# The Radio Parallax of the Crab Pulsar: A First VLBI Measurement Calibrated
with Giant Pulses

Rebecca Lin Department of Astronomy and Astrophysics, University of Toronto,
50 St. George Street, Toronto, ON M5S 3H4, Canada  Marten H. van Kerkwijk
Department of Astronomy and Astrophysics, University of Toronto, 50 St. George
Street, Toronto, ON M5S 3H4, Canada  Franz Kirsten Department of Space, Earth
and Environment, Chalmers University of Technology, Onsala Space Observatory,
439 92, Onsala, Sweden  Ue-Li Pen Institute of Astronomy and Astrophysics,
Academia Sinica, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt
Road, Taipei 10617, Taiwan  Canadian Institute for Theoretical Astrophysics,
60 St. George Street, Toronto, ON M5S 3H8, Canada  Canadian Institute for
Advanced Research, 180 Dundas St West, Toronto, ON M5G 1Z8, Canada  Dunlap
Institute for Astronomy and Astrophysics, University of Toronto, 50 St George
Street, Toronto, ON M5S 3H4, Canada  Perimeter Institute of Theoretical
Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada  Adam T.
Deller Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, John St., Hawthorn, VIC S1RR, Australia  ARC Centre of Excellence
for Gravitational Wave Discovery (OzGrav), Australia Rebecca Lin
<EMAIL_ADDRESS>
###### Abstract

We use four observations with the European VLBI network to measure the first
precise radio parallax of the Crab Pulsar. We found two in-beam extragalactic
sources just outside the Crab Nebula, with one bright enough to use as a
background reference source in our data. We use the Crab Pulsar’s giant pulses
to determine fringe and bandpass calibration solutions, which greatly improved
the sensitivity and reliability of our images and allowed us to determine
precise positional offsets between the pulsar and the background source. From
those offsets, we determine a parallax of $\pi=0.53\pm 0.06\rm{\;mas}$ and
proper motion of $(\mu_{\alpha},\mu_{\delta})=(-11.34\pm 0.06,2.65\pm
0.14)\rm{\;mas\;yr^{-1}}$, yielding a distance of
$d=1.90^{+0.22}_{-0.18}\rm{\;kpc}$ and transverse velocity of
$v_{\perp}=104^{+13}_{-11}\rm{\;km\;s^{-1}}$. These results are consistent
with the _Gaia_ 3 measurements, and open up the possibility of far more
accurate astrometry with further VLBI observations.

Parallax (1197) — Pulsars (1306) — Radio astrometry (1337) — Radio bursts
(1339) — Supernova remnants (1667) — Very long baseline interferometry (1769)

††facilities: The European VLBI Network is a joint facility of independent
European, African, Asian, and North American radio astronomy institutes.
Scientific results from data presented in this publication are derived from
the following EVN project codes: EK036 A-D. The NRAO is a facility of the
National Science Foundation operated under cooperative agreement by Associated
Universities, Inc.††software:  astropy (Astropy Collaboration et al., 2013,
2018, 2022), Baseband (Van Kerkwijk et al., 2020), CALC10 (Ryan & Vandenberg,
1980), CASA (The CASA Team et al., 2022), numpy (Harris et al., 2020),
matplotlib (Hunter, 2007), pulsarbat (Mahajan & Lin, 2022), scipy (Gommers et
al., 2022), SFXC (Keimpema et al., 2015), tempo2 (Hobbs & Edwards, 2012).

##  1 Introduction

The Crab Pulsar (PSR B0531+21 (catalog )) is one of the youngest pulsars,
situated at the heart of the Crab Nebula (catalog ), the remnant of supernova
SN 1054 (Duyvendak, 1942; Mayall & Oort, 1942). One of the most observed
pulsars, it has been continuously monitored by the $13\rm{\;m}$ dish at the
Jodrell Bank Observatory since 1984 (Lyne et al., 1993). The mean radio
profile of the pulsar shows multiple components, with the dominant ones being
the main pulse (MP) and interpulses (IPs) which are made up of “giant pulses”,
extremely narrow and bright pulses (for a review, see Eilek & Hankins 2016).
The pulse emissions are not only bright in radio but visible up to
$\gamma$-ray energies with the MP and IPs showing strong alignment across the
full electromagnetic spectrum (Moffett & Hankins, 1996). The pulsar also
undergoes glitches, discrete changes in the pulsar rotation rate, every few
years111http://www.jb.man.ac.uk/pulsar/glitches.html (e.g., Espinoza et al.
2011; Shaw et al. 2021). This wealth of pulse phenomena offers a great
opportunity for understanding the pulsar emission mechanism and possibly
constrain the nuclear physics of neutron star interiors. Additionally, the
young age of this system ($\sim\\!1000\rm{\;yr}$) makes it the ideal
laboratory to study not only the evolution of young pulsars but also pulsar
wind nebulae and supernova remnants.

Since the discovery of the Crab Pulsar, there have been several attempts to
constrain the distance and proper motion of the pulsar. For the distance,
early attempts based on various lines of evidence including kinematic,
spectroscopic and age-related considerations placed the pulsar between $1.4$
and $2.7\rm{\;kpc}$ (Trimble, 1973). From galactic electron density
distribution models, the distance to the pulsar can be estimated as
$\sim\\!1.7\rm{\;kpc}$ with the NE2001 model (Cordes & Lazio, 2002) and
$\sim\\!1.3\rm{\;kpc}$ with the YMW16 model (Yao et al., 2017).

While these estimates give a sense of the distance, none of them are precise
and none are direct measurements. Indeed, many rely on assumptions one would
like to test. For instance, the kinematic constraints implicitly assume a
roughly spherical nebula, while the dispersion-measure based distances rely on
electron density models.

For the proper motion, similarly early measurements of the Crab Pulsar were
relatively poor (Minkowski, 1970; Wyckoff & Murray, 1977; Caraveo & Mignani,
1999). A first relatively precise measurement was derived from Hubble Space
Telescope observations spanning over a decade, of
$(\mu_{\alpha},\mu_{\delta})=(-11.8\pm 0.4\pm 0.5,4.4\pm 0.4\pm
0.5)\rm{\;mas\;yr^{-1}}$ (Kaplan et al., 2008).

While a precise parallax and proper motion measurement of the Crab Pulsar
would be important, it is impeded by complications in doing astrometry at both
radio and optical wavelengths; furthermore, due to the glitches, pulsar timing
also cannot help (for a review, see Kaplan et al. 2008). In the optical, this
changed with the _Gaia_ mission, which presented the first precise astrometry
in its second data release (_Gaia_ DR2, Gaia Collaboration et al. 2018):
$\pi=0.27\pm 0.12\rm{\;mas}$ and $(\mu_{\alpha},\mu_{\delta})=(-11.8\pm
0.2,2.65\pm 0.17)\rm{\;mas\;yr^{-1}}$, respectively. The precision was
improved in the third data release, _Gaia_ DR3 (Gaia Collaboration et al.,
2022; Antoniadis, 2020): $\pi=0.51\pm 0.08\rm{\;mas}$ and
$(\mu_{\alpha},\mu_{\delta})=(-11.51\pm 0.10,2.30\pm
0.06)\rm{\;mas\;yr^{-1}}$, respectively.

While impressive, the difference in measured parallax between the two data
releases is somewhat worrying. It might be related to the fact that the
measurements are affected by the Crab Pulsar not being a typical optical
source, being embedded in an optically bright nebula and producing variable
emission near itself that would be only marginally resolved. Hence, it would
be best to have an independent measurement.

At radio wavelengths, Very Long Baseline Interferometry (VLBI) has been very
successful in measuring accurate parallaxes and proper motions for pulsars
both weaker and further away than the Crab Pulsar (e.g., Chatterjee et al.
2009; Deller et al. 2019). For the Crab Pulsar, a difficulty is that it is
embedded in a large, $\sim\\!6\arcmin\times 4\arcmin$, radio-bright nebula.
The high brightness effectively raises the overall system temperature in any
observation, making the average emission of the Crab Pulsar hard to detect.
This particularly affects observations at higher frequencies, where the
angular resolution is better but the pulse emission fainter
($f_{\nu}\propto\nu^{-3.1}$, Lorimer et al. 1995). But at lower frequencies,
where the pulsar is brighter, the ionosphere hinders astrometry, especially in
the absence of an extragalactic source that can be used as an in-beam
calibrator – which has to be outside the nebula, since otherwise it would be
severely broadened by scattering.

The problem of a lack of an in-beam calibrator has recently been solved: the
Wide-field VLBA Calibrator Survey (WFCS; Petrov 2021) lists a suitable nearby
source (one which we also discovered independently; see Section 2). With such
an in-beam extragalactic source, one avoids uncertainties in extrapolating
phasing solutions for a phase calibrator that is multiple degrees away. And
even if the in-beam calibrator is not very bright, a parallax measurement to
within a $0.1\rm{\;mas}$ should be possible if one can self-calibrate on the
pulsar (Fomalont et al., 1999; Deller et al., 2019).

For the Crab Pulsar, the bright nebula prevents self-calibration on the
regular pulse emission (e.g., Lobanov et al. 2011 used an external phase
calibrator for their VLBI imaging). In principle, the Crab Pulsar’s giant
pulses can help, as they are extremely bright and can be detected with single
dishes. Because they occur randomly in time, however, even pulsar gating on
the corresponding phase windows does not give very good signal-to-noise (S/N)
ratios.

In this paper, we present a technique using only the Crab Pulsar’s giant
pulses to model ionospheric and instrumentation variations for self-
calibration, and show that with the newly found nearby extragalactic reference
sources this enables precise parallax and proper motion measurements. In the
following, we first describe in Section 2 the VLBI data we took, as well as
the archival Very Large Array (VLA) dataset we used to search for
extragalactic references. In Section 3, we describe how we correlated our VLBI
data to form visibilities, calibrated the visibility data with the giant
pulses, and extracted positions of our sources. In Section 4, we derive the
parallax and proper motion from the positions. We compare with the _Gaia_
results in Section 5, and discuss ramifications and future prospects in
Section 6.

##  2 Observations

Figure 1:  VLA image of the Crab Pulsar with our ten candidate reference
sources marked (black circles). Overlaid are the approximate fields of view of
Effelseberg (orange dotted circle) and Badary (green dotted circle), as
implied by the full-width-half-maximum (FWHM) at our central observing
frequency (using effective apertures of $78\rm{\;m}$ and $30.5\rm{\;m}$,
respectively). All candidates are outside the field of view of Effelseberg,
but within the fields of view of most other dishes, which have diameters
comparable to Badary.

Our observations were taken with the European VLBI Network (EVN) at four
epochs between 2015 Oct and 2017 May, using a total of $10\rm{\;hr}$ (see
Table 1). Real-sampled data in left and right circular polarizations were
recorded in either 2-bit MARK 5B or VDIF format at each telescope, except for
the $70\rm{\;m}$ at the Robledo Deep Space Station (Ro) where only left
circular was available. The frequency range of $1594.49$–$1722.49\rm{\;MHz}$
was covered, in either eight contiguous $16\rm{\;MHz}$ or four contiguous
$32\rm{\;MHz}$ wide bands.

Individual scans on the Crab Pulsar lasted $\sim\\!5$ (EK036 C-D) to
$\sim\\!25\rm{\;min}$ (EK036 A-B), and were interleaved with observations of
J0530+1331 ($\sim\\!5$ to $\sim\\!10\rm{\;min}$, bandpass calibrator source at
$8\fdg 5$ from the target) and/or J0518+2054 ($\sim\\!0.5$ to
$\sim\\!1\rm{\;min}$, phase calibrator source at $4\fdg 0$ from the target).
The unusually long integration times on the target (in particular in EK036 A
and B) and short integrations on the phase calibrator were chosen because we
only intended the phase calibrator to provide a first crude calibration, just
enough to later perform self-calibration on the target; we realized phase
calibration at the level required for accurate astrometry would be impossible
given the large separation between target and phase calibrator and the bright
emission from the Crab Nebula.

After a first inspection of the data from EK036 A, however, it became clear
that our initial approach led to phase errors that were too large to perform
traditional self-calibration on the Crab Pulsar (see Figure 4). Hence, we
reduced the integration times on the target in the subsequent EK036 C and D
observations (e.g., Lobanov et al. 2011 were able to transfer phase solutions
from J0518+2054 using a much shorter calibrator/target cycle of
$2/5\rm{\;min}$.).

At the time of these observations, no extragalactic sources near the Crab
Pulsar were known that would be suitable as in-beam calibrators. Therefore,
the Crab Pulsar pointings were centered on the pulsar itself (see Table 2), in
the hope that suitable in-beam references could be found within the field of
view of the smaller participating stations.

Table 1: Observation and Giant Pulse Log Observation | Date | MJD |  $t_{\text{exp}}$aaTotal observing time, including telescope setup and calibration. |  $t_{\text{target}}$bbTotal exposure on target. | Telescopes usedcc We omit telescopes where data were corrupt, where significant RFI occurred and/or where we were unable to determine reliable fringe solutions. Asterisks beside a telescope indicate that the telescope was unable to see the source for the full observing time; furthermore, Ro had left-circular polarization only. Abbreviations are: Ef: the $100\rm{\;m}$ Effelsberg telescope; Bd: the $32\rm{\;m}$ at Badary; Hh: the $26\rm{\;m}$ in Hartebeesthoek; Jb: the $76\rm{\;m}$ Lovell telescope; Jb-II: the $25\rm{\;m}$ Mark II Telescope at the Jodrell Bank Observatory; Mc: the $32\rm{\;m}$ at Medicina; O8: the $25\rm{\;m}$ at Onsala; Ro: the $70\rm{\;m}$ at the Robledo Deep Space Station; Sv: the $32\rm{\;m}$ at Svetloe; T6: $65\rm{\;m}$ at Tianma; Tr: the $32\rm{\;m}$ at Toruń; Wb: the $25\rm{\;m}$ RT1 telescope at Westerbork; and Zc: the $32\rm{\;m}$ at Zelenchukskaya.  | DMddInferred from the giant pulses. | Giant PulseseeTotal number and rate of giant pulses (including both MP and IP) found using a detection threshold of $50\sigma$ on incoherently summed data (for details, see Lin et al. 2023).
---|---|---|---|---|---|---|---
code |  |  | (h) | (h) |  | ($\rm{pc\;cm^{-3}}$) | $N$ |  $r$ ($\rm{min^{-1}}$)
EK036 A | 2015 Oct 18 | 57313.96 | 4 | 3.27 | Ef Bd Hh Jb Mc O8 Ro* Sv T6* Tr Wb Zc | 56.7772 |  686 | 3.50
EK036 B | 2016 Oct 31 | 57692.98 | 2 | 1.65 | Ef Bd Hh Mc O8 Sv Wb Zc | 56.7668 |  1067 | 10.81
EK036 C | 2017 Feb 25 | 57809.67 | 2 | 1.15 | Ef Bd Hh Jb Mc O8 Sv Ur Wb Zc | 56.7725 |  281 | 4.08
EK036 D | 2017 May 28 | 57901.40 | 2 | 1.24 | Ef Bd Hh Jb-II Mc O8 Sv Tr Ur Wb Zc | 56.7851 |  740 | 9.94
Table 2: Target and Calibrator Scan Pointing Centers

Source | Right Ascension | Declination | Sep.
---|---|---|---
| ($\alpha$) | ($\delta$) | (°)
PSR B0531+21 | $05^{\rm{h}}34^{\rm{m}}31.934^{\rm{s}}$ | 22°00′52.191″ |  $\cdots$
J0530+1331 | $05^{\rm{h}}30^{\rm{m}}56.4167465^{\rm{s}}$ | 13°31′55.149516″ | 8.5
J0518+2054 | $05^{\rm{h}}18^{\rm{m}}03.8245128^{\rm{s}}$ | 20°54′52.497365″ | 4.0

Note. — Coordinates listed here are in the J2000 FK5 frame. The separation
between the calibrator sources and the Crab Pulsar is given in the last
column.

Table 3: Candidate Extragalactic Sources from VLA Data

Source | Right Ascension | Declination | Peak | Sep.
---|---|---|---|---
| ($\alpha$) | ($\delta$) | S/NaaAs found in the VLA data. | (′)
NE_CAND1 | $05^{\rm{h}}34^{\rm{m}}50.43^{\rm{s}}$ | 22°03′37.58″ | 16.3 | 5.1
NE_CAND2 | $05^{\rm{h}}34^{\rm{m}}50.93^{\rm{s}}$ | 22°06′39.79″ | 8.4 | 7.3
NE_CAND3 | $05^{\rm{h}}34^{\rm{m}}50.80^{\rm{s}}$ | 22°04′46.41″ | 4.9 | 5.9
NE_CAND4 | $05^{\rm{h}}35^{\rm{m}}14.05^{\rm{s}}$ | 22°04′07.43″ | 42.6 | 10.3
SE_CAND1 | $05^{\rm{h}}34^{\rm{m}}55.31^{\rm{s}}$ | 21°55′18.94″ | 14.8 | 7.8
SE_CAND2 | $05^{\rm{h}}34^{\rm{m}}40.74^{\rm{s}}$ | 21°55′15.31″ | 6.5 | 6.0
SE_CAND3 | $05^{\rm{h}}35^{\rm{m}}06.34^{\rm{s}}$ | 21°56′49.11″ | 463.0 | 8.9
NW_CAND1 | $05^{\rm{h}}34^{\rm{m}}07.34^{\rm{s}}$ | 22°08′45.63″ | 19.8 | 9.7
SW_CAND1 | $05^{\rm{h}}34^{\rm{m}}11.62^{\rm{s}}$ | 21°58′53.38″ | 18.7 | 5.1
SW_CAND2 | $05^{\rm{h}}34^{\rm{m}}15.14^{\rm{s}}$ | 21°57′12.71″ | 14.0 | 5.3

Note. — Coordinates listed here are in the J2000 FK5 frame. SE_CAND3 and
NE_CAND4 where later confirmed to be visible in the EVN data. The separation
between the candidate sources and the Crab Pulsar is given in the last column.

Given the high resolution of the EVN data, an untargeted search for in-beam
sources is nearly intractable. Instead, we searched for candidates in an
archival VLA dataset (project code 12B-380), taken in A-array configuration on
2012 Nov. 26 & 27 at $3\rm{\;GHz}$ (S band, covering 2–4 GHz). We used the
standard Common Astronomy Software Applications VLA calibration pipeline (casa
5.1.1, The CASA Team et al. 2022) to perform automatic flagging and
calibration of the two datasets. No careful flux calibration was applied.
After inspection of the data, we decided to focus only on the later run, from
2012 Nov. 27. We used the casa task tclean for imaging, limiting ourselves to
the lower half of the frequency band, i.e., 2–3$\rm{\;GHz}$. Moreover, we
limited the $uv$ range, excluding visibilities from baselines
$<\\!75\rm{\;k\lambda}$ in order to filter out the extended emission from the
Crab Nebula itself. Since our aim was to find compact sources within the field
of view of the VLA, we generate an image of $8192\times 8192$ pixels at an
angular resolution of $0.15\rm{\;arcsec/pixel}$, oversampling by about a
factor 3 the $477\times 470\rm{\;mas}$ beam (position angle $-60\arcdeg$). The
rms in the final image varies by a factor of up to 4 between the central
region and the outer region of the image because of the Crab Nebula’s
emission.

We exported the cleaned image as a FITS file and searched for radio sources by
normalizing the image relative to a median-filtered version to pick out
outliers. In this way, we found ten candidates, which we list in Table 3 and
show in Figure 1.

For all of the candidates, we created images from our EVN data, finding that
the two brightest ones were detected: SE_CAND3 and NE_CAND4 (see Section 3.3
and Figure 5). We were able to find a source at the location of SE_CAND3 in
the new Wide Field Very Long Baseline Array (VLBA) calibrator survey (Petrov,
2021), which lists it as the compact source WFCS J0535+2156, and in the Very
Large Array Sky Survey (VLASS) (Gordon et al., 2021) as VLASS1QLCIR
J053506.32+215649.3. Candidate NE_CAND4 was also seen in VLASS as VLASS1QLCIR
J053514.04+220407.7, but not in the VLBA catalogue.

Looking through other catalogues, we found sources matching the position of
SE_CAND3 in the Wide-field Infrared Survey Explorer Data Release (Cutri & et
al., 2012) and in the UKIRT Infrared Deep Sky Survey (Lawrence et al., 2012).
It also has a counterpart in _Gaia_ DR3 (Gaia Collaboration et al., 2022),
with a parallax and proper motion consistent with zero. Thus, it seems likely
that SE_CAND3 is an active galactic nucleus.

##  3 Correlation, Calibration, Images, Positions and Uncertainties

###  3.1 Visibilities

We correlated the data from the different telescopes using the publicly
available Super FX Correlator (sfxc 5.1; Keimpema et al. 2015), which prior to
preforming the correlations, corrects for station clock offsets and rates, as
well as for geometric delays using CALC10222https://space-
geodesy.nasa.gov/techniques/tools/calc_solve/calc_solve.html (Ryan &
Vandenberg, 1980). At this stage, no additional station-specific delays or
atmospheric distortions of the wavefront are taken into account.

For each observation, two correlation passes were performed. The first pass
correlated on the Crab Pulsar in pulsar gating mode (described below), and on
the bandpass and phase calibrators in ungated mode. The correlation centers in
this pass are the same as the antenna pointing centers given in Table 2. In
the second correlator pass, we correlated all target scans again, but now
ungated and centred on the locations of our candidate sources (see Table 3),
using sfxc’s multi-phase center mode.

For the pulsar gating, we created polyco files using Tempo2 (Hobbs & Edwards,
2012) with the Crab Pulsar ephemeris, starting from the ephemeris provided by
Jodrell Bank Observatory333http://www.jb.man.ac.uk/~pulsar/crab.html (Lyne et
al., 1993) and then adjusting the phase and dispersion measure to values found
in Lin et al. (2023) for the same data (see Table 1). With these, we used sfxc
to incoherently de-disperse444sfxc version 5.1 does not have coherent de-
dispersion capabilities., fold, and gate the pulsar observations on the MP
phase window ($2.1\%$ of the $\sim\\!33\rm{\;ms}$ pulse period).

With the gated mode, we gain S/N by removing time ranges when little if any
pulsar signal is present. However, since giant pulses are short in duration
(most of the signal is within the scattering timescale of $\sim\\!5\rm{\;\mu
s}$ at our observing frequency, see Lin et al. (2023) for examples of giant
pulses from these datasets) and occur only in some pulse rotations, one could
in principle get much better S/N ratio by only including pulse rotations in
which giant pulses occur. Furthermore, one could also include IP giant pulses
and possibly other pulse components. We did not pursue these potential
improvements, since we find below (in Section 3.3) that the S/N ratio of the
images created from the MP gated visibilities is much larger than that of the
in-beam candidate sources, and thus does not limit the accuracy of the
astrometry.

For all correlations, we used a spectral resolution of $4096$ channels across
the total bandwidth, limiting any dispersive in-channel smearing to
$3\rm{\;\mu s}$ (of order a giant pulse width). We used a temporal resolution
of $0.5\rm{\;s}$ to have the option of, in post-processing, select only time
integrations where particularly bright giant pulses occurred (but we did not
use this, as the S/N ratio sufficed). In total, for each observation eleven
visibility sets were created, one for the Crab Pulsar and one each for our
candidate sources. Calibrator visibility data were included in each visibility
set, hence each set contained three sources.

###  3.2 Calibration

We calibrated our visibilities with the help of casa 6.5, writing custom
calibration scripts to ensure that our calibrations are consistent across all
observations and to help track our configurations.

In preparation, we first converted visibility data to casa Measurement Sets
using Joint Institute for VLBI in Europe (JIVE) post-processing tools, and set
up antenna tables with diameters and axis offsets from the station summary
files. We also set up amplitude calibration tables with system temperature,
gain curve and primary beam corrections. Since system temperature and gain
curve measurements from the telescope logs were affected by the bright Crab
Nebula and thus unreliable (and some were simply missing), we instead used
nominal values taken from the EVN status
table555http://old.evlbi.org/user_guide/EVNstatus.txt and included the flux
density of the Crab Nebula $S_{\text{CN}}=955\nu^{-0.27}\rm{\;Jy}$, where
$\nu$ is our observing frequency in $\rm{GHz}$ (Bietenholz et al., 1997). As
our goal is precise astrometry, the true flux density of our sources is of
little importance and this flux scaling is sufficient for estimating which of
the candidate sources will likely be visible in the EVN datasets. We flagged
times and frequencies where the signal was poor (i.e., before the start and
end of each scan, and at passband edges), as well as particularly strong radio
frequency interference (RFI) previously detected in the baseband data (see Lin
et al. 2023), taking care to ensure that giant pulse signals were not
accidentally removed. Finally, to account for the reduced sensitivity away
from the antenna pointings, we applied a primary beam correction for our in-
beam candidate correlation centers (assuming an Airy disk with effective
aperture sizes provided by the JIVE team, separately for each of our eight
spectral windows).

For calibration, we started by determining phase and delay corrections due to
instrument and atmospheric variations towards our calibrator sources: we use
casa’s fringefit task to determine solutions in $60\rm{\;s}$ intervals for
each spectral window and polarization independently, with Effelsberg as the
reference antenna. We attempted transferring the fringe solutions to the Crab
Pulsar, but found relatively poor results (see Figure 4 and Section 3.3
below). This was not unexpected given that our calibrators are far from the
Crab Pulsar and that the target scans are relatively long compared to the
timescale of a few minutes of ionospheric variations.

Figure 2:  Amplitude (top), fringe delay (middle) and fringe phase (bottom) of
Badary relative to Effelsberg derived from giant pulses of EK036 A for the
frequency band $1626.49\\!-\\!1642.49\rm{\;MHz}$ in left circular polarization
(blue points). The opacity of the individual blue points scales with the
square root of the S/N of the giant pulse. The red lines shows our fits. The
gray shaded regions indicate when the telescope was not observing the Crab
Pulsar. The fringe rate increases strongly near the end as the Crab Pulsar is
setting at Badary. Figure 3:  Complex bandpass amplitude (top) and phase
(bottom) of Lovell relative to Effelsberg derived from giant pulses of EK036 A
for the frequency band $1626.49\\!-\\!1642.49\rm{\;MHz}$ in left circular
polarization (blue points). The red lines shows our fits. The gray shaded
regions indicate were we flagged data in casa as little signal is detected and
the passband rolls off. Figure 4:  Dirty images of the Crab Pulsar for our
four observations, comparing calibration solutions transferred from calibrator
sources with those using giant pulses for self-calibration. Top: In general,
transfer of the calibrator solutions to the Crab Pulsar resulted in poor
images, particularly for EK036 A and EK036 B where the calibrator/target cycle
was $\sim\\!0.5/25\rm{\;min}$ and ionospheric variations could not be modeled
well. In EK036 C and EK036 D, the calibrator/target cycle was
$\sim\\!1/5\rm{\;min}$ resulting in somewhat better dirty images. Bottom:
Using calibration solutions derived directly from the Crab Pulsar’s giant
pulses significantly improves the dirty images, with the Crab Pulsar clearly
located in the center.

Since the Crab Pulsar is the brightest of the in-beam sources, we instead used
it to self-calibrate. We first tried using the gated pulsar visibilities from
sfxc, but these do not have sufficient S/N on short integrations and hence the
resulting fringe solutions obtained using casa’s fringefit task were
unreliable, showing extreme variations without any discernible pattern.

Instead, we follow Lin et al. (2023) and use giant pulses to model the delays,
amplitude and phase rotations in each spectral window, and write these models
to casa compatible fringe and amplitude tables. Specifically, we use all giant
pulses (both MP and IP) detected with a S/N ratio of 50 in the incoherently
summed telescope data (a cut-off that ensures no false detections; we find no
pulses outside of the MP and IP phase windows, see Lin et al. 2023 for details
on the data reduction giant and giant pulse detection). We show an example of
an extreme fringe solution in Figure 2 (for Badary in the EK036 A observation,
where the Crab Pulsar is setting, causing a rapid increase in path length
through the ionosphere thus a large increase in fringe rate). We applied these
solutions to both the Crab Pulsar and the candidate in-beam source visibility
sets.

Since our detection rate is high, at $\sim\\!4\\!-\\!11$ every minute (see
Table 1), we can easily follow ionospheric variations towards the Crab Pulsar
and thus, unlike many calibration pipelines, do not apply archival global
ionosphere models such as the ionosphere vertical total electron content (TEC)
maps from NASA’s Crustal Dynamic Data Information System
(CDDIS)666https://cddis.nasa.gov/Data_and_Derived_Products/GNSS/atmospheric_products.html#iono
to the pulsar. We thus avoid uncertainties associated with the coarse
resolution of the TEC maps (5° in longitude by 2.5° in latitude and
$2\rm{\;hr}$ temporal resolution), the accuracy of $\sim\\!2$ to
$8\rm{\;TECU}$, and modeling assumptions required in using it (in casa, that
the ionosphere is a thin shell at a constant height of $450\rm{\;km}$).
Indeed, unreliable TEC information can sometimes produce negative parallaxes
with smaller errors (e.g. Deller et al. 2009), and recent analysis by Petrov
(2023) found that while TEC maps certainly can help improve measurements,
corrections to the default values and implementation were needed to obtain the
best absolute astrometry.

Still, even for our close in-beam reference source, the ionospheric
contributions will differ slightly between it and the Crab Pulsar. Thus, while
for our main analysis described below we do not use the TEC maps, we describe
a separate analysis applying a relative correction based on them in Appendix
A. We find that this gives consistent astrometric results with those of
Section 4 below.

For bandpass calibration (i.e., time-independent frequency calibration), we
again use our giant pulses, this time creating visibilities (for details, see
Lin et al. 2023), which we then averaged across time to solve for the complex
bandpass. As before, we use Effelsberg as our reference antenna. The complex
bandpass was smoothed with a median filter to remove any remaining RFI
contributions, then modeled with a simple cubic spline interpolation and
normalized to unity to preserve the flux density scale. We wrote our solutions
to casa compatible bandpass tables and applied the corrections to both the
Crab Pulsar and the candidate in-beam source visibility sets generated by
sfxc. We show a typical bandpass solution in Figure 3.

To verify our solutions, we also determined bandpass solutions using casa’s
bandpass task on the calibrator sources (using Effelsberg as the reference
antenna). We found that there were no significant differences between these
solutions and the ones determined from the giant pulses. Since we do not use
the calibrators elsewhere in our analysis, we decided to stick with the giant-
pulse bandpass solutions.

Lastly, we used casa’s gaincal task with a solution interval of
$\sim\\!5\rm{\;mins}$ to refine our amplitudes on the gated Crab Pulsar
visibilities. Overall, we find that this final amplitude correction showed no
variations related to scintillation as in Deller et al. (2009), as expected
given that the Crab Pulsar’s scintillation decorrelation bandwidth is much
smaller that the width of our subbands. As before, this adjustment to the
absolute flux density scaling should not affect positions (which we confirmed
by omitting this step), but does improve the S/N of images slightly, by
$\lesssim 5\%$. We again apply these solutions to both the Crab Pulsar and
candidate reference sources.

After calibration, we reduced the data to a more manageable size by lowering
the spectral resolution to $1024$ channels and the temporal resolution to
$2\rm{\;s}$.

###  3.3 Images

Figure 5:  Clean images of the Crab Pulsar and candidate reference sources
SE_CAND3 and NE_CAND4. The correlation centers for the pulsar are at the
origin by construction. The centers for the other two are relative to those of
the pulsar and thus these sources show reflex motion (for details, see Section
3.1).

All imaging was done using casa’s tclean task. For all sources, we started
with our full bandwidth, and used natural weighting to optimize S/N. The
synthesized beam is similar in all observation, with full width at half
maximum of roughly $4\rm{\;mas}\times 12\rm{\;mas}$, elongated in declination.
To adequately sample this beam, we use a pixel size of $0.5\rm{\;mas}$ for our
images. All our generated images are $4096\times 4096$ pixels in size.

We first formed dirty images for all our visibility sets. For the Crab Pulsar,
after applying the initial calibrations to the visibilities, we applied
further calibrations in two separate ways: one using only solutions inferred
from the calibrator sources (i.e., phase-referencing), and one using the
solutions obtained from giant pulses (i.e., self-calibration). We compare the
resulting dirty images of the Crab Pulsar in Figure 4. One sees that our giant
pulse self-calibration provides much better results. Thus, for the candidate
reference sources, after applying the initial calibrations, we applied further
calibrations using only solutions obtained from giant pulses (i.e.,
effectively phase-referenced relative to the Crab Pulsar).

From these dirty images, we were only able to confidently see two of the in-
beam candidate sources, SE_CAND3 and NE_CAND4. This is perhaps unsurprising,
since the other candidate sources are much weaker (see Table 3) and our
average sensitivity limit is quite poor: even away from the nebula, the rms is
$\sim\\!0.25\rm{\;mJy/beam}$. Another possibility is that some of these
sources are extended beyond our largest angular scales
($\sim\\!140\rm{\;mas}$) and hence resolved out.

For SE_CAND3, we measured fluxes between $\sim\\!13$ and $24\rm{\;mJy}$ in our
four epochs, while for NE_CAND4, we found fluxes between $\sim\\!3$ and
$5\rm{\;mJy}$, For comparison, for SE_CAND3, Petrov (2021) gives $4.3$ and
$7.6\rm{\;GHz}$ fluxes of $\sim\\!42$ and $\sim\\!36\rm{\;mJy}$, respectively,
in the WFCS, while Gordon et al. (2021) finds $3\rm{\;GHz}$ fluxes of
$\sim\\!39$ and $\sim\\!4\rm{\;mJy}$ in VLASS for SE_CAND3 and NE_CAND4,
respectively. These fluxes seem roughly consistent, taking into account our
approximate flux calibration, differences in observing frequency and
resolution, as well as possible source variability and structure.

The positions of SE_CAND3 and NE_CAND4 are within $50\rm{\;mas}$ of their
correlation centers (see Figure 5) and well within the uncertainties of
positions measured in the VLA data. Thus, phase drifts resulting from the
sources not being exactly at their correlation center are negligible (Fomalont
et al., 1999) and we do not re-correlate any data.

Table 4: Relative Positions between the Reference Sources and the Crab Pulsar

Observation | SE_CAND3 | NE_CAND4
---|---|---
code |  $\Delta\alpha^{\ast}$ ($\rm{mas}$) |  $\Delta\delta$ ($\rm{mas}$) |  $\Delta\alpha^{\ast}$ ($\rm{mas}$) |  $\Delta\delta$ ($\rm{mas}$)
EK036 A | $-478472.866\pm 0.013\pm 0.00\pm 0.04$ | $243085.54\pm 0.04\pm 0.07\pm 0.14$ | $-585690.43\pm 0.13$ | $-195268.6\pm 0.3$
EK036 B | $-478484.651\pm 0.02\phantom{0}\pm 0.02\pm 0.04$ | $243088.25\pm 0.06\pm 0.14\pm 0.14$ | $-585702.27\pm 0.2\phantom{0}$ | $-195265.4\pm 0.5$
EK036 C | $-478489.143\pm 0.03\phantom{0}\pm 0.03\pm 0.04$ | $243089.04\pm 0.09\pm 0.11\pm 0.14$ | $-585706.66\pm 0.18$ | $-195264.3\pm 0.4$
EK036 D | $-478491.763\pm 0.02\phantom{0}\pm 0.03\pm 0.04$ | $243089.84\pm 0.09\pm 0.09\pm 0.14$ | $-585708.76\pm 0.19$ | $-195262.1\pm 0.5$

Note. — All right ascension offset are calculated at the declination of the
pulsar. For SE_CAND3, we provide the measurement errors inferred from the fits
to the cleaned images, and estimates of the intra-epoch (see Section 3.4) and
inter-epoch errors (see Section 4) , respectively. These should be added in
quadrature to obtain the total uncertainty. For NE_CAND4, we list only the
errors from the position fit, since they are substantially larger than any
systematic effects.

To clean our images of the Crab Pulsar, SE_CAND3 and NE_CAND4, we apply a
single elliptical mask the size and orientation of the synthesized beam
centered on the peak flux in the dirty images to guide the cleaning. The
cleaning was stopped when the residual reached an rms equal to that of a
$4096\times 4096$ pixel dirty map centered $\sim\\!2\arcsec$ West from the
source (this is far enough away that there are no sources in the map and side-
lobe effects do not affect the field significantly so the average rms
measurement is more accurate; the noise level was measured using casa’s imstat
task).

We show our clean images of the Crab Pulsar and the two in-beam candidates
SE_CAND3 and NE_CAND4 in Figure 5. Since we self-calibrated on the pulsar, its
position is fixed to the antenna pointing position (see Table 2). Assuming an
extragalactic origin of the in-beam candidate sources, one expects them to
move slightly between epochs. As can be seen in Figure 5, this is indeed the
case.

###  3.4 Positions and their Uncertainties

We first tried fitting the cleaned source images with elliptical Gaussians
using the casa task imfit which is based on the procedure of Condon (1997).
However, we found that the position errors provided by imfit were odd – we
expected errors in right ascension and declination to scale with their
respective beam sizes, but found that the ratio was substantially different
(with errors in declination a factor 7-10 times larger than those in right
ascension, instead of the expected factor of $\sim\\!3$). We compared casa’s
imfit results with those from the jmfit task from the Astronomical Image
Processing System (aips; Associated Universities 1999) which is also based on
Condon (1997). The fitted positions were consistent, but the uncertainties
from aips’s jmfit do have the expected scaling with beam size.

To investigate this discrepancy in position uncertainties, we implemented our
own elliptical Gaussian fit routine in python. We discovered that the
discrepancy between casa and aips comes from how the noise and restoring beams
are used when determining S/N. We concluded that for a point source, the
procedure of aips’s jmfit task is the logical one: calculate the S/N from the
ratio of fitted peak amplitude and measured rms, and then estimate position
uncertainties as usual for correlated noise, by dividing the fitted beam sizes
by the S/N, and rotate to right ascension and declination (in our case, the
beam is nearly aligned, so the effects of rotation are tiny).

To derive our final positions, we used our fitting routine, taking a large
$128\times 128$ pixel window centered on the peak of each image to ensure a
good fit. The rms fluctuations were measured from the whole image with the
$128\times 128$ pixel window centered on the peak removed. We confirmed that
our fitted positions were in agreement with those from casa and aips and our
errors were consistent with those from aips, but different from those of
casa777Our final parallax value and uncertainty do not change if we use the
casa uncertainties, since the differences in the error estimates end up being
absorbed by the intra-epoch errors we add later.

From Figure 1, we see that both SE_CAND3 and NE_CAND4 are outside the FWHM of
the Effelsberg beam. To confirm that we have applied our primary beam
corrections correctly and Effelsberg does not affect the positions of the
candidate sources, we remove visibilities with baselines involving Effelsberg
and verified that the positions remain unchanged. As all images are calibrated
to the pulsar, the positions are relative to it, and thus the inferred
position of the pulsar should by definition be equal to the pointing center.
We confirmed that this was indeed the case (to well within nominal
uncertainties) by fitting the Crab Pulsar’s cleaned images as well.

The position uncertainties calculated this way may be slightly underestimated,
since we are fitting a zero level offset instead of fixing one, and errors
made in cleaning our images may not have fully propagated. In addition, errors
from EK036 B-D may be underestimated a bit more than those of EK036 A because
of their sparser coverage of the $uv$ plane (EK036 A was twice as long as the
other epochs and more EVN stations participated in the observation). Finally,
beyond fitting errors, there could be other residual cleaning artifacts, as
well as unmodeled ionospheric and instrumental effects.

To estimate such errors for each epoch individually (“intra-epoch error”), we
compare the position offsets of SE_CAND3 inferred from the full bandwidth with
offsets measured across spectral windows (similar to Deller et al. 2009; we
omitted NE_CAND4 as its S/N ratio in the images from the whole bands was
already rather poor). For this purpose, we made cleaned images of the sources
by splitting the total bandwidth into four, $32\rm{\;MHz}$ wide parts, and
fitted those to infer positions888We tried making images for every spectral
window (i.e., eight $16\rm{\;MHz}$ bands) but found the S/N to be too low for
reliable position measurements.. To account for this additional source of
uncertainty, we added intra-epoch errors for SE_CAND3 by the amount, added in
quadrature to each relative position measurement in an epoch, required to
produce a $\chi^{2}_{\rm red}=1$ (separately for right accession and
declination; see Table 4). This is a somewhat more conservative approach than
simply scaling the errors to obtain a $\chi^{2}_{\rm red}=1$, but ignores that
with only four measurements there is a reasonable probability to find either
smaller or larger $\chi^{2}_{\rm red}$ values by chance. It would be
worthwhile to explore this further for a larger data set.

In order to check the effect of duration, we also tried splitting the EK036 A
observation in half, such that the duration and $uv$ coverage are similar to
what we have in our other observations. We find that the intra-epoch errors in
both halves of the EK036 A observation increase and become comparable to those
in the other observations, suggesting that increased sampling in the $uv$
plane helps minimize systematic errors.

Our final adopted positions and the associated uncertainties are listed in
Table 4.

##  4 Astrometry

We use the position offsets from Table 4 to fit for the parallax ($\pi$),
proper motion ($\mu_{\alpha^{\ast}}$, $\mu_{\delta}$)999We denote differences
in right ascension multiplied by $\cos{\delta}$ with $\ast$. in right
ascension and declination respectively, and residual positional offset
($\Delta\alpha^{\ast}_{0}$, $\Delta\delta_{0}$), again in right ascension and
declination respectively. In terms of these parameters, the observed offsets
are fit to,

$\displaystyle\begin{aligned} \Delta\alpha^{\ast}_{i}&=\pi
f_{\alpha^{\ast},i}+\mu_{\alpha^{\ast}}(t_{i}-t_{0})+\Delta\alpha^{\ast}_{0},\\\
\Delta\delta_{i}&=\pi
f_{\delta,i}+\mu_{\delta}(t_{i}-t_{0})+\Delta\delta_{0},\end{aligned}$ (1)

where $t_{0}$ is a reference time – which we chose to be the average time over
our observations (MJD 57680) to minimize covariance between the proper motion
and the position offsets – and $f_{\alpha^{\ast}}$ and $f_{\delta}$ are the
parallax factors, given by

$\displaystyle f_{\alpha^{\ast}}(t)$
$\displaystyle=X(t)\sin(\alpha_{0})-Y(t)\cos(\alpha_{0}),$ (2) $\displaystyle
f_{\delta}(t)$
$\displaystyle=[X(t)\cos(\alpha_{0})+Y(t)\sin(\alpha_{0})]\sin(\delta_{0})$
$\displaystyle\phantom{=\ }-Z(t)\cos(\delta_{0}),$ (3)

where $X(t)$, $Y(t)$, and $Z(t)$ are the components of the barycentric
position of the Earth at time $t$, and $\alpha_{0}$ and $\delta_{0}$ are the
approximate position of the Crab Pulsar (i.e., we neglect differences between
the precise and approximate positions of the Crab Pulsar in the sine and
cosine terms). We use astropy (Astropy Collaboration et al., 2013, 2018, 2022)
to calculate the barycentric positions.

As mentioned in Section 2, SE_CAND3 is identified also in the VLBA calibrator
survey and is likely an active galactic nucleus. We compared the differences
in relative positions of SE_CAND3 and NE_CAND4 between the epochs. We found
these to be roughly consistent with zero and thus conclude NE_CAND4 likely
also is extragalactic in origin. As NE_CAND4 is much weaker than SE_CAND3 and
its position measurements are much less reliable, we will only use SE_CAND3 in
our parallax and proper motion fits below.

Our preliminary fit, including intra-epoch errors (see Section 3.4 and Table
4), yielded a parallax $\pi=0.54\pm 0.03\rm{\;mas}$ and proper motion of
$(\mu_{\alpha^{\ast}},\mu_{\delta})=(-11.31\pm 0.03,2.65\pm
0.08)\rm{\;mas\;yr^{-1}}$. We find $\chi^{2}_{\rm red}=2.3$, larger than the
expected unity. This could simply reflect that we have very few degrees of
freedom: in particular, the parallax fit is dominated by the four right
ascension offsets, to which three parameters are fitted, leaving only a single
degree of freedom. Indeed, Reid et al. (2017) showed that with four epochs and
one effective degree of freedom, the uncertainty on the uncertainty in the
parallax can be significant. Still, we will assume conservatively that,
instead, there are unmodeled systematic errors between epochs (“inter-epoch
errors”). We estimate these at $0.04\rm{\;mas}$ and $0.14\rm{\;mas}$, for
right accession and declination respectively, the value that, added in
quadrature to the measurement errors of both right ascension and declination
in all epochs, gives a $\chi^{2}_{\rm red}=1$ (see Table 4). The inter-epoch
errors in right ascension and declination were taken to be roughly
proportional to the beam size, as might be expected if the systematic effects
are due to phasing errors101010Our results suggest the error in declination
may be overestimated. If we take errors that are the same in each coordinate,
we find we require these to be $0.04{\rm\;mas}$. With these, we find identical
results except for a somewhat reduced final error in the proper motion in
declination.. With these, we derive the final fit results presented in Table 5
and shown in Figures 6 and 7.

Table 5: Astrometric Parameters

Parameter | EVN |  _Gaia_ DR3
---|---|---
$\pi\rm{\;(mas)}$ | $0.53\pm 0.06$ | $0.51\pm 0.08$
$\mu_{\alpha^{\ast}}\rm{\;(mas\;yr^{-1})}\ldots$ | $-11.34\pm 0.06$ | $-11.51\pm 0.10$
$\mu_{\delta}\rm{\;(mas\;yr^{-1})}$ | $2.65\pm 0.14$ | $2.30\pm 0.06$
$\alpha_{\rm J2000}$ | $5^{\rm{h}}34^{\rm{m}}31.93357^{\rm{s}}$ | $5^{\rm{h}}34^{\rm{m}}31.933561(5)^{\rm{s}}$
$\delta_{\rm J2000}$ | $22\arcdeg 00\arcmin 52.1927\arcsec$ | $22\arcdeg 00\arcmin 52.19236(6)\arcsec$
$d\rm{\;(kpc)}$ | $1.90^{+0.22}_{-0.18}$ | $1.96^{+0.36}_{-0.26}$
$v_{\perp}\rm{\;(km\;s^{-1})}$ | $104^{\phantom{.0}+13}_{\phantom{.0}-11}$ | $109^{\phantom{.0}+21}_{\phantom{.0}-15}$

Note. — Shown are both our results and those from _Gaia_. Distances are
calculated directly from the parallax measurements and the transverse velocity
$v_{\perp}$ is inferred from the proper motion and inferred distance.
Coordinates listed here are in the J2000 ICRS frame at MJD 57680 (our
reference epoch), with the uncertainties in our EVN results dominated by the
uncertainty in the position of our reference source ($\sim\\!1\rm{\;mas}$, see
text), and those for _Gaia_ given by the values in parentheses.

We also split the EK036 A observation in half in time and use the source
position fits obtained from each half as independent measurements in a new fit
for the parallax and proper motion. We find no significant changes in our fit
parameters; however, the error on the parallax reduces a little and there is
less of a need for an inter-epoch contribution. Since this may just be a
statistical fluke, we continue with our regular solution below.

One possible cause of systematic errors between epochs might be residual
ionospheric errors between the pulsar and SE_CAND3. To give a sense of the
size of the error from differences between the mean path length through the
ionosphere, we find from CDDIS TEC maps that the average residual vertical TEC
between antennas for SE_CAND3 relative to the pulsar is
$\sim\\!0.02\rm{\;TECU}$. Though the resolution and accuracy of the TEC maps
are poor, if we take the residual vertical TEC at face value, this translates
to an extra path length of $\sim\\!0.3\rm{\;cm}$, which, if systematic over
all telescopes, might induce position offsets of up to
$\sim\\!0.06\rm{\;mas}$, comparable to the inter-epoch errors we infer.
Indeed, in Appendix A, we show that the inclusion of residual ionosphere
correction from TEC map information results in shifts in position of this
order of magnitude. We also find this leads to somewhat smaller inferred
inter-epoch error and thus smaller uncertainties in the astrometric
parameters, but do not feel confident enough in these results to use them (see
Appendix A).

Another source of systematic error may come from refraction in the
interstellar medium. This will affect both the calibrators and the pulsar, but
differently. For an estimate, we use that Rudnitskii et al. (2016) measured an
scattering disk with full width at half maximum varying between
$0.5\\!-\\!1.3\rm{\;mas}$ at $18\rm{\;cm}$. The variability suggests that at
times the screen is asymmetric, which would lead to position offsets if not
accounted for. If this induces relative position shifts of order 10% of the
width, which seems not unreasonable, it would induce offsets of
$\sim\\!0.05\rm{\;mas}$, the right order of magnitude to account for the
possible systematic errors between epochs.

Figure 6:  Motion of the Crab Pulsar on the sky. Shown are 95.4% confidence
ellipses (in blue) of our four position offsets between the Crab Pulsar and
SE_CAND3 (see Table 4), after subtracting the best-fit offset at our reference
epoch, MJD 57680 (see Table 5). The modeled trajectory of the Crab Pulsar
based our best-fit astrometry (see Table 5) is shown by the orange line, with
the orange crosses corresponding to the modeled positions at the times of our
four observations.  Figure 7: Position residuals in right ascension (top) and
declination (bottom). The best-fit proper motion and relative offset between
the Crab Pulsar and SE_CAND3 have been removed, leaving just the signal due to
parallax. Different colors in the error bars indicate the effects of the
additional contributions to the uncertainties (see Table 4).  Figure 8: Error
ellipses of the parallax and proper motion from our work (in red) and _Gaia_
DR3 (in blue). The 68.3% and 95.4% confidence ellipses are shown by the solid
and dashed ellipses, respectively.

Finally, in our source images we see no apparent jets or other structures that
could induce positional errors. However, we note that Koryukova et al. (2022)
found that for SE_CAND3, the measured angular core size appeared to vary
between $\sim\\!0.07$ and $1.56\rm{\;mas}$ at $4-8\rm{\;GHz}$ for two
observations separated by $2.6\rm{\;yr}$. If real, this variability might also
change the centroid by amounts comparable to the systematic errors we infer
between epochs.

##  5 Results

We measure a parallax of $\pi=0.53\pm 0.06\rm{\;mas}$ for the Crab Pulsar and
infer a distance of $d=1.90^{+0.22}_{-0.18}$ by taking the reciprocal of the
measured parallax (we do not attempt to correct for Lutz & Kelker (1973) bias,
as it is not clear what the prior likelihood of finding a supernova remnant at
a given height above the galactic plane would be). From our best-fit proper
motion and inferred distance, we also derive a transverse velocity of
$v_{\perp}=104^{+13}_{-11}\rm{\;km\;s^{-1}}$. Using the coordinates of
SE_CAND3 (WFCS J0535+2156; Petrov 2021),

$\displaystyle\begin{aligned} \alpha_{\rm
J2000}&=5^{\rm{h}}35^{\rm{m}}06.34125^{\rm{s}},\\\ \delta_{\rm
J2000}&=21\arcdeg 56\arcmin 49.1045\arcsec,\end{aligned}$ (4)

in the J2000 International Celestial Reference System (ICRS) frame, we
determine the absolute position of the Crab Pulsar, in the same reference
frame, at MJD 57680, as,

$\displaystyle\begin{aligned} \alpha_{\rm
J2000}&=5^{\rm{h}}34^{\rm{m}}31.93357^{\rm{s}},\\\ \delta_{\rm
J2000}&=22\arcdeg 00\arcmin 52.1927\arcsec.\end{aligned}$ (5)

The uncertainty in our position for the Crab Pulsar is dominated by the
uncertainty in the position of SE_CAND3. The formal errors are $\pm
0.6\rm{\;mas}$ in each coordinate, but those are for the positions measured at
$4\\!-\\!8\rm{\;GHz}$ and we have not accounted for possible frequency
dependent core-shifts, which typically are of order $1\rm{\;mas}$ (Sokolovsky
et al., 2011). Hence, we estimate the uncertainties in the position at
$\sim\\!1\rm{\;mas}$ in each coordinate. Our measured and derived values are
presented in Table 5.

Comparing our results with those of _Gaia_ DR3, listed also in Table 5, we
find good agreement for the parallax but some tension for the proper motion.
To investigate this further, we show confidence ellipses of our parallax and
proper motion along with those from _Gaia_ DR3 in Figure 8. One sees that the
main discrepancy is for the proper motion in declination. Our measurements are
less sensitive in declination, since most EVN telescopes are spread East-West,
with most of the North-South constraint coming from Hartebeesthoek. Thus, we
may still underestimate the uncertainty of the proper motion in declination.
Fortunately, this should not affect the parallax: since the Crab Pulsar is
near the ecliptic, the parallax barely correlates with the proper motion in
declination. It has some correlation with proper motion in right ascension,
and, taking our error ellipse and that of _Gaia_ DR3 at face value, a slightly
lower parallax might be inferred.

We note that systematic effects may affect not just our measurement (see
above), but also the _Gaia_ DR3 astrometry of the Crab Pulsar. Indeed, the
values for the proper motion presented in _Gaia_ DR2 and DR3 differ
significantly (see Section 1). For the parallax, there is a possible overall
zero-point correction, but this is a small effect: applying the correction of
$-0.03\rm{\;mas}$ from Lindegren et al. (2021a) to the raw _Gaia_ DR3 parallax
from Table 5, yields $\pi=0.54\pm 0.08\rm{\;mas}$ (and an inferred distance of
$d=1.86_{-0.24}^{+0.32}$), which still agrees well with our measured and
inferred results. Another possible systematic effect is due to source color.
In _Gaia_ DR3, a 6-parameter fit including the pseudo-color was used for the
astrometry, and the solution shows fairly strong covariance between the
pseudo-color and the proper motion. According to Lindegren et al. (2021b), for
cases where strong correlation is seen, independent colour information may
significantly improve precision and accuracy. Here, one would have to be
somewhat careful, since the Crab Pulsar’s spectrum is not like that of regular
stars, for which the color corrections are calibrated. Finally, it also seems
possible that the variable optical emission surrounding the Crab Pulsar, such
as the wisp-like structures moving outwards from the pulsar, and halos and
knots close to it (Hester et al., 2002; Tziamtzis et al., 2009), might induce
positional offsets that could affect the astrometry. We conclude that in both
optical and radio it will be useful to analyze further observations and try to
carefully account for potential biases and systematic effects.

##  6 Future Work

Our pilot study shows that it is possible to measure the parallax of the Crab
Pulsar with VLBI. It should be relatively straightforward to improve the
measurement down to the $\lesssim\\!5\%$ level with further VLBI observations.
The EVN’s extended East-West baseline is particularly useful for constraining
the parallax of the Crab Pulsar as the synthesized beam is narrower in right
ascension and the pulsar is very close to the ecliptic. More observations
should be scheduled around October and March when the parallax signature would
peak in right ascension.

Future observations should try to include more small dishes to give maximum
sensitivity for the in-beam extragalactic reference sources. Furthermore, the
pointing center can be shifted towards SE_CAND3 and NE_CAND4 (e.g. centroid of
all sources) to boost the signal of those sources. With the higher
sensitivity, NE_CAND4 should become more useful in helping to constrain and
verify the astrometry. The addition of NE_CAND4 may also allow one to use the
MultiView technique (Rioja et al., 2017), or variants thereof (e.g. Reid et
al. 2017; Hyland et al. 2022), which has shown success in improving residual
spatial ionospheric corrections.

We have shown that our technique of using giant pulses to determine fringe and
bandpass solutions works exceedingly well for self-calibration, removing the
need to observe phase calibrators. Our estimates of systematic effects between
epochs suggests it is better to have a larger number of observations rather
than to have longer ones. However, since the intra-epoch error for EK036 A is
quite a bit smaller than in EK036 B-D, one would not want to reduce the time
too much.

With more observations and better time coverage, the error analysis could be
improved, e.g., using a bootstrap fit like was done by Deller et al. (2019).
Overall, we suggest at least 8 to 9 observations, each lasting at least
$2\rm{\;hr}$ in order to ensure sufficient $uv$ coverage. If the detection
rate of strong giant pulses remain high enough for self-calibration, it may be
better to observe at slightly higher frequencies, say $\sim\\!2\rm{\;GHz}$, to
reduce the effects of ionospheric variations and interstellar scattering.
Calibration of both scattering and residual ionospheric effects would be
helped by simultaneous dual-frequency or wide-band
($\gtrapprox\\!350\rm{\;MHz}$) observations (Brisken et al., 2000; Petrov,
2023). These wider-band observations may allow for an alternative measurement
of the small differences in contributions from the ionosphere between the Crab
Pulsar and the in-beam calibrators and improve on the application of TEC maps
described in Appendix A. Of course, ionospheric errors can also be reduced by
trying to schedule observations when the solar cycle is at its minimum.

Our technique of self-calibration using giant pulses should also help future
studies of the Crab Pulsar’s environment, such as the flaring regions within
the Crab Nebula studied by Lobanov et al. (2011). Furthermore, the technique
may also be useful for measuring distances to other giant pulse emitters such
PSR J1824$-$2452A (Bilous et al., 2015) and PSR J1823$-$3021A (Abbate et al.,
2020), as well as to bright rotating radio
transients111111http://astro.phys.wvu.edu/rratalog/ such as PSR J1819$-$1458
and PSR J1840$-$1419, which have bursts every $\sim\\!3.4\rm{\;min}$ and
$\sim\\!1.3\rm{\;min}$, respectively (McLaughlin et al., 2006). For PSR
J1824–2452A and PSR J1823-3021A, which are both in globular clusters, using
their pulses for phase calibration would also aid searches of further globular
cluster pulsars and other radio emitters. Similarly, applying our technique to
the Crab Pulsar twin PSR J0540-6919, which also exhibits giant pulses (Geyer
et al., 2021), may help searches of new radio sources in the Large Magellanic
Cloud.

Our cleaned images in FITS format are made available as a dataset
at10.5281/zenodo.7910778 (catalog 10.5281/zenodo.7910778). The raw baseband
data along with our custom scripts are available upon request121212As the
baseband data were correlated by us and not by the JIVE team, the visibility
products are not available on the EVN Data Archive..

## Acknowledgements

We thank Cees Bassa for his contribution to the EVN proposal and the anonymous
referee for useful comments. R.L. thanks Aard Keimpena, Bob Campbell, Benito
Marcote, and Marjolein Verkouter for useful advice on using sfxc and JIVE
post-processing tools. R.L. thanks the National Radio Astronomy Observatory
(NRAO) helpdesk for advice on using casa, including details of the structure
of the various tables, and for providing access to the NRAO computing
facilities. Computations were performed on the New Mexico Array Science Center
(NMASC) cluster and the Niagara supercomputer at the SciNet HPC Consortium
(Loken et al., 2010; Ponce et al., 2019). SciNet is funded by: the Canada
Foundation for Innovation; the Government of Ontario; Ontario Research Fund -
Research Excellence; and the University of Toronto. M.Hv.K. is supported by
the Natural Sciences and Engineering Research Council of Canada (NSERC) via
discovery and accelerator grants, and by a Killam Fellowship. F.K.
acknowledges support from the Onsala Space Observatory for the provisioning of
its facilities/observational support. The Onsala Space Observatory national
research infrastructure is funded through Swedish Research Council grant No
2017-00648. U.-L.P. receives support from Ontario Research Fund-Research
Excellence Program (ORF-RE), NSERC [funding reference Nos. RGPIN-2019-067, CRD
523638-18, 555585-20], Canadian Institute for Advanced Research (CIFAR), the
National Science Foundation of China (grant No. 11929301), Alexander von
Humboldt Foundation, and the National Science and Technology Council (NSTC) of
Taiwan (111-2123-M-001, -008, and 111-2811-M-001, -040).

##  Appendix A Differential Ionosphere Corrections using TEC Maps

A source of error in our position measurements of the Crab Pulsar relative to
our in-beam calibrators arises from slight differences in the total electron
column (TEC) in the ionosphere between the different sources. Estimates of
these differences can be made from TEC maps, as is becoming common in VLBI
astrometry. While this use of TEC maps, including the underlying assumptions
about the ionosphere, have not been fully validated (see Petrov 2023), we
follow it to get a sense of the improvement that may be attainable. Since our
giant-pulse based fringe solutions already include the contribution of the
ionosphere towards the Crab Pulsar (along with delays introduced by antenna
location, electronics, geometric models, etc.), we only need to apply a
differential correction for the extragalactic sources.

To determine the residual ionospheric corrections, we first download CDDIS TEC
maps using casa’s tec_maps function. We then use casa’s gencal task to
estimate the line-of-sight TEC from each antenna to each of our sources across
each observation (gencal models the ionosphere as a thin shell at a constant
height of $450\rm{\;km}$). Using custom scripts, we then calculate the
differential TEC between the Crab Pulsar and extragalactic sources for each
antenna and write the residuals into our own casa compatible calibration
tables. These new calibration tables are applied to the visibilities data of
SE_CAND3 and NE_CAND4 using casa’s applycal task (after applying the
calibrations described in Section 3.2). We then create images and extract
position offsets as in Sections 3.3 and 3.4. We list the resulting offsets in
Table 6, and fit these to our astrometric model (including intra- and inter-
epoch errors estimated like in Sections 3.4 and 4).

We find a parallax of $\pi=0.49\pm 0.04\rm{\;mas}$ and proper motion of
$(\mu_{\alpha},\mu_{\delta})=(-11.41\pm 0.05,2.54\pm
0.11)\rm{\;mas\;yr^{-1}}$, i.e., values consistent with our results in Section
5 and with _Gaia_ DR3. We note that the uncertainties are slightly reduced, a
consequence of the fit to the offsets being somewhat better, thus reducing the
estimated inter-epoch error contribution to the uncertainties. While
encouraging, we caution that with the small number of data points, a reduction
by chance is not unlikely, in particular in the presence of possible other
sources of systematic error such as refraction in the interstellar medium and
source variability (see Section 4).

As a further check on the reliability, we also tried applying TEC corrections
when transferring calibrator solutions to the pulsar as above, but this time
applying a differential correction for the pulsar. As the angular separations
of the calibrator sources and pulsar are quite large and the calibrator/pulsar
cycle is quite long, we also tried removing the ionospheric contributions
towards the calibrators using the TEC maps before solving for the calibrator
fringes (in the hopes that these new calibrator fringe solutions with slower
time variations can be better extrapolated to the pulsar). We then applied TEC
corrections towards the Crab Pulsar and the new calibrator fringe solution to
the pulsar. Both methods resulted in similar quality images. If the
corrections were good, we expect that with these solutions, the dirty images
would improve, i.e., that we would see the Crab Pulsar becoming more point-
like. However, we found that with the TEC corrections, the dirty images were
of poorer quality (more smeared) than those shown in the top panels of Figure
4. Given this contradictory result, we concluded that without better
understanding it was best not to use the above TEC-map assisted astrometry,
even though it gave notionally better results. Since our “ionosphere
corrected” SE_CAND3 and NE_CAND4 images may still be useful for future
astrometry of the Crab Pulsar, we provide these (along with those from Figure
5) at 10.5281/zenodo.7910778 (catalog 10.5281/zenodo.7910778).

Table 6: Relative Positions between the Reference Sources and the Crab Pulsar,
with Ionosphere Corrections applied.

Observation | SE_CAND3 | NE_CAND4
---|---|---
code |  $\Delta\alpha^{\ast}$ ($\rm{mas}$) |  $\Delta\delta$ ($\rm{mas}$) |  $\Delta\alpha^{\ast}$ ($\rm{mas}$) |  $\Delta\delta$ ($\rm{mas}$)
EK036 A | $-478472.634\pm 0.015\pm 0.00\pm 0.03$ | $243085.61\pm 0.04\pm 0.08\pm 0.09$ | $-585690.03\pm 0.14$ | $-195268.3\pm 0.3$
EK036 B | $-478484.582\pm 0.02\phantom{0}\pm 0.02\pm 0.03$ | $243088.27\pm 0.06\pm 0.14\pm 0.09$ | $-585702.12\pm 0.2\phantom{0}$ | $-195265.3\pm 0.5$
EK036 C | $-478489.033\pm 0.03\phantom{0}\pm 0.02\pm 0.03$ | $243088.92\pm 0.09\pm 0.10\pm 0.09$ | $-585706.43\pm 0.18$ | $-195264.2\pm 0.4$
EK036 D | $-478491.501\pm 0.02\phantom{0}\pm 0.03\pm 0.03$ | $243089.78\pm 0.09\pm 0.09\pm 0.09$ | $-585708.26\pm 0.19$ | $-195261.8\pm 0.5$

Note. — Values and uncertainties are as for Table 4, except that here they
were derived from data for which we tried to correct for differential
ionospheric effects using TEC maps.

## References

  * Abbate et al. (2020) Abbate, F., Bailes, M., Buchner, S. J., et al. 2020, MNRAS, 498, 875, doi: 10.1093/mnras/staa2510
  * Antoniadis (2020) Antoniadis, J. 2020, Research Notes of the American Astronomical Society, 4, 223, doi: 10.3847/2515-5172/abd189
  * Associated Universities (1999) Associated Universities, I. 1999, AIPS: Astronomical Image Processing System, Astrophysics Source Code Library, record ascl:9911.003.  http://ascl.net/9911.003
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Bietenholz et al. (1997) Bietenholz, M. F., Kassim, N., Frail, D. A., et al. 1997, ApJ, 490, 291, doi: 10.1086/304853
  * Bilous et al. (2015) Bilous, A. V., Pennucci, T. T., Demorest, P., & Ransom, S. M. 2015, ApJ, 803, 83, doi: 10.1088/0004-637X/803/2/83
  * Brisken et al. (2000) Brisken, W. F., Benson, J. M., Beasley, A. J., et al. 2000, The Astrophysical Journal, 541, 959, doi: 10.1086/309492
  * Caraveo & Mignani (1999) Caraveo, P. A., & Mignani, R. P. 1999, A&A, 344, 367.  https://arxiv.org/abs/astro-ph/9901031
  * Chatterjee et al. (2009) Chatterjee, S., Brisken, W. F., Vlemmings, W. H. T., et al. 2009, ApJ, 698, 250, doi: 10.1088/0004-637X/698/1/250
  * Condon (1997) Condon, J. J. 1997, PASP, 109, 166, doi: 10.1086/133871
  * Cordes & Lazio (2002) Cordes, J. M., & Lazio, T. J. W. 2002, arXiv e-prints, astro.  https://arxiv.org/abs/astro-ph/0207156
  * Cutri & et al. (2012) Cutri, R. M., & et al. 2012, VizieR Online Data Catalog, II/311
  * Deller et al. (2009) Deller, A. T., Tingay, S. J., & Brisken, W. 2009, ApJ, 690, 198, doi: 10.1088/0004-637X/690/1/198
  * Deller et al. (2019) Deller, A. T., Goss, W. M., Brisken, W. F., et al. 2019, ApJ, 875, 100, doi: 10.3847/1538-4357/ab11c7
  * Duyvendak (1942) Duyvendak, J. J. L. 1942, PASP, 54, 91, doi: 10.1086/125409
  * Eilek & Hankins (2016) Eilek, J. A., & Hankins, T. H. 2016, Journal of Plasma Physics, 82, 635820302, doi: 10.1017/S002237781600043X
  * Espinoza et al. (2011) Espinoza, C. M., Lyne, A. G., Stappers, B. W., & Kramer, M. 2011, MNRAS, 414, 1679, doi: 10.1111/j.1365-2966.2011.18503.x
  * Fomalont et al. (1999) Fomalont, E. B., Goss, W. M., Beasley, A. J., & Chatterjee, S. 1999, AJ, 117, 3025, doi: 10.1086/300877
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1, doi: 10.1051/0004-6361/201833051
  * Gaia Collaboration et al. (2022) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2022, arXiv e-prints, arXiv:2208.00211, doi: 10.48550/arXiv.2208.00211
  * Geyer et al. (2021) Geyer, M., Serylak, M., Abbate, F., et al. 2021, MNRAS, 505, 4468, doi: 10.1093/mnras/stab1501
  * Gommers et al. (2022) Gommers, R., Virtanen, P., Burovski, E., et al. 2022, scipy/scipy: SciPy 1.8.0, v1.8.0, Zenodo, Zenodo, doi: 10.5281/zenodo.595738
  * Gordon et al. (2021) Gordon, Y. A., Boyce, M. M., O’Dea, C. P., et al. 2021, ApJS, 255, 30, doi: 10.3847/1538-4365/ac05c0
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hester et al. (2002) Hester, J. J., Mori, K., Burrows, D., et al. 2002, ApJ, 577, L49, doi: 10.1086/344132
  * Hobbs & Edwards (2012) Hobbs, G., & Edwards, R. 2012, Tempo2: Pulsar Timing Package, Astrophysics Source Code Library, record ascl:1210.015.  http://ascl.net/1210.015
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Hyland et al. (2022) Hyland, L. J., Reid, M. J., Ellingsen, S. P., et al. 2022, ApJ, 932, 52, doi: 10.3847/1538-4357/ac6d5b
  * Kaplan et al. (2008) Kaplan, D. L., Chatterjee, S., Gaensler, B. M., & Anderson, J. 2008, ApJ, 677, 1201, doi: 10.1086/529026
  * Keimpema et al. (2015) Keimpema, A., Kettenis, M. M., Pogrebenko, S. V., et al. 2015, Experimental Astronomy, 39, 259, doi: 10.1007/s10686-015-9446-1
  * Koryukova et al. (2022) Koryukova, T. A., Pushkarev, A. B., Plavin, A. V., & Kovalev, Y. Y. 2022, MNRAS, 515, 1736, doi: 10.1093/mnras/stac1898
  * Lawrence et al. (2012) Lawrence, A., Warren, S. J., Almaini, O., et al. 2012, VizieR Online Data Catalog, II/314
  * Lin et al. (2023) Lin, R., van Kerkwijk, M. H., Main, R., et al. 2023, ApJ, 945, 115, doi: 10.3847/1538-4357/acba95
  * Lindegren et al. (2021a) Lindegren, L., Bastian, U., Biermann, M., et al. 2021a, A&A, 649, A4, doi: 10.1051/0004-6361/202039653
  * Lindegren et al. (2021b) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021b, A&A, 649, A2, doi: 10.1051/0004-6361/202039709
  * Lobanov et al. (2011) Lobanov, A. P., Horns, D., & Muxlow, T. W. B. 2011, A&A, 533, A10, doi: 10.1051/0004-6361/201117082
  * Loken et al. (2010) Loken, C., Gruner, D., Groer, L., et al. 2010, Journal of Physics: Conference Series, 256, 012026, doi: 10.1088/1742-6596/256/1/012026
  * Lorimer et al. (1995) Lorimer, D. R., Yates, J. A., Lyne, A. G., & Gould, D. M. 1995, Monthly Notices of the Royal Astronomical Society, 273, 411, doi: 10.1093/mnras/273.2.411
  * Lutz & Kelker (1973) Lutz, T. E., & Kelker, D. H. 1973, PASP, 85, 573, doi: 10.1086/129506
  * Lyne et al. (1993) Lyne, A. G., Pritchard, R. S., & Graham Smith, F. 1993, MNRAS, 265, 1003, doi: 10.1093/mnras/265.4.1003
  * Mahajan & Lin (2022) Mahajan, N., & Lin, R. 2022, theXYZT/pulsarbat: pulsarbat 0.0.7, v0.0.7, Zenodo, doi: 10.5281/zenodo.6934356
  * Mayall & Oort (1942) Mayall, N. U., & Oort, J. H. 1942, PASP, 54, 95, doi: 10.1086/125410
  * McLaughlin et al. (2006) McLaughlin, M. A., Lyne, A. G., Lorimer, D. R., et al. 2006, Nature, 439, 817, doi: 10.1038/nature04440
  * Minkowski (1970) Minkowski, R. 1970, PASP, 82, 470, doi: 10.1086/128929
  * Moffett & Hankins (1996) Moffett, D. A., & Hankins, T. H. 1996, ApJ, 468, 779, doi: 10.1086/177734
  * Petrov (2021) Petrov, L. 2021, AJ, 161, 14, doi: 10.3847/1538-3881/abc4e1
  * Petrov (2023) Petrov, L. 2023, The Astronomical Journal, 165, 183, doi: 10.3847/1538-3881/acc174
  * Ponce et al. (2019) Ponce, M., van Zon, R., Northrup, S., et al. 2019, CoRR, abs/1907.13600.  https://arxiv.org/abs/1907.13600
  * Reid et al. (2017) Reid, M. J., Brunthaler, A., Menten, K. M., et al. 2017, AJ, 154, 63, doi: 10.3847/1538-3881/aa7850
  * Rioja et al. (2017) Rioja, M. J., Dodson, R., Orosz, G., Imai, H., & Frey, S. 2017, AJ, 153, 105, doi: 10.3847/1538-3881/153/3/105
  * Rudnitskii et al. (2016) Rudnitskii, A. G., Karuppusamy, R., Popov, M. V., & Soglasnov, V. A. 2016, Astronomy Reports, 60, 211, doi: 10.1134/S1063772916020116
  * Ryan & Vandenberg (1980) Ryan, J. W., & Vandenberg, N. R. 1980, in Bulletin of the American Astronomical Society, Vol. 12, 457.  https://ui.adsabs.harvard.edu/abs/1980BAAS...12..457R
  * Shaw et al. (2021) Shaw, B., Keith, M. J., Lyne, A. G., et al. 2021, MNRAS, 505, L6, doi: 10.1093/mnrasl/slab038
  * Sokolovsky et al. (2011) Sokolovsky, K. V., Kovalev, Y. Y., Pushkarev, A. B., & Lobanov, A. P. 2011, A&A, 532, A38, doi: 10.1051/0004-6361/201016072
  * The CASA Team et al. (2022) The CASA Team, Bean, B., Bhatnagar, S., et al. 2022, arXiv e-prints, arXiv:2210.02276.  https://arxiv.org/abs/2210.02276
  * Trimble (1973) Trimble, V. 1973, PASP, 85, 579, doi: 10.1086/129507
  * Tziamtzis et al. (2009) Tziamtzis, A., Lundqvist, P., & Djupvik, A. A. 2009, A&A, 508, 221, doi: 10.1051/0004-6361/200912031
  * Van Kerkwijk et al. (2020) Van Kerkwijk, M. H., Zhu, C. C., Guo, S. G., et al. 2020, mhvk/baseband, v4.0.3, Zenodo, doi: 10.5281/zenodo.4292543
  * Wyckoff & Murray (1977) Wyckoff, S., & Murray, C. A. 1977, MNRAS, 180, 717, doi: 10.1093/mnras/180.4.717
  * Yao et al. (2017) Yao, J. M., Manchester, R. N., & Wang, N. 2017, ApJ, 835, 29, doi: 10.3847/1538-4357/835/1/29