Energy budget of the bifurcated component in the radio pulsar profile of PSR J1012+5307
aa r X i v : . [ a s t r o - ph . H E ] N ov Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 28 August 2018 (MN L A TEX style file v2.2)
Energy budget of the bifurcated component in the radiopulsar profile of PSR J1012 + J. Dyks and B. Rudak
Nicolaus Copernicus Astronomical Center, Rabia´nska 8, 87-100, Toru´n, Poland
Accepted 2013 July 1. Received 2013 June 11; in original form 2013 February 25
ABSTRACT
The bifurcated emission component (BEC) in the radio profile of the millisecond pul-sar J1012+5307 can be interpreted as the signature of the curvature radiation beampolarised orthogonally to the plane of electron trajectory. Since the beam is intrin-sically narrow ( ∼ ◦ ), the associated emission region must be small for the observedBEC to avoid smearing out by spatial convolution. We estimate whether the energyavailable in the stream is sufficient to produce such a bright feature in the averagedprofile. The energy considerations become complicated by the angular constraintsimposed by the width of the microbeam, and by the specific spectrum of the BECwhich is found to have the spectral index ξ BEC ≈ − . ξ ≈ − erg/s, whereas the maximum-possible beam-size-limited powerof the stream is L ∆Φ ≈ erg/s. This implies the minimum energy-conversionefficiency of η ∆Φ ≈ − . The BEC’s luminosity does not exceed any absolute limitsof energetics, in particular, it is smaller than the power of primary electron and/orsecondary plasma stream. However, the implied efficiency of energy transfer into theradio band is extreme if the coherently emitting charge-separated plasma density islimited to the Goldreich-Julian value. This suggests that the bifurcated shape of theBEC has macroscopic origin, however, several uncertainties (eg. the dipole inclinationand spectral shape) make this conclusion not firm. Key words: pulsars: general – pulsars: individual: J1012+5307 – Radiation mecha-nisms: non-thermal.
Bifurcated emission components (BECs) have so far beenobserved in integrated radio profiles of J0437 − − ∼
32 per cent the BEC’s peak flux at 820 MHz. Ifthe shape of this feature is mostly determined by the intrin-sic shape of the curvature radiation microbeam, the extentof the associated emission region must be small enough sothat the spatial convolution of the curvature emission beamsdoes not smear out the BEC.At high frequencies ( ν & ∼ ◦ for typically expected pa-rameters), it is worth to verify if the energy supplied by theGoldreich-Julian density in such a narrow stream is sufficientto produce the observed flux of the BEC.After introducing some energetics-related definitions inSection 2 we estimate the BEC’s luminosity (Section 3). InSection 4 we estimate the maximum energy flux that can be c (cid:13) J. Dyks, and B. Rudak confined in the plasma stream, the width of which is limitedby the resolved form of the BEC. In Sect. 4.2.1 we compareour result to another published estimate, and reiterate ourmain conclusions in Sect. 5.
The radio luminosity of a pulsar beam cannot be accu-rately determined, because we do not know if our line ofsight samples representative parts of the beam. Withoutthe a priori knowledge of the emission beam and viewinggeometry, we cannot tell how much the observed flux dif-fers from the flux averaged over the full solid angle of pulsaremission. The missing information needs to be provided bysome model of the beam and viewing geometry. The sim-plest model assumes a uniform emission beam of solid angle∆Ω( ν ), with the uniform emissivity determined by the ob-served flux S on ( ν ) averaged within the ‘on-pulse’ interval ofpulse longitude. This implies the pseudo-luminosity: L = d Z ∆Ω( ν ) S on ( ν ) dν, (1)where the integration is within the frequency band of inter-est (between ν min and ν max ). Hereafter, the term ‘pseudo’,which expresses our assumption that the measured fluxrepresents the beam-integrated flux, will be neglected. Formany pulsars the observed pulse width does not change withfrequency or it changes slowly enough to consider the solidangle as ν -independent, and to extract ∆Ω from the inte-grand (Gould & Lyne 1998; Hankins & Rankin 2010). Inour case ∆Ω and S on ( ν ) must be determined for the BECof J1012+5307. The BEC’s spectrum will be calculated fur-ther below for a ν -independent pulse longitude interval of35 ◦ , marked in Fig. 1. The solid angle ∆Ω will accordinglybe considered fixed ( ν -independent). Eq. (2) then becomes: L BEC = d ∆Ω Z S BEC ( ν ) dν, (2)where S BEC is the mean flux of the BEC.The luminosity of eq. (2) cannot exceed the maxi-mum power which is theoretically available for the emittingstream: L ∆Φ = e ∆Φ pc c n GJ ( r ) A ( r ) , (3)where e ∆Φ pc ≡ E max is the energy corresponding to the po-tential drop above the polar cap, n GJ is the Goldreich-Juliandensity of the stream, and A is the crossectional area of thestream, measured at the same radial distance r as n GJ ( r ).Note that eqs. (2) and (3) are not independent: the emittingarea A in L ∆Φ refers to the same emission region as the solidangle ∆Ω in L . The choice of the emission region simulta-neously determines both A and ∆Ω in these equations.The accelerating potential drop is approximated by thepotential difference between the center and the edge of thepolar cap, as derived for a perfectly conducting neutronstar with vacuum magnetosphere and no dipole inclination:∆Φ pc = 6 . V B pc, R /P , where B pc, is the polarmagnetic induction in TG, R is the neutron star radiusin units of 10 km and P is the pulsar period (Goldreich &Julian 1969). To compare the BEC’s luminosity with the power givenby eq. (3) one can define the efficiency: η ∆Φ ≡ L BEC L ∆Φ (4)which is expected to be much less than unity. Since the pro-duction of coherent radio emission is not understood in de-tail, it is also useful to compare the BEC’s luminosity to thepower carried by the outflowing stream of particles (primaryelectrons and secondary e ± -pairs): L part = L pr + L ± = (5)= ( γ pr + n ± γ ± ) mc n GJ ( r ) A ( r ) , (6)where γ pr denotes the Lorentz factor acquired by an ac-celerated primary electron, γ ± is the initial Lorentz fac-tor of e ± pairs, and n ± is the number of pairs producedper one primary electron in the cascade which is responsi-ble for the BEC. Eq. (6) defines the power carried by theprimary electrons ( L pr = γ pr mc n GJ A ) and the secondary e ± plasma ( L ± = n ± γ ± mc n GJ A ). We will also use L ± , ,which is equal to L ± taken for n ± = 1. Below we also discussthe radio-emission power L R , determined by the minimumLorentz factor γ R required for the curvature radio spectrumto extend at least up to the upper integration limit ν ineq. (2). The stream’s radio power can be expressed by: L R = γ R mc n GJ ( r ) A ( r ) . (7)where a contribution due to n ± is neglected, because it iselectromagnetically difficult to separate plasma into chargedensity exceeding n GJ (Gil & Melikidze 2010, hereafterGM10). For all the afore-described power-related quanti-ties, we define their corresponding efficiencies in the wayanalogical to eq. (4), eg.: η pr ≡ L BEC /L pr , η ± ≡ L BEC /L ± , η ± , ≡ L BEC /L ± , , η R ≡ L BEC /L R . The distance d to J1012+5307 is estimated to 520 pc (Nicas-tro et al. 1995). To estimate S BEC , we calculate the mean flux density withina 35 ◦ -wide interval centered at the BEC, and compare itwith the mean flux density for the total profile (Fig. 1).This is done by assuming that emission from J1012+5307 isnegligibly low at the pulse longitude of the minimum flux(around 190 ◦ in Fig. 1). The result, illustrated in Fig. 1, is: S BEC = 0 . S mean at 0 . S BEC = 1 . S mean at 1 . , (9)where S mean = 14 mJy at 0 . S mean = 3mJy at 1.4 GHz (Kramer et al. 1998). The mean flux den-sity within the BEC is then roughly the same as the meanflux density of full profile around 1 GHz: S BEC ≈ S mean .The case of the BEC of J1012+5307 is then considerablydifferent from the standard case of luminosity estimate fornormal pulsars with narrow beams. In the latter case, S on isan order of magnitude larger than S mean . This is because todetermine the mean flux S mean , the energy contained within c (cid:13) , 000–000 nergy budget of J1012 + Figure 1.
Average pulse profiles and their mean flux forJ1012+5307 at 0 .
82 (solid) and 1 . φ obs = 71 ◦ mark the mean flux of theBEC alone, ie. the flux averaged within the 35 ◦ -wide phase in-terval centered at the BEC. The long horizontal lines mark themean flux S mean for the total profiles. The minimum flux near φ = 190 ◦ is assumed to present the zero level (dashed line). Theprofiles are phase-aligned to match the BECs at both frequencies.This also aligns the trailing side of the main pulse. Data courtesy:Paul Demorest (DRD10). the narrow pulse is attributed to the full rotation period.Equation 3.41 in Lorimer & Kramer (2005, hereafter LK05)is sometimes used to quickly estimate radio pulsar lumi-nosities. It is based on the exemplificative assumption that S peak = 25 S mean for narrow profiles of typical pulsars (withduty cycle 0 .
04 = 1 / S BEC ∼ . S peak (com-pare the level of bars at φ obs = 71 ◦ in Fig. 1 with the peaksof the BEC at the corresponding ν ). The narrow duty cycleexpressed by the equation S peak = 25 S mean , would then im-ply S BEC ∼ S mean , to be compared with eqs. (8) and (9).Thus, if eq. 3.41 from LK05 is directly used to calculate theluminosity of the BEC, the result becomes overestimated byone order of magnitude.Kramer et al. (1999) find that between 0.4 and 5 GHz,the spectra of many MSPs can be well approximated by apower-law function of: S mean ( ν ) = S mean , (cid:16) νν (cid:17) ξ , (10)where S mean , ≡ S mean ( ν ) is the mean flux density at afrequency ν . To perform the integration in eq. (2) the in-tegration limits are set to ν = 10 MHz and ν = 100 GHz,which is a much larger interval than has been observation-ally explored so far for J1012+5307. Data points in Fig. 2present the νF ν spectrum of total profile of J1012+5307based on available literature (Nicastro et al. 1995; Krameret al. 1998; Kramer et al. 1999; Stairs et al. 1999). Solid linepresents the power-law of eq. (10) fitted to the six pointsin Fig. 10, with the index of ξ = − . ± . νF ν slope of − . ξ , and the knownflux-contributions of the BEC at 0 .
82 and 1 . Figure 2. νF ν radio spectrum of J1012+5307. The horizon-tal axis covers the full range of integration in eq. (2). Solid linepresents the fitted spectral slope ( ξ = − . ξ = − .
87. The dotted line presents theBEC’s spectrum implicitly assumed by GM10. The dashed spec-trum contains two times more energy than the dotted one. Nu-merical values for all indices are given in the F ν convention. and 9), we have determined the spectral index of the BEC ξ BEC = − . ± .
58. The BEC’s spectrum, shown in Fig. 2with dashed line, is then completely different from the totalone. At 5 GHz the flux density contained within the BEC( S BEC ) is ∼ ν and ν , is 14 times lower than the one calculated for thetotal profile with the ‘total’ spectral index of − .
0. This iscaused by disparate levels of the BEC’s and total spectra atlow frequencies (see Fig. 2).
Interestingly, the BEC’s spectral index ξ BEC = − . ± . − / N e ∝ γ − p with p = 4 (see eq. 3 in Rudak & Dyks1999). The observed value of ξ BEC implies p = 4 . ± . ξ cr = − / ν ≈
400 MHz, which isthe lowest ν at which the BEC has been detected so far),only when the curvature radiation can reduce the electronLorentz factor down to γ ∼
40, as implied by eq. (14) with ρ = 10 cm. This can occur provided the characteristic timescale of particle escape from the emission region: t esc ∼ ρ/c ∼ . − s ρ (11)is longer than the timescale of the curvature radiation cool-ing: t cr ∼ γ | ˙ γ cr | = 32 mce κ ρ γ = 1 . s κ − ρ γ , (12) c (cid:13)000
40, as implied by eq. (14) with ρ = 10 cm. This can occur provided the characteristic timescale of particle escape from the emission region: t esc ∼ ρ/c ∼ . − s ρ (11)is longer than the timescale of the curvature radiation cool-ing: t cr ∼ γ | ˙ γ cr | = 32 mce κ ρ γ = 1 . s κ − ρ γ , (12) c (cid:13)000 , 000–000 J. Dyks, and B. Rudak where κ has been introduced to take into account the in-crease of energy loss above the vacuum value as a resultof unknown coherency mechanism. The condition t esc & t cr with γ = 40 requires κ & . , an apparently enormousvalue. Thus, the observed spectrum of the BEC can be un-derstood as the curvature radiation from an initially-narrowelectron energy distribution, provided that the radiative en-ergy loss rate ˙ γ cr is larger by the factor κ than the nonco-herent value. However, the uncertainty of p is large, and onecannot exclude the possibility that other factors are respon-sible for the observed spectral slope. The value of solid angle ∆Ω depends on the beam associatedwith the observed BEC. In the ‘stream-cut’ model, the BECis observed when the line of sight is traversing through a nar-row but elongated, fan-shaped emission beam (see figs. 1, 2,and 4 in DR12). The transverse width of this beam corre-sponds to the angular size of the curvature radiation mi-crobeam. In what follows the word ‘microbeam’ is used tomean the elementary pattern of radiation characteristic ofthe coherent emission process operating in pulsar magneto-sphere. It should be discerned from the ‘pulsar beam’ whichis observed at the Earth and results from spatial convolu-tion of many microbeams. The BEC of J1012+5307, at leastaround ν ∼ The beam of curvature radiation emitted in vacuum hasa mostly filled-in, pencil-like shape. Deep in the magneto-sphere, however, the ordinary-mode part of the beam can bedamped and absorbed by plasma. The remaining part, whichis the X mode polarised orthogonal to the plane of electrontrajectory, has the two-lobed form which we associate withthe BEC (DRD10).The microbeam then consists of two lobes that point ata small angle ψ with respect to the plane of electron tra-jectory, with no emission within the plane itself. The anglebetween the lobes is:2 ψ = 0 . ◦ ( ρ ν ) / (13)where ρ = 10 cm × ρ is the curvature radius of electrontrajectory, and ν = 10 Hz × ν . Hence for typical parameters( ρ ∼ ν ∼
1) the microbeam size is ten times smallerthan the observed separation of maxima in the BEC around1 GHz: ∆
BEC = 7 . ◦ . We assume that the large apparentwidth of the BEC results from the very small cut angle δ cut between the beam and the trajectory of the line of sight. When we walk across a railway track at a decreasingly smallangle, the distance between two points at which we crosseach rail increases. Small δ cut increases the apparent widthof the BEC in a similar way (see fig. 2 in DR12, with theangle δ cut marked on the right-hand side). A BEC producedby the beam of size 2 ψ , effectively has the observed width of Further below we will discuss wider beams with ρ ≪ ∆ ≈ ψ/ (sin ζ sin δ cut ), where ζ is the viewing angle betweenthe rotation axis and the line of sight.There are several important reasons for why we useeq. (13) instead of the popular result of ψ ≃ /γ , where γ is the Lorentz factor of the radio-emitting electrons:1) Eq. (13) is valid for any frequency smaller than, or com-parable to, the characteristic frequency of the curvature ra-diation spectrum: ν crv = 3 c π γ ρ = 7 GHz γ ρ [cm] . (14)Eq. (13) does include the result of ψ ∼ /γ as a special casewhen ν = ν crv but it also holds true for any ν ν crv . Thiscan be immediately verified by inserting ν crv into eq. (13),which gives ψ = 0 . γ − .2) The approximation given by eq. (13) is fairly accuratefor frequencies extending all the way up to the peak ofthe curvature spectrum. The maximum of this spectrumoccurs at ν pk = 0 . ν crv . At this spectral peak, eq. (13)overestimates 2 ψ by only a factor of 1 .
05. At ν = ν crv , theangle is still overestimated only by a modest factor of 1 . ψ ∼ /γ is valid only in twocases: i) for a frequency-integrated BEC; ii) at the peak ofthe curvature spectrum: ν ≃ ν crv . Case ii) does not have toapply for the actual, frequency-resolved BEC observed at afixed ν by a real radio telescope.4) The most important reason: within the validity range ofeq. (13), ie. for ν ν crv , the angle ψ does not depend on γ . Therefore, when the formula ψ ∼ /γ is used instead ofeq. (13), one may misleadingly invoke that for, eg., γ = 10 ,the angle ψ at ν = 1 GHz is equal to 10 − rad = 0 . ◦ .This is in general wrong, because at ν = 1 GHz (fixed bythe properties of a radio receiver) the angle ψ depends onthe curvature radius only, and for ρ = 10 cm is of theorder of 1 ◦ regardless of how high value of γ is assumed. The curvature radiation has then this interesting propertythat as long as the curvature spectrum extends up to thereceiver frequency ν , the detected beam has the angularwidth which is fully determined by the curvature radius ρ only. This angular width is with good accuracy the samefor all values of γ that ensure ν ν crv .
5) The separation of maxima in the BEC of J1012+5207evolves with ν in a way expected in the limit of ν ≪ ν crv (see fig. 7 in DR12). The use of eq. (13), which is also validin this limit, ensures consistency.6) The formula ψ ∼ /γ is blind to the question of whetherthe curvature spectrum for a chosen γ extends up to theobserved frequency band. Eg., for γ ∼
10, which neatlyfits the observed ∆
BEC in the absence of any geometrical The frequency ν crv is sometimes defined to be twice largerthan in eq. (14). In such a case the curvature spectrum peaks at0 . ν crv , ie. at a frequency almost one order of magnitude lowerthan ν crv . The value of ν crv itself is then located at the onsetof the exponential high-frequency cutoff of the spectrum, wherethe flux has already dropped down to ∼
30 per cent the peak flux.Note that the energy spectrum ( F ν convention) is assumed in thisdiscussion of spectral peak location. It may be worth to note here that in the low frequency limit ν ≪ ν crv , the ν -resolved intensity of the curvature radiation andthe shape of the microbeam do not depend on the Lorentz factoreither. c (cid:13) , 000–000 nergy budget of J1012 + magnification, the curvature spectrum does not reach 1GHz at all (if ρ ∼ cm). Whereas in the case of eq. (13)it is immediately visible that an extremely small ρ ∼ cm is required to get ∆ BEC ≃ ◦ at 1 GHz. Since the BEC’s beam is emitted by the stream, the pro-jection of the beam on the sky can be approximated by anelongated rectangle, described by two dimensions: one in thetransverse direction orthogonal to plane of the stream (di-rection of the magnetic azimuth φ ), and the other parallelto the stream (direction of the magnetic colatitude θ ).The flux contained in the BEC has been estimated inSection 3.1 through the integration over the pulse-longitudeinterval of 35 ◦ , which is 4 . BEC ∼ ◦ ). Since thepeak separation itself is interpreted as the angle 2 ψ givenby eq. (13), we assume that the transverse size of the solidangle is equal to 4 . × ψ = 0 .
06 rad / ( ρ ν ) / . The lineof sight may cut the beam at a small angle δ cut ≪ δ cut is small whereas the viewing angle ζ is not, the beamneeds to extend in the θ direction by an angle compara-ble to the BEC’s phase interval itself ( ∼ . ≃ .
03 sr / ( ρ ν ) / . This value is similar to the solidangle of a typical polar beam of normal pulsar (LK05 as-sume 0 .
034 srad).The BEC’s spectrum has been determined above byflux-integration within the same phase interval at differ-ent frequencies. For consistency, therefore, the solid an-gle is assumed to be ν -independent by setting ν = 1,ie. ∆Ω ≃ .
03 srad /ρ / . For the specific spectral indexof the BEC ( ξ BEC ≃ − .
87, see Fig. 2) the resulting lu-minosity changes only by a factor of 1 . ν -dependentsolid angle is used in eq. (2). Also note that a choice of widerlongitude interval for the BEC does not change the resultmuch, because the values of S BEC given by eqs. (8) and (9)decrease for wider intervals. This compensates the increaseof ∆Ω.
Taking d = 520 pc, ν = 1 . S mean , = 3 mJy, S BEC =1 . S mean , , ν = 10 MHz, ν = 100 GHz, ξ BEC = − . .
03 srad /ρ / we get: L BEC = 4 10 ρ − / erg / s . (15)The only previously-known estimate of the luminosityof the BEC is the one by GM10, which has not yet beenpublished in any astronomical journal, but is being widelybroadcasted on most recent pulsar conferences. The valueobtained by GM10 is 15 times larger than 4 10 . The mainreason for this is that GM10 used eq. 3.41 from LK05,which assumes that because of the usually narrow duty cy-cle δ = 0 .
04, the peak flux S peak is 25 times larger than themean flux S mean . In the case of the BEC of J1012+5307, wehave S peak ≃ . S mean at ν ≃ . /δ , where δ is the MSP’s duty cycle (or to use their eq. 3.40 instead of3.41).The other difference is that GM10 used the ‘global’spectral index of the total pulsar population ( ξ = − . smaller than ours. Therefore, GM10obtain the luminosity which is approximately larger by afactor 25 / The transverse crossection of the stream is assumed to ex-tend laterally through the distance ∆ l φ (in the direction ofmagnetic azimuth φ ) and meridionally through ∆ l θ (in thedirection of magnetic colatitude θ ). Then the area of thecrossection A = ∆ l φ ∆ l θ . The size of ∆ l φ is limited by thespread of magnetic field lines within the emitting area A ,which must not be too large in comparison with 2 ψ (eq. 13)to not blur the BEC. Let θ B denotes the angle between thetangent to a dipolar B -field line and the magnetic dipoleaxis. For two points separated azimuthally by ∆ φ , and lo-cated at the same r and θ , the dipolar B -field lines divergeby the angle of δ B , (the angle between the tangents to thefield lines) given by:sin δ B θ B sin ∆ φ . (16)For emission orthogonal to the dipole axis ( θ B = 90 ◦ ),eq. (16) gives δ B = ∆ φ . For two points on the oppositesides of the polar cap ( θ B = 1 . θ pc and ∆ φ = 180 ◦ ), it gives δ B = 3 θ pc ≪ ∆ φ . As can be seen, a specific difference ∆ φ inthe magnetic azimuth results in the B -field-line divergencethat is smaller than ∆ φ by the factor of sin θ B . This is be-cause for small θ B , B -field lines become almost parallel toeach other (and to the dipole axis) irrespective of ∆ φ .The divergence δ B of the B -field lines within the emis-sion region is allowed to comprise a fraction ǫ φ of the mi-crobeam’s width: δ B = ǫ φ ψ, (17)where ǫ φ < δ B and ∆ φ are small, from the last two equations weget: ǫ φ ψ = sin θ B ∆ φ. (18)This gives the following limitation on the transverse size ofthe stream:∆ l φ = r ⊥ ∆ φ = r ⊥ ǫ φ ψ/ sin θ B , (19) c (cid:13) , 000–000 J. Dyks, and B. Rudak
Figure 3.
Convolution of the curvature radiation microbeamwith a rectangular distribution of emitting plasma density (a),and a Gaussian distribution (b). The lowermost curve in eachpanel presents the unconvolved microbeam with the peak sepa-ration of 2 ψ , marked by the vertical lines. Different curves corre-spond to different widths of a stream ǫ φ = 1 . , . , . , ..., . , . ǫ φ are marked explicitly for theupper curves), where ǫ φ is a fraction of 2 ψ . The grey rectangleat the center presents the level of flux at the center of the BECof J1012+5307, as measured relative to the BEC’s peaks, in thefrequency range between 0 .
82 and 1 . where r ⊥ is the distance of the stream from the dipole axis. For the rim of the polar cap of J1012+5307, we have: θ B =17 . ◦ which allows ∆ l φ to be 3.3 times larger than the valueof r ⊥ ǫ φ ψ , expected for orthogonal viewing. The BEC ofJ1012+5307 is, however, observed 50 ◦ away from the phaseof the interpulse (IP), which may suggest θ B ∼ ◦ , forwhich (sin θ B ) − ∼ .
3. Since geometric effects can make theobserved BEC-IP separation both smaller and larger than θ B , below we assume that (sin θ B ) − = 1 . ǫ φ of the beam size, that can be occupiedby the stream can be estimated by making convolutions ofvarious density profiles with the shape of the elementarymicrobeam, given by eq. (11) in DR12.Fig. 3 presents such results for the rectangular (top hat)density distribution (Fig. 3a) and the Gaussian distribution(Fig. 3b). The grey rectangle in the center of both panelspresents the observed level of the central minimum between0 .
82 and 1 . For consistency, r ⊥ needs to refer to the same radial distance r from the star centre, as the Goldreich-Julian density does ineq. (3). . ǫ φ .
74. In the Gaussian case we have assumed that ǫ φ = 1 corresponds to the width of the Gauss function at thehalf power level (1 . σ ). The observed depth of the centralminimum then implies 0 . ǫ φ .
57. Thus, the maxi-mum allowed width of the stream depends on the sharpnessof the density distribution. Below we assume ǫ φ = 0 . φ , but different colatitude θ , we have δ B = ∆ θ B ≃ (3 / θ = (3 / l θ /r , where r is the radialdistance of the emission points from the neutron star cen-tre. This implies ∆ l θ which is larger than ∆ l φ by the factorof (2 / r/r ⊥ ) sin θ B . However, the spread of B -field lines inthe colatitude is not limited to a fraction of 2 ψ , because theextent of the emission region within the plane of the B -fieldlines does not smear out the BEC. Effects of the colatitudeextent are degenerate with the effect of motion of electronsalong the B -field lines and do not (directly) smear out theBEC. This can make misleading impression that the colati-tudinal extent is not limited at all by the unsmeared shapeof the BEC. However, because the emission beam is instan-taneously narrow, the parts of the emission region that arefar from the line of sight do not contribute to the detectableflux. This may lead one to think that ∆ l θ still needs to beconstrained by ∼ ψ so that ∆ θ B in the stream does notconsiderably exceed the beam size 2 ψ . This is not the case,because the emission from the poleward part of the stream(located closer to the dipole axis) becomes tangent to theline of sight at a slightly larger r (as a result of the curva-ture of B -field lines). For an arbitrarily large colatitudinalextent ∆ l θ there exists some radial distance r , at which thepoleward extremes of the stream can become visible to theobserver. Therefore, the extent ∆ l θ can still appear to beunconstrained by the beam size. However, because of the r -dependent effects of aberration and retardation (Blask-iewicz, Cordes & Wasserman 1991; Kumar & Gangadhara2012), the radial extent that is related to ∆ l θ produces apulse-longitude spread of ∆ φ obs ≃ r/R lc (Dyks, Rudak& Harding 2004). This spread must be a small fraction ofthe size of the beam:∆ φ obs ≃ r/R lc ǫ r ψ, (20)where ǫ r <
1. For ǫ r = ǫ φ = 0 . R lc = 25 10 cm,this constrains ∆ r to 8 . cm. The extent in colatitude∆ l θ is thereby also constrained to a value that can be deter-mined as follows. Consider the aforesaid two points at thesame r and φ , one of them located at θ , whereas the otherat a slightly smaller colatitude of θ (1 − ∆ s ), where ∆ s ≪ θ , irre-spective of their radial distance r . Therefore, the B -field linethat crosses the second (poleward) point becomes tangentto the line of sight at a slightly higher position ( r + ∆ r, φ, θ )determined by the equation of dipolar field lines:sin ( θ (1 − ∆ s )) r = sin θr + ∆ r , (21) Prompted by the referee we explain that dipolar B -field at lo-cations with small magnetic colatitude θ is inclined at the angle(3 / θ with respect to the dipole axis. Hence the 3 / (cid:13) , 000–000 nergy budget of J1012 + which in the small angle approximation gives:∆ r ≃ s r. (22)The limit of eq. (20) on ∆ r then translates to:∆ s . ǫ r ψ R lc r . (23)For ǫ r = 0 .
5, 2 ψ = 0 . ◦ = 0 .
014 rad, and r = 10 cm oneobtains ∆ s < . θ :∆ l θ = r ∆ θ = r ∆ s θ ≃ r ⊥ ∆ s . ǫ r ψ R lc r r ⊥ . (24)The value of ∆ l θ = (sin θ B / R lc /r )∆ l φ may then be afew times larger than ∆ l φ . Taking r ⊥ = r pc (rim of thepolar cap), one obtains ∆ l θ = 0 . r pc , which is twice aslarge as ∆ l φ . The apparent bifurcation of the BEC does nottherefore put equally tight constraints on the stream size incolatitude, as in the azimuth. Actually, it is possible to con-sider streams with elliptical crossection, with the longer axisof the ellipse pointing towards the magnetic pole. Because ofthe curvature of magnetic field lines, pair production indeedtends to spread the pairs in the θ direction. Let ∆ θ ± denotesthe range of colatitudes over which e ± -pairs associated to asingle primary electron were produced. Detailed numericalsimulations of type such as in Daugherty & Harding (1982)suggest that ∆ θ ± does not exceed few hundredths of angularpolar cap radius θ pc , ie. ∆ θ ± is comparable to ∆ θ given byeq. (24). For the sake of simplicity and minimalism, however,we will assume below that the stream has the same narrowsize in both directions: ∆ l θ = ∆ l φ , as given by eq. (19).Using r ⊥ = r pc = 2 10 cm, ǫ φ (1 GHz) = 0 .
5, and(sin θ B ) − = 1 . l φ = 1 . cm, andassuming ∆ l θ = ∆ l φ , the stream’s crossection has the areaof A = ∆ l φ ∆ l θ = 3 . cm ρ − / . The electric potential difference between the center andthe edge of the polar cap of J1012+5307 is e ∆Φ pc =1 .
45 10 eV = 232 ergs which sets up the upper limit tothe Lorentz factor: γ max = 2 . .Using eq. (3) with the surface value of the Goldreich-Julian density n GJ = 4 .
43 10 cm − ( ˙ P /P ) / this value of γ max corresponds to the following maximum kinetic lumi-nosity of the stream: L ∆Φ = 2 10 erg s − ρ − / . (25)The corresponding value of minimum radio emission effi-ciency, calculated using L BEC from eq. (15), is: η ∆Φ = 2 10 − ρ / , (26)which fits the reasonable range expected for radio emis-sion. Thus, even with the available energy limited by thenarrowness of the beam, the stream has enough energy topower the bright BEC observed in the average pulse profileof J1012+5307.We have retained dependence on ρ , because for ρ = 1the stream must be observed at a very small angle (0 . ρ ≪ ρ = 10 cm, η ∆Φ = 10 − , whereas for ρ = r pc = 2 10 cm, η ∆Φ = 6 10 − .Note that except for E max , we have been conservative inour estimates, so some parameters may still be set to makethe energy requirements even less demanding. For exam-ple, the spectrum of the BEC (Fig. 2) may be integratedonly between 100 MHz and 10 GHz, which is already awider interval than has ever been explored for J1012+5307.The stream may be assumed to have an elliptical shapewith ∆ l θ = 0 . r pc , and the ‘multipolar’ ρ = 0 . η ∆Φ = 1 . − . The energy contained in the BEC is then anegligible fraction of the maximum energy that can possiblybe attributed to the particle stream.However, the efficiency is larger when the BEC’s lu-minosity is compared to the energy of primary electronsor secondary pairs. Let us define the efficiency: η ( γ ) = L BEC / ( γmc n GJ A ). By setting the upper limit of η ( γ ) = 1one can calculate the minimum Lorentz factor that theradio-emitting particles need to have, to supply the energyobserved in the BEC: γ min ≃ ( ν ρ ) / , (27)where ν = ν max / (100GHz) is the upper limit of thefrequency-integration in eq. (15). The energy transformedinto the radio BEC needs to outflow at least at a rate ex-ceeding L min ≈ γ min mc n GJ A .In the strongly curved B -field lines of MSPs, a bal-ance between the radiative cooling and acceleration will con-strain the Lorentz factor of primaries to 9 . B / P − / (eg. Rudak & Ritter 1994), hence γ pr = 2 . . This isclearly larger than γ min , and implies η pr ≃ . − , ie. only0 .
2% of the primary electron energy is needed to explain theenergy of the BEC. The value obtained in GM10 is η pr ≃ γ ± ≃ . P / B − . For J1012+5307, γ ± ≈ . whichis six times larger than γ min . Thus, it is enough that onlyone secondary particle (out of n ± pairs produced per eachprimary electron) transfers 17% of its initial energy into theBEC.However, the secondary electron with so high Lorentzfactor will loose almost all its energy in the form of syn-chrotron X-rays, not the radio waves. As shown in the ap-pendix, the remaining energy of parallel motion is γ k ≈ P / , thus γ k ≈
57 for J1012+5307. This would have im-plied η k ≡ L BEC / ( γ k mc n GJ A ) ∼ , however, such a valueof γ k is too low for the curvature spectrum to reach the up-per limit ν max of our integration range (if ρ ∼ cm). Thismeans that either the secondaries are accelerated or the cur-vature radius ρ is much smaller than dipolar. Therefore, toestimate the upper limit for radio emission efficiency, it isnecessary to calculate the minimum Lorentz factor γ R forwhich the peak of CR spectrum reaches the radio band. For ν max = 100 GHz and ρ = 1, eq. (14) gives γ R = 522. Hence η R ≈ η R is independent of curva-ture radius of electron trajectory ρ , because both γ min and γ R are proportional to ρ / . GM10 obtain η R ≈ in c (cid:13) , 000–000 J. Dyks, and B. Rudak their optimistic case, or η R ∼ for parameters that theycall more realistic.Thus, although no absolute energy limit is exceeded(there is initially 6 n ± times more energy in the pair plasmathan L min ), to explain the observed flux of the BEC, theavailable energy would have to be transformed into radiowaves at an extreme rate. If the charge-separated (bunched)secondaries loose most of their energy in the form of X-rays,some process is required to draw the energy from anothersource, eg. from the primaries or the charge-unseparatedplasma, which outnumbers the Goldreich-Julian energy fluxby the factor of n ± . Alternatively, super-Goldreich-Juliancharge densities would have to emit coherently, ie. n GJ ineq. (7) would have to be replaced by n > n GJ . Contrary to GM10, we find that the energy content of theBEC does not break any strict upper limits. Eg. we find η ∆Φ ≃ − , η pr ≈ − (in GM10 η pr ≈ η ± , ≈ . η R ≈ , ie. weconfirm the need for extremely efficient energy transportinto the radio band. However, GM10 using a similar methodestimate η R ≈ − , which is in a notable disagreementwith our result. There are several reasons for this difference:1) GM10 have overestimated L BEC by a factor of 15, be-cause the duty cycle of J1012+5307 is much larger than 0 . γ pr = 5 10 , ie. for unspecified reasonthey assume that only 1 .
8% of available potential drop canbe used up for powering the stream. We use the maximumLorentz factor that the primary electrons can reach in theradiation-reaction-limited acceleration. We emphasize, how-ever, that the energy available for radio-emitting e ± pairsmay actually be larger than γ pr , because when primary elec-trons are moving up with a fixed Lorentz factor (balancedby the energy losses to the curvature radiation), the energyis anyway being produced in the form of curvature pho-tons that can produce the radio-emitting electron-positronplasma. It is then possible to produce the energy in form ofthe electron-positron plasma without any change of electronenergy (or even while the electron energy is increasing). Forthis reason the energy available for the stream may havemore to do with the maximum potential drop rather thanwith the maximum achievable Lorentz factor.3) GM10 neglect the factor (sin θ B ) − in eq. (19), whichat the polar cap’s rim of J1012 can increase the maximumallowed width of the stream 3 . θ B = 50 ◦ and (sin θ B ) − = 1 . L ∆Φ should bedecreased by a factor of 7 because the double-lobed,orthogonally-polarised part of the curvature microbeamcomprises only 1 / γ ± = 400, the stream size of ǫ φ × (1 /γ ± ), and ǫ φ =0 . L R which is several orders of magnitude lowerthan L BEC . However, their set of parameters ( γ ± = 400, ǫ φ = 0 .
1) is self-inconsistent because the BEC has the well- resolved double form at ν ≃ ρ = 1, the value of γ ± = 400 corresponds to ν crv = 45 GHz,which is in the range where the BEC is unresolved and ǫ φ can considerably exceed 1. Around 1 GHz the flux observedat the minimum between the peaks increases quickly up andit is reasonable to expect that the BEC is fully merged at ν ≫ ǫ φ = 0 . ν . ν the stream may well bewider than the microbeam ( ǫ φ > ψ ∼ /γ ± ) is onlyvalid at ν ≃ ν crv . Therefore, both ǫ φ and the beam size(hence, the value of γ ± ) must refer to ν ≃ γ ± = 400 can only be made consistentwith ν crv = 1 GHz if ρ = 448 10 cm (as implied by eq. 14with γ ± = 400 and ν crv = 1 GHz). The value of ρ implicitlypresent in their beam-size calculation is ρ = 18 R lc , where R lc = 25 10 cm is the light cylinder radius. Thus, to jus-tify their pessimistic values of L R , GM10 assume parametersthat imply the curvature radius several times larger than thelight cylinder radius. This should not be practiced to findthe maximum available kinetic luminosity.6) GM10 increase γ ± to decrease the stream crossection A ∝ (1 /γ ± ) , while keeping the BEC’s luminosity fixed,which again implies η R ≫
1. One should remember, how-ever, that the beam size also determines the size of the solidangle ∆Ω in eq. (2) for the BEC’s luminosity. For the nar-rowing split-fan beam, our line of sight must cut it at asmaller angle δ cut so that the 8 ◦ -wide BEC is observed. Thevalue of the solid angle ∆Ω, which is proportional to thebeam’s width 2 ψ (or, in the case of GM10, to 1 /γ ± ), shouldtherefore be decreased accordingly. For ∆ l θ ∝ ∆ l φ ∝ ψ theratio L BEC /L R is then proportional to ψ − instead of beingproportional to ψ − . It is then neccessary to treat eqs. (2)and (3) as related to the same beam opening angle to avoidthe overestimate of η R . We have calculated the radio luminosity of a single compo-nent selected from an average radio pulsar profile: the bifur-cated component of J1012+5307. This luminosity, equal to4 10 erg/s has been attributed to a narrow stream of radio-emitting plasma. The width of this stream is limited by theopening angle of the curvature radiation microbeam at 1GHz, as determined by the well-resolved double-peak formof the BEC at this frequency. It has been shown that the ef-ficiency of converting the stream’s energy into the BEC’sradio luminosity is of the order of η ∆Φ ≈ . − ρ / , η pr ≈ − , η ± , ≈ .
17, when the cross-cap potentialdrop, maximum energy of primary electrons, or initial en-ergy of pairs is taken as a reference, respectively. Thus, noabsolute energy limits are violated, and there is no energydeficit that can definitely be considered ‘fatal’ (as phrasedby GM10) for the microbeam model of the BEC.However, this result implies that large fraction of ini-tial energy of a single secondary electron (per one primary)needs to be transferred to the radio BEC. This is unlikely,because such pairs loose most of their energy in the formof synchrotron X-ray photons. To power the radio emission,therefore, the energy would have to be drawn from the pri- c (cid:13) , 000–000 nergy budget of J1012 + Figure 4.
Peak separation ∆ of pulsar double features as mea-sured at 1 GHz (horizontal lines). The solid curve presents therelation ∆( ρ ) = 2 ψ/ (sin ζ sin δ cut ) for the viewing angle ζ = 60 ◦ ,stream-cut angle δ cut = 45 ◦ , and 2 ψ given by eq. (13) (curva-ture radiation). The other curves have ( ζ, δ cut ) = (90 ◦ , ◦ ) (dot-ted), and (60 ◦ , ◦ ) (dashed). Most features are consistent with ρ . cm for statistically average geometry (solid curve). Notethe isolated location of J1012+5307. mary electrons or from the remaining charge-unseparatedplasma (there is an extra energy of n ± − α ∼ ◦ , and a large viewing angle ζ , measuredbetween the star’s rotation axis and the line of sight. Theorthogonal geometry is supported by the presence of the in-terpulse separated by half of the rotation period from theMP, as well as by the width of the MP itself (about 40 ◦ ),which is very close to the opening angle of the surface polarbeam (35 ◦ ). For small α and ζ , which we consider unlikely,both the luminosity of the BEC, and the radio efficiency η R become smaller than quoted above. This is because theobserved width of the BEC (a few times larger than ∆ BEC ) corresponds to the intrinsic solid angle ∆Ω that becomessmaller by a factor of sin ζ .Our results can additionally be affected by the un-certainty in the spectral index, bulk shape of spectral en-ergy distribution (spectral breaks or cut-offs in the yet-unexplored frequency range), distance and scintillation-affected mean flux. However, since most of these factors canbias the result in both directions, it is unlikely that they canconsiderably decrease the energy requirements.The value of the polar cap radius r pc that enterseqs. (19) and (24) is known at best with the accuracy 16%,which is the difference between the vacuum and force freecase (fig. 4 in Bai & Spitkovsky 2010). This implies a 36%uncertainty in the stream area A . If a more narrow spec-tral band is assumed (0 . .
01 – 100GHz) the BEC’s luminosty estimate decreases by a fac-tor of 2 .
1. However, widening the band up to the range(1 MHz − L BEC by only a factor of1 .
6. If the spectral index of the BEC is increased or de-creased by 0 . ∼
2. This is because in both these cases the spectrumbecomes steep in comparison to the present slope of +0 . νF ν convention.The possibility of spectral breaks beyond the GHzband can reduce the energy requirements. Most millisec-ond pulsars do not exhibit any spectral breaks above 100MHz (Kuzmin & Losovsky 2001, hereafter KL01; Krameret al. 1999) For J1012+5307 however, the Pushchino mea-surements at 102 MHz (KL01; Malofeev et al. 2000) suggesta break at ∼
600 MHz. This break is not included in ouranalysis (and not shown in Fig. 2), because the BEC is notdiscernible in the low-frequency profiles (see fig.2 in Kon-dratiev et al. 2012). It is therefore not possible to determinethe fractional energy content of the BEC below the spectralbreak. However, for the BEC spectrum shown in Fig. (2)with the dashed line, the mean flux of the BEC becomescomparable to the mean flux of total profile near 100 MHz(30 mJy, KL01). The lack of strong BEC in the 100 MHzprofiles implies that the actual spectrum of the BEC at lowfrequencies is steeper than in the GHz range.The Goldreich Julian density in the emission region,which is proportional to ~ Ω · ~B can be much smaller than weassume, if the local ~B is orthogonal to ~ Ω. This can happenbecause we assume rather large viewing angle. However, al-though we fix ζ to 90 ◦ for practical reasons, any values in thebroad range of ζ = 90 ◦ ± ◦ are possible. Moreover, localnon-dipolar enhancements of B are capable of increasing thedensity to a level considerably higher than the dipolar one. Ifthe dipole inclination is not orthogonal, the magnetic fieldderived from the dipolar radiation energy loss should fur-thermore be increased by the factor (sin α ) − . Other break-ing mechanisms introduce additional uncertainty. For exam-ple, the magnetospheric currents in PSR B1931+24 change˙ P by a factor of 1 . B .Contrary to GM10 we find that the BEC’s flux com-prises a tiny part of the maximum limit for the stream en-ergy ( η pr ≈ − , as compared to η pr ≈ η R ≈ ), yet it is 2–4 orders of magnitude smaller than inGM10. The determination of luminosity and efficiency forisolated components in pulsar profiles is more complicated c (cid:13) , 000–000 J. Dyks, and B. Rudak than standard energy considerations. Special care is requiredbecause: 1) Such an isolated component may have consider-ably different spectrum than the total pulsar spectrum andit may have the flux density which can be at any ratio withthe mean flux density of the total profile. 2) Fraction of thepolar cap outflow that is responsible for the observed compo-nent needs to be carefully estimated, with the constraint ofnot blurring the component which is observed sharp and re-solved at a frequency ν . In the case of the split-fan beam, this‘sharp view’ condition is different in the transverse ( φ -) di-rection than in the meridional direction of θ . 3) For ν ν crv ,the size of the curvature microbeam at a fixed frequency ν is independent of the Lorentz factor γ of the radio emittingplasma. The popular formula ψ ∼ γ − can only be usedwhen ν crv ( γ, ρ ) = ν , where ν is the center frequency of theobserved bandwith. If the ratio ν/ν crv is not known, andthe spectrum extends up to the observation band, it may bemore safe to use eq. (13). 4) Both the radio luminosity ofthe BEC, through ∆Ω, as well as the maximum power of thestream, through A , depend on the width of the microbeam.Any decrease of available power imposed by the decreasingwidth of the beam is alleviated by the simultaneous decreaseof BEC’s luminosity. APPENDIX
The energy of parallel motion of e ± pairs can be estimatedin the following way. Let us consider a secondary electronwith initial Lorentz factor γ ± . Let v k be the component ofthis electron’s velocity parallel to the magnetc field, and γ k = (1 − ( v k /c ) ) − / is the corresponding Lorentz fac-tor. Now consider a primed Lorentz frame which movesalong ~B with the velocity v k . In this frame our electronhas purely transverse velocity v ′⊥ and a Lorentz factor γ ′⊥ = (1 − ( v ′⊥ /c ) ) − / . The Lorentz transformation of ve-locities implies: γ k = γ ± γ ′⊥ . (28)In the case of millisecond pulsars, the Sturrock’s conditionfor pair creation is: χ ≡ ǫmc BB Q sin ψ ≈ . , (29)(Sturrock 1971), where ǫ is the energy of the pair-producingphoton, mc is the electron rest energy, B Q ≈
44 TG is thecritical magnetic field value, and ψ is the angle at which thephoton crosses the magnetic field. For classical pulsars thenumber on the right hand side is closer to 1 /
15. In the caseof χ ≪
1, each component of the created pair takes up halfenergy of the parent photon: γ ± mc ≈ ǫ γ ± ≫
1, it holdsthat cos ψ = v k /v ± ≈ v k /c , hence: 1 /γ k ≈ sin ψ . Eqs. (29)and (30) then give: γ ′⊥ = 3 . (cid:16) rR ns (cid:17) B − pc, , (31)where the local B field is B = B pc ( r/R ns ) − . The Lorentz factor γ ± can also be estimated from (30)and (29), by noting that a photon emitted in dipolar field at( r, θ ) encounters the largest value of B sin ψ = 0 . θB ( r )(Rudak & Ritter 1994). This is approximately the placewhere the one-photon absoption coefficient is maximum, andthe pair production is most likely and efficient. By insertingthe last formula into (29) one can derive so called ‘escapeenergy’, which is the minimum photon energy required toproduce pairs in pulsar magnetosphere: ǫ esc ≈ . MeV R − / ns, ( r/R ns ) / B − pc, P / , (32)where R ns, = R ns / (10 cm), P = P/ (10 − s), and θ ≈ ( r/R lc ) / was assumed to correspond to the polar cap rim.By inserting (32) into (30) we get: γ ± = 9 . ( r/R ns ) / P / B − pc, . (33)From (28), (31), and (33) we obtain γ k = 25 ( r/R ns ) − / P / , (34)which, in the limit of near-surface emission and pair pro-duction ( r ∼ R ns ) is used in the main text. The derivedestimates well reproduce the results of exact numerical sim-ulations (see the distributions of γ k and γ ′⊥ for a normal andmillisecond pulsar in fig. 1 of Rudak & Dyks 1999). Theyare also useful in semi-analytical modelling of pair cascades(Zhang & Harding 2000). ACKNOWLEDGEMENTS
We thank Paul Demorest for providing us with GBT dataon J1012+5307. This work was supported by the NationalScience Centre grant DEC-2011/02/A/ST9/00256 and thegrant N203 387737 of the Polish Ministry of Science andHigher Education.
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