You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We investigate polarization of high-energy emissions from the Crab pulsar in the frame work of the outer gap accelerator, following previous works of Cheng and coworkers. The recent version of the outer gap, which extends from inside the null charge surface to the light cylinder, is used for examining the synchrotron radiations from the secondary and the tertiary pairs, which are produced outside the gap. We calculate the light curve, the spectrum and the polarization characteristics, simultaneously, by taking into account gyration motion of the particles. The polarization position angle curve and the polarization degree are calculated to compare with the Crab optical data. We demonstrate that the radiations from inside the null charge surface make outer-wing and off-pulse emissions in the light curve, and the tertiary pairs contribute to bridge emissions. The emissions from the secondary pairs explain the main features of the observed light curve and spectrum. On the other hand, both emissions from inside the null charge surface and from the tertiary pairs are required to explain the optical polarization behavior of the Crab pulsar. The energy dependence of the polarization features is expected by the present model. For the Crab pulsar, the polarization position angle curve indicates that the viewing angle of the observer measured from the rotational axis is greater than $90^{\circ}$.'
author:
- 'J. Takata'
- 'H.-K. Chang'
- 'K.S. Cheng'
title: 'Polarization of high-energy emissions from the Crab pulsar'
---

INTRODUCTION
============

The *Compton Gamma-Ray Observatory* (CGRO) had shown that young pulsars are strong $\gamma$-ray sources, and had detected seven $\gamma$-ray pulsars (Thompson et al. 1999). The CGRO revealed that the light curve with double peaks in a period and the spectrum extending to above GeV are typical features of the high-energy emissions from the $\gamma$-ray pulsars. Although these data have constrained proposed models, the origin of the $\gamma$-ray emission is not yet conclusive. One important reason is that various models have successfully explained the features of the observed spectra and/or light curves. For example, in the frame works of the polar cap model (Daugherty & Harding 1996), the two-pole caustic model (Dyks & Rudak 2003) and the outer gap model (Romani & Yadigaroglu 1995; Cheng et al. 2000, hereafter CRZ00), the main features of the observed light curve such as two peaks in a period and the emissions between the two peaks are all expected. So, we cannot discriminate the three different models using the observed light curve. Furthermore, both the polar cap and outer gap models have explained the observed $\gamma$-ray spectrum (Daugherty & Harding 1996; Romani 1996).

Polarization measurement will play an important role to discriminate the various models, because it increases the number of observed parameters, namely, polarization degree (p.d.) and position angle (p.a.) swing. So far, only the optical polarization data for the Crab pulsar is available (Smith et al. 1988; Kanbach et al. 2005) in high energy bands. For the Crab pulsar, the spectrum is continuously extending from optical to $\gamma$-ray bands. In addition, the pulse positions in the wide energy bands are all in phase, which would indicate that the optical emission mechanism is related to higher energy emission mechanisms. In the future, the next generation Compton telescope will probably be able to measure polarization characteristics in MeV bands. These data will be useful for discriminating the different models.

Chen et al. (1996) considered the polarization characteristics in the peaks of the light curve for the Crab pulsar with an outer gap model. In that model, the synchrotron radiation was used. The model assumed that the charge particles are distributed with a Gauss function in the azimuthal direction to guarantee the formation of the peaks in the light curve. Romani & Yadigaroglu (1995) calculated the polarization characteristics predicted by the curvature radiation process in the frame work of the one pole outer gap model. In that model, however, the optical polarization data of the Crab pulsar was reproduced by very specialized selection of the model parameters such as the inclination angle and the viewing angle of the observer (Dyks et al. 2004, hereafter DHR04). DHR04 showed that the two-pole caustic geometry, in which the acceleration region extends from the stellar surface to near the light cylinder, explains the pattern of the p.d. and the fast swing of the p.a. at both peaks in the light curve. DHR04 also showed that the effect of depolarization due to overlap of the emissions from the different magnetic field lines is not strong so that the intrinsic level of the polarization degree at each radiating point remains in the bridge and off-pulse phases of the light curve. Recently, Pétri & Kirk (2005) proposed that the optical emission originates from outside the light cylinder and calculated the polarization characteristic predicted by the pulsar striped wind model (Kirk et al. 2002). However, the observations show that the pulse peaks of the radio, optical, X-ray and $\gamma$-ray are all in phase, and it is not clear how the pulsar striped wind can radiate in multi frequency. Furthermore, all of the previous models have not considered the spectrum, the light curve and the polarization all together.

In this paper, we examine the optical polarization characteristics of the Crab pulsar with the spectrum and the light curve predicted by modifying the 3-D outer gap model in CRZ00. CRZ00 has calculated the synchrotron self-inverse Compton scattering process of the secondary pairs produced outside the outer gap and has explained the Crab spectrum from X-ray to $\gamma$-ray bands. Zhang & Cheng (2002) reconsidered CRZ00 model and calculated the energy dependent light curves and the phase resolved X-ray spectrum. In CRZ00, however, the outer-wing and the off-pulse emissions of the Crab pulsars cannot be reproduced, because the traditional outer gap geometry, which extends from the null charge surface of the Goldreich-Julian charge density (Goldreich & Julian 1969) to the light cylinder along the magnetic field lines and about $180^{\circ}$ in azimuthal direction, is assumed. Furthermore, the spectrum in the optical band was not considered. In this paper, on these grounds, we modify the CRZ00 geometrical model into a more realistic model, following recent 2-D electrodynamical studies (Takata et al. 2004, 2006; Hirotani 2006), and we examine the light curve, the spectrum and the polarization characteristics of the Crab pulsar.

In section \[model\], we present the high-energy emission model and the calculation method for the polarization. In section \[result\], we compare the polarization characteristics in the optical band with the Crab data, and demonstrate that the present model reproduces the observed light curve, the spectrum and the polarization characteristics all together. We diagnose the viewing angle for various inclination angle and for various emission height by comparing the model with the Crab optical data. In section \[discuss\], we predict the polarization characteristics in higher energy bands.

EMISSION MODEL {#model}
==============

The outline of the outer gap model for the Crab pulsar is as follows. The charge particles are accelerated by the electric field (section \[oustruc\]) parallel to the magnetic field lines in so called gap, where the charge density is different from the Goldreich-Julina charge density (Goldreich & Julian 1969), $\rho_{GJ}\sim -\mbox{\boldmath$\Omega$}\cdot\mbox
{\boldmath$B$}/2\pi c$, where $\mbox{\boldmath$\Omega$}$ is the angular velocity, $\mbox{\boldmath$B$}$ is the local magnetic field and $c$ is the speed of light. The high energy particles accelerated in the gap emit the $\gamma$-ray photons (called primary photons) via the curvature radiation process. For the Crab pulsar, most of the primary photons escaping from the outer gap will convert into secondary pairs outside the gap, where the accelerating electric field vanishes, by colliding with surface and/or synchrotron X-rays emitted by the secondary pairs. The secondary pairs emit optical - MeV photons via the synchrotron process (section \[synemi\]) and photons above MeV with the inverse Compton process. The high-energy photons emitted by the secondary pairs may convert into tertiary pairs at higher altitude by colliding with the soft X-ray from the stellar surface. The tertiary pairs emit the optical-UV photons via the synchrotron process (section \[tertiary\]). This secondary and tertiary photons appear as the observed radiations from the Crab pulsar. In section \[result\], we will show that although the observed main features of the light curve and the spectrum are explained by the emissions from the secondary pairs, the observed polarization characteristics are explained with the emissions from the tertiary pairs, which were not considered in CRZ00.

In section \[oustruc\], we describe the outer gap structure. In section \[synemi\], we discuss the synchrotron emission from the secondary and the tertiary pairs. The calculation method of the polarization are described in section \[stokes\], In section \[para\], we introduce the model parameters of the present study.

Outer gap structure {#oustruc}
-------------------

Because the Crab pulsar has a thin gap, we describe the accelerating electric field (Cheng et al. 1986a, 1986b) with

$$E_{||}(r)=\frac{\Omega B(r)f^2(r)R_{lc}^2}{c s(r)},
\label{electric}$$

where $f(r)$ is the local gap thickness in units of the light radius, $R_{lc}=c/\Omega$, and $s(r)$ is the curvature radius of the magnetic field line. This traditional outer gap model assumes that the outer gap starts from the null charge surface. However, it has been well known that the traditional outer gap geometry cannot reproduce the off-pulse emission of the Crab pulsar. On the other hand, recent 2-D electrodynamical studies (e.g. Takata et al. 2004; Hirotani 2006) for the outer gap accelerator have demonstrated that the inner boundary of the outer gap is shifted toward the stellar surface from the null charge surface by the current through the gap. Therefore, we take into account the radiation and the pair-creation processes inside the null charge surface. In such a case, the emissions between the stellar surface and the null charge surface contribute to the light curves as the outer-wing and the off-pulse emissions.

As demonstrated by the 2-D electrodynamical model, the electric field inside the null charge surface rapidly decreases along the magnetic field line because of the screening effects of the pairs produced near the inner boundary. To simulate the accelerating electric field inside null charge surface in the present geometrical study, we assume that the strength of the accelerating field changes quadratically along the magnetic field line as $$E_{||}(r)=E_{n}\frac{(r/r_{i})^2-1}{(r_{n}/r_{i})^2-1},~~r_i\leq r\leq r_n, 
\label{electeq}$$ where $E_{n}$ is the strength of the electric field at the null charge surface and $r_{n}$ and $r_{i}$ are the radial distances to the null charge surface and the inner boundary of the gap, respectively. For the inclined rotator, the radial distance to the null charge surface varies for different field lines so that the $r_n(\phi)$ is a function of the azimuthal angle ($\phi$). In this paper, the ratio $r_i(\phi)/r_n(\phi)$ is assumed to be a constant for each field line (that is, no azimuthal dependence of $r_i/r_n$), and is treated as a model parameter. The local Lorentz factor of the accelerated particles (called primary particles) in the outer gap is described by $\Gamma_p(r)=[3s^2(r)E_{||}/2e]^{1/4}$ with assuming the force balance between the acceleration and the curvature radiation back reaction.

Rhe primary photons emitted in the outer gap may make the pairs inside and outside the gap with the soft X-ray from the stellar surface or the synchrotron radiation of the secondary pairs. CHRb (1986) considered the pair creation process in the gap between the primary curvature photons and the soft photons emitted by the secondary pairs, which was produced outside of the gap by the pair-creation process of the primary photons, and estimated the typical fractional gap size as $f\sim33.2B^{-13/12}_{12}P^{33/20}$, where $B_{12}$ is the strength of the stellar magnetic field in unit of $10^{12}$ G and $P$ is the rotational period. However, the soft photons emitted from the secondary pairs may be beamed out of the outer gap, because the secondary pairs are created just outside the gap and emit the photons to the convex side of the field lines. The screening pairs will be created by the pair-creation process between the primary curvature photons and the surface X-rays. In such a case, Zhang & Cheng (1997) estimated typical fractional size of the outer gap as $$f(R_{lc}/2)~\sim 5.5B_{12}^{-4/7}P^{26/21}.
\label{fsize}$$ The local fractional size of the outer gap is estimated by $f(r)\sim f(R_{lc}/2)(2r/R_{lc})^{1.5}$ (CRZ00). In this paper, we use the fractional gap size $f(R_{lc})=0.11$ with $B_{12}=3.7$, which is inferred from the dipole radiation model of the pulsar spin down.

The present 3-D geometrical model assumes that the outer gap extends around the whole polar cap, because we have not had any reliable model for the 3-D geometry of the acceleration region. If we consider the emission region extending very close to the light cylinder $\rho\sim R_{lc}$, the expected light curve with a moderately changing emissivity along the field line may have triple or more peaks. Therefore, we expect that the emissivity near the light cylinder is declined and/or the radiations from near the light cylinder are beamed out of line of sight due to the magnetic bending. In the calculation, we constrain the boundaries of the axial distance and radial distance for the emission regions with $\rho_{\max}=0.9R_{lc}$ and $r=R_{lc}$, respectively.

Synchrotron emission from the pairs {#synemi}
-----------------------------------

### secondary pairs

The primary photons escaping the outer gap convert into the secondary pairs by colliding the non-thermal X-ray photons, which were emitted by the synchrotron process of the secondary pairs. The photon spectrum of the synchrotron radiation by the secondary pairs is described by (CRZ00) $$F_{syn}(E_{\gamma},r)=\frac{3^{1/2}e^3B(r)\sin\theta_{p}(r)}{mc^2h
E_{\gamma}}\int\left[\frac{dn_e(r)}{dE_e}\right]F(x)dE_edV_{rad},
\label{emis}$$ where $x=E_{\gamma}/E_{syn}$, $E_{syn}(r)=3he\Gamma_s^2(r)
B(r)\sin\theta_p(r)/4\pi m_ec$ is the typical photon energy of the secondary pairs, $\Gamma_s$ represents Lorentz factor of the secondary pairs, $\theta_p$ is the pitch angle of the particle, $F(x)=x\int_x^{\infty} K_{5/3}(y)dy$, where $K_{5/3}$ is the modified Bessel function of order 5/3, and $dV_{rad}$ is the volume element of the radiation region considered. The distribution of the pairs is given by $$\frac{dn_e}{dE_{e}}\sim\frac
{l_{cur}n_{GJ}\mathrm{ln}(E_{cur}/E_e)}{\dot{E}_eE_{cur}},
\label{dist}$$ where $l_{cur}=eE_{||}c$ is the local power of the curvature radiation, $n_{GJ}=\Omega B/2\pi ce$ is the Goldreich-Julian number density, $E_{cur}(r)=3h\Gamma_{p}^3(r)c/4\pi s(r)$ is the characteristic energy of the curvature photons emitted by the primary particles and $\dot{E}_{e}= 2e^4B^2(r)\sin^2\theta_p(r)
\Gamma_s^2/3m^2c^3$ is the energy loss rate of the synchrotron radiation of the secondary pairs.

The pitch angle of the secondary pairs is estimated from $\sin\theta_p(R_{lc})\sim \lambda/s(R_{lc})$, where $\lambda$ is the mean free path of the pair-creation between the primary $\gamma$-rays and the non-thermal X-rays from the secondary pairs. The mean free path is estimated from $\lambda^{-1}\sim  n_X\sigma_{\gamma\gamma}$, where $n_X$ is the typical non-thermal X-ray number density and $\sigma_{\gamma\gamma}$ is the pair-creation cross section, which is approximately given by $\sigma_{\gamma\gamma}\sim \sigma_{T}/3$, where $\sigma_{T}$ is the Thomson cross section. For the Crab pulsar, the typical number density becomes $n_{X}\sim L_{X}(<E_X>)/\delta\Omega R_{lc}^2
c <E_X>\sim 8\times10^{17}~\mathrm{cm}^3$, where we used the typical energy $<E_X>\sim (2m_ec^2)^2/10
~\mathrm{GeV}\sim100$ eV, the typical non-thermal X-ray luminosity $L_X\sim
10^{35}\mathrm{erg/s}$, and the solid angle $\delta\Omega=1$ radian. As a result, the mean free path becomes $\lambda\sim
10^{7}$ cm so that the pitch angle is estimated by $\sin\theta_{p}\sim\lambda/s(R_{lc})\sim 0.06$, where we used the curvature radius $s(R_{lc})=R_{lc}$. In this paper, therefore, we adopt $\sin\theta_p(R_{lc})=0.06$. The local pitch angle is calculated from $\sin\theta_p(r)=\sin\theta_p(R_{lc})(r/R_{lc})^{1/2}$.

The outer gap extends above the last-open lines with the thickness $f(R_{lc})$. And then, we assume that the secondary pair region extends just above the outer gap with the thickness $\lambda$.

### tertiary pairs {#tertiary}

Some high-energy photons emitted by the inverse Compton process of the secondary pairs may convert into tertiary pairs at higher altitude by colliding with thermal X-ray photons from the star. The energy of the new born tertiary pairs will be described by the most energetic secondary photons from the secondary pairs. According to the study by CRZ00, the most energetic ($\sim$1GeV) secondary photons via the inverse Compton process of the secondary pairs are about one order magnitude smaller than that of the primary photons ($\sim$10GeV), which make the secondary pairs. Therefore, we expect that the tertiary pairs are produced with a Lorentz factor of one order magnitude smaller than that of the secondary pairs. The optical depth of the pair-creation between the high-energy photons emitted by the secondary pairs and the thermal X-ray photons from the stellar surface is estimated as $\tau\sim
n_X\sigma_{\gamma\gamma}R_{lc}\sim 0.1$, with the typical thermal X-ray number density $n_X\sim4\pi R_*^3\sigma T^4/4\pi R_{lc}^2 c
k_BT\sim 5\times 10^{15}~\mathrm{/cm^3}$, where $R_*=10^{6}~\mathrm{cm}$ is the stellar radius, $\sigma$ is the Stefan-Boltzmann constant, $k_B$ is the Boltzmann constant, and $T$ is the surface temperature, for which we adopt the reasonable value $T=2\cdot10^{6}$ K (Yakovlev & Pethick 2004). In this paper, therefore, we use that the maximum energy of and the local number density of the tertiary pairs are smaller than about $10\%$ of those of the secondary pairs. Because the pitch angle of the pairs increases with altitude, we use $\sin\theta_p=0.1$ for the pitch angle of the tertiary pairs. In fact, the results are not sensitive to the pitch angle of the tertiary pairs.

The tertiary pairs are produced above the region of the secondary pairs. As we will see in section \[lcurve\], the emissions from the secondary pairs make the main features of the light curve such as the two peaks in a period, and the emissions from the tertiary pairs contribute to the bridge phase.

### synchrotron cooling {#cooling}

For the Crab pulsar, the observed spectrum has a spectral break around 10 eV. This feature was not considered in CRZ00. One possibility of the explanation of the observed break is due to the effect of the synchrotron cooling. The damping length due to the synchrotron cooling is given by $l_{syn}\sim3\cdot10^{4}(B/10^7\mathrm{G})^{-2}(\Gamma_{\perp}/10^2)^{-1}
(\sin\theta_p/0.06)$ cm, where $\Gamma_{\perp}=\Gamma/\Gamma_{||}$ and $\Gamma_{||}=1/\sin\theta_p$. The charge particles quickly lose their perpendicular momentum via the synchrotron radiation. The minimum Lorentz factor of the pairs in the magnetosphere may be described by $\Gamma\sim \Gamma_{||}\sim 17$, which is corresponding to the synchrotron characteristic energy $E_{c}\sim 3 (\Gamma/17)^2(B/10^7~\mathrm{G})(\sin\theta_p/0.06)$ eV. In the present model, therefore, the spectral index $s_{\nu}$ of the synchrotron photons varies from $s_{\nu}=(p-1)/2$ with $p=2$ of equation (\[dist\]) in X-ray bands to $s_{\nu}\sim-1/3$, which is reflecting the single particle emissivity, below the energy $E_c$.

### emission direction of the pairs {#emid}

We use the rotating dipole field in the inertial observer frame (hereafter IOF). On the other hand, the previous works such as CRZ00 and DHR04 used the rotating dipole field in the co-rotating frame, in which the emission direction coincides with the local magnetic field direction, and performed the Lorentz transformation to calculate the emission direction in IOF. As a result, the configuration of the magnetic field in IOF is different between the present model and the previous works, although the difference is small except for near the light cylinder.

For a high Lorentz factor, we can anticipate that the emission direction of the particles coincides with the direction of the particle’s velocity. In IOF, the motion of the pairs created outside of the gap may be described by $$\mbox{\boldmath$n$}=\beta_0\cos\theta_p\mbox{\boldmath$b$}
+\beta_0\sin\theta_p\mbox{\boldmath$b$}_
{\perp}+\beta_{co}\mbox{\boldmath$e$}_{\phi},
\label{pmotion}$$ where the first term in the right hand side represents the particle motion along the field line, $\mbox{\boldmath$b$} 
=\mbox{\boldmath$B$}/B$ (or $-\mbox{\boldmath$B$}/B$) for the particles migrating parallel (or counter parallel) to the direction of the magnetic field. In this paper, we consider only outgoing particles because the photons emitted by ingoing particles will be much fainter than that by the outgoing particles (CRZ00). The second term in equation (\[pmotion\]) represents gyration motion around the magnetic field line and the third term is co-rotation motion with the star, $\beta_{co}=\rho\Omega/c$. The unit vector $\mbox{\boldmath$b$}_{\perp}$ perpendicular to the magnetic field line becomes $$\mbox{\boldmath$b$}_{\perp}\equiv\pm(\cos\delta\phi\mbox{\boldmath$k$}+\sin\delta\phi\mbox{\boldmath$k$}
\times\mbox{\boldmath$b$}),
\label{gyration}$$ where $\pm$ represents the gyration of the positrons ($+$) and the electrons ($-$), $\delta\phi$ refers the phase of gyration motion and $\mbox{\boldmath$k$}=(\mbox{\boldmath$b$}\cdot\nabla)\mbox{\boldmath$b$}/
|(\mbox{\boldmath$b$}\cdot\nabla)\mbox{\boldmath$b$}|$ is the unit vector of the curvature of the magnetic field lines. The gyration phase $\delta\phi$ is defined so that the value increases in the direction of the gyration motion of the positrons and so that the pairs with $\delta\phi=0$ emit the photons in the plane spanned by the directions of the local magnetic field and it’s curvature if there were no co-rotation motion in equation (\[pmotion\]). The co-rotation motion affects the emission direction as the aberration effect. The value of $\beta_0$ at each point is determined by the condition that $|\mbox{\boldmath$n$}|=1$.

The emission direction of equation (\[pmotion\]) is described in terms of the viewing angle measured from the rotational axis, $\xi=\cos^{-1}n_z$, and the rotation phase, $\Phi=-\Phi_{n}-\mbox{\boldmath$r$}\cdot\mbox{\boldmath$n$}$, where $n_z$ is the component of the emission direction parallel to the rotational axis, $\Phi_{n}$ is the azimuthal angle of the emission direction and $\mbox{\boldmath$r$}$ is the emitting location in units of the light radius.

Because the particles distribute on the gyration phase $\delta\phi$, the emitted beam at each point must become cone like shape with opening angle $\theta_p (r)$. For each radiating point, the emission directions of the different particles on the gyration phase are projected onto the different points in $(\xi,\Phi)$ plane. Furthermore, the polarization plane of the radiations also depends on the gyration phases of the radiating particles. Taking into account the dependence on the gyration phase, therefore, we calculate the radiations from the particles for all of the gyration phase $\delta\phi= 2\pi i/n$ ($i=1,\cdots,n-1$). Figure \[map\] shows the emission projection onto $(\xi,\Phi)$ plane for different gyration phases $(\delta\phi=0^{\circ},
~90^{\circ},~180^{\circ}$ and $270^{\circ})$ for the positrons. For the electrons, we find from the equation (\[gyration\]) that the emission projection map of the electrons is identical with Figure \[map\] but the gyration phase is different by $180^{\circ}$; for example, the panels for $\delta\phi=0^{\circ}$ and $90^{\circ}$ for the positrons in Figure \[map\] also describe the projection maps of emission from the electrons with $\delta\phi=180^{\circ}$ and $270^{\circ}$, respectively.

With the projection map of the emissions, the expected pulse profile is determined by choosing the viewing angle $\xi$ of the observer and collecting all photons from the possible emitting points and the gyration phases with the emissivity of equation (\[emis\]).

The Stokes parameters {#stokes}
---------------------

We assume that the radiation at each point linearly polarizes with degree of $\Pi_{syn}=(p+1)/(p+7/3)$, where $p$ is the power law index of the particle distribution, and circular polarization is zero, that is, $V=0$ in terms of the Stokes parameters. The direction of the electric vector of the electro-magnetic wave toward the observer is parallel to the projected direction of the acceleration of the particle on the sky (Blaskiewicz et al. 1991). The magnitude of the microscopic acceleration of the gyration motion is much larger than that of the macroscopic acceleration of the co-rotation motion so that the ratio of the magnitude of the two accelerations becomes $\omega_B/\Omega\sim 10^8 (B/10^6\mathrm{G})
(\Gamma/10^3)^{-1}$, where $\omega_B$ and $\Omega$ are the gyration and the co-rotation frequencies, respectively. Unless the pitch angle is very small, the acceleration with equation (\[pmotion\]) is approximately written by $$\mbox{\boldmath$a$}\sim \beta_{0}\omega_B\sin\theta_p(-\sin\delta\phi\mbox{\boldmath$k$}
+\cos\delta\phi\mbox{\boldmath$k$}\times\mbox{\boldmath$b$}).
\label{accele}$$ The electric vector $\mbox{\boldmath$E$}_{em}$ emitted in the direction $\mbox{\boldmath$n$}$ becomes $\mbox{\boldmath$E$}_{em}\propto \mbox{\boldmath$a$}
-(\mbox{\boldmath$n$}\cdot\mbox{\boldmath$a$})\mbox{\boldmath$n$}$.

To calculate the Stokes parameters $Q^i$ and $U^i$ for each radiating point, we define the position angle $\chi^i$ to be angle between the electric field $\mbox{\boldmath$E$}_{em}$ and the projected rotational axis on the sky, $\mbox{\boldmath$\Omega$}_p=\mbox{\boldmath$\Omega$}-
(\mbox{\boldmath$n$}\cdot\mbox{\boldmath$\Omega$})\mbox{\boldmath$n$}$. The Stokes parameters $Q^{i}$ and $U^{i}$ at each radiation is represented by $Q^{i}=\Pi_{syn}I^{i}\cos2\chi^{i}$ and $U^{i}=\Pi_{syn}I^{i}\sin2\chi^{i}$, where $I^{i}$ is the intensity. After collecting the photons from the possible points for each rotation phase $\Phi$ and a viewing angle $\xi$, the expected p.d. and p.a. are, respectively, obtained from $$P(\xi,\Phi)=\Pi_{syn}\frac{\sqrt{Q^2(\xi,\Phi)
+U^2(\xi,\Phi)}}{I(\xi,\Phi)},$$ and $$\chi(\xi,\Phi)
=0.5\mathrm{atan}
\left[\frac{U(\xi,\Phi)}{Q(\xi,\Phi)}\right],$$ where $Q(\xi,\Phi)=\Sigma Q^i$ and $U(\xi,\Phi)=\Sigma U^i$ are the Stokes parameters after collecting the photons.

Finally, we describe the difference between polarization characteristics predicted by the curvature emission and synchrotron emission models. If we ignore the effects of the aberration due to the co-rotation motion, the direction of the electric vector of the wave for the curvature and synchrotron cases are, respectively, parallel to and perpendicular to the magnetic field projected on the sky (Rybicki & Lightman 1979). Secondly, in the curvature radiation model the photons are radiated only one direction at each point, which coincides with the direction of the local magnetic field line if we ignore the aberration effect. Furthermore, the radiations from neighboring positions polarize in similar directions. In such a case, the intrinsic level of the polarization degree at the each radiating point remains in the bridge and off-pulse phase of the light curve (DHR04). Therefore, to explain the observed polarization degree $\sim10\%$ at the bridge phase of the Crab pulsar, the curvature emission model may require the radiation linearly polarized with about 10% at each radiating points. As we have mentioned for the synchrotron case, on the other hand, the photons are emitted along the surface of the cone with opening angle $\theta_p$ at each radiating position. Furthermore, the polarization plane of the radiation depends on the gyration phase of the radiating particle. In such a case, the observed radiation consists of the radiations from the different particles on the gyration phase. This overlap of the radiations causes a strong depolarization, and as a result a lower polarization degree is expected for the synchrotron case. We need not assume the radiations with a low polarization degree at each radiating point to explain the Crab data. In the present model, the intrinsic polarization degree of the radiation at the each radiating point is $\sim70\%$ using the particle distribution $p=2$ described by equation (\[dist\]).

Model parameters {#para}
----------------

In this subsection, we introduce the model parameters. The inclination angle $\alpha$ and the viewing angles $\xi$ measured from the rotational axis are the model parameters. Because the inner boundary of the outer gap is determined by the current through the gap (Takata et al. 2004), we consider the position of inner boundary located inside of the null charge. In this paper, the ratio of the radial distances to the inner boundary $r_i$ and the null charge surface of the rotating dipole $r_n$, which is a function of the azimuthal angle, is teated as a model parameters, and is assumed to be constant for each field line as described in section \[oustruc\].

Since the magnetic field must be modified by the rotational and the plasma effects near the light cylinder, the last-open field line must be different with the traditional magnetic surface that is tangent to the light cylinder for the vacuum case. For example, Romani (1996) defined the last open lines as the field lines parallel to the rotational axis at $r=R_{lc}/2^{1/2}$. where the corotational velocity equals the Alfvén speed. To specify the gap upper surface, therefore, it is convenient to refer the footpoints of the magnetic field lines on the stellar surface. With the assumption that the gap upper surface coincides with a magnetic surface, we parameterize the fractional polar angle $a=\theta_u/\theta_{lc}$, where $\theta_u$ is the polar angle of the footpoints of the magnetic field lines of the gap upper surface and $\theta_{lc}$ is the polar angle of the field lines which are tangent to the light cylinder for the vacuum case.

Finally, we describe how the model parameters ($\alpha$, $\xi$, $r_i$ and $a$) are diagnosed by the present model and the Crab data. The model parameters are chosen so that the expected light curve, the spectrum and the polarization characteristics are simultaneously consistent with the Crab data such as the phase separation $\delta\Phi\sim0.4$ phase between two peaks and the large position angle swings at the both peaks. As we will demonstrate in section \[lcurve\], the features of the expected light curve is sensitive to the viewing angle $\xi$ but not to the position of the inner boundary $r_i$, if we fix the inclination angle $\alpha$ and the fractional angle $a$. Therefore, the viewing angle $\xi$ is determined by comparing the model and the observed light curves. With the determined viewing angle, on the other hand, the radial distance to the inner boundary $r_i$ affects sensitively to the polarization characteristics at the off-pulse phase. Therefore, if the inclination angle $\alpha$ and the gap upper surface $a$ are determined in some way, the viewing angle $\xi$ of the observer and the position of the inner boundary of the outer gap $r_i$ are diagnosed by the present model and the Crab data. The present geometrical model produces a consistent spectrum with the Crab data (section \[polari\]) using the viewing angle determined from the observed light curve. We need not introduce another model parameter for fitting the spectrum. It is difficult to constrain both the inclination angle $\alpha$ and the upper surface $a$ with the present model. As we will show in section \[inmag\], however, if either inclination angle $\alpha$ or the altitude of the upper surface $a$ is determined in some way, the other may be diagnosed by the present model.

RESULTS {#result}
=======

Light curve {#lcurve}
-----------

Figure \[compari\] compares the polarization characteristics at 1 eV predicted by three different emission geometries. The left column summarizes the results for the traditional outer gap geometry, in which the inner boundary of the gap is located at the null charge surface of the Goldreich-Julian charge density. Although the traditional model in CRZ00 and the present model assume the different extensions of the outer gap in the azimuthal direction, that is around half (in CRZ00) and whole (in the present model) polar-cap region, we find that the predicted polarization characteristics are not so different between the two azimuthal extension of the outer gap geometries as long as the inner boundary is located at the null charge surface. In this section, therefore, the gap geometry that stars from the null charge surface and extends around whole polar cap region is also called as “traditional geometry”.

The middle and right in Figure \[compari\] columns show the results for the radial distance of 67% of the distance to null charge surface $r_i=0.67r$, and the right column is taking into account also the emissions from the tertiary pairs. The other model parameters are $\alpha=50^{\circ}$, $a=0.94$, and $\xi\sim100^{\circ}$, where the viewing angle is chosen so that the predicted phase separation between the two peaks is consistent with the observed value $\delta\Phi\sim0.4$ phase. In the figure, we define zero of the rotation phase at the main peak.

By comparing the pulse profiles between the light curves of the left and middle columns, we find that the radiations from the secondary pairs inside the null charge surface contribute to the outer-wing and the off-pulse emissions. In the present case, the off-pulse emissions are $<10^{-1}\%$ of the peak flux, because the line of sight marginally passes through the emission regions in the off-pulse phase as the horizontal lines show in Figure \[map\]. Near the inner boundary, because the accelerating electric field and resultant the energy of emitted primary photons in the gap are small, the secondary pairs are produced with a lower energy, and emit the synchrotron photons with a smaller emissivity.

As Figure \[map\] indicates, the emerging radiations in the light curve originate from the two poles; one pole contributes to the light curve with the two peaks and the bridge photons emitted beyond the null charge surface, and the other contributes with the outer-wing and the off-pulse photons emitted inside the null charge surface. Dyks & Rudak (2003) have proposed the radiations associated with two magnetic pole. In that model, with the constant emissivity along the magnetic field lines, the two peaks are associated with the different poles and the different emission regions, and are formed by the caustic effect near the stellar surface. In the present outer gap model, on the other hand, the electric field, and the resultant emissivity of the synchrotron radiation of the secondary pairs quickly decreases inside the null charge surface. Therefore, although the caustic effect near the stellar surface is strong, the emissions inside the null charge surface do not make a strong peak compared with the present main peak, which is formed by the radiations near the light cylinder. In the present case, therefore, the two peaks in the light curve is associated with the one magnetic pole.

By comparing the flux levels of the bridge emissions between the light curves in the middle and right columns, we find that the tertiary pairs mainly contribute to the emissions at the bridge phase. This is because the tertiary pairs are born and emit photons at higher altitude than the secondary pairs, which make two peaks in the light curve.

Polarization {#polari}
------------

Middle and lower panels in Figure \[compari\] show the predicted polarization position angle (p.a.) and the polarization degree (p.d.), respectively. For reference, the light curve is overplotted in each frame.

As seen in the polarization characteristics by the traditional model, we find that the secondary emissions beyond the null charge surface make the polarization characteristics such that the polarization degree takes a lower value at the bridge phase and a larger value near the peaks. In the synchrotron case, the cone like beam is radiated at each point, and an overlap of the radiations from the different particles on the gyration phase causes the depolarization. For the viewing angle $\xi\sim100^{\circ}$, the radiations from all gyration phases contribute to the observed radiation at the bridge phase as the vertical dotted lines at $\Phi=0.2$ phase in Figure \[map\] show. In such a case, the depolarization is strong, and as a result, the emerging radiation from the secondary pairs polarizes with a very low p.d. ($<10\%$). Near the peaks, on the other hand, the radiations from the some gyration phase are not observed as the vertical dotted-dashed lines in Figure \[map\] show. For example, the observer with the viewing angle $\xi\sim100^{\circ}$ detects the photons from the gyration phase $\delta\phi=0^{\circ}$ at the rotation phase $\Phi=0.4$ phase (second peak), but does not detect from $\delta\phi=180^{\circ}$. In such a case, the depolarization is weaker and the emerging radiation highly polarizes.

As the polarization degrees in the middle and right columns in Figure \[compari\] show, the radiations from the inside the null charge surface may be observed with a large polarization degree at the off-pulse phase. This is because the line of sight $\xi\sim 100^{\circ}$ passes through marginally the edge of the radiating region with $r_i=0.67r_n$ at the off-pulse phase as horizontal lines in Figure \[map\] show. The observer can not detect the radiations from the particles within a range of the gyration phase; for example, at the rotational phase $\Phi=0.6$ phase (off-pulse phase) in Figure \[map\], the observer with viewing angle $\xi\sim100^{\circ}$ detects the radiations from the particles with the gyration phases $\delta\phi=180^{\circ}$ and $\delta\phi=270^{\circ}$, but does not with $\delta\phi=0^{\circ}$ and $\delta\phi=90^{\circ}$. In such a case, the depolarization in the off-pulse phase is weak and the expected p.d. exhibits a larger value. Actually, as we will show in section \[inner\], the p.d. at the off-pulse phase is sensitive to the position of the inner boundary $r_i$.

We can see the effects of the tertiary pairs on the polarization characteristics at the bridge phase. By comparing the p.d. between middle and right panels, we find that tertiary pairs produce the radiations with $\sim 10\%$ of the p.d. at the bridge phase.

Comparison with observations {#comob}
----------------------------

Figure \[datmod\] compares the predicted polarization characteristics at 1 eV with the Crab optical data. Left and middle columns are, respectively, the Crab optical data for the total emissions and for the emissions after subtraction of the DC level, which has the constant intensity at the level of 1.24% of the main pulse intensity (Kanbach et al. 2005).

In the total emissions (left column), the impressive polarization feature from the Crab pulsar is that the off-pulse and bridge phases have the fixed value of the p.a. These polarization features of the observation are not predicted by the present model, which predicts about $90^{\circ}$ difference on the p.a. between the off-pulse and the bridge phases as the right column shows.

The constant p.a. in the total emissions may suggest that the Crab optical emissions consist of two components, that is, constant and pulsed components. The DC level emissions may include both the magnetospheric component and the background components (e.g. the pulsar wind and the nebula components). After the subtraction of the DC level, the large p.a. swings larger than $\sim100^{\circ}$ appears at the both peaks, and the constancy of the p.a. at both bridge and off-pulse phase disappears. We can see that the polarization characteristics predicted by the present model are more consistent with the Crab optical data after the subtraction of the DC level. Especially, the model reproduces the most striking feature in the observed p.a. that the large swing at both peaks, and the observed low p.d. at bridge phase $\sim 10\%$. Also, the pattern of the p.d. are reproduced by the present model.

In the off-pulse phase, it is not clarified that which radiation component, that is , the magnetospheric or back ground (e.g the pulsar wind and nebula components) components dominates the other one at the off-pulse phase, where the flux level of the magnetospheric component is much smaller than the peak flux. Furthermore, the data for emissions after subtraction of the DC level has a few photons in the off-pulse phase so that the polarization behavior in the off-pulse phase may not be determined. The present model (e.g. the polarization characteristics in the right column in Figure \[compari\]) predicts that radiation from the magnetospheric component at the off-pulse phase has a relatively constant position angle, which is about $90^{\circ}$ difference from that in the bridge phase. This constancy of the p.a. in the off-pulse phase is not sensitive to the model parameters such as the viewing angle $\xi$ and the position of the inner boundary $r_i$ (Figures \[compari1\] and $\ref{compari2}$).

Although the present model has successfully explained the main features of the Crab optical polarization data, it also has some disagreements with the Crab data. For example, the model predicts the small p.a. swing at the leading-wing of the second peak before appearing the large p.a. swing at the second peak. In the p.d., furthermore, the another peaks in the p.d. at both peaks are predicted.

Figure \[spectrum\] compares the model spectrum with the Crab data in optical-MeV bands. The model parameters are same with that in the right column in Figure \[datmod\]. In this case. we assume that the pairs escape from the light cylinder with the Lorentz factor $\Gamma\sim17$ (section \[cooling\]), which predicts the spectral break around 10 eV. The model spectrum also explains the general features of the data. Therefore, the outer gap model can explain the general features of the observed light curve, the spectrum and the polarization characteristics in optical band for the Crab pulsar, simultaneously.

Dependence on the model parameters {#depend}
----------------------------------

In this section, we discuss the dependence of the polarization characteristics on the model parameters and diagnose that for the Crab pulsar.

### viewing angle {#viewing}

Figure \[compari1\] summarizes the dependence of the polarization characteristics on the viewing angle. With $\alpha=50^{\circ}$, $a=0.94$ and $r_{i}=0.67r_n$, the left, middle and right columns show, respectively, the polarization characteristics for the viewing angle of $\xi\sim95^{\circ}$, $\xi\sim100^{\circ}$ and $\xi\sim105^{\circ}$. For $\xi\sim100^{\circ}$, the phase separation of the two peaks in the light curve is $\delta\Phi\sim0.4$ phase similar with the observation.

We can see that the phase separation becomes wider (or narrower) with decreasing (of increasing ) the viewing angle from $\xi\sim100^{\circ}$. For example, as the light curve of the left column shows, the phase separation between two peaks for $\xi\sim 95^{\circ}$ becomes obviously wider than $\delta\Phi\sim0.4$ phase. On the other hand, the phase separation for $\xi\sim 105^{\circ}$ becomes narrower than the data. Furthermore, in the light curve for $\xi\sim105^{\circ}$, we can see a conspicuous peak in the leading-wing of the main peak. This leading small peak is formed by the radiations inside the null charge surface of the radiation region connecting to the other pole, while the main peak is formed by the radiations near the light cylinder. For $\xi\sim 95^{\circ}$ and $\sim100^{\circ}$, these two peaks are observed as a single main peak, because the phase separation of the two peaks is very narrow.

By comparing the p.d. for three cases, we find that p.d. in off-pulse phase increases with the viewing angle; e.g. typical p.d. in the off-pulse phase is $\sim20\%$ for $\xi\sim95^{\circ}$ and $\sim60\%$ for $\xi\sim100^{\circ}$. As we have mentioned in section \[polari\], the present model predicts highly polarized radiations at the off-pulse phase, if the line of sight marginally passes through the emission region. Increasing the viewing angle with a specific position of the inner boundary $r_i$, the line of sight approaches the inner boundary from inside the emission region as we can expect from Figure \[map\], and then the radiations from wider range of the gyration phases become to be beamed out of the line of the sight. Therefore, the p.d. in off-pulse phase increases with the viewing angle. Finally, if the line of sight passes through outside the emission region, there are no emissions in the off-pulse phase such as the light curve of the right column in Figure \[compari1\].

As we have seen, the viewing angle affects sensitively to the model light curve and the polarization degree in the off-pulse phase. In the present model, especially, the observed phase separation of the two peaks $\delta\Phi\sim0.4$ restricts the viewing angle with $\pm5^{\circ}$ uncertainty.

Let us consider the two viewing angles mutually symmetric with respect to the rotational equator (e.g. the viewing angles $80^{\circ}$ and $100^{\circ}$). For such symmetric viewing angles, the light curves, the spectra and the p.d. curves are identical. However, the p.a. curves are mirror symmetry with respect to the rotational equator because of the difference directions of the projected magnetic field on the sky. Figure \[paangle\] shows the polarization position angles for the viewing angles $\xi\sim80^{\circ}$ (left column) and $\xi\sim100^{\circ}$ (right column) with $\alpha=50^{\circ}$, $a=0.94$ and $r_i=0.67r_n$. By comparing the swing pattern at the both peaks between the model results in Figure \[paangle\] and the data in Figure \[datmod\], we find that the p.a. for $\xi\sim100^{\circ}$ is more consistent with the Crab data. The behavior of the p.a. swing does not change for different viewing angles in the same hemisphere. Therefore, the viewing angle larger than $90^{\circ}$ are preferred for the Crab pulsar.

Although we can distinguish the two viewing angle mutually symmetric with respect to the rotational equator, the present model does not allow to distinguish the two inclination angles mutually symmetric with respect to the rotational equator, that is, $\alpha$ and $180^{\circ}-\alpha$. This is because the present model has considered the only outgoing electron and positron pairs. We have not used the information of the magnetic polarity for the emissions from the pairs.

### inner boundary, $r_i$ {#inner}

We consider the dependence of the polarization characteristics on the position of the inner boundary of the outer gap by fixing the inclination angle, $\alpha=50^{\circ}$, the gap upper surface $a=0.94$ and the viewing angle $\xi\sim100^{\circ}$. As mentioned in section \[oustruc\], we assume the ratio of the radial distances to the inner boundary and the null charge surface, $r_i(\phi)/r_n(\phi)$, is not a function of the azimuthal angle $\phi$, although $r_n(\phi)$ and $r_i(\phi)$ depend on the azimuthal angle.

Figure \[compari2\] summarizes the polarization characteristics for $r_i=0.60r_n$ (right column), $0.67r_n$ (middle column) and $0.74r_n$ (right column). From Figure \[compari2\], we find that the position of the inner boundary hardly affects the expected light curve. This is because the radiations from inside of the null charge surface contribute to the off-pulse emissions with a small flux (Figure \[compari\]).

From Figure \[compari2\], we find that the p.d. in the off-pulse phase increases with shifting the inner boundary from the stellar surface toward the null charge surface; e.g. typical p.d. in the off-pulse phase is $\sim30\%$ for $r_i=0.60r_{n}$ and $\sim60\%$ for $r_i=0.67r_{n}$. The reason of the increase is the same with the results in section \[viewing\], where we discussed the dependence of the viewing angle $\xi$ with a fixed position of the inner boundary $r_i$. In the present case, the line of sight $\xi\sim100^{\circ}$ passes through the radiating points of $r\sim 0.7r_n$ in the off-pulse phase. In such a case, for $r_i=0.6r_n$ the observer detects the photons from most of all gyration phase, and emerging radiation polarizes with a low p.d. For $r_i=0.67r_n$, on the other hand, the line of sight passes through near the inner boundary and therefore, the radiation at off-pulse phase appears with a large p.d as discussed in \[viewing\]. For $r_i=0.74r_n$ (right column), the line of the sight passes through outside the emission region at the off-pulse phase.

In the middle panels for each column, we see that the main features of the p.a. (e.g. the large swings at both peaks) do not depend the position of the inner boundary. Therefore, the dependence of the position of the inner boundary mainly appears as the difference of the p.d. at the off-pulse phase. In the observation, however, both the emissions from the magnetosphere and the back ground (e.g. the wind region, and probably nebula) components would contribute to the off-pulse emissions. It is not clear which component dominates the emissions in the off-pulse phase, while the magnetospheric component must dominate in the pulse and the bridge phases. To constrain the inner boundary of the emission region with the present magnetospheric radiation model, the model requires the data of the p.d. of the off-pulse emissions from the magnetosphere. If the p.d. of the magnetospheric component are measured at the off-pulse phase, the present model will be able to restrict the radial distance of the inner boundary better. Furthermore, we may be able to diagnose how large current runs through the gap with the present geometrical model and the Crab optical data, because the position of the inner boundary of the gap is related to the current through the gap (Takata et al. 2004).

As we have seen in sections \[viewing\] and \[inner\], the polarization characteristics are sensitive to the both viewing angle $\xi$ and the position of the inner boundary $r_i$, on the other hand, the expected light curve is sensitive to only the viewing angle $\xi$. Therefore, the viewing angle is restricted by the observed light curve rather than the polarization characteristics. With the viewing angle determined by the light curve, the position of the inner boundary is restricted by the polarization characteristics.

### inclination angle and gap upper surface {#inmag}

As we have shown in sections \[viewing\] and \[inner\], if the inclination angle and the altitude of the gap upper surface could be determined, the viewing angle $\xi$ are restricted by the observed light curve, and then the position of the inner boundary of the gap $r_i$ are determined by the Crab optical polarization data. In this section, we examine how $\xi$ and $r_i$ for explaining the Crab data are changing with the inclination angle $\alpha$ and the altitude of the gap upper surface (in other words the fractional angle $a$), above which the secondary pairs are produced and the emit the observed photons. For each inclination angle $\alpha$ and the altitude of the gap upper surface $a$, the viewing angle $\xi$ is chosen to explain the observed characteristics that the light curve has two peaks, first peak is stronger than the second peak (in optical band) and the phase separation between two peaks is $\sim0.4$ phase. The position of the inner boundary $r_i$ is determined so that the p.a. has a large swings at the both peaks, and the p.d. in the off-pulse phase becomes about $\sim60\%$ because we do not have accurate data of the p.d. for the magnetospheric radiations at the off-pulse phase.

Table 1 summarizes the expected viewing angle $\xi$ and the radial distance to the inner boundary $r_i$ for the inclination angles $\alpha=40^{\circ}$, $50^{\circ}$ and $60^{\circ}$ and the various altitude of the gap upper surface. In the table, the increasing of the value of the fractional angle $a$ means the decreasing of the altitude of the upper surface of the gap with $f(R_{lc})=0.11$, and of the emissions region of the secondary pairs. We find that there is a critical altitude of the upper surface ($\equiv a_c$) for each inclination angle; e.g $a_c\sim0.91$ for $\alpha=40^{\circ}$, $a_c\sim0.93$ for $\alpha=50^{\circ}$ and $a_c\sim0.95$ for $\alpha=60^{\circ}$. With a fractional angle $a$ smaller than the critical value (in other words, with a higher upper surface than the critical altitude), there are no viewing angle that produces the light curve consistent with data, and the expected light curve has triple peaks (leading peak, main peak and second peak) such like the light curve in the right column of Figure \[compari1\].

For a specific inclination angle $\alpha$, the expected viewing angle $\xi$ increases with decreasing the altitude of the gap upper surface (or with the increasing the fractional angle $a$). The reason is explained as follows. Firstly, the phase separation of the two peaks increases with decreasing emission height because the area of the magnetic surface for the radiation regions becomes wider for lower altitude; for example, the viewing angle $\xi\sim100^{\circ}$ produces the phase separation $\delta\Phi\sim 0.4$ phase in the light curve with $\alpha=40^{\circ}$ and $a=0.91$, and therefore predicts a wider phase separation of the two peaks than $\delta\Phi=0.4$ phase for $a=0.92$. Secondarily, the phase separation of the two peaks becomes narrower with increasing the viewing angle as shown in Figure \[compari1\]. As a result, the suitable viewing angle $\xi\sim102.5^{\circ}$ of $a=0.92$ is larger than $\xi\sim100^{\circ}$ of $a=0.91$.

We also find that the critical altitude of the upper surface decreases when the inclination angle $\alpha$ increases. As we have discussed in section \[lcurve\], if the phase separation between the main peak and the small leading peak, which originates of the emissions inside null charge surface, is enough narrow, these two peaks appear as a single main peak in the light curve. If it is not, the light curve has a small peak in the leading-wing of the main peak. With a fixed altitude of the upper surface $a$, greater inclination angle $\alpha$ produces a wider phase separation between the main peak and the leading small peak. On the other hand, a lower altitude of the emission region with a fixed inclination angle produces a narrower phase separation of the two peaks. As a result, to have a single main peak without the leading small peak, a lower altitude of the emission regions of the secondary pairs, in other words, a larger value of the fractional angle $a$ of the gap upper surface is required for a greater inclination angle $\alpha$.

The position of the inner boundary $r_i$, which is chosen so that the p.d. at the off-pulse phase becomes $\sim 60\%$, approaches to the stellar surface with decreasing (or increasing) of the altitude of the gap upper surface (or the fractional angle $a$) . As we discussed in the third paragraph of this section, a larger viewing angle $\xi$ is preferred for explaining the phase separation of the two peaks for a lower altitude of the gap upper surface. And, the larger viewing angle (for $\xi>90^{\circ}$) detects the photons emitted from the positions nearer the stellar surface. Therefore, the inner boundary is shifted toward stellar surface to hold a constant p.d. at the off-pulse phase with decreasing (or increasing) of the altitude of the upper surface (or the fractional angle $a$).

Finally, the present local model cannot identify the inclination angle $\alpha$ and the altitude of the upper gap surface, where may be determined by the observation or other ways. However if either the inclination angle or the gap upper surface is determined, the present model can constrain the other one; for example, if the inclination angle of $\alpha\sim50^{\circ}$ were determined, the Crab pulsar should have the gap upper surface, which is located at lower altitude than that referred by $a\sim0.93$. If the inclination angle $\alpha$ and the altitude of the upper surface are determined, the viewing angle $\xi$ and the position of the inner boundary $r_i$ are determined by the observed light curve and the polarization characteristics as we discussed in sections \[viewing\] and  \[inner\].

SUMMARY & DISCUSSION {#discuss}
====================

We have considered the light curve and the spectrum for the Crab pulsar predicted by the outer gap model which takes into account the emissions from the inside the null charge surface and from the tertiary pairs. We have also calculated the polarization characteristics in the optical band. We have shown that the emissions from the inside the null charge surface contribute to the outer-wing and the off-pulse phase. On the other hand, the radiations from the tertiary pairs contribute to the bridge emissions of the light curve. We find that the expected polarization characteristics are consistent with the Crab optical data after subtraction of the DC level. The general features of the polarization characteristics, the light curve and the spectrum in optical bands have been reproduced simultaneously. For the Crab pulsar, the observed position angle swing indicates that the viewing angle of the observer measured from the rotational axis is greater than $90^{\circ}$.

Although the present model has explained the observed light curve, spectrum and the polarization characteristics in the optical band for the Crab pulsar, we also find that the small peak, which leads the main peak and originates from the emissions inside null charge surface, becomes to be conspicuous with increasing the energy bands if we fixed the model parameters to produce the phase separation $\delta\Phi\sim0.4$ phase between the main and the second peaks. The main peak consists of the radiations from the near the light cylinder, while the leading peak consists of the radiations near the stellar surface. The spectrum of the photons of the leading peak becomes harder than that of the main peak, because the magnetic field near the stellar surface is much stronger than that near the light cylinder. The flux ratio of the leading peak and the main peak decreases with increasing the energy bands. As a result, the expected light curve in higher energy bands may have triple peaks although the light curve in the optical has only two peaks.

If we are allowed a small discrepancy on the phase separation between the main and the second peaks with the Crab data $\delta\Phi\sim0.4$, the present model can successful explain the general features of other observed features. Figure \[enecompari\] summarizes the results for the viewing angles $\xi\sim95^{\circ}$ and $r_{i}=0.85r_{n}$ with the inclination angle $\alpha=50^{\circ}$ and the gap upper surface of $a=0.94$. We see that although the phase separation $\delta\Phi\sim 0.45$ phase of the two peaks is slightly larger than $\delta\Phi\sim 0.4$ phase for the Crab pulsar, the leading peak originating from the emissions near the stellar surface keeps a low profile in the light curves in wide energy bands. Furthermore, we note that the model light curves reproduce the energy dependent features of the observed light curves that the flux levels of the second peak and bridge phase increase relative to that of the main peak, and the flux levels of the two peaks become to be the same at around 10 keV (Kuiper et al. 2001). Because the phase separation of the main and the second peaks depends on the configuration of the magnetic field, the present results may predict that actual magnetic field in the pulsar magnetosphere is modified on some level from the vacuum dipole field, which has been assumed in the present paper, by the plasma effects (Muslimov & Harding 2005).

From Figure \[enecompari\], we can see that the polarization characteristics depend on the energy bands. Especially, the polarization degree in the bridge phase decreases from about $10\%$ in optical band with increasing the energy bands. In the optical band, the tertiary pairs contribute to the bridge emissions with about $10\%$ of the p.d. For higher energy bands, on the other hand, the emissions from the secondary pairs dominate in the bridge phase, because the tertiary pairs contribute to the bridge emissions with a smaller Lorentz factor and emissivity via the synchrotron process. In such a case, the lower p.d. than $10\%$ is expected in the bridge phase as we have seen in section \[polari\]. Therefore, the present model predicts that the polarization characteristics in the bridge phase depends on the energy bands. In the pulse and the off-pulse phases, the polarization characteristics do not depend the energy bands very much, because the synchrotron radiation from the secondary pairs takes a main contribution from optical to soft X-ray emissions. Above hard X-ray bands, the inverse Compton process of the secondary pairs will contribute to the emissions. Because the next generation Compton telescope will probably be able to measure the polarization characteristics in MeV bands, it will be required to model prediction for it, which will be the issue in the subsequent papers.

The authors appreciate fruitful discussion with K.Hirotani, S.Shibata and R.Taam. We thank L.Kuiper for providing COMPTEL, Bepposax and OSSE data, and the anonymous referee for his/her helpful comments on improvements to the paper. This work was supported by the Theoretical Institute for Advanced Research in Astrophysics (TIARA) operated under Academia Sinica and the National Science Council Excellence Projects program in Taiwan administered through grant number NSC 94-2112-M-007-002, NSC 94-2752-M-007-002-PAE and NSC 95-2752-M-007-006-PAE (J.T. and H.-K. C.), and by a RGC grant number HKU7015/05P (K.S.C.).

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![Emission projection onto ($\xi,~\Phi)$ plane for the magnetic surface, $a=0.94$, of the upper boundary of the outer gap. The inclination angle is $\alpha=50^{\circ}$. The emission region extends from $r=0.67r_n$ to $r=R_{lc}$ or $\rho=0.9R_{lc}$. Each panel shows the radiations from the positrons with the gyration phase of $\delta\phi=0^{\circ},~90^{\circ},~180^{\circ}$ and $270^{\circ}$, respectively. The thick solid circles in the figure show the shape of the polar cap. []{data-label="map"}](f1.eps){width="15cm" height="15cm"}

![Polarization characteristics for three different emission geometries. The left column shows the result for the traditional model, which considers the radiation from the secondary pairs and emission regions extending from the null charge surface, $r_i=r_n$. The middle and right column take into account the radiations from the inside the null charge surface, $r_i=0.67r_n$, and furthermore the right column considers the effects of the emissions from the tertiary pairs. The upper, middle and lower panels in each column show, respectively, the light curve, the position angle and the polarization degree. The model parameters are $\alpha=50^{\circ}$, $\xi=100^{\circ}$ and $a=0.94$.[]{data-label="compari"}](f2.eps){width="15cm" height="15cm"}

![Optical polarization for the Crab pulsar. Left:Polarization characteristics for the total emissions from the Crab pulsar. Middle:Polarization characteristics of the emissions after subtraction of the DC level (Kanbach et al. 2005). Right:Predicted polarization characteristics at 1 eV for $\alpha=50^{\circ}$, $a=0.94$, $\xi\sim100^{\circ}$ and $r_{i}=0.67r_n$. The figures for the Crab optical data was transcribed from DHR04 and was arranged by authors.[]{data-label="datmod"}](f3.eps){width="15cm" height="15cm"}

![The optical-X ray spectrum for the Crab pulsar. Thin solid line shows the expected spectrum for $\alpha=50^{\circ}$, $a=0.94$, $r_i=0.67$ and $\xi_i\sim100^{\circ}$. The X-ray data are taken from Kuiper et al. (2002) and reference therein, and the optical data from Sollerman et al. (2000).[]{data-label="spectrum"}](f4.eps){width="10cm" height="10cm"}

![The polarization characteristics for three different viewing angles, $\xi\sim95^{\circ}$ (left column), $100^{\circ}$ (middle column) and $105^{\circ}$ (right column), for $\alpha=50^{\circ}$, $a=0.94$ and $r_i=0.67r_n$.[]{data-label="compari1"}](f5.eps){width="15cm" height="15cm"}

![The polarization position angle for viewing angles, $\xi\sim80^{\circ}$ and $100^{\circ}$, which are mutually symmetric with respect to the rotational equator. The calculations are for $\alpha=50^{\circ}$, $a=0.94$ and $r_i=0.67r_n$. For the reference, the light curve are overplotted.[]{data-label="paangle"}](f6.eps){width="15cm" height="7cm"}

![The polarization characteristics for three different positions of the inner boundary, $r_{i}=0.60r_n$ (left column), $0.67r_n$ (middle column) and $0.74r_n$ (right column), for $\alpha=50^{\circ}$, $a=0.94$ and $\xi\sim100^{\circ}$.[]{data-label="compari2"}](f7.eps){width="15cm" height="15cm"}

  ---------------------------- -------------------------------- --------------------------- -------------------------- -------------------------- -------------------------- -------------------------- ---------------------- -----------------
                                                                                                                                                                                                                               
                                                                                                       0.91                       0.92                       0.93                       0.94                     0.95                0.96   
                                \[0pt\]\[0pt\][$40^{\circ}$]{}   \[0pt\]\[0pt\] [$\ast$]{}     $\xi\sim100^{\circ}$         $102.5^{\circ}$             $105^{\circ}$             $107.5^{\circ}$           $110^{\circ}$       $112.5^{\circ}$
                                                                                                 $r_{i}\sim0.75$                  0.65                       0.58                       0.50                     0.43                0.37
   \[0pt\]\[0pt\][$\alpha$]{}   \[0pt\]\[0pt\][$50^{\circ}$]{}   \[0pt\]\[0pt\][$\ast$]{}    \[0pt\]\[0pt\][$\ast$]{}   \[0pt\]\[0pt\][$\ast$]{}     $\xi\sim97^{\circ}$           $100^{\circ}$            $105^{\circ}$       $107.5^{\circ}$
                                                                                                                                                        $r_i\sim0.78$                   0.67                     0.46                0.37
                                \[0pt\]\[0pt\][$60^{\circ}$]{}   \[0pt\]\[0pt\][$\ast$]{}    \[0pt\]\[0pt\][$\ast$]{}   \[0pt\]\[0pt\][$\ast$]{}   \[0pt\]\[0pt\][$\ast$]{}   \[0pt\]\[0pt\][$\ast$]{}   $\xi\sim100^{\circ}$    $105^{\circ}$
                                                                                                                                                                                                            $r_i\sim0.53$            0.30
  ---------------------------- -------------------------------- --------------------------- -------------------------- -------------------------- -------------------------- -------------------------- ---------------------- -----------------

  : Expected viewing angle and the position of the inner boundary for various inclination angle $\alpha$ and the fraction angle of the gap upper surface $a=\theta_u/\theta_{cl}$, where $\theta_u$ is the polar angle of the footpoint of the magnetic surface of the gap upper boundary and $\theta_{cl}$ is that of the magnetic surface tangent to the light cylinder for the rotating dipole field. For each $\alpha$ and $a$, the viewing angle $\xi$ is chosen to explain the phase separation of the two peaks, and the position of the inner boundary $r_i$ is determined so that the p.a. has large swings at the both peaks and the p.d. in the off-pulse phase becomes about $\sim60\%$.

\[table1\]

![The polarization characteristics of three energy bands, 1 eV (left column), 100 eV (middle column) and 10 keV (right column) for $\xi\sim95^{\circ}$, $\alpha=50^{\circ}$, $a=0.94$ and $r_i=0.85r_n$. The present model predicts the energy dependent polarization characteristics.[]{data-label="enecompari"}](f8.eps){width="15cm" height="15cm"}