# High-Time-Resolution Measurements of the Polarization of the Crab Pulsar at
1.38 GHz

Agnieszka Słowikowska11affiliation: Kepler Institute of Astronomy, University
of Zielona Góra, Lubuska 2, 65-265 Zielona Góra, Poland , Benjamin W.
Stappers22affiliation: Jodrell Bank Centre for Astrophysics, University of
Manchester, Manchester M13 9PL, UK , Alice K. Harding33affiliation:
Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA , Stephen L. O’Dell44affiliation: Astrophysics Office, NASA
Marshall Space Flight Center, ZP12, Huntsville, AL 35812, USA , Ronald F.
Elsner44affiliation: Astrophysics Office, NASA Marshall Space Flight Center,
ZP12, Huntsville, AL 35812, USA , Alexander J. van der Horst55affiliation:
Astronomical Institute, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands , Martin C. Weisskopf44affiliation: Astrophysics
Office, NASA Marshall Space Flight Center, ZP12, Huntsville, AL 35812, USA

###### Abstract

Using the Westerbork Synthesis Radio Telescope (WSRT), we obtained high-time-
resolution measurements of the full (linear and circular) polarization of the
Crab pulsar. Taken at a resolution of 1/8192 of the 34-ms pulse period (i.e.,
$4.1~{}\mu{\rm s}$), the 1.38-GHz linear-polarization measurements are in
general agreement with previous lower-time-resolution 1.4-GHz measurements of
linear polarization in the main pulse (MP), in the interpulse (IP), and in the
low-frequency component (LFC). We find the MP and IP to be linearly polarized
at about $24\%$ and $21\%$, with no discernible difference in polarization
position angle. However, and contrary to theoretical expectations and
measurements in the visible, we find no evidence for significant variation
(sweep) in polarization position angle over the MP, the IP, or the LFC.
Although, the main pulse exhibits a small but statistically significant
quadratic variation in the degree of linear polarization. We discuss the
implications which appear to be in contradiction to theoretical expectations.
In addition, we detect weak circular polarization in the main pulse and
interpulse, and strong ($\approx 20\%$) circular polarization in the low-
frequency component, which also exhibits very strong ($\approx 98\%$) linear
polarization at a position angle about $40^{\circ}$ from that of the MP or IP.
The pulse-mean polarization properties are consistent with the LFC being a
low-altitude component and the MP and IP being high-altitude caustic
components. Nevertheless, current models for the MP and IP emission do not
readily account for the observed absence of pronounced polarization changes
across the pulse. Finally, we measure IP and LFC pulse phases relative to the
MP that are consistent with recent measurements, which have shown that the
phases of these pulse components are evolving with time.

neutron stars: general — pulsars: individual — Crab pulsar (catalog ) (PSR
B0531+21) (catalog ) — polarization

††slugcomment:  Version 16: 2014.10.20

##  1 Introduction

The Australia Telescope National Facility Pulsar Catalog (Manchester et al.,
2005) lists over 2300 radio pulsars. Several radio studies (e.g., Gould &
Lyne, 1998; Karastergiou & Johnston, 2006; Weltevrede & Johnston, 2008) have
measured the polarization for many of these pulsars. Radio pulsars typically
show moderate-to-strong linear polarization ($p_{L}$), being stronger for
those of higher spin-down energy-loss rate (Weltevrede & Johnston, 2008,
Figure 8). The linear polarization sometimes exhibits a characteristic swing
or sweep of the position angle in an S-like shape near the pulse center, which
is routinely interpreted in terms of the rotating vector model (RVM,
Radhakrishnan & Cooke, 1969). For this model the point of emission is assumed
to be in the polar cap region of the pulsar where a dipolar magnetic-field
line points with a small angle (beamwidth) towards the observer. The two free
parameters of this simple model are the angle between the axes of rotation and
the orientation of the magnetic dipole, and the view angle between the line of
sight and the rotation axis. The variation of the radio position angle from
some pulsars (e.g., Lyne & Graham-Smith, 2006, and references therein) can be
described by this model.

The Crab pulsar, the compact remnant of SN1054, and its pulsar wind nebula
(PWN) are amongst the most intensively studied objects in the sky. The pulsar
is one of the youngest and most energetic and its pulsed emission has been
detected from 10 MHz (Bridle, 1970) up to 400 GeV by VERITAS (Aliu et al.,
2011) and by MAGIC (Aleksić et al., 2012). The PWN is detected at energies up
to 100 TeV (Aharonian et al., 2004, 2006; Allen & Yodh, 2007; Abdo et al.,
2012). Both the pulsar and nebula are predominantly sources of non-thermal
radiation (synchrotron, curvature, and Compton processes), indicated not only
by the broadband spectral continua but also by strong polarization in many
wavelength bands (Lyne & Graham-Smith, 2006; Bühler & Blandford, 2013).

In the visible band, spatially-resolved polarimetry of the nebula, which began
over a half century ago (Oort & Walraven, 1956; Woltjer, 1957), continues
(e.g., Moran et al., 2013b, and references therein). Owing to its brightness,
phase-resolved optical polarimetry of the pulsar has also been possible (Jones
et al., 1981; Smith et al., 1988; Słowikowska et al., 2009). However, phase-
resolved X- and $\gamma$-ray polarimetry measurements of the Crab pulsar
require space-based instruments, which have had limited sensitivity. OSO-8
observations (Silver et al., 1978) of the Crab established only upper limits
to the X-ray (2.6-keV and 5.2-keV) polarization of the pulsed emission.
INTEGRAL IBIS observations (Forot et al., 2008; Moran et al., 2013a) also
detect no significant pulsed $\gamma$-ray (200–800-keV) polarization, although
the off-pulse emission appears highly linearly polarized and is possibly
associated with structures close to the pulsar rather than with the pulsar
itself.

The Crab pulsar’s light curve exhibits different features at different
wavelengths, but it is currently the only pulsar for which the principal
features persist over all wavelengths, from radio to $\gamma$-ray. There are
two principal components—the main pulse (MP) and the interpulse (IP). This
double-peak structure remains more-or-less phase-aligned over all spectral
bands (Moffett & Hankins, 1996; Kuiper et al., 2001). One of several
additional features in the radio band is the low-frequency component (LFC,
e.g., Moffett & Hankins 1996; 1999), having very low amplitude and occurring
$\approx 0.10$ fractional pulse phase ($36^{\circ}$) before the MP. This
component is most prominent around 1.4 GHz, in contrast with the “precursor”
component (Moffett & Hankins, 1996), which precedes the MP by $\approx 0.05$
fractional pulse phase ($19^{\circ}$) at 0.327 and 0.610 GHz (Table 2 of
Backer, Wong & Valanju, 2000).

The MP and IP appear at roughly the same pulse phase from radio to
$\gamma$-ray wavelengths, suggesting that their emission originates from a
similar location in the magnetosphere at all wavebands. Modeling of
$\gamma$-ray light curves from the many pulsars observed by the Fermi Gamma-
ray Space Telescope (Abdo et al., 2013) strongly indicates that the high-
energy emission originates in the outer magnetosphere, at altitudes comparable
to the light-cylinder radius (Romani & Watters, 2010; Pierbattista et al.,
2014; Bai & Spitkovsky, 2010). Outer magnetosphere emission models, such as
the outer-gap (Romani & Yadigaroglu, 1995), slot-gap (Muslimov & Harding,
2004), and current-sheet (Pétri & Kirk, 2005) had been proposed and studied
prior to the Fermi observations, but their emission geometry seems to account
for the characteristics and variety of observed $\gamma$-ray light curves. In
addition, Fermi has discovered a number of $\gamma$-ray millisecond pulsars
whose radio peaks are nearly aligned with their $\gamma$-ray peaks (e.g.,
Espinoza et al., 2013), like the Crab. Modeling both $\gamma$-ray and radio
light curves of these pulsars with the same outer magnetosphere emission
models used to model young pulsars has suggested that their radio emission may
originate from very high altitudes (Venter et al., 2012). Thus, in this paper
we compare the phase-resolved radio polarization observations (§2) that we
have analyzed (§3) with such models (§4).

Manchester (1971) measured the linear polarization of the Crab pulsar’s MP and
precursor components at two radio frequencies. The 0.410-GHz measurements
found the MP to be $20\%$ linearly polarized at position angle $140^{\circ}$
and the precursor to be $80\%$ linearly polarized at position angle
$140^{\circ}$. The 1.664-GHz measurements found the MP to be $24\%$ linearly
polarized at position angle $60^{\circ}$ and the precursor to be completely
absent. As these measurements had rather large uncertainties and were obtained
with a time resolution $1/256$ of the pulse period, they were quite limited
for detecting variation of the linear polarization degree or position angle
within a feature. However, Manchester noted a suggestion of rotation of the
1.664-GHz polarization position angle by about $30^{\circ}$ through the MP.

More recently, Moffett & Hankins (1999) examined the pulse-profile morphology
and polarization properties at three radio frequencies—1.424 GHz, 4.885 GHz,
and 8.435 GHz—with a time resolution of $256~{}\mu{\rm s}$ (about $1/130$ of
the pulse period). The 1.424-GHz measurements found the MP to be $25\%$
linearly polarized at position angle $120^{\circ}$; the IP, $20\%$ at position
angle $120^{\circ}$; and the LFC, $45\%$ at position angle $155^{\circ}$.
Moffett & Hankins note that the polarization position angle “changes across
the full period, although not significantly within components”.

Here we first report our observations (§2), using the Westerbork Synthesis
Radio Telescope (WSRT) in the Netherlands, of the full (linear and circular)
1.38-GHz polarization of the Crab pulsar, at high time resolution. We then
describe the polarimetry analysis and results (§3 and Appendix A) for the
three pulse components, with a primary objective of determining the sweep of
the position angle across each. Next we discuss the implications (§4) of our
measurements and analysis upon theoretical models for the pulsar emission.
Finally, we summarize our conclusions (§5).

##  2 The Observations

The WSRT observations, on 2011 August 8, used 14 25-m-diameter dishes combined
coherently to form the equivalent of a 94-m dish for pulsar observations.
Owing to the interferometric nature of the WSRT, the observations partially
resolve out the radio-bright Nebula, thus improving sensitivity over typical
single-dish observations. Moreover, as the WSRT is an equatorially mounted
telescope, there is no need to correct for parallactic angle.

To combine coherently the dishes, correlated data from observation of a bright
calibrator source is used to determine phase delays amongst dishes. This is
accomplished using initially an unpolarized calibrator to determine delays
between the two linear polarizations separately, followed by observation of a
polarized calibrator to determine any residual delays between the two
polarizations. These procedures accurately calibrate the relative fluxes in
the four Stokes parameters—hence, the polarization properties—but not the
absolute flux. Consequently, we express the Stokes measurements (e.g., Figure
2) in arbitrary units.

The PuMa-II (Karuppusamy et al., 2008) pulsar back-end was used to record
Nyquist-sampled voltages at 8-bit resolution, across a 160-MHz band centered
on 1380 MHz, for PSRs B0531+21 (Crab) and B0355+54, for a total of 144 and 18
minutes, respectively. The data were subsequently coherently de-dispersed and
folded using the DSPSR (van Straten & Bailes, 2011) software package.
Polarization profiles were formed after correcting for (frequency-dependent)
interstellar Faraday rotation (rotation measure RM = $-42.3\pm
0.5~{}\rm{rad~{}m^{-2}}$) of the position angle, using the PSRCHIVE software
package (van Straten et al., 2012). The polarization calibration was already
carried out when forming the coherent sum of the dishes, nevertheless PSR
B0355+54 was observed to verify that no further polarization calibration was
required. Comparison with the profile observed by Gould & Lyne (1998) showed
that the polarization calibration matched exactly. The Crab-pulsar profile was
folded using the Jodrell Bank
Ephemeris111http://www.jb.man.ac.uk/pulsar/crab.html with 8192 bins (about 4.1
$\mu$s/bin) across the pulse profile, matching the time resolution of the data
after dividing into frequency channels and coherently de-dispersing. This time
resolution was chosen also to match approximately the minimum broadening
caused by scattering of the Crab pulsed emission by free electrons in the Crab
Nebula (e.g., Backer, Wong & Valanju, 2000; Kuzmin et al., 2008).

Four Stokes parameters $I\times 10^{3}$, $Q\times 10^{4}$, $U\times 10^{4}$,
and $V\times 10^{4}$ (arbitrary units) as functions of pulse phase $\varphi$,
where the peak of the main pulse (MP) defines $\varphi=0$. The coordinate
system for the Stokes parameters here sets $U=0$ and $Q<0$ for the MP.

Figure 2 displays our measurement of the four Stokes parameters $I$, $Q$, $U$,
and $V$—which fully characterize the polarization—folded on the pulse period.
Unfortunately, we were unable to determine the absolute polarization position
angle for the Crab pulsar observation. Instead, we selected a coordinate
system for the Stokes parameters such that the MP has $U=0$ and $Q<0$.
Inspection of Figure 2 immediately shows that our 1.38-GHz observations detect
the flux and polarization of three components—MP, IP, and LFC. Like the MP,
the IP has $U\approx 0$ and $Q<0$; but the LFC has $U<0$ and $Q\approx 0$:
Thus, the polarization position angles for the MP and the IP are roughly equal
but differ from that of the LFC by about $40^{\circ}$ (cf. Eq. 2). Similarly,
but less obviously, the circular polarization of the MP and the IP are
comparable, but that of the LFC has opposite polarity.

##  3 Analysis and Results

The Stokes parameters have the virtues that they are statistically
independent, typically exhibit Gaussian errors, and are directly
superposable—i.e., each Stokes component ($I$, $Q$, $U$, or $V$) for multiple
sources is the sum of the respective Stokes component for each source. These
properties follow from the fact that the Stokes parameters describe the
polarization state in Cartesian-like coordinates. This has the added virtue
that there is no coordinate singularity at the origin, as occurs for polar-
like coordinates—such as linear-polarization degree $p_{L}$ and position angle
$\psi$. Consequently, we perform all statistical analyses and model fitting
(Appendix A) on (pre-processed, Appendix A.1) raw Stokes data.

It is, of course, straightforward to transform to more customary
parameters—e.g., linear-polarization degree $p_{L}$ (Eq. 1), position angle
$\psi$ (Eq. 2), and circular-polarization (signed) degree $p_{C}$ (Eq. 3):

$p_{L}=\sqrt{(Q^{2}+U^{2})}/I;$ (1) $\psi=\frac{1}{2}\tan^{-1}(U/Q);$ (2)
$p_{C}=V/I.$ (3)

For the three pulse features (MP, IP, and LFC), we estimate
$p_{L}(\varphi_{n})$, $\psi(\varphi_{n})$, and $p_{C}(\varphi_{n})$ at each
datum $n$ by substituting the measured $I_{n}$, $Q_{n}$, $U_{n}$, and $V_{n}$
into Equations 1, 2, and 3.

Direct estimate of customary polarization parameters of the main pulse versus
the phase-angle offset $\Delta\varphi$ from the MP center. From the top, the
plots display measured intensity $I$ data and then directly calculated
fractional linear polarization $p_{L}$, position angle $\psi$, and fractional
circular polarization $p_{C}$. The smooth solid lines show the best-fit phase-
dependent polarization properties based upon forward modeling of the Stokes
data (Table 1).

Direct estimate of customary polarization parameters of the interpulse versus
the phase-angle offset $\Delta\varphi$ from the IP center. From the top, the
plots display measured intensity $I$ data and then directly calculated
fractional linear polarization $p_{L}$, position angle $\psi$, and fractional
circular polarization $p_{C}$. The smooth solid lines show the best-fit phase-
dependent polarization properties based upon forward modeling of the Stokes
data (Table 1).

Figures 3 and 3 display the direct estimates of $I_{n}$, $p_{L\,n}$,
$\psi_{n}$, and $p_{C\,n}$ over the MP and IP, respectively. As the LFC is
quite weak relative to the MP and the IP, the plots for the LFC are too noisy
to display legibly. Even for the stronger features—MP and IP—the RMS noise in
the directly calculated polarization parameters ($p_{L\,n}$, $\psi_{n}$, and
$p_{C\,n})$, which serves as an estimator of the statistical error,
substantially increases away from the pulse center due to the low signal-to-
noise ratio per sample in the pulse wings. In order to deal effectively with
low-signal-to-noise data in the wings of the MP and IP and throughout the
(weaker) LFC, we adopt a more rigorous forward-modeling approach to fit the
measured Stokes data to the modeled $I(\varphi)$, $Q(\varphi)$, $U(\varphi)$,
and $V(\varphi)$:

$Q(\varphi)=I(\varphi)p_{L}(\varphi)\cos(2\psi(\varphi));$ (4)
$U(\varphi)=I(\varphi)p_{L}(\varphi)\sin(2\psi(\varphi));$ (5)
$V(\varphi)=I(\varphi)p_{C}(\varphi).$ (6)

Appendix A describes in some detail our approach for fitting polarization
models to the Stokes data. As Figures 3 and 3 indicate that neither
$p_{L}(\varphi)$, $\psi(\varphi)$, nor $p_{C}(\varphi)$ varies rapidly across
the pulse profile, the approach simply models $p_{L}(\varphi)$,
$\psi(\varphi)$, and $p_{C}(\varphi)$ as Taylor-series expansions in the
phase-angle offset $\Delta\varphi\equiv(\varphi-\varphi_{0})$ from the center
$\varphi_{0}$ of the respective pulse feature (MP, IP, or LFC). Table 1
tabulates the best-fit Taylor-expansion coefficients for the polarization
dependence upon phase-angle offset:

$p_{L}(\varphi)=p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2};$
(7)
$\psi(\varphi)=\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2};$
(8)
$p_{C}(\varphi)=p_{C0}+p^{\prime}_{C0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C0}(\varphi-\varphi_{0})^{2}.$
(9)

Table 1: Best-fit polarization coefficients for the MP, IP, and LFC, using a single Gaussian for each pulse profile and up-to-quadratic variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$.  Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.389\pm 0.027$ | $-37.75\pm 0.19$
$p_{L0}$ | $\%$ | $22.98\pm 0.30$ | $21.3\pm 1.0$ | $98.2\pm 6.7$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.31\pm 0.19$ | $1.02\pm 0.62$ | $-0.8\pm 2.2$
$p^{\prime\prime}_{L0}$ | $\%/^{\circ}/^{\circ}$ | $0.88\pm 0.22$ | $-0.02\pm 0.63$ | $0.0\pm 1.3$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $-0.1\pm 1.3$ | $40.8\pm 1.5$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.16\pm 0.20$ | $0.82\pm 0.78$ | $-0.16\pm 0.49$
$\psi^{\prime\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}/^{\circ}$ | $-0.06\pm 0.21$ | $1.00\pm 0.89$ | $-0.21\pm 0.28$
$p_{C0}$ | $\%$ | $-1.25\pm 0.20$ | $-3.15\pm 0.94$ | $20.5\pm 4.9$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $0.01\pm 0.13$ | $0.38\pm 0.56$ | $0.3\pm 1.7$
$p^{\prime\prime}_{C0}$ | $\%/^{\circ}/^{\circ}$ | $-0.20\pm 0.15$ | $0.47\pm 0.57$ | $-0.49\pm 0.97$

An important conclusion of this study is that the Stokes data are
consistent—within statistical uncertainties—with constant polarization
position angle $\psi$ across each of the three pulse features (MP, IP, and
LFC) individually. However, the MP does exhibit a small but statistically
significant quadratic variation in the linear-polarization degree $p_{L}$.
While our 1.380-GHz polarimetry of the Crab pulsar has finer time resolution
and better statistical accuracy than previous 1.424-GHz polarimetry (Moffett &
Hankins, 1999), measured values for the polarization degree and position angle
(relative to MP) are mostly similar for the MP and for the IP. The only
significant difference is for the LFC’s linear polarization degree and
position angle. We measured nearly total ($98\%\pm 7\%$) linear polarization
at a $+40.8^{\circ}\pm 1.5^{\circ}$ position-angle offset from the MP, whereas
Moffett & Hankins (1999) found the LFC to be $\approx 40\%$ linearly polarized
at a $\approx+30^{\circ}$ position-angle offset from the MP. We also detect
circular polarization, which is moderately strong in the LFC ($20.5\pm 4.9\%$)
but weak and opposite polarity in the MP ($-1.3\%\pm 0.2\%$) and in the IP
($-3.2\%\pm 0.9\%$). In contrast with Moffett & Hankins, we find no
significant variation in the circular polarization across any of the three
pulse components MP, IP, and LFC.

Another important conclusion—albeit peripheral to the polarimetry—relates to
substructure in the pulse profile of the MP. The fine time resolution and
better statistical accuracy of our radio observation of the Crab pulsar
resulted in measurement of statistically significant substructure (Appendix
A.3) in the profile of the main pulse (Figure 3). The typical width of the
substructure is roughly $10\ \mu$s—i.e., $\leq 0.1$ the width of the MP
profile. As the current analysis utilizes the sum of all data collected during
the observation at a single epoch (2011 August 8), we have not assessed the
temporal behavior of the profile. However, we presume that this substructure
results from sporadic, very strong giant radio pulses (Bhat et al., 2008;
Karuppusamy et al., 2010; Majid et al., 2011; Hankins et al., 2003) occurring
during the 144-minute observation. Although the substructure is readily
apparent in the $I$ profile of the MP, the discernible subpulses contribute
only about $5\%$ of the fluence in the MP over the observation. However, they
likely result from only the strongest giant radio pulses in a distribution of
pulse amplitudes. Note that our conclusions as to the average pulse-phase
dependences of the polarimetry are effectively independent of the precise
modeling of the intensity profile of the MP. On the other hand, inspection of
the Stokes parameters (Figure 3) or polarization parameters (Figure 3)
indicates that the polarization of some of the subpulses (e.g., at phase
offset $\Delta\varphi\approx-2.3^{\circ}$) differs substantially from the
average polarization of the MP.

We also note that our WSRT-measured pulse-phase offsets of the IP and of the
LFC from the MP are in good agreement with contemporaneous measurements at
Jodrell Bank (Lyne et al., 2013). This tends to support the conclusion of Lyne
et al. (2013) that the phase separations of the IP and of the LFC from the MP
are evolving with time. Furthermore, the evolution of phase separations might
contribute to the difference between our measurement of the LFC’s polarization
and earlier measurements (Moffett & Hankins, 1999).

Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the main pulse (MP). The lines represent the best-fit
(minimum-$\chi^{2}$) Stokes functions for a multi-Gaussian profile and up-to-
second-order variations in linear-polarization degree, position angle, and
circular-polarization degree. The pulse profile comprises 2 broad and 4 narrow
Gaussians.

##  4 Implications for Theoretical Models

Emission at altitudes comparable to the light-cylinder radius produce caustic
peaks, formed by cancellation of phase differences due to aberration and
retardation with that due to field-line curvature of radiation along the
trailing magnetic-field lines (Dyks & Rudak, 2003). In outer-magnetosphere
models, peaks in the light curves form when the observer’s sight line sweeps
across one or more bright caustic. The caustics display distinct linear-
polarization characteristics (Dyks et al., 2004), including fast sweeps of
position angle and dips in polarization degree at the peaks, which are caused
by piling up radiation emitted over a large range of altitudes and magnetic-
field directions into the caustics. These characteristics are in fact seen in
the optical polarization of the Crab pulsar (Słowikowska et al., 2009), which
exhibits rapid swings of position angle across both the MP and IP, as well as
dips in polarization degree to the 5% level on the trailing edge of each peak.

From the results presented in this paper, however, the characteristics of the
radio linear polarization of the MP and IP resemble neither those of caustics
in existing geometric models nor those observed in the optical emission. The
lack of position-angle swing in the radio MP and IP is in stark contrast to
the rapid position-angle swings in the optical. The very low circular
polarization and moderate linear polarization observed here in the radio MP
and IP are consistent with caustics, but the observed linear-polarization
values ($\approx 22\%$) in the radio are significantly higher than those in
the optical, and there is only a small variation with phase in the MP. On the
other hand, the radio pulses are much narrower than the optical pulses,
indicating that the radio MP and IP may originate along a smaller range of
altitudes and/or in a subset of field lines.

We have modeled the caustic emission and corresponding linear-polarization
degree $p_{L}$ and position angle $\psi$ for the Crab pulsar, with a
simulation using geometric renditions of standard slot-gap and outer-gap
emission. These geometric emission models assume constant emissivity in the
corotating frame along a set of field lines within the gaps, defined by a gap
width $w$ across field lines in open-volume coordinates (Dyks et al., 2004),
where the width is a fraction of radius of open magnetic field lines. As in
Dyks et al. (2004), the emission is assumed to occur over a fixed radius
range, from minimum $r_{\rm min}$ to maximum $r_{\rm max}$. For the
simulations of Crab polarization here, we explored gap widths
$w=0.002,0.01,0.02,0.05$, $r_{\rm min}=0.3-0.9\,R_{\rm LC}$ and $r_{\rm
max}=0.5-1.2\,R_{\rm LC}$, where $R_{\rm LC}=c/\Omega$ is the light-cylinder
radius. These are smaller ranges of altitude and smaller gap widths than in
standard slot-gap or outer-gap models used in Dyks et al. (2004), which were
$r_{\rm min}=R_{\rm NS}$, $r_{\rm max}=0.95\,R_{\rm LC}$ for the slot gap and
$r_{\rm min}=R_{\rm NC}$, $r_{\rm max}=0.97\,R_{\rm LC}$ for the outer gap.
Here $R_{\rm NS}$ is the neutron star radius and $R_{\rm NC}$ is the radius of
the null-charge surface, at which the magnetospheric charge density in the
corotating frame $\rho_{0}=\mathbf{\Omega\cdot B}/(2\pi c)$ vanishes.

We simulated emission using both retarded-vacuum-dipole (Deutsch, 1955), as in
Dyks et al. (2004), and force-free (Contopoulos & Kalapotharakos, 2010)
magnetic-field geometries, as in Harding et al. (2011). Then we computed light
curves and Stokes parameters for magnetic inclination angles
$\alpha=45^{\circ}-80^{\circ}$, with $5^{\circ}$ resolution for vacuum and
$15^{\circ}$ resolution for force-free magnetospheres, and observer viewing
angles $\zeta=55^{\circ}-80^{\circ}$ (both with respect to the rotation axis).
These ranges of $\alpha$ and $\zeta$ bracket the viewing angle of
$60^{\circ}-65^{\circ}$ suggested by modeling of the X-ray torus (Ng & Romani,
2008). Following Dyks et al. (2004), Blaskiewicz et al. (1991), and Hibschman
& Arons (2001), we assume that the photon electric-field vector is parallel to
the electron acceleration at each point along the field line to determine the
Stokes parameters.

Although simulated light curves for the smaller gap widths produce narrower
caustic peaks with less position-angle swing and depolarization, it is
difficult to produce both $\psi(\varphi)$ and $p_{L}(\varphi)$ curves with no
variation through the peaks. We compared a range of simulated light curves,
$p_{L}$ and $\psi$ to the ones observed, and found that none of the models
agree with the data. For the vacuum magnetospheres, the slot-gap model can
produce appropriately narrow peaks for $w<0.01$, but there is always some
change in $\psi$ through both the MP and IP. At $\zeta=60^{\circ}$, there are
dips in $p_{L}$ at only the first peak for $\alpha<75^{\circ}$ and dips at
both peaks for $\alpha>75^{\circ}$. The outer-gap model produces a change in
$\psi$ mostly in the IP but dips in $p_{L}$ in both peaks. While the force-
free geometry, whose poloidal field lines are straighter than those in vacuum,
can give a flatter position angle for certain inclination and viewing angles,
the model’s $p_{L}$ shows strong variation through the peaks in contradiction
with the data. For the force-free magnetospheres, the slot-gap model produces
much less change in $\psi$ at the peaks for $\zeta=55^{\circ}-65^{\circ}$ and
$\alpha=45^{\circ}-75^{\circ}$, but still not constant as observed. There is
also a high level of depolarization in both peaks but $p_{L}$ is not constant
through the peaks, as in the data. The outer gap in the force-free
magnetosphere also produces changes in $\psi$ and $p_{L}$ in both peaks for
these same ranges of $\alpha$ and $\zeta$.

For comparison with our measurement of the phase-resolved polarization
properties of the Crab pulsar, we simulated 66 total (48 vacuum and 18 force-
free) cases. Based upon inspection of the results of these numerous simulated
cases, the model light curve and polarization characteristics that seem to
resemble most the Crab pulsar radio data is for the case of the slot-gap model
in the force-free magnetosphere with $\alpha=45^{\circ}$ and
$\zeta=60^{\circ}$. Figure 4 displays the results for this model for the MP.
Note that this model does predict a rapid swing in polarization position angle
and degree which we do not see; however, these swings occur on the preceding
wing of the pulse, when the intensity is very low.

Predicted relative variation through the MP of the intensity $I$ (red), linear
polarization degree $p_{L}$ (green), and position angle $\psi$ (blue) for the
slot-gap model, with a force-free magnetosphere. For this case, the magnetic
inclination angle $\alpha=45^{\circ}$ and observer viewing angle
$\zeta=60^{\circ}$ with respect to the spin axis. The ordinate range 0–1
corresponds to zero to peak intensity for $I$, 0%–100% polarization for
$p_{L}$, and $-90^{\circ}$ to $90^{\circ}$ for $\psi$.

In order to explore the possibility that the linear-polarization degree
$p_{L}$ or position angle $\psi$ changes sharply in the preceding wing of the
MP (as in Figure 4), we fit the Stokes data to a simple model of a step jump
in the values of $p_{L}$ and of $\psi$ at a pulse phase $\varphi_{\rm step}$.

$p_{L}(\varphi)=p_{L0}+\Delta p_{L}\;\Theta(\varphi_{\rm step}-\varphi);$ (10)
$\psi(\varphi)=\psi_{0}+\Delta\psi\;\Theta(\varphi_{\rm step}-\varphi).$ (11)

Here, $p_{L0}$ and $\psi_{0}$ are the best-fit values for constant linear-
polarization degree and position angle; $\Delta p_{L}$ and $\Delta\psi$, the
pre-step differences in the value of each; and $\Theta(\varphi_{\rm
step}-\varphi)$, the unit step distribution ($=1$ for $\varphi<\varphi_{\rm
step}$, 0 otherwise). Figure 4 shows the best-fit differences and their
(1-sigma) uncertainties as functions of pulse phase of the step (relative to
pulse center). From this analysis, we conclude that any position-angle swing
must be small—$|\Delta\psi|<10^{\circ}$ for $\varphi_{\rm step}>-3.5^{\circ}$.
A large position-angle swing—$|\Delta\psi|>45^{\circ}$, say—is consistent with
the data (but not required) only for $\varphi_{\rm step}<-4^{\circ}$. Note
that the analysis requires $\Delta p_{L}>0$ for $\varphi_{\rm
step}\geq-2.5^{\circ}$ (and allows it for earlier $\varphi_{\rm step}$), as
this analysis does not include the small positive second derivative
$p^{\prime\prime}_{L0}$ in the linear-polarization degree, which the Taylor-
expansion fit to the MP Stokes data requires (cf. Table 1).

Constraints on a sharp step in the MP linear-polarization degree
($p_{L}(\varphi)$, left) and position angle ($\psi(\varphi)$, right) versus
the putative step’s pulse phase $\varphi_{\rm step}$ (relative to the MP
center). Large position-angle swings ($|\Delta\psi|>45^{\circ}$, say) are
allowed (but not required) only very early ($\varphi_{\rm step}<-4^{\circ}$)
in the pulse—i.e., where the signal-to-noise ratio is low.

It is possible that the radio linear polarization in the MP and LP is very
sensitive to the magnetic-field structure. Existing models explored only the
two extremes of vacuum (accelerating fields but no plasma) and force-free
(plasma but no accelerating fields), neither of which describe real pulsars.
More realistic, dissipative magnetosphere models with finite conductivity now
exist (Kalapotharakos et al., 2012; Li et al., 2012) and should be used to
model light curves and polarization characteristics. It is also possible that
the radio emission in the MP and IP occurs along sets of field lines that lie
deeper within the open/closed field boundary or the current sheet and have
different polarization properties.

The low-frequency component (LFC) is substantially weaker than the MP and IP
at 1.4 GHz. As its name suggests, the LFC is not detected at radio frequencies
higher than a few GHz and has no corresponding component in the visible band.
The nearly complete radio polarization ($p_{L}\approx 98\%$ and $p_{C}\approx
20\%$) of the LFC support the hypothesis that it is a highly coherent, low-
altitude component. Note that the (lower frequency) precursor is also believed
to be a highly coherent, low-altitude component, due to its high polarization
and steep spectrum (Rankin, 1990).

##  5 Conclusions

Our 1.38-GHz observations of the Crab pulsar measured significant linear and
circular polarization in the three most prominent pulse components—the main
pulse (MP), inter pulse (IP), and low-frequency component (LFC). These results
are mostly in agreement with previous measurements of linear polarization at
similar radio frequencies (cf. Moffett & Hankins, 1999). The MP and IP are
moderately linearly polarized ($p_{L}\approx 23\%$ and $21\%$, respectively)
at the same position angle ($\psi_{\rm IP}-\psi_{\rm MP}\approx 0$); they are
weakly circularly polarized ($p_{C}\approx-1.3\%$ and $-3.2\%$, respectively).
In contrast, the LFC is very strongly linearly polarized ($p_{L}\approx
98\%$), at a position angle $+40^{\circ}$ from that of the MP or IP, and
moderately circularly polarized ($p_{C}\approx 20\%$).

The fine time resolution (Period/8192 = 4.1 $\mu$s) and good sensitivity of
the measurements at the Westerbork Synthesis Radio Telescope (WSRT) enabled a
meaningful search for changes in linear-polarization degree $p_{L}$, in
position angle $\psi$, and in circular-polarization degree $p_{C}$ across each
of the three pulse components. Neither the MP, IP, nor LFC exhibits a
statistically significant change in the polarization position angle or
circular polarization across the pulse. For the MP, the linear term (“sweep”)
is well constrained: $\psi^{\prime}_{0\,\rm MP}=(-0.16\pm 0.20)^{\circ}{\rm
PA}/^{\circ}$. Likewise, neither the IP nor LFC displays a statistically
significant change in the polarization degree. However, the MP does show a
small but statistically significant quadratic variation in linear-polarization
degree—$p^{\prime\prime}_{L0\,\rm MP}=(0.88\pm 0.22)\%/^{\circ}/^{\circ}$
about its central value—$p_{L0\,\rm MP}=(23.0\pm 0.3)\%$—for a pulse-average
linear polarization $\overline{p}_{L\,\rm MP}=(23.7\pm 0.3)\%$.

Our analysis of the radio Stokes data shows no strong sweep of the linear-
polarization position angle. This lack of strong position-angle swings
contrasts with the rapid swings observed in the visible band. Current models
for pulsar emission geometries do not readily account for the absence of
substantial variations in both polarization degree and position angle across a
pulse component (§ 4). Thus, alternative models—e.g., dissipative
magnetopheres—should be considered in modeling the radio polarization of the
Crab pulsar’s MP and IP. The nearly complete polarization of the LFC suggest
that it originates at a different location and via a different mechanism than
do the stronger MP and IP.

Finally, the fine time resolution and high signal-to-noise ratio in the MP
data led to detection of statistically significant substructure in its pulse
profile. We surmise that this substructure results from giant radio pulses
occurring during the 144-minute observation.

Acknowledgments

The Westerbork Synthesis Radio Telescope (WSRT) is operated by ASTRON, the
Netherlands Institute for Radio Astronomy, with support from NWO, the
Netherlands Foundation for Scientific Research. AS acknowledges grant
DEC-2011/03/D/ST9/00656 from the Polish National Science Centre; BWS, a
Consolidated Grant from the UK Science and Technology Facilities Council; AKH,
NASA grants Astrophysics Theory 12-ATP12-0169 and Fermi Guest Investigator
11-FERMI11-0052; AJvdH, Advanced Investigator Grant 247295 (PI: R. A. M. J.
Wijers) from the European Research Council; and SLO, RFE and MCW, support by
NASA’s Chandra Program.

##  Appendix A Statistical analysis

###  A.1 Procedures

As Figure 2 shows, the main pulse (MP), interpulse (IP), and low-frequency
component (LFC) are well separated in the 1.38-GHz data folded on the Crab
pulsar’s period. Consequently, we choose to analyze each of these three
features individually, using phase ranges $(-7.2^{\circ},7.2^{\circ})$ for the
MP, $(134.6^{\circ},156.2^{\circ})$ for the IP, and
$(-52.1^{\circ},-23.3^{\circ})$ for the LFC, where the center of the MP
defines pulse-phase angle $\varphi=0^{\circ}$. We use data over the remaining
phase ranges to measure the off-pulse mean and the root-of-mean-square (RMS)
noise in $I$, $Q$, $U$, and $V$. Upon measuring the off-pulse mean values for
$I$, we noticed that its off-pulse value near the MP is depressed with respect
to the remaining phase ranges. Specifically, in phase ranges
$(-14.4^{\circ},-7.2^{\circ})$ and $(7.2^{\circ},14.4^{\circ})$, the mean $I$
is 0.0273 ($\times 1000)$ less than in other off-pulse ranges. Taking this
into account lowered $\chi^{2}$ by about 300 in fitting the $I$ pulse profile,
but did not significantly alter the fitted polarization properties.

For convenience, we pre-process the raw data by subtracting the respective
off-pulse mean value, under the assumption that the expectation values for
$I$, $Q$, $U$, and $V$ are zero away from pulse features. Furthermore, we take
the RMS noise levels—0.0324, 0.0310, 0.0311, and 0.0307 (each $\times
1000$)—as estimators of the statistical standard deviations $\sigma_{I}$,
$\sigma_{Q}$, $\sigma_{U}$, and $\sigma_{V}$, respectively.

In order to fit the model to the data for each pulse feature, we minimize the
chi-square statistic of the combined Stokes data

$\displaystyle\chi^{2}(\varpi)=\chi_{I}^{2}(\varpi)+\chi_{Q}^{2}(\varpi)+\chi_{U}^{2}(\varpi)+\chi_{V}^{2}(\varpi)=$
$\displaystyle\sum_{n=1}^{N}\left[\frac{(I_{n}-I(\varphi_{n};\varpi))^{2}}{\sigma_{I}^{2}}+\frac{(Q_{n}-Q(\varphi_{n};\varpi))^{2}}{\sigma_{Q}^{2}}+\frac{(U_{n}-U(\varphi_{n};\varpi))^{2}}{\sigma_{U}^{2}}+\frac{(V_{n}-V(\varphi_{n};\varpi))^{2}}{\sigma_{V}^{2}}\right],$

with respect to a set $\varpi$ of $K$ model parameters, leaving $\nu=N-K$
degrees of freedom. We obtain the statistical uncertainty in each parameter,
based upon $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}$. To perform the
$\chi^{2}$ analysis, we used the MathematicaTM (Wolfram, 2013) function
NonlinearModelFit222http://reference.wolfram.com/mathematica/ref/NonlinearModelFit.html,
which finds best-fit model parameters, their errors, correlation matrix
amongst them, etc.

Modeling the Stokes data requires parameterized functions for the pulse
profile $I(\varphi)$, the linear-polarization fraction $p_{L}(\varphi)$, the
polarization position angle $\psi(\varphi)$, and the circular-polarization
fraction $p_{C}(\varphi)$ (cf. Equations 4, 5, and 6 for $Q(\varphi)$,
$U(\varphi)$, and $V(\varphi)$, respectively). As there is no evidence for
rapid changes in polarization degree or position angle over a pulse feature
(cf. Figures 3 and 3), simple Taylor-series expansions suffice:

$\displaystyle p_{L}(\varphi)$ $\displaystyle=$ $\displaystyle
p_{L}(\varphi_{0})+p^{\prime}_{L}(\varphi_{0})(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L}(\varphi_{0})(\varphi-\varphi_{0})^{2}+\cdots$
(A2) $\displaystyle\equiv$ $\displaystyle
p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2}+\cdots;$
$\displaystyle\psi(\varphi)$ $\displaystyle=$
$\displaystyle\psi(\varphi_{0})+\psi^{\prime}(\varphi_{0})(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}(\varphi_{0})(\varphi-\varphi_{0})^{2}+\cdots$
(A3) $\displaystyle\equiv$
$\displaystyle\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2}+\cdots;$
$\displaystyle p_{C}(\varphi)$ $\displaystyle=$ $\displaystyle
p_{C}(\varphi_{0})+p^{\prime}_{C}(\varphi_{0})(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C}(\varphi_{0})(\varphi-\varphi_{0})^{2}+\cdots$
(A4) $\displaystyle\equiv$ $\displaystyle
p_{C0}+p^{\prime}_{C0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C0}(\varphi-\varphi_{0})^{2}+\cdots.$

To parameterize the pulse profile, we use a Gaussian (§A.2) for each pulse
feature (MP, IP, or LFC) or multiple Gaussians (§A.3) for the MP.

###  A.2 Single-Gaussian fits to the MP, the IP, and to the LFC

To complete the parameterized model for the four Stokes functions, we assume a
Gaussian profile:

$I(\varphi)=I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right),$
(A5)

with $I_{0}$ the value of $I(\varphi)$ at pulse center, $\sigma_{\varphi}$ the
Gaussian width, and $\varphi_{0}$ the phase at the pulse center. Combining
this parameterization with Equations 4, 5, 6, A2, A3, A4, the full model for
the other three Stokes functions follows:

$\displaystyle Q(\varphi)$ $\displaystyle=$ $\displaystyle
I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right)[p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2}]$
(A6)
$\displaystyle\times\cos(2[\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2}]);$
$\displaystyle U(\varphi)$ $\displaystyle=$ $\displaystyle
I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right)[p_{L0}+p^{\prime}_{L0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{L0}(\varphi-\varphi_{0})^{2}]$
(A7)
$\displaystyle\times\sin(2[\psi_{0}+\psi^{\prime}_{0}(\varphi-\varphi_{0})+\frac{1}{2}\psi^{\prime\prime}_{0}(\varphi-\varphi_{0})^{2}]);$
$\displaystyle V(\varphi)$ $\displaystyle=$ $\displaystyle
I_{0}\exp\left(-\frac{(\varphi-\varphi_{0})^{2}}{2\sigma^{2}_{\varphi}}\right)[p_{C0}+p^{\prime}_{C0}(\varphi-\varphi_{0})+\frac{1}{2}p^{\prime\prime}_{C0}(\varphi-\varphi_{0})^{2}].$
(A8)

Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the main pulse (MP). The lines represent the best-fit
(minimum-$\chi^{2}$) Stokes functions for a single-Gaussian profile and up-to-
second-order variations in polarization degree and in position angle.

Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the inter pulse (IP). The lines represent the best-fit
(minimum-$\chi^{2}$) Stokes functions for a single-Gaussian profile and up-to-
second-order variations in polarization degree and in position angle.

Stokes data $I$, $Q$, $U$, and $V$ versus pulse phase offset $\Delta\varphi$
from the center of the low-frequency component (LFC). The lines represent the
best-fit (minimum-$\chi^{2}$) Stokes functions for a single-Gaussian profile
and up-to-second-order variations in polarization degree and in position
angle.

Figures A.2, A.2, and A.2 display Stokes data for the MP, IP, and LFC,
respectively. The lines represent best-fit (minimum-$\chi^{2}$) Stokes
functions (Equations A5, A6, A7, and A8) for a single-Gaussian profile
$I(\varphi)$ and up-to-quadratic variations in linear-polarization degree
$p_{L}(\varphi)$, in position angle $\psi(\varphi)$, and in circular-
polarization degree $p_{C}(\varphi)$. Tables 2, 3, and 4 tabulate the results
of the $\chi^{2}$ analysis for a Gaussian profile and retaining polarization
terms (Equations A6, A7, and A8) through, zeroth, first, and second order,
respectively. For each pulse feature—MP, IP, and LFC—the tables list the
minimum $\chi^{2}$ and degrees of freedom $\nu$ for $I$, $Q$, $U$, and $V$
data sets combined and separately, followed by best-fit estimators and
(1-sigma) uncertainties for the 3 pulse-profile parameters ($I_{0}$,
$\sigma_{\varphi}$, $\varphi_{0}$) and for the relevant polarization
coefficients ($p_{L0}$, $p^{\prime}_{L0}$, $p^{\prime\prime}_{L0}$;
$\psi_{0}$, $\psi^{\prime}_{0}$, $\psi^{\prime\prime}_{0}$; $p_{C0}$,
$p^{\prime}_{C0}$, $p^{\prime\prime}_{C0}$). Note that these tables reference
the pulse-phase angles ($\varphi_{0}$) and polarization position angles
($\psi_{0}$) to the MP, as we set $\varphi_{\rm MP}\equiv 0$ and were unable
to obtain an absolute measurement of position angle $\psi_{\rm MP}$.

Table 2: Best-fit parameters for the MP, IP, and the LFC, using a simple Gaussian for each profile and no variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$.  Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\chi^{2}/\nu$ |  | $3081./1302$ | $2022./1962$ | $2518./2618$
$\chi^{2}_{I}/\nu_{I}$ |  | $1910./324$ | $561./489$ | $577./653$
$\chi^{2}_{Q}/\nu_{Q}$ |  | $460./322$ | $534./487$ | $674./651$
$\chi^{2}_{U}/\nu_{U}$ |  | $441./322$ | $463./487$ | $603./651$
$\chi^{2}_{V}/\nu_{V}$ |  | $269./323$ | $463./488$ | $664./652$
$I_{0}$ | $\times 1000$ | $1.9894\pm 0.0046$ | $0.4414\pm 0.0044$ | $0.0668\pm 0.0031$
$\sigma_{\varphi}$ | ∘ | $1.7801\pm 0.0047$ | $1.947\pm 0.022$ | $3.40\pm 0.14$
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.399\pm 0.023$ | $-37.79\pm 0.14$
$p_{L0}$ | $\%$ | $23.67\pm 0.19$ | $21.24\pm 0.81$ | $98.3\pm 5.7$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $1.0\pm 1.1$ | $40.3\pm 1.2$
$p_{C0}$ | $\%$ | $-1.40\pm 0.18$ | $-2.70\pm 0.78$ | $19.0\pm 4.0$
Table 3: Best-fit parameters for the MP, the IP, and the LFC, using a simple Gaussian for each profile and up-to-linear variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$.  Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\chi^{2}/\nu$ |  | $3076./1299$ | $2017./1959$ | $2517./2615$
$\chi^{2}_{I}/\nu_{I}$ |  | $1910./324$ | $561./489$ | $577./653$
$\chi^{2}_{Q}/\nu_{Q}$ |  | $456./320$ | $532./485$ | $674./649$
$\chi^{2}_{U}/\nu_{U}$ |  | $440./320$ | $462./485$ | $602./649$
$\chi^{2}_{V}/\nu_{V}$ |  | $269./322$ | $463./487$ | $664./651$
$I_{0}$ | $\times 1000$ | $1.9894\pm 0.0046$ | $0.4415\pm 0.0044$ | $0.0668\pm 0.0031$
$\sigma_{\varphi}$ | ∘ | $1.7801\pm 0.0047$ | $1.946\pm 0.022$ | $3.40\pm 0.14$
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.389\pm 0.023$ | $-37.74\pm 0.20$
$p_{L0}$ | $\%$ | $23.67\pm 0.19$ | $21.25\pm 0.81$ | $98.3\pm 5.7$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.32\pm 0.15$ | $1.09\pm 0.59$ | $-0.9\pm 2.4$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $0.9\pm 1.1$ | $40.3\pm 1.2$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.15\pm 0.18$ | $0.91\pm 0.78$ | $-0.18\pm 0.48$
$p_{C0}$ | $\%$ | $-1.40\pm 0.18$ | $-2.70\pm 0.78$ | $19.0\pm 4.0$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $-0.01\pm 0.14$ | $0.38\pm 0.57$ | $0.3\pm 1.7$
Table 4: Best-fit parameters for the MP, for the IP, and for the LFC, using a simple Gaussian for each profile and up-to-quadratic variations in polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$.  Parameter | Units | MP | IP | LFC
---|---|---|---|---
$\chi^{2}/\nu$ |  | $3049./1296$ | $2016./1956$ | $2517./2612$
$\chi^{2}_{I}/\nu_{I}$ |  | $1909./324$ | $561./489$ | $577./653$
$\chi^{2}_{Q}/\nu_{Q}$ |  | $432./318$ | $531./483$ | $674./647$
$\chi^{2}_{U}/\nu_{U}$ |  | $440./318$ | $461./483$ | $603./647$
$\chi^{2}_{V}/\nu_{V}$ |  | $268./321$ | $462./486$ | $664./650$
$I_{0}$ | $\times 1000$ | $1.9927\pm 0.0047$ | $0.4414\pm 0.0045$ | $0.0666\pm 0.0034$
$\sigma_{\varphi}$ | ∘ | $1.7742\pm 0.0048$ | $1.947\pm 0.023$ | $3.42\pm 0.20$
$\varphi_{0}-\varphi_{\rm MP}$ | ∘ | $\equiv 0$ | $145.389\pm 0.023$ | $-37.75\pm 0.20$
$p_{L0}$ | $\%$ | $22.99\pm 0.23$ | $21.24\pm 0.99$ | $98.1\pm 7.0$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.32\pm 0.15$ | $1.03\pm 0.59$ | $-0.9\pm 2.4$
$p^{\prime\prime}_{L0}$ | $\%/^{\circ}/^{\circ}$ | $0.86\pm 0.17$ | $-0.04\pm 0.61$ | $0.1\pm 1.4$
$\psi_{0}-\psi_{\rm MP}$ | ${}^{\circ}{\rm PA}$ | $\equiv 0$ | $-0.1\pm 1.3$ | $40.8\pm 1.4$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.16\pm 0.17$ | $0.82\pm 0.79$ | $-0.16\pm 0.48$
$\psi^{\prime\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}/^{\circ}$ | $-0.06\pm 0.18$ | $1.07\pm 0.80$ | $-0.21\pm 0.28$
$p_{C0}$ | $\%$ | $-1.25\pm 0.22$ | $-3.15\pm 0.96$ | $20.5\pm 4.9$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $0.01\pm 0.15$ | $0.38\pm 0.57$ | $0.3\pm 1.7$
$p^{\prime\prime}_{C0}$ | $\%/^{\circ}/^{\circ}$ | $-0.20\pm 0.16$ | $0.47\pm 0.59$ | $-0.49\pm 0.96$

Table 3 documents that, to within statistical uncertainties,
$p^{\prime}_{L0}=0$, $\psi^{\prime}_{0}=0$, and $p^{\prime}_{C0}=0$ for each
of the three pulse features—MP, IP, or LFC. Equivalently, including the three
linear coefficients $p^{\prime}_{L0}=0$, $\psi^{\prime}_{0}=0$, and
$p^{\prime}_{C0}=0$, does not result in a statistically significant reduction
in the value of $\chi^{2}_{\rm min}$ (cf. Tables 2 and 3). In contrast,
including the quadratic parameter $p^{\prime\prime}_{L0}$ does significantly
reduce the value of $\chi^{2}_{\rm min}$ for the MP (cf. Table 4 with Table 3
or 2), but not for the IP nor for the LFC.

###  A.3 Comparison of model fits to MP

Table 2 shows that a single-Gaussian profile and constant polarization degree
and position angle provide a statistically adequate fit to the Stokes data for
the IP and for the LFC. However, the simple model does not provide a
statistically adequate fit to the Stokes data for the MP, at least in part due
to the higher signal-to-noise ratio in the MP Stokes data. Consequently, we
here investigate more complicated models in order to improve the goodness of
the $\chi^{2}$ fits to the MP Stokes data. In particular, we investigate using
a multi-Gaussian function for the MP pulse profile. Table 5 lists the minimum
$\chi^{2}$ and degrees of freedom $\nu$ for $I$, $Q$, $U$, and $V$ data sets
combined and separately, followed by best-fit estimators and (1-sigma)
uncertainties for the 9 polarization coefficients ($p_{L0}$,
$p^{\prime}_{L0}$, $p^{\prime\prime}_{L0}$; $\psi_{0}$, $\psi^{\prime}_{0}$,
$\psi^{\prime\prime}_{0}$; $p_{C0}$, $p^{\prime}_{C0}$,
$p^{\prime\prime}_{C0}$) of the Taylor expansion through second order.

Table 5: Comparison of results of fitting the main pulse (MP) profile with a simple Gaussian, with a multi-Gaussian, and with a simple Gaussian after adjusting weightings. The models retain up-to-quadratic variations in the polarization functions $p_{L}(\varphi)$, $\psi(\varphi)$, and $p_{C}(\varphi)$. Parameter | Units | 1-Gaussian | 6-Gaussian | 1-Gaussian (Adj.)
---|---|---|---|---
$\chi^{2}/\nu$ |  | $3049./1296$ | $1823./1281$ | $1281./1296$
$\chi^{2}_{I}/\nu_{I}$ |  | $1909./324$ | $688./309$ | $324./324$
$\chi^{2}_{Q}/\nu_{Q}$ |  | $432./318$ | $430./303$ | $318./318$
$\chi^{2}_{U}/\nu_{U}$ |  | $440./318$ | $438./303$ | $318./318$
$\chi^{2}_{V}/\nu_{V}$ |  | $268./321$ | $267./306$ | $321./321$
$p_{L0}$ | $\%$ | $22.99\pm 0.23$ | $22.91\pm 0.24$ | $22.98\pm 0.30$
$p^{\prime}_{L0}$ | $\%/^{\circ}$ | $-0.32\pm 0.15$ | $-0.29\pm 0.15$ | $-0.31\pm 0.19$
$p^{\prime\prime}_{L0}$ | $\%/^{\circ}/^{\circ}$ | $0.86\pm 0.17$ | $0.89\pm 0.19$ | $0.88\pm 0.22$
$\psi_{0}$ | ${}^{\circ}{\rm PA}$ | $-89.34\pm 0.27$ | $-89.38\pm 0.29$ | $-89.34\pm 0.32$
$\psi^{\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}$ | $-0.16\pm 0.17$ | $-0.19\pm 0.17$ | $-0.16\pm 0.20$
$\psi^{\prime\prime}_{0}$ | ${}^{\circ}{\rm PA}/^{\circ}/^{\circ}$ | $-0.06\pm 0.18$ | $0.05\pm 0.20$ | $-0.06\pm 0.21$
$p_{C0}$ | $\%$ | $-1.25\pm 0.22$ | $-1.27\pm 0.23$ | $-1.25\pm 0.20$
$p^{\prime}_{C0}$ | $\%/^{\circ}$ | $-0.01\pm 0.15$ | $-0.02\pm 0.14$ | $-0.01\pm 0.13$
$p^{\prime\prime}_{C0}$ | $\%/^{\circ}/^{\circ}$ | $-0.20\pm 0.16$ | $-0.18\pm 0.18$ | $-0.20\pm 0.15$

Comparison of the column “MP” in Table 3 with that in Table 4 (or,
equivalently, with the column “1-Gaussian” in Table 5) finds that inclusion of
the three quadratic polarization coefficients—especially
$p^{\prime\prime}_{L0}$—reduces $\chi^{2}_{Q}$ by 42 (from 473 to 431). While
$\psi^{\prime\prime}_{0}=0$ and $p^{\prime\prime}_{C0}=0$ within statistical
uncertainties, $p^{\prime\prime}_{L0}\approx(0.9\pm 0.2)\%/^{\circ}/^{\circ}$
is statistically significant but small.

The main cause of the poor fit of the 1-Gaussian model to the MP data,
however, has nothing to do with polarization. Figure 3 illustrates that, for
the fine time resolution and the high signal-to-noise ratio of the MP data,
substructure in the pulse profile is quite evident. Using a 6-Gaussian (2
broad and 4 narrow) profile for $I(\varphi)$ substantially improves the fit.
Comparing the column “6-Gaussian” with “1-Gaussian” in Table 5 finds that
inclusion of $15=5\times 3$ additional (Gaussian) parameters reduces
$\chi^{2}_{I}$ by 1221 (from 1909 to 688). Even so, the fit to the Stokes data
is not formally acceptable.

It is important to note that the best-fit expectation values and uncertainties
for the polarization coefficients ($p_{L0}$, $p^{\prime}_{L0}$,
$p^{\prime\prime}_{L0}$; $\psi_{0}$, $\psi^{\prime}_{0}$,
$\psi^{\prime\prime}_{0}$; $p_{C0}$, $p^{\prime}_{C0}$,
$p^{\prime\prime}_{C0}$) are rather insensitive to details of the pulse
profile. Thus, we compensate for fine substructure in the pulse profile by
increasing the estimators for the measurement standard deviations until a
statistically acceptable fit is achieved. That is, we adjust $\sigma_{I}$,
$\sigma_{Q}$, $\sigma_{U}$, and $\sigma_{V}$ until (Eq. A.1)
$\chi_{I}^{2}/\nu_{I}$, $\chi_{Q}^{2}/\nu_{Q}$, $\chi_{U}^{2}/\nu_{U}$, and
$\chi_{V}^{2}/\nu_{V}$, respectively, are close to unity. The column
“1-Gaussian (Adj.)” in Table 5 shows the best-fit polarization parameters for
a single-Gaussian profile, with weightings adjusted as described. The only
noticeable effect of this adjustment upon the best-fit polarization parameters
is a small change—typically an increase—in their uncertainties. The
uncertainties quoted in Table 1 (§3) are the typically more conservative
values obtained using the single-Gaussian profiles and adjusted weightings.

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