# A Comprehensive Study of the Spectral Variation and the Brightness Profile
of Young Pulsar Wind Nebulae

Chin-Ping Hu Department of Physics, National Changhua University of Education,
Changhua, 50007, Taiwan  Wataru Ishizaki Yukawa Institute for Theoretical
Physics, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502,
Japan  C.-Y. Ng Department of Physics, The University of Hong Kong, Pokfulam
Road, Hong Kong  Shuta J. Tanaka Department of Physical Sciences, Aoyama
Gakuin University, Kanagawa, Sagamihara, 252-5258, Japan  Y.-L. Mong  School
of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia
C.-P. Hu<EMAIL_ADDRESS>
(Accepted Jan. 18, 2022)

###### Abstract

We present a systematic study of particle transport by diffusion in young
pulsar wind nebulae (PWNe). We selected nine bright sources that are well
resolved with the _Chandra X-ray Observatory_. We analyzed archival data to
obtain their radial profiles of photon index ($\Gamma$) and surface brightness
($\Sigma$) in a consistent way. These profiles were then fit with a pure
diffusion model that was tested on Crab, 3C 58 and G21.5$-$0.9 before. In
addition to the spectral softening due to the diffusion, we calculated the
synchrotron power and built up the theoretical surface brightness profile. For
each source, we performed separate fits to the $\Gamma$ and the $\Sigma$
profiles. We found these two profiles of most PWNe are similar, except for
Crab and Vela.  Both profiles can be well described by our model, suggesting
that diffusion dominates the particle transport in most sampled PWNe. The
discrepancy of parameters between the $\Gamma$ profile and $\Sigma$ profiles
is relatively large for 3C 58 and G54.1+0.3. This difference could be
attributed to the elongated shape, reflecting boundary, and the non-uniform
magnetic field. Finally, we found no significant correlations between
diffusion parameters and physical parameters of PWN and pulsar.

Interstellar synchrotron emission (856); Supernova remnants (1667); Neutron
Stars (1108); Rotation powered pulsars (1408); X-ray sources (1822)

††facilities: _CXO_ (ACIS)††software: CIAO (Fruscione et al., 2006), Sherpa
(Freeman et al., 2001)

##  1 Introduction

An isolated pulsar releases most of its rotational energy through high-energy
relativistic particle outflow called pulsar wind. When the pulsar wind
interacts with the surrounding medium and is over-pressured, a pulsar wind
nebula (PWN) is formed. This drives a wind-termination shock, at which the
particles are heated and re-accelerated to ultra-relativistic regime (see
reviews by Gaensler & Slane, 2006; Slane, 2017). The shocked particles move
through the magnetic field in a PWN. They cool down and emit synchrotron
radiation from radio to X-rays, and inverse-Compton radiation in
$\gamma$-rays.

Observationally, the injected particle distribution could be inferred from the
broadband spectral-energy distribution of PWNe. A one-zone model without
spatial information can reproduce the observed properties (see, e.g. Pacini &
Salvati, 1973; Tanaka & Takahara, 2010; Bucciantini et al., 2011; Vorster et
al., 2013). The improvement of spatial resolution of instruments makes it
possible to investigate the detailed structure and the evolution of PWNe. When
transporting outward, the high-energy particles lose their energies on a fast
timescale and result in a softening of the X-ray spectrum. At the same time,
the synchrotron flux decreases. The radial profiles in the radio band have
been first modeled as a diffusion nebula with a particle source injected at
the center (Gratton, 1972; Wilson, 1972). A one-dimensional
magnetohydrodynamic (MHD) model with a time-independent steady flow has been
established from this idea and tested on Crab (Kennel & Coroniti, 1984a, b;
Porth et al., 2014, 2016).

However, this model is not applicable for a few other PWNe under the
unprecedented spatial resolution of _Chandra X-ray Observatory_ (see e.g.,
Slane et al., 2004). Combining the photon index ($\Gamma$) profile and the
surface brightness ($\Sigma$) profile, Ishizaki et al. (2017) found that the
simple MHD model with steady stream fails to reproduce observed properties in
3C 58 and G21.5$-$0.9, indicating that spatial diffusion of particles, which
has been modeled by Tang & Chevalier (2012), should be taken into account. The
pure diffusion model was first proposed to explain the flux and the spectral
index of Crab in the optical band (Wilson, 1972). The morphology of a young
PWN is approximated by a sphere that results from the diffusion of an input
particle source into the surrounding space. Later X-ray observations on 3C 58
and G21.5$-$0.9 show similar $\Gamma$ distribution and suggest the importance
of diffusion in these PWNe (Bocchino et al., 2005; Slane et al., 2000, 2004;
Guest et al., 2019). Moreover, Porth et al. (2016) and Ishizaki et al. (2018)
conclude that the particle transfer in these two objects is dominated by
diffusion through modeling the surface brightness. In addition, the
$\gamma$-ray halo observed in Geminga also indicates that diffusion plays an
important role in PWNe (Linden et al., 2017; Abeysekara et al., 2017; Bao et
al., 2021).

Although these works suggest the importance of diffusion, a systematic test of
the pure diffusion model is lacking. We aim to test whether this model can be
applied to the $\Gamma$ and $\Sigma$ profiles in young PWNe that are observed
with _Chandra_. The sample selection is described in Section 2, in which nine
PWNe are selected. In Section 3, we describe the X-ray data reduction and the
spectral analysis. We perform joint fit of multiple _Chandra_ observations and
calculate the observational $\Gamma$ and $\Sigma$ profiles. The detailed
theoretical modeling is described in Section 4. We follow the model derived by
Tang & Chevalier (2012) with transmitting boundary ,i.e., particles are not
reflected by the boundary and the emission from particles beyond the boundary
is not taken into account, and use the full synchrotron spectrum to calculate
the $\Gamma$ and the $\Sigma$ profiles (Ishizaki et al., 2017, 2018). The
fitting results are shown in Section 5 and discussed in Section 6. Finally, we
summarize our work in Section 7.

Table 1: Physical properties and observational parameters for the young PWNe
in this study

PWN | PSR | Distance | $\tau_{\rm{PWN}}^{a}$ | $B_{\rm{PWN}}^{a}$ | log $L_{\rm{PWN}}$ | log $L_{\rm{sd}}$ | Exp.
---|---|---|---|---|---|---|---
|  | (kpc) | (kyr) | ($10^{-5}$ G) | (erg s-1) | (erg s-1) | (ks)
3C 58 | J0205+6449 | 2.0 | 2.5 | 1.7 | 31.1 | 37.4 | 391.9
G21.5$-$0.9 | J1833$-$1034 | 4.4 | 1.0 | 4.7 | 33.1 | 37.75 | 749.0
G54.1+0.3 | J1930+1852 | 4.9 | 1.7 | 1.0 | 32.0 | 37.1 | 325.6
G291.0$-$0.1b | unknown | 3.5 | 1.2 | 7.0 | 30.5 | 37.7 | 472.6
G310.6$-$1.6 | J1400$-$6325 | 7.0 | 0.92 | 1.7 | 33.0 | 37.7 | 231.5
Kes 75 | J1846$-$0258 | 5.8 | 0.7 | 2.0 | 33.3 | 36.9 | 383.1
MSH 15$-$52 | B1509$-$58 | 5.2 | 1.7 | 1.5 | 32.5 | 37.5 | 244.2
Vela | B0833$-$45 | 0.287 | 6.5 | 0.49 | 30.3 | 37.5 | 630.1
Crab | B0531+21 | 2.0 | 0.95 | 8.5 | 34.6 | 38.7 | 23.4

aafootnotetext: These two quantities are adopted from Tanaka & Takahara (2011,
2013).

bbfootnotetext: An X-ray source CXOU J111148.6$-$603926 is suggested to be the
pulsar powers G291.0$-$0.1 although the timing properties remains unknown. The
spin-down power is estimated from the broadband spectral fit of the PWN (Slane
et al., 2012).

##  2 Sample Selection

Our PWN sample consists of young and bright sources obtained from Table 1 in
Kargaltsev & Pavlov (2010) that are well resolved by _Chandra_ with angular
size over 20″. Nine PWNe, including 3C 58, G21.5$-$0.9, G54.1+0.3,
G291.0$-$0.1, G310.6$-$1.6, Kes 75, MSH 15$-$52, Vela, and Crab, are selected
in this study. Their basic properties are listed in Table 1. The distances,
$B$-field strengths, PWN radius, and age are adopted from literatures (see
e.g., Tanaka & Takahara, 2011, 2013, and references therein). We obtain all
the pulsar properties from the ATNF pulsar
catalog111https://www.atnf.csiro.au/research/pulsar/psrcat/ (Manchester et
al., 2005).

We analyzed the _Chandra_ observations taken with the Advanced CCD Imaging
Spectrometer (ACIS) and operated in the timed-exposure (TE) mode for each PWN.
For G21.5$-$0.9, we excluded those observations for calculation purposes, with
the focal plane offset or the CCD not at optimal temperature. Moreover, we
only analyze recent _Chandra_ observations (after 2012) of Crab to reduce the
computing time. All the data sets used in this analysis are summarized in
Table A1. The radial profile of Crab, 3C 58, and G21.5$-$0.9 have been
investigated before (see, e.g., Slane et al., 2004; Bamba et al., 2010; Tang &
Chevalier, 2012; Guest et al., 2019). We, however, re-analyzed the data on our
own to ensure the latest calibration is applied and consistent with other PWNe
in our sample.

Note that G10.65+2.96 fits our selection criterion. However, we cannot obtain
the difference in spectral properties of inner and outer regions.  G11.2$-$0.3
is one of the youngest PWN (Borkowski et al., 2016). It has an elongated shape
and most of the PWN emission overlapped the jet viewing from Earth. After
removing the jet (see, e.g., Figure 5 in Borkowski et al., 2016), the
remaining photons are not statistically significant to measure the spectral
difference between the inner and outer PWN.  We excluded them from our
samples.

The central pulsar of G291.0$-$0.1 remains controversial. A young pulsar
J1105$-$6107 was suggested as the central pulsar of this PWN (Kaspi et al.,
1997). However, a later broadband study suggested  that J1105$-$6107 is well
outside the PWN and unlikely related (Slane et al., 2012). A spin-down
luminosity of $\sim 5.7\times 10^{37}$ erg s-1 of the central object CXOU
J111148.6$-$603926 is suggested although this is estimated from the broadband
spectral fit to the PWN (Slane et al., 2012). We note that this is not a
precise measurement since a large uncertainty is possible.

##  3 Spectral Analysis

All the data reduction and basic analysis were carried out using the Chandra
Interactive Analysis of Observations (CIAO) version 4.11 with the calibration
database of version 4.8.3. We reprocessed the data using the task
chandra_repro. We use reproject_obs to reproject all data sets of a PWN to
create a merged event file, and use flux_obs to create an exposure-corrected
image. These images were used to detect the position of pulsars and other
point sources. The innermost circular region within 2.5″ radius of the pulsar
was removed in the subsequent spectral analysis. We then extracted the PWN
spectra from circular annulus regions centered on the pulsar. We adjusted the
width of annuli to obtain a sufficient signal-to-noise ratio in each bin
(Figure A1). Several targets have jets , which are known to have different
spectral behaviors compared to other parts of PWNe and may have different
mechanisms of particle transfer (see, e.g., Slane et al., 2004; Temim et al.,
2010). Moreover, a few objects like MSH 15$-$52 have knots or blobs at the
opposite direction of jets. We followed previous analysis to remove those
structures as well as background point sources from our spectral analysis
(see, e.g., Yatsu et al., 2009).

The source and background spectra, corresponding weighted response matrix
files (RMFs), and ancillary response files (ARFs) of each annulus region are
extracted using specextract command. The background is selected from a nearby
source-free region on the same chip. Following the standard analysis procedure
for extended sources, we generated weighted ARFs by setting weight=yes and
apply no point-source aperture correction by setting correctpsf=no.

We performed all the spectral analysis using the Sherpa environment. The
energy range of the spectral fitting is restricted to 0.5–8 keV to optimize
the signal-to-noise ratio except for 3C 58, where we fit its high-energy tail
in the 1–8 keV range to avoid the contribution of a soft thermal component
below $\sim 1$ keV (Slane et al., 2004; Gotthelf et al., 2007). All the
spectra are grouped to have at least 30 X-ray photons in each energy bin. We
use an absorbed power-law (PL) model to fit the spectra jointly across all
observations for each PWN. We chose absorption model tbabs and set the
interstellar abundance according to Wilms et al. (2000), in which uses the
cross-section presented in Verner et al. (1993).

We first fit the entire PWN with a single absorbed PL model to constrain the
hydrogen column density $N_{\rm{H}}$. This is justified if we assume that the
$N_{\rm{H}}$ is generally the same among the entire PWN and the spectral
behavior of each PWN is steady. Then, we fixed $N_{\rm{H}}$ at the best-fit
value and fit the spectra of each annulus by linking $\Gamma$ and the
normalization across all observations. The purpose is to reduce the
computation time. A few time-variable softening/hardening areas of some PWNe
have been reported, but they do not significantly change the overall spectrum
of each annulus (see, e.g., Guest & Safi-Harb, 2020). We calculated the 0.5–8
keV unabsorbed flux using sample_flux function with $10^{4}$ iterations to
obtain the $\Sigma$ profile of each PWN.  For each iteration, this command
simulates parameter values from a normal distribution around best-fit
parameters.

##  4 Pure Diffusion Model

A model based on diffusion was proposed to improve the MHD model, in which the
cross-field scattering is small and the particle diffusion can be ignored (de
Jager & Djannati-Ataï, 2009). In the pure diffusion model, the diffusion
coefficient ($D$) is energy-independent. We further assume that $D$ and the
$B$-field strength ($B_{\rm{PWN}}$) are constants in the PWN and set the outer
boundary to be transparent for particles.  Previous studies have suggested
that additional advection of particles could improve the fit, but this only
dominates the innermost region of Crab (Tang & Chevalier, 2012; Ishizaki et
al., 2017). On the other hand, diffusion dominates most of the volume of the
PWN.

Figure 1: The radial profile of $\Gamma$ and the unabsorbed surface brightness
of for our PWN sample.For each PWN, the upper panel shows the the radial
profiles of $\Gamma$, and the lower panel shows the $\Sigma$ profile in units
of erg cm-2 s-1 arcsec-2. The purple curve denotes the best  $\Gamma$-fit for
the $\Gamma$ profile. The green curve denotes the best  $\Sigma$-fit that fits
the $\Sigma$ profile.

We assume power-law distributed particles $\dot{N}(E,r=0)=KE^{-\alpha}$
injected from the center of a PWN, where $E$ is the particle energy and
$\alpha$ is the power-law index. The particles diffuse from the center and
distribute spherically symmetry. As the particles extend to a radius $R$,
which corresponds to the outer bound of a PWN, we adopt a simple case that the
particles transmit out and leave the system. The $B$-field strength and the
PWN radius $R$ are pre-determined parameters using the broadband spectral-
energy distribution fitting results obtained from Tanaka & Takahara (2011,
2013). (see Table 1). Following Gratton (1972), the particle number density
distribution is

$N(E,r,t)=\frac{K}{4\pi rD}E^{-\alpha}f_{\alpha}(u,v),$ (1)

where $f_{\alpha}(u,v)$ is an integral

$f_{\alpha}(u,v)=\frac{1}{\sqrt{\pi}}\int_{v}^{\infty}\left(1-\frac{u}{x}\right)^{\alpha-2}\frac{e^{-x}}{\sqrt{x}}dx,$
(2)

and $u$ is defined as

$u=\frac{r^{2}QE}{4D},$

where

$Q=\frac{4\sigma_{T}}{3m^{2}c^{3}}\frac{B^{2}}{8\pi}$

is the factor of synchrotron energy loss. The lower limit of the integral in
Equation 2 is

$v=\begin{cases}\displaystyle\frac{r^{2}}{4Dt}&\displaystyle
t<\frac{1}{QE}\\\\[8.5359pt] \displaystyle\frac{r^{2}QE}{4D}&\displaystyle
t>\frac{1}{QE}\end{cases}$

where $t$ is the age of the PWN. Once we obtain the energy distribution of
particles, we calculate the synchrotron spectrum at any radius with an assumed
magnetic field. The normalization constant $K$ can be derived from the
luminosity of the PWN. We assume a fraction $\eta$ of the spin-down energy
that is converted to the energy of particles above 1 TeV. This can be written
as

$\int_{E_{0}}^{E_{\rm{max}}}E\dot{N}(E,r=0,t)dE=\eta L_{\rm{sd}},$ (3)

where $E_{0}=1~{}{\rm TeV}$ is the lower bound of the integration, and
$E_{\rm{max}}$ is the maximum value of the electrostatic energy in the polar
cap of the central pulsar. By performing an integration to
$\dot{N}(E,r=0,t)=KE^{-\alpha}$, we obtain

$K=\frac{\eta L_{\rm sd}\left(\alpha-2\right)}{1-\left(E_{\rm
max}/E_{0}\right)^{2-\alpha}}E_{0}^{\alpha-2}.$ (4)

Figure 2: $\Gamma$ and unabsorbed $\Sigma$ profiles of all PWNe. The x-axis is
the physical size in units of parsec. The size is converted from the angular
size and assumed distances listed in Table 1.

In our analysis, we use the full synchrotron spectrum to normalize the energy
distribution of particles for the emission calculation. We followed the
procedure presented in Tanaka & Takahara (2010, 2011) and Ishizaki et al.
(2017) to calculate the synchrotron power. The spectral flux density
$j_{\nu}(r)$ can be written as

$j_{\nu}(r)=\int N(E,r,t)p_{\nu}(E)dE,$ (5)

where

$p_{\nu}(E)=\frac{\sqrt{3}e^{3}B}{mc^{2}}F\left(\frac{\nu}{\nu_{c}}\right),$
(6) $\nu_{c}=\frac{3}{4\pi}\frac{eB}{mc}\gamma^{2},$ (7)

$F(x)=x\int_{x}^{\infty}K_{5/3}(y)dy$ with $x=\nu/\nu_{c}$, and $K_{5/3}$ is
the modified Bessel function of order $5/3$ (Rybicki & Lightman, 1986). The
specific surface brightness $M_{\nu}$ can be integrated as

$M_{\nu}(r_{\perp})=\int_{C}\frac{j_{\nu}(r)}{4\pi}ds,$ (8)

where $ds$ is a line element along the line-of-sight $C$, $r_{\perp}$ is the
distance perpendicular to the line of sight from the central pulsar, i.e.
angular distance. In a spherically symmetric system, this line integral can be
written explicitly as

$M_{\nu}(r_{\perp})=2\int_{r_{\perp}}^{r_{N}}\frac{j_{\nu}(r)}{4\pi}\frac{rdr}{\sqrt{r^{2}-r_{\perp}^{2}}},$
(9)

where $r_{N}$ is a radius of the nebula.  The surface brightness
$\Sigma(r_{\perp})$ can be obtained by integrating over the energy range from
$\nu_{1}$ to $\nu_{2}$ as follows:

$\Sigma(r_{\perp})=\int_{\nu_{1}}^{\nu_{2}}M_{\nu}(r_{\perp})d\nu.$ (10)

Finally, the spectral index $\Gamma$ (from $\nu_{1}$ to $\nu_{2}$) can be
calculated as

$\Gamma=\frac{\log[M_{\nu_{1}}(r_{\perp})/M_{\nu_{2}}(r_{\perp})]}{\log(\nu_{1}/\nu_{2})}.$
(11)

Table 2: Fitting Result of Selected PWNe

PWN Name | Modela | $\alpha$ | $D$ | $\eta$ | $DB^{-3/2}$
---|---|---|---|---|---
|  |  | ($10^{25}$ cm2 s-1) | ($\times 10^{-3}$) | ($10^{32}$ cm2 s-1 G-3/2)
3C 58 | $\Gamma$ | 2.6 | $12$ | 12.3 | $17$
| $\Sigma$ | 2.9 | $30$ | 37.8 | $43$
G21.5$-$0.9 | $\Gamma$ | 1.75 | $16$ | 56.1 | $4.9$
| $\Sigma$ | 1.9 | $20$ | 50 | $6.2$
G54.1+0.3 | $\Gamma$ | 2.5 | $11$ | 66.8 | $35$
| $\Sigma$ | 2.3 | $45$ | 35.3 | $14$
G291.0$-$0.1 | $\Gamma$ | 1.7 | $220$ | 0.6 | $38$
| $\Sigma$ | 1.8 | $350$ | 0.8 | $60$
G310.6$-$1.6 | $\Gamma$ | 2.46 | $5$ | 68.5 | $7.1$
| $\Sigma$ | 2.55 | $7.8$ | 84.7 | $11$
Kes 75 | $\Gamma$ | 2.6 | $6.5$ | 537 | $7.3$
| $\Sigma$ | 2.5 | $4$ | 362 | $4.4$
MSH 15$-$52 | $\Gamma$ | 1.6 | $19$ | 52.6 | $33$
| $\Sigma$ | 1.5 | $13$ | 40 | $22$
Vela | P | 1.4 | $0.025$ | 0.62 | $0.23$
| $\Sigma$ | 1.2 | $0.0085$ | 0.59 | $0.078$
Crab | $\Gamma$ | 2.01 | $30$ | 340.2 | $3.8$
| $\Sigma$ | 2.25 | $50$ | 349.8 | $6.4$

aafootnotetext: The $\Gamma$ model fits the $\Gamma$ profile and the $\Sigma$
model fits the $\Sigma$ profile.

##  5 Fitting results

We fit the radial profiles of $\Gamma$ and $\Sigma$ with the pure diffusion
model described in Section 4.  Three parameters, $\alpha$, $D$, and $\eta$ are
tuned to fit the profiles, while others parameters are fixed. Our objective is
to determine if the simple diffusion could reproduce the characteristic of the
profile. Therefore, we did not perform a formal $\chi^{2}$-fit because it is
difficult to discuss the goodness of fit statistically owing to the large
systematic uncertainty of this simple model. The best-fit results are listed
in Table 2. We performed two fits for each source: one fits the $\Gamma$
profile ( $\Gamma$ fit) and another fits the $\Sigma$ profile ( $\Sigma$ fit).
Both profiles for each PWN are shown in Figure 1.  Note that these two fits
are not independent. The shape of both profiles are determined once $\alpha$
and $D$ are chosen. The parameter $\eta$ only tunes the normalization of the
$\Sigma$ profile. We discuss the parameter dependence in Section 6.

The pure diffusion model generally well describes the sample PWNe. Both
$\Gamma$\- and $\Sigma$-fits yield consistent results in G21.5$-$0.9,
G291.0$-$0.1, G310.6$-$1.6, and Kes 75. These PWNe have no complex structure
compared to Crab, Vela, and MSH 15$-$5 _2_. However, some PWNe are relatively
faint with large uncertainties in $\Gamma$ that could smear detailed
structures in the profile. On the other hand, large discrepancies between the
parameters yielded from the  $\Gamma$\- and $\Sigma$-fits are seen in 3C 58,
G54.1+0.3, and possibly MSH 15$-$52. There are however a couple of poor fits,
in particular Crab and Vela. They have significant changes in slope in both
profiles, and their brightness profiles are concave down.  Moreover, these are
especially observed in the innermost region, which the advection may play an
important role (Tang & Chevalier, 2012; Ishizaki et al., 2017).

In Figure 2 we plot the $\Gamma$ and $\Sigma$ profiles versus the physical
size. All the PWNe have a similar shape in both profiles except for Crab and
Vela. The $\Gamma$ monotonically and smoothly increases with distance from the
pulsar in nearly all sources. The overall slope resides in a narrow range. All
the sampled PWNe have similar $\Sigma$ profiles except for Crab and Vela. We
discuss these apparent features and the physical interpretation using the
diffusion model in Section 6.

##  6 Discussion

###  6.1 Applicability of the Pure Diffusion Model

Our result shows that the pure diffusion model can explain the radial profiles
of $\Gamma$ and $\Sigma$ in soft X-rays for most young PWNe, except for a few
cases. We suggest the pure diffusion model could be applied to other young
PWNe in general. Except for Crab, none of the sampled PWNe show an abrupt
change in $\Gamma$ profile, suggesting that the steady flow assumed in the MHD
model cannot describe the observed phenomena (Slane et al., 2004; Tang &
Chevalier, 2012; Ishizaki et al., 2017). A few PWNe have large discrepancies
between the  $\Gamma$\- and $\Sigma$-fits parameters, implying further
improvements to the model may be needed. Moreover, the profiles of Crab and
Vela cannot be described with the diffusion model. Here we discuss the
discrepancy between the $\Gamma$ and the $\Sigma$ profiles and the detailed
features of individual PWNe.

The spin-down power of the central pulsar of G291 remains unclear. With the
assumption of $L_{sd}=5.7\times 10^{37}$ erg s${}^{-}1$, both the $\Gamma$ and
$\Sigma$ profiles can be fit with $\alpha\sim 1.7$ and $\eta\lesssim 0.001$.
This suggests a low energy conversion fraction comparable to that of Vela.
Similar low ratio of the X-ray luminosity and the spin-down luminosity was
also suggested by Slane et al. (2012). This is low but remains within a
reasonable range of known pulsars. We also tried to use different values of
$L_{sd}$ in the range between $\sim 10^{36}$ erg s${}^{-}1$ and $5.7\times
10^{37}$ erg s${}^{-}1$. All values gives the same fitting with the same
parameters except for $\eta$.

The discrepancy between the  $\Gamma$\- and $\Sigma$-fits is significant in 3C
58 and G54.1+0.3. The best-fit $D$ of  $\Gamma$ and $\Sigma$ fits differ from
each other at a factor larger than 2. We note that these sources are more
elongated than others, such that the $B$-field could be different in the axial
and equatorial directions. It was suggested that the viewing geometry of 3C 58
is close to edge-on, while G54.1+0.3 has an inclination angle close to
$\approx 147^{\circ}$ between the torus axis and the line of sight (Ng &
Romani, 2004, 2008). The non-spherical diffusion and the viewing geometry
could result in the discrepancy between the $\Gamma$ and the $\Sigma$
profiles. This hints at the limitation of our simple spherical symmetric PWN
under the pure diffusion approximation for the synchrotron emissivity
calculation.  Moreover, the age of 3C 58 remains controversial, as its
connection to SN 1181 is not clear (Kothes, 2013; Ritter et al., 2021). If 3C
58 is not as young as other sampled PWNe, the interaction between the PWN and
the supernova remnant has to be taken into account, especially for the outer
region of the PWN. Finally, the energy-dependence of $D$, the reflection
effect of the PWN boundary, the advection flow, and the inhomogeneity of the
$B$ field may be further considered to improve the fit (Tang & Chevalier,
2012).

Crab and Vela are the only two cases that clearly cannot be described by our
diffusion model. It was argued that the advection is significant in Crab (Tang
& Chevalier, 2012). If this is true, the magnetic field is expected to change
spatially. On the other hand, Vela shows an opposite trend as Crab in the
$\Gamma$ profile, suggesting a different mechanism. Considering that the
distance to Vela is at least one order of magnitude closer than other PWNe in
our sample, the observed emission region could only represent the very central
part of the entire PWN (see, e.g., Kargaltsev & Pavlov, 2008). Therefore, it
may not be easy to interpret it in the same way as other sources. Moreover,
Vela is old comparing with the other PWNe.  The interaction between the
supernova remnant (SNR) and the PWN has to be taken into account. However, the
jet and torus structure is not distorted by the reverse shock (see, e.g., Ng &
Romani, 2004). A more plausible possibility is that the advection dominates
the particle transfer in the innermost region of a PWN. In addition, the
spatial structure of the magnetic field could be another possibility to make
it difficult to describe the profile with our simple model. Similar to Crab
and Vela, MSH 15$-$52 shows clear inner-ring and torus structures. A valley is
seen in the brightness profile at 20–40″ (0.5–1 pc), which was reported by
Yatsu et al. (2009). Therefore, a simple spherically symmetric or constant
$B$-field approximation may not be justified.

Figure 3: Parameter dependence of model profiles in $\Gamma$ (a)–(d) and
$\Sigma$ (e)–(h) in units of erg cm-2 s-1 arcsec-2. Data points are results
from 3C 58.  Figure 4: Correlation between model parameters and apparent
parameters of the $\Gamma$ and $\Sigma$ profiles. Gray circles denote the
parameters derived from the $\Gamma$ fit, while the red triangles are
parameters from the $\Sigma$ fit. We use blue line to link the parameters
obtained by  $\Gamma$ and $\Sigma$ fits for each PWN.

###  6.2 Degeneracy of Model Parameters

Based on our pure diffusion model with transmitting boundary, three parameters
are critical for the $\Gamma$ profile: $B$ field in the PWN, $D$, and the
injected power-law index $\alpha$ of the particle energy. Moreover, $\eta$
plays an important role in the $\Sigma$ profile.  However, the $B$ field
strength is a pre-defined value and varies dramatically among studies, see,
e.g., G21 (Tanaka & Takahara, 2011, 2013; Guest et al., 2019). This parameter
is coupled with $D$ and therefore it is meaningless to directly compare $D$
among PWNe. Taking 3C 58 as an example, we demonstrate how the model profile
changes as we change these parameters in Figure 3. From Figures 3 (a) and (b),
the effect of $B$ and $D$ are coupled with each other (Tang & Chevalier,
2012). A flatter $\Gamma$ profile, which implies a lower energy dissipation,
can be achieved by both decreasing $B$ and increasing $D$, and vice versa.
This is because the spatial extent of the X-ray emitting particles is defined
by the distance over which they spread before exhausting their energy via
synchrotron radiation. The ratio of the diffusion length of a particle
emitting a photon of a given frequency $\nu$ to a radius $R$ can be written as
$\sqrt{4Dt_{\rm cool}}/R\propto\left(R^{-2}\nu^{-1/2}DB^{-3/2}\right)^{1/2}$,
where $t_{\rm cool}=1/QE$ is the cooling time scale of the synchrotron
emission. The $\Sigma$ profile is the same if $DB^{-3/2}$ remains constant.
Therefore, $DB^{-3/2}$ is the key parameter for the shape of the $\Gamma$
profile. Since the PWN $B$-field strength is difficult to estimate and it
often depends on many assumptions, we will not discuss the effect of $D$.
Instead, we focus on $DB^{-3/2}$ in the following discussions. The $DB^{-3/2}$
values of the sampled PWNe are listed in Table 2. The $\Gamma$ fitting results
of 3C 58 and G21.5$-$0.9 are in the same order as that derived in Tang &
Chevalier (2012) with full synchrotron spectrum.

We further test the sensitivity of the effect of parameters on the $\Sigma$
profile (Figure 3 (e) and (f)). The $DB^{-3/2}$ value changes not only the
shape but also the displacement (or interception) of model profiles. Compare
to the $\Gamma$ profile, we notice that the shape of the $\Sigma$ profile
becomes insensitive to $DB^{-3/2}$ as long as $DB^{-3/2}$ is large enough.
From Figures 3 (c), the offset of $\Gamma$, which implies the general hardness
of the PWN, is dominated by $\alpha$ of the injected high-energy particles. On
the other hand, the displacement of the $\Sigma$ profile could be affected by
$DB^{-3/2}$, $\alpha$, and $\eta$. Finally, $\eta$ only affects the
displacement in the $\Sigma$ profile because it represents the energy
transferring efficiency from the spin-down power of the pulsar to the PWN.

Figure 5: Correlation between model parameters and physical parameters. Gray
circles denote the parameters derived from the $\Gamma$ fit, while red
triangles are parameters from the $\Sigma$ fit. We use blue line to link the
parameters obtained by  $\Gamma$ and $\Sigma$ fits of each PWN. The ages of
PWNe and $L_{sd}$ are adopted from Table 1.

###  6.3 Profile Characteristics and Model Parameters

Based on the discussion in Section 6.2, we test whether the characteristics,
i.e. slope and mean value of the $\Gamma$ and $\Sigma$ profiles, can be
interpreted with the pure diffusion parameters. To achieve this goal, we
simply fit the $\Gamma$ profile of each PWN with a straight line. Crab and
Vela are not included in this analysis since their profiles cannot be
described with our model. The slope can roughly represent the $\Gamma$
gradient. We take the mean of $\Gamma$ as the value at the half of the PWN
radius predicted from the best-fit straight line. For the $\Sigma$ profile, we
define the same slope and the mean $\Sigma$ as we did for the $\Gamma$
profile. The difference is that we use the logarithmic scale for the surface
brightness.

Then, we plot the apparent features (slope and mean value of the $\Gamma$ and
the $\Sigma$ profiles) versus the model parameters in Figure 4. The most
significant correlation is, not surprisingly, between the mean of $\Gamma$ and
$\alpha$. The linear correlation coefficient is $\rho=0.89$ with a null
hypothesis probability $p=0.006$ for  $\Gamma$ fit, where the
$\Sigma$-fitting result is $\rho=0.96$ with $p=6\times 10^{-4}$. A relatively
weak correlation can be seen between $DB^{-3/2}$ and the slope of the $\Gamma$
profiles with $\rho=-0.85$ and $p=0.02$ for the  $\Gamma$ fit. The correlation
is insignificant for the  $\Sigma$ fit with $\rho=-0.40$ and $p=0.3$. This is
caused by the large discrepancy between the  $\Gamma$ and  $\Sigma$ fitting
results of a few PWNe where the  $\Sigma$ fit cannot well describe the
$\Gamma$ profile.

The slope of the $\Sigma$ profile is correlated with $DB^{-3/2}$, where the
correlation analysis suggests $\rho=0.75$ and $p=0.05$ for both the
$\Sigma$\- and $\Gamma$\- fits. The relation between these two parameters
might be non-linear. Therefore, we also calculate Kendall’s rank correlation
and obtain a Kendall’s $\tau=0.81$ with $p=0.01$ for the  $\Gamma$ fit, and
$\tau=0.52$ with $p=0.1$ for the  $\Sigma$-fit. This suggests a possible
correlation and consistent with the result in Section 6.2, in which we show
that a larger $D$ (or smaller $B$) result in a flatter $\Sigma$ profile.
Interestingly, the slope of the $\Sigma$ profile anti-correlates with $\eta$
with $\rho=-0.84$ and $p=0.01$ for the  $\Sigma$ fit ($\rho=-0.88$ and
$p=0.009$ for the  $\Gamma$ fit), suggesting a similar significance as that
between $DB^{-3/2}$ and the $\Sigma$ profile. This is not expected as we
discussed in Section 6.2 and could indicate physical connection underneath.
Finally, the mean surface brightness likely anti-correlates with $DB^{-3/2}$
and positively correlates with $\eta$, these are expected in Section 6.2.
However, the mean surface brightness does not correlate with $\alpha$,
suggesting that $\alpha$ is not the dominant factor for the $\Sigma$ prodile.

###  6.4 Correlation between Model Parameters and Physical Parameters

We tested whether the best-fit parameters and physical parameters correlate
with each other. We performed correlation analysis between four parameters,
including $DB^{-3/2}$, $\alpha$, $\eta$, and the PWN age. The results are
shown in Figure 5. It is worth mentioning that $\eta$ of Kes 75 is the largest
among selected PWNe. Given that PSR J1846$-$0258 is a high $B$-field pulsar
that shows magnetar-like burst and outburst behavior, it is possible that the
dissipation of the extremely strong magnetic field results in additional
energy output from the pulsar (Gavriil et al., 2008; Ng et al., 2008; Kumar &
Safi-Harb, 2008).

In general, we found no significant correlations between these parameters,
indicating that these parameters are independent of each other. We found that
$DB^{-3/2}$ may weakly anti-correlate with $\log(\eta)$ with $\rho=-0.7$ and
$p=0.07$ for the  $\Gamma$ fit. This possible correlation is not observed for
the  $\Sigma$ fit with $\rho=-0.5$ and $p=0.3$, suggesting no significance. We
could not draw a strong conclusion with our current sample. However, this
could be connected to the anti-correlation between the slope of the brightness
profile and $\eta$ in Section 6.3. A larger $DB^{-3/2}$ implies a steeper
$\Gamma$ gradient in the radial profile, i.e. stronger synchrotron energy
loss. Therefore, a high energy conversion rate might not be needed.

##  7 Summary

We applied a pure diffusion model to fit the $\Gamma$ and $\Sigma$ profiles of
nine young PWNe, including well studied 3C 58 and G21.5$-$0.9, a few faint
sources G54.1+0.3, G291.0$-$0.1, G310.6$-$1.6, Kes 75, and three bright PWNe
Crab, MSH 15$-$5 _2_ , and Vela. We analyzed archival _Chandra_ observations
and fit the spectra of each PWN with a power-law model. Except for Crab and
Vela, all the sampled PWNe show similar $\Gamma$ and $\Sigma$ profiles.
$\Gamma$ increases slowly with the distance from the central pulsar. This
behavior cannot be described by the MHD model that suggests an abrupt increase
in $\Gamma$ (Slane et al., 2004; Tang & Chevalier, 2012; Ishizaki et al.,
2017) and therefore suggests that particle diffusion dominates the particle
transfer rather than a steady flow for most PWNe. We derived the brightness
profile from the pure diffusion model and found that the observed profiles are
consistent with the model prediction. However, the parameters that can
describe the $\Gamma$ profile and those for the $\Sigma$ profiles are not
consistent with each other in a few cases like 3C 58 and G54.1+0.3. The
discrepancy implies a simple diffusion model may not be enough to produce the
entire distribution of PWNe and future improvement is necessary. The
discrepancy could be attributed to the intrinsic spherical asymmetry, the
particle reflection by the PWN boundaries, energy-dependent diffusion, and
spatially-dependent $B$ field. The latter two could be specified with multi-
wavelength studies. A model with a spatially varying magnetic field could be
constructed in the future. We further investigate the relationship between the
model parameters and the apparent feature of both profiles. The effect of the
diffusion coefficient and the magnetic field are coupled with each other, and
the observed $\Gamma$ and the $\Sigma$ gradient highly depend on the
$DB^{-3/2}$ value. The mean hardness of a PWN significantly correlates with
the distribution of the injected particles, while the brightness of a PWN is
dominated by the energy conversion efficiency $\eta$. Finally, we found no
significant correlations between model parameters and the physical quantities
of PWNe.

We thank the anonymous reviewer for valuable comments that improved this
paper. This research is based on the data obtained from the Chandra Data
Archive and has made use of the software provided by the _Chandra_ X-ray
Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. C.-P.H.
acknowledges support from the Ministry of Science and Technology in Taiwan
through grant MOST 109-2112-M-018-009-MY3. W.I. is supported by JSPS KAKENHI
Grants No. 21J01450. C.-Y.N. is supported by a GRF grant of the Hong Kong
Government under HKU 17305416. S.J.T. acknowledges support by the Aoyama
Gakuin University-Supported Program “Early Eagle Program”.

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##  Appendix A Data Sets Used in This Research

We list all data sets used in this analysis in Table A1.

Table A1: _Chandra_ observation log of selected PWNe.  PWN Name | ObsID | Date | Exposure (ks)
---|---|---|---
3C 58 | http://cda.harvard.edu/chaser/viewerContents.do?obsid=728 (catalog 728) | 2000-09-14 | 38.7
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=4383 (catalog 4383) | 2003-04-22 | 38.7
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=4382 (catalog 4382) | 2003-04-23 | 167.4
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=3832 (catalog 3832) | 2003-04-26 | 135.8
G21.5$-$0.9 | http://cda.harvard.edu/chaser/viewerContents.do?obsid=159 (catalog 159) | 1999-08-23 | 14.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1230 (catalog 1230) | 1999-08-23 | 14.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1433 (catalog 1433) | 1999-11-15 | 15.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1716 (catalog 1716) | 2000-05-23 | 7.7
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1717 (catalog 1717) | 2000-05-23 | 7.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1718 (catalog 1718) | 2000-05-23 | 7.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1769 (catalog 1769) | 2000-07-05 | 7.4
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1770 (catalog 1770) | 2000-07-05 | 7.2
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1771 (catalog 1771) | 2000-07-05 | 7.2
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1838 (catalog 1838) | 2000-09-02 | 7.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1839 (catalog 1839) | 2000-09-02 | 7.7
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1553 (catalog 1553) | 2001-03-18 | 9.8
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1554 (catalog 1554) | 2001-07-21 | 9.1
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2873 (catalog 2873) | 2002-09-14 | 9.8
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=3693 (catalog 3693) | 2003-05-16 | 9.8
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=3700 (catalog 3700) | 2003-11-09 | 9.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=5166 (catalog 5166) | 2004-03-17 | 10.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=5159 (catalog 5159) | 2004-10-27 | 9.8
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=6071 (catalog 6071) | 2005-02-26 | 9.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=6741 (catalog 6741) | 2006-02-22 | 9.8
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=8372 (catalog 8372) | 2007-05-25 | 10.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10646 (catalog 10646) | 2009-05-29 | 9.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14263 (catalog 14263) | 2012-08-08 | 9.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16420 (catalog 16420) | 2014-05-07 | 9.6
G54.1+0.3 | http://cda.harvard.edu/chaser/viewerContents.do?obsid=1983 (catalog 1983) | 2001-06-06 | 38.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=9886 (catalog 9886) | 2008-07-08 | 65.3
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=9108 (catalog 9108) | 2008-07-10 | 34.7
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=9109 (catalog 9109) | 2008-07-12 | 162.3
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=9887 (catalog 9887) | 2008-07-15 | 24.8
G291.0$-$0.1 | http://cda.harvard.edu/chaser/viewerContents.do?obsid=2782 (catalog 2782) | 2002-04-08 | 49.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16497 (catalog 16497) | 2013-10-28 | 37.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14822 (catalog 14822) | 2013-11-03 | 51.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14824 (catalog 14824) | 2013-11-04 | 74.1
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14823 (catalog 14823) | 2013-11-10 | 67.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16512 (catalog 16512) | 2013-11-13 | 63.4
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16541 (catalog 16541) | 2014-01-21 | 29.2
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16566 (catalog 16566) | 2014-02-15 | 34.6
G310.6$-$1.6 | http://cda.harvard.edu/chaser/viewerContents.do?obsid=9058 (catalog 9058) | 2008-06-29 | 5.1
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=12567 (catalog 12567) | 2010-11-17 | 52.8
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=17905 (catalog 17905) | 2016-11-25 | 13.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=19919 (catalog 19919) | 2016-11-29 | 79.7
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=19920 (catalog 19920) | 2016-12-05 | 39.46
Kes 75 | http://cda.harvard.edu/chaser/viewerContents.do?obsid=748 (catalog 748) | 2000-10-15 | 37.3
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=7337 (catalog 7337) | 2006-06-05 | 17.4
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=6686 (catalog 6686) | 2006-06-07 | 54.1
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=7338 (catalog 7338) | 2006-06-09 | 39.3
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=7339 (catalog 7339) | 2006-06-12 | 44.1
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10938 (catalog 10938) | 2009-08-10 | 44.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=18030 (catalog 18030) | 2016-06-08 | 85.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=18866 (catalog 18866) | 2016-06-11 | 61.5
MSH 15$-$52 | http://cda.harvard.edu/chaser/viewerContents.do?obsid=754 (catalog 754) | 2000-08-14 | 19.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=3834 (catalog 3834) | 2003-04-21 | 9.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=4384 (catalog 4384) | 2003-04-28 | 9.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=3833 (catalog 3833) | 2003-10-17 | 19.1
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=5334 (catalog 5334) | 2004-12-28 | 49.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=5335 (catalog 5338) | 2005-02-07 | 42.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=6116 (catalog 6116) | 2005-04-29 | 47.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=6117 (catalog 6117) | 2005-10-18 | 45.6
Vela | http://cda.harvard.edu/chaser/viewerContents.do?obsid=128 (catalog 128) | 2000-04-30 | 10.6
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=1987 (catalog 1987) | 2000-11-30 | 18.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2813 (catalog 2813) | 2001-11-25 | 17.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2814 (catalog 2814) | 2001-11-27 | 19.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2814 (catalog 2815) | 2001-12-04 | 27.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2816 (catalog 2816) | 2001-12-11 | 19.0
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2817 (catalog 2817) | 2001-12-29 | 18.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2818 (catalog 2818) | 2002-01-28 | 18.7
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2819 (catalog 2819) | 2002-04-03 | 19.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=2820 (catalog 2820) | 2002-08-06 | 19.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=3861 (catalog 3861) | 2003-08-21 | 4.9
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10132 (catalog 10132) | 2009-07-09 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10133 (catalog 10133) | 2009-07-17 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10134 (catalog 10134) | 2009-07-25 | 38.1
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10135 (catalog 10135) | 2010-06-28 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10136 (catalog 10136) | 2010-07-08 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10137 (catalog 10137) | 2010-07-17 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10138 (catalog 10138) | 2010-07-26 | 38.8
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=10139 (catalog 10139) | 2010-08-04 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=12073 (catalog 12073) | 2010-08-15 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=12074 (catalog 12074) | 2010-08-24 | 39.5
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=12075 (catalog 12075) | 2010-09-04 | 41.8
Crab | http://cda.harvard.edu/chaser/viewerContents.do?obsid=14458 (catalog 14458) | 2012-11-26 | 1.24
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14679 (catalog 14679) | 2013-01-02 | 0.66
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14680 (catalog 14680) | 2013-02-25 | 0.66
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14685 (catalog 14685) | 2013-03-07 | 0.61
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14681 (catalog 14681) | 2013-04-23 | 0.61
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14682 (catalog 14682) | 2013-08-16 | 0.61
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=14678 (catalog 14678) | 2013-10-08 | 0.63
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16245 (catalog 16245) | 2013-10-19 | 1.29
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16257 (catalog 16257) | 2013-11-04 | 1.29
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16357 (catalog 16357) | 2014-01-29 | 1.3
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16358 (catalog 16358) | 2014-04-19 | 1.32
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16359 (catalog 16359) | 2014-08-16 | 1.32
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16258 (catalog 16258) | 2014-11-06 | 1.34
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16360 (catalog 16360) | 2015-01-18 | 1.37
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16361 (catalog 16361) | 2015-04-10 | 1.42
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16362 (catalog 16362) | 2015-08-60 | 1.44
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16259 (catalog 16259) | 2015-11-06 | 1.44
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16363 (catalog 16363) | 2016-01-31 | 1.44
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16364 (catalog 16364) | 2016-04-20 | 1.43
| http://cda.harvard.edu/chaser/viewerContents.do?obsid=16365 (catalog 16365) | 2016-08-14 | 1.51

##  Appendix B _CHANDRA_ images and spectral extraction regions

Figure A1 shows the _Chandra_ 0.5–8 keV images of all the sampled PWNe. To
study the radial profile, we use multiple annulus centered on the central
pulsar to select X-ray photons and collect X-ray spectra.

Figure A1: _Chandra_ 0.5–8 keV image of (a) 3C 58, (b) G21.5$-$0.9, (c)
G54.1+0.3, (d) G291.0$-$0.1, (e) G310.6$-$1.6, (f) Kes 75, (g) MSH 15$-$5 _2_
, (h) Vela, and (i) Crab. The green annulus denote the selected areas for
spectral analysis. The jets, clumps, and background stars are removed.