Constraint on Pulsar Wind Properties from Induced Compton Scattering off Radio Pulses

Pulsar winds have longstanding problems in energy conversion and pair cascade processes which determine the magnetization $\sigma$, the pair multiplicity $\kappa$ and the bulk Lorentz factor $\gamma$ of the wind. We study induced Compton scattering by a relativistically moving cold plasma to constrain wind properties by imposing that radio pulses from the pulsar itself are not scattered by the wind as was first studied by Wilson&Rees. We find that relativistic effects cause a significant increase or decrease of the scattering coefficient depending on scattering geometry. Applying to the Crab, we consider uncertainties of an inclination angle of the wind velocity with respect to the radio beam $\theta_{\rm pl}$ and the emission region size $r_{\rm e}$ which determines an opening angle of the radio beam. We obtain the lower limit $\gamma\gtrsim10^{1.7}r^{1/2}_{\rm e,3}\theta^{-1}_{\rm pl}(1+\sigma)^{-1/4}$ ($r_{\rm e}=10^3r_{\rm e,3}$ cm) at the light cylinder $r_{\rm LC}$ for an inclined wind $\theta_{\rm pl}>10^{-2.7}$. For an aligned wind $\theta_{\rm pl}<10^{-2.7}$, we require $\gamma>10^{2.7}$ at $r_{\rm LC}$ and an additional constraint $\gamma>10^{3.4}r^{1/5}_{\rm e,3}(1+\sigma)^{-1/10}$ at the characteristic scattering radius $r_{\rm c}=10^{9.6}r^{2/5}_{\rm e,3}$ cm within which the `lack of time' effect prevents scattering. Considering the lower limit $\kappa\gtrsim10^{6.6}$ suggested by recent studies of the Crab Nebula, for $r_{\rm e}=10^3$ cm, we obtain the most optimistic constraint $10^{1.7}\lesssim\gamma\lesssim10^{3.9}$ and $10^{6.6}\lesssim\kappa\lesssim10^{8.8}$ which are independent of $r$ when $\theta_{\rm pl}\sim1$ and $1+\sigma\sim1$ at $r_{\rm LC}$.

Pulsar winds have longstanding problems in energy conversion and pair cascade processes which determine the magnetization σ, the pair multiplicity κ and the bulk Lorentz factor γ of the wind. We study induced Compton scattering by a relativistically moving cold plasma to constrain wind properties by imposing that radio pulses from the pulsar itself are not scattered by the wind as was first studied by Wilson & Rees. We find that relativistic effects cause a significant increase or decrease of the scattering coefficient depending on scattering geometry. Applying to the Crab, we consider uncertainties of an inclination angle of the wind velocity with respect to the radio beam θ pl and the emission region size r e which determines an opening angle of the radio beam. We obtain the lower limit γ 10 1.7 r 1/2 e,3 θ −1 pl (1 + σ) −1/4 (r e = 10 3 r e,3 cm) at the light cylinder r LC for an inclined wind θ pl > 10 −2.7 . For an aligned wind θ pl < 10 −2.7 , we require γ > 10 2.7 at r LC and an additional constraint γ > 10 3.4 r 1/5 e, 3 (1 + σ) −1/10 at the characteristic scattering radius r c = 10 9.6 r 2/5 e,3 cm within which the 'lack of time' effect prevents scattering. Considering the lower limit κ 10 6.6 suggested by recent studies of the Crab Nebula, for r e = 10 3 cm, we obtain the most optimistic constraint 10 1.7 γ 10 3.9 and 10 6.6 κ 10 8.8 which are independent of r when θ pl ∼ 1 and 1 + σ ∼ 1 at r LC .

Introduction
Pulsar magnetospheres create pulsar winds through pair creation and particle acceleration [1]. Because pulsar winds are radiatively inefficient, it is difficult to constrain their properties. However, their properties are inferred from observations of surrounding pulsar wind nebula (PWN) and pulsed emissions of the pulsar itself. Interestingly, a secular increase of their pulse period tells us their total energy output L spin . Because most of the spin-down power is converted into the pulsar wind, L spin constrains its properties as (see also Equation (22)) κγ(1 + σ) = 1.4 × 10 10 L spin 10 38 erg s −1 1 2 , where κ is the pair multiplicity (e ± number flux normalized by the Goldreich-Julian number fluxṄ GJ ), γ is the bulk Lorentz factor, and σ is the magnetization parameter (the ratio of the Poynting to the kinetic energy fluxes) of the pulsar wind, respectively. We usedṄ GJ ≡ 2πr 2 pc cn GJ (r pc ) = 6cL spin /e, where r pc is the polar cap radius, n GJ (r pc ) is the Goldreich-Julian density at an magnetic pole and the numerical factor two comes from the north and south magnetic poles. Pair cascade models within the magnetosphere of the Crab pulsar (L spin = 4.6 × 10 38 erg s −1 ) predict κ ∼ 10 4 with σ ∼ 10 4 and γ ∼ 10 2 in the vicinity of the light cylinder r LC [e.g., [2][3][4]. On the other hand, magnetohydrodynamic (MHD) models of the Crab Nebula reproduce its non-thermal emission from optical to γ-ray with κ ∼ 10 4 , σ ∼ 10 −3 − 10 −2 and γ ∼ 10 6 [5][6][7][8]. Although κ ∼ 10 4 in both models is consistent with particle number conservation, σ (and also γ) differs by many orders of magnitude, which is called the 'σ-problem' [c.f., 9].
It is noted that there is an additional problem of the pulsar wind properties [c.f., 10,11]. Because the MHD models of the Crab Nebula do not explicitly account for the origin of radio emitting particles, they may underestimate the pair multiplicity. Recent studies of spectral evolution of PWNe showed κ > 10 6 for the Crab Nebula and κ > 10 5 for other PWNe [e.g, [11][12][13][14]. Although the origin of the low energy particles that are responsible for the radio emission of PWNe is still an open problem, they originate most likely from the pulsar because of the continuity of the broadband spectrum and because of the radio structures apparently originating from the pulsar [15][16][17]. Thus there arises another problem on κ besides the σ-problem, while only the combination of κγ(1 + σ) in Equation (1) is firm.
In view of the σ-and κ-problems, it is interesting to consider other independent constraints on the physical conditions of pulsar winds. Wilson & Rees (1978, hereafter WR78) [18] considered induced Compton scattering off radio pulses by a pulsar wind. So far, it is thought that we have not observed a signature of scattering in radio spectra of pulsars, although we do not fully understand how scattering changes the radio spectrum (e.g., scattering by a nonrelativistic plasma was studied by [19,20]). Observations suggest that the optical depth to induced Compton scattering is less than unity, and the radio spectrum is not changed. Based on this consideration, WR78 obtained the lower limit of the bulk Lorentz factor of the Crab pulsar wind γ > 10 4 at 10 3 r LC ∼ 10 11 cm away from the pulsar. Substituting Equation (1), only for (1 + σ) ∼ 1 at 10 3 r LC , their conclusion is marginally consistent with the conclusion of κ 10 6.6 ≡ κ PWN obtained from the study of the Crab Nebula spectrum by Tanaka & Takahara (2010 [11,13]. Induced Compton scattering process has been studied for the application to high brightness temperature radio sources, such as the pulsars [e.g., 18,[21][22][23], active galactic nuclei [e.g., 19,24,25] and other sources [e.g., [26][27][28]. Induced Compton scattering is about a factor of θ 4 bm k B T b (ν)/m e c 2 times effective compared with spontaneous one in the rest frame of the plasma, where θ bm (< 1) and T b are a half-opening angle and a brightness temperature of a radio beam, respectively (see Equation (15)). Note that the value of k B T b (ν)/m e c 2 can be larger than 10 15 for the Crab pulsar (see Equation (24)). However, for scattering by relativistically moving electrons, the scattering coefficient is modified by relativistic effects and, as we will see below, either an increase or a decrease is possible depending on situations considered, e.g., the velocity u = γβ of the electrons and an inclination between an electron motion u and a radio beam k, where k is the wavenumber vector.
In this paper, we reconsider induced Compton scattering by a relativistically moving plasma and reevaluate a lower limit of the bulk Lorentz factor. Despite strong dependence on scattering geometry, WR78 considered a specific scattering geometry where the pulsar wind is completely aligned with respect to the radio pulse beam and where θ bm of the radio beam is the widest value inferred from the observations. We consider rather general geometries of the system, such as the direction of the wind being inclined with respect to the radio pulse 2/26 beam. Even if the direction of pulsed radio emission is almost radial, the pulsar wind is likely to have a significant toroidal velocity just outside r LC , or its motion in the meridional plane is not strictly radial. As already noted by WR78, the scattering coefficient may be significantly reduced if the pulsar wind inclines with respect to the radio beam. For θ bm , the scattering coefficient is reduced when the radio beam is narrow in the rest frame of the plasma. If this is the case, the lower limit of the bulk Lorentz factor of the pulsar wind may be reduced so as to be consistent with recent studies of the Crab Nebula spectrum.
While we focus on geometrical effects in this paper, we ignore effects of the magnetic field and background photons following WR78. The magnetic field effect may be important when the frequency of the photon at the plasma rest frame ν ′ is smaller than the electron cyclotron frequency ν ce [e.g., 21,29]. For the Crab pulsar, although the magnetic field in the observer frame is about B obs ∼ 10 6 G at the light cylinder (ν ce = 5.8 × 10 12 Hz for the magnetic field of B ′ = 10 6 G in the plasma rest frame), ν ce strongly depends on the magnetic field configuration and a direction of plasma motion in the observer frame. For example, if B obs ⊥ u, we find B ′ = B obs /γ and ν ′ = ν/δ D where δ D is the Doppler factor. Basically, the magnetic field effect reduces the scattering cross section, i.e., smaller γ would be allowed. For the effect of background photons, Lyubarsky & Petrova (1996) [21] discussed that scattering off the background photons induced by the beam photons may be important. They discussed that the occupation number of the background photons increases exponentially, i.e., the beam photons may decrease accordingly, when the scattering optical depth to the background photons well exceeds unity, say 10 2 . In this paper, we ignore background photons (θ bm < θ ≤ π) assuming that the occupation number of the beam photons is much larger than that of the background photons. If scattering off the background photons is efficient, scattering would be more efficient and larger γ would be required. These processes will be discussed in a separated paper.
In Section 2, we describe the scattering coefficient of induced Compton scattering by a relativistically moving plasma in a general geometry. We also show simple analytic forms of the scattering coefficient in some specific geometries. In general geometry, the scattering coefficient is written in an integral form and is obtained numerically in Appendix A. In Section 3, we consider induced Compton scattering at pulsar wind regions, specifically applying to the Crab pulsar. We show the resultant lower limits of γ and also discuss the corresponding upper limits of the pair multiplicity κ. We summarize the present results in Section 4.

INDUCED COMPTON SCATTERING OFF A PHOTON BEAM
Here, we express the scattering coefficient at a certain position x and see that the scattering coefficient strongly depends on geometry of scattering. The kinetic equation for a photon occupation number n(x, k, t) is expressed as [e.g., 21, 30] where Ω = k/k, f (p) is the distribution function of plasma and dσ/dΩ is the differential scattering cross section, respectively. Note that when the electron is initially at rest, the recoil 3/26 g is expressed as g(k, ξ) = k/(1 + kλ e (1 − cos ξ)), where λ e = /m e c represents the Compton wavelength for an electron and ξ is the angle between incident and scattered photons. We omit arguments x and t in Equation (2) and in this section. The terms 1 + n represent spontaneous and induced scattering terms, and we only consider the induced process below, assuming n ≫ 1.

Scattering Coefficient
The scattering coefficient of induced Compton scattering is the right-hand side of Equation (2) divided by n(k) [e.g., 31]. Equation (2) is simplified by following three approximations.
(I) Plasma is cold, and moves with the velocity u = γβ (the bulk Lorentz factor γ = (1 − The magnetic field is weak enough to satisfy the condition ν ce < ν ′ , where ν ce and ν ′ are the electron cyclotron frequency and the frequency of an incident photon in the plasma rest frame, respectively [e.g., 21]. (III) Photons are in the Thomson regime, i.e., kλ e ≪ 1 [c.f., 30]. The condition (III) is a good approximation for scattering off radio photons by plasma of γ ≪ 10 10 . In the observer frame, Equation (2) then becomes, [e.g., 18,21], and n pl is a number density of plasma. R(Ω, Ω 1 , u) is order unity (1 ≤ R ≤ 2) and σ T is the Thomson scattering cross section. The scattering coefficient contains the integral which depends on the occupation number itself and on scattering geometry at x, i.e., directions of photons (Ω and Ω 1 ) and a velocity of the plasma u. While WR78 performed this integral on a specific scattering geometry, we reevaluate it in more general geometries.

Geometry
Scattering geometry at a certain position x in the observer frame is depicted in Figure 1. The photon beam with a half-opening angle θ bm directs to an observer on z-axis. An inclination angle of the plasma velocity is θ pl . Note that the plasma should be depicted as a line rather than a cone on Figure 1, i.e., zero opening angle, because we assume that the plasma is cold. However, we will see that there is the characteristic angle γ −1 around the plasma velocity and then we associate the plasma with the cone of its half-opening angle γ −1 in the figures in this paper. For the plasma, we express the velocity u as u = γβ(sin θ pl e x + cos θ pl e z ). Although the cold plasma beam should be described as a line, we associate it with the cone of a half-opening angle of γ −1 in all the sketches for explanatory convenience.
We assume that the occupation number of photons is uniform inside the beam and is expressed as where H is the Heaviside's step function. The spectrum n(ν) is assumed to be a broken power-law form where p 1 and p 2 are power-law indices of low and high frequency parts and n 0 is the occupation number at a break frequency ν 0 , respectively. Observed pulsar radio spectra correspond to −7 p 2 −3, and we require p 1 > −3 for the number density of photons to be finite at ν → 0. For the application in Section 3, we take p 2 = −5 and ν 0 = 10 MHz considering the radio observations. Adopting p 1 = 3, the brightness temperature k B T b (ν) = hνn(ν) to be maximum at ν 0 . We consider scattering off photons toward the observer, i.e., Ω = e z . The scattering coefficient χ at x is expressed as As is the conventional definition of the optical depth dτ = χdl for a path l along z-axis, we include a minus sign, where the occupation number decreases along the path for a positive 5/26 value of χ and vice versa. The sign of χ can change with the sign of the function

Analytic Estimates
It is convenient to rewrite Equation (11) by introducing the normalization The scattering coefficient becomes where the integral I(ν, θ bm , θ pl , γ) represents a geometrical effect. Note that χ contains a factor of γ −3 which is independent of scattering geometries. The value of I(ν, θ bm , θ pl , γ) is obtained numerically in general and can take a wide range of values even for a fixed frequency. The numerical results of the integral I(ν) for different parameter sets (θ bm , θ pl , γ) are described in Appendix A and are also shortly summarized in the last paragraph of this section. Below, we describe simple analytic forms of the integral I(ν) for some special cases. They help understanding of dependence on (θ bm , θ pl , γ) and turn out to be useful for applications in the next section.
To see relativistic effects, we expand sin θ, cos θ and β to second-order in θ 1 , θ pl and γ −1 , i.e., we concern the situations 0 ≤ (θ pl , θ bm ) 1 and γ ≫ 1. The integrand is composed of following three factors. (I) The solid angle (and the recoil) factor originates from the solid angle element dΩ 1 and from the recoil term 1 − µ, and is expressed as This factor already appeared in the non-relativistic case (Equation (15)). (II) The aberration factor originates from the Lorentz transformation of a solid angle element from the plasma rest frame to the observer frame, and is expressed as where we introduced an angle ψ 1 between β and Ω 1 , given by the approximation ψ 2 1 = θ 2 1 − 2θ 1 θ pl cos φ 1 + θ 2 pl . (III) The frequency shift factor also originates from the Lorentz 6/26  (14) (left) and a sketch of scattering geometry (right) in the 'Narrow' case (1 > Θ 2 bm + Θ 2 pl ). The plot shows absolute values |I(ν)| versus ν with γ = 10 2 , p 1 = 3 and p 2 = −5 (lines a, b and c) together with T b (ν)/T b (ν 0 ) for comparison. Each line is for a different value of (Θ bm , Θ pl ): 'line a' for (10 −1 , 10 −1 ), 'line b' for (10 −1 , 10 −2 ), and 'line c' for (10 −2 , 10 −1 ), respectively. Note that 'line a' and 'line b' are overlapped since I(ν) is primarily determined by Θ bm as is seen in Equation (19). A discontinuity found in each line is the frequency where the sign of I(ν) changes and the high frequency side has a positive sign, while the low frequency side has a negative sign for all lines. Note also that, in the right panel, the opening angle of the plasma cone (red in color) represents γ −1 cone and does not represent the velocity distribution (see Figure 1 and the text).
transformation of a frequency, and is expressed as Analytic forms of the integral I(ν) presented below are explained by a simple combination of these three factors. We also show numerical results of the integral I(ν) for these cases in Figures 2 − 4, where we adopt p 1 = 3, p 2 = −5 and γ = 10 2 . Introducing normalized angles Θ bm ≡ γθ bm and Θ pl ≡ γθ pl , it is easy to find that the integral I(ν) depends on (Θ bm , Θ pl ) rather than separately on θ bm , θ pl and γ. We first consider the case 1 > Θ 2 bm + Θ 2 pl where the narrow photon beam and Ω = e z are well inside the γ −1 cone associated with the plasma as shown in the right panel of Figure 2. We call this case 'Narrow'. In this case, we obtain D −2 1 ≈ 4γ 4 and D/D 1 ≈ 1, and then the integral I(ν) is approximated as where we use R(e z , . This expression with γ → 1 (Θ bm → θ bm ) is almost the same as that of the non-relativistic case (Equation (15)). For the 'Narrow' case, the aberration factor increases the integral I(ν) by a factor of D −2 1 ≈ 4γ 4 compared with I NR (ν) because the opening angle increases by a factor of ∼ γ in the plasma rest frame, while the frequency shift is negligible (D/D 1 ≈ 1). Note that χ Narrow (ν) is a factor of γ larger than χ NR (ν) accounting for the factor of γ −3 in Equation (14). In the left 7/26  (14) (left) and a sketch of scattering geometry (right) in the 'Inclined' case (Θ 2 pl > Θ 2 bm + 1). Each line is for a different value of (Θ bm , Θ pl ): 'line a' for (1, 10), 'line b' for (1, 10 2 ), and 'line c' for (10 −1 , 10), respectively and the other parameters are the same as in Figure 2. Note that 'line b' and 'line c' are overlapped since I(ν) is primarily determined by the ratio Θ bm /Θ pl as seen in Equation (20).
Next case is Θ 2 pl > Θ 2 bm + 1 where u is inclined with respect to Ω and the associated cones do not overlap with Ω as shown in the right panel of Figure 3. We call this case 'Inclined'. The integral I(ν) also suffers from little frequency shift (D/D 1 ≈ 1) and the aberration factor is approximated as D −2 1 ≈ 4θ −4 pl . We obtain an approximated form of where we use R(e z , In the left panel of Figure 3, we show numerical results for the 'Inclined' case. The aberration factor decreases the integral I(ν) by a factor of Θ −4 pl compared with I Narrow (ν). Note that χ Inclined (ν) can be smaller than χ NR (ν), as The scattering geometry satisfying Θ bm > 1 > Θ pl is sketched in the right panel of Figure  4 where the γ −1 cone of plasma contains Ω and is well within the photon beam. We call this case 'Wide'. Note that although we take θ pl = 0 in Figure 4 and in Equation (21), we will find that the integral I(ν) behaves in a similar way for Θ bm > 1 > Θ pl = 0 in Appendix A. For θ pl = 0, the frequency shift factor is approximated as D/D 1 ≈ (1 + Θ 2 1 ) −1 ≤ 1. The aberration factor behave as D −2 1 ≈ 4γ 4 /(1 + Θ 2 1 ) 2 and makes the angular distribution of the photon beam almost isotropic in the plasma rest frame. Simple analytic form is found for the frequency range ν > (1 + Θ 2 bm )ν 0 ≈ Θ 2 bm ν 0 , where we use the expressions T b (ν 1 ) ≈ 8/26  (14) (left) and a sketch of scattering geometry (right) in the 'Wide' case (Θ 2 bm > 1 > Θ 2 pl = 0). Each line is for a different value of (Θ bm , Θ pl ): where 'line a' for (10, 0), 'line b' for (3, 0), and 'line c' for (1, 0), respectively and the other parameters are the same as shown in Figure 2. Note that the discontinuity frequency shifts to higher frequency for larger Θ bm , which matches Equation (21) well.
There remains the geometry Θ bm > Θ pl > 1 where the cone of plasma does not contain Ω but is within the photon beam. We do not find an analytic form of the integral I(ν) in this case. The numerical calculation in Appendix A shows that |I(ν)| takes between |I Inclined (ν)| and |I Wide (ν)| for the frequency range ν > ν 0 in which we are interested in Section 3. Note that |I Inclined (ν)| gives the smallest value and |I Wide (ν)| gives the largest value in any geometries (Θ bm , Θ pl ) for ν > ν 0 . We give a detailed discussion including this exceptional geometry in Appendix A.

APPLICATION TO THE CRAB PULSAR
We evaluate the optical depth to induced Compton scattering applying to the Crab pulsar. We require that the optical depth |τ (ν)| is less than unity and then we constrain the Crab pulsar wind properties κ, γ, and σ. 9/26

Setup
We describe assumptions to estimate the normalization χ 0 for the Crab pulsar. For a pulsar wind, three assumptions are made. (I) Almost all of the spin-down power L spin goes to the pulsar wind. (II) The pulsar wind is a cold magnetized e ± flow whose bulk Lorentz factor is γ. (III) The number density of the pulsar wind decreases with r −2 , and we ignore structures in the pulsar wind, such as the current sheet [e.g., 32]. Now, the number density of the pulsar wind in the observer frame is where we assume the radial velocity β r ∼ 1. Note that we obtain Equation (1) from Equation (22) by normalizing 4πr 2 cβ r n pl (r) withṄ GJ . Note also that a product γ(1 + σ) does not depend on r because we expect no particle production outside the light cylinder r LC , i.e., n pl ∝ r −2 .
For radio pulses, uncertainty of the brightness temperature arises from an opening angle of the radio emission θ bm . Following WR78, we assume that the emission is isotropic at r = r e where r e is an emission region size. The opening angle θ bm (r) is written as We adopt Equation (23) for the opening angle of the radio pulse throughout this paper.
The brightness temperature is expressed as [e.g., 34] where F ν and d are a flux density at a frequency ν and a distance to the object, respectively. WR78 adopted r e = 10 7 cm which is estimated from the integrated pulse width W 50 = 3 msec [33,34]. We study dependence on r e in Section 3.5. In Section 3.5, we will take r e = 10 3 cm considering the 'microbursts' of which individual pulses from the Crab pulsar show nano − microsecond duration structures [35]. Note that r e = 10 3 cm would also be considered as almost the minimum size of plasma to emit the coherent electromagnetic wave of the frequency ν = 100 MHz (c/ν = 3 × 10 2 cm). Figure 5 shows the radio spectrum of the Crab pulsar. We assume F ν ∼ 50 (ν/100 MHz) p2+3 Jy for ν 0 ≤ ν ≤ 100 MHz with ν 0 = 10 MHz and p 2 = −5. Adopting d = 2 kpc, L spin = 4.6 × 10 38 erg s −1 and the light cylinder radius r LC = 1.6 × 10 8 cm for the Crab pulsar, we obtain the normalization Although we used ν 0 = 10 MHz, we require |τ (ν)| < 1 at ν = 100 MHz because the Crab pulsar spectrum ( Figure 5) is obviously unaffected by scattering in a range ν ≥ 100 MHz.
On the assumptions made in this section, the scattering coefficient χ(ν, r) is considered to be a rapidly decreasing function of r. We introduce the exponents a and b ((a, b) > 0) 10 Crab Flux The observed spectrum of the Crab pulsar in radio. Note that the emission at ν < 100 MHz is not observed to be pulsed anymore most probably because of the interstellar scattering. So that apparently rising spectrum around 100 MHz is not real. Since the high frequency radio flux of the Crab pulsar is F ν = 646(ν/400 MHz) −3.1 mJy for ν > 400 MHz [33,36], there seems a spectral break around 100 MHz. The low frequency spectrum extends down to at least 5.6 MHz with a spectral index α = −2.09 [37,38]. Fitted line in this range is F ν ∼ 50(ν/100 MHz) −2 Jy for ν < 100 MHz. Observational data are taken from [33,37,38].
characterizing the r-dependence of the velocity u(r) as γ ∝ r a and θ pl ∝ r −b . Now, the r-dependence of χ(ν, r) (Equation (14)) is expressed as where I Wide (ν) ≈ −1 is used in this section because ν 0 ν < Θ 2 bm ν 0 (ν 0 = 10 MHz and ν = 100 MHz) is mostly attainable for the 'Wide' case (Θ bm > 1). In Equation (26), b − a < 1.25 is sufficient for χ(ν, r) to be considered as a rapidly decreasing function of r. Otherwise we consider moderate values of a and b, say, 0 < (a, b) 1.25 below. Therefore, the choice of the innermost scattering radius is important to evaluate the optical depth.
Here, we consider scattering beyond the light cylinder r ≥ r LC , because we do not know where the electron-positron plasma and the radio emission are produced inside the magnetosphere and because we do not take into account magnetic field effects which may be important close to the pulsar. We evaluate the optical depth as where r in and ∆r are the innermost scattering radius and the path length, respectively. In Equation (27), we should not simply put r in = ∆r = r LC because the path length ∆r has a lower limit originating from the 'lack of time' effect which we will discuss in the next subsection.

Characteristic Scattering Length
The 'lack of time' effect introduced by WR78 should be taken into account for the evaluation of r in and ∆r in Equation (27). This is similar to the concept of the 'coherence radiation length' [e.g., 39,40]. The normal treatment of scattering breaks down when an electron does not see one cycle of the electric field oscillation of radio waves. We determine this characteristic length l c as follows. A cycle of the incident and scattered photons in the plasma rest frame is described as ∆t ′ = δ D /ν where δ D = (γD) −1 or (γD 1 ) −1 is the Doppler factor. The characteristic length l c is the speed of light multiplied by the time interval ∆t = γ∆t ′ in the observer frame. Using D −1 ≈ 2γ 2 /(1 + Θ 2 pl ) and D −1 l c is considered as a function of only r through γ(r) or θ pl (r) for the given frequency ν = 100 MHz. On the other hand, for the geometry Θ bm > Θ pl > 1, we obtain We find l c for this case is equal to or larger than that for the 'Inclined' case. Because l c depends on Ω 1 , we cannot separate integrals over Ω 1 and r in Equations (14) and (27). In this subsection, we limit the discussion about the 'Narrow', 'Inclined' and 'Wide' cases. Now, we describe how we determine r in and ∆r taking into account the r-dependence of l c (r). Although we describe only for the 'Narrow' and 'Wide' cases (l c ∝ γ 2 ), the same discussion is applicable to the 'Inclined' case (l c ∝ θ −2 pl ) by replacing γ with θ −1 pl . We set γ(r) = γ LC (r/r LC ) a where γ LC is the Lorentz factor at r LC . Substituting it into Equation (28), we obtain In Figure 6, we show the l c (r) − r diagram. We do not consider the region r < r LC . The region r > r LC is divided into two regions by the line l c (r) = r which corresponds to γ LC ≈ 10 2.7 and a = 0.5. Scattering off the radio pulse should be considered when l c (r) < r so that three different choices of r in are possible for different values of γ LC and the exponent a, corresponding to points 'A', 'B' and 'C' in Figure 6. Point 'A' corresponds to γ LC < 10 2.7 with any values of the exponent a. Since l c (r LC ) < r LC in this case, we take r in = ∆r = r LC . Point 'B' corresponds to γ LC > 10 2.7 with a < 0.5. The radio pulse is not scattered at r LC but beyond r LC . Here, we introduce the characteristic scattering radius r c which satisfies r c = l c (r c ) > r LC so that we take r in = ∆r = r c = (10 2.8 γ 2 LC r −2a LC ) 1/(1−2a) cm. For γ LC > 10 2.7 with a ≥ 0.5, we obtain l c (r) > r everywhere beyond r LC , i.e., the electron never sees one 12 Fig. 6 The l c − r diagram for the 'Narrow' and 'Wide' cases (Equation (30)). The region r < r LC (grey in color) is not considered in this paper. When l c (r) > r (light blue in color), scattering does not occur because of the 'lack of time' effect, while scattering should be considered in the region l c (r) < r (pink in color). Three cases for r in (points 'A', 'B' and 'C') are possible by different behaviors of l c (r), i.e., γ LC and the exponent a (see also Equation (30)). r in becomes point 'A' when γ LC < 10 2.7 . For γ LC > 10 2.7 , r in is point 'B' when a < 0.5. While no r in exists for a > 0.5, because γ, i.e., l c (r), cannot be infinitely large, there must be point 'C' where l c (r) = r is satisfied. cycle of radio waves (dot-dashed line: red in color). However, γ(r) cannot be infinitely large so that there should exist the radius satisfying r in = l c (r in ) > r LC corresponding to point 'C'. In this case, we also take r in = ∆r = r c whose expression is different from that for a < 0.5. Therefore, γ LC = 10 2.7 or θ pl,LC = 10 −2.7 is a critical value in determining which to adopt as r in .
We consider whether the radio pulse can escape from scattering at the two radii r LC and r c . Rather than using the exponents a and/or b, it is convenient to introduce γ c ≡ γ(r c ) and θ pl,c ≡ θ pl (r c ). We evaluate the optical depth by treating the velocities u LC and u c , i.e., (γ LC , θ pl,LC ) and (γ c , θ pl,c ), as free parameters. Relation between the exponent a (b) and γ c (θ pl,c ) will be discussed shortly in Section 3.3.3. Note that we indirectly obtain the characteristic scattering radius r c from Equation (28) once γ c or θ pl,c is obtained.

Escape from scattering at the light cylinder.
Here, we are interested in whether the radio pulse can escape from scattering at r LC . Figure 7 shows the resultant γ − θ pl diagram which tells us whether the radio pulses can escape from scattering or not at a given point on the diagram, i.e., a given velocity u LC of the pulsar wind (see also Table 1). Since we obtain θ bm (r LC ) ≈ 10 −1.2 from Equation (23), the scattering geometries are divided by the lines γ = 10 1.2 (Θ bm,LC = 1), θ pl = 10 −1.2 (Θ pl = Θ bm,LC ) and γ = θ −1 pl (Θ pl = 1). Areas above the thick lines |τ LC | = 1 correspond to the pulsar wind structures which allow the radio pulses to escape, where τ LC is the optical depth for r in = r LC . At the upper left corner on the diagram, the region satisfies l c (r LC ) > r LC and the radio pulses also escape from scattering at r LC due to the 'lack of time' effect. The lines |τ LC | = 1 and l c (r LC ) = r LC are different for different scattering geometries as described below and summarized in Table 1.
First, we consider the 'Narrow' case (1 > Θ 2 bm + Θ 2 pl ) corresponding to the lowermost area on the diagram. The optical depth of τ LC ∼ 10 14.9 (1 + σ LC ) −1 obtained from Equations (19), (25) and (31) is independent of both γ LC and θ pl,LC . Therefore, a region |τ LC | < 1 does not appear for 1 + σ LC 10 4 and then we conclude that this case is not realized for the Crab pulsar.
Next, we consider the 'Inclined' case (Θ 2 pl > Θ 2 bm + 1) corresponding to the rightmost area on the diagram. In this case, the optical depth is expressed as τ LC ∼ 10 14.9 γ −4 LC θ −4 pl,LC (1 + σ LC ) −1 . The condition for |τ LC | < 1 is equivalent to γ LC The γ − θ pl diagram at r LC when r e = 10 7 cm (θ bm (r LC ) ≈ 10 −1.2 ). Choosing one point on the diagram specifies the pulsar wind velocity u LC . Four areas divided by three lines γ = θ −1 pl , γ = 10 1.2 and θ pl = 10 −1.2 correspond to different scattering geometries, the 'Narrow' (lowermost area: red in color), 'Inclined' (rightmost area: green in color) and 'Wide' (left triangle area: yellow in color) cases and the geometry Θ bm > Θ pl > 1 (upper triangle area: blue in color). The region above the |τ LC | = 1 line (light blue in color) corresponds to |τ LC | < 1, i.e., where the pulsar wind does not scatter the radio pulses at and beyond r LC . The upper left corner which satisfies γ > 10 2.7 and θ pl < 10 −2.7 (gray in color) corresponds to l c (r LC ) > r LC , i.e., the radio pulses are not scattered at r LC because of the 'lack of time' effect and we also require |τ c | < 1 in Figure 8. The |τ LC | = 1 lines (thick lines) at the 'Inclined' and 'Wide' areas are determined by γ LC = 10 3.7 θ −1 pl,LC (1 + σ LC ) −1/4 and γ LC = 10 5.8 (1 + σ LC ) −1/4 , respectively and depend on σ LC (see also Table 1). We adopt 1 < 1 + σ LC ≪ 10 4 in the diagram, for example, y-intercept of the |τ LC | = 1 line in the 'Inclined' area is γ ∼ 10 3.2 for 1 + σ LC ∼ 10 2 . Note that the line in the geometry Θ bm > Θ pl > 1 is drawn in a dashed line because it is an interpolated ones (see text). On the other hand, the shaded region (pink in color) is the forbidden region. θ pl,LC 10 −1.2 where the painted area above |τ LC | = 1 line in the 'Inclined' area on the diagram. We find that the radio pulses can escape for reasonable parameters when the pulsar wind has a significant non-radial motion. For example, the pulsar wind of γ LC > 10 2.7 with θ pl,LC ∼ 1 and 1 + σ LC ≈ 10 4 can escape from scattering at r LC .
Lastly, we mention the geometry of Θ bm > Θ pl > 1 which appears in the upper triangle area on the diagram. The l c (r LC ) = r LC and |τ LC | = 1 lines (dashed lines) are not calculated 15/26 Table 2 The optical depth |τ (100 MHz)| at r c (γ LC > 10 2.7 and θ pl,LC < 10 −2.7 ). Scattering geometries are classified by u c . We take r e = 10 7 r e,7 cm.  (14) controls the singularity τ LC = 0. When we neglect such a singular region, the allowed region would be above the thick dashed line and the lower limit of γ LC is clearly larger than the 'Inclined' case.

Escape from scattering beyond the light cylinder.
We investigate whether the radio pulse can escape from scattering at r c further than r LC . Because r c > r LC , we have only to consider a region of γ > 10 2.7 and θ pl < 10 −2.7 . The behaviors of γ(r) and θ pl (r) at r LC < r < r c will be discussed in Section 3.3.3. Figure 8 shows the resultant γ − θ pl diagram at r c . We set θ bm (r c ) ≈ 10 4.2 γ −2 c for the 'Narrow' and 'Wide' cases or θ bm (r c ) ≈ 10 4.2 θ 2 pl,c for the 'Inclined' case from Equations (23) and (28). The scattering geometries are divided by the lines γ = 10 4.2 (Θ bm,c = 1), θ pl = 10 −4.2 (Θ pl = Θ bm,c ) and γ = θ −1 pl (Θ pl = 1) (see Table  2). It should be noted that each scattering geometry appears in a different layout on the γ − θ pl diagram compared with Figure 7 because θ bm (r c ) depends on γ c or θ pl,c . The pulsar wind velocity u c which allows the radio pulses to escape corresponds to the area satisfying γ c ≥ 10 4.2 and θ pl,c ≤ 10 −4.2 corresponding to the 'Narrow' or 'Inclined' cases. Except for the extrapolated line in the geometry Θ bm > Θ pl > 1 (thick dashed line), the |τ c | = 1 line is not drawn on the diagram as described below, where τ c is the optical depth for r in = r c .
Next, we consider the 'Inclined' case (Θ 2 pl > Θ 2 bm + 1) corresponding to the right triangle area on the diagram. For the optical depth, we require |τ c | ∼ pl,c ). We show only the region which satisfies both γ > 10 2.7 and θ pl < 10 −2.7 because we consider the case r c > r LC . The pulsar wind velocity u c is specified by choosing one point on the diagram. Four areas divided by three lines γ = θ −1 pl , γ = 10 4.2 and θ pl = 10 −4.2 correspond to different scattering geometries, the 'Narrow' (left triangle area: red in color), 'Inclined' (upper triangle area: green in color) and 'Wide' (lowermost area: yellow in color) cases and the geometry Θ bm > Θ pl > 1 (rightmost area: blue in color). Note that each scattering geometry appears in a different layout compared with Figure 7. The painted region (light blue in color) satisfies |τ c | < 1, i.e., the radio pulses are not scattered at r c 10 11.2 cm ≈ 10 3 r LC . The |τ c | = 1 line (dashed thick line) appears only in the geometry Θ bm > Θ pl > 1 and is an extrapolated one (see text). On the other hand, the shaded region (pink in color) is forbidden region because |τ c | > 1 or, in other words, r c < 10 11.2 cm. the boundary line γ = θ −1 pl . Note that large θ pl,c does not reduce |τ c | as |τ LC | is reduced by large θ pl,LC (see the 'Inclined' area in Figure 7) because r c is a rapidly decreasing function of θ pl,c . Therefore, whole of the 'Inclined' geometry area θ pl,c ≤ 10 −4.2 is allowed for radio pulses to escape and we obtain r c 10 11.2 cm again.
The 'Wide' case (Θ bm > 1 > Θ pl ) corresponding to the lowermost area on the diagram. The condition to be |τ c | < 1 is γ c 10 4.8 (1 + σ c ) −1/6 . In this case, a region |τ c | < 1 does not appear in the 'Wide' area for 1 + σ c < 10 4 and then we conclude that this case is not realized for the Crab pulsar.

Summary.
There exist two possible cases of u LC where the radio pulses are not scattered at r LC . First, when u LC is significantly inclined with respect to the radio pulses 10 −1.2 < θ pl,LC 1 and has the Lorentz factor satisfying γ LC θ pl,LC (1 + σ LC ) 1/4 10 3.7 , we obtain τ LC < 1. In this case, the radio pulses reach the observer without scattering because χ(ν, r) decreases rapidly with r for 0 < (a, b) 1.25 as discussed in Equation (26). The second corresponds to the 'lack of time' effect, i.e., u LC is almost aligned with respect to the radio pulses θ pl,LC < 10 −2.7 with γ LC > 10 2.7 . In this case, r in = ∆r = r c , we require |τ c | < 1 when an electron reaches r c and also require l c (r) > r at r LC < r < r c . Using the result γ c > 10 4.2 and θ pl,c < 10 −4.2 for |τ c | < 1 (r c 10 11.2 cm ≈ 10 3 r LC ), γ(r) at the range of r LC < r < 10 11.2 cm should be changed with r as follows (see also Equation (30) and Figure  6). For the 'Narrow' and 'Wide' cases, we require that the point 'B' (a < 0.5) or point 'C' (a ≥ 0.5) in Figure 6 is more distant than 10 11.2 cm. For example, if γ has a constant value (a = 0), we require γ > 10 4.2 at r LC . On the other hand, if a ≥ 0.5 with γ LC > 10 2.7 , γ should have a terminal value of γ > 10 4.2 . Although the 'Inclined' case is a bit complicated, we can constrain the behavior of γ by replacing γ with θ −1 pl in the above discussion and using the condition γ > θ −1 pl (Θ pl > 1) for the 'Inclined' case. Required values of the exponents a and b change with the value of u LC , σ LC and σ c .
Lastly, we mention the result obtained by WR78. Essentially, the 'Wide' geometry with scattering at r c ∼ 10 11.2 cm of ours corresponds to the situation which they considered, although their setup is not exactly the same as ours in the radial variations of γ(r) and n pl (r). Our result of γ c 10 4.8 (1 + σ c ) −1/6 obtained in Section 3.3.2 is close to their result of γ > 10 4.4 (see their Equation (16)). Note that we did not consider the 'Wide' case with scattering at r c because γ c < 10 4.2 is also required for the geometry to be 'Wide'. Also note that they did not account for the constraint at r LC , although we require γ LC > 10 2.7 and θ pl,LC < 10 −2.7 for r c > r LC .

Constraints on Pair Multiplicity
In the last section, we obtain lower limits of γ for a given inclination angle θ pl and a magnetization σ of the pulsar wind. Here, we consider corresponding upper limits of κ using Equation (1). Note that the combination of κγ(1 + σ) = 10 10.5 is independent of r from energy conservation law and that κ alone is also expected to be independent of r from the law of conservation of particle number. Below, we consider the upper limits of κ for the two possible u LC of the pulsar wind and we do not consider constraint for the geometry Θ bm > Θ pl > 1 for simplicity. 18/26 When the pulsar wind is inclined with respect to the radio pulses at r LC (10 −1.2 < θ pl,LC 1), we obtain an upper limit of κ by eliminating γ LC from γ LC θ pl,LC (1 + σ LC ) 1/4 10 3.7 with the use of Equation (1) (κγ(r)(1 + σ(r)) = 10 10.5 ). We obtain κ 10 6.8 θ pl,LC (1 + σ LC ) − 3 4 .
Because κ conserves along the flow, κ should satisfy both of the two inequalities. Even for 1 + σ c ∼ 1, κ 10 6.3 at r c ∼ 10 3 r LC is marginal for κ > κ PWN . For customarily used magnetization σ LC ∼ 10 4 , an upper limit is κ 10 3.8 ≪ κ PWN . The results are summarized in Table 3. A little bit larger κ is allowed for the inclined u LC (θ pl,LC ∼ 1) than for the aligned u LC with respect to the radio pulse beam.

Dependence on the Size of Emission Region
We assume r e = 10 7 cm in the above calculations. Here, we discuss the constraints on γ and κ assuming Equation (23) with r e = 10 3 cm for example. The dependence on r e (10 3 ≤ r e ≤ 10 7 cm) is described explicitly in Tables 1 and 2. When we take a different value of r e , the brightness temperature T b (Equation (24)) and the integrals I Narrow and I Inclined (Equations (19) and (20)) are changed. In Tables 1 and 2, we find that the optical depth for the 'Narrow' and 'Inclined' cases is proportional to r 2 e . This is because I Narrow and I Inclined are proportional to r 4 e and T b is proportional to r −2 e . On the other hand, for the 'Wide' case, the optical depth is proportional to r −2 e because I Wide (ν) ∼ −1 whose value does not depend on θ bm in the range of ν 0 ν < Θ 2 bm ν 0 . Note that the layout of scattering geometry on the γ − θ pl diagrams ( Figure 9) is also changed where the 'Narrow' and 'Inclined' areas spread on the planes compared with those in Figures 7 and 8.
We obtain the lower limits of γ and the upper limits of κ in the same manner as the case of r e = 10 7 cm. Figure 9 shows the resultant γ − θ pl diagrams both at r LC (left) and r c (right). Obtained lower limits of γ and upper limits of κ are summarized in Table 4.
At r LC (θ bm (r LC ) ≈ 10 −5.2 ), we find two allowed regions on the diagram in the left panel of Figure 9. First is when the pulsar wind has a significant non-radial motion 10 −2.7 < θ pl,LC 1. We require γ LC θ pl,LC (1 + σ LC ) 1/4 10 1.7 r 1/2 e,3 for |τ LC | < 1 and no scattering occurs beyond r LC for the moderate values of the exponents a and b. We also find that the non-relativistic pulsar wind β LC ≪ 1 is unfavorable even for such a small opening angle of the radio beam θ bm,LC = 10 −5.2 with 1 + σ LC ≈ 10 4 .
Secondly, the region which satisfies γ LC > 10 2.7 and θ pl,LC < 10 −2.7 is also allowed to escape from scattering at r LC due to the 'lack of time' effect. In this case, in addition, we require |τ c | < 1 at r c (> r LC ). The right panel of Figure 9 shows the γ − θ pl diagram at r c . We It is important to note that the constraint at r c very weakly depends on r e as r 1/5 e . Accordingly, we obtain upper limits of κ with the help of Equation (1). When the pulsar wind is inclined with respect to the radio pulse at r LC (10 −2.7 < θ pl,LC 1), we obtain We require σ LC 10 3 ≪ 10 4 for κ > κ PWN . When the pulsar wind is aligned with respect to the radio pulse at r LC (θ pl,LC < 10 −2.7 and γ LC > 10 2.7 ), we obtain for 'Inclined'. (35) κ > κ PWN is attainable for both the 'Narrow' and 'Inclined' cases again. We obtain the lower limits of γ and the upper limits of κ for different sizes of the emission region r e . Basically, as is found from Table 4, the smaller the emission region size becomes, the easier the radio pulses escape from scattering, i.e., small γ and large κ are allowed. We obtain the most optimistic constraint for large κ (κ 10 8.8 at the uppermost row of Table 4), when θ pl,LC ∼ 1 (inclined u LC ), 1 + σ LC ∼ 1 and r e = 10 3 cm. Combined with κ κ PWN = 10 6.6 , we can write the pulsar wind properties as 10 1.7 γ 10 3.9 and κ PWN κ 10 8.8 . Although all these constraints are at r LC , the radio pulse can escape from scattering and κ κ PWN is satisfied beyond r LC because γ(r)(1 + σ(r)) ≈ γ(r) = constant beyond r LC for 1 + σ LC ∼ 1 from Equation (1) and conservation of particle number (κ = constant). Note that we obtain 10 1.2 γ LC 10 1.9 and κ PWN κ 10 7.3 for 1 + σ LC ∼ 10 2 , and we require γ(r)(1 + σ(r)) = constant and also κ = constant beyond r LC .

Summary
To constrain the pulsar wind properties, we study induced Compton scattering by a relativistically moving cold plasma. Induced Compton scattering is θ 4 bm k B T b (ν)/m e c 2 times significant compared with spontaneous scattering for the non-relativistic case. However, for scattering by the relativistically moving plasma, scattering geometry of the system changes the scattering coefficient significantly. We consider fairly general geometries of scattering in the observer frame and obtain the scattering coefficient for induced Compton scattering off the photon beam. On the other hand, we do not take into account the magnetic field effects and the scattering off the background photons in this paper.
We obtain approximate expressions of the scattering coefficient for three geometries corresponding to the 'Narrow' (1 > Θ 2 bm + Θ 2 pl ), 'Inclined' (Θ 2 pl > 1 + Θ 2 bm ) and 'Wide' (Θ bm > 1 > Θ pl ) cases, while the scattering coefficient for Θ bm > Θ pl > 1 is obtained numerically. Behavior of the scattering coefficient against a given scattering geometry is governed by a simple combination of four factors. In addition to the solid angle factor θ 4 bm appearing even for the non-relativistic case, there exist three relativistic effects; the factor independent of scattering geometry γ −3 and the other two factors depending on geometry, the aberration factor D −2 1 and the frequency shift factor D/D 1 . When the photon beam is inside the γ −1 cone of the plasma beam (the 'Narrow' case), the aberration factor increases the scattering coefficient by a factor of ∼ γ 4 (up to γθ bm ∼ 1). On the other hand, when the plasma velocity is significantly inclined with respect to the photon beam (the 'Inclined' case), this factor of 21/26 γ 4 does not appear. The frequency shift factor is important when the photon beam is wider than the γ −1 cone of the plasma beam (the 'Wide' case) and is rather complex and mostly increases the absolute value of the scattering coefficient compared with the non-relativistic case. Basically, the 'Inclined' case gives the smallest and the 'Wide' case gives the largest scattering coefficient, i.e., the Θ bm > Θ pl > 1 case is in between.
We apply induced Compton scattering to the Crab pulsar, where the high T b (ν) radio pulses go through the relativistic pulsar wind and constrain the pulsar wind properties by imposing the condition of the optical depth being smaller than unity. We introduce the characteristic scattering radius r c where the 'lack of time' effect prevents scattering at r < r c . We evaluate the scattering optical depth for both r in = r LC and r in = r c cases. We consider more general scattering geometries than WR78 and also study the dependence on the size of the emission region 10 3 ≤ r e ≤ 10 7 cm which directly affects the opening angle of the radio pulses θ bm (r). Allowable pulsar wind velocities at r LC (u LC ) and at r c (u c ) are explored assuming the canonical value of the magnetization 1 < 1 + σ 10 4 .
The two pulsar wind velocities u LC are allowed for radio pulses to escape from scattering at r LC . One is that the plasma velocity is inclined with respect to the photon beam (θ pl,LC ∼ 1). When γ LC 10 1.7 r 1/2 e,3 θ −1 pl,LC (1 + σ LC ) −1/4 is satisfied, the radio pulses reach the observer without scattering for moderate radial variation of γ(r) and θ pl (r) where γ ∝ r a and θ pl ∝ r −b with 0 < (a, b) 1.25. The other is when the plasma velocity is aligned with respect to the photon beam (θ pl,LC < 10 −2.7 ). We require the lower limit γ LC 10 2.7 for the 'lack of time' effect preventing scattering at r LC . In this case, we also require the optical depth at r c 10 9.6 r 2/5 e,3 cm = 10 1.4 r 2/5 e,3 r LC to be less than unity, where r c (= l c ) depends on γ c or θ c (Equation (28)). For example, we require γ c 10 3.4 r 1/5 e,3 (1 + σ c ) −1/10 for the completely aligned case θ pl = 0. Basically, the smaller the emission region size and the larger the inclination angle of the pulsar wind become, the smaller γ is allowed.
We discussed upper limits of the pair multiplicity using obtained constraints on the velocities of the Crab pulsar wind and Equation (1). In principle, κ κ PWN ≡ 10 6.6 [11,13] is possible although we require 1 + σ LC ≪ 10 4 , i.e., customarily used value 1 + σ LC ≈ 10 4 contradicts κ > κ PWN . The most optimistic constraint which allows large κ is obtained when θ pl,LC ∼ 1 and r e = 10 3 cm (Equation (34)). In this case with κ κ PWN , we can write the pulsar wind properties as 10 1.7 γ 10 3.9 and κ PWN κ 10 8.8 for 1 + σ LC ∼ 1 and 10 1.2 γ 10 1.9 and κ PWN κ 10 7.3 for 1 + σ LC ∼ 10 2 . Note that all these constraints are at r LC and we also require moderate radial variation of θ pl (r) and γ(r) (∝ (1 + σ(r)) −1 ) beyond r LC . normalized angles Θ bm ≡ γθ bm and Θ pl ≡ γθ pl rather than on θ bm , θ pl and γ, separately. As seen in Section 2.3, the behavior of I(ν) is very different for the value of Θ bm and Θ pl , i.e., different scattering geometries. To obtain the results of Figures A1 and A2, we set γ = 10 2 and adopt the broken power-law spectrum with p 1 = 3 and p 2 = −5 (Equation (10)). The figures show absolute values of the integral I(ν) versus frequency ν for different sets of parameters Θ bm and Θ pl . All the lines in these figures have a discontinuity where the sign of the integral I(ν) changes. The sign of the integral I(ν) is positive at high frequency side where the photon number decreases and vice versa.
Before describing details of Figures A1 and A2, we mention that the approximated forms studied in Section 2.3 can describe behaviors of most of lines in the figures. Behaviors of lines with no frequency shift is described by I Narrow and I Inclined and behaviors of lines whose discontinuity point shifted to ν > ν 0 is described by I Wide . Only behaviors of 'line d' and 'line e' in the bottom-left panel in Figure A1 and of 'line e' in the bottom-left panel in Figure  A2 are not explained by these three approximated forms corresponding to Θ bm > Θ pl > 1 which we will discuss later. 23 Figure A1 shows how the integral I(ν) changes with Θ pl (0 ≤ Θ pl ≤ 10) for fixed Θ bm . Three panels in Figure A1 correspond to different fixed values of Θ bm and the bottom-right sketch describes scattering geometry when Θ bm = 10 corresponding to the bottom-left panel in Figure A1, for example. It is common for all the panels that 'line a' is very close to 'line b', i.e, we can approximate that the photon and plasma are completely aligned (Θ pl = 0) even for Θ pl < 1. It is also common for all the panels that 'line a' is larger than other lines for ν > ν 0 and |I(ν)| decreases in order from 'line a' to 'line e', i.e., |I(ν)| is large when the photons and the plasma are aligned at least the frequency range ν > ν 0 . The top-left panel (Θ bm = 0.1) shows the case when the photon beam is considered as narrow (compared with γ −1 cone associated with the plasma) and shows little frequency shift D/D 1 ≈ 1 corresponding to I Narrow and I Inclined studied in Section 2.3. The bottom-left panel in Figure A1 is the case when the photon beam is considered as wide (Θ bm = 10: the bottom-right sketch of Figure  A1). In this case, the frequency shift effect is extreme and the absolute values |I(ν)| is almost unity at broad frequency range. Figure A2 shows how the integral I(ν) changes with Θ bm (0.1 ≤ Θ bm ≤ 10) for fixed Θ pl . Three panels in Figure A2 correspond to different fixed values of Θ pl and the bottom-right 24/26 sketch describes scattering geometry when Θ pl = 1 corresponding to the top-right panel in Figure A2, for example. Note that some lines are the same parameter set with Figure A1. It is common for all the panels that |I(ν)| decreases with the smaller values of Θ bm . 'Line d' and 'line e' on the top-left panel (Θ pl = 0) and top-right panel (Θ pl = 1) show I Wide studied in Section 2.3.