Mechanism of generation of the emission bands in the dynamic spectrum of the Crab pulsar
aa r X i v : . [ a s t r o - ph ] A p r Mon. Not. R. Astron. Soc. , 1– ?? (2008) Printed 24 October 2018 (MN L A TEX style file v2.2)
Mechanism of generation of the emission bandsin the dynamic spectrum of the Crab pulsar
Houshang Ardavan, Arzhang Ardavan, John Singleton, Mario Perez Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK National High Magnetic Field Laboratory, MS-E536, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Space Sciences and Applications, ISR-1, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
18 April 2008
ABSTRACT
We show that the proportionately spaced emission bands in the dynamic spectrum ofthe Crab pulsar (Hankins T. H. & Eilek J. A., 2007, ApJ, 670, 693) fit the oscillationsof the square of a Bessel function whose argument exceeds its order. This function hasalready been encountered in the analysis of the emission from a polarization currentwith a superluminal distribution pattern: a current whose distribution pattern rotates(with an angular frequency ω ) and oscillates (with a frequency Ω > ω differing froman integral multiple of ω ) at the same time (Ardavan H., Ardavan A. & Singleton J.,2003, J Opt Soc Am A, 20, 2137). Using the results of our earlier analysis, we findthat the dependence on frequency of the spacing and width of the observed emissionbands can be quantitatively accounted for by an appropriate choice of the value ofthe single free parameter Ω /ω . In addition, the value of this parameter, thus impliedby Hankins & Eilek’s data, places the last peak in the amplitude of the oscillatingBessel function in question at a frequency ( ∼ Ω /ω ) that agrees with the positionof the observed ultraviolet peak in the spectrum of the Crab pulsar. We also showhow the suppression of the emission bands by the interference of the contributionsfrom differring polarizations can account for the differences in the time and frequencysignatures of the interpulse and the main pulse in the Crab pulsar. Finally, we putthe emission bands in the context of the observed continuum spectrum of the Crabpulsar by fitting this broadband spectrum (over 16 orders of magnitude of frequency)with that generated by an electric current with a superluminally rotating distributionpattern. Key words: pulsars: individual (Crab Nebula pulsar)—radiation mechanisms: non-thermal.
Very soon after the discovery of pulsars, it was realized thatthe very stable periodicity of the mean profiles of their pulsescould only result from a source that rotates , and which there-fore posesses a rigidly rotating radiation distribution (Gold1968). In this paper, we show that this source rotation isnot only responsible for the periodicity of the pulses, butalso determines the detailed frequency dependence of theemitted radiation. By inferring the values of two adjustableparameters from observational data (values that are consis-tent with those of plasma frequency and electron cyclotronfrequency in a conventional pulsar magnetosphere), and bymildly restricting certain local properties of the source, weare able to account quantitatively for the emission spectrumof the Crab pulsar over 16 orders of magnitude of frequency.The rigid rotation of the overall distribution pattern of the radiation from a pulsar is described by an electromag-netic field whose distribution depends on the azimuthal an-gle ϕ only in the combination ( ϕ − ωt ), where ω is the angularfrequency of rotation of the pulsar and t is the time. As weshow in Appendix A, Maxwell’s equations demand that thecharge and current densities that give rise to this radiationfield should have the same time dependence. Therefore, theobserved motion of the radiation pattern of pulsars can onlyarise from a source whose distribution pattern rotates rigidly,i.e. a source whose average density depends on ϕ only in thecombination ( ϕ − ωt ). Furthermore, if a plasma distributionhas a rigidly rotating pattern in the emission region, thenit must have a rigidly rotating pattern everywhere; we showin Appendix B that a solution of Maxwell’s equations withthe time dependence ∂/∂t = − ω∂/∂ϕ applies either to theentire volume of the magnetospheric plasma distribution or c (cid:13) H. Ardavan et al. to a region whose boundary is an expanding wave front thatwill eventually encompass the entire magnetosphere of thepulsar.Unless the plasma atmosphere surrounding the pulsaris restricted to an unrealistically small volume, it is there-fore an inevitable consequence of the observational data thatthe macroscopic distribution of electric current in the mag-netosphere has a rigidly rotating pattern whose linear speedexceeds the speed of light in vacuo c for r > c/ω , where r is the radial distance from the axis of rotation. Althoughspecial relativity does not allow a charged particle with anon-zero inertial mass to move faster than c , there is nosuch restriction on the speed of propagation of the varia-tions in a macroscopic charge or current distribution. Forinstance, the distribution pattern of a polarization currentwith a propagation speed that exceeds c can be created bythe coordinated motion of aggregates of particles that moveslower than c (Bolotovskii & Ginzburg 1972; Ginzburg 1972;Bolotovskii & Bykov 1990). Such a polarization-currentdensity is on the same footing as the current density of freecharges in the Amp`ere-Maxwell equation, so that its propa-gating distribution pattern radiates as would any other mov-ing source of electromagnetic fields (Bolotovskii & Bykov1990); indeed, such superluminal polarization currents havebeen demonstrated to be efficient emitters of radiation inthe laboratory (Bessarab et al. 2004; Ardavan et al. 2004b;Singleton et al. 2004; Bolotovskii & Serov 2005).Once it is acknowledged that the electric cur-rent emitting the observed pulses from pulsars has asuperluminally-rotating distribution pattern, results fromthe published literature on the electrodynamics of super-luminal sources (Ardavan 1998; Ardavan et al. 2003, 2004a,2007; Schmidt et al. 2007; Ardavan et al. 2008a,b) can beapplied to pulsars. In this paper, we explain the recently-observed emission bands in the dynamic spectrum of theCrab pulsar (Hankins & Eilek 2007) using the calculationsof Ardavan et al. (2003). Both the oscillations of the inten-sity and the frequency dependence of the spacing and widthof these bands are described by a Bessel function with asingle parameter that is already constrained by other ob-servational data on the spectrum of the Crab pulsar. ThisBessel function is characteristic of the spectrum of the ra-diation by a superluminal polarization current whose distri-bution pattern rotates (with an angular frequency ω ) andoscillates (with a frequency Ω > ω differing from an integralmultiple of ω ) at the same time. It differs from the Besselfunction encountered in the analysis of synchrotron radia-tion only in the relative magnitudes of its argument andits order: while the Bessel function describing synchrotronradiation has an argument smaller than its order and sodecays exponentially with increasing frequency, the Besselfunction encountered in Ardavan et al. (2003), whose argu-ment exceeds its order, is an oscillatory function of frequencywith an amplitude that decays only algebraically. It is thisslower decay of the Bessel function in question that endowsthe emission from a superluminally-rotating source with a A corollary of this implication is that the magnetosphericplasma of a pulsar cannot be fully charge-separated, as is com-monly assumed in the works based on the Goldreich-Julian model(Goldreich & Julian 1969). broad spectrum. The physical mechanism underlying thebroadband nature of this emission is focusing in the timedomain (Ardavan et al. 2003, 2008a): contributions towardthe intensity of the radiation made over an extended periodof emission time are received during a significantly shorterperiod of observation time.This paper is organized as follows. Section 2 de-scribes how the most radiatively efficient parts of a pul-sar magnetosphere are thin filaments within the super-luminally rotating part of its current distribution pat-tern (Ardavan et al. 2007). A knowledge of the morphol-ogy of these filaments (Ardavan et al. 2007), and the abilityof a source that travels faster than its own waves to makemultiple contributions to the signal received by an observerat a given instant (Ardavan et al. 2004a), are necessary tounderstand many of the traits of pulsar observations de-scribed later in the paper. Section 3 summarizes our ear-lier analysis (Ardavan et al. 2003) on the frequency spec-trum of the radiation from a rotating superluminal source.Using these results, we derive various features of the emis-sion bands in Section 4 and compare them to the frequencybands seen in the interpulses of the Crab pulsar [Figs. 6–8, Hankins & Eilek (2007)]; using a single input parameter,related to the pulsar’s plasma density, we reproduce the ob-servational bands and predict the final ultra-violet emissionpeak of the Crab pulsar. Section 5 discusses the suppressionof the bands by interference; this is relevant to the much lessfrequency-banded microbursts and nanoshots of the Crab’smain pulses [Figs. 2 and 3, Hankins & Eilek (2007)]. Sec-tion 6 describes the continuum spectrum of the Crab pul-sar; by introducing a further input parameter related to theplasma dynamics we are able to account quantitatively forthe whole emission spectrum over 16 orders of magnitude offrequency. Section 7 gives a short discussion and summary.The mathematical details of our arguments are presented inAppendices A, B and C.
At large distances from the source, the radiation field of asuperluminally rotating extended source at an observationpoint P is dominated by the emissions of its volume ele-ments that approach P along the radiation direction withthe speed of light and zero acceleration at the retardedtime (Ardavan et al. 2004a, 2007). These elements consti-tute a filamentary part of the source whose radial andazimuthal widths become narrower (as δr ∼ R P − and δϕ ∼ R P − , respectively), the larger the distance R P ofthe observer from the source, and whose length is of theorder of the length scale l z of the source parallel to theaxis of rotation (Ardavan et al. 2007). (Here, r , ϕ , and z are the cylindrical polar coordinates of the source points.)For an observation point P with spherical polar coordinates( R P , ϕ P , θ P ) that is located in the far zone, this contribut-ing part of the source lies at ˆ r = csc θ P , ϕ = ϕ P + 3 π/
2, andis essentially a straight line parallel to the rotation axis, asshown in Fig. 1 (Ardavan et al. 2007). (The dimensionlesscoordinate ˆ r stands for rω/c , where ω , the angular frequency c (cid:13) , 1– ?? ynamic spectrum of the Crab pulsar Figure 1.
Schematic illustration of the light cylinder r = c/ω ,the filamentary part of the distribution pattern of the source thatapproaches the observation point with the speed of light and zeroacceleration at the retarded time, the orbit of this filamentarysource, and the subbeam formed by the bundle of cusps thatemanate from the constituent volume elements of this filament[after Ardavan et al. (2007)]. The subbeam is diffractionless in thepolar direction. The figure represents a snapshot correspondingto a fixed value of the observation time t P . of rotation of the distribution pattern of the source, is thesame frequency as that with which the pulsar rotates.)Once a source travels faster than its emitted waves,it can make more than one retarded contribution to thefield observed at any given instant (Ardavan et al. 2003,2004a, 2008a). This multivaluedness of the retarded timemeans that the wave fronts emitted by each of the contribut-ing elements of the source possess an envelope, which inthis case consists of a two-sheeted, tube-like surface whosesheets meet tangentially along a spiraling cusp curve; seeFig. 2 (Ardavan et al. 2004a). For moderate superluminalspeeds, the field inside the envelope receives contributionsfrom three distinct values of the retarded time, while thefield outside the envelope is influenced only by a single in-stant of emission time (Schmidt et al. 2007). Coherent su-perposition of the emitted waves on the envelope (where twoof the contributing retarded times coalesce) and on its cusp(where all three of the contributing retarded times coalesce)results in not only a spatial but also a temporal focusing ofthe waves: the contributions from emission over an extendedperiod of retarded time reach an observer who is located onthe cusp during a significantly shorter period of observationtime (Ardavan et al. 2008a).The field of each contributing volume element of thesource is strongest, therefore, on the cusp of the envelopeof wave fronts that it emits [see Ardavan et al. (2007) andreferences therein]. The bundle of cusps generated by thecollection of the contributing source elements (i.e. by thefilamentary part of the source that approaches the observerwith the speed of light and zero acceleration) constitutesa radiation subbeam whose widths in the polar and az- imuthal directions are of the order of δθ P ∼ R P − and δϕ P ∼ R P − , respectively (Ardavan et al. 2007). The over-all radiation beam generated by the source consists of a(necessarily incoherent ) superposition of such subbeams.This beam’s azimuthal width is the same as the azimuthalextent of the source, and its polar width, arccos(1 / ˆ r < ) | θ P − π/ | arccos(1 / ˆ r > ), is determined by the radialextent 1 < ˆ r < ˆ r ˆ r > of the superluminal part ofthe source (Ardavan et al. 2007). This will be important inSecs. 4 and 5; the ϕ P dependence of the radiation intensitywithin the overall beam (i.e. what is observed as the mainpulse, interpulse, and other components of the mean pulse)thus reflects the distribution of the source density aroundthe cylindrical surface ˆ r = csc θ P from which the main con-tributions to the field arise.Since the cusps represent the loci of points at which theemitted spherical waves interfere constructively [i.e. repre-sent wave packets that are constantly dispersed and recon-structed out of other waves (Ardavan 1998)], the subbeamsgenerated by a superluminal source need not be subject todiffraction as are conventional radiation beams. Neverthe-less, they have a decreasing angular width (i.e. are non-diffracting) only in the polar direction (Ardavan et al. 2007).Their azimuthal width δϕ P decreases as R P − with distancebecause they receive contributions from an azimuthal extent δϕ of the source that likewise shrinks as R P − . They wouldhave had a constant azimuthal width had the azimuthal ex-tent of the contributing part of the source been independentof R P . On the other hand, the solid angle occupied by thecusps has a thickness δz P in the direction parallel to therotation axis that remains of the order of the height l z ofthe source distribution at all distances (see Fig. 1). Con-sequently, the polar width δθ P of the particular subbeamthat goes through the observation point decreases as R P − ,instead of being independent of R P (Ardavan et al. 2007).Because it has a constant linear width parallel to therotation axis, an individual subbeam subtends an area of theorder of R P , rather than R P . In order that the energy fluxremain the same across all cross sections of the subbeam,therefore, it is essential that the Poynting vector associ-ated with this radiation correspondingly decay more slowlythan that of a conventional, spherically decaying beam: as R P − , rather than R P − , within the bundle of cusps thatemanate from the constituent volume elements of the sourceand extend into the far zone (Ardavan et al. 2004a). Thisresult, which also follows from the superposition of theLi´enard-Wiechert fields of the constituent volume elementsof a rotating superluminal source (Ardavan et al. 2007), hasbeen demonstrated experimentally (Ardavan et al. 2004b;Singleton et al. 2004).The fact that the observationally-inferred dimensionsof the plasma structures responsible for the emission frompulsars are less than 1 metre in size (Hankins et al. 2003) The superposition of the subbeams is necessarily incoherent be-cause the subbeams that are detected at two neighbouring pointswithin the overall beam arise from two distinct filamentary partsof the source with essentially no common elements. The inco-herence of this superposition would ensure that, though the fieldamplitude within a subbeam, which narrows with distance, decaysnonspherically, the field amplitude associated with the overall ra-diation beam, which occupies a constant solid angle, does not.c (cid:13) , 1– ?? H. Ardavan et al.
Figure 2. (a) Cross section of the ˇCerenkov-like envelope (boldcurves) of the spherical Huygens wave fronts (fine circles) emit-ted by a volume element S within the distribution pattern of anextended, rotating superluminal source of angular frequency ω [after Ardavan et al. (2004a)]. The point S is on a circle of ra-dius r = 2 . c/ω , or ˆ r ≡ rω/c = 2 .
5, i.e. has an instantaneouslinear velocity rω = 2 . c . The cross section is with the plane of S ’s orbit; dashed circles designate the light cylinder r P = c/ω (ˆ r P = 1) and the orbit of S . (b) Three-dimensional view of thelight cylinder, the envelope of wave fronts emanating from S , andthe cusp along which the two sheets of this envelope meet tan-gentially. The cusp is tangent to the light cylinder in the plane ofthe orbit and spirals outward into the far zone. reflects, in the present context, the narrowing (as R P − and R P − , respectively) of the radial and azimuthal dimensionsof the filamentary part of the source that approaches theobserver with the speed of light and zero acceleration at theretarded time. Not only do the nondiffracting subbeams thatemanate from such filaments account for the nanostructure,and so the brightness temperature, of the giant pulses, butthe nonspherical decay of the intensity of such subbeams (as R P − instead of R P − ) explains why their energy densitiesat their source appear to exceed the energy densities of boththe plasma and the magnetic field at the suface of a neutronstar when estimated on the basis of the inverse square law(Soglanov et al. 2004). In this section we summarize the results of Ardavan et al.(2003) relevant to pulsars; the original intent of that paperwas to calculate the spectrum of the radiation from a genericsuperluminal source that has been implemented in the lab-oratory (Ardavan et al. 2004b; Singleton et al. 2004). Thissource comprises a polarization-current density j = ∂ P /∂t for which P r,ϕ,z ( r, ϕ, z, t ) = s r,ϕ,z ( r, z ) cos( m ˆ ϕ ) cos(Ω t ) , − π < ˆ ϕ π, (1)withˆ ϕ ≡ ϕ − ωt, (2)where P r,ϕ,z are the components of the polarization P ina cylindrical coordinate system ( r, ϕ, z ) based on the axisof rotation, s ( r, z ) is an arbitrary vector that vanishes out-side a finite region of the ( r, z ) space, and m is a positiveinteger. For a fixed value of t , the azimuthal dependence of the polarization (1) along each circle of radius r withinthe source is the same as that of a sinusoidal wave trainwith the wavelength 2 πr/m whose m cycles fit around thecircumference of the circle smoothly. As time elapses, thiswave train both propagates around each circle with the ve-locity rω and oscillates in its amplitude with the frequencyΩ. This is a generic source: one can construct any distri-bution with a uniformly rotating pattern, P r,ϕ,z ( r, ˆ ϕ, z ) or j r,ϕ,z ( r, ˆ ϕ, z ), by the superposition over m of terms of theform s r,ϕ,z ( r, z, m ) cos( m ˆ ϕ ). In the following discussion, weassume that the modulation frequency Ω is positive and dif-ferent from an exact integral multiple of the rotation fre-quency ω [for the significance of this incommensurablity re-quirement, see Ardavan et al. (2003)].Although formulated in terms of a polarization current(for which there is manifestly no restriction on the propa-gation speed of the variations in the current density), theresults that follow hold true for any current distributionwhose density depends on ϕ as ϕ − ωt in r > c/ω . Theelectromagnetic fields E = −∇ P A − c − ∂ A /∂t P , B = ∇ P × A , (3)that arise from such a source are given, in the absence ofboundaries, by the following classical expression for the re-tarded four-potential: A µ ( x P , t P ) = c − Z d x d t j µ ( x , t ) δ ( t P − t − R/c ) /Rµ = 0 , · · · , . (4)Here, ( x P , t P ) = ( r P , ϕ P , z P , t P ) and ( x , t ) = ( r, ϕ, z, t ) arethe space-time coordinates of the observation point and thesource points, respectively, R stands for the magnitude of R ≡ x P − x , and µ = 1 , , A and j , of A µ and j µ in a Cartesian coordinatesystem (Jackson 1999).In Ardavan et al. (2003), we first calculated theLi´enard-Wiechert field that arises from a circularly movingpoint source (representing a volume element of an extendedsource) with a superluminal speed rω > c , i.e. considered ageneralization of the synchrotron radiation to the superlu-minal regime. We then evaluated the integral representingthe retarded field of the extended source (1) by superpos-ing the fields generated by the constituent volume elementsof this source, i.e. by using the generalization of the syn-chrotron field as the Green’s function for the problem [seeequation (19) of Ardavan et al. (2003)]. For a source thattravels faster than c , this Green’s function has extended sin-gularities arising from the constructive interference of theemitted waves on the envelope of wave fronts and its cusp(see Fig. 2).Correspondingly, the fields E and B are given, in thefrequency domain, by radiation integrals with rapidly oscil-lating integrands whose phases are stationary on the lociof the coherently contributing source elements: those sourceelements that approach the observer, along the radiation di-rection, with the speed of light and zero acceleration at theretarded time. For a radiation angular frequency nω that ap-preciably exceeds the angular velocity ω , these integrals can When discussing pulsars in the following sections, we shall beconsidering radiation frequencies in the GHz-ultaviolet range,c (cid:13) , 1– ?? ynamic spectrum of the Crab pulsar be evaluated by the method of stationary phase to obtainthe following expression for the electric field of the emittedradiation outside the plane of the source’s orbit: E = ℜ ( ˜ E + 2 ∞ X n =1 ˜ E n exp( − i n ˆ ϕ P ) ) , (5)in which˜ E n ≃ ˆ r − P exp {− i[ n ( ˆ R P + π ) − (Ω /ω )( ϕ P + π )] }× Q ˆ ϕ ¯ Q r ¯ Q z + { m → − m, Ω → − Ω } , (6) Q ˆ ϕ = ( − n + m sin (cid:16) π Ω ω (cid:17) (cid:18) µ +2 n − µ + + µ − n − µ − (cid:19) , (7)¯ Q r ≃ ˆ r > − ˆ r < , n ≪ π ˆ R P / (ˆ r > − ˆ r < ) , (8)or¯ Q r ≃ (2 π ˆ R P /n ) / exp( − i π/ , n ≫ π ˆ R P / (ˆ r > − ˆ r < ) , (9)and¯ Q z = (cid:2) ¯ s r J n − Ω /ω ( n ) + i¯ s ϕ J ′ n − Ω /ω ( n ) (cid:3) ˆ e k + (cid:2) (¯ s ϕ cos θ P − ¯ s z sin θ P ) J n − Ω /ω ( n ) − i¯ s r cos θ P J ′ n − Ω /ω ( n ) (cid:3) ˆ e ⊥ , (10)with¯ s r,ϕ,z ≡ Z ∞−∞ dˆ z exp(i n ˆ z cos θ P ) s r,ϕ,z (cid:12)(cid:12) ˆ r =csc θ P (11)[see equations (15), (23), (46) and (66)–(68) ofArdavan et al. (2003)]. Here, (ˆ r, ˆ z ; ˆ r P , ˆ z P ) stand for( rω/c, zω/c ; r P ω/c, z P ω/c ), ( R P , θ P , ϕ P ) are the spher-ical polar coordintes of the observation point P , µ ± ≡ (Ω /ω ) ± m , ˆ r < < r > > s r,ϕ,z are non-zero, J and J ′ are the Anger functionand the derivative of the Anger function with respect to itsargument, respectively, and ˆ e k ≡ ˆ e z × ˆ n / | ˆ e z × ˆ n | (which isparallel to the plane of rotation) and ˆ e ⊥ ≡ ˆ n × ˆ e k comprise apair of unit vectors normal to the radiation direction ˆ n (ˆ e z is the base vector associated with the coordinate z ). Thesymbol { m → − m, Ω → − Ω } designates a term exactly likethe one preceding it but in which m and Ω are everywherereplaced by − m and − Ω, respectively.The radiated power per harmonic per unit solid angleis therefore given byd P n / dΩ P = (2 π ) − cR P | ˜ E n | ≃ (8 π ) − ( c /ω ) csc θ P | ¯ Q r | Q ˆ ϕ | ¯ Q z | , (12)where dΩ P denotes the element sin θ P d θ P d ϕ P of solid an-gle in the space of observation points. The contribution ofthe term { m → − m, Ω → − Ω } in equation (6) has beenignored here because the Anger functions for − Ω < J ν ( χ ), and a Bessel function of the first kind, J ν ( χ ), when ν is an integer. Even for a non-integral value of ν , the dif-ference between these two functions vanishes, as χ − , if whereas the rotation frequency of the Crab pulsar is ω/ π ≃
30 Hz. The condition nω ≫ ω will therefore always hold. χ ≫
1. In the regime n ≫
1, where the radiation frequency nω appreciably exceeds the rotation frequency, therefore,the Anger functions J n − Ω /ω ( n ) and J ′ n − Ω /ω ( n ) in Equa-tion (10) can be replaced by the Bessel functions J n − Ω /ω ( n )and J ′ n − Ω /ω ( n ), respectively. If the radiation frequency nω appreciably exceeds also the modulation frequency Ω, thenthese Bessel functions can in turn be approximated by thefollowing Airy functions: J n − Ω ω ( n ) ≃ (cid:16) n (cid:17) / Ai (cid:20) − (cid:16) n (cid:17) / Ω ω (cid:21) , (13) J ′ n − Ω ω ( n ) ≃ − (cid:16) n (cid:17) / Ai ′ (cid:20) − (cid:16) n (cid:17) / Ω ω (cid:21) , (14)where Ai ′ stands for the derivative of the Airy function withrespect to its argument (see Appendix C).Hence, the radiated power is given byd P n dΩ P ≃ c π ω csc θ P | ¯ Q r | Q ˆ ϕ × (cid:16) n (cid:17) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:2) ¯ s r ˆ e k + (¯ s ϕ cos θ P − ¯ s z sin θ P )ˆ e ⊥ (cid:3) × Ai (cid:20) − (cid:16) n (cid:17) / Ω ω (cid:21) − i(¯ s ϕ ˆ e k + ¯ s r cos θ P ˆ e ⊥ ) × (cid:16) n (cid:17) / Ai ′ (cid:20) − (cid:16) n (cid:17) / Ω ω (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (15)in the regime where the radiation frequency ( nω ) apprecialyexceeds both the rotation ( ω ) and modulation (Ω) frequen-cies of the source distribution (1).Asymptotic values of the amplitudes ofAi[ − (2 /n ) / (Ω /ω )] and Ai ′ [ − (2 /n ) / (Ω /ω )] are respec-tively given by (2 /n ) − / (Ω /ω ) − / and (2 /n ) / (Ω /ω ) / ,when n ≪ (Ω /ω ) , and by the constants Ai(0) and Ai ′ (0)when n ≫ (Ω /ω ) (Abramowitz & Stegun 1970). Moreover,the quantity Q ˆ ϕ , which appears in equation (15), is inde-pendent of n ( ∼ /ω ) if n ≪ µ + , decays as n − if n ≫ µ + ,and is of the order of n if n ≃ m ≫ Ω /ω [see equation(7)].Depending on the relative magnitudes of n , m and Ω /ω ,therefore, the amplitude of the radiated power d P n / dΩ P de-cays with the harmonic number n as n − σ | ¯ s r,ϕ,z ( n ) | , wherethe exponent σ can assume one of the values 1 /
2, 2 /
3, 5 / /
3, and ¯ s r,ϕ,z ( n ) denotes the frequency dependence of theFourier components of the source densities s r,ϕ,z . The wayin which ¯ s r,ϕ,z ( n ) decay with n , for large n , is determinedby the distribution of the source density s ( r, z ) with respectto z [see equation (11)], and so is model-dependent. How-ever, the spacings between the emission bands predicted byequation (15) turn out to be essentially independent of therate of decay of the amplitudes of these bands (see below).Having assembled the relevant equations, we now com-pare their predictions with established observational dataon the Crab pulsar. Note that the electric susceptibility of the magnetosphericplasma (contained in the factor s ) does not influence these re-sults, because it depends on the source frequencies µ ± ω , not onthe radiation frequency nω [see Ardavan et al. (2008b)]c (cid:13) , 1– ?? H. Ardavan et al. d P n / d Ω P nν (GHz) Figure 3.
The power d P n / dΩ P radiated per harmonic per unitsolid angle (in arbitrary units), into the signal linearly polarizedparallel to the plane of rotation, versus the harmonic number n (lower axis) for Ω /ω = 1 . × , a k = 1 . × − and b k =1 . × . The upper axis gives observation frequency nω/ π for the case of the Crab pulsar [ ω/ (2 π ) ≃
30 Hz]. The spacing,the width, and the number of oscillations in this figure are thesame as those of the bands seen in the interpulse data from theCrab pulsar over the frequency interval 8 < ν < . One of the most remarkable features of the observationaldata from the Crab pulsar is the presence of well-defined fre-quency bands within the midpulses (Hankins & Eilek 2007).We now show that these are a natural consequence of a ro-tating superluminal source and use their spacing to predictthe high-frequency emission of the Crab pulsar.When the absolute value of their argument appreciablyexceeds unity, the Airy functions in equation (15) are rapidlyoscillating functions of the harmonic number n with peaksthat spread further apart with increasing n (Fig. 3). The rateof increase of the spacing between the peaks in question isdetermined solely by the value of Ω /ω . We now derive anexplicit expression for the spacing ∆ n of the maxima of theradiated power d P n / dΩ P as a function of n and comparethe result with the proportionately spaced emission bandsobserved by Hankins & Eilek (2007).Within the range Ω /ω ≪ n ≪ (Ω /ω ) of harmonicnumbers, where they are oscillatory, the Airy functions inequation (15) can be further approximated byAi[ − (2 /n ) / (Ω /ω )] ≃ π − / ( n/ / (Ω /ω ) − / cos ψ (16)andAi ′ [ − (2 /n ) / (Ω /ω )] ≃ π − / ( n/ − / (Ω /ω ) / sin ψ (17)where ψ ≡ (2 /n ) / (Ω /ω ) / − π/ P n dΩ P ≃ c csc θ P π ω | ¯ Q r | Q ˆ ϕ (cid:16) ωn Ω (cid:17) / (cid:12)(cid:12) P k ˆ e k + P ⊥ ˆ e ⊥ (cid:12)(cid:12) , (19) in which P k ≡ ¯ s r cos ψ − i (cid:16) nω (cid:17) / ¯ s ϕ sin ψ (20)and P ⊥ ≡ (¯ s ϕ cos θ P − ¯ s z sin θ P ) cos ψ − i 2Ω nω ¯ s r cos θ P sin ψ (21)represent the contributions towards the radiated field ˜ E n from differing polarizations.Let us first consider the contribution towards the radi-ated power from the component of the field parallel to theplane of rotation. It follows from equation (20) that |P k | = sec(2 γ k ) (cid:8) − | ¯ s r | sin γ k + 2Ω nω | ¯ s ϕ | cos γ k +[ | ¯ s r | − nω | ¯ s ϕ | ] cos ( ψ − γ k ) (cid:9) , (22)where γ k = arctan [2Ω / ( nω )] / ℑ{ ¯ s r ¯ s ∗ ϕ }| ¯ s r | − [2Ω / ( nω )] | ¯ s ϕ | , (23)with ℑ and ∗ denoting the imaginary part and a complexconjugate, respectively. Two successive oscillations of |P k | in the neighbourhood of any given n arise, according to equa-tion (22), from a change in the value of the argument ofcos ( ψ − γ k ) by π . Therefore, the spacing between two suc-cessive peaks of this function is locally given by∆ n = π (cid:12)(cid:12) ∂∂n ( ψ − γ k ) (cid:12)(cid:12) − , (24)an equation that yields ∆ n versus n for all n . If the source densities ¯ s r,ϕ,z have a weak dependence on n , then the high-frequency ( n ≫
1) limits of equations (22)and (23) assume the simple forms |P k | ≃ | ¯ s r | cos ( ψ − γ k ) , (25)and γ k ≃ (cid:16) nω (cid:17) / ℑ{ ¯ s r ¯ s ∗ ϕ }| ¯ s r | ≃ [2Ω / ( nω )] / [ a k + b k ( n − n )] , (26)where a k ≡ ℑ{ ¯ s r ¯ s ∗ ϕ }| ¯ s r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n = n , (27)and b k ≡ ∂∂n (cid:20) ℑ{ ¯ s r ¯ s ∗ ϕ }| ¯ s r | (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n = n . (28)denote the first two coefficients in the Taylor expansion of ℑ{ ¯ s r ¯ s ∗ ϕ } / | ¯ s r | about a local value n of n .In this case, equation (24), together with equations (18)and (25)–(28), yields explicit expressions both for the radi-ated power |P k | and for the spacing ∆ n between its suc-cessive peaks. These expressions are plotted in Figs. 3 and 4 At any point such equations may be readily translated intothe frequency units ∆ ν versus ν employed in Fig. 10 ofHankins & Eilek (2007) by multiplying both ∆ n and n by ω/ π ≃
30 Hz, the rotation frequency relevant for the Crab pulsar.c (cid:13) , 1– ?? ynamic spectrum of the Crab pulsar over the interval 2 × n . × . In the case of theCrab pulsar, where the rotational frequency ω/ (2 π ) is ap-proximately 30 Hz, the interval of harmonic numbers shownin Figs. 3 and 4 corresponds to a frequency [ ν = nω/ (2 π )]interval covering 6 to 10 . . ± . × for the morefundamental free parameter Ω /ω that renders the sloped∆ n/ d n predicted by equation (24) the same as the slope0 . ± .
01 of the line passing through the observationaldata (Hankins & Eilek 2007) even when the phase γ k iszero. We have then fitted the ∆ n intercept of the resultingline to that of the data, without changing its slope, by fixingthe values a k = (1 . ± . × − and b k = (1 . ± . × of the parameters that appear in the expression for γ k .The Airy functions in equation (15) stop oscillat-ing once the value of their argument falls below unity(Abramowitz & Stegun 1970). We shall see in Sec. 6 thatthe value (1 . ± . × of the parameter Ω /ω thusimplied by the data of Hankins & Eilek (2007) places thelast peaks of these functions at a frequency [ ∼ (Ω /ω ) ω/π ]that agrees well with the position ν ∼ Hz of the ul-traviolet peak in the observed spectrum of the Crab pul-sar (Lyne & Graham-Smith 2006). The widths of the emis-sion bands shown in Fig. 3 also agree with the widths ofthose that are observed: the number of bands falling withinthe interval 2 . × n . × in Fig. 3 is thesame as the number of bands that occur within the corre-sponding frequency interval (8 . . |P ⊥ | = sec(2 γ ⊥ ) (cid:8) − | (¯ s ϕ cos θ P − ¯ s z sin θ P ) | sin γ ⊥ + 2Ω nω | ¯ s r | cos θ P cos γ ⊥ + (cid:2) | ¯ s ϕ − ¯ s z sin θ P | − nω | ¯ s r | cos θ P (cid:3) cos ( ψ − γ ⊥ ) (cid:9) , (29)where γ ⊥ = arctan [2Ω / ( nω )] / ℑ{ (¯ s ϕ cos θ P − ¯ s z sin θ P )¯ s ∗ r cos θ P }| ¯ s ϕ cos θ P − ¯ s z sin θ P | − [2Ω / ( nω )] | ¯ s r | cos θ P . (30)These expressions imply a band spacing with essentially thesame characteristics as those discussed above, differing fromthose in equations (22) and (23) only in that ¯ s r and ¯ s ϕ exchange their roles with ¯ s ϕ cos θ P − ¯ s z sin θ P and ¯ s ϕ cos θ P ,respectively.It will therefore be seen that the salient parameter re-sponsible for the frequency bands in the Crab pulsar is theratio Ω /ω = (1 . ± . × . Using ω/ π ≃
30 Hz relevantfor the Crab yields a modulation frequency Ω / π ≃
570 kHz.A plausible cause for this modulation is plasma oscilla-tions within the pulsar’s atmosphere. A plasma frequency of570 kHz implies a free-electron density N − ≃ × cm − ,a value that is consistent with the inferred properties of theatmospheres of neutron stars. Note that the non-integral value of Ω /ω can be approximatedby an integer everywhere except in the factor sin( π Ω /ω ) thatappears in equation (7). ∆ ν ( G H z ) ∆ n nν (GHz) Figure 4.
The spacing ∆ n (right axis) of the emission bandsshown in Fig. 3 versus their mean harmonic number n (loweraxis). The solid line represents the function given in equation(24) for Ω /ω = 1 . × , a k = 1 . × and b k = 1 . × − .The upper axis shows observation frequecy ν = nω/ π relevantfor the Crab pulsar; the left axis shows ∆ ν . Data points for theCrab pulsar are taken from Fig. 10 of Hankins & Eilek (2007). We now turn to the main pulses of the Crab pulsar, inwhich the frequency-banded structure is much less appar-ent (Hankins & Eilek 2007). If the components of the elec-tric current within the emitting region (the region in whichthe rotating distribution pattern of the current approachesthe observer with the speed of light and zero acceleration)are such that P k and P ⊥ simultaneously contribute towardsthe value of the field, destructive interference of the contri-butions with differing polarizations could result in the sup-pression of the emission bands.This can be easily seen from equation (19) if we notethat, in this case, the radiated power for n ≫ P n dΩ P ∝ Q ˆ ϕ n − / [cos ( ψ − γ k ) + κ ( n ) cos ( ψ − γ ⊥ )] , (31)where κ ( n ) ≡ (cid:12)(cid:12)(cid:12) ¯ s ϕ cos θ P − ¯ s z sin θ P ¯ s r (cid:12)(cid:12)(cid:12) (32)is a coefficient that, like γ k and γ ⊥ , could be expanded intoa Taylor series about a local value n of n .Figure 5 shows the n dependence of the right-hand sideof equation (31) for a case in which Q ˆ ϕ decays as n − ,Ω /ω = 1 . × (as in Figs. 3 and 4), a k = 1 . × , b k = 1 . × − (as in Fig. 4), γ ⊥ = γ k + π/
2, and κ canbe approximated by 0 . . n/n −
1) with n = 2 × .The emission bands are thus replaced by small-amplitudemodulations of the radiation intensity that gradually dieout. The behaviour of the radiated power shown in Fig. 5over the interval 2 . × n . × is consistentwith those of the short-lived microbursts that are observedover the corresponding frequency interval (8 . . c (cid:13) , 1– ?? H. Ardavan et al. d P n / d Ω P nν (GHz) Figure 5.
The radiated power d P n / dΩ P per harmonic per unitsolid angle (in arbitrary units) versus n from a region of the mag-netosphere in which the contributions from differing polarizationsare comparable in magnitude. The dependence of the right-handside of equation (31) on n is shown for Q ˆ ϕ ∼ n − , γ ⊥ = γ k + π/ κ = 0 . . n/n − /ω , a k , b k ,and n as in Fig. 4. The top axis of this Figure gives frequency ν relevant for observations of the Crab pulsar. liptically polarized if either ¯ s ϕ or ¯ s r is dominant, and lin-early polarized if ¯ s z is dominant. The high degree of lin-ear polarization of the interpulse in the frequency range8 . − . e ⊥ , while the main pulsearises from a region in which more than one polarizationcontributes toward the observed field.This explains not only the difference between thefrequency structures of the main pulse and the inter-pulse, but also the difference between their temporal struc-tures, i.e. the fact that the intensity of the interpulsehas a broad and smooth distribution in time, while thatof the main pulse is composed of randomly distributednanoshots (Hankins & Eilek 2007). It would be possible toidentify the individual nondiffracting subbeams describedin Sec. 2 only in the case of a source whose length scaleof spatial variations is comparable with ˆ R − P (in the case,e.g. of a turbulent plasma with a superluminally rotatingmacroscopic distribution). The overall beam within whichthe nonspherically decaying radiation is detectable wouldthen consist of an incoherent superposition of coherent, non-diffracting subbeams with widely differing amplitudes andphases, that would be observed as randomly distributednanoshots. Otherwise, the smooth transition between theadjacent filamentary parts of the source that generate neigh-bouring subbeams would result in an overall beam that islikewise smoothly distributed. The filamentary sources sam-pled from an emitting region in which a single componentof the electric current dominates would not be sufficientlydifferent from one another in either amplitude or phase fortheir associated subbeams to be distinguishable.The observed difference between the dispersion mea-sures of the main pulse and the interpulse (Hankins & Eilek2007) further supports the fact that these pulses arise from distinct regions of the magnetosphere. As we have seen inSec. 2, an observer who is located at the polar angle θ P sam-ples, in the course of a rotation, the distribution of the den-sity of the electric current around the cylinder ˆ r = csc θ P .Thus the profile of the mean pulse reflects the azimuthaldistribution pattern of the source. Since they only involve a limited range of frequencies, thefeatures of the band emission we have discussed above do notdepend on the spectral distributions of the Fourier trans-forms ¯ s r,ϕ,z of the source densities s r,ϕ,z sensitively. To putthe emission bands in the context of the continuum spec-trum of the Crab pulsar, however, we would need both amore realistic model of the emitting current distribution anda more explicit description of the dependence of the densityof emitting current on frequency. This notwithstanding, thefrequencies across which the observed spectrum of the Crabpulsar undergoes sharp changes can be identified with thefrequencies at which the Airy function and the coefficients Q ˆ ϕ and ¯ Q r in the spectrum of the radiation described byequation (15) have their critical points. We have alreadyseen that, for the value Ω /ω = (1 . ± . × impliedby the observational data of Hankins & Eilek (2007), theposition Ω / ( πω ) of the last peak of the Airy function inequation (15) coincides with the ultraviolet peak in the ob-served spectrum of the Crab pulsar (Lyne & Graham-Smith2006) (Fig. 6).The coefficient Q ˆ ϕ in equation (15) is independent of n ( ∼ /ω ) when n ≪ µ + , decays as n − when n ≫ µ + , andis of the order of n when n ≃ m ≫ Ω /ω [see equation(7)].Hence, if the frequency mω/ (2 π ) of the azimuthal variationsof the source in the Crab pulsar [see equation (1)] falls inthe terahertz range, i.e. m ∼ , then the order of mag-nitude of Q ˆ ϕ would change from Ω /ω ∼ to n ∼ across the terahertz gap in its observed spectrum. The cor-responding enhancement of the radiated power by a factorof order (∆ Q ˆ ϕ ) ∼ is in fact consistent with observa-tional data (see below and Fig. 6). A plausible cause for afrequency mω/ π ≃ G. A mag-netic field of this order at, or close to, the light cylinder, isconsistent with the spin-down properties of the Crab pul-sar (Lyne & Graham-Smith 2006).Moreover, it can be seen from Equations (8) and (9)that the coefficient ¯ Q r changes from being independent of n to decaying as n − / when n increases past π ˆ R P / (ˆ r > − ˆ r < ) ,and so the observation point falls within the Fresnel zone.Given that the Crab pulsar is at the distance R P ≃ . × cm and has a light cylinder with the radius c/ω ≃ . × cm (Lyne & Graham-Smith 2006), this would accountfor the corresponding steepening of the observed spectrumof the Crab pulsar at n ≃ × (Fig. 6) if the radial extentof the emitting plasma observed from Earth has the valueˆ r > − ˆ r < ≃ . × − , i.e. 6 . c (cid:13) , 1– ?? ynamic spectrum of the Crab pulsar l og ( d P ν / d Ω P )( W m − H z − ) log ν (Hz) Figure 6.
The spectral distribution log(d P ν / dΩ P ), predictedby equation (33), versus log ν for ν = nω/ (2 π ) ≃ n Hz ap-propriate for the Crab pulsar. The points show observationaldata (where available) of the spectrum of the Crab pulsar(Lyne & Graham-Smith 2006). In the model, the recovery of in-tensity at the ultraviolet peak at ∼ Hz is caused by res-onant enhancement due to the azimuthal modulation frequency mω/ π ≃ ∼ Hz reflects atransition across the boundary of the Fresnel zone (see text). nents ¯ s r,ϕ,z of the source densities, are adequately describedby the following simplified version of equation (15):d P n dΩ P ∝ | ¯ Q r | Q ˆ ϕ S ( n ) (cid:16) n (cid:17) / Ai (cid:20) − (cid:16) n (cid:17) / Ω ω (cid:21) , (33)where S ( n ) stands for the dependence of the source densityon n . For Ω /ω = 1 . × and a ¯ Q r ( n ) that steepens bythe factor n − / across n ≃ × , equation (33) yieldsthe continuum spectrum shown in Fig. 6 if we assume that S ( n ) dominates over Q ˆ ϕ everywhere, except across n = 10 where Q ˆ ϕ increases by the factor 10 (due to the above-mentioned resonance with mω/ π ≃ n − / in n < and as n − / in n > . Note theclose correspondence, in Fig. 6, of equation (33) with theobservational data on the spectrum of the Crab pulsar over16 orders of magnitude of frequency (Lyne & Graham-Smith2006). We must stress the model-independent nature of the resultswe have reported here. The only global property of the mag-netospheric structure of the Crab pulsar we have invoked isits quasi-steady time dependence: that the cylindrical com-ponents j r,ϕ,z ( r, ϕ, z ; t ) of the density of the magnetosphericelectric current depend on ϕ only in the combination ϕ − ωt .Not only does this property follow from the observationaldata (as rigorously shown in Appendices A and B), but itis one that has been widely used in the published literatureon pulsars [see, e.g., Mestel et al. (1976)].Calculating the retarded field that is generated by theFourier component associated with the frequency Ω of such aquasi-steady current distribution, we have derived a broad-band radiation spectrum with oscillations whose peaks are proportionately spaced in, and decay algebraically with, fre-quency. By inferring the values of the parameters Ω / π ≃
570 kHz and mω/ π ≃ s r,ϕ,z that appear in the derived expression we have quantitatively accounted for the following features in the ob-served spectrum of the Crab pulsar:(i) the spacing of the emission bands,(ii) the frequency at which the extrapolated spacing be-tween these bands would reduce to zero,(iii) the widths of the emission bands (their numberwithin a given frequency interval),(iv) the salient features of the continuum spectrum (itssharp rise across the terahertz gap, its ultraviolet peak, thechange by 1 of the value of its spectral index at 10 Hz),and(v) the differences in the polarizations, dispersion mea-sures, and the time and frequency signatures of the mainpulse and the interpulse.Given that the derived band spacing ∆ n increases as n / with the harmonic number n [equations (18) and(24)], we predict that the repetition of the observations ofHankins & Eilek (2007) at higher frequencies would resultin a dependence of the band spacing on frequency that hasa local slope steeper than 0 . n/ d n ≃ . / π ≃
570 kHz, tentatively attributed to plasma oscilla-tions of electrons in the pulsar’s atmosphere and mω/ π ≃ G. The lattergives rise to the resonant increase in spectral weight in theultraviolet (Fig. 6).At this point, it is worth considering briefly the in-fluence of the ratio Ω /ω on pulsar spectra in general.Given that typical features of the pulsar’s spectrum scaleas (Ω /ω ) , we might expect that pulsars with slower ro-tational frequencies (and therefore small values of ω ) willhave observed spectral intensities weighted towards higherfrequencies. By contrast, so-called millisecond pulsars (withlarge ω ) might be expected to have emission concentratedat lower frequencies than the Crab. These predictions seemto be borne out both by Geminga ( ω/ π ≃ . ω/ π ≃
642 Hz) and 1957+20 ( ω/ π ≃
622 Hz), both ofwhich show no emission in the GHz range but strong pulsesat radiofrequencies (Lyne & Graham-Smith 2006).Furthermore, once it is acknowledged that (as shownin Appendices A and B) the current emitting the observedpulses from pulsars has a superluminally-rotating distribu-tion pattern, the results reported in the published literatureon the electrodynamics of superluminal sources (Ardavan1998; Ardavan et al. 2003, 2004a, 2007; Schmidt et al. 2007;Ardavan et al. 2008a,b) can be used to explain the ex-treme values of the giant pulses’ brightness temperature( ∼ K) (Ardavan et al. 2007), temporal width ( ∼ ∼ c (cid:13) , 1– ?? H. Ardavan et al. polarization (their occurrence as concurrent ‘orthogonal’modes with swinging position angles and with nearly 100per cent linear or circular polarization) (Schmidt et al. 2007)and spectra (the range − − / ϕ inthe combination ( ϕ − ωt ). This constraint may be inferreddirectly from the simplest observation, that pulsars have arigidly-rotating radiation distribution, and Maxwell’s equa-tions (see Appendices A and B); it leads to a current dis-tribution with a superluminally rotating pattern at a radius r > c/ω responsible for the unique features of pulsar emis-sion. The explicit form of the current distribution plays arole only in accounting for the pulse-to-pulse variations ofthe features discussed above (i.e. for the variable ‘weather’in the pulsar magnetosphere, not its stable ‘climate’). Theplasma processes by means of which the magnetic field ofthe pulsar couples the rotational motion of the central neu-tron star to the observationally implied rigid rotation of thedistribution pattern of the emitting current have no directbearing on the results reported in this paper. The salient fea-tures of the observational data can be understood in terms ofthe superluminal emission mechanism without a knowledgeof the magnetospheric structure of the pulsar. We are grateful to Jean Eilek, Tim Hankins, Jim Sheckard,John Middleditch, Joe Fasel, and Andrea Schmidt for helpfuldiscussions. This work is supported by U.S. Department ofEnergy Grant LDRD 20080085DR, “Construction and use ofsuperluminal emission technology demonstrators with appli-cations in radar, astrophysics and secure communications”.A.A. also thanks the Royal Society for support.
REFERENCES
Abramowitz M., Stegun I.A., 1970,
Handbook of Mathe-matical Functions (Dover)Ardavan H., 1998, Phys Rev E, 58, 6659Ardavan H., Ardavan A., Singleton J., 2003, J Opt Soc AmA, 20, 2137Ardavan H., Ardavan A., Singleton J., 2004a, J Opt SocAm A, 21, 858Ardavan A., Hayes W., Singleton J., Ardavan H., FopmaJ., Halliday D., 2004b, J Appl Phys, 96, 7760Ardavan H., Ardavan A., Singleton J., Fasel J., Schmidt,A., 2007, J Opt Soc Am A, 24, 2443Ardavan H., Ardavan A., Singleton J., Fasel J., SchmidtA., 2008a, J Opt Soc Am A, 25, 543Ardavan H., Ardavan A., Singleton J., Fasel J., SchmidtA., 2008b, J Opt Soc Am A, 25, 780Bessarab A.V., Gorbunov A.A., Martynenko S.P., PrudkoyN.A., 2004, IEEE Trans Plasma Sci, 32, 1400Bolotovskii B.M., Ginzburg V.L., 1972, Sov Phys Usp, 15,184Bolotovskii B.M., Bykov V.P., 1990, Sov Phys Usp, 33, 477 Bolotovski B.M., Serov A.V., 2005, Phys Usp, 43, 903Courant R., Hilbert D., 1962,
Methods of MathematicalPhysics , Vol. II (Interscience)Ginzburg V.L., 1972, Sov Phys JETP, 35, 92Gold T., 1968, Nature, 218, 731Goldreich P., Julian W.H., 1969, Ap J, 157, 869Hankins T.H., Kern J.S., Weatherall J.C., Eilek J.A., 2003,Nature, 422, 141Hankins T.H., Eilek J.A., 2007, Ap J, 670, 693Jackson J.D., 1999,
Classical Electrodynamics , 3rd ed. (Wi-ley)Lyne A.G., Graham-Smith F., 2005,
Pulsar Astronomy (Cambridge U Press)Mestel L., Wright G.A.E., Westfold K.C., 1976, MNRAS,175, 257Moffett D.A., Hankins T.H., 1999, ApJ, 522, 1046Schmidt, A., Ardavan H., Fasel J., Singleton J., ArdavanA., 2007,
Proceedings of the 363rd WE-Heraeus Seminaron Neutron Stars and Pulsars , eds. Becker W., HuangH.H., 124Singleton J., Ardavan A., Ardavan H., Fopma J., HallidayD., Hayes W., 2004, in
Digest of the 2004 Joint 29th In-ternational Conference on Infrared and Milimeter Wavesand 12th International Conference on Terahertz Electron-ics (IEEE), 591Soglanov V.A., Popov M.V., Bartel N., Cannon W.,Novikov A.Y., Kondratiev V.I., Altunin V.I., 2004, ApJ, 616, 439
APPENDIX A: RIGID ROTATION OF THERADIATION PATTERN IMPLIES A RIGIDROTATION OF THE SOURCE’SDISTRIBUTION PATTERN
As reflected in the highly stable periodicity of the meanprofiles of the pulses detected on Earth, the pulsar radiationfield has a rigidly rotating distribution pattern on average.That is to say, the cylindrical components of the receivedradiation fields E and B depend on the azimuthal angle ϕ only in the combination ϕ − ωt : E r,ϕ,z ( r, ϕ, z ; t ) = E r,ϕ,z ( r, ϕ − ωt, z, t ) , (A1) B r,ϕ,z ( r, ϕ, z ; t ) = B r,ϕ,z ( r, ϕ − ωt, z, t ) , (A2)where ( r, ϕ, z ) are the cylindrical polar coordinates basedon the axis of rotation, and ω is the angular frequency ofrotation of the observed radiation pattern. An equivalentstatement is that the radiation fields E and B have a quasi-steady time dependence: (cid:18) ∂∂t + ω ∂∂ϕ (cid:19) E r,ϕ,z = 0 , (A3)and (cid:18) ∂∂t + ω ∂∂ϕ (cid:19) B r,ϕ,z = 0; (A4)for the general solutions of these partial differential equa-tions are given by the expressions on the right-hand sides ofequations (A1) and (A2). c (cid:13) , 1– ?? ynamic spectrum of the Crab pulsar In the Lorenz gauge, the electromagnetic fields appear-ing in equation (3) are given by a four-potential A µ thatsatisfies the wave equation ∇ A µ − c ∂ A µ ∂t = − πc j µ , (A5)where µ = 1 , , A and j of the potential A µ and the current density j µ (Jackson 1999). The retarded solution to this equation inunbounded space is given by equation (4), i.e. by A µ ( x P , t P ) = c − Z d x d t j µ ( x , t ) G ( x , t ; x P , t P ) , (A6)where G ( x , t ; x P , t P ) = δ ( t P − t − R/c ) R (A7)is the corresponding Green’s function (Jackson 1999).Employing equation (3) to write out B r,ϕ,z in termsof the cylindrical components A r,ϕ,z of the vector poten-tial, we see that the potential also has the time depen-dence expressed in equations (A1)–(A4). It follows from B r = r − ∂A z /∂ϕ − ∂A ϕ /∂z , for instance, that L B r = 1 r ∂∂ϕ L A z − ∂∂z L A ϕ . (A8)where L stands for the differential operator appearing inequations (A3) and (A4): L ≡ ∂∂t + ω ∂∂ϕ . (A9)This and the corresponding equations for B ϕ and B z showthat equation (A4) is satisfied if L A r,ϕ,z = 0, i.e. if A r,ϕ,z depend on ϕ and t only in the combination ϕ − ωt .On the other hand, the cylindrical components of thevector potential are related to the cylindrical components j r,ϕ,z of the current density via the following spatial part ofequation (A6): " A r P A ϕ P A z P = c − Z r d r d ϕ d z d t G × " cos( ϕ P − ϕ ) j r + sin( ϕ P − ϕ ) j ϕ cos( ϕ P − ϕ ) j ϕ − sin( ϕ P − ϕ ) j r j z . (A10)Applying the operator L P ≡ ∂∂t P + ω ∂∂ϕ P (A11)to both sides of equation (A10) and making use of the factthat G depends on ϕ P and t P in the combinations ϕ P − ϕ and t P − t [equation (A7)], and so L P G = −L G , we findthat the resulting equation can be cast in the form L P " A r P A ϕ P A z P = c − Z r d r d ϕ d z d t G × " cos( ϕ P − ϕ ) L j r + sin( ϕ P − ϕ ) L j ϕ cos( ϕ P − ϕ ) L j ϕ − sin( ϕ P − ϕ ) L j r L j z (A12)by means of integrations by parts with respect to ϕ and t .Hence, L A r P ,ϕ P ,z P vanishes if and only if L j r,ϕ,z is zero. It follows from equations (A9) and (A12), therefore,that a necessary and sufficient condition for the rigid ro-tation of the distribution pattern of the radiation field,i.e. L B r P ,ϕ P ,z P = 0, is the corresponding rigid rotation L j r,ϕ,z = 0 of the distribution pattern of the source den-sity. APPENDIX B: RIGID ROTATION OF THESOURCE’S DISTRIBUTION PATTERNEXTENDS BEYOND THE LIGHT CYLINDER
Our purpose here is to show that a solution of Maxwell’sequations that has the quasi-steady time dependence ∂∂t + ω ∂∂ϕ = 0 , (B1)i.e. has a rotating distribution pattern, applies either tothe entire magnetosphere or to an expanding region whoseboundary propagates at the speed of light. By consider-ing the initial-boundary-value problem for the wave equa-tion (A5) with Cauchy data satisfying equation (B1), weestablish that such a solution cannot be smoothly matchedto other types of solutions of Maxwell’s equations across aboundary that is confined to a localized region of the mag-netosphere.The change of variables ξ = r, ξ = z, ξ = ϕ − ωt, ξ = ψ ( r, ϕ, z, t ) , (B2)where ψ is, for the moment, to be considered an arbitraryfunction, transforms the z -component of the wave equation(A5) into1 ξ (cid:18) ξ ∂A z ∂ξ (cid:19) + ∂ A z ∂ξ + (cid:18) ξ − ω c (cid:19) ∂ A z ∂ξ + (cid:16) |∇ ψ | − c ψ t (cid:17) ∂ A z ∂ξ + 2 ψ r ∂ A z ∂ξ ∂ξ +2 ψ z ∂ A z ∂ξ ∂ξ + 2 (cid:16) r ψ ϕ + ωc ψ t (cid:17) ∂ A z ∂ξ ∂ξ + (cid:16) ∇ ψ − c ψ tt (cid:17) ∂A z ∂ξ = − πc j z , (B3)where ψ t ≡ ∂ψ/∂t , ψ r ≡ ∂ψ/∂r , etc. The Jacobian of theabove transformation is ∂ ( ξ , ξ , ξ , ξ ) ∂ ( r, z, ϕ, t ) = ∂ψ∂t + ω ∂ψ∂ϕ , (B4)which remains non-zero as long as ψ is not a function thatdepends on ϕ and t as in ϕ − ωt .Now suppose that at a given time t the surface ψ ( r, ϕ, z, t ) = 0 represents the boundary of a region withinwhich the solution to equation (B3) satisfies the symmetry(B1), i.e. within which A z is a function of ξ , ξ and ξ only.At all points on this boundary, we then have ∂A z ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = 0 (B5)and1 ξ ∂∂ξ (cid:18) ξ ∂A z ∂ξ (cid:19) + ∂ A z ∂ξ + (cid:18) ξ − ω c (cid:19) ∂ A z ∂ξ + 4 πc j z (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = 0 , (B6) c (cid:13) , 1– ?? H. Ardavan et al. where (B6) is the wave equation under symmetry (B1); (B5)and (B6) hold at the boundary of the region in question byvirtue of holding inside that region.Hence, any other type of solution of the wave equa-tion that would match the symmetric solution across ψ = 0smoothly has to be sought by solving the following initial-value problem: given that the Cauchy data on the hyper-surface ξ = 0 are those expressed in (B5) and (B6), whatis the solution to the hyperbolic partial differential equa-tion (B3) beyond ξ = 0? Note that since ∂A z /∂ξ van-ishes at all points of the hypersurface ξ = 0, its derivatives ∂ A z /∂ξ ξ , ∂ A z /∂ξ ξ and ∂ A z /∂ξ ξ , which are in di-rections interior to this hypersurface, also vanish at ξ = 0.Thus, equation (B3) in conjunction with the data (B5)and (B6) demands that (cid:16) |∇ ψ | − c ψ t (cid:17) ∂ A z ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = 0 . (B7)There are two ways in which this requirement could be met:either ψ = 0 is a characteristic surface of the wave equation,and hence the first factor in (B7) vanishes, or at ψ = 0the derivative of ∂A z /∂ξ in the direction normal to theboundary is also zero. In the first case, the boundary of thedomain in which symmetry (B1) is satisfied, ψ = 0, willconsist of an expanding wave front that propagates at speed c . In the second case, the solution outside ψ = 0 will alsobe symmetric, according to the Cauchy-Kowalewski theorem[cf. Courant & Hilbert (1962)]. This is because the extension(by means of a Taylor series) of the data into an integralstrip next to the boundary ψ = 0 will yield a solution that isagain independent of ξ . Since the above argument appliesalso to the new boundary of the region thus extended, itfollows that A z will also be be independent of ξ outside thesurface ψ ( r, ϕ, z, t ) = 0, i.e. will be a function of ( r, ϕ − ωt, z )throughout the magnetosphere.The corresponding results for A r and A ϕ may be de-rived in the same way. APPENDIX C: ASYMPTOTIC EXPANSION OFTHE SPECTRUM FOR HIGH RADIATIONFREQUENCIES
In this Appendix, we evaluate the leading term in the asymp-totic expansion of the Anger functions that appear in equa-tion (6) for a radiation frequency ( nω ) that appreciably ex-ceeds both the rotation ( ω ) and modulation (Ω) frequenciesof the source, though not necessarily the frequency mω ofits spatial oscillations.In this regime, the Anger functions J n − Ω /ω ( n ) and J ′ n − Ω /ω ( n ) can be approximated by the Bessel functions J n − Ω /ω ( n ) and J ′ n − Ω /ω ( n ), respectively [see equation (41)of Ardavan et al. (2003)]. Once we cast these Bessel func-tions into the canonical forms J µ ( µ + ζµ / ) and J ′ µ ( µ + ζµ / ) by introducing the new variables A discontinuity in the value of ∂A z /∂ξ that represents a fieldcomponent is not permitted here, for it would entail the introduc-tion of surface charges and currents with infinite densities withinthe magnetosphere. µ ≡ n − Ω ω , ζ ≡ Ω ω (cid:16) n − Ω ω (cid:17) − / , (C1)we can write the leading terms in their asymptotic expan-sions for large µ as J µ ( µ + ζµ / ) ≃ (2 /µ ) / Ai( − / ζ ) , (C2)and J ′ ( µ + ζµ / ) ≃ − (2 /µ ) / Ai ′ ( − / ζ ) (C3)[see equations (9.3.23), (9.3.27) of Abramowitz & Stegun(1970)]. Taking the limits n ≫ Ω /ω of the coefficients andarguments of the above Airy functions, we obtain the ex-pressions in equations (13) and (14).For (Ω /ω ) ≪ n ≪ (Ω /ω ) , the argument of the result-ing Airy function and its derivative that appear in equations(13) and (14) is large, so that we can further use the asymp-totic approximationsAi( − χ ) ≃ π − / χ − / cos( χ / − π/ , (C4)andAi ′ ( − χ ) ≃ π − / χ / sin( χ / − π/ , (C5)[see equations (10.4.60) and (10.4.62) ofAbramowitz & Stegun (1970)] to replace them by thetrigonometric functions given in equations (16) and (17).Note that if Ω is replaced by − Ω, the argument of theabove Airy functions assume a positive value, and hence theresulting Anger functions J n +Ω /ω ( n ) and J ′ n +Ω /ω ( n ) decayexponentially, rather than algebraically, with increasing n [see Abramowitz & Stegun (1970)]. c (cid:13) , 1–, 1–