High Frequency Cutoff and Change of Radio Emission Mechanism in Pulsars
Abstract
Pulsars are the fast rotating neutron stars with strong magnetic field, that emit over a wide frequency range. In spite of the efforts during 40 years after the discovery of pulsars, the mechanism of their radio emission remains to be unknown so far. We propose a new approach to solving this problem for a subset of pulsars with a highfrequency cutoff of the spectrum from the Pushchino catalogue (the ”Pushchino” sample). We provide a theoretical explanation of the observed dependence of the highfrequency cutoff from the pulsar period. The dependence of the cutoff position from the magnetic field is predicted. This explanation is based on a new mechanism for electron radio emission in pulsars. Namely, radiation occurs in the inner (polar) gap, when electrons are accelerated in the electric field that is increasing from zero level at the star surface. In this case acceleration of electrons passes through a maximum and goes to zero when the electron velocity approaches the speed of light. All the radiated power is located within the radio frequency band. The averaging of intensity radiation over the polar cap, with some natural assumptions of the coherence of the radiation, leads to the observed spectra. It also leads to an acceptable estimate of the power of radio emission.
Institute of Radio Astronomy of National Academy of Sciences of Ukraine, 4 Chervonopraporna Str., Kharkov 61002, Ukraine.
Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine.
Keywords pulsars: radio emission, spectra
1 Introduction
Pulsars are magnetized neutron stars that have a magnetosphere filled with an electronpositron plasma of about the GJ density (Smith, 1977; Manchester & Taylor, 1977; Beskin, et al., 1993). New discoveries of double pulsar system (Lane, et al., 2004) and intermittent pulsars (Kramer, et al., 2006; Lorimer, et al., 2012; Camilo, et al., 2012) give the direct observational support to that idea. It is thought that this plasma in the region of open magnetic field lines over the magnetic polar cap is generated by particles (through gamma quanta production) accelerating in a gap under the magnetosphere (Sturrock, 1971; Ruderman & Sutherland, 1975; Arons, 1981; Beskin, 2010). The acceleration of electrons occurs in the gap in the electric field that is longitudinal with respect to the magnetic field and induced by the rotation of the magnetized star. Directed coherent electromagnetic radiation of relativistic particles from the region of open lines creates the beacon effect that results in the pulses observed (the most popular explanation).
In explanation of radio emission of pulsars (see reviews (Malov, 2004; Manchester, 2009) and addition references in (Malov & Machabeli, 2009; Kontorovich, 2009; Beskin & Philippov, 2012)) the instabilities of plasma flow, beam instabilities and similar effects in the magnetospheric plasma^{1}^{1}1Note apart the plasmabeam (see as example (Usov, 1987; Kazbegi, et al., 1992)), also the cyclotron, drift, modulation instabilities, Zakharov’s wave collapse and magnetic reconnection for GP, lowfrequency ”tails” of the synchrotron, Cherenkov, Doppler and curvature radiation in the relativistic electronpositron plasma. have been discussed. Apparently, various mechanisms of radio emission are actually realized and may in certain circumstances succeed each other.
We show in this paper that for the observed pulsar radio emission a coherent radiation produced in a polar gap may be responsible, at least for pulsars of the Pushchino sample^{2}^{2}2The sample is based on Pushchino catalogue (Malofeev, 1996, 1999). Catalogue gathers simultaneous and compiled radio spectra for 340 pulsars (on more than three frequencies) with more than 120 references on measurements on different instruments including Bonn 100m and Ukrainian decameter radio telescopes. The correlation between frequency maximum, cutoff frequency and period have been found for some part of the spectra (Malofeev & Malov, 1980). We don’t touch here the more high frequency values in catalogue (Maron, 2000) with new results that ask for separate investigation, see also discussion in (Sieber, 2002). with cutoff in the radio spectrum. The relationships we have received we will also use for the pulsar in the Crab Nebula, given its peculiarities. The radiation has been emitted with the acceleration of electrons in the gap. It is quite essential that the accelerating longitudinal electric field in the gap slowly increases from zero at rising off the star surface. This mechanism has never been discussed earlier for pulsars.
In the wellknown RudermanSutherland model the strong electric field, non vanishing at the star surface, accelerates electrons so quickly that their radiation due to acceleration in the gap fully comes to the hard energy (Xray) region with no radio emission. Only in an electric field slow rising from zero level on the star surface the radiation of accelerating electrons comes to the radio band.
Resulting radiation is limited by a cutoff frequency found in this study. It coincides with the highfrequency cutoff (Malofeev & Malov, 1980; Malov, 2004) in the pulsar spectra of the Pushchino sample. Such frequency limitation is due to the fact that the electron acceleration in the electric field, vanishing on the star surface, passes through a maximum and decreases as the electron velocity approaches the relativistic limit.
There is the allnew specification of our point of view and the main result of this work.
2 Acceleration of electrons in the speed up process
We start from the equation for the electron Lorentz factor (Landau & Lifshitz, 1994).
(1) 
where is the altitude of the electron above the star surface. The equation describes the change of electron energy in a nonuniform electric field . (Only component that is parallel to the strong magnetic field directed on z axes is essential). Energy losses (neither by curvature radiation nor by inverse Compton scattering) are insignificant for relevant for us values of Lorentzfactors of order of some units. We also ignore, within the polar cap, the deviation of the magnetic field lines from orthogonality to the surface of the pulsar.
The lowfrequency radiation occurs at the small altitudes when the accelerating electric field rises from zero level on the surface of the pulsar . The vanishing of the electric field on the star surface is related to a small electron work function (Jones, 1986) of the surface. At altitudes , where h is the height of the gap, the field
increases linearly (Arons, 1981; Harding & Muslimov, 1998; Dyks & Rudak, 2000; Beskin, 2010) with the altitude z. The velocity and the acceleration , expressed in terms of Lorentz factor , are equal to , and
where . In a linear field we have
The acceleration increases from zero when , passes through a maximum (Fig.1) when
(not dependent on the field) and
(for ) and tends to zero at approaching of electrons to relativistic velocities. The value of particle acceleration at the maximum is
The height of the gap falls from these relations when used henceforward the field of the form (Muslimov & Tsygan, 1992)
Physically, this estimate is quite obvious since on the scale of the gap the characteristic velocity due to rotation is . We omit here a factor of order of unity that will be partly considered below, which contains the dependence of on the position on the polar cap. We do not discuss here the influence of general relativity effects and dependence on the angle between the axis of rotation and magnetic axis of the pulsar (Beskin, 2010), which does not affect the estimates. (The exception is PSR B0531+21 which is close to an orthogonal rotator.) The estimate
confirms the legality of the conditions which we use, as for normal pulsars.
3 Radiation at acceleration in the gap
For the radiation field of accelerated electrons, we proceed from the retarded LienardWiechert potentials (Jackson, 1962; Landau & Lifshitz, 1994). In the problem considered the particle acceleration w is directed along its velocity V. Then for the Fourier component of the wave magnetic field we have (at large distances from the radiating electron)
(2) 
Here , is the electron radius vector, is the distance from the origin on the star surface to the field observation point, .
A connection of the time with the electron coordinate is given by the integral
(3) 
where is governed by (1).
To eliminate the logarithmic divergence at zero one should take into account in the difference that the initial velocity and therefor . Here is the thermal velocity of electrons on the surface of the pulsar polar cap. It is convenient to represent the Fourier component , which determines the emission spectrum, as an integral over coordinate:
where and
(4) 
Accordingly, we have for the spectral and angular radiation intensity density interesting us (Jackson, 1962; Landau & Lifshitz, 1994)
(5) 
Here is the interval of frequencies, is the interval of solid angles. It is seen from the integral that due to rapid oscillations the field decreases exponentially for , where the cutoff frequency is (Fig.2). This estimate can be obtained numerically from Eq.(5) and independently from the physical consideration, supposing that the electron movement becomes relativistic: and . We obtain , , =. The coefficient , where is the wavelength corresponding to the cutoff.
To move to the average spectra we take into account the dependence of the field from its position on the polar cap ( cf. (Dyks & Rudak, 2000)) of the form:
(6) 
( is the star radius). Assuming for the maximum field at the center of the polar cap , we obtain the value of the cutoff frequency , which is independent of the height gap. The cutoff frequency dependence on the pulsar magnetic field is shown in Fig.3.
At this frequency, the emission spectrum for the discussed mechanism must cutoff and be replaced by the next highest, forming a kink. The cutoff frequencies for a sample of normal pulsars had been found in (Malofeev & Malov, 1980). The dependence on the period for coincides for
(7) 
(this theory)
(Malofeev & Malov, 1980).
(The tilde over frequency means the experimental value). The dependence on and other parameters, (see as example (Dyks & Rudak, 2000)) determining the electric field in the gap, is available for testing. It can also serve as a criterion for the correctness of this description. Below we use the cutoff frequency corresponding decrease of intensity radiation of a single electron to times (see Fig.2). The actual frequency of the cutoff may be less than the estimated cutoff frequency both due to the fact that the actual accelerating field in the gap is less than the adopted estimates and the cutoff corresponds to higher values of gammafactor.
4 Average spectra
Now we consider the emission spectrum of a large number of electrons accelerated in the electric field of the gap. From the intensity of a single particle (taking into account that in the electric field, linearly increasing from zero on the star surface, the acceleration maximum is proportional to the square root of )
(8) 
and changing the spectrum in the Fig.2 by the stepfunction, we turn to the spectral density for :
(9) 
where
Accordingly, the frequency range of radiation of the single particle has the form
(10) 
therefore for the spectrum (in an incoherent case) we obtain
(11) 
where is the number of emitting particles measured through the current across the polar cap. Integration over the polar cap gives
(12)  
i.e. the spectrum is flat and has a cut off at the frequency .
Let us now consider the coherence of the emission (Ginzburg et al., 1969; Ginzburg, 1971; Kuzmin & Solov’ev, 1986; Golgreich & Keeley, 1971). In the discussed mechanism only thin a disk with emits and the disk thickness is less then any wavelengths emitted. Therefore, in our case there is no need to raise additional assumptions about formation in the longitudinal direction coherently emitting electron bunches, which represents up to now a significant difficulty in all theories of pulsar radio emission. However, we must assume that in the disk plane a fragmentation takes place to coherently emitting regions, just as it is supposed in explaining the drift of subpulses (Lyne & GrahamSmith, 1998). Here we restrict ourselves to the case when fragmentation occurs in a region with a transverse dimension less than the length of the radiated wave^{3}^{3}3In the case of a thin disk it is possible, in principle, a coherent radiation of considerably larger in transverse dimension regions, bounded by the scale (Smith, 1977), which gives for the upper boundary of power a significantly large value. When it gives about a dozen of coherently emitting regions on the polar cap, comparable with the number of observed ones in the drift of subpulses (Deshpande & Rankin, 1999; Smith, 1977; Palfreyman et al., 2011). See also (Young, 2003).. In this case, we find the upper limit of radiated power at a given fragmentation. For this the total number of particles N is divided into coherent blocks (which are different for different wavelengths and depend on blocks position)
(13) 
Inside of the blocks the intensity is proportional to the square of the particle number . The required limitation (Golgreich & Keeley, 1971) of block height is provided by the condition , where is the height to which the radiation mechanism acts. The blocks are summed additively
(14) 
The condition for all emitted waves is fulfilled automatically in our case that is very important for realization a coherent emission mechanism. As for division of the polar cap on coherent areas, at this stage we must restrict the discussion only by assumptions. Dividing polar cap on the regions of order lambda, we obtain, in coherent case, the lower estimates for intensity of radiation due to the minimum square of the number of particles in each region.
Obviously, the number of blocks equals to , and the number of particles in the block is . Then for radiation spectrum we have
(15) 
where is given by Eq. (11) and the frequency is determined accordingly to Eq. (12). We have obtained a powerlaw spectrum with spectral index 2 (Fig.5), which is typical for the majority of pulsars. Let us note, that Eq.(12) and (15) are only the asymptotics of the exact spectra. In reality there is a low frequency break (or turnover) at the frequency (Malov, 2004). Such spectrum behaviour can be obtained in the discussed model^{4}^{4}4Note, that there is a dynamical but not a dissipative reason (cf. (Ochelkov & Usov, 1980)) for a turnover. and we have considered in detail the physical meaning of that break and its position in the separate paper (Kontorovich, Flanchik, 2013). Near the cutoff frequency the emission spectrum can not be considered strictly as a power law. For a small number of measured values it may be perceived as a kink. The spectral index depends on transverse dimensions of coherent blocks, which it seems also reveals themselves in the geometry of the subpulse drift.
The coherence provides a reasonable estimate of the intensity of radio emission. The rough estimate of the power of radio emission has the form
(16) 
where is the radiation power of a single particle at the maximum of acceleration. Here , where is a wavelength corresponding to the maximum in the spectrum of radio emission. Writing an estimate for the number of particles in a coherent block in the form , where is the average density of particles near the surface, we have , whence the estimate results. For the parameters , it leads to , which agrees well with the data on the radio luminosity of pulsars. For the fast rotating pulsar B0531+21 with high magnetic field this estimate reaches 10 erg/s taking into account its proximity to the orthogonal rotator.
Note, that the assumption about change of radiation mechanisms in the high frequency cutoff region allows in principle to explain (Kontorovich & Flanchik, 2011) the main pulse disappearance of PSR B0531+21 at frequencies near 8 GHz (Hankins & Eilek, 2007). Really, from our point of view at lower frequencies the radiation at longitudinal acceleration of subrelativistic electrons gives the principal contribution to the main pulse. This emission vanishes at the cutoff frequency that may lead to disappearance of the pulse. The radiation due to lowfrequency tail of narrow directed aberrational relativistic mechanism remains. This anisotropic radiation does not fall to the main pulse window but reveals itself in the interpulse if its line of sight is more close to the magnetic axis.
Note also that the discussed mechanism of radio emission (for which the place of radiation is definite and occupies a thin layer near the surface of the star) may use for checking the new methods to determine the generation location by a dispersion delay of the signal in the magnetospere (Hassall, et al., 2012). The accuracy of such methods is limited now by the lack of the adequate knowledge of magnetosphere properties.
5 Conclusions
We have found above that the radiation of accelerating electrons in the electric field, slow increasing from zero on the star surface, entirely comes to the radio spectral band. This is the main result of the work.
The considered approach allows us explain the radio emission for the pulsars from Pushchino sample, including the position of the spectrum cutoff. As a result, we obtain a powerlaw spectrum, which arise at averaging over the polar cap due to dependence of the accelerating electric field on its position. The density radiation in the gap is large without any assumptions about the gap as a cavity (cf. (Kontorovich, 2009)). It explains also the part of the gammaray emission from pulsars and its correlation (Bilous, et al., 2011) with giant pulses (Kontorovich & Flanchik, 2008).
The radiation with the linear acceleration was considered in (Melrose et al., 2009a, b) for acceleration in a so strong electric field on the star surface, when an electron reaches relativistic velocities in a time shorter then the period of the wave emitted. In this case the considered effects are absent.
The obtained results make it also possible to compare the theories of accelerating fields in the inner gap (Harding & Muslimov, 1998; Dyks & Rudak, 2000) (that is the foundation of all physics of pulsars) with observations, transforming them from a ”thing in itself” that is not available to direct observations, in the ”thing for us”.
Acknowledgements We are grateful to our colleagues from RI NAS of Ukraine and to participants of PRAO2011, JENAM2011 and NS2011 conferences for discussions, to M. Azbel’, O. Ulyanov and V. Usov for useful comments and to N. Kisilova, Y. Schukin and V. Tsvetkova for their help in translation of the text. The authors are extremely grateful to reviewer for a kindly and qualified critique of this work.
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