Cannonball or Bowling Ball: Proper Motion and Parallax for PSR J0002+6216

Abstract

We report the results of careful astrometric measurements of the cannonball pulsar J0002+6216 carried out over 3 yr using the High Sensitivity Array. We significantly refine the proper motion to μ = 35.3 ± 0.6 mas yr ⁻¹ and place new constraints on the distance, with the overall effect of lowering the velocity and increasing the inferred age to 47.60 ± 0.80 kyr. Although the pulsar is brought more in line with the standard natal kick distribution, this new velocity has implications for the morphology of the pulsar wind nebula that surrounds it, the density of the interstellar medium through which it travels, and the age of the supernova remnant (CTB 1) from which it originates.

Full text

Cannonball or Bowling Ball: Proper Motion and Parallax for PSR J0002+6216
S. Bruzewski
1
, F. K. Schinzel
2,4
, G. B. Taylor
1
, P. Demorest
2
, D. A. Frail
2
, M. Kerr
3
, and P. Kumar
1
1
Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA; bruzewskis@unm.edu
2
National Radio Astronomy Observatory, P.O. Box O, Socorro, NM 87801, USA
3
Space Science Division, US Naval Research Laboratory, Washington, DC 20375, USA
Received 2023 September 26; revised 2023 October 26; accepted 2023 October 27; published 2023 November 22
Abstract
We report the results of careful astrometric measurements of the cannonball pulsar J0002+6216 carried out over
3 yr using the High Sensitivity Array. We significantly refine the proper motion to μ=35.3 ±0.6 mas yr
−1
and
place new constraints on the distance, with the overall effect of lowering the velocity and increasing the inferred
age to 47.60 ±0.80 kyr. Although the pulsar is brought more in line with the standard natal kick distribution, this
new velocity has implications for the morphology of the pulsar wind nebula that surrounds it, the density of the
interstellar medium through which it travels, and the age of the supernova remnant (CTB 1)from which it
originates.
Unified Astronomy Thesaurus concepts: Supernova remnants (1667);Interstellar medium (847);Pulsar wind
nebulae (2215);Proper motions (1295);Pulsars (1306);Parallax (1197);Astrometry (80)
1. Introduction
Identifying compact objects associated with supernova
remnants (SNRs)provides a unique window into the diverse
outcomes of core-collapse supernovae. A veritable zoo of such
young neutron stars has been found (Popov 2023)including
central compact objects (CCOs)and various types of magnetar,
such as soft gamma-ray repeaters (SGRs)and anomalous X-ray
pulsars (AXPs; Gaensler et al. 2001). Young radio pulsars
comprise the majority of neutron stars found in SNR
associations (Ferrand & Safi-Harb 2012; Green 2019). The
study of these systems has helped constrain initial pulsar
periods, magnetic fields, and beaming fractions, in addition to
their kick velocities at birth (Frail & Moffett 1993; Hansen &
Phinney 1997; Johnston et al. 2005; Igoshev et al. 2022; Kapil
et al. 2023).
Schinzel et al. (2019, henceforth Paper I)noted that the
115 ms γ-ray and radio pulsar PSR J0002+6216 was found at
the head of a bow-shock pulsar wind nebula (PWN). Follow-up
radio and X-ray observations resolved the PWN and showed
that it was consistent with a high-Mach shock formed from the
wind from the energetic high-velocity PSR J0002+6216
coming into ram pressure balance with its surrounding medium
(Kumar et al. 2023). Remarkably, the tail of the PWN extends
at least 7′–10′from PSR J0002+6216, pointing backward
toward the geometric center of the Galactic SNR CTB-1
(G116.9+0.2)28′away. The authors argued that CTB 1 is the
remnant of the supernova that produced PSR J0002+6216,
indicating that PSR J0002+6216 was born with an unusually
high velocity (V
PSR
>1000 km s
−1
)that allowed it to escape
the parent SNR.
Not all of the young pulsars found in or near SNRs are real
associations. Often the case is made on the basis of positional
coincidence and some agreement between ages and distances.
The gold standard for making such claims is accurate pulsar
proper motion and parallax measurements. Often both the
magnitude and direction of the proper motion can reinforce
the claimed association by showing that the pulsar likely
originated near the geometric center of the SNR (Deller et al.
2012; Van Etten et al. 2012;Shterninetal.2019;Johnston&
Lower 2021;Longetal.2022). In other cases, proper motion
measurements either exclude an association or require more
complex scenarios to remain tenable (Brisken et al. 2006;de
Vries et al. 2021; Espinoza et al. 2022).Pulsars(or PWNs)
with large offsets from the geometric center of the SNR,
including pulsars outside the SNR (e.g., Ng et al. 2012; Motta
et al. 2023), are especially important for constraining the high
birth velocity tail of PSRs (Frail et al. 1994;Hobbsetal.
2005; Verbunt et al. 2017). The burden of proof is high, since
in these cases the probability of a chance association is
greatly increased (Gaensler & Johnston 1995). Proper motion
measurements of such associations have given mixed results
at best (Zeiger et al. 2008; Hales et al. 2009). In the case of
PSR J0002+6216/CTB-1, Paper Ibolstered their case by
using nearly a decade of data from the Fermi Gamma-ray
Space Telescope to infer a proper motion of μ=115 ±
33 mas yr
−1
and θ
μ
=121°±13°. This value, although of
modest significance, agrees in magnitude and direction with
the PSR–SNR angular offset, assuming an age for the system
of 10 kyr.
In this work, we have undertaken very long baseline
interferometry (VLBI)observations over the course of
several years toward providing an accurate measure for the
proper motion and parallax of PSR J0002+6216. Having
such values allows us to provide a more definitive association
between PSR J0002+6216 and CTB-1, as well as to probe
the implications of the SNR age implied from the more
accurately measured velocity of the pulsar. Section 2will
discuss our observational setup, data processing, and the
fitting of parallax and proper motion from our data. Section 3
describes the best-fit parameters, and Section 4discusses
their implications. Finally, Section 5concludes and sum-
marizesthiswork.
The Astrophysical Journal, 958:163 (9pp), 2023 December 1 https://doi.org/10.3847/1538-4357/ad07e4
© 2023. The Author(s). Published by the American Astronomical Society.
4
An Adjunct Professor at the University of New Mexico.
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

2. Data
2.1. Observations
Observations were performed under project code BS278
(VLBA/19B-048)with the High Sensitivity Array (HSA)
5
comprised of NRAO’s Very Long Baseline Array (VLBA)and
the phased Karl G. Jansky Very Large Array (VLA), as well as
the Effelsberg 100 m radio telescope. The observations
spanned a period of just over two years (2020 February
10–2022 July 30). The antennas unavailable during a specific
observation, together with additional summarizing character-
istics, are noted in Table 1.
Initially, we performed a test observation (BS278Z)to check
for suitable phase reference calibrators, both for VLBA-only
in-beam calibration (selected from the VLA L-band image
presented in Paper I)and nearby bright calibrator sources
selected from the VLBA calibrator catalog.
6
This test did not
yield a suitable in-beam calibrator and only identified one
suitable phase calibrator, J0003+6307, separated by 0°.87
with a flux density of 46.5 ±1.3 mJy and a peak of 10.48 ±
0.24 mJy beam
−1
. Given the weak nature of this calibrator, we
included a secondary phase calibrator J2339+6010, which is at
a distance of 3°. 53 from our target with a flux density of about
70 mJy and a peak of about 36 mJy beam
−1
. In addition, we
observed J0014+6117 at a distance of 1°.71 (about 30 mJy/
15 mJy beam
−1
peak)as a VLA phasing calibrator. J0014
+6177 can also be used as cross-check calibrator. However, it
seems to be resolved out on the longest baselines. J0319+
4130 and J0137+3309 served as fringe finders, bandpass
calibrators, and flux references. The calibrators and phase
centers are summarized in Table 2.
The Z and A segments of BS278 were observed using the
18 cm receiver and a frequency range of 1.568–1.824 GHz and
the RDBE personality of the VLBA back end with a recording
data rate of 2 Gbps. This resulted in nondetection of our target,
primarily due to strong radio interference at some of the
antenna sites. Subsequent observations (BS278C onward)thus
used a lower frequency range centered on a mostly inter-
ference-free part of the spectrum, 1.268–1.524 GHz in the
21 cm wavelength band. For Effelsberg we utilize the center
horn of the seven-beam 21 cm receiver. In addition, phased
VLA was used to record the full allowable bandwidth of 1 GHz
(starting with segment C)to determine the pulsar ephemerides
using VLA’s WIDAR correlator for later binned correlation of
the HSA observations. The phased-VLA observations yielded
detections in all cases with an average flux density of ∼30 μJy.
The pulsar is linearly polarized and shows a rotation measure of
−178.5 ±2.5 radians m
−2
.
The HSA correlation was performed using the VLBA DiFX
correlator (Deller et al. 2011)using 256 correlation channels
per one of the eight dual-polarization spectral windows with
32 MHz bandwidth each and a 0.5 s dump time. This allows
full coverage of 256 MHz in bandwidth centered on
1.396 GHz. As mentioned before, for each phased-VLA
observation, pulsar timing data were recorded to obtain pulsar
ephemerides of our target pulsar, which were then supplied to
DiFX for binned correlation, where 10 bins in the pulsar spin
phase were formed to increase the signal-to-noise ratio of the
pulsar emission, with the first bin (“ON”)centered on the main
pulse of the pulsar. The VLBA DiFX correlator is set such that
the flux density remains the same as the unbinned equivalent.
In addition, multiple phase centers were placed on locations of
known bright sources that were identified from the wide-field
VLA L-band observation mentioned above. In addition, we
updated the target phase center four times throughout the
campaign to account for the pulsar’s proper motion and keep
the pulsar close to the correlation phase center. Correlation
phase centers are provided in Table 2.
2.2. Calibration and Model Fitting
The correlated data were calibrated using the Astronomical
Image Processing System (AIPS; Greisen 2003)using the
included VLBAUTIL procedures. Standard astrometric calibra-
tion was performed by applying corrections for Earth’s
orientation, the ionosphere, for digital sampling, and delay.
After this, bandpass, sampler, gain, and parallactic angle
corrections were applied, followed by determination and
application of phase corrections using the calibrator, J0003
+6307. For applying calibration solutions to the different phase
centers and bins the VLBAMPHC procedure was used. After
this, the calibration corrections were applied to the correlated
data and split by observed source, and written to separate
UVFITS formatted files for further analysis.
Then we used the Difmap package (Shepherd 1997)for
additional manual flagging of radio frequency interference,
primarily from Global Navigation Satellite System interference,
such as GPS, Galileo, or GLONASS in the 21 cm wavelength
band. We then used u,vplane model fitting of a delta function,
as a good representation for an unresolved point source, to
determine the position of our target and calibrators for each
epoch. The detections were typically at the 5σlevel, which did
not allow for robust fitting of a Gaussian function to the
visibilities. We also used Difmap to produce restored clean
images for the calibrators and target.
One issue arising from this approach is that while Difmap’s
model fitting excels in this particular case of faint sources, it
does not provide much in the way of uncertainties for the
calculated best-fit coordinates. Ideally, we would like to have
an estimate at each epoch for how well the position of a source
of a certain flux density can be localized amid some level of
background noise. We do this by making a copy of our original
Table 1
HSA Observation Summary
Date of Obs. S A Missing Issues
2019–07–20
a
Z 11 EF, Y L
2019–10–09
a
A12 LNL, EF gaps
2020–02–10
a
B 11 MK EB, Y gaps
2020–05–04 C 11 SC KP, FD, OV gaps
2020–08–08 D 12 LSC, KP, NL, LA gaps
2020–10–05 E 11 HN KP, SC gaps
2020–12–14 F 12 LOV high winds
2021–03–28
a
G 11 OV SC, PT, MK 1h lost
2021–06–18 H 11 SC Y late, FD timing
2021–09–20 I 12 LL
2022–02–14 J 11 HN MK missing data
2022–07–30 K 10 KP, MK L
Notes. The S column denotes the segment of BS278 and the A column counts
the number of participating antennas. Issues: gaps are noted where a significant
amount of data are missing from correlation.
a
Nondetections.
5
https://science.nrao.edu/facilities/vlba/HSA
6
https://obs.vlba.nrao.edu/cst
2
The Astrophysical Journal, 958:163 (9pp), 2023 December 1 Bruzewski et al.

visibility data, then using CASA (CASA Team et al. 2022)to
replace all data points with an artificial point source of known
position and flux density (set to approximately match the
properties of PSR J0002+6216). Random simple noise is then
added to the data, which we rescale using the existing weights.
Difmap then provided this artificial data, as well as a
perturbation of the position to use as an initial starting model,
and made to perform model fitting for 50 steps, after which the
final model is recorded to disk.
Conveniently, none of the algorithms involved in this
simulation are all that intensive, and so the entire pipeline
can be run using a single script on a single core. Given that the
only difference between two simulations will be the seed of the
random noise, the problem is embarrassingly parallelizable,
and so we perform 1000 such simulations at each epoch. Once
the final models of all simulations have been collected and
tabulated, we simply need to characterize the scatter of the
models to establish our fitting uncertainty. We note that the
scatter in R.A. and decl. seem to be highly correlated in all
epochs, likely due to our u,vcoverage, such that we cannot
disregard the off-diagonal terms of the covariance matrix at this
stage. Instead, we opt to record the covariance matrix at each
epoch, measured over the coordinates aaa d
D
=- cos
*(¯)· ¯
and
ddd
D
=-(¯)
.
7
2.3. Atmospheric Effects
For our observations, the positional uncertainties on a given
date will be dominated by the effects of the ionosphere. This
can be corrected for at some level by ionospheric corrections
performed during calibration, but for corrections at the
milliarcsecond level, typically further refinements are required.
Similar work (Deller et al. 2016,2019; Ding et al. 2023)has
typically made use of one or more relatively bright in-beam
calibration sources, which (if truly extragalactic and stationary)
can be simultaneously phase calibrated, effectively removing
atmospheric effects by providing an absolute position offset
between the two sources at the submilliarcsecond level.
Unfortunately, such calibration is not possible in the case of
J0002+6216. The area immediately surrounding the pulsar is
devoid of any other sources out to the edges of the VLBA field
of view. Furthermore, as this source is quite faint (∼22–30 μJy
in single dish/phased-VLA observations), we require the
additional sensitivity from the inclusion of Effelsberg and the
phased VLA, which have an even smaller field of view, further
ruling out this option. The typical detection in the “ON”bin of
our observations resulted in a flux density of ∼14 μJy,
suggesting that a significant fraction is lost due to residual
ionospheric phase errors that dominate in this frequency range.
Lacking a suitable source in the field of view, we opt to
leverage a nearby but out-of-beam source. This secondary
source, dubbed SRC2, has a peak flux density near 700 μJy
beam
−1
, meaning its own positional uncertainties (as measured
via simulations described in Section 2.2)are more than 1 order
of magnitude better constrained than those of our target. Its
position seems to experience ionospheric variations on the
same scale as other calibrators and has no apparent proper
motion across our observations. As SRC2 is fairly close to our
target, we can assume that it lies behind roughly the same
region of the ionosphere (as described approximately by Taylor
et al. 1999), and should have its position perturbed similarly.
With this in mind, we use the displacement of SRC2 from its
median position to apply a correction to the position of J0002
+6216 at each epoch. This should very roughly approximate
the results of the aforementioned simultaneous phase calibra-
tion, albeit with higher uncertainties. These updated positions
are reported in Table 3.
In order to treat the uncertainty fully, we need a probe of the
systematic uncertainty produced by the ionosphere. As a proxy
for this, we choose to measure the scatter of each of our
Table 2
Calibrators and Phase Centers
Name R.A. Decl. Dist. Description
h:m:s d:m:s (deg)
J0001+6051 00:01:07.099891 +60:51:22.79829 1.43 Position check calibrator
J0003+6307 00:03:36.511479 +63:07:55.87126 0.87 Primary phase calibrator
J0014+6117 00:14:48.792109 +61:17:43.54262 1.71 VLA phasing calibrator
J0137+3309 01:37:41.299543 +33:09:35.13377 3.99 Position check calibrator
J0319+4130 03:19:48.160114 +41:30:42.10568 LFringe check/bandpass calibrator
J2236+2828 22:36:22.470849 +28:28:57.41328 LFlux check/fringe check
J2339+6010 23:39:21.125227 +60:10:11.84951 3.53 Secondary phase calibrator
J0003+6219 00:02:53.527 +62:19:16.940 LCorrelation phase center
J0002+6205 00:01:44.120 +62:04:30.690 LCorrelation phase center
J0001+6228 00:00:43.195377 +62:27:59.722930 LCorrelation phase center
J0000+6222 00:00:17.959930 +62:22:20.034252 LCorrelation phase center
J0000+6215 00:00:24.397047 +62:15:21.717124 LCorrelation phase center
J0000+622A 00:00:17.753 +62:22:14.354 LCorrelation phase center
TARGET 00:02:58.17 +62:16:09.4 LCorrelation phase center for segment B
TARGET 00:02:58.205 +62:16:09.510 LCorrelation phase center for segments D–E
TARGET 00:02:58.207 +62:16:09.500 LCorrelation phase center for segments F–H
TARGET 00:02:58.210 +62:16:09.4958 LCorrelation phase center for segments I–K
Note. List of calibrators and correlation phase centers for BS278C-K, including updated target positions. Coordinates are in J2000.
7
Note that this cosine correction will appear fairly regularly throughout
this work.
3
The Astrophysical Journal, 958:163 (9pp), 2023 December 1 Bruzewski et al.

calibrators relative to each of their mean positions across all
epochs. We then measure the covariance of this distribution
and add it to the simulated positional covariance at each epoch.
The scale of this systematic uncertainty is approximately 1 mas
(found from the diagonal elements in the covariance matrix),
and is only very mildly correlated in R.A. and decl. The
uncertainties reported in Table 3include this systematic
atmospheric term.
2.4. Parallax and Proper Motion Fitting
For fitting parallax and proper motion, we adopt the model
used by Gaia as described in Lindegren et al. (2016)Equation
(2), similarly assuming that the effect of radial proper motion is
small enough as to be undetectable in our observations. Using
that equation, a position can be found as a function of time,
given the set of variables {α
0
,δ
0
,ω,
m
a
*,μ
δ
}and choosing a
reference epoch. For our analysis, we choose a reference epoch
of MJD 58,849, which corresponds to 2020 January 1, slightly
prior to our first observation.
We implement a Gaussian prior on the total proper motion of
115 ±33 mas yr
−1
, as reported in Paper I, though we note that
the final parameter estimates do not seem to be strongly
effected by this prior. Because our position uncertainties are
covariant, the equation for log-likelihood takes on the form
åp=- + +
=
-
VxVxln 1
22ln2 ln det , 1
i
n
ii
T
ii
0
1
[() ]()
where at a given epoch i,x
i
=(m
i
−o
i
)represents the
difference between the model position ad=m,
imm
i
()
and
observed position ad=o,
ioo
i
()
, and V
i
represents the
observed/estimated covariance. We then pass the data,
uncertainties, prior, and log-likelihood functions to the emcee
Python package. We use 200 walkers initialized in a small
space around initial approximate estimates provided by least-
squares fitting of a straight line, and the scatter relative to that.
The walkers progress for 20,000 steps, after which we calculate
the maximum auto-correlation time τ(typically of order 50
steps)and use that to discard the first 3τsteps (prior to burn-in)
and thin by a factor of τ/2. These sample chains are flattened
and then can be evaluated to find the best-fit values and
uncertainties for each of our parameters as shown in Table 4.
3. Results
Our best-fit model is illustrated in Figure 1. Here, we mainly
focus on the estimates of parallax and proper motion, as those
have the largest physical implications. From the best-fit values
for proper motion along each axis, we calculate the total proper
motion magnitude
m
mm=+= 
ad
35.30 0.60
tot
22
*mas yr
−1
and position angle
q
mm==

ad
arctan 112 . 86 0 . 8
3
PA *
() .
As the total proper motion is generally of more interest than its
individual components, we illustrate the distribution in this
parameter against parallax in Figure 2. As the full five-parameter
corner plot is quite large and does not show any correlations of
note, we opt to make it available upon request.
If we compare our total proper motion and position angle
with the initial estimates of these parameters found from
Fermi data in Paper I(μ
tot
=115 ±33 mas yr
−1
and θ
PA
=
121°±13°),wefind differences of 2.37σand 0.59σ,
respectively. The agreement in position angle is expected, as
the prior estimate of that value also matched well with the
direction of the orientation of the PWN as described in Paper I
and later in Kumar et al. (2023), with the later finding a
position angle of θ
PA
=111°.13±0°.52.
Next, we examine the parallax of the source, for which a
value of ω=0.63 can be naively inverted to find a distance and
velocity of 1.59 kpc and 244 km s
−1
respectively. That said,
such a naive inversion will suffer from Lutz–Kelker bias, and
can produce meaningless or unphysical values in cases where
the parallax is not incredibly well constrained (Verbiest et al.
2010). Ultimately a proper measurement of the distance to
J0002+6216 via parallax is thwarted by the effects of the
ionosphere. The scale of its perturbation on our observations is
above the scale we expect to see the parallax at, and in fact
Paper Ishowed that we can largely rule out parallaxes larger
than 1 mas or so.
Perhaps the best estimate for distance currently possible
comes from the association with the remnant CTB-1. The new
estimates of proper motion trace back a path directly through
the geometric center of the SNR (as seen in Figure 3and
discussed more in Section 4), making a chance association
quite unlikely. This geometric argument has been sufficient for
other systems such as the recently discovered “Mini Mouse”
(Motta et al. 2023). If we assume this association to be true,
then our best constraints for the distance to the pulsar are once
Table 3
Target Positions
Date Position (J2000)
s
a
*σ
δ
MJD (mas)(mas)
58973.44 00
h
02
m
58 20495 +62d16m09 50991 1.29 1.11
59069.70 00
h
02
m
58 20652 +62
d
16
m
09 50328 1.18 1.07
59127.89 00
h
02
m
58 20686 +62
d
16
m
09 50229 1.19 1.09
59197.80 00
h
02
m
58 20773 +62
d
16
m
09 49950 1.21 1.22
59383.63 00
h
02
m
58 21090 +62
d
16
m
09 49124 1.28 1.12
59477.80 00
h
02
m
58 21150 +62
d
16
m
09 49155 1.17 1.10
59624.17 00
h
02
m
58 21284 +62
d
16
m
09 48623 1.13 1.05
59790.60 00
h
02
m
58 21578 +62
d
16
m
09 47532 1.23 1.12
Note. The position of J0002+6216 at each epoch it was detected. Uncertainties
are derived from simulations of position fitting to our u,vdata, as well as
systematic uncertainty from the ionosphere. Here we report only the diagonal
terms of the covariance matrix as our uncertainties, as the correlation is fairly
small.
Table 4
Estimated and Calculated Values
Parameter Estimate
α
0
00
h
02
m
58 20346 ±0.89 mas
δ
0
+62
d
16
m
09 51294 ±0.81 mas
ω0.63 ±0.45 mas
m
a
*32.52 ±0.59 mas yr
−1
μ
δ
–13.71 ±0.53 mas yr
−1
μ
tot
35.30 ±0.60 mas yr
−1
θ
PA
112.86°±0.83°
Age 47.60 ±0.80 kyr
v
2kpc
a
334.90 ±5.66 km s
−1
v
3kpc
a
502.35 ±8.4 km s
−1
Notes. Values provided are the median of all samples for each parameter.
Uncertainties are reported at the 1σlevel.
a
These velocities assume distances with no uncertainties.
4
The Astrophysical Journal, 958:163 (9pp), 2023 December 1 Bruzewski et al.

again approximately 1.5–4 kpc (Landecker et al. 1982; Hailey
& Craig 1994; Fesen et al. 1997; Yar-Uyaniker et al. 2004),
plus an upper limit d7 kpc derived from X-ray and γ-ray
efficiencies (Wu et al. 2018; Zyuzin et al. 2018).
4. Discussion
Given our significantly improved measurement of the proper
motion, we can begin to discuss the implications of the pulsar’s
spatial velocity. This velocity can now be written as
=-
vd
334.90 2 kpc km s , 2
1
⎜⎟
⎛
⎝⎞
⎠()
where we have chosen to use 2 kpc as our reference point to
match the assumptions in Paper I. Our new estimate has the
effect of decreasing the speed by a factor of more than three
from the original estimate of over 1000 km s
−1
. For the range
1.5–4 kpc we find velocities 251–670 km s
−1
, and more
specifically for the value of 3.1 ±0.4 kpc obtained by Hailey &
Craig (1994)we find 519.09 ±67.56 km s
−1
. While any of
these values would still place PSR J0002+6216 among a rare
category of high-velocity pulsars compared to the overall
distribution (Hansen & Phinney 1997; Verbunt et al. 2017),it
could perhaps now be considered for demotion from the ranks
of other cannonball pulsars (see Hobbs et al. 2005; Shternin
et al. 2019)to something more akin to a vigorously thrown
bowling ball.
Next, we examine the implications of our proper motion
result for PSR J0002+6216, using three different measures of
the pulsar age. This approach does not require estimates of other
(uncertain)parameters such as the distance, and thus should
yield secure results. In the first instance, we assume that the age
of PSR J0002+6216 is equal to its characteristic age (τ
c
=
306 kyr). In this case, our measured proper motion places the
pulsar birthplace more than 2°away, well outside the boundaries
of CTB 1. In this instance, there is no physical PSR–SNR
association, just a line-of-sight coincidence. There are at least
two weaknesses with this hypothesis. Ng et al. (2012)have
shown that long PWN tails that point back toward SNRs have a
very low posterior probability of occurrence. If anything, the
chance probability is even smaller for this system since both our
proper motion result and the PWN tail of PSR J0002+6216 can
be accurately traced back to the geometric center of CTB 1.
Another objection could be making the (common)assump-
tion that τ
c
can be used as a proxy for the actual age of the
system. This has been shown to have questionable accuracy, at
least for young pulsars in which τ
c
is seen to be substantially
lower than or greater than the quoted age of the associated SNR
(Blazek et al. 2006; Popov & Turolla 2012; Johnston &
Lower 2021). This discrepancy with characteristic age implies
Figure 1. The data points along with the median fit produced by our program. Here, color corresponds to time, such that purple (the beginning of the viridis scale)
represents the beginning of the year. The closer a data point is to a similar color in the fit, the better the fit. Ellipses surrounding each data point represent the 1σ,2σ,
and 3σpositional uncertainties.
Figure 2. A 2D histogram of the relevant parameter distributions. Here we
show the total proper motion, calculated using its individual components.
5
The Astrophysical Journal, 958:163 (9pp), 2023 December 1 Bruzewski et al.

a relatively long birth spin period, close to the present period of
115 ms, and has marked implications for kick models which
predict a close association between the direction of proper
motion and the rotation axis (e.g., Ng & Romani 2007;
Coleman & Burrows 2022). Examining this alignment would
require a more sensitive and full Stokes measurement of the
pulsar profile to improve existing measurements of both the
rotation and magnetic axes relative to the line of sight (Wu
et al. 2018). Figure 3illustrates the path of the pulsar over time,
out to the distance that would be covered under this assumption
of age.
Motivated by these findings, we consider the implications if
the pulsar age is equal to the age of the SNR of 10 ±2 kyr,
derivedinPaperIfrom several sources in the literature.
Using the measured proper motion, we find that the pulsar will
have traveled only
¢
6from its current location, placing it
outside the southeastern edge of CTB 1 and thus is not
associated. However, the difficulty in adopting this age is that
wewouldneedtoexplainhowPSRJ0002+6216 was born
inside the tail of its own PWN. In our original discovery
image (Paper I)the tail is at least
¢7
in length and in the lower
resolution Canadian Galactic Plane Survey (Kothes et al.
2006)images it can be traced
¢
11 back to the edge of CTB 1. It
seems more likely that the true age of PSR J0002+6216 is
greaterthan10kyr.
Finally, we can examine the kinematic age of the system. If
we assume that the pulsar originated from the geometric center
of the remnant, then we can use the ¢
¢2
81
offset between the
pulsar and the geometric center to find a kinematic age for the
system of approximately 47.6 ±0.8 kyr. This has the effect of
implying CTB 1 is significantly older than the estimates
discussed above. However, this is not completely unexpected,
as there is substantial evidence that CTB 1 is in its radiative
phase, and more specifically belongs to a class of mixed-
morphology remnants, as evidenced by its ring-like radio
emission and centrally located X-ray emission (Lazendic &
Slane 2006). This means that typical Sedov–Taylor scaling
relationships (Taylor 1950; Sedov 1959)one might use for
younger remnants may not produce accurate results from X-ray
properties and could vary significantly depending on various
assumptions.
To illustrate the effect of this increased age on the parameters
of the system, we can model the approximate evolution of
CTB 1 over time, following the procedure roughly described in
Vink (2012). The remnant initially follows the Sedov–Taylor
equations up until radiative losses become important, and
further evolution is directed by momentum conservation.
Adopting the equation for the latter phase from Toledo-Roy
et al. (2009), our model can be written as
»
<
-
Rt
En t t t
rt
t
tt
5.03 pc, ,
1.58 0.58 , , 3a
51 0 1525 rad
rad
rad
14
rad
⎜⎟
⎧
⎨
⎪
⎩
⎪⎛
⎝⎞
⎠
///
/
()
()
()
»tEn44.6 kyr, 3b
rad 51 0 13
//
() ()
Figure 3. The possible history of the pulsar, accounting for uncertainties in position and proper motion. While PSR J0002+6216 is likely to have originated at or near
the geometric center of CTB 1 (G116.9+0.2 in this figure), here we illustrate the possible positions if we assume that it originated somewhere beyond the SNR, out to
approximately the characteristic age of the pulsar. The black circles represent the possible 1σ,3σ, and 5σpositional scatter at each epoch, whereas the shaded regions
represent the paths between those epoch contours. Colored circles represent remnants found in Green (2019).
6
The Astrophysical Journal, 958:163 (9pp), 2023 December 1 Bruzewski et al.

»rEn23 pc, 3c
rad 5 1 0 13
//
() ()
where t
rad
and r
rad
are, respectively, the time and radius at
which the remnant transitions into the radiative phase. Note
that this evolution is dependent on the term (E
51
/n
0
), the ratio
of the supernova energy (in units of 10
51
erg)and the preshock
hydrogen density (in units of cm
−3
). This means that for a
given choice of energy, density, and distance, one can
accurately describe the angular radius as a function of time,
allowing constraints on those parameters knowing the current
angular size of CTB 1 (~¢17.5)and its presumed age (∼48 kyr).
From this we can see that from reasonable choices of distance,
(E
51
/n
0
)≈0.01–0.5, which implies a high-density medium, a
weak supernova, or some combination of the two.
Furthermore, for mixed-morphology remnants Cox et al.
(1999)derived that the time/radius at which the radio shell will
begin to form can be written as
»-
tEn53 kyr, 4
shell 51
314
0
47// ()
»-
rEn24.9 pc, 5
shell 51
27
0
37// ()
allowing one to place constraints similarly using the inner edge
of the radio emission ring (which has an angular size of ~
¢
13 ).
Figure 4shows such constraints for the inner and outer radii,
assuming a distance of 3 kpc so as to probe the supernova
energy and local preshock density. Most choices of distance
seem to require fairly typical energies, and the high value of the
density could be expected given that most mixed-morphology
remnants seem to form in regions of denser than the typical
interstellar medium (ISM). At the 3 kpc distance, the best match
to the constraints comes when E
51
≈0.7 and n
0
≈4cm
−3
.
Interestingly, this result of a higher-than-expected density
also has implications for the bow shock of PSR J0002+6216.
As described in Kumar et al. (2023), the pulsar is moving
rapidly through the local ISM, generating a bow shock around
it and trailing a long and highly collimated tail of emission for
several arcminutes. Paper Iestimated a Mach number
»200from initial measurements and comparisons with
systems with similar morphology (Kargaltsev & Pavlov 2008;
Ng et al. 2012). Assuming the distance and density used as an
example above (and a 10% helium mixture as in Paper I),we
estimate that the pulsar would have a Mach number »110,
still broadly consistent with the observed collimation of the tail.
Without an increase in density, the decrease in velocity due to
the updated proper motion would lower the overall Mach
number, resulting in a less striking collimation. Changes in
various parameters also have the effect of lowering the standoff
distance of the shock, derived by equating the wind pressure
and the ram pressure of the pulsar (Frail et al. 1996)to
=-
--
rnd
0.0015 3.5 cm 3 kpc pc, 6
SO 0
3
12 1
⎜⎟
⎛
⎝⎞
⎠⎛
⎝⎞
⎠()
which would be a few times smaller than the best estimates from
Kumar et al. (2023), and among the smallest standoffs known.
5. Conclusions
Using several years worth of HSA data, we have significantly
refined the estimate of the proper motion of the “cannonball”
pulsar J0002+6216, to a new value of μ
tot
=35.30 ±
0.60 mas yr
−1
. This new estimate is more than three times
smaller than the previous best estimates made in Paper Iusing
Fermi data, and as such would act to lower the spatial velocity
perpendicular to the line of sight. This new proper motion also
provides for a precise distance-independent kinematic age
measurement for the system of 47.60 ±0.80 kyr, significantly
older than previously estimated from the pulsar/remnant, but also
still significantly younger than the pulsar’s characteristic age.
Wealsoattempttoprovideamoreaccuratemeasureofthe
distance to the source, as that will also play a role in determining
the kinematic properties. While a value for the parallax of
ω=0.63 ±0.45 mas was obtained, the high uncertainties make a
definitive distance difficult to determine. These uncertainties are
dominated by scattering in the ionosphere, but we also see
uncertainty simply from trying to fit a position to a fairly weak
source. Refinement of this value would require both increased
sensitivity and more complex observing strategies, which could be
used to isolate and remove these ionospheric effects. We note that
this is likely to be an excellent target for the ngVLA in the coming
decade, as it will have baselines similar to the present-day VLBA,
but with significantly more antennas allowing for a dramatic
increase in sensitivity.
As an example, in 1 hr the HSA
8
can be expected to reach a
thermal noise level of approximately 13 μJy at 1.4 GHz, which
then necessitates longer and more difficult-to-schedule obser-
vations to provide enough signal to noise for a 14 μJy source
such as PSR J0002+6216. In comparison, a 1 hr observation
on the ngVLA
9
at 2.4 GHz should reach a thermal noise of
0.24 μJy. Even if we assume that the pulsar will dim at this
higher frequency by a factor of about 2, following the standard
Figure 4. Constraints in the density–energy space derived from the observed
outer and inner radii of CTB 1, assuming a distance of 3 kpc (near that found in
Hailey & Craig 1994). The red region represents outer radii which are within
¢1
of ¢
1
7.5, while the blue region represents inner/shell radii which are within
¢2
of
¢1
3. The space where both regions overlap is shown for 3 kpc as well as a few
other distances.
8
http://old.evlbi.org/cgi-bin/EVNcalc.pl
9
https://ngvla.nrao.edu/page/performance
7
The Astrophysical Journal, 958:163 (9pp), 2023 December 1 Bruzewski et al.

pulsar spectral index of α≈−1.4 (Bates et al. 2013),we
should still expect an approximate signal-to-noise ratio of 27
after just 1 hr. Combine this with the possibility of more
advanced observation and calibration techniques (such as
simultaneous subarray observations of a calibrator source), and
it is likely that ngVLA could provide an estimate of the
parallax at a significantly reduced uncertainty.
Beyond the parallax, further questions also remain for this
system. The proper motion, now reduced in magnitude but with
substantially less uncertainty, combined with the expansion of
CTB 1 seem to be indicative of a fairly large ISM density. The
implied value would be beyond the low density (∼0.1–1.0 cm
−3
)
which has typically been inferred from the remnant’s relatively
weak Hαemission (Landecker et al. 1982)or from the
nondetection of tracers such as CO (Zhou et al. 2023). While
this can be somewhat mitigated if one assumes a weaker
supernova energy (E
51
<1), the collimation of the pulsar’s bow
shock is decoupled from this term and still requires a higher
density given this lower velocity. Further work will be necessary
to constrain the excess density and identify its origin properly.
Acknowledgments
S.B., F.K.S., and G.B.T. acknowledge support from NASA
Fermi Guest Investigator Program, grants 80NSSC18K1725,
80NSSC19K1508, and 80NSSC22K1643. Work at NRL is
supported by NASA. The National Radio Astronomy Obser-
vatory is a facility of the National Science Foundation operated
under cooperative agreement by Associated Universities, Inc.
Support for this work was provided by the NSF through the
Grote Reber Fellowship Program administered by Associated
Universities, Inc. National Radio Astronomy Observatory and
the Student Observing Support Program. We thank Tom
Maccarone, Adam Deller, and David Green for insightful
discussions toward the goals of this work, as well as Jason Wu
for his supporting of early HSA observations from Effelsberg.
This work has made use of data from the European Space
Agency (ESA)mission Gaia (https://www.cosmos.esa.int/
gaia), processed by the Gaia Data Processing and Analysis
Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/
dpac/consortium). Funding for the DPAC has been provided
by national institutions, in particular, the institutions participat-
ing in the Gaia Multilateral Agreement.
This research has made use of NASA's Astrophysics Data
System and has made use of the NASA/IPAC Extragalactic
Database (NED), which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract
with the National Aeronautics and Space Administration. This
research has made use of data, software and/or web tools
obtained from NASA's High Energy Astrophysics Science
Archive Research Center (HEASARC), a service of Goddard
Space Flight Center and the Smithsonian Astrophysical
Observatory, of the SIMBAD database, operated at CDS,
Strasbourg, France.
Software:Astropy(http://www.astropy.org; Astropy Colla-
boration et al. 2022),CASA(http://casa.nrao.edu/;CASATeam
et al. 2022), corner.py (http://corner.readthedocs.io/en/latest/;
Foreman-Mackey 2016),Difmap(http://www.cv.nrao.edu/adass/
adassVI/shepherdm.html;Shepherd1997),emcee(http://emcee.
readthedocs.io; Foreman-Mackey et al. 2013), matplotlib (http://
www.matplotlib.org; Hunter 2007), Numpy (http://www.numpy.
org;Harrisetal.2020),andScipy(http://www.scipy.org; Virtanen
et al. 2020).
ORCID iDs
S. Bruzewski https://orcid.org/0000-0001-7887-1912
F. K. Schinzel https://orcid.org/0000-0001-6672-128X
G. B. Taylor https://orcid.org/0000-0001-6495-7731
P. Demorest https://orcid.org/0000-0002-6664-965X
M. Kerr https://orcid.org/0000-0002-0893-4073
P. Kumar https://orcid.org/0000-0003-0101-1986
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