A Broadband X-Ray Study of the Rabbit Pulsar Wind Nebula Powered by PSR J1418-6058

Abstract

We report on broadband X-ray properties of the Rabbit pulsar wind nebula (PWN) associated with the pulsar PSR J1418−6058 using archival Chandra and XMM-Newton data, as well as a new NuSTAR observation. NuSTAR data above 10 keV allowed us to detect the 110 ms spin period of the pulsar, characterize its hard X-ray pulse profile, and resolve hard X-ray emission from the PWN after removing contamination from the pulsar and other overlapping point sources. The extended PWN was detected up to ∼20 keV and is described well by a power-law model with a photon index Γ ≈ 2. The PWN shape does not vary significantly with energy, and its X-ray spectrum shows no clear evidence of softening away from the pulsar. We modeled the spatial profile of X-ray spectra and broadband spectral energy distribution in the radio to TeV band to infer the physical properties of the PWN. We found that a model with low magnetic field strength ( B ∼ 10 μ G) and efficient diffusion ( D ∼ 10 ²⁷ cm ² s ⁻¹ ) fits the PWN data well. The extended hard X-ray and TeV emission, associated respectively with synchrotron radiation and inverse Compton scattering by relativistic electrons, suggest that particles are accelerated to very high energies (≳500 TeV), indicating that the Rabbit PWN is a Galactic PeVatron candidate.

Full text

A Broadband X-Ray Study of the Rabbit Pulsar Wind Nebula Powered by PSR
J1418-6058
Jaegeun Park
1
, Chanho Kim
1
, Jooyun Woo
2
, Hongjun An
1
, Kaya Mori
2
, Stephen P. Reynolds
3
, and
Samar Safi-Harb
4
1
Department of Astronomy and Space Science, Chungbuk National University, Cheongju, 28644, Republic of Korea; hjan@cbnu.ac.kr
2
Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA
3
Physics Department, NC State University, Raleigh, NC 27695, USA
4
Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
Received 2022 May 22; revised 2023 February 5; accepted 2023 February 6; published 2023 March 8
Abstract
We report on broadband X-ray properties of the Rabbit pulsar wind nebula (PWN)associated with the pulsar
PSR J1418−6058 using archival Chandra and XMM-Newton data, as well as a new NuSTAR observation.
NuSTAR data above 10 keV allowed us to detect the 110 ms spin period of the pulsar, characterize its hard X-ray
pulse profile, and resolve hard X-ray emission from the PWN after removing contamination from the pulsar and
other overlapping point sources. The extended PWN was detected up to ∼20 keV and is described well by a
power-law model with a photon index Γ≈2. The PWN shape does not vary significantly with energy, and its
X-ray spectrum shows no clear evidence of softening away from the pulsar. We modeled the spatial profile of
X-ray spectra and broadband spectral energy distribution in the radio to TeV band to infer the physical properties
of the PWN. We found that a model with low magnetic field strength (B∼10 μG)and efficient diffusion (D∼10
27
cm
2
s
−1
)fits the PWN data well. The extended hard X-ray and TeV emission, associated respectively with
synchrotron radiation and inverse Compton scattering by relativistic electrons, suggest that particles are accelerated
to very high energies (500 TeV), indicating that the Rabbit PWN is a Galactic PeVatron candidate.
Unified Astronomy Thesaurus concepts: High-energy astrophysics (739);Pulsar wind nebulae (2215);Rotation-
powered pulsars (1408);Spectral energy distribution (2129);Astronomy data analysis (1858);Gamma-ray
sources (633)
1. Introduction
Pulsar wind nebulae (PWNe)are bubbles of relativistic
particles powered by the rotational energy released from a
pulsar, and they generally emit electromagnetic radiation from
radio to TeV gamma-ray energies (for a review, see
Slane 2017). It is widely accepted that the pulsar wind particles
are accelerated to very high energies at a termination shock
(Kennel & Coroniti 1984), and their interactions with magnetic
fields and ambient low-energy photons result in broadband
emission from the radio to gamma-ray bands. In particular, the
detection of many PWNe in the very high-energy (VHE;
>0.1TeV)gamma-ray band (e.g., H.E.S.S. Collaboration et al.
2018)suggests that these sources are strong Galactic PeVatron
candidates—the particle acceleration and transport mechan-
isms in PWNe are essential for understanding the origin of
TeV–PeV cosmic-ray electrons and positrons detected on Earth
(e.g., Fiori et al. 2022). Furthermore, multiwavelength studies
of PWNe can provide insights into relativistic shock physics
(e.g., Sironi et al. 2015)and magnetohydrodynamic (MHD)
flow of high-energy particles (e.g., Kennel & Coroniti 1984).
Broadband spectral energy distributions (SEDs)of PWNe
are characterized well by two components originating from the
same population of relativistic particles: synchrotron radiation
(E
γ
MeV)and inverse Compton scattering (ICS)of ambient
photon fields in the gamma-ray band (e.g., Bednarek &
Bartosik 2003). Because the energy distribution of particles is
imprinted in the emission spectra, both obtaining and modeling
SED data accurately allow us to understand the particle
acceleration processes in PWNe. Characterizing the synchro-
tron emission through imaging and spectroscopic data in the
hard X-ray band (E
γ
>10 keV)is particularly useful, since it
allows us to directly probe the highest-energy (sub-PeV)
particle distributions, whereas the ICS component is mainly
affected by the properties of the ambient photon fields.
In general, young rotation-powered PWNe (10
3
yr)such as
the Crab Nebula and G21.5−0.9, compared to older nebulae,
are bright in X-rays and relatively faint in the VHE band,
implying that magnetic fields in young PWNe are likely strong
(B∼100 μG; Meyer et al. 2010; Guest et al. 2019). Particles in
these PWNe lose energy efficiently via synchrotron radiation,
which is observed as a spectral break and/or a PWN size
decrease with increasing photon energy (e.g., Reynolds 2016).
The X-ray-to-VHE flux ratios of numerous PWNe in different
evolutionary stages (e.g., young, middle-aged, and relic PWNe)
have been observed to decrease with their ages (Kargaltsev
et al. 2013). This implies that magnetic fields are lower (e.g.,
B10 μG; Kargaltsev et al. 2013; Zhu et al. 2018)in the older
PWNe (>10
4
yr)that can supply the interstellar medium (ISM)
with highly energetic particles, due to the weaker synchrotron
cooling rates (e.g., Giacinti et al. 2020).
The Rabbit PWN was discovered in an X-ray study of a
region around a bright EGRET source (Roberts & Romani
1998). A radio counterpart was discovered (Roberts & Romani
1998; Roberts et al. 1999), and extended TeV emission was
observed ∼8′west of the PWN (Aharonian et al. 2006; H.E.S.
S. Collaboration et al. 2018). The central engine of the PWN
was later identified to be a relatively young gamma-ray pulsar
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 https://doi.org/10.3847/1538-4357/acba0e
© 2023. The Author(s). Published by the American Astronomical Society.
Original content from this work may be used under the terms
of the Creative Commons Attribution 4.0 licence. Any further
distribution of this work must maintain attribution to the author(s)and the title
of the work, journal citation and DOI.
1

(PSR J1418−6058; Abdo et al. 2009; J1418 hereafter)with a
characteristic age of τ
c
≈10
4
yr, and X-ray pulsations were
suggested
5
and later confirmed (Kim & An 2020a); in this
work, the pulsar’s X-ray spectrum was fitted by a power law
with a hard photon index Γ=1.0 ±0.6. Distance to the PWN
was estimated to be 3.5–5.6 kpc based on a column density and
ISM morphology study (Voisin et al. 2019).
Kishishita et al. (2012)carried out an X-ray study of the
Rabbit PWN using Suzaku data and found that the PWN
spectra extracted from annular regions soften with increasing
distance from the pulsar, indicating significant synchrotron
cooling in the Rabbit PWN as has been observed in young
PWNe (e.g., An et al. 2014a; Nynka et al. 2014). However, a
high-resolution Chandra image (e.g., Figure 1)found a handful
of point sources and a torus or jet-like structure around the
pulsar (Ng et al. 2005; Kim & An 2020a). Because these
features as well as the (hard)pulsar emission were not resolved
in the Suzaku data, their X-ray emission has likely biased the
Suzaku analysis, making the X-ray spectra appear hard in the
inner regions where the contamination is more severe. Hence,
revisiting an SED study of the Rabbit PWN by scrutinizing the
contaminating sources is warranted.
In this paper, we present a recently acquired NuSTAR
observation and use archival Chandra and XMM-Newton data
to reveal the X-ray emission properties of the Rabbit PWN,
independently of its pulsar and contaminating sources. We
present our data reduction in Section 2.1, and show detailed
timing analysis in Section 2.2. We determine X-ray spectra of
the pulsar and other contaminating sources within the PWN
and assess their impact on measuring the PWN emission in
Section 2.3. We then carry out an imaging analysis
(Section 2.4), and investigate spatially integrated and resolve-
demission of the PWN in Section 2.5. We inspect data taken
by the Fermi Large Area Telescope (LAT; Atwood et al. 2009)
to confirm the GeV measurements reported in the 4FGL DR-2
catalog (gll_psc_v27.fit; Abdollahi et al. 2020)in
Section 3. After collecting multiband spectral data, we modeled
the broadband SED and X-ray spectral variation of the PWN
(Section 4). We discuss the results from our model fitting and
infer the physical properties of the PWN in Section 5. Note that
all errors are at the 1σlevel and quoted flux values are
absorption-corrected ones throughout the paper.
2. X-Ray Data Analysis
2.1. Data Reduction
We use the 70 ks Chandra and 120 ks XMM-Newton
archival data taken on 2007 June 14 (Obs. ID 7640)and on
2009 February 21 (Obs. ID 0555700101), respectively, and
hard X-ray data obtained through the NuSTAR campaign of
TeV PWNe (Mori et al. 2022)on 2021 April 20 for 140 ks.
Figure 1. Background-subtracted and exposure-corrected X-ray images of the Rabbit PWN measured with Chandra (a), XMM-Newton (b), and NuSTAR (c). The
inset in panel (b)is a zoomed-in image of the central region. The pulsar (J1418)position is marked by a blue cross, and contaminating point sources (
R
3
<
¢
~
)are
denoted in white circles. The GeV and TeV counterparts of the PWN are denoted in a white and a yellow ellipse in panel (c), respectively, and
R
3=
¢
and
R
5=
¢
circles (cyan dashed)are shown for reference. Notice that X1 (9″north of S1; panels (a)and (b)) was detected in the XMM-Newton image but not in the Chandra
image. The figures are smoothed, and scales are adjusted for better legibility.
5
https://cxc.harvard.edu/cdo/snr09/pres/Roberts_Mallory.pdf
2
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

We processed the Chandra data using chandra_repro of
CIAO 4.13 along with the most recent calibration database
(version 4.9.6). The Chandra data are useful to identify and
characterize contaminating sources within the PWN, despite its
coverage of the PWN being limited due to the chip gaps. The
XMM-Newton MOS data were processed with the emproc task
of SAS 20211130_0941 along with the most recent calibration
database (updated in 2021 November). Note that we analyzed
only the MOS data because the PN exposure did not cover the
PWN and those data were already analyzed for the pulsar (Kim &
An 2020a). The XMM-Newton MOS data were further cleaned
following the standard flare-removal procedure.
6
We processed
the NuSTAR data with nupipeline integrated in HEASOFT
v6.29 (CALDB 20211020)using the SAA_MODE = strict
flag as recommended by the NuSTAR science operation center.
Net exposures after this initial reduction are 70 ks, 100 ks, and
55 ks for Chandra, XMM-Newton, and NuSTAR, respectively.
X-ray images of the Rabbit PWN and its surrounding regions
are displayed in Figure 1.
2.2. NuSTAR Timing Analysis
The 110 ms X-ray pulsations of J1418 were detected in the
XMM-Newton PN data, but the detection significance was not
very high, with a chance probability p≈10
−7
(Kim &
An 2020a). Hence, independent confirmation of the X-ray
pulsations would be very useful. The high-energy sensitivity
and superb timing resolution of NuSTAR (Harrison et al. 2013)
are beneficial for the detection of pulsations of this rapidly
spinning pulsar, characterized by a hard X-ray spectrum.
We inspected the NuSTAR hard-band images (10–30 keV)
and identified a point source in each of the FPMA and FPMB
images. Assuming that the point source is J1418 (see also
Section 2.4.1), we extracted source events within an R=30″
circle centered at the point source, applied a barycenter correction
to the event arrival times using the pulsar position (R.A., decl.)=
(214°.677695, −60°.967483), and performed timing analysis
employing an Htest (de Jager et al. 1989)in the 3–30 keV
band. Extrapolating the pulsar timing solution (Kerr et al. 2015)
derived from the Fermi-LAT data predicts a spin frequency
f=9.03857 Hz at the epoch of the NuSTAR observation.
However, it is possible that the extrapolated frequency may be
inaccurate due to large timing noise and (undetected)glitches.
7
Hence, we searched for pulsations in a broad range around the
extrapolated spin frequency (f=9.03845–9.03865 Hz)after
fixing the frequency derivative f
to the LAT-measured value of
−1.383 ×10
−11
s
−2
. We detected significant pulsations at
f=9.03863 Hz on MJD 59328 with an Hstatistic of H=33,
corresponding to p=4×10
−5
(Figure 2top)after considering
a trial factor of 22 (i.e., the number of independent frequency
bins). The significance increases slightly (H=37)in the
3–79 keV band. Due to the harder X-ray emission of the pulsar
(relative to other X-ray sources including the PWN), we found
a more significant detection of the pulsation in higher-energy
bands, e.g., 9–30 keV and 9–79 keV, with H=45 and H=54,
respectively. The latter corresponds to p=9×10
−9
(red curve
in Figure 2top).
The pulse profile (Figure 2bottom)is measured well with
the NuSTAR data—a sharp spike and a broad bump with a
phase separation of ∼0.5 are apparent. Note that the spike
(f=0.05)was seen, but the broad bump (f∼0.5)was not
detected well in the previous XMM-Newton profile (e.g., Kim
&An2020a). The NuSTAR profile allows a selection of on-
(Δf=0–0.11 and 0.42–0.63)and off-pulse phase intervals for
pulsar and PWN studies, respectively. Small features are also
visible in the profile (e.g., f∼0.2). However, they disappear
when using different energy bands or regions, and therefore
they are likely caused by statistical fluctuations.
2.3. Assessment of the Point Source Contamination
To accurately measure the spectrum of the PWN in the broad
X-ray band with the Chandra, XMM-Newton and NuSTAR
data, we need to adequately account for the contamination by
point sources in the PWN. Although the pulsar contamination
can be reduced in phase-resolved NuSTAR spectra by selecting
the off-pulse phases, constant (off-pulse)pulsar emission may
still be present. Furthermore, other X-ray sources, as seen in the
Chandra and the XMM-Newton images (Figure 1), may affect
the NuSTAR spectral analysis. Here, we assessed contamina-
tion by the point sources in the NuSTAR’s PWN spectrum and
then characterized the broadband X-ray spectra of the PWN
(Section 2.5).
Figure 2. Top: H-test results in the 3–30 keV (black)and the 9–79 keV band
(red). Bottom: A 3–30 keV pulse profile constructed by folding the source
events within R=30″circles (FPMA and FPMB combined)on the best
frequency of 9.03863 Hz. A folded background light curve is presented in red,
and blue vertical lines show on- and off-pulse intervals.
6
https://www.cosMOS.esa.int/web/xmm-newton/sas-thread-epic-
filterbackground
7
https://www.slac.stanford.edu/~kerrm/fermi_pulsar_timing/
3
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

2.3.1. Contamination by the Off-pulse Emission of J1418
X-ray spectra of the pulsar in the Chandra and XMM-
Newton band (0.5–10 keV)were fit to a power-law model with
Γ≈1 and Γ≈1.5 for the pulsed and total emission,
respectively, in our previous XMM-Newton study (Kim &
An 2020a). We also determined the on-pulse spectrum of the
pulsar using the NuSTAR data. We extracted photon events
within an R=30″circle in the on- and off-pulse data for the
source and the background spectra, respectively. Response files
were computed with the nuproducttool. After grouping the
spectrum to have a minimum of 30 events per bin, we fit the
3–30 keV spectrum with an absorbed power-law model in
XSPEC v12.12, holding the hydrogen column density fixed at
N
H
=2.78 ×10
22
cm
−2
(see Section 2.5.2). It should be noted
that we used the tbabsmodel along with the vern cross
section (Verner et al. 1996)and angr abundance (Anders &
Grevesse 1989)for the Galactic absorption throughout this
paper, to compare with the previous Suzaku study of Rabbit
(Kishishita et al. 2012, see Section 5.1).
8
The best-fit
parameters are Γ=0.94 ±0.33 and 2–10 keV flux of
F2.61 10 erg cm s
210keV 0.69
0.94 13 2 1
=´
-
+---
–. The latter corre-
sponds to 8.36 10 erg cm s
2.21
3.00 14 2 1
´
-
+---
when averaged over
a spin cycle.
We next estimated the off-pulse emission of the pulsar by
analyzing the Chandra data. We extracted a source spectrum
using an R=2″circular region centered at the pulsar position,
and a background spectrum was extracted from an R=21–5″
annular region around the pulsar. Response files were
generated with the specextract tool. The Chandra
spectrum was grouped to have a minimum of five events per
bin, and we used lstat
9
(Loredo 1992)in XSPEC. We fit
the spectrum with an absorbed power-law model and
reproduced the earlier results for the pulsar’s total (pulsed +
off-pulse)emission (Kim & An 2020a). To estimate the
off-pulse spectrum, we fit the Chandra data with a blackbody
plus power-law or a double power-law model with the
parameters of the second model component (power law)held
fixed at the best-fit values obtained for the pulsed spectrum in
the NuSTAR analysis above. Both provided an acceptable fit,
but the blackbody plus power-law model yielded an unreason-
ably high kT =0.66 ±0.14 keV (χ
2
/dof =28/43)for thermal
emission from isolated neutron stars. The double power-law
fit resulted in Γ=2.21 ±0.51 and F2.98
210keV 0.80
1.10
=´
-
+
–
10 erg cm s
14 2 1---
(χ
2
/dof =27/43)for the first power-law
component (with frozen N
H
=2.78 ×10
22
cm
−2
). Additional
uncertainties due to the on-pulse spectral model were estimated
to be 0.57
0.81
D
G=-
+and F10 erg cm s
210keV 1.31
1.71 14 2 1
D
=´
-
+---
–.
Although the large error bars associated with the pulsed
spectral parameters and cross-calibration uncertainties of
Chandra and NuSTAR (Madsen et al. 2015)do not allow a
precise estimation of the off-pulse emission, its soft and faint
spectrum seems not to significantly contaminate the NuSTAR
PWN spectrum at >3 keV (see Table 1). We further investigate
below (Section 2.5.2)any contamination by this off-pulse
emission in the NuSTAR analysis.
2.3.2. Contamination by Other Sources
In the high-resolution Chandra data, we identified several
contaminating sources within the PWN (e.g., S1–S8 within the
R
3=
¢
circle shown in Figure 1(a)). In addition, another
variable source (X1)in the north of S1 appeared only in the
XMM-Newton data (Figure 1(b)) as reported in our previous
study (Kim & An 2020a). While S1–S8 are quite faint, X1 was
brighter than the pulsar in the XMM-Newton observation.
However, X1 is highly variable; it was very faint or undetected
in other Chandra/XMM-Newton and our NuSTAR observa-
tions (Section 2.4.1). It should also be noted that there is
another bright source (X2 in Figure 1(b)) at 3
¢
east of J1418.
This source was also seen in the Suzaku data and can affect our
spectral analysis of
R
3>
¢
regions.
We first measured an X-ray spectrum of X1 using the XMM-
Newton data. We extracted the source spectrum within a
R=10″circle and grouped it to have at least 30 counts per
spectral bin. A background spectrum was extracted from an
R=20″circle within the PWN in order to properly account for
the PWN background within the source region. Response files
for the point source were produced with the rmfgen and
arfgen tasks of SAS. We fit the XMM-Newton EPIC spectra
with an absorbed power-law model. The fit was acceptable,
with χ
2
/dof =45/40 (p=0.26), yielding the best-fit para-
meters of N
H
=(1.70 ±0.32)×10
22
cm
−2
,Γ=1.02 ±0.17,
and F3.27 10 erg cm s
2 10keV 0.19
0.20 13 2 1
=´
-
+---
–(Table 1).A
single blackbody model fit was also acceptable with
kT =1.72 ±0.12 keV and N
H
=(0.37 ±0.20)×10
22
cm
−2
(χ
2
/dof =48/40)without requiring an additional component
with an F-test probability of ≈0.2. Below, we assumed the
power-law spectrum for X1 as a conservative estimate for the
hard X-ray band contamination (Section 2.5.2).
X2 was detected in both the XMM-Newton and Chandra
data. For the XMM-Newton data analysis, we extracted source
and background spectra using R=16″and R=32″circles,
respectively. We grouped the source spectrum to have at least
30 events per bin, fit the spectrum with a power-law model,
and inferred the best-fit parameters of N
H
=(4.52 ±0.40)×
10
21
cm
−2
,Γ=3.05 ±0.13, and F4.98
210keV 0.40
0.44
=´
-
+
()
–
10 erg cm s
14 2 1---
. We also analyzed the Chandra data
using an R=3″circle and an R=31–6″annulus for the
source and background spectrum, respectively. A power-law fit
to the Chandra spectrum resulted in N
H
=(2.55 ±0.74)×
10
21
cm
22
,Γ=2.35 ±0.16, and F8.86
210keV 0.79
0.87
=´
-
+
()
–
10 erg cm s
14 2 1---
(Table 1). The Chandra and XMM-Newton
results are significantly different, meaning that this source is
variable. The source is outside the
R
3=
¢
circle, and its
emission is weak and spectrally soft. Therefore, its influence on
the NuSTAR analysis should not be substantial.
We analyzed Chandra spectra for S1–S8 extracted from
R=2″circles around the source positions, collecting 20–60
counts for each source. Although their spectral parameters (N
H
and Γ)are not constrained well, due to the paucity of counts,
contamination from these faint sources to the NuSTAR PWN
spectra above 3 keV can be ignored. We verify this in
Section 2.5.2 using the Chandra data.
2.4. Image Analysis
2.4.1. Pulsar Image in the NuSTAR Data
We first inspected the NuSTAR image of the central region.
This is particularly important because the variable source X1,
8
Using the newer wilm abundance table, we obtain a larger N
H
of
(4.14 ±0.18)×10
22
cm
−2
for the
R
3=
¢
PWN (Section 2.5.2), but the other
parameter values did not change significantly.
9
https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/
XSappendixStatistics.html
4
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

which had a hard spectrum (see Section 2.3.2)and was brighter
than the pulsar in the XMM-Newton data, is only ∼20″west of
the pulsar (Figure 1(b)) and might have been bright during the
NuSTAR observation, contaminating the PWN emission. The
detection of the pulsations from J1418 (Section 2.2)in the
NuSTAR data already suggests that X1 was not bright during
the NuSTAR observation.
In order to identify hard X-ray point sources (e.g., pulsar and
X1), we inspected on- and off-pulse NuSTAR images in the
10–30 keV band and clearly detected only one point source in
each of FPMA and FPMB in the on-pulse intervals. We aligned
FPMA and FPMB images using the point-source positions and
produced a 10–30 keV “on−off”image of the source (Figure 3
left). Assuming that X1 is as bright as it was in the XMM-
Newton observation, we investigated the effect of X1 using an
image simulation. We estimated 10–30 keV count rates of
J1418 and X1 using the measured spectra (Table 1), and
simulated a NuSTAR image by convolving the two point
sources (offset by 21″with each other)with the NuSTAR PSF.
A simulated image is displayed in the right panel of Figure 3.
Our simulation shows that X1 should have been resolved from
the pulsar in the NuSTAR images, noting that NuSTAR’s
FWHM is 18″(e.g., An et al. 2014b). Comparing the observed
(left)and the simulated (right)images suggests that X1 was not
bright during the NuSTAR observation. We further investigate
the impact of (fainter)X1 emission in the spectral analysis
below (Section 2.5.2).
2.4.2. NuSTAR PWN Image Analysis
We used the off-pulse NuSTAR data to produce a PWN
image in the 3–20 keV band because the background dominates
over the PWN emission above 20 keV. The NuSTAR back-
ground is dominated by a nonuniform aperture component that
produces chip-to-chip variation. We simulated background
images using the nuskybgd
10
tool to account for the aperture
background component. While the simulated images reflect
well the aperture and the detector background components, we
found that the simulated counts for each chip differ from the
observed ones by 16%, 16%, and −24% for detector chips 1, 2,
and 3, respectively. We renormalized the simulated background
level to match the observed background counts by adjusting the
usernorm parameters for the aperture, CXB, and GRXE
components in nuskybgd; after this process, the differences
between the observational and simulated counts in the back-
ground regions were 1%. Then the simulated backgrounds
were subtracted from the observed images. The resulting
3–20 keV image (FPMA and FPMB combined)is displayed in
Figure 1(c). The NuSTAR image also clearly shows extended
X-ray emission, which is well contained within an
R
3=
¢
circle.
Given the broadband NuSTAR data, we investigated whether
the PWN size varies with energy. To capture the southwest
extension, we rotated the combined NuSTAR image by 38°from
west to north, with the origin at the pulsar position (Figure 1(c)),
and defined the horizontal and vertical directions as the x- and y-
axes, respectively. We then selected two energy bands in which
the numbers of source counts are approximately the same. We
adopted a 12 10¢´
¢
box and projected the box image onto the x-
and y-axes. The projected profiles are displayed in Figure 4.The
low- and high-energy profiles projected onto the x-axis display a
small difference; the latter appears to be slightly narrower than the
former (Figure 4top), possibly indicating an energy-dependent
size shrinkage in the southwest (tail)direction. However, the
Table 1
Power-law Fit Results for Point Sources in the PWN and for the PWN
Data Instrument
a
Energy Range N
H
ΓF
2–10keV
χ
2
/dof
(keV)(10
22
cm
−2
)(10
−13
erg s
−1
cm
−2
)
PSR pulsed NuSTAR 3–30 2.78
b
0.94 ±0.33
2
.61 0.69
0.9
4
-
+8/14
PSR off-pulse CXO+NuSTAR 0.5–10 2.78
b
2.21 ±0.51 0.30 0.0
8
0.11
-
+27/43
X1 XMM 0.5–10 1.70 ±0.32 1.02 ±0.17
3
.27 0.19
0.2
0
-
+45/40
X2 XMM 0.5–10 0.45 ±0.04 3.05 ±0.13 0.50 ±0.04 71/63
X2 CXO 0.5–10 0.26 ±0.07 2.35 ±0.16 0.89 0.08
0.09
-
+24/23
PWN
c
XMM 0.5–10 3.11 ±0.38 ±0.17
d
2.16 ±0.14 ±0.07
d
28.9 ±1.0 ±0.4
d
324/324
PWN
c
CXO 0.5–10 2.45 ±0.19 ±0.16
d
1.78 ±0.12 ±0.05
d
3
5.4 1.0 1.9
2.
0
-
+
d
137/164
PWN
c
NuSTAR 5–20 2.78
b
2.05 ±0.08 ±0.04
d
41.2 ±2.3 ±1.3
d
111/128
PWN
c
CXO+XMM+NuSTAR 0.5–20 2.78 ±0.12 2.02 ±0.05 34.8 ±0.08 688/642
Notes.
a
CXO: Chandra, XMM: XMM-Newton.
b
Fixed at the value measured from a joint fit of Chandra, XMM-Newton, and NuSTAR data of the PWN.
c
Measured within an
R
3=
¢
circular region.
d
Second errors are systematic uncertainties estimated by varying background regions.
Figure 3. An observed “on−off”image (left)and a simulated image of J1418
+X1 (right). Left: An exposure-scaled off-pulse image is subtracted from an
on-pulse one in the 10–30 keV band. Right: A simulated image of the pulsar
(center)and X1 (west)assuming that X1 was as bright as was seen in the
XMM-Newton data. X1 is placed at ∼21″northwest of J1418 (as observed by
XMM-Newton).R=30″white circles are shown for reference, the figures are
smoothed, and scales are adjusted for better legibility.
10
https://github.com/NuSTAR/nuskybgd
5
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

difference is not statistically significant, and we conclude that
X-ray spectral softening is not observed in the Rabbit PWN with
the current observation.
2.4.3. XMM-Newton and Chandra PWN Images on Large Scales
To better identify the extended emission of the PWN, we
produced images on large scales using the XMM-Newton and
Chandra data. For this, we used filter wheel closed (FWC)data
and blank-sky data for the XMM-Newton and Chandra
analysis, respectively, to adequately remove the particle-
induced background.
For the XMM-Newton data analysis, we generated the filter
wheel closed (FWC)data using the evqpb task of SAS. The
FWC image appeared similar to the observed one in source-free
regions, but some (energy-dependent)differences were notice-
able, especially at low energies. In addition, the exposure-time
ratio of the observational and FWC data was different from the
measured count ratios in the source-free regions, perhaps
because of the temporal variation of particle flares. To derive a
normalization factor appropriate for the measured count ratios,
we compared the observational and FWC data taken outside the
field of view (FoV). The spectral shapes of the observational
and FWC data agreed well at 1–2 keV. By comparing the
>2 keV spectra, we derived normalization factors of 0.45 and
0.33 for MOS1 and MOS2, respectively, which are smaller
than the exposure ratio of 0.48 (see also Section 2.5). We then
generated images of the full FoV, subtracted the FWC image
from the observed one, and divided the FWC-subtracted image
by the exposure map. An MOS1+MOS2 image in the 2–8 keV
band is displayed in Figure 1(b). The source is extended
primarily in the NE–SW direction, and the bright tail emission
is contained within an
R
3~
¢
region. There appears some
emission slightly outside the
R
5=
¢
circle (north and north-
west). While this may be the PWN emission, we found that the
brightness profiles in the north and northwest directions do not
follow a monotonic trend; the brightness decreases to 4′–5′and
then increases at 
R
5
¢
. This outer emission might have been
produced by some other sources or by imperfect subtraction of
the FWC background near the chip boundaries (e.g., Kuntz &
Snowden 2008).
We adopted the above procedure for the Chandra data
analysis using the blank-sky data. We generated the blank-sky
events with the blanksky task of CIAO. For the Chandra
data, we found that the exposure-time ratio adequately explains
the measured count ratios of the observational and blank-sky
data in low-brightness regions of the former. We produced
observational and blank-sky images in the 2–7 keV band,
subtracted the latter from the former, and divided the resulting
image by the exposure map. The image is displayed in Figure 1
(a). The Chandra image resembles the XMM-Newton one, and
we found that the brightness in the north and northwest
directions show a non-monotonic trend as was seen in the
XMM-Newton image.
2.5. Spectral Analysis
In this section, we measure the spatially integrated and
resolved spectra of the Rabbit PWN, taking into account the
point-source contamination.
2.5.1. Background Spectra for the Extended PWN
Because the source emission extends to large distances from
J1418 (Figure 1), we need to extract background spectra from
source-free regions far away from the PWN. In this case, the
detector and particle-induced backgrounds may not represent
well the source-region background. To mitigate this, we used
the FWC and blank-sky data for the XMM-Newton and
Chandra analysis, respectively, as was done for the image
analysis. For NuSTAR data analysis, we employed the
nuskybgd simulations (Wik et al. 2014).
For the XMM-Newton data, we selected background regions
at 7>
¢
and collected spectra from the observational and FWC
data. These spectra showed a prominent instrumental line at
∼1.5 keV. We compared the continuum-subtracted lines
measured in the observational and FWC data, and we verified
the normalization factors obtained by comparing the out-of-
FoV continuum spectra (Section 2.4.3). We subtracted the
FWC spectrum from the observed one, to remove the particle-
induced background, and produced a “sky background”
spectrum. We did the same for the source-region spectra,
using the same normalization factors obtained above, and
constructed the source spectra. The “sky background”spectrum
collected far away from the source region would be an
underestimation of the source-region background, due to the
optics vignetting effect. To correct for this, we multiplied the
background spectrum by the energy-dependent effective area
(i.e., ancillary response files; ARF)ratio of the source and
Figure 4. 3–7 keV and 7–20 keV NuSTAR profiles projected onto the x- and y-
axes defined in Figure 1(c).
6
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

background regions. In these processes, we carefully propa-
gated the counting uncertainties.
The XMM-Newton spectra constructed following the
aforementioned procedure exhibited small but noticeable
residuals around the instrumental line complex at 1.5–2 keV;
the residuals (e.g., dashed black and red lines in Figure 5top)
show a structure that is broader than individual instrumental
lines. This is possibly caused by temporal variation of the line
emissions and likely imperfect subtraction of the FWC
background (e.g., Kuntz & Snowden 2008). Therefore, we
added a broad Gaussian line along with diagonal response files
to our spectral model (Section 2.5.2–2.5.4); the typical central
line energy and width of the Gaussian were estimated to be
∼1.5 keV and ∼0.15 keV, respectively.
A similar procedure was applied to the Chandra analysis
with the blank-sky data. We constructed the source and
background spectra within the PWN (see below)and the
source-free regions of the observation data, respectively. We
then produced the blank-sky spectra within the same source
and background regions and subtracted the blank-sky emis-
sions from the observed source and background spectra. We
checked that the blank-sky-subtracted spectra did not show any
noticeable instrumental line. We then scaled the background
spectra by the effective area (ARF)ratio of the source and
background regions. For the NuSTAR data analysis, we used
the off-pulse data to construct the source spectra and performed
nuskybgd simulations using annular regions at
R
5>
¢
from
J1418 to generate the background spectra appropriate for the
source regions (e.g., Wik et al. 2014).
Using this method, we measure the PWN spectra within a
few representative regions, namely
R
3=
¢
,
R
5=
¢
,and
R
7=
¢
circles. The latter two are to be compared with
previous Suzaku and ASCA results. We adopted an absorbed
power-law model for our spectral analyses and fit the data in
the 0.5–10 keV (XMM-Newton and Chandra)and 5–20 keV
bands (NuSTAR).
2.5.2. PWN Spectra of the Bright
R
3=
¢
Region
To measure the PWN spectra within the
R
3=
¢
region
(Figure 1), we excluded J1418, S1, S2, X1 (using an R=40″
circle),X2(slightly outside the
R
3=
¢
circle), and three faint
sources (S4, S5, and S8; R=16″circles)from the XMM-
Newton data (Figure 1(a)and (b)). We excised nine point
sources (S1–S8 and J1418; R=2″circles)from the Chandra
data but did not excise any region from the NuSTAR off-pulse
data. We verified that the contamination by S1–S8 affects only
the estimation of the flux (∼4%), not the spectral slope (Γ),by
comparing the Chandra spectra measured with and without
S1–S8.
We first fit the XMM-Newton, Chandra, and NuSTAR
spectra separately after grouping them to have at least 100, 50,
and 50 counts per spectral bin, respectively. From the XMM-
Newton data, we inferred the best-fit parameters of N
H
=
(3.11 ±0.38)×10
22
cm
−2
,Γ=2.16 ±0.14, and F
2–10 keV
=
(2.89 ±0.10)×10
−12
erg cm
−2
s
−1
. The fit was formally
unacceptable with χ
2
/dof of 435/347, and a residual trend at
low energies (<1 keV)was noticeable. We verified that
ignoring the low-energy data improved the fit(χ
2
/dof =
324/324)without altering the best-fit parameter values
significantly. Note that the best-fit values change depending
on the background-region selection by ΔN
H
=±0.17 ×
10
22
cm
−2
,ΔΓ =±0.07, and thus F210keV 4.25
4.32
D
=´
-
+
–
10 erg cm s
14 2 1---
(standard deviations). The changes of N
H
and Γare due to their covariance, and we find ΔΓ =±0.02 for
frozen N
H
(=2.78 ×10
22
cm
−2
).
The best-fit parameters obtained from the Chandra data
are N
H
=(2.45 ±0.19)×10
22
cm
−2
,Γ=1.78 ±0.12, and
F
2–10keV
=(3.54 ±0.10)×10
−12
erg cm
−2
s
−1
(χ
2
/dof =137/
164). The parameter values vary depending on the background
selection by ΔN
H
=±0.16 ×10
22
cm
−2
,ΔΓ =±0.05, and
F10 erg cm s
210keV 1.88
1.98 13 2 1
D
=´
-
+---
–(standard devia-
tions).ΔΓ was estimated to be ±0.05 for frozen N
H
.The
XMM-Newton and Chandra results are discrepant because of
different excision regions and parameter covariance (N
H
versus
Γ);byfreezingN
H
to a common value, we found consistent Γin
the XMM-Newton and Chandra fits.
We used the nuskybgd simulations for the NuSTAR data
analysis. Because the simulations may be less accurate below
5 keV
11
and background dominates above 20 keV, we fit the
NuSTAR spectra in the 5–20 keV band. The best-fit parameters
were estimated to be Γ=2.05 ±0.08 and F
2–10keV
=(4.12 ±
0.23)×10
−12
erg cm
−2
s
−1
for frozen N
H
=2.78 ×10
22
cm
−2
(χ
2
/dof =111/128). To assess the systematic effects of the
background selection for the nuskybgd simulations, we tried
different background regions. In this case, the best-fit parameter
values change by ΔΓ =±0.04 and ΔF
2–10keV
=±1.3 ×
10
−13
erg cm
−2
s
−1
. We cross-checked these results by an
analysis performed with “in-flight”backgrounds. The best-fitΓ
value did not alter significantly, but the flux value varied by
∼±10% depending on the background region. As noted
above, the NuSTAR-measured flux includes contamination
from S1 to S8.
Additional uncertainties can be introduced in the NuSTAR
analysis by contamination from the pulsar (off-pulse)and X1.
We assessed their effects by simultaneously modeling the
emission in the NuSTAR off-pulse data fits. We first varied the
best-fit parameters of the pulsar’s off-pulse spectrum
Figure 5. X-ray spectra of the PWN (crosses; 1–20 keV)extracted within an
R
3=
¢
circle, and those of the pulsar (empty circles; 1–30 keV). The best-fit
instrumental Gaussian lines for the PWN spectra measured by XMM-Newton
are presented in dashed black (MOS1)and red (MOS2)lines in the top panel.
The pulsar spectrum measured by Chandra (cyan circles in the top panel)
includes both the pulsed and off-pulse components, and those measured by
NuSTAR are only for the pulsed component (blue and green circles). The
bottom panel shows residuals after subtracting the best-fit power-law and
instrumental Gaussian models.
11
https://github.com/NuSTAR/nuskybgd
7
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

(Section 2.3)within their 68% confidence intervals, consider-
ing the uncertainties in the on-pulse spectral model and
the covariance between Γand F
2–10 keV
. We then fit the
PWN spectra with two power laws, holding the parameters
for the second power law (pulsar emission)at the varied
values, and found that the influence of the pulsar emission
is negligible (e.g., ΔΓ =+0.03 and ΔF
2–10 keV
=−3.33 ×
10
−14
erg cm
−2
s
−1
). We did the same for the X1 emission. The
variable source was faint, but its brightness was not measured
well at the NuSTAR epoch. Hence, we assumed that X1 was
30% as bright as it was at the XMM-Newton epoch. In these
cases, ΔΓ and ΔF
2–10 keV
were estimated to be +0.02 and
−2.5%, respectively, values that are smaller than the statistical
uncertainties.
To characterize the
R
3=
¢
PWN emission better, we jointly
fit the XMM-Newton, Chandra, and NuSTAR spectra,
and inferred the best-fit parameters to be N
H
=(2.78 ±
0.11)×10
22
cm
−2
,Γ=2.02 ±0.05, and F
2–10keV
=(3.48 ±
0.08)×10
−12
erg cm
−2
s
−1
. The fit was acceptable with
χ
2
/dof of 688/642, but improved significantly
(χ
2
/dof =577/619)without altering the best-fit parameter
values when we ignored the low-energy (<1 keV)XMM-
Newton data (Figure 5left). The cross-normalization factors
(set to 1 for Chandra)were estimated to be 0.82 ±0.02
and 1.17 ±0.06 for MOS1 and FPMA, respectively.
The central R=40″region excised from the XMM-Newton
data has a spectrum that is characterized well by a power
law with Γ=1.96 ±0.08 and F
2–10 keV
=(6.79 ±0.27)×
10
−13
erg cm
−2
s
−1
(χ
2
/dof =46/54), containing ∼20% flux
of the
R
3=
¢
PWN. Noting that larger regions were excised
from the XMM-Newton data and the NuSTAR data included
point sources (S1–S8, the pulsar’s off-pulse and putative X1
emission), the estimated normalization factors seem reasonable.
Additionally, imperfect cross-calibration of the instruments
would introduce some uncertainties (Madsen et al. 2015);
differences in the flux calibration would be included in the
cross-normalization factors, and differences in the spectral-
slope calibration would affect the fit-inferred N
H
. In the spectral
analyses below (Sections 2.5.3 and 2.5.4), we held N
H
fixed at
2.78 ×10
22
cm
−2
.
2.5.3. Spectra of the More Extended Diffuse Regions
We next characterize the source emission within larger
regions (
R
5=
¢
and
R
7=
¢
circles)using the XMM-Newton
and Chandra data to compare with previous ASCA and Suzaku
measurements (Roberts et al. 2001; Kishishita et al. 2012).We
excised J1418, S2 and X1 (central 40″circle),X2(24″), and
other point sources (e.g., S4, S5, and S8 using 16″circles)from
the XMM-Newton data. We removed the point sources (e.g.,
X1, X2, and S1–S8)from the Chandra analysis using R=2″
apertures. We collected events within
R
5=
¢
to construct the
source spectra, grouped them to have >100 events per bin, and
jointly fit the Chandra and XMM-Newton data. The best-fit
parameters were inferred to be Γ=2.11 ±0.05 and
F
2–10keV
=(5.72 ±0.18)×10
−12
erg cm
−2
s
−2
. The MOS1
cross-normalization factor was measured to be 0.78 ±0.03
with respect to that of Chandra; this can be ascribed to the
larger excision apertures (e.g., the central R=40″)used for the
XMM-Newton analysis. The 2–10 keV flux we measured is
still lower by 9% than the previous Suzaku measurement of
F
2–10keV
=(6.27 ±0.13)×10
−12
erg cm
−2
s
−2
. We suspect
that this difference is due to the inclusion of point sources in
the Suzaku analysis. Indeed, by using all the emissions within
the aperture except for X2, as was done for the Suzaku data
analysis of Kishishita et al. (2012), we found Γ=2.02 ±0.05
and F
2–10keV
=(6.14 ±0.18)×10
−12
erg cm
−2
s
−2
. The sys-
tematic uncertainties on the flux estimations due to the
background selection are 2% and 9% for XMM-Newton and
Chandra, respectively, and our results agree with the Suzaku
one at <1σlevels.
We also measured the larger
R
7=
¢
region spectrum to
compare with the previous ASCA result (Roberts et al.
2001). For the comparison, we included all sources within
the aperture as was done for the ASCA data analysis. The
XMM-Newton+Chandra spectra were fit with a power
law having Γ=2.35 ±0.05 and F
2–10keV
=(7.39 ±0.24)×
10
−12
erg cm
−2
s
−1
for frozen N
H
=2.78 ×10
22
cm
−2
.The
flux value is consistent with the F
2–10keV
=(7.33 ±0.17)×
10
−12
erg cm
−2
s
−1
measured by ASCA. It should be noted that
the exact region size for the ASCA flux was not reported and the
ASCA-inferred N
H
was substantially smaller.
2.5.4. Spatially Resolved Spectrum of the PWN
We searched for any spatial variation of the X-ray spectrum
due to the synchrotron burn-off effect, even though the
NuSTAR imaging analysis did not show strong evidence for it
(Section 2.4.2).
For the XMM-Newton data, we used 11 annular regions
within
R
5=
¢
. These regions were selected to have different
sizes depending on the brightness. After excising the central
40″and point sources, we constructed a spectrum for each
region and grouped the spectrum to have at least 50 counts per
spectral bin. We jointly fit the 11 spectra with an absorbed
power law. The other parameters were optimized separately for
each region (i.e., untied Γ). We also assessed systematic
uncertainties due to background selection and added them in
quadrature to the statistical errors. The results are displayed in
Figure 6. The brightness monotonically decreases with
increasing radius. Γdoes not show a monotonic trend out to
∼35, and then it slowly increases. The fit was formally
unacceptable with χ
2
/dof of 1751/1494 (p≈4×10
−6
), but
ignoring low-energy data (<1 keV)improved the fit
(χ
2
/dof =1452/1366 and p=0.05)without altering the best-
fit parameter values. We compared the above results with ones
obtained by a model with tied Γfor the 11 spectra, and we
found that the untied Γmodel provides a better fit with an F-
test probability of 3 ×10
−3
.
In our further investigation, we found that the large χ
2
was
caused mostly by the low-energy spectra of outer regions. In
addition, the cross-normalization factors between MOS1 and
MOS2 significantly deviated from 1 in outer regions; the
factors are consistent with 1 out to
R
3.5Ȣ
, but they are
1.30 ±0.13 and 1.69 ±0.22 in the outermost two regions (Reg.
11 in Figure 6left). These are probably because the faint
emission in the R=35–
5¢
regions is strongly affected by the
background and thus cannot be measured reliably. Further-
more, these regions are near the chip boundary where the
particle-induced background is strong (Kuntz & Snow-
den 2008), and there seemed to be some contamination at
R
4>
¢
in the north and northwest directions (Section 2.4.3).
Hence, we do not use the last two data points (
R
3.5¢)in
Figure 6(right)for our modeling below. In this case, the untied
Γmodel was not significantly favored over the tied Γone with
an F-test probability of 17%.
8
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

We found similar results with the Chandra data (Figure 6
right). The Chandra results agree well with the XMM-Newton
ones but have larger uncertainties. In the outermost region,
Chandra data implied a smaller Γ, but it cannot be
discriminated from the XMM-Newton results due to the large
uncertainty; the faint emission spread over a large region was
difficult to precisely characterize.
We also analyzed NuSTAR’s off-pulse data to measure
spatial variation of the spectrum within an
R
3=
¢
region
(within the FoV). We extracted spectra using a R=30″circle
and two annular regions having widths of 60″and 90″.We
grouped the NuSTAR spectra to have a minimum of 30 counts
per bin and fit the 5–20 keV spectra with a power-law model
having separate Γfor each spectrum. The fit was acceptable
with χ
2
/dof =221/214, and the results are presented in
Figure 6. The NuSTAR results broadly agree with the XMM-
Newton and Chandra ones, but some difference is noticeable
(e.g., the bottom right panel of Figure 6)perhaps because of
contamination from the point sources and cross-calibration
issues. We also tried to fit the data with a model having a
common photon index, and we found that the model provided
an equally good fit(χ
2
/dof =224/216)with the best-fit photon
index of 2.03 ±0.07. An F-test comparison of the tied-Γand
untied-Γmodels gives p∼0.24, implying insignificant spectral
softening.
It should be noted that our results are discrepant with the
significant spectral softening measured by Suzaku (Kishishita
et al. 2012). We speculated that this is because the pulsar (and
X1)emission was not removed in that analysis. To confirm, we
analyzed the XMM-Newton, Chandra, and NuSTAR data using
the same annular regions as the Suzaku ones without excising
any point source. Spectral softening trends, similar to those in
the Suzaku measurement, were measured in our analysis. In the
innermost zone (
R
1<
¢
), the Chandra and NuSTAR spectra
were measured to be consistent with the Suzaku spectrum
(Γ≈1.8)while the XMM-Newton spectrum is significantly
harder (Γ≈1.6); this is likely caused by contamination from
X1. In outer zones, the photon indices were measured to be
consistent with the Suzaku results (Γ=2.1–2.2). We note
again that, in the outermost zone (
R
3>
¢
), the cross-normal-
ization factor between MOS1 and MOS2 significantly deviated
from 1.
3. Fermi-LAT Data Analysis
We analyzed gamma-ray data taken with the Fermi LAT. We
extracted 100 MeV–1 TeV events acquired between 2008
August 4 and 2022 February 10 spanning approximately
13.5 yr. The data were analyzed with Fermipy v1.0.1 (Wood
et al. 2017)along with the P8R3_SOURCE_V3 instrument
response.
12
We selected the Front+Back event type in the
SOURCE class within a 10°×10°square region of interest
(RoI)centered at 4FGL J1417.7−6057 (LAT counterpart of
Rabbit; J1417 hereafter)and reduced the data using the zenith
angle <90°, DATA_QUAL>0, and LAT_CONFIG=1. We
further analyzed the data as described below.
We performed a binned likelihood analysis using 0°. 05 bins
in the 100 MeV–1 TeV band to measure the source spectrum.
Because our data are not very different from those used for the
4FGL DR-2 catalog (gll_psc_v27.fit; Abdollahi et al.
2020), the 4FGL values are expected to be accurate. Never-
theless, we verified the parameter values below. We created an
XML model including all the 4FGL DR-2 sources within a
square of 30°×30°using the parameters given in the catalog.
We started by optimizing parameters for J1417 and normal-
izations for the diffuse emissions (gll_iem_v07 and
iso_P8R3_SOURCE_V3_v1).
13
We then gradually increased
the number of sources to fit until no excess or deficit was
identifiable in the residual plot. Our optimized parameters were
fully consistent with the 4FGL DR-2 values.
We next inspected a LAT image in the high-energy band
(>30 GeV)where the 4FGL DR-2 catalog found significant
emission (i.e., J1417). We produced a count map in the
>30 GeV band and found that there were excess counts in the
R∼0°. 1 region of the H.E.S.S counterpart (HESS J1418−609;
Figure 7top). To verify the excess, we generated a TS map of
the region. We removed J1417 from our optimized model and
ran the tsmap tool of Fermipy to generate a >30 GeV TS
map (TS map), which is displayed in Figure 7bottom. The TS
map showed significant emission in the same region as that of
the count map. Most of the excess in the TS map was resolved
when J1417 was included in the model. However, it was not
Figure 6. Results of the spatially resolved spectral analysis. Left: XMM-Newton spectra of the 11 annular regions. The spectrum of the innermost region (reg. 1)is
shown in black, and the spectra of the outer regions are displaced for visibility. The MOS1 and MOS2 spectra are displayed as solid and dashed lines, respectively.
Right: radial profiles of brightness (top)and photon index (bottom).
12
https://fermi.gsfc.nasa.gov/ssc
13
https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html
9
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

possible to accurately determine the position or the extension
with the data, due to poor photon statistics and/or broad LAT
PSF (i.e., 68% containment radius of 0°.1 at >30 GeV).
14
4. SED Modeling
4.1. Construction of a Broadband SED and Radial Profiles of
X-Ray Properties
We constructed a broadband SED of the Rabbit PWN by
adding the published radio and VHE measurements to our
X-ray and LAT data. The radio flux densities measured for
“Rabbit”with ATCA observations were taken from Roberts
et al. (1999), and we used the VHE results for HESS J1418
−609 reported by Aharonian et al. (2006). Note that the X-ray
and VHE fluxes are measured from different regions. We used
an
R
3=
¢
region for the X-ray SED, but the VHE SED was
measured from a larger region (e.g., H.E.S.S. size). In our
modeling, we integrate the model emission over the sizes
appropriate for the X-ray and VHE SEDs and compare it with
the observed data. The radio flux-density measurements may
not be accurate, due to possible contamination from the
Kookaburra complex and incomplete UV coverage. For SED
modeling, we took the radio data at face value, although model
conclusions are not strongly influenced by those constraints.
4.2. Multizone Emission Model for the PWN
We describe a multizone SED model (e.g., Kim &
An 2020b)that we use to infer the particle energies and the
flow properties in the Rabbit PWN by simultaneously fitting
the radial profiles and the broadband SED (Figure 8). The
observed morphology of the PWN is asymmetric, and thus our
model with the assumption of spherical or conical flow is only
approximate (see Section 5.2).
In the model, electrons characterized by a power-law energy
distribution
dN
ddt N,,1
e
e
peee0 ,min ,max
1
ggg gg=<<
-()
where γ
e
is the electron Lorentz factor, are injected into the
termination shock at R
TS
and flow in the PWN. The particle-
injection power
Emc
dN
ddt
d,2
eee
e
e,inj 2
e
e
,min
,max
òggg=
g
g
()
where m
e
is mass of an electron and cis the speed of light, is a
fraction of the pulsar’s luminosity (
E
Lt
e,inj
h=()
), which is
assumed to decrease with time following
Lt L t
1, 3
0
0
n
n
1
1
t
=+
-+
-
⎜⎟
⎛
⎝⎞
⎠
() ( )
where τ
0
=2τ
c
/(n−1)−t
age
and nis the braking index
(assumed to be 3; Gaensler & Slane 2006; Gelfand et al. 2009).
The properties within the radio/X-ray PWN are prescribed as
power laws:
Br B r
R4
0
TS
B
=
a
⎜⎟
⎛
⎝⎞
⎠
() ()
for the magnetic field,
Vr V
r
R5
flow 0
TS
V
=
a
⎜⎟
⎛
⎝⎞
⎠
() ()
for the bulk flow speed, and
DD B
G100 10 6
e
0
1
9
1
3
m
g
=
-
⎜⎟
⎛
⎝⎞
⎠⎛
⎝⎞
⎠()
for the diffusion. We assumed α
B
+α
V
=−1, which is valid
for spherical (or conical)flow and transverse Bwith magnetic
flux conservation. Note that this relation could be different for
other flow geometries or Bconfigurations (Reynolds 2009).In
our model, R
TS
and R
PWN
are constant in time, and thus Band
V
flow
are also constant in time. However, particles with
different ages experience different B, and V
flow
in our model,
as the particles are at different radial positions (see below).
Figure 7. A Fermi-LAT image (top)and a TS map (TS map; bottom)of a
∼1°×1°region at >30 GeV. The TS map was produced by omitting the
model for 4FGL J1417.7−6057. The ellipses show the Rabbit radio/X-ray
PWN, and the dashed circles denote the TeV source HESS J1418−609
associated with Rabbit. The crosses mark a few LAT point sources:
4FGL J1417.7−6057 (green), PSR J1420−6048 (white), and J1418 (cross
within the ellipses). The images are smoothed, and the scales are adjusted for
legibility.
14
https://www.slac.stanford.edu/exp/glast/groups/canda/lat_
Performance.htm
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Larger VHE emission regions compared to their radio/X-
ray emission zones observed in some PWNe (as in Rabbit)
indicate that particles outside the compact X-ray PWN can
produce VHE emission via ICS. To account for this, we
assumed that the flow bulk motion (V
flow
)carries particles and
Bonly out to the boundary of the X-ray PWN (at r=R
PWN
):
i.e., we set V
flow
(r>R
PWN
)=0 and assign a small value
for B(r>R
PWN
)≡B
ext
(e.g., 1.5 μG). As a result, such B
and V
flow
values cause a discontinuity at the outer boundary of
the X-ray PWN (e.g., B(R
PWN
)=5.7μG and V
flow
(R
PWN
)=
530 km s
−1
for the parameters in Table 2), and so we connected
the parameter values between the inside and outside regions,
using a rapidly decreasing function. We verified that the exact
functional form (e.g., step, logistic, or exponential function)
does not alter the resulting emission significantly as long as
R
PWN
covers the emission zones that we model.
R
PWN
is prescribed to represent the PWN region with intense
X-ray emission. Without a sharp edge in the X-ray image, it is
very difficult to determine the R
PWN
value observationally, but
at the same time, the value does not have a large influence on
the model as long as the “bright”X-ray emission zone is
included within R
PWN
(see Section 4.3). The sudden drop of B
is necessary (and sufficient)to match the size of bright X-ray
emission of PWNe with a sharp boundary (e.g., Crab)by
suppressing synchrotron emission at r>R
PWN
, but the
assumption of V
flow
(r>R
PWN
)=0 does not have a large
impact on the emission in our model. The particles can
propagate farther out (into the ISM)via diffusion and produce
ICS emission.
The particle flow was computed using Monte-Carlo simula-
tions (e.g., Tang & Chevalier 2012). At each time step (dt),
particles moved radially outward by V
flow
(r)dt, randomly
diffused by Ddt2in each direction, and cooled via adiabatic
Figure 8. Measurements and models for broadband SED and radial profiles of X-ray photon index and brightness. Panel (a)shows a broadband SED (data points)and
emission models. The green dotted line is the synchrotron emission model, and the blue, red, and purple dotted lines are models for ICS emission from CMB and IR
background in the PWN and in the ISM regions, respectively. The cyan solid line shows the summed model. Panel (b)shows electron distributions in the inner
(r<100″; blue), the outer (100″<r<270″; red), and the whole region (black). Panels (c)and (d)show radial profiles of the photon index (c)and brightness (d)in
the X-ray band.
Table 2
Parameters for the Multizone SED Model
Parameter Symbol Value
Spin-down power (today)L
SD
5×10
36
erg s
−1
Characteristic age of the pulsar τ
c
10,400 yr
Age of the PWN t
age
7000 yr
Size of the PWN R
pwn
4.6 pc
Radius of termination shock R
TS
0.1 pc
Distance to the PWN d3.5 kpc
Index for the particle distribution p
1
2.27
Minimum Lorentz factor e,min
g10
4.36
Maximum Lorentz factor e,max
g10
9.3
Magnetic field B
0
12.3μG
Magnetic field at r>R
PWN
B
ext
1.5μG
Magnetic index α
B
−0.2
Flow speed V
0
0.038c
Speed index α
V
−0.8
Diffusion coefficient D
0
1.1 ×10
27
cm
2
s
−1
Energy fraction injected into particles η0.88
Energy fraction injected into Bfield η
B
0.0036
Temperature of IR seeds T
IR
20 K
Energy density of IR seeds u
IR
2.2 eV cm
−3
CMB temperature T
CMB
2.7 K
CMB energy density u
CMB
0.26 eV cm
−3
11
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expansion, synchrotron radiation in randomly oriented B, and
ICS radiation off of the CMB (T
CMB
=2.7 K and
u
CMB
=0.26 eV cm
−3
)and ambient IR photons (T
IR
and u
IR
).
This process was repeated over the assumed age of the PWN
(t
age
). We assumed reflecting and transmitting boundary
conditions at the inner (r=R
TS
)and the outer boundaries
(r=R
PWN
)of the X-ray PWN, respectively. The synchrotron
and ICS emissions of the isotropic particles were computed
using the formulae given in Finke et al. (2008)at each time and
position. We then calculated integrated SED and radial profiles
by projecting the emission onto the tangent plane of the
observer.
In the model, there are many free parameters: e.g., R
TS
,
R
PWN
,p
1
,
e,min
g
,
e,max
g
,t
age
,B
0
,α
B
,V
0
,α
V
(=−1−α
B
),D
0
,
T
IR
, and u
IR
. Some of the parameters can be tightly constrained
by the observation data (e.g., R
PWN
if a sharp X-ray edge is
detected). The shapes of the synchrotron and ICS SEDs are
primarily controlled by the injected particle distribution (p
1
,
e,min
g
, and ;
e,max
g
Equation (1)),B, and V
flow
(Equations (4)and
(5)). The synchrotron cooling is dominant for the highest-
energy electrons (near
e,max
g
)and thus determines the
synchrotron SED shape in the hard X-ray band, whereas
adiabatic cooling is dominant for low-energy electrons and
therefore is relevant to the low-energy SED. The parameters for
the ambient IR field (T
IR
and u
IR
)were adjusted to match the
amplitude of the TeV SED, and V
0
(α
B
and α
V
)and D
0
are
adjusted to match the radial profiles of the X-ray brightness and
photon index.
For the parameter optimization, we iteratively adjusted the
model parameters until a good match between the model and
the measurements was achieved (visual inspection). We then
carried out pair scans of important parameters to refine our
parameter estimations, employing the χ
2
statistic (Figures 8
and 9). Because it is uncertain whether or not all of the radio
and TeV emissions are associated with the X-ray PWN, and the
radio and TeV data can be matched relatively easily by simply
adjusting
e,min
g
and u
IR
without altering the other important
parameters (e.g.,
e,max
g
and B
0
), we used only the X-ray
measurements for the fit. In principle, a change of
e,min
g
results
in a slight change in the total particle energy (η). Then, to keep
ηconstant while matching the X-ray data, we need to adjust B
0
,
but the required change of B
0
is small for a modest change of
e,min
g
. We note that a sharp cutoff in the electron energy
distribution below some γ
e
is not expected in most theories of
particle acceleration at shocks. More detailed radio observa-
tions of K3 at 1 GHz and below could be of significant value in
improving our understanding of PWN radio emission in
general. We further note that the radial profiles measured by
XMM-Newton, Chandra, and NuSTAR (Figure 6right)show
large scatter due to the cross-normalization issues
(Section 2.5.2), and thus matching them all with a model is
not possible. Therefore, we used the XMM-Newton measure-
ments of the radial profiles for the pair scans. Because the
broadband X-ray SED, especially the NuSTAR measurement,
is crucial for the estimation of
e,max
g
, we used all the spectral
measurements in Figure 5after normalizing the flux to the
XMM-Newton flux of the R
4
03< <
¢
region.
4.3. Application of the Model to the Rabbit PWN
In a previous study, a one-zone time-dependent model (Zhu
et al. 2018)was used to explain broadband SEDs of several
PWNe (including the Rabbit PWN). We used their model
parameters as a guide to our model input. However, the one-
zone modeling did not account for the spatial variation of the
X-ray spectra within the PWNe, and hence the parameters
needed to be modified in our multizone model. The true age of
the Rabbit PWN is unknown, but a correlation between the
X-ray-to-gamma-ray luminosity ratio and the age of a large
ensemble of PWNe (Kargaltsev et al. 2013; Zhu et al. 2018)
suggests several kyr for Rabbit’s true age. We assumed an age
of 7000 yr, R
TS
=0.1 pc, and R
PWN
=4.6 pc (i.e., an X-ray
emission region of 4.5Ȣ for an assumed d=3.5 kpc). Multi-
zone emission models computed with 10
5
temporal, 1000
spatial, and 10
4
energy bins are plotted in Figure 8, and the
model parameters are presented in Table 2.
The spectral shape of the synchrotron emission by uncooled
electrons (Figure 8(a)), which is relevant to p
1
, is not measured,
well, but the LAT SED suggests a hard power law for the
electron distribution and p
1
=2.27 adequately explains both the
X-ray and VHE SEDs (Figure 8). The similar radio and X-ray
sizes of the Rabbit PWN already suggest that Bis weak in the
source (as in G21.5−0.9; Matheson & Safi-Harb 2010). The
insignificant softening of the X-ray spectrum (Sections 2.4.2
and 2.5.4)further implies that diffusion is efficient in the
source. Our best-fitB
0
of 12 μG is similar to that inferred from
the one-zone modeling (4.4 μG; Zhu et al. 2018). For this B
0
,
the synchrotron cooling timescale for the X-ray emitting
electrons (γ
e
≈10
8
–10
9
)is 10
3
yr, and hence they cool
substantially over 7000 yr. However, because the efficient
diffusion compensates for the cooling, the particle spectral
variation with distance from the pulsar is not large (Figure 8
(b)). Alternatively, the insignificant spectral softening may be
explained by a pure advection model that predicts nearly
constant Γout to a certain radius and a rapidly increasing trend
at large radii in a 1D case (e.g., Reynolds 2003)if the cooling
break is above the observed X-ray band in the inner regions.
This requires low B(<3μG)for the assumed age of 7000 yr of
Rabbit, and then N
0
and thus ηshould increase by more than an
order of magnitude to fit the X-ray SED. We do not consider
this case, because it is very difficult to substantially increase the
injected particle energy for the given L
SD
of J1418.
Adiabatic cooling of the low-energy particles accounts for
the hard radio SED without requiring an intrinsic spectral break
in the particle distribution. Radio fluxes are controlled
primarily by
e,min
g
, and the radio SED slope is related to
particle cooling (e.g., Band V
flow
). Because of uncertainties in
the radio measurements, we matched them with our model only
by visual inspection. As noted above, the radial profile of the
X-ray photon index (Figure 8(c)) suggests fast diffusion in the
PWN (see also Van Etten & Romani 2011; Tang &
Chevalier 2012).Wefind that D
0
≈10
27
cm
2
s
−1
can ade-
quately explain the measured Γprofile. The brightness profile
(Figure 8(d)) implies that the radial decrease of Bis small
(α
B
=−0.2).
Because the X-ray PWN does not have a sharp boundary,
our choice of the R
PWN
value is rather arbitrary, although the
bright part of the PWN is included well within the radius. As
noted above, this is related to the sudden drop of Band V
flow
in
our model. The former has some influence on the results. For
larger R
PWN
, the Band brightness in outer zones are higher,
making the brightness profile flatter; this can be accommodated
by our model with changes of the other parameters (e.g., B
0
,
and α
B
). Because Band brightness in outer zones are already
low, the required changes of the parameters to preserve a data
12
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versus model match are not large. For an increase of R
PWN
by a
factor of ∼1.5, the other parameters need to be changed 20%.
Note that the parameter dependence on R
PWN
is not linear,
because emission in far outer regions is much weaker.
To match the TeV SED, we adjusted the temperature (e.g.,
T
dust
=10–30 K; Zhu & Huang 2014)and density of the IR
field, as has been often done in SED modelings (e.g., Torres
et al. 2013; Zhu et al. 2018). The TeV emission of Rabbit is
detected over a more extended region than the lower-energy
counterparts (H.E.S.S. Collaboration et al. 2018), indicating
that particles that escaped from the X-ray PWN (e.g.,
R
4. 5
PWN =¢)into the ISM upscatter IR photons there. The
computed TeV SEDs within the X-ray PWN (red)and in the
ISM (purple)are separately presented in Figure 8(a). Note that
we assumed the IR field is spatially homogeneous. However,
bright mid-IR emission at 8–20 μm was detected ∼2′–3′west
from the pulsar in the WISE and Glimpse images. If this mid-
IR source is associated with or at the same das the Rabbit
PWN, the ICS emission would have an SED bump at
∼100 GeV, which we do not see in the VHE data. Moreover,
the VHE emission would be stronger at the location of the mid-
IR source because its emission is intense. The LAT and TeV
counterparts of Rabbit are far away from the mid-IR source.
These suggest that the mid-IR source may not be associated
with the PWN. It is also possible that some of the VHE
emission is produced by sources other than the PWN, e.g., a
putative supernova remnant (SNR)shell. In our model, this
would imply a lower external IR density (u
IR
)or alternatively
higher Band a smaller number of particles for fixed u
IR
.
4.4. Model Parameter Covariance
Optimization of the model parameters and inspection of
covariance between them require multidimensional parameter
scans, which are computationally infeasible. Hence, we instead
carried out pair scans for several important parameters. Note
that the other parameters are frozen because they do not
significantly influence the X-ray emission (e.g., R
TS
,
e,min
g
,T
IR
,
and u
IR
)or could be determined by images (e.g., R
TS
and
R
PWN
)in principle. For a pair of parameters, we varied their
values around those determined by visual inspection and
computed χ
2
by fitting the X-ray data. The results of the pair
Figure 9. χ
2
contours obtained by pair scans. The 68%, 90%, and 99% contours are displayed. The designated pairs of the model parameters are simultaneously
scanned while holding the other parameters fixed at their best-fit values (Table 2), and χ
2
is computed with the X-ray SED and radial profiles of Γand brightness. The
crosses mark the parameter values reported in Table 2.
13
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scans are presented in Figure 9. Because we did not
simultaneously optimize the parameters, the reported (our
optimized)parameters are off-centered in some of the panels
but within the 68% contours.
The complex interplay among the parameters is not fully
captured by the pair scans, but they show some general
covariances that could be qualitatively understood as follows.
The integrated flux is mainly determined by particle residence
time and B. For small V
0
,α
V
(large negative value), and D
0
, the
residence time is long and thus emission within the PWN
volume is large. α
V
and D
0
further control the radial profiles,
with some complications due to the α
V
+α
B
=−1 relation and
energy-dependent diffusion; in general, smaller values produce
more rapidly falling brightness and Γprofiles. The X-ray
spectral shape is affected by p
1
,B
0
, and
e,max
g
, as the
synchrotron emission frequency (ν
syn
)is proportional to
B
e
02
g.
e,max
g
is weakly correlated with D
0
,p
1
,V
0
and α
V
, and
anticorrelated with B
0
. Larger
e,max
g
does not degrade the SED
fit because there are no high-energy data (>20 keV). In this
case, however, the Γprofile is flattened by diffusion/advection
of the higher-energy particles to outer regions. To maintain the
fit quality, these higher-energy particles need to be pushed out
to low-Bregions by more rapid propagation (larger D
0
,V
0
,or
α
V
), or alternatively, the injected particle spectrum should be
softer (larger p
1
). The B
0
–
e,max
g
anticorrelation is obvious,
given that Be
SY 2
n
gµ.D
0
is correlated with B
0
but antic-
orrelated with α
V
,V
0
, and p
1
. The decrease of the total PWN
emission caused by shorter residence time (larger D
0
)is
balanced by larger B
0
and/or smaller α
V
and V
0
. An increased
loss of high-energy particles in outer regions, due to stronger
diffusion, makes the X-ray spectrum softer, which is
compensated by smaller p
1
(harder injection spectrum). The
p
1
–V
0
,p
1
–B
0
, and p
1
–α
V
correlations are seen because larger p
1
means less X-ray emitting particles for given η; to explain the
observed X-ray flux, higher B
0
or longer residence time is
necessary. The V
0
–B
0
,V
0
–α
V
, and B
0
–α
V
correlations are also
related to the total emission within the PWN.
5. Discussion and Conclusions
We determined broadband X-ray spectra of the Rabbit PWN
using archival Chandra and XMM-Newton data, as well as a
new NuSTAR observation. NuSTAR’s high temporal resolu-
tion allowed us to detect 110 ms X-ray pulsations of J1418
with high significance, characterize the pulse profile in the hard
X-ray band and clearly distinguish between the on- and off-
pulse emissions. By jointly analyzing Chandra, XMM-Newton,
and NuSTAR’s off-pulse data, we found that the X-ray
spectrum of the PWN is described well by a power-law model
with Γ≈2. We then applied a multizone emission model to
investigate flow properties in the PWN and found that the
electrons are accelerated to very high energies (500 TeV)in
the Rabbit PWN.
5.1. Observed X-Ray and VHE Properties of the Rabbit PWN
While the detection of X-ray pulsations of J1418 was
claimed in a previous XMM-Newton study (Kim & An 2020a),
the significance was not very high and the pulse profile was not
characterized well. The NuSTAR confirmation of the pulsa-
tions from J1418 firmly established its association with the
Rabbit PWN. Furthermore, we found that the X-ray pulse
profile of J1418 exhibits a sharp peak and a broad bump with a
phase separation of 0.5. The pulsar’s gamma-ray light curve
also shows two peaks with the same phase separation. A
comparison between the gamma-ray and X-ray pulse profiles
can lead to determining the spin orientation of the pulsar (e.g.,
Wang et al. 2013); the previous XMM-Newton study found
that the sharp X-ray peak in the profile phase-aligns well with a
GeV peak. A further investigation with our NuSTAR
measurement requires an accurate LAT timing solution that
covers the NuSTAR observation epoch.
The accurate characterization of the X-ray pulse profile with
the NuSTAR data allowed an investigation of the hard X-ray
emission properties of the PWN. Joint spectral analyses of the
XMM-Newton, Chandra, and NuSTAR data demonstrated that
the spatially integrated X-ray spectrum of the PWN is
described well by an absorbed power law with N
H
=
(2.78 ±0.12)×10
22
cm
−2
and a photon index Γ=2.02 ±
0.05. These are consistent with those measured by Suzaku
(Γ=2.00 ±0.06 for N
H
=2.7 ×10
22
cm
−2
; Kishishita et al.
2012). However, that they did not report the abundance and
cross sections used for their Galactic absorption model, and
hence we assumed that they used the angr abundance and
vern cross section (the default in XSPEC). In the Suzaku data,
Kishishita et al. (2012)found a significant spectral softening,
with Γgradually growing from 1.77 in the inner region
(
R
0.8<¢
)to 2.12 in the outer region (
R
1.7=¢
–3′), which is not
consistent with the results of our imaging analysis (Section 2.4)
nor our spatially resolved spectral analysis (Section 2.5.4).
Most likely, the contamination by the pulsar (and possibly X1)
was not adequately removed in the Suzaku data analysis, and
the hard pulsar emission may have contaminated their PWN
spectra and caused the spectral softening (see Section 2.5.4).
We found that the high-energy (>30 GeV)emission in the
Rabbit regions appears to overlap well with the R=0°. 11 H.E.
S.S source (Figure 7)and that the >30 GeV SED connects well
to the VHE one (Figure 8(a)). This verifies that the LAT and
the H.E.S.S sources are indeed associated with each other.
Thus, the LAT source should also be extended because the ICS
mechanism produces gamma-ray emission in the LAT and the
VHE bands, although the LAT extension could not be clearly
constrained with the current data, due to the paucity of counts
and the broad LAT PSF.
5.2. Modeling of the Broadband Emission Properties
It has been suggested that the one-sided morphology and
the offset TeV emission of Rabbit might be caused by a
reverse shock interaction that diverts the particles in the
direction opposite to the interaction site with respect to the
pulsar (e.g., Aharonian et al. 2006). The interaction would
complicate the flow geometry and thus require MHD
simulations that incorporate detailed physics of the reverse
shock interaction and the subsequent flow, which are beyond
the scope of our phenomenological emission model. For
nonspherical (or nonconical)flow, the adiabatic cooling and
the α
B
–α
V
relation may differ (e.g., Reynolds 2009), and thus
different values for
e,min
g
,α
B
,andα
V
may be inferred. In
addition, particle propagation and PWN properties in the
direction toward the presumed reverse shock (i.e., the
northeast direction for Rabbit)may be different from those
in the X-ray PWN (i.e., the southwest tail of the Rabbit
nebula). Our model assumes that the same electron population
produces both the X-ray synchrotron nebula and TeV
emission, after accounting for energy losses and diffusion.
14
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

In reality, it could occur that, for instance, the relative motion
between the pulsar and its surroundings sweeps the particle
flow toward the southwest, elongating the PWN in that
direction as observed (e.g., Kolb et al. 2017;Slaneetal.
2018), and subsequently the TeV emission further down-
stream. Moreover, some of the observed emissions may be
contaminated by sources (e.g., a putative SNR)other than the
Rabbit PWN. Since our simple spherical model does not
account for these complex phenomena, the reported values of
the parameters need to be taken with caution. Adequate
treatments of the aforementioned complexities await further
theoretical studies, which we defer to future work.
Zhu et al. (2018)modeled a Rabbit SED using a one-zone
time-dependent scenario, but this one-zone model did not
account for the radial profiles of the X-ray brightness and
photon index. Our multizone emission model with a single
power-law electron spectrum and spatially varying PWN
properties reproduced the measured SED and radial properties
of the photon index and brightness well. Although a unique set
of model parameters may not be derived from the current
observational data alone, due to covariance among the
parameters, there are a few interesting parameters that we
could infer from the modeling.
The large extension of the VHE emission of Rabbit
compared to the radio/X-ray PWN (Figure 1(b)) can be
attributed to rapid diffusion (e.g., Van Etten & Romani
2011); particles that escaped from a compact X-ray PWN can
give rise to TeV emission by ICS of ambient IR photons in a
much larger region. This scenario seems plausible for the
Rabbit PWN, as its TeV emission lies in the direction of the
X-ray tail (along the particle outflow). Intriguingly, the
diffusion length scale of
R
Dt2 4 10 c
m
diff age 19
~»´
estimated for the VHE emitting γ
e
≈10
7
electrons corre-
sponds to 0°. 23 for the assumed distance of 3.5 kpc, which is
in accord with the extension of the TeV source from J1418
(Figure 1(b)).
Our measurements and multizone modeling provide further
insights into the Rabbit PWN. In particular, the photon index
and brightness profiles provide important clues to understanding
the source. The flat X-ray photon-index profile is hard to explain
without rapid diffusion (≈10
27
cm
2
s
−1
), because the
synchrotron and ICS cooling are severe even for low B
(∼10 μG). We note again that a pure advection model (e.g.,
Reynolds 2003)with very low B
0
may also explain the flat Γ
profile, but in this case, the required particle-injection power
would be greater than L
SD
of J1418. While this may be remedied
by changes in the other parameters,
e,max
g
is unlikely to change
substantially in such an alternative model. The flat photon-index
profile (i.e., insignificant softening)and the hard X-ray emission
of Rabbit helped to constrain the maximum energy of electrons
in the PWN to be 500 TeV (10
e,max 9
g
»). A smaller
e,max
g
and a larger Bmay account for the hard X-ray emission, but then
faster particle cooling will make it difficult to match the radial
profiles of the photon index and brightness. On the other hand, a
model with a larger
e,max
g
and a smaller Boverpredicts the
brightness at large distances.
To investigate the possibility of lower
e,max
g
or D
0
, we held
the parameter fixed at smaller values and optimized the other
parameters by visual inspection. Models with lower
e,max
g
or D
0
are displayed in Figure 10. The lower
e,max
g
models predicted
the synchrotron cutoff at lower energies, making the predicted
X-ray spectra softer (larger effective Γvalues)and thus fits to
the X-ray SED and Γprofile poorer. Models with smaller D
0
values could match the X-ray SED, but the Γprofile showed a
significant softening that is different from the measured flat
trend when D
0
was significantly lower (Figure 10 (e)).
Moreover, the particle residence time is large, and thus the
ICS emission of the PWN is very strong. To match the VHE
SED, we need to reduce the IR seeds (i.e., u
IR
≈0 for
D
0
=10
26
cm
2
s
−1
case), but then the Fermi-LAT SED is
underpredicted at low energies (Figure 10 (d)).
While the above study added some credence to our
e,max
g
estimation, the model parameter degeneracy was not fully
explored in this estimation of
e,max
g
. It can be directly inferred
by measuring a spectral cutoff of the synchrotron emission
(e.g., An 2019), which was not detected in the NuSTAR data
we analyzed. Sensitive hard X-ray and MeV observations
beyond the NuSTAR band, with near-future observatories such
as FORCE, HEX-P, and COSI (Madsen et al. 2018; Nakazawa
et al. 2018; Tomsick et al. 2019), will be needed.
Figure 10. Optimized models for different e,max
g(a–c)and D
0
(d–f): broadband SEDs (a and d), and radial profiles of Γ(b and e)and brightness (c and f). Either e,max
g
or D
0
was held fixed at the designated values, and the other parameters were optimized by visual inspection.
15
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

5.3. Comparison with Another Middle-aged PWN
We compare the properties of Rabbit to the archetypal Vela
X PWN, which also exhibits one-sided tail emission and is
powered by a middle-aged pulsar with τ
c
=11.3 kyr and
E
710ergs
SD 36 1
=´ -.Γwithin the Vela X PWN seems
unchanged in the inner regions (to 50~
¢
), but an overall
increase (to 90~
¢
)is noticeable along the tail (e.g., Slane et al.
2018). TeV emission was detected, and H.E.S.S. Collaboration
et al. (2019)inferred that Bin the Vela X PWN decreases from
8.6 to 5.4 μGover a distance of ≈3 pc. These Bvalues imply
α
B
≈−0.24 (Equation (4)). From the TeV size of the source,
H.E.S.S. Collaboration et al. (2019)inferred a diffusion
coefficient of D10
27
cm
2
s
−1
for 1 TeV electrons, which is
consistent with the 10
28
cm
2
s
−1
at 10 TeV suggested by
Huang et al. (2018).
While these properties are generally similar to those we
measured or inferred for Rabbit, some differences are notice-
able in detailed comparisons. In the Rabbit PWN, X-ray
spectral softening is less significant, the inferred Bprofile is
flatter, and the inferred maximum particle energy (i.e.,
e,max
g
)is
higher (by a factor of 2–3)than in the Vela X PWN. In
addition, the Vela X PWN shows a compact region in the
vicinity of the pulsar, which is connected to a diffuse PWN
region by a narrow structure, whereas the Rabbit PWN exhibits
a broad and continuous tail. We speculate that these differences
are related to the evolutionary stage of the sources. HD
simulations (Kolb et al. 2017; Slane et al. 2018)have shown
that one-sided morphologies are produced when the reverse
shock (RS)disrupts the PWN; the predicted morphology of a
PWN in this stage appears similar to that of Rabbit (e.g., the
7500 yr case in Figure 11 of Slane et al. 2018). At later times,
the RS sweeps the pulsar wind and creates a relic PWN; the
Vela X PWN was suggested to be in the relic-PWN stage
(Figure 12 of Slane et al. 2018). Then, the differences in the
X-ray spectral softening and Bprofile for the Rabbit and Vela
X PWNe could be explained as due to larger Bcontrast in the
more evolved Vela X PWN between the inner fresh-wind zone
and the outer relic PWN.
The most significant difference between the two sources is
whether or not their SNRs (e.g., ejecta and shell emission)were
detected; emission of the host SNR of the Vela X PWN was
identified (e.g., Slane et al. 2018), whereas the emission
signature of the Rabbit SNR has not been found yet. With the
lack of SNR emission, the formation of the tail in Rabbit is
puzzling in the PWN–SNR evolution scenarios (e.g., Kolb
et al. 2017; Slane et al. 2018). Is the one-sided morphology of
Rabbit produced by supersonic motion of the pulsar as in bow-
shock nebulae, and not by the RS interaction? Then the small
characteristic age of J1418 and the strong TeV emission from
the Rabbit PWN are unusual compared to other bow-shock
nebulae (e.g., Kargaltsev et al. 2017). A proper motion
measurement of J1418 and/or detection of SNR emission
around Rabbit, with sensitive X-ray observatories (e.g., Lynx
or AXIS; Gaskin et al. 2019; Mushotzky et al. 2019), will be
needed to address this issue, and dedicated radio studies might
also cast light on the system, refining the fluxes attributable to
the PWN and perhaps locating the SNR shell. Furthermore,
observational and theoretical studies of these two and other
middle-aged PWNe could provide insights into the evolution of
PWNe and their interaction with the SNR and ambient
medium.
6. Summary
We characterized the emission properties of J1418 and
Rabbit, and applied a multizone model to the measurements.
Below, we summarize our main conclusions.
1. We found that the X-ray pulse profile of J1418 exhibits a
sharp peak and a broad bump separated by ≈0.5 phase.
2. We found out that the 0.5–20 keV spectrum of the Rabbit
PWN is modeled well by a Γ≈2 power law and does not
significantly soften with increasing distance from the
pulsar.
3. Our multizone modeling of the broadband SED and the
radial profiles of Γand brightness of the PWN suggests
that its magnetic field is low (∼10 μG), and the particles
are accelerated to very high energies (>
~
500 TeV)and
diffuse out efficiently (D∼10
27
cm
2
s
−1
).
As noted above, the flat radial profile of the photon index
requires both low Band efficient diffusion. We have assumed
magnetic flux conservation (α
V
+α
B
=−1), but the magnetic
field could either be dissipated by reconnection or perhaps
amplified by some mechanism in internal shocks. The magnetic
energy may have significantly dissipated in Rabbit, such that
particles could diffuse out more efficiently, as speculated based
on the TeV emission outside the X-ray PWN. There are other
PWNe whose TeV emission is more extended than the radio/
X-ray emitting regions. It will be intriguing to see if these
PWNe also exhibit a flat radial profile of their X-ray photon
index with deep X-ray observations. Further multiwavelength
observations of other PWNe detected in the TeV band,
including NuSTAR hard X-ray observations, will be presented
in our forthcoming papers, offering a good opportunity to
explore the PWN origin of Galactic PeVatrons (Mori et al.
2022).
This work used data from the NuSTAR mission, a project
led by the California Institute of Technology, managed by the
Jet Propulsion Laboratory, and funded by NASA. We made use
of the NuSTAR Data Analysis Software (NuSTARDAS)
jointly developed by the ASI Science Data Center (ASDC,
Italy)and the California Institute of Technology (USA). This
research was supported by Basic Science Research Program
through the National Research Foundation of Korea (NRF)
funded by the Ministry of Science, ICT & Future Planning
(NRF-2022R1F1A1063468). Support for this work was
partially provided by NASA through NuSTAR Cycle 6 Guest
Observer Program grant NNH19ZDA001N. S.S.H. acknowl-
edges support from the Natural Sciences and Engineering
Research Council of Canada (NSERC)through the Discovery
Grants and Canada Research Chairs programs and from the
Canadian Space Agency (CSA). We thank the referee for a
detailed reading of the manuscript and constructive comments
that helped strengthen the paper.
Facilities: CXO, XMM-Newton, NuSTAR, Fermi/LAT.
Software: HEAsoft (v6.29; HEASARC 2014), CIAO (v4.13;
Fruscione et al. 2006), XMM-SAS (v20180620; Gabriel 2017),
XSPEC (v12.12; Arnaud 1996), FermiPy (v1.0.1; Wood et al.
2017).
ORCID iDs
Jaegeun Park https://orcid.org/0000-0002-9103-506X
Chanho Kim https://orcid.org/0000‐0003‐0226‐9524
Hongjun An https://orcid.org/0000-0002-6389-9012
16
The Astrophysical Journal, 945:66 (17pp), 2023 March 1 Park et al.

Kaya Mori https://orcid.org/0000-0002-9709-5389
Stephen P. Reynolds https://orcid.org/0000-0002-5365-5444
Samar Safi-Harb https://orcid.org/0000-0001-6189-7665
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