1. There are some contaminated data points in this fit, it is feasible that the slope is slightly flatter, especially if one includes two COS-B points below 0.5 GeV, cf. [44], but not by much. Anyway, we will use s≈1.9 as tachyonic spectral index in this range. The slope fMeV(E)=E2NMeV′(E) joins the GeV plateau at E1≈380 MeV, where fTeV(E1)=fMeV(E1). Hence, fTeV(E) applies above 380 MeV, up to at least 50 TeV. Below 380 MeV, the slope fMeV(E) intersects a second plateau value defined by the COMPTEL data points, which give Nctel′(E)≈1.9×10−4(E/2.6 MeV)−2.0 cm−2 s−1 MeV−1 for the unpulsed flux in the 1–10 MeV range [47 and 48]. This plateau is reproduced by fMeV(E), extended below the spectral break at Eb1≈11 MeV according to (6.3). The break energy is obtained by solving Eb12Nctel′(Eb1)≈1.18fMeV(Eb1); we find Eb12Nctel′(Eb1)≈1.3 keV cm−2 s−1. Finally, most of the keV range is covered by a broken power-law derived from HEAO 1 and earlier observations [49];NX′(E)≈9.23(E/1 keV)−2.13 cm−2 s−1 keV−1 (unpulsed) applies from 17 keV up to the spectral break at 150 keV, and NkeV′(E)≈2.14×10−4(E/150 keV)−2.54 cm−2 s−1 keV−1 holds above 150 keV. The slope E2NX′(E) can be continued with broken power-laws to soft X-ray energies and into the UV, optical, IR and radio bands as done in [50]. The high-energy slope E2NkeV′(E) hits the COMPTEL plateau at E≈1.7 MeV.

The spectrum up to 1.7 MeV is photonic, but the COMPTEL plateau from 1.7 to 11 MeV and the subsequent EGRET slope from 11 to 380 MeV is unlikely to be of electromagnetic origin. A power-law slope preceded by a plateau value of f(E) is characteristic for tachyon radiation. For this reason, I also suggest that the EGRET plateau from 380 MeV to 0.6 TeV and the subsequent power-law slope up to 50 TeV is tachyonic. The EGRET points give a clear picture of the GeV γ-ray spectrum, though they have been criticized on the grounds of contamination and large error bars, as they fail to provide evidence for inverse Compton scattering and pion decay, for the downward slope in the GeV region that is, in all remnants investigated. Plots of the multi-band spectrum with the mentioned flux points can be found in [42, 44, 45, 47 and 48].

The energy range of the electrons producing the tachyonic spectral slopes and plateaus can be read off from the tachyonic break energies var epsilonb in Table 1. The tachyonic slope in the 0.6–50 TeV interval corresponds to 5.15less-than-or-equals, slantkless-than-or-equals, slant7.1, from which we infer electron energies between 0.14 and 13 PeV. The electronic power-law index coincides with the tachyonic spectral index of 1.6. The exponent k parametrizes the electron energy in Table 1. This is to be compared to the ‘knee’ of the cosmic ray spectrum at k≈6.5 (≈3 PeV) [52]. The second tachyonic slope, between 11 and 380 MeV, corresponds to 0.42less-than-or-equals, slantkless-than-or-equals, slant1.96, which translates into electron energies from 2.6 to 91 GeV with index 1.9.

We turn to the remnant Cas A; γ-rays have been detected in the 1–10 TeV range, with an integral flux of 5.8×10−13 cm−2 s−1 and a slope NTeV′(E)∝(E/1 TeV)−2.5 (HEGRA Collab. [53]), from which we infer a proportionality constant of 9.0×10−13 cm−2 s−1 TeV−1. At the break energy Eb1≈1 TeV, we thus find fTeV(Eb1)≈0.90 eV cm−2 s−1, which extends as plateau fTeV(E)≈1.0 eV cm−2 s−1 into the GeV and MeV range according to (6.3) (with s≈1.5). As for the X-ray slopes [54], the interval 1–120 keV is covered by a broken power-law with spectral break at 16 keV. The differential number count NX′(E)≈0.038(E/1 keV)−1.8 cm−2 s−1 keV−1 applies below 16 keV, and above it steepens to NkeV′(E)≈2.6×10−4 (E/16 keV)−3.0 cm−2 s−1 keV−1. The E2NkeV′(E) slope of the photonic high-energy X-ray tail joins the tachyonic fTeV(E)-plateau at E≈1.1 MeV, provided that the X-ray count NkeV′(E) extends as unbroken power-law to this energy. The tachyon spectrum spreads from 1.1 MeV to 1 TeV with a flat fTeV(E), and continues above the spectral break at 1 TeV with a slope fTeV(E)≈0.90(E/1 TeV)−0.5 eV cm−2 s−1, cf. after (6.3). The energy of the electrons producing this fTeV(E) can be inferred from the tachyonic break energies var epsilonb in Table 1, like for the Crab above. The tachyonic 1–10 TeV range corresponds to 5.38less-than-or-equals, slantkless-than-or-equals, slant6.38, resulting in electron energies of 0.24–2.4 PeV, with a power-law index of 1.5.

As for the remnant SN 1006, an integral X-ray flux of 89 eV cm−2 s−1 in the 0.1–2.0 keV range is quoted in [56], and the differential number count between 0.1 and 1.85 keV scales as NSX′(E)≈0.027(E/1 keV)−2.1 cm−2 s−1 keV−1. Above the spectral break at 1.85 keV, up to 17 keV, the flux scales as NHX′(E)≈0.047(E/1 keV)−3.0 cm−2 s−1 keV−1, cf. [56 and 57]. The observed γ-ray spectrum extends to about 20 TeV (CANGAROO Collab. [58 and 59]), the spectral index is as yet not well determined. fTeV(E) is nearly flat between 1 and 5 TeV, indicating a spectral index s≈1, so that in this range NTeV′(E)≈9×10−12(E/1 TeV)−2 cm−2 s−1 TeV−1. We thus infer fTeV(E)≈9 eV cm−2 s−1 below 5 TeV, and this plateau extends down to 1 GeV, cutting through the EGRET upper limits in the 1–10 GeV range; a plot of the broadband spectrum with the EGRET flux can be found in [59]. The EGRET data indicate a spectral break at 1 GeV and a downward slope ranging from about 50 MeV to 1 GeV. Below 50 MeV, down to high-energy X-rays, there are no flux data available yet. In any case, the fTeV-plateau in the GeV range followed by a spectral break in the low TeV region is also present in this remnant, like in Cas A and the Crab, though the evidence is scantier.

The fourth remnant detected in TeV γ-rays is RX J1713.7–3946; the CANGAROO differential flux is NTeV′(E)≈1.6×10−11(E/1TeV)−2.8 cm−2 s−1 TeV−1 in the 0.4–8 TeV range [60]. We find fTeV(Eb1)≈19 eV cm−2 s−1 at the break energy of 0.8 TeV, followed by the plateau fTeV(E)≈22 eV cm−2 s−1 in the GeV region, cf. (6.3) (with s≈1.8). This plateau cuts horizontally through the data points of the unidentified γ-ray source 3EG J1714–3857 located in the range 0.5–10 GeV. The EGRET spectrum is flat in this region, but starts to steepen below 500 MeV indicating a spectral break, cf. the Crab. A plot of the EGRET points in the wideband spectrum can be found in [61]. The spectrum from 10 MeV down to hard X-rays has not been measured yet. In the 0.5–10 keV interval, the photonic energy flux scales as ENX′(E)∝(E/1 keV)−1.4, and the integral flux in this range is 0.10 keV cm−2 s−1, so that E2NX′(E)≈0.043(E/1 keV)−0.4 keV cm−2 s−1, as inferred from ASCA & ROSAT data [62 and 63]. RXTE observations indicate a steepening of this slope in the 10–30 keV range [64]. The RXTE count does not smoothly join the ROSAT count, there is a discontinuity at 2 keV extending over one order in the count rate. This could be due to background radiation from the Galactic Ridge, but there can also be a tachyonic f(E)-plateau extending down to the tachyon mass at 2.15 keV, cf. (6.3), which is not to be subtracted as background radiation.

Tachyonic γ-rays in the 0.8–8 TeV range correspond to electron energies from 0.2 to 2 PeV, parametrized by 5.3less-than-or-equals, slantkless-than-or-equals, slant6.3 in Table 1. The fTeV-plateau in the GeV range cutting through the 3EG J1714–3857 flux points again suggests that the GeV and TeV radiation is tachyonic; such an extended plateau is unlikely to be produced by inverse Compton scattering or π0 decay. The TeV γ-ray flux of this remnant is quite on the Crab scale, whereas the X-ray flux is much weaker and comparable with Cas A, whose TeV radiation is by one order weaker. TeV γ-rays from other remnants such as IC 443 and Tycho have escaped detection. This suggests that there are two energetically different electron populations in remnants with TeV spectra, one emitting tachyonic MeV to TeV γ-rays, the other generating the photonic synchrotron radiation in the radio-to-X-ray bands. I also note that the tachyonic spectral densities in ((6.1) and (6.2)) stem from the linear densities (4.24), in particular the rescaled flux density f(E), key to the preceding discussion, is independent of the magnetic field; the tachyonic curvature radiation is negligible for the low magnetic field strengths in the remnants, cf. Table 3 and Table 4.

The tachyonic luminosity and magnetospheric electron population of γ-ray pulsars can be inferred from COMPTEL and EGRET observations. Due to the strong surface fields, the electromagnetic synchrotron radiation in the γ-ray band is suppressed by a quantum cutoff, but not so tachyonic γ-radiation. This provides an exceptional opportunity to search for tachyon radiation, unspoiled by electromagnetic emission. However, the asymptotics developed here, ωcbmuch less-than1, cf. (4.2) and Table 1, does not apply to excessive magnetic field strengths, the opposite limit is realized in the surface fields of γ-ray pulsars.

Very promising sources of superluminal γ-rays are TeV blazars; multi-band spectra of Mrk 421 and Mrk 501 are compiled in [21, 65, 66, 67 and 68]. The EGRET flux points, however, do not stem from the specific flares for which the broadband spectra are assembled, and there are also rather large variations in the amplitudes and exponents of the TeV γ-ray fluxes, let alone spectral curvature and cutoff energies. The tachyonic interpretation of the γ-ray spectra again hinges on the GeV radiation, which is not known for the individual flares, although the EGRET points in the spectral maps are quite suggestive to that effect. Finally, tachyons do not interact with IR background photons, so that there is no absorption of tachyonic TeV γ-rays from extragalactic sources by electron–positron creation.


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, the Institute of Mathematical Sciences, Madras, and the Tata Institute of Fundamental Research, Bombay, are likewise gratefully acknowledged.


1. A. Proca. J. Phys. 7 (1936), p. 347.

2. G. Wentzel. Quantum Theory of Fields, Interscience, New York (1949).

3. A.S. Goldhaber and M.M. Nieto. Rev. Mod. Phys. 43 (1971), p. 277. 

4. R. Tomaschitz. Class. Quantum Gravity 18 (2001), p. 4395. 

5. R. Tomaschitz. Eur. Phys. J. B 17 (2000), p. 523. 

6. R. Tomaschitz. Physica A 320 (2003), p. 329.

7. H. Motz. J. Appl. Phys. 22 (1951), p. 527. 

8. J. Schwinger. Phys. Rev. 75 (1949), p. 1912.

9. V.L. Ginzburg and S.I. Syrovatskii. Ann. Rev. Astron. Astrophys. 3 (1965), p. 297.

10. V.L. Ginzburg and S.I. Syrovatskii. Ann. Rev. Astron. Astrophys. 7 (1969), p. 375.

11. A.A. Sokolov and I.M. Ternov. Radiation from Relativistic Electrons, Hilger, Bristol (1986).

12. R. Tomaschitz. Chaos Solitons Fractals 9 (1998), p. 1199.

13. R. Tomaschitz. Physica A 307 (2002), p. 375. 

14. R. Tomaschitz. Chaos Solitons Fractals 20 (2004), p. 713.

15. R. Tomaschitz. Physica A 293 (2001), p. 247. 

16. S. Krinsky, M.L. Perlman and R.E. Watson. E.E. Koch, Editor, Handbook on Synchrotron Radiation Vol. 1, Elsevier, Amsterdam (1983).

17. G. Feinberg. Phys. Rev. 159 (1967), p. 1089. 

18. N.M. Lloyd and V. Petrosian. Astrophys. J. 543 (2000), p. 722. 

19. N.M. Lloyd-Ronning and V. Petrosian. Astrophys. J. 565 (2002), p. 182. 

20. V. Petrosian. Astrophys. J. 557 (2001), p. 560. 

21. F.A. Aharonian et al.. Astron. Astrophys. 366 (2001), p. 62. 

22. S.P. Reynolds and J.W. Keohane. Astrophys. J. 525 (1999), p. 368. 

23. J. Schwinger. Proc. Natl. Acad. Sci. 40 (1954), p. 132.

24. J.A. Van Allen. In: T. Gehrels, Editor, Jupiter, University of Arizona Press, Tucson (1976).

25. J.A. Van Allen and C.L. Grosskreutz. J. Geophys. Res. 94 (1989), p. 8731.

26. I. de Pater. Astron. J. 102 (1991), p. 795. 

27. J.A. Van Allen. Icarus 122 (1996), p. 209. 

28. S. Sukumar, I. de Pater and R.G. Strom. Icarus 131 (1998), p. 358.

29. M.H. Wong et al.. Icarus 121 (1996), p. 457. 

30. O. Funke et al.. Planet. Space Sci. 45 (1997), p. 1177. 

31. I. de Pater and S.H. Brecht. Icarus 151 (2001), p. 39. 

32. S.J. Bolton et al.. Nature 415 (2002), p. 987. 

33. P.H.M. Galopeau, E. Gerard and A. Lecacheux. Planet. Space Sci. 45 (1997), p. 1197.

34. I. de Pater et al.. Science 268 (1995), p. 1879.

35. G.A. Dulk, R.J. Sault and Y. Leblance. Planet. Space Sci. 45 (1997), p. 1231.

36. M.G. Kivelson and D.J. Southwood. Planet. Space Sci. 51 (2003), p. 891.

37. J.H. Waite et al.. J. Geophys. Res. 99 (1994), p. 14799. 

38. J.H. Waite et al.. Adv. Space Res. 26 (2000), p. 1453. 

39. A. Bhardwaj and G.R. Gladstone. Rev. Geophys. 38 (2000), p. 295.

40. A.E. Metzger et al.. J. Geophys. Res. 88 (1983), p. 7731.

41. F. A. Aharonian et al.. Astrophys. J. 539 (2000), p. 317.

42. A. M. Hillas et al.. Astrophys. J. 503 (1998), p. 744.

43. T. Tanimori et al.. Astrophys. J. 492 (1998), p. L33. 

44. O. C. de Jager et al.. Astrophys. J. 457 (1996), p. 253.

45. M. de Naurois et al.. Astrophys. J. 566 (2002), p. 343. 

46. P. L. Nolan et al.. Astrophys. J. 409 (1993), p. 697.

47. R. Much et al.. Astron. Astrophys. 299 (1995), p. 435.

48. R.D. van der Meulen et al.. Astron. Astrophys. 330 (1998), p. 321. 

49. G.V. Jung. Astrophys. J. 338 (1989), p. 972.

50. M.P. Véron-Cetty and L. Woltjer. Astron. Astrophys. 270 (1993), p. 370.

51. J.J. Hester et al.. Astrophys. J. 448 (1995), p. 240. 

52. M. Nagano and A.A. Watson. Rev. Mod. Phys. 72 (2000), p. 689. 

53. F. Aharonian et al.. Astron. Astrophys. 370 (2001), p. 112. 

54. G.E. Allen et al.. Astrophys. J. 487 (1997), p. L97.

55. M.C. Anderson and L. Rudnick. Astrophys. J. 456 (1996), p. 234. 

56. G.E. Allen, R. Petre and E.V. Gotthelf. Astrophys. J. 558 (2001), p. 739.

57. K. Koyama et al.. Nature 378 (1995), p. 255. 

58. T. Tanimori et al.. Astrophys. J. 497 (1998), p. L25. 

59. T. Naito et al.. Astron. Nachr. 320 (1999), p. 205.

60. R. Enomoto et al.. Nature 416 (2002), p. 823. 

61. O. Reimer and M. Pohl. Astron. Astrophys. 390 (2002), p. L43. 

62. K. Koyama et al.. Publ. Astron. Soc. Japan 49 (1997), p. L7. 

63. P. Slane et al.. Astrophys. J. 525 (1999), p. 357. 

64. T.G. Pannuti and G.E. Allen. In: P.O. Slane and B.M. Gaensler, Editors, Neutron Stars in Supernova Remnants, ASP, San Francisco (2002).

65. J. Kataoka et al.. Astrophys. J. 514 (1999), p. 138. 

66. T. Takahashi et al.. Astrophys. J. 542 (2000), p. L105. 

67. F. Krennrich et al.. Astrophys. J. 575 (2002), p. L9. 

68. K. Okumura et al.. Astrophys. J. 579 (2002), p. L9. 

69. G.N. Watson. A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge (1996).

70. D.E. Aspnes. Phys. Rev. 147 (1966), p. 554. 

71. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC (1972).

72. E.T. Copson. Asymptotic Expansions, Cambridge University Press, Cambridge (1965).

Appendix A. Superluminal spectral and power asymptotics

We start with the stationary phase asymptotics employed in Section 3. The Schott identities ((3.7), (3.8) and (3.9)) are a convenient way of rewriting the powers in the multipole expansion (3.4), and the notation customary in synchrotron radiation theory is based on them [8 and 11]. These identities can be derived by using the power series expansion of Jn2 and term by term integration; the Jn2 integration is settled by


The asymptotics of electromagnetic synchrotron radiation theory also applies to tachyon radiation. Contrary to the photonic case, however, we have to consider two asymptotic regimes, since xn/n in (3.1) can be larger or smaller than one. The Bessel functions in ((3.7), (3.8) and (3.9)) are replaced by their Nicolson asymptotics, which applies for large positive n and z, so that z/n≈1, cf. [69]. If z>n,




in leading order of a 1/n and z/n−1 double series expansion. We also note Image for n>−1 and n>0, respectively, from which the asymptotics of Image can be recovered. (A.3) and (A.4) readily follow from (A.2) and the identities

(J−1/3+J1/3)′=J−2/3J2/3−(J−1/3+J1/3)/(3z) ,

(J−2/3J2/3)′=−(J−1/3+J1/3)−2(J−2/3J2/3)/(3z) ,

J−5/3+J5/3=J−1/3+J1/3+2(J−2/3J2/3)′ .
In the powers ((3.5) and (3.6)), we can thus approximate, if xn/ngreater-or-equal, slanted1, cf. (3.1),



In (A.7), the lower plus-sign refers to the perpendicular-subscript. In this way, we arrive at the spectral densities ((3.13) and (3.15)) of the low-frequency regime. The relation F0=2G0L0 between the spectral functions (3.14) is obtained by integrating the third identity in (A.6). We note the expansions


where Image . In the opposite limit,

A very different asymptotics applies for z<n, familiar from electromagnetic synchrotron radiation theory [11],




This is again the leading order in a 1/n and 1−z/n expansion. (A.13) and (A.14) follow from (A.12) and the identities

K1/3′=−K2/3−(1/(3z))K1/3, K2/3′=−K1/3−(2/(3z))K2/3 ,

K1/3=−2K2/3′−K5/3, K5/3=K1/3+(4/(3z))K2/3 .
To derive the asymptotics in ((A.5) and (A.15)), we use the identity Jn/z=(1/(2n))(Jn+1+Jn−1) and write, within the accuracy of ((A.2) and (A.12)), Jn+k(z)≈Jn(z(1−k/n)), so that ((A.5) and (A.15)) follow by expanding to second order in k/n. One can check by differentiation that these formulas are valid up to terms of O((z/n−1)2).

Hence, if xn/nless-than-or-equals, slant1, cf. (3.1), we approximate the powers ((3.5) and (3.6)) by


with ξ as in (A.9). In (A.17), the lower plus-sign again refers to the perpendicular-subscript, and we thus recover the spectral densities ((3.16) and (3.18)) of the high-frequency regime. The relation F=2GL between the spectral functions (3.17) follows by integrating the third identity in (A.16). These functions admit the expansions


and rapidly decay for ξ→∞,

(F,G,L)(ξ)not, vert, similar(6πξ)−1/2e−ξ .

The representation of the spectral densities by Airy functions, cf. (3.21), demonstrates analyticity at the break frequency ωb. To derive this, we write ξ=(2/3)z3/2, so that the Airy integral and its (anti-)derivatives read [70 and 71],



all entire functions, and Ai″(z)=z Ai(z). The integration boundaries may be replaced by Image , and we note Image . Accordingly, the spectral functions ((3.14) and (3.17)) relate to the Airy integral as


where z=(3ξ/2)2/3. (By the way, Ai′(−z) is the derivative of Ai(z) taken at −z.) The expansions ((A.10) and (A.19)) follow from those of the (anti-)derivatives of Ai(±z), listed in [71]. By substituting (A.22) into the transversal densities ((3.13) and (3.16)), we easily see that they are the analytic ω-continuation of each other, cf. ((3.20) and (3.21)). Similarly, the longitudinal densities ((3.15) and (3.18)) define the same analytic function, determining their high/low-frequency counterpart by analytic continuation.

We turn to the frequency integration of the spectral densities ((3.13) and (3.15)). The power radiated as curvature radiation in the low-frequency regime, cf. ((4.4) and (4.7)), and the corresponding number counts can be assembled from the integrals

The spectral functions are defined in (3.14), with ξ(ω) and ωb in (3.11). After rescaling with ωb, we introduce a new integration variable, ω4x2=(1−ω2)3, so that

up to terms of O(γ−2). The parameter κ is defined in (3.11); we need the κ→∞ asymptotics of these integrals, cf. the discussion following (4.2). To this end, we expand in (A.24),

ω(x)=1minus-or-plus sign(1/2)x2/3+(5/24)x4/3minus-or-plus sign(1/16)x2+cdots, three dots, centered ,

ωβω′=minus-or-plus sign(1/3)x−1/3(1minus-or-plus sign(3β+5)x2/3/6+(β+1)(β+3)x4/3/8minus-or-plus signcdots, three dots, centered) .
Only the upper signs are relevant at this point, the lower ones will be used in (A.29). The asymptotic expansion of integrals (A.24) is thus effected by (A.25) combined with the Bessel integrals [69],

in analytic continuation. Hence,


3F0(1)/4=G0(1)=3L0(1)/2=−4/9 .
L0(−1/3) is calculated from (A.26) by means of an var epsilon-regularizer, μ=−1/3+var epsilon. The use of expansion (A.25) over the whole integration range, the interchange of summation and integration, and the analytic continuation in μ can be justified as follows. We split the integration range in (A.24) into two intervals, [0,δ] and [δ,∞], where the cut δ lies within the convergence radius of (A.25). In the lower range, the integrals (A.24) can be reduced, by means of (A.25), to integrals of type Image , which admit antiderivatives in terms of Lommel functions and thus straightforward asymptotic expansions for κ→∞, cf. [69]. The integration over [δ,∞] is settled by replacing in (A.24) the spectral functions (A.22) by their asymptotic expansions for large argument [71], instead of expanding ω(x). The resulting Fourier integrals admit a standard asymptotic expansion [72] which entirely hinges on the lower integration boundary δ (where (A.25) still applies), since at the upper end ω(x→∞)not, vert, similarx−1/2. Collecting terms, we find the expansion indicated in ((A.24), (A.25) and (A.26)).

The expansion of the radiant powers in the high-frequency regime, cf. (4.13), is settled analogously. The integration of the transversal and longitudinal spectral densities ((3.16) and (3.18)) requires integrals of type

with the spectral functions (3.17). We change the integration variable, ω4x2=(ω2−1)3, so that

and substitute expansion (A.25) with the lower signs. The use of this expansion beyond its convergence radius is justified by Watson's Lemma [72], because of the exponential decay of the integrands, cf. (A.20); we have ω(x→∞)not, vert, similarx in this case. A partial integration with regard to F and L gives

to be compared with (A.27),

L(−1/3)=L0(−1/3), (F,G,L)(1/3)=−(F,G,L)0(1/3) ,

L(−1/3) is obtained by var epsilon-expansion of the first Γ-function, otherwise there is no analytic continuation necessary in (A.30). The asymptotic expansion of integrals (A.28) is thus obtained by substituting (A.25) (with lower signs) into (A.29), followed by an interchange of summation and integration, and a subsequent application of (A.30). The curvature radiation, that is, the radiated powers and the number counts discussed in Section 4, can be assembled from integrals ((A.23) and (A.28)).

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