---
abstract: 'The theory of magnetized induced scattering off relativistic gyrating particles is developed. It is directly applicable to the magnetosphere of a pulsar, in which case the particles acquire gyration energies as a result of resonant absorption of radio emission. In the course of the radio beam scattering into background the scattered radiation concentrates along the ambient magnetic field. The scattering from different harmonics of the particle gyrofrequency takes place at different characteristic altitudes in the magnetosphere and, because of the rotational effect, gives rise to different components in the pulse profile. It is demonstrated that the induced scattering from the first harmonic into the state under the resonance can account for the so-called low-frequency component in the radio profile of the Crab pulsar. The precursor component is believed to result from the induced scattering between the two states well below the resonance. It is shown that these ideas are strongly supported by the polarization data observed. Based on an analysis of the fluctuation behaviour of the scattering efficiencies, the transient components of a similar nature are predicted for other pulsars.'
author:
- |
    S. A. Petrova [^1]\
    Institute of Radio Astronomy, NAS of Ukraine, 4, Chervonopraporna Str., 61002 Kharkov, Ukraine
date: 'Received…'
title: 'Interpretation of the Low-Frequency Peculiarities in the Radio Profile Structure of the Crab Pulsar'
---

\[firstpage\]

pulsars: general – pulsars: individual (the Crab pulsar) – radiation mechanisms: non-thermal – scattering

Introduction
============

Radio emission components outside of the main pulse
---------------------------------------------------

The Crab pulsar is known for its complex radio profile [e.g. @mh96]. It is built of a total of seven components, which are spread out over the whole pulse period and exhibit substantially distinct spectral and polarization properties. At the lowest frequencies, $\la 600$ MHz, the profile consists of the three components: the main pulse (MP), the precursor (PR) $\sim
15^\circ$ ahead of the MP and the interpulse (IP), which lags the MP by $\sim 150^\circ$ and is connected to it by a weak emission bridge [@r70; @mht72; @v73]. The PR component is distinguished by its complete linear polarization and extremely steep spectrum. At frequencies $\ga 1$ GHz, where the PR has already vanished, there appears another component $\sim 36^\circ$ in advance of the MP [@mh96]. This so-called low-frequency component (LFC) is broader and weaker than the PR. Moreover, the percentage of linear polarization in the LFC is less, $\sim 40\%$ [@mh98], though still markedly exceeds that in the MP and IP ($\sim 25\%$ and $15\%$, respectively).

The IP and LFC become invisible at $\sim 3$ and 5 GHz, and at still higher frequencies the profile structure changes drastically [@mh96; @mh98]. In addition to the MP, there is the interpulse (IP’), which re-appears some $10^\circ$ earlier in phase, and two high-frequency components (HFC1 and HFC2), $\sim 70^\circ$ and $130^\circ$ behind the IP’. All these new components are characterized by high linear polarization and relatively flat spectra, so that at frequencies $\ga 8$ GHz the MP disappears. The fluctuation properties of these components are also worth noting. According to the recent high-frequency studies, all the components show occasional giant pulses [@h05; @slow05], the temporal and frequency structure of the giant MPs and IP’s being essentially different [@hc07].

The components outside of the MP are present in other pulsars as well. About $40\%$ of the millisecond pulsars and $2\%$ of the normal ones are known to exhibit IPs [@k99]. Besides that, a handful of pulsars have firmly established PRs. Note that the PRs are met in the profiles with IPs. Thus, the structure of the Crab profile at the lowest frequencies is similar to that in some other pulsars (e.g., PSR B1055-52, @m76 and PSR B1822-09, @f81). However, in these cases the profile structure is preserved over a wide frequency range, though the component spectra are also somewhat different. Such components of the Crab profile as the LFC, HFC1 and HFC2 as well as the high-frequency shift of the IP to earlier pulse longitudes are unique. It should be noted, however, that some millisecond pulsars have even more complex structure (e.g., PSR J0437-4715, @mj95), but it is not well studied and classified.

The mechanisms of the IP and PR emissions are still a matter of debate, while the nature of other components of the Crab profile is completely obscure. The IP components are usually interpreted in terms of geometrical models. It is assumed that the IP emission originates in a distinct region (e.g. in the outer magnetosphere or at the opposite magnetic pole) and can only be observable due to a specific geometry of the pulsar (in cases of approximate alignment or approximate orthogonality of the rotational and magnetic axes). Recently @d05 have developed a generalized geometrical model for the PSR B1822-09, which includes the formation of the PR as well. It has been suggested that the PR component originates well above the MP and the backward emission from this region forms the IP. Such a pattern can be observable if the pulsar is a nearly orthogonal rotator.

It should be noted that the geometrical models are insufficient to account for the bulk of observational facts. Firstly, the components outside of the MP show peculiar polarization and spectral properties. Secondly, the fluctuation behaviour of the components strongly testifies to their physical connection with the MP. In particular, the subpulse modulation in the MP and IP of the PSR B1702-19 has been found to be intrinsically in phase [@welt07]. All this calls for a physical interpretation.

Recently we have proposed a physical mechanism of the PR and IP components based on propagation effects in pulsar magnetosphere [@p07a; @p07b]. These components are suggested to result from induced scattering of the MP emission into the background. In case of efficient scattering, the scattered radiation grows substantially and concentrates in the direction corresponding to the maximum scattering probability. In the regime of a superstrong magnetic field, the scattered component is directed along the ambient field and can be identified with the PR. In a moderately strong magnetic field, the radiation is predominantly scattered in the opposite direction, giving rise to the IP. Within the framework of this model, the basic features of the components as well as their connection to the MP are explained naturally. Our theory can be elaborated further to explain the complicated radio emission pattern of the Crab pulsar. The present paper is devoted to the generalized mechanism of the two components, the PR and LFC, which precede the MP and develop at relatively low frequencies. The formation of the high-frequency components of the Crab will be addressed in our forthcoming paper. It will be argued that the IP’ results from the backward scattering of the PR, while the HFC1 and HFC2 present the backscattered emission of the LFC.

Statement of the problem
------------------------

The magnetosphere of a pulsar contains the ultrarelativistic electron-positron plasma, which streams outwards along the open magnetic field lines and leaves the magnetosphere as a pulsar wind. The pulsar radio emission is generally believed to originate deep in the open field line tube, and on its way in the magnetosphere it should propagate through the plasma flow. As the brightness temperatures of the pulsar radio emission are extremely high, one can expect that induced scattering off the plasma particles is significant. Deep in the magnetosphere the magnetic field is strong enough to affect the scattering process considerably by modifying both the scattering cross-section and the particle recoil. This happens for the waves below the cyclotron resonance, as long as the frequency in the particle rest frame remains much less than the electron gyrofrequency, $\omega^\prime\ll\omega_G\equiv eB/mc$. The magnetized induced scattering in pulsars is known to be efficient [@bs76] and is suggested to have a number of observational consequences [@lp96; @p04a; @p04b; @p07a; @p07b]. As the magnetic field strength decreases with distance from the neutron star, in the outer magnetosphere the radio waves pass through the resonance. The scattering by the pulsar wind holds in the non-magnetic regime and can also be efficient in pulsars [@wr78; @lp96]. Note that in the resonance region itself the radio waves are subject to resonant absorption rather than scattering [@bs76; @lp98; @p02; @p03; @melr_a; @melr_b].

Close to the neutron star surface, the magnetic field is so strong that any perpendicular momentum of the particles is almost immediately lost via synchrotron re-emission. Hence, the particles are confined to the magnetic field lines, and it is usually assumed that they perform ultrarelativistic rectilinear motion throughout the open field line tube. However, in the outer magnetosphere, where synchrotron re-emission is already inefficient, the particles can easily gain relativistic gyration energies as a result of resonant absorption of the radio emission [@lp98; @p02; @p03]. As has been shown in @p02 [@p03], the absorbing particles reach relativistic gyration at the very bottom of the resonance region for radio waves, in the course of absorption of the highest-frequency waves, $\nu\ga 10$ GHz. Then the lower-frequency waves, $\nu\ll 10$ GHz, which are still below the resonance, $\omega^\prime\ll \omega_G$, are scattered by the particles performing relativistic helical motion.

The spontaneous scattering off a gyrating electron in application to pulsars has recently been studied in @p07c. The process has been found to differ substantially from the scattering by the particle at rest. The total cross-section of the former process appears much larger, and the scattered waves predominantly concentrate at high harmonics of the gyrofrequency, close to the maximum of the synchrotron emission of the scattering electron, whereas the scattering by the particle at rest leaves the wave frequency unchanged. Although the spontaneous scattering by gyrating particles appears weak enough to markedly suppress pulsar radiation, except for the lowest frequencies, $\nu <100$ MHz, it supplements the synchrotron re-emission in reprocessing the radio photons into the high-energy emission of the optical or soft X-ray band.

All the previous studies of induced scattering in pulsars have assumed rectilinear motion of the scattering particles. It has been demonstrated that even in the regime of a superstrong magnetic field the induced scattering is most efficient well above the emission region, at distances roughly comparable with the radius of cyclotron resonance [@p07a]. Therefore the gyration of the scattering particles needs to be taken into account. In the present paper, we examine the induced scattering off the spiraling particles. In contrast to the spontaneous scattering, the induced scattering between the states corresponding to close harmonics of the gyrofrequency is more efficient. We consider the induced scattering between the zeroth-harmonic states, when the incident and final frequencies are both below the resonance, study the scattering from several first harmonics to the zeroth one and conclude that these processes can be efficient in pulsars. The kinetic equations derived are applied to the problem of the radio beam scattering into the background. It is found that the radiation is predominantly scattered along the ambient magnetic field. Since the scattering regions lie at different altitudes above the emission region, the rotational aberration places the scattered components at different pulse longitudes ahead of the MP. Proceeding from the numerical estimate of the radius of cyclotron resonance in the Crab pulsar we show that the PR component should result from the scattering under the resonance and the LFC from the first-harmonic scattering.

The plan of the paper is as follows. Basic formalism of induced scattering off the spiraling particles is developed in Sect. 2. The problem on the radio beam scattering into the background is examined in Sect. 3. Applications to the the Crab pulsar are presented in Sect. 4. Our results are discussed in Sect. 5 and briefly summarized in Sect. 6.

Basic formalism of induced scattering
=====================================

Let us consider the induced scattering of transverse electromagnetic waves by the system of particles in the presence of an external magnetic field. The scattering particles are assumed to perform relativistic helical motion. The rate of change of the photon occupation number $N$ as a result of the scattering is given by $$\frac{\partial N}{\partial t}=n_e\int wNN_1\frac{{\rm d}^3{\bmath
k_1 }}{(2\pi)^3}\frac{\partial f}{\partial{\bmath p}}\Delta{\bmath
p}{\rm d^3}{\bmath p},$$ where $N=N(\bmath k)$, $N_1=N_1(\bmath k_1)$, $\bmath k$ and $\bmath k_1$ are the photon wavevectors in the initial and final states, $w=w({\bmath k},{\bmath k_1},{\bmath p})$ is the scattering probability, $n_e$ is the number density of the scattering particles, $f(\bmath p)$ is the particle distribution function in momenta normalized as $\int f(\bmath p){\rm
d}^3{\bmath p}=1$, $\Delta\bmath p$ is the momentum increment in the scattering act. In case of the scattering by the particles at rest, the recoil is extremely small, $\Delta k/k\sim \hbar k/mc<<<
1$, so that the photon frequency is approximately unaltered in the scattering act, $\omega_1^{(\rm r)}\approx \omega^{(\rm r)}$. In case of the scattering by the spiraling particles, the situation is essentially distinct: The frequencies of the scattered photons present a discrete set. In the guiding centre frame, $$\omega_1^{(\rm c)}=\omega^{(\rm c)}+s\omega_H/\gamma_0,\quad
s=0,\pm 1,\pm 2,\dots ,$$ where $\gamma_0$ is the Lorentz-factor of the circular motion of the particle, $\gamma_0\equiv (1-\beta_0^2)^{-1/2}$, $\beta_0$ is the particle velocity in units of $c$. In the laboratory frame, equation (2) is written as $$\omega_1\gamma_\Vert\eta_1=\omega\gamma_\Vert\eta+s\omega_H/\gamma_0,\quad
s=0,\pm 1,\pm 2,\dots ,$$ where $\eta=1-\beta_\Vert\cos\theta$, $\eta_1=1-\beta_\Vert\cos\theta_1$, $\beta_\Vert$ is the longitudinal component of the particle velocity normalized to $c$, $\theta$ and $\theta_1$ are the tilts of the initial and final wavevectors to the magnetic field, $\gamma_\Vert\equiv(1-\beta_\Vert^2)^{-1/2}$.

For a given harmonic of the gyrofrequency, $s$, the scattering probability is related to the differential cross-section per elementary solid angle ${\rm d}O_1$ as $$w_s=(2\pi)^3c^4\frac{\omega}{\omega_1^3}\frac{{\rm
d}\sigma_s}{{\rm
d}O_1}\delta(\omega_1-\omega\eta/\eta_1-s\omega_H/\gamma\eta_1)$$ and the relativistic transformation of the cross-section from the guiding-centre frame reads $$\frac{{\rm d}\sigma}{{\rm d}O_1}=\left (\frac{{\rm d}\sigma}{{\rm
d }O_1}\right )^{({\rm c})}\frac{\eta^2}{\gamma_\Vert^2\eta_1^3}$$ [for more details see @p07c]. In equation (4) above, $\gamma$ stands for the total Lorentz-factor of the particle, and using the invariance of the transverse momentum, $p_\perp=\beta_\perp\gamma mc=\beta_0\gamma_0mc$, it is easy to see that $\gamma=\gamma_0\gamma_\Vert$. A general form of the scattering cross-section in the guiding centre frame is given by eqs. (11)-(12) in @p07c. It actually includes four cross-sections, which correspond to one of the two possible polarizations of the photons in the initial and final states. The photons are polarized either in the plane of the wavevector and the ambient magnetic field (A-polarization) or perpendicularly to this plane (B-polarization).

Given that $\beta_\Vert=0$, $s=0$ and $\omega=\omega_1\ll
\omega_H/\gamma_0$, the cross-sections are reduced to [see also eq. (17) in @p07c] $$\frac{{\rm d}\sigma_0^{AA}}{{\rm
d}O_1}=\frac{r_e^2\sin^2\theta\sin^2\theta_1}{\gamma_0^2},$$ $$\frac{{\rm d}\sigma_0^{AB}}{{\rm
d}O_1}=\frac{r_e^2\omega^2}{\omega_H^2}\left
[\cos\theta\cos\Delta\phi -\frac{\beta_0^2\sin\theta
\sin\theta_1\cos\theta_1}{2}\right ]^2,$$ $$\frac{{\rm d}\sigma_0^{BA}}{{\rm
d}O_1}=\frac{r_e^2\omega^2}{\omega_H^2}\left
[\cos\theta_1\cos\Delta\phi -\frac{\beta_0^2\sin\theta
\sin\theta_1\cos\theta}{2}\right ]^2,$$ $$\frac{{\rm d}\sigma_0^{BB}}{{\rm
d}O_1}=\frac{r_e^2\omega^2}{\omega_H^2}\sin^2\Delta\phi,$$ where $\Delta\phi$ is the difference of the azimuthal coordinates of the wavevectors in the initial and final states. Besides that, we are interested in the induced scattering between the states, one of which still corresponds to the low frequency, $\omega_1\ll\omega_H/\gamma_0$, while in another one $\omega\approx s\omega_H/\gamma_0$. The induced scattering is believed to be efficient for small enough $s$, $s\sim 1$, and the scattering between the states with $\omega\sim n\omega_H/\gamma_0$ and $\omega_1\sim l\omega_H/\gamma_0$, where $n,l\gg 1$, should be much weaker. In the case of interest, the cross-sections take the form $$\frac{{\rm d}\sigma_s^{AA}}{{\rm
d}O_1}=\frac{r_e^2}{\gamma_0^2}J_s^2(s\beta_0\sin\theta)
\sin^2\theta\sin^2\theta_1\cot^4\theta,$$ $$\frac{{\rm d}\sigma_s^{AB}}{{\rm
d}O_1}=\frac{r_e^2\omega_1^2}{\omega_H^2}J_s^2(s\beta\sin\theta)\cot^4\theta$$ $$\times\left\{\sin\theta\cos\Delta\phi+\sin\theta_1\left [
1-\beta_0^2(1-\cos\theta\cos\theta_1)/2\right ]\right\}^2,$$ $$\frac{{\rm d}\sigma_s^{BA}}{{\rm
d}O_1}=\frac{r_e^2}{\gamma_0^2}J_s^{\prime^2}(s\beta_0\sin\theta)
\beta_0^2\cos^2\theta\sin^2\theta_1,$$ $$\frac{{\rm d}\sigma_s^{BB}}{{\rm
d}O_1}=\frac{r_e^2\beta_0^2\omega_1^2}{\omega_H^2}J_s^{\prime^2}(s\beta\sin\theta)$$ $$\times\left\{\sin\theta\cos\Delta\phi+\sin\theta_1\left [
1-\beta_0^2(1-\cos\theta\cos\theta_1)/2\right ]\right\}^2.$$ For the inverse scattering, $\omega\ll\omega_H/\gamma_0$, $\omega_1\approx s\omega_H/\gamma_0$, the cross-sections are given by eq. (18) in @p07c. Comparing those results with equation (7) above, one can see that the scattering probability (4) is symmetrical with respect to the initial and final photon states. This is known to be the fundamental property of this quantity.

The momentum increment $\Delta\bmath p$ can be found from the conservation laws in the elementary scattering act. In the presence of an external magnetic field, the energy and the momentum component parallel to the field are conserved, $$\Delta\gamma mc^2=\hbar (\omega-\omega_1),$$ $$\Delta p_\Vert=\hbar (k\cos\theta-k_1\cos\theta_1).$$ Using equation (3) and differentiating the relation $\gamma^2m^2c^4\equiv m^2c^4+(p_\perp^2+p_\Vert^2)c^2$, one can find that $$\Delta p_\perp =-s\hbar\omega_H\frac{m}{p_\perp}.$$ Thus, the perpendicular momentum is changed if $s\neq 0$.

Induced scattering of pulsar radio beam into background
=======================================================

Now let us concentrate on the problem of induced scattering into background based on the general theory presented above. Pulsar radiation is known to be highly directional: At any point of the emission cone it is concentrated into a beam of the opening angle $\la 1/\gamma$, which is typically much less than the angular width of the emission cone. Therefore at any point of the scattering region the incident radiation can be approximately presented by a single wavevector $\bmath k$. Far enough from the emission region, the radiation propagates quasi-transversely with respect to the ambient magnetic field, $1/\gamma\ll\theta\la 1$.

Since the rate of induced scattering depends on the photon occupation number in the final state, the scattering out of the beam is impossible unless there are some background photons which trigger the scattering process. Such photons should indeed be present. In particular, they may result from the spontaneous scattering from the radio beam. Although the occupation numbers of the background are extremely small, they may still stimulate efficient induced scattering from the beam. Then the beam photons are predominantly scattered into the state $\bmath k_1$ corresponding to the maximum scattering probability, and the photon occupation number in this state may become comparable with the original occupation number of the beam photons. Thus, we come to the problem on induced scattering between the two photon states, $\bmath k$ and $\bmath k_1$; the parameters of the background state corresponding to the maximum scattering probability will be specified below based on an analysis of the concrete kinetic equations.

The case $s=0$
--------------

We start from the case when the frequency in the guiding centre frame remains unchanged in the scattering act, i.e. $s=0$ and $\omega\eta=\omega_1\eta_1$. Then $\Delta p_\perp =0$ and the scattering is analogous to that off the rectilinearly moving particles. The latter has been examined in @p07a, and now we are interested if there are any quantitative differences between the two processes. It is convenient to replace the photon occupation numbers $N$ and $N_1$ with the intensities $i_\nu$ and $i_{\nu_1}$ defined as $$i_\nu (\nu,\theta,\phi){\rm d}\nu{\rm d}O\equiv
2\hbar\omega cN({\bmath k})\frac{{\rm d}^3{\bmath k}}{(2\pi)^3},$$ $$i_{\nu_1} (\nu_1,\theta_1,\phi_1){\rm d}\nu_1{\rm d}O_1\equiv
2\hbar\omega_1 cN({\bmath k_1})\frac{{\rm d}^3{\bmath
k_1}}{(2\pi)^3},$$ and to perform trivial integration over the solid angle, introducing the spectral intensities $I_\nu\equiv\int i_\nu{\rm
d}O$ and $I_{\nu_1}\equiv\int i_{\nu_1}{\rm d}O_1$. The scattering is assumed to be stationary: The intensity varies only on account of the beam propagation through the scattering region, whereas the parameters of the scattering process do not depend on time, and therefore $\partial /\partial t=c{\rm d}/{\rm d}r$.

Recall that the scattering cross-sections for gyrating particles given by equation (6) are obtained in the approximation $\omega^{(\rm c)}\ll\omega_H/\gamma_0$. One can see that $\left
({\rm d }\sigma^{AB}_0/{\rm d}O_1,{\rm d }\sigma^{BA}_0/{\rm
d}O_1,{\rm d }\sigma^{BB}_0/{\rm d}O_1\right )/\left ({\rm d
}\sigma^{AA}_0/{\rm d}O_1\right )
\sim\omega^{(c)^2}\gamma_0^2/\omega_H^2\ll 1$, i.e. the scattering in the polarization channel $A\to A$ strongly dominates those in the other channels. Note that the same polarization signature is also characteristic of the scattering by the particle at rest.

The cross-section ${\rm d }\sigma^{AA}_0/{\rm d}O_1$ can be rewritten in terms of the quantities of the laboratory frame using the relativistic transformations $\sin\theta^{(\rm
c)}=\sin\theta/\gamma_\Vert\eta$ and $\sin\theta_1^{(\rm
c)}=\sin\theta_1/\gamma_\Vert\eta_1$. Then, substituting equations (4) and (5) into equation (1) and performing integration over the wavenumber with the help of delta-function, one can obtain $$\frac{{\rm d}I_\nu}{{\rm d}r}=I_\nu
I_{\nu_1}r_e^2n_ec\int\frac{\cos\theta
-\cos\theta_1}{2\nu_1^2\eta_1^2}\frac{\sin^2\theta\sin^2\theta_1}
{\eta^2\eta_1^2}$$ $$\times\frac{m^2c^2p_\perp^4}{p_\Vert^6}\frac{\partial f }{\partial
p_\Vert}{\rm d}p_\Vert p_\perp{\rm d}p_\perp .$$ Here it is taken into account that $\gamma_\Vert=\gamma/\gamma_0$, $p_\Vert =\beta_\Vert\gamma mc\approx \gamma mc$ and $p_\perp
=\beta_0\gamma_0mc\approx\gamma_0mc$. For the inverse scattering we find analogously $$\frac{{\rm d}I_{\nu_1}}{{\rm d}r}=I_\nu
I_{\nu_1}r_e^2n_ec\int\frac{\cos\theta_1
-\cos\theta}{2\nu^2\eta^2}\frac{\sin^2\theta\sin^2\theta_1}
{\eta^2\eta_1^2}$$ $$\times\frac{m^2c^2p_\perp^4}{p_\Vert^6}\frac{\partial f }{\partial
p_\Vert}{\rm d}p_\Vert p_\perp{\rm d}p_\perp .$$ Keeping in mind that $\nu\eta=\nu_1\eta_1$, one can see that the right-hand sides of equations (11) and (12) are equal in absolute value and have opposite signs. Performing integration by parts and noticing that the scattering probability peaks at $\theta_1^{\rm
max }\sim 1/\gamma_\Vert$, we come to the system $$\frac{{\rm d}I_\nu}{{\rm d}r}=-a_0I_\nu
I_{\nu_1},$$ $$\frac{{\rm d}I_{\nu_1}}{{\rm d}r}=a_0I_\nu I_{\nu_1},$$ where $$a_0\sim\frac{24n_er_e^2\gamma_0^2}{m\nu^2\theta^4\gamma^5},$$ $\gamma$ and $\gamma_0$ stand for the characteristic values of these quantities for a given particle distribution. It should be noted that the system (13) along with equation (14) are literally the same as those for the scattering off the rectilinearly moving particles, except for the factor $\gamma_0^2$ entering equation (14) [cf. eqs. (8)-(9) in @p07a]. Generally speaking, one can write that $\gamma_0^2/\gamma^5=1/\gamma_\Vert^5\gamma_0^3$ and conclude that the scattering by spiraling particles with the longitudinal Lorentz-factor $\gamma_\Vert$ is $\gamma_0^3$ times less efficient than that by the particles streaming along the magnetic field with the same Lorentz-factor $\gamma_\Vert$. However, in pulsar case the situation is somewhat different. As has been shown in @lp98, the synchrotron absorption, which determines the evolution of the particle distribution function, acts to slow down the longitudinal motion and increase the transverse momenta in such a way that $\gamma=\gamma_0\gamma_\Vert$ keeps constant as long as the particle pitch-angle is small enough, $\gamma_0/\gamma <\theta$. Thus, it is reasonable to compare the scatterings by the original and evolved distributions of particles, in which case $\gamma$ is the same. Then the spiraling particles scatter $\gamma_0^2$ times more efficiently than the streaming particles. Note also that in the two processes the directions of predominant scattering of the radio beam photons differ, $\theta_1^{\rm max}\sim
1/\gamma_\Vert=\gamma_0/\gamma$ and $\theta_1^{\rm max}\sim
1/\gamma$, respectively, implying different relations between the interacting frequencies, $\nu_1\sim\nu\theta^2\gamma^2/\gamma_0^2$ and $\nu_1\sim\nu\theta^2\gamma^2$.

The system (13) has the first integral, $$I\equiv I_\nu+I_{\nu_1}={\rm const},$$ and the solution is written as $$I_\nu=\frac{I/x}{1+1/x},$$ $$I_{\nu_1}=\frac{I}{1+1/x},$$ where $x\equiv [I_{\nu_1}^{(0)}/I_\nu^{(0)}]\exp (\Gamma_0)$ and $\Gamma_0\equiv Ia_0r$. The latter quantity characterizes the scattering efficiency, whereas the former one the extent of intensity transfer from the radio beam to the background. As long as $x\ll 1$, $I_{\nu_1}\sim I_{\nu_1}^{(0)}\exp (\Gamma_0)$ and $I_\nu\approx I_\nu^{(0)}$. At $x\sim 1$ $I_{\nu_1}$ becomes comparable with $I_\nu^{(0)}$ and enters the stage of saturation. Given that $x\gg 1$, $I_{\nu_1}\approx I_\nu^{(0)}$ and $I_\nu\sim
I_\nu^{(0)}/x$. Since initially the intensity ratio of the background and the beam is extremely small, $I_{\nu_1}^{(0)}/I_\nu^{(0)}<<<1 $, $\Gamma_0\sim n\times 10$ is necessary to provide $x\sim 1$, in which case a substantial part of the beam intensity is transferred to the background. As has been shown in @p07a, in case of the scattering by the streaming particles this condition may well be satisfied in pulsars. For the scattering by the spiraling particles $\Gamma_0$ is $\gamma_0^2$ larger and, correspondingly, the growth of the scattered component should be even more significant.

The case $\omega^{(\rm c)}\sim s\omega_H/\gamma_0$, $\omega_1^{(\rm c)}\ll \omega_H/\gamma_0$
---------------------------------------------------------------------------------------------

Now we turn to induced scattering between such two states that one of the frequencies is close to the resonance, $\omega\gamma\eta\sim s\omega_H$, and another one is well below the resonance, $\omega_1\gamma\eta_1\equiv\omega\gamma\eta
-s\omega_H\ll\omega_H$. The intensity evolution in the two states is given by $$\frac{{\rm d}I_\nu^i}{{\rm d}r}=I_\nu^iI_{\nu_1}^j\frac{n_ec}{2}
\int\frac{\eta^2\nu}{\eta_1^3\gamma_\Vert^2\nu_1^4}\left
(\frac{{\rm d}\sigma_{\nu\to\nu_1}^{ij}}{{\rm d}O_1}\right )^{(\rm
c )}\frac{s\omega_H}{2\pi}$$ $$\times\left [\frac{\partial f}{\partial p_\perp}\frac{mc}{p_\perp}+\frac{\partial f}
{\partial p_\Vert}\frac{mc\cos\theta}{p_\Vert\eta}\right ]p_\perp
{\rm d}p_\perp{\rm d}p_\Vert,$$ $$\frac{{\rm d}I_{\nu_1}^j}{{\rm
d}r}=-I_\nu^iI_{\nu_1}^j\frac{n_ec}{2}
\int\frac{\eta_1^2\nu_1}{\eta^3\gamma_\Vert^2\nu^4}\left
(\frac{{\rm d}\sigma_{\nu_1\to\nu}^{ji}}{{\rm d}O}\right )^{(\rm c
)}\frac{s\omega_H}{2\pi}$$ $$\times\left [\frac{\partial f}{\partial
p_\perp}\frac{mc}{p_\perp}+\frac{\partial f} {\partial
p_\Vert}\frac{mc\cos\theta}{p_\Vert\eta}\right ]p_\perp {\rm
d}p_\perp{\rm d}p_\Vert,$$ where the subscripts $i,j$ denote the polarization states of the initial and final photons, $({\rm d}\sigma_{\nu\to\nu_1}^{ij}/{\rm
d}O_1)^{(\rm c )}$ is the cross-section of the scattering from $\nu$ to $\nu_1$ in the guiding centre frame (it is given by eq. (7) and should be expressed via the quantities of the laboratory system), the components of the momentum increment are given by equations (8) and (9) and it is taken into account that $\omega_1\gamma\eta_1\ll\omega_H,\omega\gamma\eta$.

The symmetry of the scattering probability (4) with respect to direct and inverse scatterings implies that $({\rm
d}\sigma_{\nu_1\to\nu}^{ji}/{\rm d}O)^{(\rm c )}=(\nu^4/\nu_1^4)
({\rm d}\sigma_{\nu\to\nu_1}^{ij}/{\rm d}O_1)^{(\rm c )}$, and one can see that $\vert{\rm d}I_\nu^i/{\rm d}r\vert=\alpha\vert{\rm
d}I_{\nu_1}^j/{\rm d}r\vert$, where $\alpha\equiv\nu\eta/\nu_1\eta_1\gg 1$. Comparing the increments of the transverse and longitudinal momenta we find that $\Delta
p_\perp\partial f/\partial
p_\perp\sim(\theta^2p_\Vert^2/p_\perp^2)\Delta p_\Vert\partial
f/\partial p_\Vert$. In our consideration $p_\perp/p_\Vert\ll\theta$, i.e. the particle pitch-angle is much less than the photon propagation angle, and hence the contribution of the longitudinal increment can be neglected. Note that $p_\perp$ enters equation (17) via $f$ as well as via the argument $s\xi$ of the Bessel function and its derivative in the cross-sections (7), $\xi\equiv\beta_0\sin\theta^{(c)}=p_\perp\sin\theta/p_\Vert\eta\sim
2p_\perp/p_\Vert\theta\ll 1$. One can see that integration of the kinetic equations (17) over $p_\perp$ by parts results in a change of the sign of the right-hand sides. Thus, in the course of the scattering the photons are mainly transferred from high harmonics, $\nu\sim s\omega_H/2\pi\gamma\eta$, to the zeroth one, $\nu_1\ll\omega_H/2\pi\gamma\eta_1$.

The harmonic number $s$ enters ${\rm d}I_\nu/{\rm d}r$ via $s^2J_s^2(s\xi)$ or $s^2J_S^{\prime^2}(s\xi)$, which peak at high harmonics, $s^{\rm max }\sim\gamma_0^3$. Note, however, that inside the light cylinder $s^{\rm max}$ corresponds to the frequencies lying in the optical or soft X-ray range [see, e.g., @p07c], and the intensity of pulsar radiation in these bands is much weaker than the radio intensity. Therefore the induced scattering from high harmonics, $s\sim s^{\rm max}$, is expected to be inefficient. In the present consideration, we leave aside this problem and concentrate on the induced scattering of the radio emission. As the magnetic field strength decreases with distance from the neutron star, a given radio frequency passes through the resonances of increasingly higher order. At the same time, the number density of the scattering particles and the radio intensity strongly decrease with distance, so that the scattering becomes much weaker.

Thus, we are interested in the radio beam scattering at several first harmonics of the gyrofrequency, which takes place within the light cylinder. Then $s\xi$ is still a small quantity and one can take approximately that $$J_s(s\xi)\approx\frac{1}{s!}\left (\frac{s\xi}{2}\right )^s,\quad
J_s^{\prime}(s\xi)\approx\frac{1}{2(s-1)!}\left
(\frac{s\xi}{2}\right )^{s-1}.$$ Performing integration in equation (17), taking into account that $\beta_0\approx 1$, $\theta\ll 1$, $\cos\theta^{(\rm
c)}=(\cos\theta -\beta_\Vert)/\eta\approx -1$, $\Delta\phi^{(\rm c
)}=\Delta\phi$ and noticing that $\theta_1^{\rm
max}\sim\gamma_0/\gamma$, one can come to the system $$\frac{{\rm d}I_\nu^i}{{\rm d}r}=-\alpha a_s^{ij}I_\nu^iI_{\nu_1}^j,$$ $$\frac{{\rm d}I_{\nu_1}^j}{{\rm d}r}=a_s^{ij}I_\nu^iI_{\nu_1}^j,$$ with $$a_s^{AA}=a_s^{BA}=\frac{r_e^2n_e}{8m\gamma^3\nu_1^2\theta^2}
\left (\frac{\omega_H/2\pi\gamma}{\nu_1\eta_1}\right
)\frac{s^{2s+2}}{(s!)^2}\left (\frac{\gamma_0}{\gamma\theta}\right
)^{2s-6},$$ $$a_s^{AB}=a_s^{BB}$$ $$=\frac{r_e^2n_e}{64m\gamma^3\nu_1^2\theta^2}
\left (\frac{\nu_1\eta_1}{\omega_H/2\pi\gamma}\right
)\frac{(4+2s)s^{2s+1}}{(s!)^2}\left
(\frac{\gamma_0}{\gamma\theta}\right )^{2s-6}.$$ One can see that $a_s^{AA}/a_s^{AB}=a_s^{BA}/a_s^{BB}\sim
(\omega_H/2\pi\gamma\nu_1\eta_1)^2\gg 1$, i.e. the scattering into the state with A-polarization is much more intense. Hence, the scattered component should be dominated by A-polarization, similarly to the scattering between the frequencies below the resonance (see Sect. 3.1). However, in contrast to that case, the initial intensities in the two polarizations are affected identically.

The general forms of the systems (13) and (19) differ by the presence of the factor $\alpha$ in the latter one. Taking notice that the substitution $\widetilde{I}_{\nu_1}=\alpha I_{\nu_1}$ makes equation (19) similar to equation (13), we immediately write the first integral, $$I\equiv I_\nu +\alpha I_{\nu_1}={\rm const},$$ and the solution $$I_\nu=\frac{I/y}{1+1/y},\quad I_{\nu_1}=\frac{I/\alpha}{1+1/y},$$ where $y\equiv[\alpha I_{\nu_1}^{(0)}/I_\nu^{(0)}]\exp (\Gamma_s)$ and $\Gamma_s\equiv Ia_sr$. One can see that the intensity transfer becomes significant at $y\ga 1$. Although the qualitative character of the solution (22) is the same as that in case of the scattering below the resonance, the quantitative difference is substantial. The maximum possible intensity of the scattered component appears much less, $I_{\nu_1}\sim I_\nu^{(0)}/\alpha\ll
I_\nu^{(0)}$, the rest of the energy being deposited to the scattering particles. Note that the scattered intensity may still be comparable with the original radio beam intensity at the same frequency $\nu_1$. Indeed, the conditions $\nu_1\ll\nu\eta/\eta_1\sim\nu\theta^2\gamma^2/\gamma_0^2$ and $\theta^2\gamma^2/\gamma_0^2\gg 1$ allow that $\nu_1\gg \nu$. Then, with the decreasing spectrum of the radio beam, even a small part of the beam intensity $I_\nu(\nu)$, which is transferred to the background, may be strong enough as compared to $I_\nu(\nu_1)$.

Application to the Crab pulsar
==============================

Now we turn to numerical estimates of the scattering efficiency and start from the scattering between the frequencies below the resonance. The growth of the scattered component at the frequency $\nu_1$ is characterized by the quantity $\Gamma_0=I_\nu^{(0)}a_0r$, where it is taken that $I\equiv
I_\nu^{(0)}+I_{\nu_1}^{(0)}\approx I_\nu^{(0)}$ and $a_0$ is given by equation (14) with $\nu=2\nu_1/\theta^2\gamma_\Vert^2$. The spectral intensity of the pulsar radio beam can be presented as $$I_\nu^{(0)}=I_{\nu_0}\left (\frac{\nu}{\nu_0}\right )^{-\alpha}.$$ Here $\nu_0\sim 10^8$ Hz and $$I_{\nu_0}=\frac{L}{\nu_0S},$$ where $L$ is the total radio luminosity of the pulsar, $S=\pi
r^2w^2/4$ is the cross-section of the pulsar beam at a distance $r$ and $w$ is the pulse width in the angular measure. The number density of the scattering particles can be written in terms of the multiplicity factor of the plasma, $\kappa$, $$n_e=\frac{\kappa B}{Pce},$$ where $P$ is the pulsar period. With the dipolar geometry of the magnetic field, $B\propto r^{-3}$, the scattering efficiency can be estimated as $$\Gamma_0=25\frac{0.1\,{\rm s}}{P}\frac{L}
{10^{30}\,{\rm erg\,s}^{-1}}\frac{B_\star}{10^{12}\,{\rm G}}
\frac{\kappa}{10^2}\left (\frac{10^9\,{\rm Hz}}{\nu_1}\right )^2
\left (\frac{\nu}{10^8\,{\rm Hz}}\right )^{-\alpha}$$ $$\times\left (\frac{w}{0.4}\right )^{-2}\left (\frac{r}{10^8\,{\rm
cm }}\right )^{-4}\left (\frac{\gamma}{10^2}\right )^{-5}\left
(\frac{\gamma_\Vert}{10}\right )^4\gamma_0^2,$$ where $B_\star$ is the magnetic field strength at the surface of the neutron star and the radius of the star is taken to be $10^6$ cm. The scattered component grows significantly provided that $x\equiv [I_{\nu_1}^{(0)}/I_\nu^{(0)}]\exp (\Gamma_0)\sim 1$. As has been shown in @p07a, this condition is satisfied for $\Gamma_0\sim$ 20–30. From equation (26) above one can conclude that such values of $\Gamma_0$ can indeed be characteristic of the Crab pulsar.

It should be noted that the pulsar radio beam is broadband and its angle of incidence increases with distance from the neutron star, $\theta\propto r$. Therefore at different altitudes $r$ the background component of a given frequency $\nu_1=\nu\theta^2(r)\gamma_\Vert^2/2$ is fed by the radiation of different frequencies $\nu\propto r^{-2}$. The number density of the scattering particles and the incident intensity entering $\Gamma_0$ are known to decrease with distance, $n_e\propto
r^{-3}$ and $I_\nu^{(0)}\propto r^{-2}$. However, at larger altitudes the feeding frequency $\nu$ is lower, $I_\nu^{(0)}$ is much larger, and for steep enough spectrum of the radio beam the scattering is more efficient. In case of the Crab pulsar $\alpha\approx 3$, so that $\Gamma_0\propto r^2$ and in the course of the scattering the intensity is transferred to the background from the lowest frequencies, the process taking place at distances of order of the cyclotron resonance radius.

The efficiency of the scattering from several first harmonics of the gyrofrequency, $\Gamma_s=I_\nu^{(0)}a_sr$, can be estimated analogously. Substituting equations (23)-(25) into the first line of equation (20), we obtain $$\Gamma_s=0.5\frac{s^{2s+2}}{(s!)^2}\frac{\omega_H/2\pi}
{\nu_1\eta_1\gamma}\left (\frac{\gamma_0}{\gamma\theta}\right
)^{2s-4}\frac{0.1\,{\rm s}}{P}\frac{L} {10^{30}\,{\rm
erg\,s}^{-1}}$$ $$\times\frac{B_\star}{10^{12}\,{\rm G}}
\frac{\kappa}{10^2}\left (\frac{10^9\,{\rm Hz}}{\nu_1}\right )^2
\left (\frac{\nu}{10^8\,{\rm Hz}}\right )^{-\alpha}$$ $$\times\left (\frac{w}{0.4}\right )^{-2}\left (\frac{r}{10^8\,{\rm
cm }}\right )^{-4}\left (\frac{\gamma}{10^2}\right )^{-5}\left
(\frac{\gamma_\Vert}{10}\right )^4\gamma_0^2.$$ The scattering of the lowest frequencies of the pulsar radio beam is most efficient and it occurs just beyond the corresponding resonance, at the distance satisfying the condition $\nu\theta^2\gamma/2=s\omega_H/2\pi+\nu_1\gamma/\gamma_\Vert^2$ ($\nu_1\gamma/\gamma_\Vert^2\ll\nu\theta^2\gamma/2,\,s\omega_H/2\pi$).

Comparison of equations (26) and (27) shows that $$\frac{\Gamma_s}{\Gamma_0}\sim\frac{\nu_s\eta}
{\nu_{1_s}\eta_1}\left (\frac{\gamma_0}{\gamma\theta}\right
)^{2s-4}\left (\frac{\nu_0}{\nu_s}\right )^\alpha\left
(\frac{\nu_{1_0}}{\nu_{1_s}}\right )^2,$$ where $\nu_s$ and $\nu_{1_s}$ are the frequencies involved in the scattering at the harmonics of the gyrofrequency, whereas $\nu_0$ and $\nu_{1_0}$ participate in the scattering below the resonance. Taking into account the relations $\nu_0\eta=\nu_{1_0}\eta_1$ and $\nu_s\eta\gg\nu_{1_s}\eta_1$, one can see that for the scattering of the same frequency, $\nu_s=\nu_0$, $\Gamma_s/\Gamma_0\sim(\nu_s\eta/\nu_{1_s}\eta_1)^3
(\gamma_0/\gamma\theta)^{2s-4}$. As $\gamma_0/\gamma\theta\ll 1$, we conclude that the scattering from several first harmonics of the gyrofrequency (at least for $s=1,2$) may dominate the scattering below the resonance. It should be kept in mind, however, that these scatterings take place at different altitudes in the magnetosphere, below and above the resonance, and because of the spatial dependence of $n_e$ and $I_\nu^{(0)}$ the scattering at the harmonics of the gyrofrequency is somewhat less efficient.

The original radio beam is known to be directed along the magnetic field in the emission region. Because of the magnetosphere rotation, at large enough altitudes $r$ it makes the angle $\sim
r/2r_L$ with the local magnetic field direction (here $r_L=5\times
10^9P$ cm is the light cylinder radius). In the scattering regimes considered in the present paper, the radio beam photons are predominantly scattered along the ambient magnetic field in the frame corotating with the neutron star. The rotational aberration in the scattering region shifts the wavevector of the scattered radiation by $\sim r/r_L$ in the direction of the magnetosphere rotation, so that in the laboratory frame it makes the angle $\sim
r/2r_L$ with the original radio beam. Correspondingly, in the pulse profile the scattered component precedes the MP in pulse longitude by $\Delta\lambda\sim r/2r_L$, where $r$ is the characteristic altitude of the scattering region [for more details see @p07a]. As the radio beam scattering at the harmonics of the gyrofrequency occurs at larger altitudes in the magnetosphere as compared to the scattering below the resonance, it should result in the components with larger separations from the MP.

To have a notion about the component location in the pulse profile let us estimate the radius of cyclotron resonance, $r_c$. Proceeding from the resonance condition $\nu\gamma\theta^2/2=\omega_H/2\pi$ and taking into account that $\omega_H\propto B\propto r^{-3}$ and $\theta\approx r/2r_L$, we find $$\frac{r_c}{r_L}=\left (\frac{0.1\,{\rm s}}{P}\right )^{3/5} \left
(\frac{B_\star}{10^{12}\,{\rm G}}\frac{10^9\,{\rm Hz}}{\nu}
\frac{10^2}{\gamma}\right )^{1/5}.$$ One can see that in the Crab pulsar the region of cyclotron resonance of the radio frequencies lies close to the light cylinder radius. Although for low enough frequencies the resonance is slightly beyond $r_L$, we assume that the estimate $\Delta\lambda\sim r/2r_L$ is still roughly applicable. The location of the LFC $\sim 30^\circ$ ahead of the MP implies that the scattering region is close to the light cylinder. As the PR precedes the MP by $\sim 15^\circ$, it should originate in the outer magnetosphere. Thus, we conclude that the PR component results from the scattering below the resonance, whereas the LFC arises in the course of the first-harmonic scattering.

Discussion
==========

We have examined the process of magnetized induced scattering off spiraling particles. Our consideration is restricted to the scattering of radio frequencies in the magnetic fields well below the critical value. In contrast to the classical problem on the scattering by rectilinearly moving particles, in our case the scattering at the harmonics of the particle gyrofrequency is not negligible. In the course of spontaneous scattering off the spiraling particles, the waves of the frequencies well below the first resonance are mainly scattered to high harmonics, into the range of the spectral maximum of the particle synchrotron emission [@p07c]. The induced scattering, however, appears most efficient for the pairs of states corresponding to close harmonics. We have concentrated at the low-frequency scattering, in which case one of the frequencies is well below the resonance and another one corresponds to one of the several first harmonics. It has been shown that the scattering in these regimes may dominate the scattering between the pair of frequencies below the resonance. The latter process is analogous to the scattering off the rectilinearly moving particles, but it appears $\sim
\gamma_0^2$ times more efficient provided that the scattering particles have the same total Lorentz-factor $\gamma$.

In application to pulsars, we are interested in the induced scattering of a narrow radio beam into the background. Given that the scattering is efficient, the photons are mainly scattered along the ambient magnetic field, in the direction corresponding to the maximum scattering probability, and the scattered component may grow roughly as large as the original radio beam. In the course of the scattering, the intensity is transferred from the harmonics of the gyrofrequency into the state below the resonance. The numerical estimates show that the scattering from several first harmonics and between the frequencies below the resonance can be substantial, especially for low enough frequencies of the radio beam, in which case the incident intensity is the largest because of the decreasing spectrum of the beam.

The scattered radiation makes the angle $\sim r/2r_L$ with the incident radio beam (where $r$ stands for the altitude of the scattering region) and precedes the MP in pulse longitude. The scattering from different harmonics takes place at different altitudes in the magnetosphere and therefore results in different components in the pulse profile. The scattering between the frequencies below the resonance has the lowest characteristic altitude and gives rise to the component closest to the MP. The components formed as a result of the scattering from increasingly high harmonics should have increasingly large separations from the MP. In application to the Crab pulsar, one can identify the component resulting from the scattering below the resonance with the PR and the component resulting from the first-harmonic scattering with LFC. The MP-LFC separation, $\Delta\lambda\sim
30^\circ$, implies that the first-harmonic scattering occurs close to the light cylinder. This is consistent with the estimate of the location of the resonance region in this pulsar.

The formation of the LFC close to the light cylinder is strongly supported by the polarization data: The position angle of linear polarization in the LFC is shifted by $\sim 30^\circ$ from that of the MP. Note that the position angle of the MP radiation is determined by the magnetic field direction in the emission region, whereas the position angle of the LFC radiation should reflect the magnetic field orientation in the scattering region. Because of the magnetosphere rotation, the ray emitted along the magnetic field makes the angle $\sim r/2r_L$ with the local magnetic field direction [see @p07a and Sect. 4 above]. Thus, the magnetic field orientations in the scattering and emission regions differ by $\sim r/2r_L$, the difference between the position angles of the original and scattered radiation being approximately the same. For the scattering taking place close to the light cylinder this difference is $\sim 30^\circ$. It is worthy to point out that in our consideration the position angle shift of the scattered component roughly equals its longitudinal separation from the MP. This is indeed the case for the LFC of the Crab pulsar [@mh98] and is believed to be a distinctive feature of the scattered components in other pulsars.

High percentage of linear polarization is another characteristic feature of the scattered components. In case of the scattering in a strong magnetic field, the scattered radiation is dominated by the waves of ordinary polarization, whose electric vector lies in the plane of the wavevector and the external magnetic field. In the original radio beam, only the ordinary waves are subject to the scattering below the resonance, whereas both types of waves, the ordinary and extraordinary ones, undergo equally efficient scattering at the harmonics of the gyrofrequency. In the Crab, as well as in other pulsars, the PR component is known to have almost complete linear polarization. Although the percentage of linear polarization of the LFC is lower, $\sim 40\%$, it still exceeds that of the MP.

One can expect that the LFC radiation suffers depolarization. This can be understood as follows. A noticeable sweep of the position angle across the LFC [@mh98] implies that this component is formed by the radiation coming from somewhat different altitudes in the magnetosphere, which results from the scattering of somewhat different frequencies. If one take into account the finite width of the MP, at a fixed pulse longitude within the LFC there should be radiation from different altitudes and therefore with different position angles. The superposition of the waves with different position angles may actually lead to a substantial depolarization of the resultant radiation.

The LFC and PR of the Crab pulsar are known to exhibit pronounced frequency evolution [@mh96]. The PR component is significant at the lowest frequencies, the LFC becomes strong at frequencies $\sim 1$ GHz, and at higher frequencies both components vanish. To analyze the spectral behaviour of the scattered components in our model let us turn to equation (28) and consider the ratio of the scattering efficiencies given that the frequencies of the scattered radiation are equal, $\nu_{1_0}=\nu_{1_s}$. Then we have $\Gamma_s/\Gamma_0\sim(\nu_s\eta/\nu_{1_s}\eta_1)^{1-\alpha}
(\gamma_0/\theta\gamma)^{2s-4}$, i.e. the role of the scattering at the harmonics of the gyrofrequency increases with frequency, $\Gamma_s/\Gamma_0\propto\nu_{1_s}^{\alpha-1}$. Thus, the LFC is expected to dominate at somewhat higher frequencies, which is in accordance with the observed trend.

On the way in the magnetosphere, both scattered components, the PR and LFC, may be further subject to scattering. Because of magnetosphere rotation their inclination to the ambient magnetic field rapidly increases with distance, while the magnetic field strength rapidly decreases. Similarly to the MP, the components may be involved in the induced transverse scattering in a moderately strong magnetic field, in which case the radiation is scattered backwards [for the general theory of this process see @p07b]. The consequences of the backward scattering of the components will be studied in detail in a separate paper. It will be shown that this process may give rise to the high-frequency components in the profile of the Crab pulsar: the backscattering of the PR can account for the IP’, whereas the scattering of the LFC can explain the HFC1 and HFC2. It will also be demonstrated that the efficiency of the induced transverse scattering increases with frequency and, correspondingly, at high enough frequencies the PR and LFC vanish, their intensities being almost completely transferred to the backward components.

The scattering efficiencies given by equations (26) and (27) depend on the intensity of the incident radio beam and on the number density and the characteristic Lorentz-factor of the scattering plasma particles. All these quantities may show marked pulse-to-pulse fluctuations, so that the scattering efficiencies may vary as well. If the scattering is so strong that the component growth is at the stage of saturation, $x\gg 1$ or $y\gg
1$, the fluctuations of $\Gamma$ do not affect the intensity of the scattered component significantly. On condition that $x\sim 1$ or $y\sim 1$, however, even small fluctuations of the scattering efficiency may lead to drastic variations of the scattered component. According to equations (26)-(27), this condition may be satisfied in a number of pulsars, which have large enough radio luminosities, strong magnetic fields and short periods. Such pulsars are believed to exhibit occasional activity at the pulse longitudes preceding the MP. Namely, these pulsars are expected to show the transient components with the spectral and polarization properties similar to those known for the PR and LFC of the Crab pulsar. Furthermore, the transient components resulting from the higher-harmonic scattering can also be present in pulsar profiles, in particular, in the Crab pulsar.

Conclusions
===========

The components of pulsar profiles outside of the MP are known to exhibit a number of peculiar properties, and at the same time the pulse-to-pulse fluctuations generally testify to a physical relation of these components to the MP. We believe that the components outside of the MP originate as a result of induced scattering of the pulsar radio beam into the background, with different types of the components corresponding to different scattering regimes. In the present paper, we have considered the magnetized induced scattering off the spiraling particles, which may be present in the outer magnetosphere of a pulsar. In this case the scattering at the harmonics of the particle gyrofrequency may be efficient. Our investigation is aimed at explaining, at least partially, the extremely complex radio emission pattern of the Crab pulsar. It has been demonstrated that the scattering from the first harmonic of the gyrofrequency into the state below the resonance can account for the formation of the LFC, whereas the scattering between the states below the resonance can explain the origin of the PR component.

As the scattered radiation concentrates along the local magnetic field direction in the scattering region, the scattering in different regimes, which hold at different altitudes, does form different components in the pulse profile because of the magnetosphere rotation. The observed LFC separation from the MP, $\Delta\lambda\sim 30^\circ$, implies that the first-harmonic scattering takes place close to the light cylinder. It is important to note that the formation of the LFC at large enough altitude in the magnetosphere and its orientation along the ambient magnetic field are strongly supported by the observed shift of the position angle of linear polarization with respect to that of the MP. The position angle shift appears approximately equal to the LFC-MP separation in pulse longitude. This is also expected from an analysis of the ray-magnetic field geometry in the rotating magnetosphere.

In the two regimes considered in the present paper, the scattering mainly results in the waves of the ordinary polarization, whereas both the ordinary and extraordinary waves are believed to be present in the original radio beam. Hence, the PR and LFC should be strongly polarized, which is consistent with the observational data.

As long as the scattering efficiency is large enough for the growth of the scattered component to reach the stage of saturation, the frequency dependence of $\Gamma$ and the variations of $\Gamma$ due to the pulse-to-pulse fluctuations of the incident radio beam intensity and of the parameters of the scattering plasma do not affect the scattered component substantially. This is believed to be the case for the PR and LFC of the Crab pulsar. However, the variations of the scattered component can be dramatic provided that $\Gamma$ is somewhat smaller and the scattering is about to reach the saturation stage. Therefore we conclude that a number of pulsars may have the transient components, which precede the MP and are analogous to the PR and LFC in the Crab pulsar. The presence of the transient components resulting from the higher-harmonic scattering is not excluded as well.

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[^1]: E-mail: petrova@ira.kharkov.ua
