# Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray
Flares?

Alexander Rudy11affiliation: Department of Astronomy & Astrophysics,
University of California, Santa Cruz, CA 95064, USA , Dieter
Horns22affiliation: Institut für Experimentalphysik, Universität Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany , Andrea DeLuca33affiliation:
INAF–IASF Milano, via E. Bassini 15, I-20133 Milano, Italy; INFN Pavia, via A.
Bassi 6, I-27100 Pavia, Italy , Jeffery Kolodziejczak44affiliation: NASA
Marshall Space Flight Center, Astrophysics Office (ZP12), Huntsville, AL
35812, USA , Allyn Tennant44affiliation: NASA Marshall Space Flight Center,
Astrophysics Office (ZP12), Huntsville, AL 35812, USA , Yajie
Yuan77affiliation: W. W. Hansen Experimental Physics Laboratory, Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, CA 94305, USA , Rolf Buehler55affiliation: DESY, Platanenallee 6,
D-15738 Zeuthen, Germany , Jonathon Arons66affiliation: Astronomy Department
and Theoretical Astrophysics Center, University of California, Berkeley, 601
Campbell Hall, Berkeley, CA 94720, USA , Roger Blandford77affiliation: W. W.
Hansen Experimental Physics Laboratory, Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA ,
Patrizia Caraveo33affiliation: INAF–IASF Milano, via E. Bassini 15, I-20133
Milano, Italy; INFN Pavia, via A. Bassi 6, I-27100 Pavia, Italy ,
Enrico Costa88affiliation: INFN Roma Tor Vergata, via della Ricerca
Scientifica 1, I-00133 Roma, Italy , Stephan Funk77affiliation: W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94305, USA , Elizabeth
Hays1111affiliation: NASA Goddard Space Flight Center, Astrophysics Science
Division, Greenbelt, MD 20771, USA , Andrei Lobanov77affiliation: W. W. Hansen
Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94305, USA , Claire
Max11affiliation: Department of Astronomy & Astrophysics, University of
California, Santa Cruz, CA 95064, USA ,
Michael Mayer55affiliation: DESY, Platanenallee 6, D-15738 Zeuthen, Germany ,
Roberto Mignani1010affiliation: INAF–IASF Milano, via E. Bassini 15, I-20133
Milano, Italy; Kepler Institute of Astronomy, University of Zielona G´ora,
Lubuska 2, 65-265 Zielona G´ora, Poland , Stephen L. O’Dell44affiliation: NASA
Marshall Space Flight Center, Astrophysics Office (ZP12), Huntsville, AL
35812, USA , Roger Romani77affiliation: W. W. Hansen Experimental Physics
Laboratory, Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Stanford, CA 94305, USA , Marco Tavani33affiliation: INAF–IASF
Milano, via E. Bassini 15, I-20133 Milano, Italy; INFN Pavia, via A. Bassi 6,
I-27100 Pavia, Italy ,
Martin C. Weisskopf44affiliation: NASA Marshall Space Flight Center,
Astrophysics Office (ZP12), Huntsville, AL 35812, USA

###### Abstract

One of the most intriguing results from the $\gamma$-ray instruments currently
in orbit has been the detection of powerful flares from the Crab Nebula. Such
events, with a cadence of about one per year, can be so dramatic as to make
the system the brightest source in the $\gamma$-ray sky as occurred in 2011
April. These flares challenge our understanding of pulsar wind nebulae and
models for particle acceleration. To pinpoint the production site(s) within
the Nebula, a multiwavelength campaign has been carried out using Keck, Hubble
Space Telescope (HST), and Chandra X-ray Observatory. As the short time scales
of the flares ($\lesssim 1$ day) suggest a small emitting region, the Crab’s
inner knot, located within a fraction of an arcsecond from the pulsar, has
been proposed to be just such a site. In this paper, we focus on IR, optical,
and X-ray observations of this feature to see if it might be related to the
$\gamma$-ray flares and to try to understand its nature. We find that the
knot’s radial size, tangential size, peak flux, and the ratio of the knot flux
to that of the pulsar as measured with HST are all correlated with the
projected distance of the knot from the pulsar. A new approach, using singular
value decomposition for analyzing time series of images, was introduced
yielding results consistent with the more traditional methods while some
uncertainties were substantially reduced.

We exploit the new characterization of the Crab’s inner knot to discuss
constraints on standard shock-model parameters that may be inferred from our
observations assuming the inner knot lies near to the shocked surface. These
include inferences as to wind magnetization $\sigma$, shock shape parameters
such as incident angle $\delta_{1}$ and poloidal radius of curvature $R_{c}$,
as well as the IR/optical emitting particle enthalpy fraction. We find that
while the standard shock model gives good agreement with observation in many
respects, there remain two puzzles: (a) the observed angular size of the knot
relative to the pulsar–knot separation is measured to be much smaller than
expected; (b) the variable, yet high degree of polarization reported is
difficult to reconcile with a highly relativistic downstream flow. In
contrast, the IR/optical flux of the inner knot is marginally consistent with
the scenario that the shock accelerates most of the optical emitting particles
in the nebula.

neutron stars: general — pulsars: individual — Crab nebula and pulsar (catalog
) (PSR B0531+21) (catalog ) — inner knot

††slugcomment:  Version 8: 2015.04.1799affiliationtext: Max-Planck-Institut
für Radioastronomie, auf dem Hügel 69, D-53121 Bonn, Germany

##  1 Introduction

The Crab pulsar and its nebula are one of the most studied targets in the sky
at all wavelengths. It serves as a test bed for pulsar theories as well as,
more generally, for astrophysical non-thermal processes. A very thorough
review of the Crab system, including high spatial resolution observations with
HST and Chandra, showing the complex, dynamic interaction of the pulsar wind
with the surrounding medium, has been compiled by Hester (2008). A more recent
review by Bühler & Blandford (2014) also includes the saga of the $\gamma$-ray
flares.

A crucial question implied by the observations of the $\gamma$-ray flares is
“how are the particles being accelerated to sustain the flaring behavior?”
Radiating particles must have PeV energies to emit synchrotron radiation at
the observed $\gamma$-ray wavelengths. A mechanism is needed that can
accelerate particles to such high energies in less than one gyration time in
the ambient magnetic field. For example, the large flare of 2011 April
produced a radiant energy equivalent to the energy stored in the mean magnetic
field within a region of order $2\times 10^{14}$ m subtending an angle
$\approx 0.3\arcsec$ at 2 kpc, consistent with the emission-region size
estimated from light-travel-time arguments (Weisskopf et al., 2013).

An intense theoretical effort is currently being devoted to explain the
variable Crab behavior (e.g., Uzdensky et al., 2011; Yuan et al., 2011;
Komissarov & Lyutikov, 2011; Arons, 2012; Sturrock & Aschwanden, 2012;
Lyubarsky, 2012; Lyutikov et al., 2012; Bykov et al., 2012; Clausen-Brown &
Lyutikov, 2012; Cerutti et al., 2012, 2013; Baty et al., 2013; Teraki &
Takahara, 2013). However, the mechanism driving the flares, their impulsive
nature, the $\approx 12$-month recurrence time, and the location, remain
unknown. Possible explanations include plasma instabilities in the nebula,
magnetic reconnection, discontinuity in the pulsar wind acceleration, emission
from the anvil, emission from the inner knot, or portions of the termination
shock. See, for example, the review by Hester (2008) for the association of
specific features with this nomenclature

There are a number of reasons to suspect that the feature known as the ”inner
knot” should be considered as a possible site for the origin of the flares
(e.g., Komissarov & Lyutikov, 2011). First, the luminosity of the flares
accounts for a relatively high fraction of the pulsar spin-down luminosity
(assuming a radiation efficiency of 3–5%), suggesting an origin in a region
close to the pulsar (Tavani et al., 2011). Second, the very short variability
time scales combined with causality arguments, strongly constrain the size of
the flare’s emitting region (Buehler et al., 2012; Striani et al., 2011).

A multiwavelength campaign to study the Crab, described in Weisskopf et al.
(2013), is being performed. Another $\gamma$-ray flare—the second largest
observed to date—was detected with Fermi/LAT in 2013 March (Mayer et al.,
2013). This triggered further observations with Keck, with the Hubble Space
Telescope (HST), and with the Chandra X-ray Observatory. Here we concentrate
on what was learned about the inner knot as a consequence of this campaign. We
note that very-high-energy ($E\geq 100$ GeV) contemporaneous data from the
HESS (H. E. S. S. Collaboration et al., 2014) and VERITAS (Aliu et al., 2014)
Cherenkov-telescope arrays found no significant correlation of TeV flux with
the flare observed with Fermi/LAT.

The observations are presented in §2. The measured properties of the knot are
given in §3. §4 compares observations of the inner knot at the different
wavelengths. Comparison and examination of possible correlations of the knot’s
properties with $\gamma$-rays detected using the Fermi/LAT are discussed in
§5. §6 discusses theoretical implications. §7 summarizes.

##  2 The Observations

The feature we refer to as the “inner knot” (aka “knot 1”, or “synchrotron
knot”) was discovered by Hester et al. (1995) in a detailed study of the Crab
using high-resolution images taken with HST. The feature, although resolved in
those data, is small ($\approx 0.1\arcsec$) and close to the pulsar ($\approx
0.6\arcsec$). The feature was first suggested to be associated with a shock or
instability in the jet $\approx 1500$ AU from the pulsar, due to its good
alignment with the jet and elongation perpendicular to the jet. Later
Komissarov & Lyubarsky (2003) proposed, based on two-dimensional
magnetohydrodynamic (MHD) simulation of the pulsar wind nebula, that the knot
represents Doppler-boosted emission from an oblique termination shock.
Synthetic synchrotron images from these 2D simulations (Komissarov & Lyutikov,
2011) and more recently 3D simulations (Porth et al., 2014) indeed show a
knot-like feature when the system is viewed from the proper (and reasonable)
orientation. Moreover, the observed high degree of polarization and a position
angle aligned with the symmetry axis (Moran et al., 2013) seems to lend
further support to the oblique-shock scenario.

The salient features of the inner knot deduced prior to our observations are
that it has a power-law spectrum $F_{\nu}\propto\nu^{-\alpha}$ with index
$\alpha\sim 0.8$ in the IR to optical (Sollerman (2003), Melatos et al.
(2005)) and, as most recently measured by Moran et al. (2013), the flux
varies, the position appears to change, the degree of polarization is very
large but does not appear to change with the flux (but see the discussion at
the end of §5), and the magnetic field (determined from the position angle of
the optical polarization) is orthogonal to the axis of symmetry (presumably
the spin axis) and consistent with the direction of the field for the rest of
the inner nebula.

###  2.1 Keck

We obtained K’ and H band near-infrared (NIR) images of the inner $40\arcsec$
of the Crab Nebula using the Keck Near-Infrared Camera 2 (NIRC2; K. Matthews,
PI), with laser-guide-star adaptive optics (AO; Wizinowich et al., 2006). From
2012 February to 2014 January, we obtained 11 observations of the nebula
during $\gamma$-ray quiescent periods and 1 target-of-opportunity observation
(K8) triggered by $\gamma$-ray flares. Table 1 summarizes the Keck
observations.

Data reduction was carried out using the pipeline developed by Ghez et al.
(2008). Individual frames were sky-subtracted, flat-fielded, bad-pixel-
corrected, and corrected for the NIRC2’s camera distortion. We used the
solution developed by Yelda et al. (2010), which matched the positions of
stars in the globular cluster M92 measured with HST to those measured in the
NIRC2. Frames were checked to ensure that they showed no irregularities in the
point-spread function (PSF), and that the Strehl ratio was at least $\sim
0.10$. Individual frames were then combined using the Drizzle algorithm
(Fruchter & Hook, 2002) to provide sub-pixel accuracy and high signal-to-noise
images for analysis and modeling (§3.1 and Appendix).

The observations were conducted with two different pixel scales, using the
NIRC2 “wide” camera, with $0.04\arcsec$ pixels and a $40\arcsec\times
40\arcsec$ field of view, and the NIRC2 “narrow” camera, with $0.01\arcsec$
pixels and a $10\arcsec\times 10\arcsec$ field of view. The total effective
exposure times ranged from $420$ to $\approx 2600$ seconds. The shorter
exposures result in optimal use of 1-hour Target-of-Opportunity
interrupts—including acquisition, calibration, and sky measurements.

The spatial resolution of Keck’s NIRC2 with AO proved to be very effective in
resolving the detailed structure and position of the knot. Figure
Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray
Flares? shows the 12 NIRC2 images. In the narrow camera, the center of the
pulsar and the center of the knot are separated by $\approx 60$ pixels.
Typical values for the signal-to-noise ratio of the knot are $5$. The PSF of
NIRC2 with AO varies slightly in time due to the variable performance of the
AO system. As the pulsar and knot are well resolved from each other in the
Keck images, we chose not to deconvolve the PSF from the Keck images.

###  2.2 HST

Since 2012 January, we have obtained 17 observations using the HST Advanced
Camera for Surveys (ACS) Wide-Field Channel (WFC), with $0.05\arcsec$ pixels.
Each observation comprises 4 exposures totaling 2000 seconds and uses a
standard 4-point BOX dithering pattern to fill the inter-chip gap as well as
to allow efficient cosmic-ray cleaning. We employed the F550M filter, which is
well suited to sampling continuum emission from the Crab with almost no
contamination from line emission.

Table 2 summarizes the observations, listing the date and the mean observing
time. The Appendix discusses in detail our analyses of the 17 HST/ACS images.
Figure Characterization of the Inner Knot of the Crab: The Site of the Gamma-
ray Flares? displays the processed images, which have removed effects of the
HST/ACS WFC point spread function using singular-value decomposition (SVD) and
Richardson-Lucy deconvolution.

###  2.3 Chandra

Our approach for searching for X-ray emission from the inner knot with Chandra
involved a special use of both the high time resolution and high spatial
resolution of the High-Resolution Camera for Spectroscopy (HRC-S) array in
timing mode. So doing combines the best spatial resolution and the best time
resolution afforded by the Chandra instrumentation. To achieve the time
resolution, it is vital that the counting rate be below the rate where
telemetry is saturated (see, e.g., Tennant et al., 2001). Consequently, the
Low-Energy Transmission Grating (LETG) was inserted into the optical path to
reduce the flux. As inserting the grating is of itself insufficient to avoid
telemetry saturation, the trigger threshold of the HRC-S was was raised to
reduce the rate of detected events. This approach was first successfully used
in a trial experiment (ObsID 11245) on 2010 November 16.

The next set of observations in this mode was initiated by us on 2013 March 5
as part of a sequence triggered by a $\gamma$-ray flare (Ojha et al., 2013).
Unfortunately, the gain of the HRC had dropped since 2010. Consequently, the
trigger threshold was too high during our first observation on March 5-6,
reducing the counting rate from the pulsar to about a quarter of that
expected, dramatically decreasing the number of detected counts. The next HRC
observation was to take place on March 10 and the observation had already been
fully planned with the commands to execute onboard the spacecraft. To replan
at this stage would require an extremely short turnaround, yet it was
accomplished. Finally, beginning on 2013 October 22, we triggered another
sequence of HRC observations in response to the announcement of another
$\gamma$-ray flare (Buson et al., 2013).

We also consider ObsID 9765 taken on 2008 January 22. For this observation the
HRC was operated in a more standard configuration with the threshold set at
the nominal value. However, to reduce the count rate, a blade was inserted
into the optical path, which is no longer allowed. This blade also reduced the
diffracted flux in one direction. By chance, this reduction is greatest near
the nominal location of the knot. This reduction, together with a long
integration time for this observation, results in ObsID 9765 providing the
best data for study of the knot. Table 3 summarizes the relevant Chandra
observations.

##  3 Properties of the Knot

###  3.1 Keck

Table 4 lists the best-fit values of the model parameters for the inner knot
for each of the 12 Keck observations based upon the analysis and modeling
described in the Appendix. The properties are these: $\psi_{0}$, the position
angle (E of N) from the pulsar to the center of the knot; $r_{0}$, the
projected separation between the pulsar and the knot; FWHMr, the full width at
half maximum of the knot in the radial direction; FWHMt, the FWHM of the knot
in the tangential direction; and $F_{k}/F_{p}$, the ratio of the flux in the
knot to that in the pulsar. The flux ratio is used as measuring the fluxes of
the 5 stars in in the NIRC2-wide camera’s field of view shows that those
fluxes change relative to each other from one observation to the next. This
occurs because the nebular background varies spatially across the detector in
a non-trivial way, making it difficult to determine precisely the flux of any
given source. Consequently, we measure the ratio of the knot’s flux to the
pulsar’s, rather than the knot’s absolute flux. Figure Characterization of the
Inner Knot of the Crab: The Site of the Gamma-ray Flares? plots these
properties versus the pulsar–knot separation, based upon the Keck data.

We seek possible correlations of these properties with the separation $r_{0}$
from the pulsar, of the following form: $p(r_{0})=p(r_{1})[r_{0}/r_{1}]^{q}$.
We choose a reference distance $r_{1}=0.6692\arcsec$, obtained from the
logarithmic average of the 12 values of $r_{0}$. The position angle of the
knot, $\psi_{0}$, was fit to the linear regression
$\psi_{0}=\psi_{0}(r_{1})+\psi_{0}^{\prime}(r_{1})[r_{0}-r_{1}]$. Table 5
tabulates the results of the regression analyses of the inner knot’s intrinsic
properties with its (projected) separation from the pulsar. Based upon the
F-test probability, the power-law regression analysis finds marginally
significant correlations of the radial width, tangential width, and peak
intensity (surface brightness) with separation. The correlation of the knot’s
flux to the separation is not significant.

###  3.2 HST

Table 6 lists the best-fit values of the model parameters for the inner knot
for each of the 17 HST observations based upon the modeling and analysis
described in the Appendix. The properties are these: the position angle
$\psi_{0}$ of the knot with respect to the pulsar, the (projected) distance
$r_{0}$ from the pulsar, radial width (FWHMr) and tangential width (FWHMt) of
the knot, the peak surface brightness $S_{k}$, the knot flux $F_{k}$, the
pulsar flux, $F_{p}$, and flux ratio $F_{k}/F_{p}$. Although we expected the
pulsar flux to be steady, we measured a small ($2.5\sigma$) variation.
Consequently, we prefer to work with the flux ratio $F_{k}/F_{p}$.

In the Appendix we discuss two ways of analyzing these data. Table 6 presents
the results for one of these methods. However, our general conclusions
concerning the knot’s optical properties are for the most part the same for
either analysis. Figure Characterization of the Inner Knot of the Crab: The
Site of the Gamma-ray Flares? plots the knot properties versus the pulsar–knot
separation, based upon the HST data.

As with the Keck data, we perform a regression analysis of the inner knot’s
intrinsic properties with its (projected) separation from the pulsar (Table
7). We choose a reference distance $r_{1}=0.6559\arcsec$, obtained from the
logarithmic average of the 17 measured values of $r_{0}$. The position angle
of the knot, $\psi_{0}$, was fit to a linear regression
$\psi_{0}=\psi_{0}(r_{1})+\psi_{0}^{\prime}(r_{1})[r_{0}-r_{1}]$.

Based upon the F-test probability, the power-law regression analysis finds
statistically significant correlations of the tangential and radial widths
with separation from the pulsar, as well as the previously known (Moran et
al., 2013) anti-correlations of flux and peak surface brightness with
separation. The correlation of $\psi_{0}$ with separation $r_{0}$ is the least
strong. There is a small systematic bias in the power-law index of the FWHMr
correlation, in that the radial width of the knot at times approaches the
resolution limit of the instrument and analysis procedures. Applying our
analysis procedures (see Appendix) to simulated knots of known true radial
width, we estimate that the smallest measured values of FWHMr are oversized by
about 7% and the largest by about 3%, which would increase the power-law index
for the FWHMr–$r_{0}$ correlation from $q=0.72$ to about $0.8$. Finally, we
emphasize that we find similarly strong correlations—except for the flux—using
the more traditional HST analysis methods described in the Appendix.

###  3.3 Chandra

For the Chandra data, the main question is whether X-ray emission from the
knot can be detected—especially in the vicinity of the bright, X-ray-emitting
pulsar. The answer, unfortunately, is “No” and here we set an informative
upper limit to the X-ray flux of the inner knot relative to that of the
pulsar.

Table 8 shows the number of counts in each of three extraction regions: (1)
centered on the approximate location of the optical knot, (2) centered on the
pulsar, and (3) placed on a reference region to estimate the background.
Details may be found in the Appendix and are illustrated in Figure
Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray
Flares?. The analysis was performed both for the full phase-averaged data and
for the data at pulse minimum. Consider, for example, the phase-averaged data
for ObsID 11245. Comparing the inner knot counts to the background (Table 8)
we find an excess of 167 counts. If the excess were really due to the knot, we
would accordingly expect an excess of 50 ($=167\times 0.3$) counts at phase
minimum. As this was not the case, we attribute the excess to the wings of the
PSF from the pulsar and posit that it is due to the spring/fall asymmetry
discussed in the Appendix. As the impact of that asymmetry is below the
statistical noise for the data taken at phase minimum, we then use the phase-
minimum data to derive upper limits. We convert our phase-minimum upper limit
into a phase-average upper limit by accounting for the $0.3$ phase duration of
the minimum and then divide by the phase average pulsar flux. Results are
listed in the last column in Table 8.

##  4 Comparison amongst observations of the inner knot

Our Chandra observations were unable to detect X-ray flux from the knot,
setting a 3-sigma upper limit to the ratio of knot flux to pulsar flux
$F_{k}/F_{p}<0.0022$. This upper limit is 20–40 times smaller than the ratio
measured in the optical ($F_{k}/F_{p}=0.044\pm 0.001$) and in the near-IR
($F_{k}/F_{p}=0.079\pm 0.005$), indicating that the optical–X-ray spectral
index of the knot is significantly steeper (by at least 0.5) than that of the
pulsar.

Both our infrared and optical observations detected time variability in the
pulsar–knot separation, with roughly consistent separations when measured
contemporaneously (see the lower portion of Figure Characterization of the
Inner Knot of the Crab: The Site of the Gamma-ray Flares?). Correlations of
knot properties with pulsar–knot separation are very strong for the optical
measurements but rather weak for the infrared measurements. We attribute this
difference to a combination of factors—including poorer statistics for the
much shorter NIR measurements and a number of systematic effects, resulting in
part from the NIR PSF and background varying across the field and in time. As
the lower portion of Figure Characterization of the Inner Knot of the Crab:
The Site of the Gamma-ray Flares? and Table 4 indicate, NIR-measured
separations at essentially the same epoch exhibit a scatter substantially
larger than the quoted statistical errors in the measurement, thus suggesting
the presence of unaccounted systematic errors.

##  5 Comparison with $\gamma$-ray fluxes

It is clear from the previous sections that the intrinsic properties of the
optical knot are correlated with its projected separation from the pulsar. The
correlation for the NIR knot is but marginally significant. Thus, in searching
for any relationship between the knot and the $\gamma$-ray flux, we
concentrate on the separation versus $\gamma$-ray flux. The analysis performed
to reduce the Fermi/LAT data is the same as described in Mayer et al. (2013).

Table 9 lists the dates and separations measured in the Keck and in the HST
observations, followed by the corresponding 12-hour-average Fermi/LAT $\geq
100$-MeV fluxes. The upper portion of Figure Characterization of the Inner
Knot of the Crab: The Site of the Gamma-ray Flares? shows the Fermi/LAT flux
as a function of time covering the interval when our Keck and HST observations
were made. The vertical lines identify times when observations were made with
Keck (red) or with HST (green). The lower portion of Figure Characterization
of the Inner Knot of the Crab: The Site of the Gamma-ray Flares? plots the
pulsar–knot separations on the same time scale. Figure Characterization of the
Inner Knot of the Crab: The Site of the Gamma-ray Flares? expands Figure
Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray
Flares? at the times of the largest $\gamma$-ray flares.

Figure Characterization of the Inner Knot of the Crab: The Site of the Gamma-
ray Flares? plots the $\gamma$-ray flux versus the measured radial separation
of the knot from the pulsar. The results captured in this figure clearly do
not represent a significant simple correlation of the knot properties with the
$\gamma$-ray flux: The formal correlation coefficient is only 0.2.

Inspection of the $\gamma$-ray light curve and the time series of pulsar–knot
separations (Figure Characterization of the Inner Knot of the Crab: The Site
of the Gamma-ray Flares?) indicates that the most energetic $\gamma$-ray
flares occurred at a time which is not precisely coincident with the time of
greatest pulsar–knot separation. It is reasonable to ask how probable such a
coincidence would be as a chance occurrence. A simple answer is that the
probability of a chance coincidence is just the difference in the central
times of the two events divided by the total duration of this observing
program, as described below.

We next estimate the time and duration of a $\gamma$-ray flare and the
excursion in $r_{0}$ (lower portion of Figure Characterization of the Inner
Knot of the Crab: The Site of the Gamma-ray Flares?), using a simple Gaussian
model in both cases and including a linear baseline trend for the $\gamma$-ray
flare. This is not to suggest that a Gaussian properly describes the
underlying physical processes, but simply to enable comparison of the two time
series, one for the flares, one for the separations. Fitting separation data
in the MJD range 56180-56400 and $\gamma$-ray fluxes in the range 56345-56370
places the midpoint of the inner knot excursion at 56331.8, followed by the
$\gamma$-ray flare at 56357.3, 25.5 days later. The Gaussian width is a
measure of the event duration with FWHMs of 70.7 and 7.6 days for the
separation excursion and $\gamma-$ray flare data respectively.

The time between our first and last pulsar–knot separation measurements is
826.1 days. However, on 4 occasions the gap between separation measurements
was longer than the 70.7 day duration of the observed separation excursion. We
assume then that a separation excursion would be observable if an observation
fell within the FWHM window and not observable if it fell outside the window.
To obtain the total observing program under this assumption, we remove the
time of 4 gaps (140.1 days) that exceeded 70.7 days making the total duration
of the observing program 686 days. Thus an estimate of the chance probability
is $25.5/686=3.7\%$, which is approximately at the $2\sigma$ level.

It is worth noting that the largest pulsar–knot separation incursion (toward
the pulsar) occurs near the second most energetic $\gamma-$ray flare. However,
the knot separation data are sparser at this time. Therefore, we use the
previous FWHM value of 70.7 days to establish the time of the incursion. With
this prior, fitting the separation data in the MJD range 56390-56670 and
$\gamma$-ray fluxes in the range 56575-56590 places the midpoint of the inner
knot excursion at 56566.0, followed by the $\gamma$-ray flare at 56582.6, 16.6
days later.

In this second case, the comparable chance probability estimate is
$16.6/686=2.4\%$ ($\sim 2.5\sigma$). As each variation of the knot-pulsar
separation is of the opposite sign, we must multiply the probabilities by a
factor of 2 when combining them. So our estimate of the chance probability
that the 2 largest knot excursions/incursions would occur within the observed
intervals of the 2 largest $\gamma$-ray flares is $0.037\times 0.024\times
4=0.4\%$, which is approximately at the $2.9\sigma$ level. There are
additional $\gamma$-ray flares; however, the third most energetic flare occurs
only 12 days later than the one on MJD 56582.6 and therefore is well within
the assumed 70-day range of the associated knot incursion. The other flares
are significantly weaker and less well-sampled by the optical/infrared
observations.

Although coverage is sparse, there is no compelling evidence in the
pulsar–knot separations for large ($>0.01\arcsec$) excursions from the mean
separation at time-scales comparable to the flare durations. Indeed, knot
incursion/excursion time scales appear to be $>10\times$ longer than flare
durations, so direct correlation of the $\gamma$-ray light curve with the
separation time-evolution produces correlation coefficients that are rather
difficult to interpret without associated simulations. Such simulations would
require assumptions regarding an underlying physical model relating the two
curves. As we have no such model, any correlation analysis would be purely ad
hoc.

The 0.4% value for the chance probability is based upon a rather simplistic
analysis involving only 2 flares and was, of course, determined a posteriori.
However, in continuing to monitor the Crab at multiple wavelengths with the
goal of understanding the origin of its $\gamma$-ray flares, the inner knot
will continue to be a feature of interest. Based upon the $r_{0}$ time scale
of 71 days (30-day Gaussian $1\sigma$), a monitoring program which includes
monthly optical observations is necessary to detect significant inner-knot
excursions from its mean pulsar separation. Monitoring every $\sim 3$ weeks
should be sufficient to characterize fully these variations and to verify the
small statistical sample from which these parameter estimates were derived.

We also note that an inner-knot incursion with an $\approx 2$-month FWHM
timescale and $0.075\arcsec\pm 0.025\arcsec$ pulsar–knot separation inward
amplitude was also reported by Moran et al. (2013). This event appears to be
quite similar in duration and amplitude to our observations. The reference
also reports an $\approx 2\sigma$ apparent change in the knot polarization
parameters, which occurred $22\pm 5$ days after the time of maximum incursion.
The degree of linear polarization steps from $59.9\%\pm 6.9\%$ on 2005 Nov 25
to $42.8\%\pm 6.2\%$ 2005 Dec 5 and the position angle changes from
$125.4^{\circ}\pm 3.3^{\circ}$ to $121.6^{\circ}\pm 4.2^{\circ}$.
Interestingly, the time frame of this polarization shift occurred in a window
closely matching both the 25.5 and 16.6 day $\gamma$-ray flare delays after
the peak inner-knot excursion/incursion times discussed above. While the
claimed $2\sigma$ result is not very significant, we observe that prior to the
step near the start of 2005 December, all 10 of the reported measurements of
the degree of linear polarization fall into a population with mean $60.7\%$
and small standard deviation $1.0\%$, whereas the 2 subsequent measurements
had a mean of $43.4\%$ with standard deviation of $0.8\%$. If we suppose that
the quoted errors in the reference apply to the absolute, but not relative
measured values, then the size of the step change would be $>10\sigma$ instead
of $\approx 2\sigma$. This may be a reasonable assumption, as the point-to-
point measurement variations are smaller by a factor $>6$ than the quoted
errors. Similar arguments would apply to the polarization position angle. From
this we conclude that polarization should also be of interest in future
attempts to relate inner-knot behavior to $\gamma$-ray flaring.

##  6 Implications for Theoretical Models

In the basic model of the Crab Nebula (e.g., Rees & Gunn, 1974), the pulsar
wind passes through a shock at a radius $\sim 3\times 10^{17}$ cm where the
wind momentum flux balances the nebular pressure. However, the wind should be
anisotropic. For example, the momentum flux in a split-monopole
$\propto\sin^{2}\theta$ (Bogovalov, 1999) and recent simulations suggest
$\propto\sin^{4}\theta$ (Tchekhovskoy et al., 2013; Tchekhovskoy, Philippov, &
Spitkovsky, 2015) where $\theta$ is the polar angle. As a result, the shock is
likely to be quite oblate. The sections of the shock near the poles would be
oblique and much closer to the pulsar than the equatorial part. The observed
radiation presumably comes from relativistic particles (electrons or
positrons) accelerated behind the shock. Since the outflow from an oblique
shock can remain relativistic, we would be able to see a compact, emitting
feature (the inner knot) if the relativistic outflow happens to be aligned
with our line of sight so that its emission is beamed. The inner knot should
appear to us as having some offset from the pulsar due to the deflection of
the outflow from its presumed radial motion (Figure 1). If the shock is
approximately axisymmetric as one would expect, the projected emitting site
would fall on the projected symmetry axis, leading to an alignment with the
jet. The shape of the inner knot should also be more or less symmetric about
the axis; whether it is elongated parallel or perpendicular to the axis
depends on the geometry of the oblique shock. Furthermore the synchrotron
emission should be linearly polarized (Komissarov & Lyutikov, 2011, Yuan &
Blandford 2015 in preparation).

The shock model predicts that the scaling between the observed properties of
the knot—projected pulsar–knot separation $r_{0}$, tangential angular width
FWHMt, radial angular width FWHMr and surface brightness $S_{k}$—should be
determined by the upstream magnetization $\sigma\equiv
B_{1}^{2}/\mu_{0}n_{1}\gamma_{1}^{2}mc^{2}$ (where $m$ is electron mass and
$B_{1}$, $n_{1}$, $\gamma_{1}$ are upstream magnetic field, proper density,
bulk Lorentz factor, respectively) plus three more parameters that
characterize the shape of the shock near the emitting site: The unprojected
pulsar-knot separation $r_{k}$; the incident angle $\delta_{1}$, defined to be
the angle between upstream velocity and the shock surface as shown in Figure
1; and the shock poloidal radius of curvature $R_{c}$.

Let the outflow be deflected from the radial direction by an angle $\Delta$ at
the shock. We then have $r_{0}=r_{k}\Delta$. The knot tangential size is
determined by the Doppler beaming and the shock radius of curvature in the
toroidal direction as FWHM${}_{t}\approx 2r_{k}/\gamma$, where $\gamma$ is the
downstream Lorentz factor. In the simplest shock model with an isotropic
plasma, both $\Delta$ and $\gamma$ are functions of $\sigma$ and $\delta_{1}$:
$\Delta=\delta_{1}-\arctan(\chi\tan\delta_{1})$ and
$\gamma=1/(\sin\delta_{1}\sqrt{1-\chi^{2}})$, where $\chi\equiv
v_{2\perp}/v_{1\perp}=B_{1}/B_{2}=(1+2\sigma+\sqrt{16\sigma^{2}+16\sigma+1})/6(1+\sigma)$
is the compression ratio at the shock, $v_{1,2\perp}$ being the component of
upstream/downstream velocity perpendicular to the shock (Komissarov &
Lyutikov, 2011). The radial knot size FWHMr should also be proportional to
$r_{k}/\gamma$ but will have an additional factor involving the ratio
$R_{c}/r_{k}$.

As to the intensity, we assume that the emitting particles have a power-law
distribution $\propto n^{\prime}\gamma^{\prime-p}$ in the fluid rest frame,
where $n^{\prime}$, $\gamma^{\prime}$ are particle density and Lorentz factor
in the same reference frame, and $p\sim 2.6$ from the measured spectral index
of the knot (Sollerman, 2003; Melatos et al., 2005).

The emissivity in the fluid rest frame is
$j^{\prime}_{\nu^{\prime}}(\nu^{\prime})\propto
n^{\prime}{B^{\prime}}^{(p+1)/2}{\nu^{\prime}}^{-(p-1)/2}$, where $B^{\prime}$
and $\nu^{\prime}$ are magnetic field and radiation frequency measured in this
reference frame. After transforming to the nebula frame, the emissivity has a
dependence on the flow Lorentz factor $\gamma$, particle (improper) number
density $n$, and magnetic field $B$ as follows:
$j_{\nu}\propto\mathcal{D}^{2+(p-1)/2}(n/\gamma)(B/\gamma)^{(p+1)/2}$, where
$\mathcal{D}=\nu/\nu^{\prime}=\sqrt{1-\beta^{2}}/(1-\vec{\beta}\cdot\vec{n})$
is the Doppler factor, in which $\vec{\beta}$ is the flow velocity and
$\vec{n}$ is a unit vector along the direction of line of sight. The surface
brightness corresponds to the emissivity integrated along the line of sight in
the nebular frame: $S_{\nu}=\int j_{\nu}dl$. To estimate the peak surface
brightness, one takes the maximum Doppler factor $\mathcal{D}\approx 2\gamma$,
and we get $S_{\nu,peak}\propto nB^{(p+1)/2}\ell$ where $\ell$ is an
estimation of the length of the emitting region along the line of sight. As
roughly $B\propto r_{k}^{-1}$ and $n\propto r_{k}^{-2}$, we have
$S_{\nu,peak}\propto r_{k}^{-2-(p+1)/2}\ell$.

These previous relations allow us to set the constraints discussed below using
both the steady and variable properties of the knot.

###  6.1 Variability

In the shock model, variability can arise either upstream or downstream of the
shock. Firstly, stress tensor variation in the nebula can cause the shock
radius and shape to vary over time. Three dimensional MHD simulations (Porth
et al., 2014) show that even if the upstream condition is fixed, the post-
shock flow can be quite variable: there is vortex-shedding from the
termination shock, and the shock constantly interacts with waves and vortices
in the nebula. Those variations are expected to be mostly magnetic in origin,
and the interactions between the shock and the downstream flow are quite
nonlinear. The picture is similar to earlier 2D simulations (e.g., Camus et
al., 2009) except that the kink instability sets in so that the hoop
compression is less prominent as in the 2D case; also short-term variability
is less pronounced. The 3D simulations give a typical variation time scale of
less than one year.

Now suppose that the change of downstream pressure causes the shock radius
$r_{k}$ to change. For simplicity we first assume that $\Delta$, $\ell$,
$\gamma$ and $p$ stay more or less the same, then from the above scaling
relations, we have $S_{\nu,peak}\propto r_{0}^{-2-(p+1)/2}$,
FWHM${}_{t}\propto r_{0}$, and knot flux $F_{\nu}\propto S_{\nu,peak}{\rm
FWHM}_{t}\,{\rm FWHM}_{r}\propto r_{0}^{-(p+1)/2}$, assuming FWHMr to be also
roughly proportional to $r_{k}$. In reality $\Delta$, $\ell$, $\gamma$ change
with the shock radius $r_{k}$ as well, as the shock shape and upstream
incident angle $\delta_{1}$ vary. Consequently, the observed correlation is
more complicated and less clean; however, the general trend is consistent with
our expectation of the shock model.

Variations initiated by the pulsar can also cause the knot properties to
change. In such a scenario, the shortest possible variation time scale is
$t_{v}\sim r_{k}(1-\cos(1/\gamma))/c\sim r_{k}/(2\gamma^{2}c)\sim 1\
\text{day}(r_{k}/10^{17}\,\text{cm})(\gamma/5)^{-2}$ up to 10% inaccuracy
assuming $\gamma\gtrsim 3$. However, there is no evidence in the pulsar timing
that this is actually happening.

###  6.2 Time averaged properties of the knot

The properties of the knot may be used to set constraints on the flow
composition and how much dissipation occurs at the shock. One constraint comes
from geometrical relations amongst the three measured angles $r_{0}$, FWHMt,
and FWHMr. For the simplest shock model, we have $r_{0}=r_{k}\Delta$ and
FWHM${}_{t}\approx r_{k}/\gamma$, where $\Delta$ and $\gamma$ only depend on
$\sigma$ and $\delta_{1}$, such that the ratio FWHM${}_{t}/r_{0}$ sets limit
on $\sigma$ and $\delta_{1}$. We find that
FWHM${}_{t}/r_{0}=2/(\gamma\Delta)=2\sin\delta_{1}\sqrt{1-\chi^{2}}/[\delta_{1}-\arctan(\chi\tan\delta_{1})]\geq
2.8$, with the minimum at $\sigma=0$ and $\delta_{1}=0.39$. In contrast, the
observation gives FWHM${}_{t}\sim 0.5r_{0}$, indicating that some important
ingredient is missing in the standard shock model. For example, the upstream
flow might have already been bent before arriving at the shock; and/or the
anisotropic particle distribution downstream might be important in determining
the size of the knot. Although more detailed investigations will be described
elsewhere, the above discussion on variability and the following discussion on
energetics should be valid for quite general shock models.

The shape of the knot also has some interesting implications. In both the
Hubble and Keck images, the knot is not just a simple ellipse that only
involves quadratic terms of coordinates; it has curvature and appears to be
convex away from the pulsar (“smile”), so at least a third moment of the
coordinates is needed to describe the shape. A simple intuitive picture is
based on the geometry that the shock surface is similar to a torus centered on
the pulsar and we look through the hole from the bottom. If emission from the
shock outflow roughly follows the local toroidal magnetic field, one would
imagine the knot to be convex toward the pulsar (“frown”)—contradictory to
what we see. In fact, we should not overlook the possibility that the outflow
may have significant bending downstream. If the outflow bends toward the
equator by an angle $\sim 1/\gamma$ within the emission length, we would see
the knot more extended (contracted) on the far (near) side with respect to the
pulsar, as additional flow lines enter (originally aligned flow lines leave)
the emission cone on the corresponding side.

Another constraint comes from the observed flux of the knot. In the
$K^{\prime}$ band, the typical knot-pulsar flux ratio is $\sim 0.063$ (Table
4). If we take the photometry of the pulsar from Sandberg & Sollerman (2009),
the pulsar $K_{s}$ band magnitude is $13.80\pm 0.01$, after dereddening with
the best fit value $E(B-V)=0.52$ (Sollerman et al., 2000) and the extinction
law from Fitzpatrick (1999), this corresponds to a flux of $2.2\times
10^{-26}\text{erg}\,\text{s}^{-1}\text{cm}^{-2}\text{Hz}^{-1}$. So the inner
knot flux in $K^{\prime}$ band should be roughly
$F_{\nu}^{\text{knot}}=1.4\times
10^{-27}\text{erg}\,\text{s}^{-1}\text{cm}^{-2}\text{Hz}^{-1}$. Similarly,
optical data (Table 6) on average give a dereddened flux of
$F_{\nu}^{\text{knot}}=1\times
10^{-27}\text{erg}\,\text{s}^{-1}\text{cm}^{-2}\text{Hz}^{-1}$. However, we do
not attempt here to get a spectral index using the time averaged fluxes due to
the variability of the knot and the fact that most of the multi-wavelength
observations are not simultaneous. Previous measurements by Sollerman (2003)
and Melatos et al. (2005) indicate that the spectrum of the knot in IR to
optical range is a power law $F_{\nu}\propto\nu^{-\alpha}$ with $\alpha\sim
0.8$, and Melatos et al. (2005) further measured that this power law extends
to NUV ($160-320$ nm), with a flux in NUV $F_{\nu}^{\text{knot}}=0.12\,{\rm
mJy}=1.2\times
10^{-27}\text{erg}\,\text{s}^{-1}\text{cm}^{-2}\text{Hz}^{-1}$—this is the
highest energy at which the knot has been measured. A few more recent
measurements show slightly different spectral index ranging from 0.63
(Tziamtzis et al., 2009) to 1.3 (Sandberg & Sollerman, 2009) but unfortunately
these are based on non-simultaneous data. It would be very interesting to
refine the spectral measurement using near simultaneous IR/optical images in
the future. In what follows we adopt a fiducial spectral model
$F_{\nu}^{\text{knot}}\sim
10^{-27}(\nu/10^{15}\,\rm{Hz})^{-0.8}\text{erg}\,\text{s}^{-1}\text{cm}^{-2}\text{Hz}^{-1}$
with the caveat that this could change with better observations.

Using the above spectral model, we find that the knot integrated flux in
IR/optical band ($(0.1-2)\times 10^{15}\,\text{Hz}$) is $F_{\rm
O/IR}^{\text{knot}}=2.6\times
10^{-12}\,\text{erg}\,\text{s}^{-1}\text{cm}^{-2}$. Thus, the luminosity of
the knot per steradian is
$L_{\text{O/IR},\Omega}=D^{2}F_{\text{O/IR}}^{\text{knot}}=10^{32}\,\text{erg}\,\text{s}^{-1}$,
where $D\sim 2$ kpc is the distance of the Crab. If we define $\epsilon\equiv
L_{\text{O/IR},\Omega}/(\dot{E}/4\pi)$, where $\dot{E}=5\times 10^{38}\ {\rm
erg}\,{\rm s}^{-1}$ is the pulsar spin-down power, $\epsilon$ can be used as a
measure of the radiative efficiency of the shock along the direction of line
of sight in IR to optical range. Here we find $\epsilon=2.5\times 10^{-6}$ and
the shock is highly adiabatic.

We can get an estimation of the enthalpy fraction of IR/optical emitting
particles in the downstream flow

$\eta(\gamma^{\prime})=\frac{8}{3}\,\frac{\gamma
t^{\prime}_{\text{cool}}}{t_{\text{flow}}}\,\frac{\nu
L_{\nu,\Omega}}{\dot{E}/(4\pi)}=\frac{8}{3}\,\frac{\gamma
t^{\prime}_{\text{cool}}}{t_{\text{flow}}}\,\frac{\nu
L_{\nu,\Omega}}{\dot{E}/(4\pi)},$ (1)

where $t_{\text{flow}}$ is the flow time scale and
$t^{\prime}_{\text{cool}}=6\pi\epsilon_{0}m^{3}c^{3}/(e^{4}B^{\prime
2}\gamma^{\prime})\approx 10^{2}\gamma^{2}B_{-3}^{-3/2}\nu_{14}^{-1/2}\
\rm{years}$ is the synchrotron cooling time of the particles in the fluid rest
frame (we have adopted $B=B_{-3}\,\rm{mG}$ and $\nu=\nu_{14}10^{14}\,\rm{Hz}$
for the numerical value). Thus the particle injection rate per unit steradian
in corresponding energy band is

$\dot{N}(\gamma^{\prime})=\frac{3\eta(\gamma^{\prime})}{4\gamma^{\prime}mc^{2}}\frac{\dot{E}}{4\pi\gamma}.$
(2)

Adopting typical values $\gamma=5\gamma_{5}$, $t_{\rm
flow}=r_{17}10^{17}\,{\rm cm}/c$, we find that for IR emitting particles,
$\eta_{\rm{IR}}=0.05\gamma_{5}^{3}B_{-3}^{-3/2}r_{17}^{-1}$, and
$\dot{N}_{\rm{IR}}=2\times
10^{36}\gamma_{5}^{2}B_{-3}^{-1}r_{17}^{-1}\;\text{s}^{-1}\text{sr}^{-1}$. The
spectral index of the knot indicates that the particle distribution powerlaw
index should be $p\sim 2.6$, thus the IR emitting particles should comprise
the majority of particle pressure downstream. From the above estimations, it
seems that the particle injection rate at the shock is marginally consistent
with the scenario that most of the IR/optical emitting particles are provided
by the shock.

Regarding particle acceleration mechanisms, we notice that IR emitting
particles usually go through $N=t_{\rm flow}\nu_{g}$ Larmor orbits, where
$\nu_{g}=\nu_{B}/\gamma^{\prime}$ is the relativistic gyrofrequency—the
cyclotron frequency ($\nu_{B}$) divided by the electron’s Lorentz factor
($\gamma^{\prime}$). Eliminating $\gamma^{\prime}$ in favor of the magnetic
field ($B$) and observing frequency ($\nu$), the number of orbits within the
flow timescale is $N\approx 6\times 10^{4}B_{-3}^{3/2}r_{17}\nu_{14}^{-1/2}$.
To allow sufficient time for acceleration mechanisms requiring instabilities
or stochastic processes to operate, requires that $N>>1$. However, for
particles emitting $\gamma$ rays of energy 300 MeV, $N\lesssim 1$, thus
requiring special acceleration mechanisms—if the emitting electrons are indeed
accelerated at the shock.

One more constraint comes from the polarization of the knot. Most recent
polarimetry performed by Moran et al. (2013) gives a high polarization degree
$\sim 60\%$, with position angle aligned with the symmetry axis, indicating
strongly a toroidal magnetic field. Here we need to be careful about the
depolarization effect due to relativistic kinematics (Lyutikov et al., 2003).
What happens then is that Lorentz transformation of the emission of a
relativistic plasma from the comoving frame to the lab frame produces a
rotation of the polarization vector. For a curved emitting surface,
neighboring fluid elements have slightly different velocities, thus their
polarization vectors experience different amounts of rotation. As one sums the
contribution from the visible surface (this is essentially what we observe),
the result is some degree of depolarization. We find that for the oblique
shock in the Crab Nebula, under an ultrarelativistic approximation, the upper
limit of polarization degree is similar to that in Lyutikov et al. (2003):
$56.25\%$ for particle spectral index $p=3$ and $43.4\%$ for $p=2$. Thus, the
observed high degree of polarization and its possible variation is highly
constraining on theoretical models. It would be interesting to repeat this
observation, especially with adaptive optics.

##  7 Conclusions

We have (1) introduced a new approach to analyzing time series of images; (2)
discovered that key properties of the knot (radial width, tangential width,
flux) are correlated with the time-variable separation between the knot and
the pulsar; (3) shown that, with the available data, it is not possible to
determine a strong correlation between the knot separation and the occurrence
of $\gamma$-ray flares; (4) set an upper limit to the low-energy X-ray flux
from the inner knot; (5) discussed the implications of our observations to set
constraints on particular elements of the standard shock modeling of the
relativistic outflow from the pulsar. These include inferences as to wind
magnetization $\sigma$, shock shape parameters such as incident angle
$\delta_{1}$ and poloidal radius of curvature $R_{c}$, as well as the
IR/optical emitting particle enthalpy fraction. We found that while the
standard shock model gives good agreements with observations in many aspects,
there remain two puzzles: (a) the angular size of the knot relative to the
pulsar–knot separation is much smaller than expected; (b) the variable, yet
high degree of polarization is difficult to reconcile with a highly
relativistic outflow. We also found that the IR/optical flux of the inner knot
is marginally consistent with the scenario that the shock accelerates most of
the optical emitting particles in the nebula.

Acknowledgments The Fermi LAT Collaboration acknowledges generous ongoing
support from a number of agencies and institutes that have supported both the
development and the operation of the LAT as well as scientific data analysis.
These include the National Aeronautics and Space Administration and the
Department of Energy in the United States, the Commissariat à l’Energie
Atomique and the Centre National de la Recherche Scientifique / Institut
National de Physique Nucléaire et de Physique des Particules in France, the
Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in
Italy, the Ministry of Education, Culture, Sports, Science and Technology
(MEXT), High Energy Accelerator Research Organization (KEK) and Japan
Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg
Foundation, the Swedish Research Council and the Swedish National Space Board
in Sweden. Additional support for science analysis during operations phase is
gratefully acknowledged from the Istituto Nazionale di Astrofisica and the
Centre d’Etudes Spatiales in France. The research leading to these results has
also received funding from the European Commission Seventh Framework Programme
(FP7/2007-2013) under grant agreement n. 267251. Several of the authors would
also like to acknowledge both funding and solid support from the Chandra X-ray
Center and the Hubble Space Telescope Science Institute under a number of
observing proposals: GO3-14054Z, GO3-14057Z, GO4-15058Z, GO4-15059Z, GO-13109,
GO-13196, GO-13348

##  Appendix A Appendix - Data Analysis

###  A.1 Keck

To study the knot using the high-resolution Keck adaptive-optics (AO) data, we
characterize the region around the pulsar using an empirically measured point
spread function and an analytical model of the knot geometry.

####  A.1.1 PSF Subtraction

We subtracted the PSF of the pulsar using a nearby PSF star (Figure
Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray
Flares?). During our observations, the laser guide star, and so the center of
the anisoplanatic patch, was aimed at the point half-way between the PSF star
and the pulsar. We fit the PSF star, a background plus a power-law based
seeing-disk to the pulsar. For the fit, we excluded the region in the pulsar
subimage that surrounds the knot, and used the other $270^{\circ}$. The
background was fit with a sloping plane model, to allow for the variable
nebular background in the pulsar and PSF star subimages. The fitting was
performed with a Levenburg-Marquadt fitter.

####  A.1.2 Deconvolution

We chose not to deconvolve the knot with the empirically measured point spread
function. Tests on several images showed that deconvolution of the shape of
the knot made insignificant differences to the fit values and uncertainty.

####  A.1.3 Knot Model

To fit the geometric parameters of the knot, we masked out the center
diffraction-limited core of the pulsar, and used a Levenburg-Marquadt fitting
technique to fit a two dimensional Gaussian to the shape of the Knot.

We used a model of the form

$S(r,\psi)=S_{b}+S_{k}\exp(-\frac{1}{2}((r-r_{0})^{2}/\sigma_{r}^{2}+(\psi-\psi_{0})^{2}/\sigma_{\psi}^{2}))$

We also allowed the center of the coordinate system ($x_{0},y_{0}$) to vary
slightly ($0.05\arcsec$). To determine the fit uncertainties, we used the
$1$-$\sigma$ errors from the self-covariance of the fit parameters. Results
were tabulated in Table 4.

###  A.2 HST

####  A.2.1 HST - Traditional Analysis

In order to measure the knot properties (position, flux, tangential and radial
width) we used the SExtractor package (Bertin & Arnouts, 1996). SExtractor has
been extensively used for the analysis of HST data, in particular for the ACS
camera (e.g., for the Hubble Ultra Deep Field project, Beckwith et al., 2006).
For source detection, we require a minimum of five contiguous pixels with a
detection threshold $5\sigma$ above the root-mean-square (rms) background,
with a total of 32 deblending subthresholds, and with a contrast parameter of
0.005, setting the background mesh size to 16$\times$16 ACS pixels.

The position of a source is evaluated by SExtractor as the barycenter of the
source brightness distribution. Flux is computed within an elliptical
aperture, using an implementation of the method by Kron (1980). Note that the
pulsar is not saturated and that diffraction spikes from the pulsar that might
cross the inner knot only marginally affect its flux measurements. The
parameters of this ellipse (semiaxes and orientation), evaluated using the
second moments of the object’s brightness distribution also yield a measure of
the object morphology. Indeed, for the case of the knot, the direction of the
minor axis of the ellipse turned out to be consistent with the pulsar–knot
direction in all images. Thus, the minor axis and the major axis of the
ellipse measure the RMS full widths (FWRMS) in the radial and tangential
directions, respectively.

In order to assess systematic errors, we performed simulations with the
ESO/MIDAS software111 https://www.eso.org/sci/software/esomidas/. We added to
the ACS images a “synthetic knot”. To generate such an artificial source, we
assumed a two-dimensional Gaussian brightness distribution, with the minor
axis aligned with the true pulsar–knot direction. The synthetic knot was
positioned to the NW of the pulsar, opposite but along the true pulsar–knot
direction, and at an angular distance comparable to that of the true knot. We
repeated the exercise by varying the flux, position, and morphology of the
synthetic knot and we estimated the uncertainties in the parameters recovered
using SExtractor 222http://www.astromatic.net/software/sextractor.

Table 10 lists the best-fit values of the model parameters of the inner knot
for each of the 17 observations based upon this traditional analysis and
modeling. Table 11 shows the results of a regression analysis on the listed
parameters.

####  A.2.2 HST Singular Value Decomposition (SVD)

As the inner knot is within $1\arcsec$ of the Crab pulsar and varies in
position and size, we also developed special procedures for characterizing it
using the central $121\times 121$-pixel ($6.05\arcsec\times 6.05\arcsec$)
images of the 17 HST/ACS observations. The major steps are (1) to remove the
pulsar from each image, (2) to use the residual image to generate an image of
the inner knot, and (3) to characterize the properties of the inner knot.

Remove pulsar and generate an image.

In order to remove the pulsar from each of the 17 central $121\times
121$-pixel ($6.05\arcsec\times 6.05\arcsec$) HST images, we first determine
the PSF of the HST/ACS using $121\times 121$-pixel images around each of 19
isolated stars in the field. In doing this, we account for the fact that some
observations were at a roll angle $\sim 180\deg$ opposite that of the others.
After subtracting a fitted linear gradient from the image of each (Figure
Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray
Flares?) of the 19 stars and for each of the 17 observations, we register the
resulting $323=19\times 17$ star images and use singular-value decomposition
(SVD) to generate a linear basis (lowest term shown in Figure Characterization
of the Inner Knot of the Crab: The Site of the Gamma-ray Flares?) describing
the PSF. The model pulsar image (Figure Characterization of the Inner Knot of
the Crab: The Site of the Gamma-ray Flares?) uses the first 72 components of
the SVD basis. After subtracting a linear gradient from each of the 17 central
(pulsar) images (Figure Characterization of the Inner Knot of the Crab: The
Site of the Gamma-ray Flares?), we fit each using the PSF basis to generate 17
residual (pulsar-removed) images (also Figure Characterization of the Inner
Knot of the Crab: The Site of the Gamma-ray Flares?), each now dominated by
the inner knot.

Characterize inner-knot properties.

The measured extrinsic properties of the inner knot are the projected radial
separation $r_{0}$ and polar angle $\psi_{0}$ of the peak surface brightness
with respect to the pulsar. To facilitate characterization of the intrinsic
properties of the inner knot, we map each inner-knot sub-image (Figure
Characterization of the Inner Knot of the Crab: The Site of the Gamma-ray
Flares?) onto a $\psi$–$r$ grid (Figure Characterization of the Inner Knot of
the Crab: The Site of the Gamma-ray Flares?). For an initial model of the
surface brightness (intensity) distribution of the inner knot, we use a simple
bivariate normal distribution:
$S(r,\psi)=S_{k}\exp(-\frac{1}{2}((r-r_{0})^{2}/\sigma_{r}^{2}+(\psi-\psi_{0})^{2}/\sigma_{\psi}^{2}))$.

The major intrinsic properties of the inner knot are its radial dispersion
$\sigma_{r}$ (FWHM${}_{r}=2.35\sigma_{r}$ for a Gaussian profile), azimuthal
dispersion $\sigma_{\psi}$ (or tangential dispersion
$\sigma_{t}=r_{0}\sigma_{\psi}$), and peak intensity (surface brightness)
$S_{k}$. Integrating the intensity over solid angle gives the flux
$F_{k}=2\pi\sigma_{r}\sigma_{t}S_{k}$. In fitting this model to the $\psi$–$r$
image of the inner knot, we include a constant-surface-brightness background
$S_{b}$ as a model parameter.

While this simple model provides adequate estimates to characterize the
primary properties of the knot (Table 6), as a check we also introduced a
somewhat more complicated model to deal with minor asymmetries in the
$\psi$–$r$ plane. In particular, we took the radial location and radial width
to be weak functions of the azimuthal angle, which we expand as a Taylor
series to second order: $r\approx r(\psi)\approx
r_{0}+r^{\prime}_{0}(\psi-\psi_{0})+\frac{1}{2}r^{\prime\prime}_{0}(\psi-\psi_{0})^{2}$
and
$\sigma_{r}\approx\sigma_{r}(\psi)\approx\sigma_{r_{0}}+\sigma^{\prime}_{r_{0}}(\psi-\psi_{0})+\frac{1}{2}\sigma^{\prime\prime}_{r_{0}}(\psi-\psi_{0})^{2}$.
However, the derivatives proved not to alter significantly the results.

Comparison between methods

The comparison between methods and the reasons for somewhat emphasizing the
SVD-based results considers the various measured parameters individually.
Figure Characterization of the Inner Knot of the Crab: The Site of the Gamma-
ray Flares? compares measurements of similar variables between the two methods
and illustrates the fact that the two methods. From the figure we see that,
with the possible exception of the knot flux, both methods yield measurements
that track each other apart from a scale factor. This leads to the result that
the pulsar–knot separation determines all the intrinsic properties of the
knot.

Differences for $r_{0}$ are quite small, whereas the knot flux and size
estimates differ by factors of $\approx 2$. We would expect differences in the
size estimates as the two analyses are using different measures of knot
size—the first, RMS full width (FWRMS); the second, FWHM—which are not simply
related for a non-Gaussian profile. That the fluxes differ is not perhaps so
surprising as its measurement is sensitive to the method of background
subtraction and the impact of emission in the wings of the inner knot’s
profile. A significant difference between the two methods is the smaller
uncertainty in $r_{0}$ afforded by the SVD approach, which removes those SVD
components that represent noise, thus enhancing signal to noise.

###  A.3 Chandra

Figure Characterization of the Inner Knot of the Crab: The Site of the Gamma-
ray Flares? shows the summed image at pulse minimum from the 6 “flaring”
observations (ObsIDs 14684-16247 in Table 3). Prior to the binning used to
produce the figure, we extract the counts in the 3 regions shown. It is
difficult, if not impossible, to make use of the absolute number of counts as
the different instrument configurations are all uncalibrated. However, we can
make use of the relative numbers of counts from the different regions. Thus,
the central region serves as an indicator of the number of pulsar counts which
can then be compared with the other observations to derive physical fluxes
(assuming of course that the phase averaged pulsar flux has not varied). The
region to the southeast provides the upper limit for the flux in the knot and
we use the region to the northwest to estimate the background, comprised
mainly of that due to the wings of the pulsar PSF.

The alert reader will notice a slight excess of counts roughly $0.6\arcsec$ to
the southwest of the pulsar. This feature is seen both in the phase-averaged
pulse and in the data from ObsID 11245. (The asymmetric effect of the blade
inserted for Obs ID 9765 does not allow these data to be used in making this
comparison.) Therefore, we conclude that this feature is part of the pulsar
PSF. A more subtle effect is also present as there is a slight asymmetry in
the PSF that causes the region we associate with the inner knot to be somewhat
brighter than the background during observations taken in the fall. In the
spring, when the spacecraft has (naturally) been set at a roll angle that
differs by $\approx 180\deg$ from that in the fall, the background region is
brighter than that we associate with the location of the inner knot. This
effect is mainly seen in the phase averaged data, and by comparing data from
the ObsID 9765 (spring) and ObsID 11245 (fall) observations. Both effects
described in this paragraph are relatively minor, are mainly due to the fact
we are working slightly below the spatial resolution of Chandra, and have thus
been ignored in setting the upper limits in Table 8

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Table 1: Summary of the Keck observations

# | Date | MJDa | Instrumentb | Exposure (s)
---|---|---|---|---
K1 | 2012-02-08 | 55965.285 | NIRC2-wide | 420
K2 | 2012-03-05 | 55991.319 | NIRC2-wide | 420
K3 | 2012-12-22 | 56283.408 | NIRC2-wide | 1250
K4 | 2012-12-23 | 56284.410 | NIRC2-narrow | 870
K5 | 2012-12-24 | 56285.388 | NIRC2-narrow | 1350
K6 | 2012-12-25 | 56286.390 | NIRC2-wide | 1125
K7 | 2013-02-06 | 56329.308 | NIRC2-narrow | 2020
K8 | 2013-10-22 | 56587.448 | NIRC2-narrow | 640
K9 | 2014-01-09 | 56666.274 | NIRC2-wide | 1560
K10 | 2014-01-09 | 56666.377 | NIRC2-narrow | 2580
K11 | 2014-01-17 | 56674.325 | NIRC2-wide | 2270
K12 | 2014-01-17 | 56674.394 | NIRC2-narrow | 1260

a Quoted MJD is at the midpoint of the observation.
b Plate scale is 0.04$\arcsec$/pixel for NIRC2-wide; 0.01$\arcsec$/pixel for
NIRC2-narrow.

Table 2: Summary of the HST observations

# | Date | MJDab
---|---|---
H1 | 2012-01-08 | 55934.797
H2 | 2012-02-10 | 55967.170
H3 | 2012-03-12 | 55998.414
H4 | 2012-04-22 | 56039.850
H5 | 2012-08-16 | 56155.336
H6 | 2012-09-10 | 56180.960
H7 | 2013-01-10 | 56302.992
H8 | 2013-02-24 | 56347.055
H9 | 2013-03-06 | 56357.023
H10 | 2013-04-01 | 56383.320
H11 | 2013-04-14 | 56396.113
H12 | 2013-08-13 | 56517.710
H13 | 2013-10-20 | 56585.793
H14 | 2013-10-29 | 56594.700
H15 | 2013-12-01 | 56627.273
H16 | 2014-01-20 | 56677.530
H17 | 2014-04-13 | 56760.900

a Quoted MJD is at the midpoint of the observation.
b Plate scale is 0.05$\arcsec$/pixel.

Table 3: Summary of the Chandra observations

# | ObsID | Date | MJDa | Exposure (s)
---|---|---|---|---
C1 | 09765 | 2008-01-22 | 54487.670 | 95082
C2 | 11245 | 2010-11-16 | 55516.294 | 22039
C3 | 14684 | 2013-03-05 | 56356.972 | 19485
C4 | 14686 | 2013-03-10 | 56361.338 | 20052
C5 | 14687 | 2013-03-17 | 56368.324 | 16718
C6 | 16244 | 2013-10-19 | 56584.658 | 20048
C7 | 16246 | 2013-10-22 | 56587.515 | 20013
C8 | 16247 | 2013-10-24 | 56589.007 | 19770

a Quoted MJD is at the start of the observation.

Table 4: Knot properties based upon analysis of the Keck data  | $\psi_{0}$ | $r_{0}$ | FWHMr | FWHMt | $F_{k}$
---|---|---|---|---|---
# | ∘ | $\arcsec$ | $\arcsec$ | $\arcsec$ | $F_{p}$
K1 | $118.50\pm 0.16$ | $0.6711\pm 0.0015$ | $0.3931\pm 0.0040$ | $0.4719\pm 0.0051$ | $0.0639\pm 0.0045$
K2 | $121.00\pm 0.13$ | $0.6577\pm 0.0015$ | $0.4121\pm 0.0035$ | $0.4252\pm 0.0042$ | $0.0626\pm 0.0067$
K3 | $122.64\pm 0.11$ | $0.6893\pm 0.0011$ | $0.3215\pm 0.0028$ | $0.3948\pm 0.0039$ | $0.0804\pm 0.0074$
K4 | $120.96\pm 0.18$ | $0.7118\pm 0.0021$ | $0.3401\pm 0.0051$ | $0.4081\pm 0.0059$ | $0.0473\pm 0.0003$
K5 | $121.20\pm 0.07$ | $0.6981\pm 0.0007$ | $0.2983\pm 0.0019$ | $0.3767\pm 0.0024$ | $0.0555\pm 0.0005$
K6 | $120.79\pm 0.12$ | $0.7164\pm 0.0010$ | $0.3375\pm 0.0029$ | $0.4238\pm 0.0042$ | $0.0599\pm 0.0085$
K7 | $121.93\pm 0.12$ | $0.7756\pm 0.0013$ | $0.3355\pm 0.0034$ | $0.4049\pm 0.0038$ | $0.0662\pm 0.0004$
K8 | $120.33\pm 0.06$ | $0.5722\pm 0.0004$ | $0.2216\pm 0.0011$ | $0.3172\pm 0.0015$ | $0.0621\pm 0.0005$
K9 | $121.46\pm 0.08$ | $0.6543\pm 0.0006$ | $0.2958\pm 0.0018$ | $0.4030\pm 0.0024$ | $0.0708\pm 0.0047$
K10 | $115.77\pm 0.03$ | $0.6257\pm 0.0003$ | $0.2689\pm 0.0007$ | $0.3683\pm 0.0009$ | $0.0663\pm 0.0006$
K11 | $121.49\pm 0.10$ | $0.6371\pm 0.0008$ | $0.3119\pm 0.0022$ | $0.3787\pm 0.0029$ | $0.0647\pm 0.0078$
K12 | $118.99\pm 0.09$ | $0.6438\pm 0.0009$ | $0.3248\pm 0.0023$ | $0.3766\pm 0.0025$ | $0.0574\pm 0.0001$
Table 5: Results of regression analyses for the 12 Keck observations  Power-law for $r_{1}=0.6692\arcsec$ |  |  |  |
---|---|---|---|---
Property | Unit | $p(r_{1})$ | Power-law index $q$ | Probability
FWHMr | $\arcsec$ | $0.3180\pm 0.0132$ | $1.19\pm 0.54$ | 5.4E-2
FWHMt | $\arcsec$ | $0.3941\pm 0.0093$ | $0.75\pm 0.31$ | 3.8E-2
$S_{k}$ | $F_{p}/(\arcsec)^{2}$ | $0.439\pm 0.033$ | $-2.14\pm 0.97$ | 5.2E-2
$F_{k}$ | $F_{p}$ | $0.0626\pm 0.0025$ | $-0.20\pm 0.53$ | 7.1E-1
Linear for $<r_{0}>=0.6711\arcsec$ |  |  |  |
Property |  | $\psi_{0}(<r_{0}>)$ | $d\psi_{0}/dr_{0}$ |
$\psi_{0}$ |  ∘, ${}^{\circ}/\arcsec$ | $120.42\pm 0.51$ | $15.2\pm 10.2$ | 1.7E-1
Table 6: Knot properties based upon SVD analysis of the 17 HST observations (see Appendix)  # | $\psi_{0}$ | $r_{0}$ |  FWHMr |  FWHMt | $S_{k}$ | $F_{k}$ | $F_{p}$ | $F_{k}$
---|---|---|---|---|---|---|---|---
| ∘ | $\arcsec$ | $\arcsec$ | $\arcsec$ |  e/s/$(\arcsec)^{2}$ | e/s | e/s | $F_{p}$
H1 | $120.54\pm 0.40$ | $0.6819\pm 0.0023$ | $0.1526\pm 0.0056$ | $0.3221\pm 0.0116$ | $1064\pm 33$ | $59.8\pm 3.6$ | $1425\pm 82$ | $0.0420\pm 0.0035$
H2 | $119.45\pm 0.41$ | $0.6535\pm 0.0024$ | $0.1536\pm 0.0058$ | $0.3071\pm 0.0114$ | $1008\pm 32$ | $53.8\pm 3.4$ | $1433\pm 80$ | $0.0375\pm 0.0032$
H3 | $120.37\pm 0.33$ | $0.6708\pm 0.0022$ | $0.1751\pm 0.0052$ | $0.3121\pm 0.0092$ | $902\pm 23$ | $55.0\pm 2.6$ | $1471\pm 64$ | $0.0374\pm 0.0024$
H4 | $121.74\pm 0.41$ | $0.6356\pm 0.0024$ | $0.1465\pm 0.0056$ | $0.2942\pm 0.0110$ | $1053\pm 34$ | $52.3\pm 3.2$ | $1276\pm 80$ | $0.0410\pm 0.0035$
H5 | $117.90\pm 0.39$ | $0.6166\pm 0.0020$ | $0.1406\pm 0.0048$ | $0.3043\pm 0.0102$ | $1256\pm 36$ | $61.1\pm 3.4$ | $1231\pm 84$ | $0.0497\pm 0.0044$
H6 | $117.62\pm 0.41$ | $0.6577\pm 0.0024$ | $0.1456\pm 0.0058$ | $0.3029\pm 0.0112$ | $1102\pm 36$ | $56.7\pm 3.5$ | $1184\pm 86$ | $0.0479\pm 0.0046$
H7 | $121.17\pm 0.59$ | $0.7240\pm 0.0035$ | $0.1704\pm 0.0084$ | $0.3539\pm 0.0176$ | $675\pm 28$ | $44.7\pm 3.9$ | $1422\pm 77$ | $0.0314\pm 0.0032$
H8 | $119.91\pm 0.50$ | $0.7516\pm 0.0031$ | $0.1751\pm 0.0076$ | $0.3533\pm 0.0158$ | $667\pm 25$ | $44.5\pm 3.4$ | $1421\pm 65$ | $0.0313\pm 0.0028$
H9 | $120.88\pm 0.52$ | $0.7491\pm 0.0033$ | $0.1740\pm 0.0080$ | $0.3362\pm 0.0166$ | $728\pm 29$ | $45.8\pm 3.8$ | $1359\pm 73$ | $0.0337\pm 0.0033$
H10 | $120.60\pm 0.32$ | $0.6614\pm 0.0019$ | $0.1598\pm 0.0046$ | $0.3139\pm 0.0090$ | $1098\pm 27$ | $63.3\pm 3.0$ | $1295\pm 72$ | $0.0489\pm 0.0036$
H11 | $119.39\pm 0.36$ | $0.6557\pm 0.0021$ | $0.1639\pm 0.0050$ | $0.3220\pm 0.0098$ | $1090\pm 28$ | $64.9\pm 3.3$ | $1509\pm 77$ | $0.0430\pm 0.0031$
H12 | $119.28\pm 0.41$ | $0.6324\pm 0.0023$ | $0.1691\pm 0.0056$ | $0.3299\pm 0.0110$ | $1028\pm 29$ | $63.4\pm 3.6$ | $1283\pm 83$ | $0.0494\pm 0.0042$
H13 | $117.85\pm 0.36$ | $0.5687\pm 0.0019$ | $0.1426\pm 0.0046$ | $0.2739\pm 0.0082$ | $1668\pm 44$ | $75.6\pm 3.8$ | $1222\pm 106$ | $0.0618\pm 0.0062$
H14 | $118.82\pm 0.41$ | $0.5849\pm 0.0022$ | $0.1441\pm 0.0052$ | $0.2886\pm 0.0098$ | $1583\pm 48$ | $75.6\pm 4.4$ | $1285\pm 118$ | $0.0589\pm 0.0064$
H15 | $118.59\pm 0.52$ | $0.6624\pm 0.0026$ | $0.1476\pm 0.0061$ | $0.3393\pm 0.0144$ | $1105\pm 40$ | $61.8\pm 4.4$ | $1353\pm 100$ | $0.0457\pm 0.0047$
H16 | $118.82\pm 0.41$ | $0.6299\pm 0.0026$ | $0.1640\pm 0.0064$ | $0.2961\pm 0.0106$ | $1084\pm 35$ | $62.1\pm 3.7$ | $1419\pm 98$ | $0.0438\pm 0.0040$
H17 | $120.88\pm 0.41$ | $0.6448\pm 0.0024$ | $0.1611\pm 0.0058$ | $0.3212\pm 0.0110$ | $1164\pm 35$ | $69.6\pm 4.0$ | $1177\pm 99$ | $0.0591\pm 0.0060$
Table 7: Results of regression analyses for the 17 HST observations based on the data in Table 6 Power-law for $r_{1}=0.6559\arcsec$ |  |  |  |
---|---|---|---|---
Property | Unit | $p(r_{1})$ | Power-law index $q$ | Probability
FWHMr | $\arcsec$ | $0.1575\pm 0.0022$ | $0.72\pm 0.19$ | 1.7E-3
FWHMt | $\arcsec$ | $0.3152\pm 0.0029$ | $0.80\pm 0.13$ | 1.4E-5
$S_{k}$ | $F_{p}/(\arcsec)^{2}$ | $0.782\pm 0.023$ | $-3.66\pm 0.40$ | 1.7E-7
$F_{k}$ | $F_{p}$ | $0.0440\pm 0.0012$ | $-2.39\pm 0.37$ | 9.8E-6
Linear for $<r_{0}>=0.6577\arcsec$ |  |  |  |
Property |  | $\psi_{0}(<r_{0}>)$ | $d\psi_{0}/dr_{0}$ |
$\psi_{0}$ |  ∘, ${}^{\circ}/\arcsec$ | $119.64\pm 0.27$ | $13.5\pm 5.5$ | 2.7E-2
Table 8: Results of the Chandra analysis  | Pulse Average | Pulse Minimum | $F_{k}/F_{p}$
---|---|---|---
ObsID | Pulsar | Knot | Background | Pulsar | Knot | Background | $3\sigma$
09765 | 51844 | 2963 | 3704 | 845 | 57 | 72 | $<0.0022$
11245 | 5630 | 1018 | 851 | 88 | 16 | 12 | $<0.0094$
last 6 | 13549 | 2622 | 2503 | 203 | 39 | 44 | $<0.0067$
Table 9: Fermi/LAT 12-hr-average fluxes and pulsar–knot separations as measured with Keck and HST (SVD analysis)  # | Date | $t_{\rm mid}$ | $r_{0}$ |  Fermi/LAT $\geq 100$ MeV
---|---|---|---|---
|  | MJD | $\arcsec$ |  $10^{-6}$ ph/s/cm2
H1 | 2012-01-08 | 55934.797 | $0.6819\pm 0.0023$ | 2.45 (+0.47, -0.43)
K1 | 2012-02-08 | 55965.280 | $0.6711\pm 0.0015$ | 2.58 (+0.60, -0.54)
H2 | 2012-02-10 | 55967.171 | $0.6535\pm 0.0024$ | 2.79 (+0.65, -0.58)
K2 | 2012-03-05 | 55991.311 | $0.6577\pm 0.0015$ | 2.78 (+0.46, -0.43)
H3 | 2012-03-12 | 55998.414 | $0.6708\pm 0.0022$ | 3.55 (+0.52, -0.49)
H4 | 2012-04-22 | 56039.850 | $0.6356\pm 0.0024$ | 2.18 (+0.56, -0.50)
H5 | 2012-08-16 | 56155.336 | $0.6166\pm 0.0020$ | 3.25 (+0.56, -0.52)
H6 | 2012-09-10 | 56180.960 | $0.6577\pm 0.0024$ | 3.86 (+0.50, -0.47)
K3 | 2012-12-22 | 56283.381 | $0.6893\pm 0.0011$ | 3.13 (+0.68, -0.63)
K4 | 2012-12-23 | 56284.403 | $0.7118\pm 0.0021$ | 3.51 (+0.67, -0.62)
K5 | 2012-12-24 | 56285.379 | $0.6981\pm 0.0007$ | 3.29 (+0.67, -0.62)
K6 | 2012-12-25 | 56286.370 | $0.7164\pm 0.0010$ | 2.74 (+0.62, -0.56)
H7 | 2013-01-10 | 56302.992 | $0.7240\pm 0.0035$ | 3.22 (+0.66, -0.66)
K7 | 2013-02-06 | 56329.248 | $0.7756\pm 0.0013$ | 3.93 (+0.75, -0.70)
H8 | 2013-02-24 | 56347.055 | $0.7516\pm 0.0031$ | 5.44 (+0.72, -0.68)
H9 | 2013-03-06 | 56357.023 | $0.7491\pm 0.0033$ | 10.09 (+0.52, -0.51)
H10 | 2013-04-01 | 56383.320 | $0.6614\pm 0.0019$ | 3.50 (+0.68, -0.63)
H11 | 2013-04-14 | 56396.113 | $0.6557\pm 0.0021$ | 2.89 (+0.59, -0.54)
H12 | 2013-08-13 | 56517.710 | $0.6324\pm 0.0023$ | 2.65 (+0.46, -0.43)
H13 | 2013-10-20 | 56585.793 | $0.5687\pm 0.0019$ | 3.71 (+0.26, -0.25)
K8 | 2013-10-22 | 56587.443 | $0.5722\pm 0.0004$ | 1.25 (+1.73, -1.73)
H14 | 2013-10-29 | 56594.700 | $0.5849\pm 0.0022$ | 7.46 (+1.16, -1.07)
H15 | 2013-12-01 | 56627.273 | $0.6624\pm 0.0026$ | 2.15 (+0.53, -0.47)
K9 | 2014-01-09 | 56666.316 | $0.6543\pm 0.0006$ | 3.38 (+0.53, -0.50)
K10 | 2014-01-09 | 56666.438 | $0.6257\pm 0.0003$ | 3.54 (+0.51, -0.48)
K11 | 2014-01-17 | 56674.356 | $0.6371\pm 0.0008$ | 3.84 (+0.53, -0.50)
K12 | 2014-01-17 | 56674.400 | $0.6438\pm 0.0009$ | 3.47 (+0.44, -0.42)
H16 | 2014-01-20 | 56677.530 | $0.6299\pm 0.0026$ | 2.85 (+0.43, -0.41)
H17 | 2014-04-13 | 56760.900 | $0.6448\pm 0.0024$ | 3.96 (+0.78, -0.71)
Table 10: Results of the Traditional HST Analysis  # | $r_{0}$ | FWRMSr | FWRMSt | $F_{k}$
---|---|---|---|---
| $\arcsec$ | $\arcsec$ | $\arcsec$ | e/s
H1 | $0.660\pm 0.012$ | $0.287\pm 0.019$ | $0.551\pm 0.037$ | $116.5\pm 2.9$
H2 | $0.634\pm 0.012$ | $0.283\pm 0.020$ | $0.538\pm 0.038$ | $110.9\pm 2.8$
H3 | $0.649\pm 0.013$ | $0.333\pm 0.021$ | $0.594\pm 0.038$ | $131.6\pm 3.3$
H4 | $0.615\pm 0.012$ | $0.255\pm 0.020$ | $0.489\pm 0.038$ | $90.4\pm 2.3$
H5 | $0.588\pm 0.012$ | $0.278\pm 0.019$ | $0.560\pm 0.038$ | $112.9\pm 2.8$
H6 | $0.635\pm 0.012$ | $0.290\pm 0.020$ | $0.552\pm 0.037$ | $111.6\pm 2.8$
H7 | $0.706\pm 0.010$ | $0.288\pm 0.021$ | $0.589\pm 0.044$ | $97.6\pm 2.4$
H8 | $0.734\pm 0.017$ | $0.336\pm 0.024$ | $0.658\pm 0.046$ | $109.2\pm 2.7$
H9 | $0.732\pm 0.013$ | $0.319\pm 0.022$ | $0.650\pm 0.046$ | $108.1\pm 2.7$
H10 | $0.641\pm 0.009$ | $0.299\pm 0.020$ | $0.580\pm 0.039$ | $117.6\pm 2.9$
H11 | $0.637\pm 0.011$ | $0.314\pm 0.020$ | $0.561\pm 0.036$ | $129.2\pm 3.2$
H12 | $0.613\pm 0.012$ | $0.258\pm 0.017$ | $0.578\pm 0.039$ | $121.3\pm 3.0$
H13 | $0.547\pm 0.009$ | $0.224\pm 0.015$ | $0.491\pm 0.033$ | $119.2\pm 3.0$
H14 | $0.552\pm 0.009$ | $0.236\pm 0.015$ | $0.511\pm 0.033$ | $124.8\pm 3.1$
H15 | $0.631\pm 0.012$ | $0.268\pm 0.018$ | $0.588\pm 0.040$ | $115.2\pm 2.9$
H16 | $0.614\pm 0.011$ | $0.266\pm 0.025$ | $0.519\pm 0.049$ | $116.9\pm 2.9$
H17 | $0.624\pm 0.012$ | $0.259\pm 0.024$ | $0.549\pm 0.050$ | $127.8\pm 3.2$
Table 11: Results of regression analyses for the 17 HST observations based on the traditional data analysis  Power-law for $r_{1}=0.6342\arcsec$ |  |  |  |
---|---|---|---|---
Property | Unit | $p(r_{1})$ | Power-law index $q$ | Probability
FWRMSr | $\arcsec$ | $0.2801\pm 0.0046$ | $1.14\pm 0.21$ | 6.0E-5
FWRMSt | $\arcsec$ | $0.5605\pm 0.0067$ | $0.85\pm 0.15$ | 4.9E-5
$F_{k}$ | e/s | $114.8\pm 2.6$ | $-0.39\pm 0.29$ | 1.9E-1

Individual images of the inner knot, with the pulsar removed, from Keck NIRC2
with laser-guide-star adaptive optics (AO). In each image, the PSF of the
pulsar has been subtracted using the scaled PSF of a nearby star and replaced
with a gray circle at the original location of the pulsar. Each image is
$2.2\arcsec$ by $2.2\arcsec$. Images are from the NIRC2 wide camera
($0.04\arcsec$ pixels) or the NIRC2 narrow camera ($0.01\arcsec$ pixels). All
images were taken using the $K^{\prime}$ filter. A red dot marks the average
position of the inner knot across all 12 observations.

HST-SVD-Processed images of the Crab pulsar and inner knot after removing
effects of the HST/ACS WFC point spread function. A red dot marks the average
position of the inner knot across all 17 observations.

Variation of the Keck-measured knot properties versus separation of the inner
knot from the pulsar.

Variation of the HST-SVD-measured properties with separation of the inner knot
from the pulsar.

Chandra image at pulse minimum from the sum of the last 6 flaring data sets in
Table 3. The 3 small black circles show $0.3\arcsec$-radius extraction
regions, the central of which is used to estimate the pulsar flux. The circle,
at a position angle $\psi=120^{\circ}$ east of north and $0.65\arcsec$ from
the pulsar, is roughly centered on the average optical location of the knot.
Data in the opposite small circle (the one to the NW where N is up) are used
to estimate background. Large circles show the best-fit Gaussian to the phase-
average pulsar data at intensity levels of 10, 20, 40, 80, 160, 320 and 640
cts/pixel illustrating the level of impact of the PSF at the site of the
knot—i.e., slight but non-negligible. The intensity level is shown by the
grayscale bar on the right.

Fermi/LAT 12-hr average flux (upper panel) and pulsar–knot separation (lower
panel) versus time, for Keck (red) and HST (green) observations.

Fermi/LAT 12-hr average fluxes during Keck (red) and HST (green) observations
at the time of the largest $\gamma$-ray flares in 2013 March (upper panel) and
in 2013 October (lower panel).

Fermi/LAT 12-hr-average fluxes centered on the times of Keck (red) and HST
(green) measurements of pulsar–knot separations.

Figure 1: Geometry of the termination shock on the meridional plane determined
by the pulsar rotation axis and the observer’s line of sight toward the
pulsar. The pulsar is located at point P. The angle between line of sight and
pulsar rotation axis is $\theta_{\rm ob}=2\pi/3$, consistent with that
measured by Ng & Romani (2008). The knot is projected on the plane of the sky
at an angle $r_{0}\approx 0.65^{\prime\prime}$ southeast of the pulsar
(represented by point K here). The angle between the upstream velocity and the
shock surface is $\delta_{1}$. The outflow is deflected from the radial
direction by an angle $\Delta$.

The Pulsar (left) and a nearby star (right) from Keck data taken on 2013
October 22. (a) An H filter image of the pulsar and knot (in the lower-left)
as seen by the NIRC2-narrow camera ($0.01\arcsec$ pixels and a
$10\arcsec\times 10\arcsec$ field of view). The structure of the knot is
resolved and separated from the pulsar by a statistically significant valley.
(b) A nearby ($5\arcsec$) comparison star, used to establish the point-
response-function. Both images are shown with a logarithmic color stretch.

Images illustrating the steps in determining the HST/ACS PSF for this
analysis: (a) Extract 323 ($19\times 17$) $121\times 121$-pixel
($6.05\arcsec\times 6.05\arcsec$) images and subtract a linear-gradient
background for 19 isolated stars in these 17 HST/ACS observations of the Crab;
(b) register images of stars and use singular value decomposition (SVD) to
determine a basis describing the PSF (first SVD component shown); and (c)
model the pulsar image using the first 72 SVD components.

Images further illustrating steps in processing HST/ACS images of the Crab
pulsar and inner knot: (a) Subtract a linear-gradient background in central
$121\times 121$ pixel ($6.05\arcsec\times 6.05\arcsec$) image of pulsar and
inner knot; (b) remove pulsar using SVD model of PSF leaving inner knot; (c)
remove inner knot using its SVD model leaving residual background; (d) add SVD
model of inner knot to residual background; (e) apply Richardson-Lucy
algorithm to generate deconvolved image of inner knot (and residual
background); and (f) synthesize $25\times 25$ pixel ($1.25\arcsec\times
1.25\arcsec$) sub-image of registered (unresolved) pulsar and SVD model of
inner knot.

s

Mapping of processed images of the inner knot onto a $\psi$–$r$ grid. Here
$\psi$ is the angular displacement in radian from the knot’s centroid; $r$,
the radial distance from the pulsar in ($0.05\arcsec$) pixels.

Comparison of HST-SVD and HST-traditional measured properties of the inner
knot.