**Physics Letters A**

Volume 366, Issues 4-5, 2 July 2007, Pages 289-297

### Abstract

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

**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

### Article Outline

### 1. Introduction

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

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

We consider electrons in a constant magnetic field **B**=(0,0,*B*),
*B*>0.
The equations of motion read *d*(*γ**m***v**)/*d**t*=*e***v**×**B**.
The helical orbits can be parametrized as

*α*is the pitch angle between

**B**and

**v**, 0

*α*

*π*. The speed

*υ*along the orbit is constant. A pitch angle of

*π*/2 gives circular orbits in the (

*x*,

*y*)-plane, and

*α*=0 or

*π*means straight uniform motion. We put =

*c*=1, and use the Heaviside–Lorentz system. The electric charge

*e*is defined negative for electrons,

*γ*is the Lorentz factor, and

*m*the electron mass. We also admit sign(

*e*)=0 in (1), that is, planar transversal oscillations, realizable by undulators in storage rings [3], [4], [5], [6], [7] and [8].

### 2. Superluminal radiation densities

#### 2.1. High-frequency regime

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

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

*α*

_{q}=

*q*

^{2}/(4

*π*

*c*)≈1.0×10

^{−13}is the tachyonic fine structure constant. The superluminal radiation field, a Proca field with negative mass square, is coupled by minimal substitution to the subluminal current, and

*q*is the tachyonic charge of the source particle. The tachyon mass is

*m*

_{t}≈2.15 keV/

*c*

^{2}. These estimates are obtained from hydrogenic Lamb shifts [9]. The tachyonic frequencies and wave numbers in (2) read

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

^{2}(

*e*)=1, since in this way the contributions of the two linear transversal polarizations (defined by

*ε*_{,}) to the radiated power can easily be distinguished; the term stemming from the

*ε*_{}-polarized component of the radiation field has the sign

^{2}(

*e*)-factor attached. Moreover, by putting sign

^{2}(

*e*)=0 in (4), we find the transversal power radiated by oscillating charges in undulator fields. The electromagnetic counterpart of (2), (3) and (4) is recovered in the limit of vanishing tachyon mass [10].

The integrals
in (2) give the
power transversally or longitudinally radiated in the harmonic *ω*_{n}.
We are interested in the large-*n* asymptotics,

*n*

^{2}-expansion of the wave numbers reads

and the argument

*z*

_{n}of the Bessel functions in (4) is expanded as

In (2), we expand
the integrands in *θ*−*θ*_{c},
at the stationary phase, *d**z*_{n}(*θ*_{c})/*d**θ*=0,
which reads

We introduce a new integration variable

*ψ*in (2),

where

*γ*=(1−

*υ*

^{2})

^{−1/2}1, and approximate

In the vicinity of

*θ*

_{c}, the -factors in (4) can be replaced by

and the Bessel functions in (12) by their Nicholson approximation. The latter applies for large positive

*n*and

*z*, so that

*z*/

*n*≈1 and

*z*<

*n*,

In the preceding notation, this means

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

Here, we defined

with

*ε*in (10). The transversal power radiated in

*ε*_{}-polarization is given by the sign

^{2}(

*e*) term, and the

*ε*_{}-polarized transversal radiation by the -integration in (15). The integrals in (15) and (16) can be simplified by the identities [10] and [11]

where the spectral functions on the right-hand side read

The easiest way to derive these identities is to write *K*_{1/3}
and *K*_{2/3}
in terms of Ai(*x*)
and its derivative, and then to use the standard integral
representation for Ai(*x*),
cf. Section 2.2.

The total power transversally and longitudinally radiated is
obtained by summing over the individual modes, .
We pass to continuous frequencies by identifying *ω*=*ω*_{n}≈*ω*_{B}*n*/sin^{2}*α*,
so that the continuous spectral densities read ,
with *n*=(*ω*/*ω*_{B})sin^{2}*α*.
We find *ξ* in (17) as

*ξ*(

*ω*) as in Ref. [1], rescaled by1/sin

*α*.

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

where

*ξ*and

*ω*

_{b}are defined in (20). The lower plus-sign in (22) refers to the -polarization.

*p*

^{T}(

*ω*) in (21) stands for the total transversal radiation; we have sign

^{2}(

*e*)=1. The longitudinal density is assembled with (16) and (18),

The spectral functions *F*_{∞}(*ξ*),
*G*_{∞}(*ξ*),
and *L*_{∞}(*ξ*)
are defined in (19). The pitch
angle enters in *p*^{T,L}(*ω*)
via the rescaled gyrofrequency, *ω*_{B}→*ω*_{B}sin*α*,
cf. (20).

#### 2.2. Analytic continuation into the low-frequency band

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

*z*=(3

*ξ*/2)

^{2/3}. We define the shortcut

so that

*ξ*=(2/3)|

*η*|

^{3/2}, cf. (20), and find

*ω*

*ω*

_{b}, and they also constitute the analytic continuation into the lower frequency band

*ω*

*ω*

_{b}, where

*η*(

*ω*,

*α*) is negative. The lower plus-sign in (26) refers to the -polarization. The pitch angle enters only via

*η*(

*ω*,

*α*), cf. (25).

### 3. Pitch-angle averaging

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

*λ*>−1. This density is normalized to one over the interval [0,

*π*/2]. The averaging is done by replacing in (26) and (27) with

and Ai

^{′}(

*η*)/

*η*in (26) is replaced by

Here,

*η*

_{0}(

*ω*):=

*η*(

*ω*,

*α*=

*π*/2), cf. (25), so that

*η*=

*η*

_{0}/sin

^{2/3}

*α*. The averages (29) and (30) remain unaltered, if we replace the integration over by . This also holds for the normalization of density (28), . More generally, we may average over the full pitch-angle range [0,

*π*] with density

where

*λ*

_{1,2}>−1 and 0

*A*1.

*d*

*ρ*

_{λ}is defined in (28). The distribution (31) is normalized to one, . Thus the averaging of the spectral densities (26) and (27) is effected by the substitutions

We denote the averaged densities by *p*^{T,L}(*ω*)_{σ}.

The averages ∫Ai_{λ}
and Ai^{′}_{λ}
in (29) and (30) depend on *ω*
only via *η*_{0}(*ω*),
cf. after (30). We briefly
discuss their |*η*_{0}|1 asymptotics, starting with the
low-frequency band, *ω**ω*_{b},
cf. after (27), where *η*_{0}
is negative. In leading order,

where

*β*0. 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*/|

*η*

_{0}|

^{3/2}1 for the O-term to be small. In (29), we also need

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

Thus the averaged spectral densities (as defined by (26) and (27) with substitutions (32) and (33)) converge for large |

*η*

_{0}| to the densities generated by a charge in uniform motion, cf. Section 4, irrespectively of the choice of parameters in the pitch-angle distribution (31).

We turn to the upper frequency range, *ω**ω*_{b},
where *η*_{0}(*ω*)
is positive. The asymptotics of ∫Ai_{λ}
and Ai^{′}_{λ}
is obtained from

The averaged spectral densities vanish for large (and even moderate) positive

*η*

_{0}(

*ω*), reflecting the fact that the spectral densities of a charge in uniform motion vanish identically for

*ω*

*ω*

_{b}, cf. Section 4.

The pitch-angle distribution *d**σ*_{A,λ1,λ2}(*α*)
is peaked in the interval [0,*π*/2],
at ,
provided that *λ*_{1}>0.
If −1<*λ*_{1}0,
it decreases monotonically to zero. The peak in the interval [*π*/2,*π*]
occurs at .
If −1<*λ*_{2}<0,
the density increases in this interval from zero to infinity. For
instance, *λ*_{1,2}=1
gives *α*_{p1}=*π*/4;
*λ*_{1,2}=1/3
gives *α*_{p1}=*π*/6,
and *λ*_{1,2}=3
amounts to a peak at *α*_{p1}=*π*/3.
The respective second peak in [*π*/2,*π*]
is obtained as *α*_{p2}=*π*−*α*_{p1}.
If *A*=1/2
and *λ*_{1}=*λ*_{2},
the density is symmetric around *α*=*π*/2.
If *λ*_{1,2}>0,
it vanishes at the interval boundaries and *π*/2.
These three pitch angles correspond to the limit cases of rectilinear
motion parallel and antiparallel to the magnetic field, and to circular
orbits orthogonal to **B**. If *λ*_{1,2}
is negative or close to zero, the average is dominated by pitch angles
close to zero and *π*, so that the radiation density
generated in uniform motion is prevalent. The latter is obtained by
performing the zero-pitch-angle limit sin*α*→0
in (26) and (27), cf. Ref. [12] and Fig. 1 and Fig. 2. If *λ*_{1,2}1, 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 sin*α*≈1
dominate the average.

Fig. 1. Transversal tachyonic spectral density *p*^{T}(*ω*),
cf. (26), in the
polar surface field, *B*≈3.1×10^{13} G,
of the *γ*-ray pulsar PSR B1509–58, cf. Refs. [13], [14] and [15]. 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 *α*_{q},
cf. after (2). The
gyrofrequency is *ω*_{B}≈18 eV,
the helical curvature radius *R*≈1.1×10^{−6} cm.
The break frequency is *ω*_{b}≈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 *p*^{L}(*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, *ω**ω*_{b}.

The units =*c*=1
can easily be restored. Gyroradius (magnetic bending radius, helical
curvature radius) and gyrofrequency are connected by *R*=*c*/*ω*_{B}
in the ultra-relativistic regime. The length of one helical turn is 2*π**R*,
and *R*sin*α*
is the radius of the helical base circle. Electronic energy and Lorentz
factor are related as *γ*≈1957*E*
[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 *p*^{T,L}(*ω*)_{σ},
averaged with the symmetric pitch-angle distribution *d**σ*(*A*=1/2,*λ*_{1}=*λ*_{2}=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 *p*^{T}(*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 [2]. We
demonstrated that a pitch-angle average attenuates the modulations in
the spectral slope, cf. Fig. 3, so that
the synchrotron densities approach the spectral densities of a
uniformly moving charge,

*Δ*

^{T}=1 and

*Δ*

^{L}=0 for the transversal and longitudinal radiation component, respectively. The spectral cutoff occurs at , to be identified with the break frequency

*ω*

_{b}in the ultra-relativistic limit, cf. (20). Only frequencies in the range 0<

*ω*<

*ω*

_{max}(

*γ*) can be radiated by a uniformly moving charge [16].

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

*A*

_{α,β}is determined by , where

*n*

_{1}is the electron count and

*γ*

_{1}the smallest Lorentz factor of the source population.

*β*=

*m*

*c*

^{2}/(

*k*

*T*), and

*α*is the electronic power-law index (not to be confused with the pitch angle). Here, we consider thermal distributions, defined by

*α*=−2 and

*γ*

_{1}=1. The average is carried out as [2]

The spectral fit is based on the

*E*

^{2}-rescaled flux density

*d*

*N*

^{T,L}/

*d*

*E*, which is related to the averaged energy density by

where

*d*is the distance to the pulsar. In Fig. 4, Fig. 5 and Fig. 6, we plot the flux density of two thermal electron populations

*ρ*

_{1,2}specified in Table 1. Each electron density generates a cascade

*ρ*

_{i}, the wideband spectral map is obtained by adding the cascade spectra.

Fig. 4. *γ*-ray wideband of the
binary pulsar PSR B1259–63. RXTE and ISGRI data points from [18], OSSE
points from [19], COMPTEL
and EGRET upper limits from [20], HESS
points from [21]. The solid
line T+L is the unpolarized differential tachyon flux *d**N*^{T+L}/*d**E*,
obtained by adding the flux densities *ρ*_{1,2}
of two thermal electron populations. The *χ*^{2}-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 [18], 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 *d**N*^{T+L}/*d**E*
rescaled with *E*^{2}
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 *ρ*_{1,2}
sets in at about *E*_{cut}≈(*m*_{t}/*m*)*k**T*,
implying cutoffs at 0.8 TeV for the *ρ*_{1}-cascade
and 80 GeV for *ρ*_{2}.
The unpolarized flux T+L can be split into transversal (dot-dashed) and
longitudinal (dot-dot-dashed) components *d**N*^{T,L}/*d**E*,
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 *ρ*_{1,2}.
The TeV spectral map coincides with the *ρ*_{1}-cascade,
since the *ρ*_{2}-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 *ρ*_{1,2}
generating the *γ*-ray broadband of the binary pulsar
PSR B1259–63

The cascades *ρ*_{1,2}
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 *n*^{e},
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 *χ*^{2}-fit.

As for the electron count *n*_{1},
it is convenient to use a rescaled parameter
for the fit,

*m*

_{t}≈2.15 in the spectral density (40). At

*γ*-ray energies, only a tiny

*α*

_{q}/

*α*

_{e}-fraction (the ratio of tachyonic and electric fine structure constants) of the tachyon flux is actually absorbed by the detector [17], which requires a rescaling of the electron count

*n*

_{1}, so that the actual number of radiating electrons is . We thus find the electron count as [kpc], where is determined by the tachyonic flux amplitude obtained from the spectral fit. Finally, electron temperature and cutoff parameter in the electronic Boltzmann factor are related as .

Fig. 4, Fig. 5 and Fig. 6 show the
least-squares fit to the *γ*-ray wideband of the
binary system PSR B1259–63/SS 2883, cf. Refs. [18], [19], [20], [21], [22], [23] and [24]. The
extended spectral plateau as well as the spectral curvature over the
entire frequency range can readily be reproduced by a tachyonic cascade
spectrum. (As for the high-magnetic-field pulsar PSR B1509–58, cf. Fig. 1, Fig. 2 and Fig. 3, there
are currently not enough data points available for a spectral fit [14]. A
tachyonic spectral fit to the pulsed Crab flux is given in Ref. [25], where the
spectral asymptotics of the averaged densities and the technical
details of the nonlinear *χ*^{2}-fit
are discussed as well.) We assume *d*≈1.5 kpc
as distance to the pulsar, based on optical and scintillation
observations [22], although
it could be as high as 4.5 kpc if inferred from the dispersion measure [23]. In the
latter case, the electronic source count has to be rescaled
accordingly, ;
the spectral maps are not affected by the distance estimate. The total
flux above 380 GeV is 5% of the unpulsed Crab flux [21]. The
electronic source count for the Crab pulsar at 2 kpc is 2.6×10^{49},
cf. Ref. [25], as
compared to 3.2×10^{47}
for PSR B1259–63, cf. Table 1.

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

### 5. Conclusion

We have studied superluminal radiation from ultrarelativistic electrons helically orbiting in strong magnetic fields and derived the pitch-angle scaling of the radiation densities. In the zero-magnetic-field limit, the tachyonic synchrotron densities converge to the spectral densities (40) for uniform motion. Orbital curvature induces modulations along the spectral slopes. If integrated over the pitch angles, these oscillations are averaged out, cf. Fig. 1, Fig. 2 and Fig. 3.

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

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

### Acknowledgements

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.

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