# MeerKAT Pulsar Timing Array parallaxes and proper motions

M. Shamohammadi,1,2 M. Bailes,1,2 C. Flynn,1,2 D. J. Reardon,1,2 R. M.
Shannon,1,2 S. Buchner,3A. D. Cameron,1,2 F. Camilo,3 A. Coronigu,4 M.
Geyer,3,5 M. Kramer,6,7 M. Miles,1,2 R. Spiewak7

1Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, PO Box 218, Hawthorn, VIC 3122, Australia
2ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav), Mail H29,
Swinburne University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia
3South African Radio Astronomy Observatory, 2 Fir Street, Black River Park,
Observatory 7925, South Africa (SARAO)
4INAF - Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047
Selargius (CA), Italy
5Department of Astronomy, University of Cape Town, Rondebosch, Cape Town 7700,
South Africa
6Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn,
Germany
7Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Manchester M13 9PL, UK<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We have determined positions, proper motions, and parallaxes of $77$
millisecond pulsars (MSPs) from $\sim 3$ years of MeerKAT radio telescope
observations. Our timing and noise analyses enable us to measure $35$
significant parallaxes ($12$ of them for the first time) and $69$ significant
proper motions. Eight pulsars near the ecliptic have an accurate proper motion
in ecliptic longitude only. PSR J0955$-$6150 has a good upper limit on its
very small proper motion ($<$0.4 mas yr-1). We used pulsars with accurate
parallaxes to study the MSP velocities. This yields $39$ MSP transverse
velocities, and combined with MSPs in the literature (excluding those in
Globular Clusters) we analyse $66$ MSPs in total. We find that MSPs have, on
average, much lower velocities than normal pulsars, with a mean transverse
velocity of only $78(8)$ km s-1 (MSPs) compared with $246(21)$ km s-1 (normal
pulsars). We found no statistical differences between the velocity
distributions of isolated and binary millisecond pulsars. From Galactocentric
cylindrical velocities of the MSPs, we derive 3-D velocity dispersions of
$\sigma_{\rho}$, $\sigma_{\phi}$, $\sigma_{z}$ = $63(11)$, $48(8)$, $19(3)$ km
s-1. We measure a mean asymmetric drift with amplitude $38(11)$ km s-1,
consistent with expectation for MSPs, given their velocity dispersions and
ages. The MSP velocity distribution is consistent with binary evolution models
that predict very few MSPs with velocities $>300$ km s-1 and a mild
anticorrelation of transverse velocity with orbital period.

###### keywords:

parallaxes – proper motions – stars: neutron – pulsars: general

††pubyear: 2015††pagerange: MeerKAT Pulsar Timing Array parallaxes and proper
motions–A.2

##  1 Introduction

Studying the transverse velocities of pulsars enables us to better understand
their origins in the Milky Way, the evolutionary scenarios leading up to their
birth, and the kick velocities that they receive when they are born (Lyne et
al., 1982; Cordes & Chernoff, 1997). Distances and velocities of pulsars also
enable us to constrain the dynamical models of supernovae (e.g. Gaensler &
Johnston, 1995) and understand the kinematics of binary evolution (e.g.
Gonzalez et al., 2011; Freire et al., 2011).

Transverse velocities for radio pulsars are primarily determined using two
different methods: firstly, measurements of pulsar proper motions and
parallaxes through pulsar timing or Very Long Baseline Interferometry (VLBI)
(e.g. Reardon et al., 2021; Ding et al., 2023), and secondly, using
interstellar scintillation patterns in a pulsar’s dynamic spectrum (e.g.
Bogdanov et al., 2002; Reardon et al., 2020). Slow (or young) pulsars possess
timing noise that makes simple fits for proper motion unreliable and most of
the first proper motions determined were reliant on interferometric techniques
(e.g., Lyne et al., 1982; Bailes et al., 1990; Harrison et al., 1993).

A breakthrough study of pulsar transverse velocities was made by Hobbs et al.
(2005). They collated all the pulsar proper motions from the literature and
significantly added to them with a new timing technique from the Jodrell Bank
timing data to explore a total sample of $233$ pulsar proper motions. They
found the two-dimensional velocities of young pulsars range from a few tens to
$\approx 1600$ km s-1. They used a novel deconvolution technique to derive a
mean 3D pulsar birth velocity of approximately $400\pm 40$ km s-1 for young
($<$3 Myr) pulsars, suggesting that pulsars receive large kicks at birth. Such
high kick velocities imply that the local convective instability in the
collapsed stellar core (Lai et al., 2001) is unlikely to be a pulsar kick
mechanism (the mechanism that causes the neutron star to get kicked with a
different velocity compared to its progenitor star after the supernova
explosion), and they favoured more energetic mechanisms such as global
asymmetric perturbations or neutrino emission in the supernovae. They found
that a single Maxwellian distribution adequately fit the distribution of
velocities, and derived a one-dimensional root-mean-square (rms) velocity of
$265$ km s-1, much higher than their progenitors, the OB stars which typically
only possess velocities of a few tens of km s-1.

Slowly rotating pulsars cannot be timed sufficiently accurately to yield
reliable timing parallaxes and so for many years pulsar parallaxes were rare.
In the 2000s, dedicated very long baseline arrays such as the Very Long
Baseline Array and efficient pipelines made pulsar proper motion and parallax
studies more routine, and several large-scale surveys were conducted on both
slow-spinning pulsars and MSPs that greatly expanded our knowledge of their
kinematics and distances (Chatterjee et al., 2009; Deller et al., 2019; Ding
et al., 2023). Millisecond pulsar timing arrays have also started
significantly contributing to the number of MSPs with measured timing
parallaxes. PTAs now routinely obtain sub-microsecond timing precision, and
this is permitting the measurement of many MSP parallaxes via the pulse timing
method (e.g., Desvignes et al., 2016; Arzoumanian et al., 2018; Reardon et
al., 2021). In some special cases, the non-zero apparent orbital period
derivative of pulsars enables a very model independent estimate of their
distance (e.g., Bell & Bailes, 1996).

In this work, we concentrate on measurements of transverse velocities for
MSPs, and substantially increase the sample with which to study their
distribution. For the purposes of this paper, MSPs are defined as having a
spin period of $P\leq 30$ ms and a spin-down rate of $\dot{P}\leq 10^{-17}$.

MSPs are well known to have extremely precise spin periods and are accurate
clocks. Their short spin periods and timing stability enhances the reliability
of timing models. We are often able to predict and measure the arrival times
of pulses (ToAs) of MSPs with sub-microsecond precision. By having such
precise ToAs, we can use the pulsar timing technique to measure the positions
of MSPs often to sub-milliarcsecond accuracies. Having several years of data
allows us to also measure the proper motions with the precision of better than
milliarcseconds per year and even reach sub-milliarcsecond parallaxes. Using
these proper motions and parallaxes, we can then derive the distances and
transverse velocities with a precision of sometimes the order of a few percent
and few $\rm km\,s^{-1}$, respectively (e.g., Matthews et al., 2016; Reardon
et al., 2021). To do so, we need to observe pulsars regularly over the course
of at least one year in order to correct for the Rømer delay, which causes a
nearly sinusoidal variation of the pulse’s travel time due to the Earth’s
orbit around the Sun and allows us to measure the position of the pulsar. A
transverse motion of the pulsar on the sky causes a linearly increasing
sinusoid with a period of one year to appear in the timing residuals, and
usually after about $3$ years the proper motions can be determined to high
accuracy. The curvature of the pulse train’s wavefront due to its finite
distance of origin allows us to measure a pulsar’s parallax, provided it is
not too far from the ecliptic plane — unfortunately an MSP at the ecliptic
pole has almost no discernable parallax timing signature. One pulsar in our
sample, (PSR J0711–6830) is almost 83∘ from the ecliptic plane and has never
had its timing parallax determined.

The applications enabled by possessing pulsar parallaxes are many. For
example, combining a parallax-derived distance and the pulsar’s dispersion
measure (as will be explained in Section 3.1) can be used for developing
Galactic electron density distribution models such as TC93 (Taylor & Cordes,
1993), NE2001 (Cordes & Lazio, 2002, 2003), and YMW16 (Yao et al., 2017).
These models allow us to make estimates of pulsar distances when parallaxes
are unavailable, using the DM as a proxy for distance. When combined with a
proper motion, these yield an estimate of the pulsar’s transverse velocity. In
addition, the Galactic magnetic field parallel to the line of sight can be
mapped out through the measurements of the distances, DMs and the Faraday
rotation of pulsars (e.g., Han et al., 2006; Spiewak et al., 2022). Moreover,
pulsar distances can be used to correct for the Shklovskii effect, which
otherwise is the main contaminant to the observed period and orbital period
derivatives from their intrinsic values (Shklovskii, 1970). This leads to
improved estimates of the pulsar’s characteristic age, magnetic field strength
and in tests of General Relativity that require the orbital period derivative
(Camilo et al., 1994; Bell & Bailes, 1996).

The focus of this paper is on MSP parallaxes, proper motions and hence
distances and transverse velocities. We use these to assemble the largest-ever
sample of MSPs with transverse velocities to study how they might differ from
the slowly rotating pulsars, and also as a population. Determining pulsar
transverse velocities requires measurement of pulsar distances and parallaxes,
and consequently, we have selected a sample of MSPs observed by the MeerKAT
timing program (MeerTime, Bailes et al., 2018) aimed at improving as many
proper motions and parallaxes as possible.

In Section 2, we explain how the MeerKAT observations were performed and how
the ToAs were determined. In Section 3, we describe our methods for timing
each pulsar and the modelling of timing noise, and the methods employed for
the measurement of parallaxes and proper motions. In Section 4, we present the
positions, proper motions, and parallaxes. In Section 5, we derive distances
and velocities for our sample. We also present the velocity distributions and
the dispersions of velocity components in Galactocentric coordinates. In
Section 6, we compare our results to the previous work and provide discussions
about the velocity distributions and the velocity dispersions. In Section 7,
we list our conclusions and discuss the implications of our findings.

##  2 The data sets

We used the first MeerKAT pulsar timing array (MPTA) data release provided by
Miles et al. (2023) as the foundation of our study. This data set contains 78
MSPs observed with approximately a two week cadence with the 64-dish MeerKAT
radio telescope over a period of $\sim 2.5$ years, starting from early $2019$.
We augmented the timing baseline of the data set to $\sim 3$ years by adding
all data taken after the first data release and up to May $2022$.

The observations were made with the L-band receiver, operating at the centre
frequency of $1284$ MHz with $856$ MHz of frequency bandwidth (Bailes et al.,
2020), and recorded with $1024$ frequency channels. Following Spiewak et al.
(2022), we removed the top and bottom 48 channels ($\sim 10\%$ of the total
bandwidth) from the L-band receiver where response roll-off affects the
signal-to-noise ratio, and the remaining $928$ channels were averaged in time
and polarization to form Stokes I profiles. The typical minimum length of the
observations was $4$ minutes, but depending on the brightness of the pulsars
and the orbital phase of the binary pulsars, they could be up to $4$ hours
long.

Timing baselines of observations ranged from $2.33$ yr (for J1652$-$4838) to
$3.26$ yr (for PSR J1909$-$3744). We used only $1.98$ years of PSR
J1713$+$0747 observations taken prior to a change in its pulse shape between
April $16$ and $17$, $2021$ (e.g., Jennings et al., 2022) that spoil its
timing properties. Channels affected by radio frequency interference (RFI)
were zero-weighted using MeerGuard, a modified version of CoastGuard (Lazarus
et al., 2016) for removing RFI from the MeerKAT observations. Almost no
observations were deleted due to RFI, although a small percentage had to be
removed because of observatory set-up issues such as poor phasing of the
array. These outliers were very obvious and did not present any analysis
issues.

We formed sub-banded pulse profiles by averaging frequency channels by a
factor of $58$ in order to detect any frequency-dependent variation of
dispersion measure (DM) across the band (Keith et al., 2013; Jones et al.,
2017; Donner et al., 2020; Nobleson et al., 2022) in addition to any
frequency-dependent trends that could be attributed to the intrinsic profile
shape changes across frequency (Kramer et al., 1999; Ahuja et al., 2007; Xu et
al., 2021). This left us with 16 sub-bands that could be used to model
dispersion measure variations.

We produced smoothed, high signal-to-noise, standard ‘template’ profiles for
each pulsar from the sub-banded pulse profiles for use in measuring ToAs. Each
template represents a set of denoised analytical pulse profiles that evolve
across frequency (Pennucci et al., 2014; Pennucci, 2019), using the
PulsePortraiture software
package111https://github.com/pennucci/PulsePortraiture/tree/py3. Employing a
2D portrait enabled us to calculate ToAs for each sub-band, while
simultaneously examining profile stability and correct for the frequency-
dependent trends (if they are significant) due to time-dependent dispersion
measure.

The most recently available ephemerides for the pulsars (Spiewak et al., 2022,
private communication) were used to initiate the timing analysis. Using the
pat software tool in the psrchive package (van Straten et al., 2012), we
measured the ToAs for all the observations, setting the threshold for timing
at a signal-to-noise ratio SNR $>10$. The number of ToAs ranged from $269$ for
PSR J1327$-$0757 to $3074$ for PSR J1909$-$3744\. We describe the timing
analysis of this set of ToAs in the next section.

##  3 Methods

###  3.1 Timing and Noise analysis

The astrometric, period, spin-down, DM, and (where necessary) binary
parameters of every pulsar are described in a timing model. Using the pulsar
timing software tempo2 (Hobbs et al., 2006; Edwards et al., 2006), we refined
the timing models by fitting for the par-file parameters. tempo2 uses a linear
least-squares approach to fitting parameter values, such that some of the
parameters that have non-linear forms need to be fit multiple times in order
to yield the best timing residuals. We also include systematic time jumps in
our timing models derived from the entire population. We used the JPL DE440
ephemeris as the model of the Earth’s orbit around the Solar System barycentre
(Park et al., 2021), and the ToAs were referred to the
TT(BIPM2020)222https://webtai.bipm.org/ftp/pub/tai/ttbipm/TTBIPM.2020, and the
Barycentric Coordinate Time (TCB) was used as the coordinate time standard for
calculating the orbits in the Solar System (Seidelmann & Fukushima, 1992).

Once accurate timing models had been produced for each pulsar, they were then
employed for the noise analysis using the Bayesian inference software
temponest (Lentati et al., 2014). The advantage of using temponest in our
analysis is that it uses Multinest (Feroz & Hobson, 2008; Feroz et al., 2009)
to sample both timing parameter and noise parameter spaces efficiently. The
noise parameters were modeled by three white noise terms, an achromatic red
noise term, and a dispersion noise (i.e. due to DM variations) term.

  * •

White noise: there are three contributions for the white noise as follows:

    1. 1.

EFAC: while calculating the ToAs described in Section 2, the radiometer noise
is a certain contributor to the ToA uncertainties $\sigma_{\rm i}$. EFAC is
modelled using a parameter, which is multiplied by the ToA uncertainties and
scales them. An EFAC estimate is made for each pulsar/receiver combination. If
EFAC deviates too far from unity the error estimates are not well understood.

    2. 2.

EQUAD: The EFAC parameter might not be able to fully account for the actual
error, and we can broaden the errors by defining EQUAD parameter which is
added in quadrature to EFAC. As for the EFAC parameter, an EQUAD estimate is
made for each pulsar/receiver combination. Again if EQUAD is much greater than
the rms residual, this is concerning as it means there are systematic errors
in the arrival times that are of unknown origin.

    3. 3.

ECORR: for sub-banded data, there is likely to be a correlation between ToAs
at different frequencies that are collected simultaneously because of issues
such as pulse jitter (NANOGrav Collaboration et al., 2015), and this
introduces another noise term that can be described using ECORR. In this
analysis, we used the TECORR estimator, as described in Bailes et al. (2020,
Eq. 1). ECORR is also estimated for each pulsar/receiver combination.

  * •

Red noise: this is thought to be related to the irregularities of the rotation
of neutron stars (Shannon & Cordes, 2010). Red noise is a stochastic,
achromatic term with a power-law spectrum of the form

$S_{\rm red}(f)=A_{\rm red}^{2}\left(\frac{f}{yr^{-1}}\right)^{\gamma_{\rm
red}}$ (1)

where $S_{\rm red}(f)$ is the power spectral density of red noise with
fluctuation frequency $f$, $A_{\rm red}$ is the amplitude in $\mu$s yr1/2, and
$\gamma_{\rm red}$ is the spectral index.

  * •

Dispersion noise: The ToAs are affected by frequency dispersion caused by the
ionized material in the interstellar medium (ISM), and ToAs at lower observing
frequencies arrive with a delay with respect to the higher ones. This delay,
measured in seconds, is equal to $1/(2.41\times
10^{-4})\,\textrm{DM}/\nu^{2}$, where DM is dispersion measure (in $\rm
pc\,cm^{-3}$), and $\nu$ is the observing frequency (in MHz) (Lorimer &
Kramer, 2005). Due to the space motions of the Earth and pulsars, as well as
the bulk motion of the ISM, DM shows a temporal variation (Keith et al., 2013;
Jones et al., 2017). This variation introduces a dispersion noise term that
can be modelled using a power law with the same form as the red noise. The
only difference is the frequency-dependent amplitude.

For every pulsar, we performed four noise analyses with the noise models
including (1) DM, red, and white noise parameters (2) DM and white noise
parameters only (3) red and white noise parameters only and (4) white noise
parameters only. We calculated a log Bayesian evidence for every noise model.
By subtracting the log Bayesian evidence values of each of two noise models we
can measure the log Bayes factor of $\ln{\mathcal{B}}$. If $\mathcal{B}>10$
(corresponding to the log Bayes factor of $2.3$) (Kass & Raftery, 1995) the
noise model with higher Bayesian evidence is more significant than the other
and is preferred as the final noise model. The prior ranges for the noise
parameters are listed in Table 1.

Table 1: Uniform prior distributions of noise parameters are adopted in the ranges given in the second column Parameter | Prior
---|---
$\log_{10}$[EFAC] | [$-1$,$1.5$]
$\log_{10}$[EQUAD] | [$-10$,$-5$]
$\log_{10}$[ECORR] | [$-10$,$-5$]
$\log_{10}$[Ared] | [$-20$,$-10$]
$\gamma_{\rm red}$ | [$0$,$6$]
$\log_{10}$[ADM] | [$-20$,$-10$]
$\gamma_{\rm DM}$ | [$0$,$6$]

###  3.2 Astrometry

####  3.2.1 Position, Proper Motion, and Parallax Measurements

After determining a preferred noise model, we measured the pulsar’s
astrometric parameters using the ToAs, and ascertained which are measured with
sufficient statistical significance for use in the later tangential velocity
analysis. For each pulsar the astrometric parameters are the equatorial
coordinates for the position ($\alpha,\delta$), the proper motions
($\mu_{\alpha}\equiv\dot{\alpha}\cos\delta,\mu_{\delta}$) and the parallax.
Due to the high sensitivity of MeerKAT observations, the position of pulsars
can be obtained extremely well, and we did not need to sample for the position
parameter; so, these remained fixed. We performed three analyses using
temponest sampling for (1) proper motion, (2) parallax, and (3) proper motion
and parallax. Bayes factors produced by temponest showed that, for all
pulsars, the proper motion and parallax were detectable (with
$\mathcal{B}>10$) and superior to models without them. In addition, we used
ecliptic coordinates for the position ($\lambda,\beta$) and proper motion
($\mu_{\lambda}\equiv\dot{\lambda}\cos\beta,\mu_{\rm\beta}$), and repeated the
analyses and compared the Bayesian evidence for these solutions with the
equatorial ones, and found them to be similar. Note that covariance between
position and proper motion parameters was minimized by setting the epoch of
the positions to be near the middle of the data set’s baseline in time.
Matthews et al. (2016) demonstrated that positional uncertainties reported in
ecliptic coordinates are less than the equatorial coordinates. Therefore, we
used timing analyses in ecliptic coordinates to derive the Galactic position
($l,b$) and Galactic proper motion ($\mu_{l}\equiv\dot{l}\cos b,\mu_{b}$) of
all pulsars using the Astropy package (Astropy Collaboration et al., 2013).

###  3.3 Distances

####  3.3.1 Distances from parallax measurement

The most direct method for measuring distance is parallax $d=\varpi^{-1}$,
where $\varpi$ is trigonometric parallax in mas and $d$ is in kpc. This method
does not depend on any intrinsic physical property of the object and is purely
geometric, and thus model-independent. For pulsars, the two most common
methods of measuring parallax are using pulsar timing solutions and VLBI
observations. As will be seen, these two methods dominate the distances used
in section 5 for our transverse velocity analysis.

###  3.4 Velocities

The full 3D velocity of a celestial object relative to the Sun can be broken
into two components: a radial velocity component and a transverse (or
tangential) velocity component. The radial velocity for stars in the Milky Way
is typically estimated using the Doppler effect, although this is not possible
for pulsars, as no spectral lines are available unless they are in a binary
system with a star. The transverse velocity (in km s-1), is the velocity
perpendicular to the line of sight and is given by

$v_{\perp}=4.74\,{\rm km\,s}^{-1}\left(\frac{\mu}{\rm
mas\,yr^{-1}}\right)\left(\frac{d}{\rm kpc}\right),$ (2)

where $\mu$ is the total proper motion, and $d$ is the distance (Lorimer et
al., 1997).

####  3.4.1 Correcting for the Local Standard of Rest and the Galactic
differential rotation

Proper motions are usually measured relative to the Solar System’s barycentre,
but if we want to understand their origins as Milky Way objects, we need to
adjust for the reference frame. To obtain the velocity of pulsars with respect
to the Galactic Center (GC), we need to correct the relative proper motions
for two effects: (1) the peculiar velocity of the Sun with respect to the
local standard of rest (LSR) and (2) differential rotation of the Galaxy
(Hobbs et al., 2005). There are two types of LSR, both of which are defined as
reference frames at the location of the Sun. The first is the ‘kinematic LSR’
which orbits the Galaxy with the mean velocity of the neighbouring stars
($\bar{v}_{\rm\phi}$). The second is the ‘dynamic LSR’ which rotates around
the Galaxy in a perfectly circular orbit with a velocity that is exerted by
the gravitational potential of the Galaxy ($v_{\rm c}$). The difference
between the two ($v_{\rm c}-\bar{v}_{\rm\phi}$) is defined as the “asymmetric
drift” (Golubov et al., 2013; Ferreras, 2019).

The Galactic disk rotates differentially, and the Galactic orbital periods of
stars are dependent on their distances from the GC. The circular rotation
speed of stars at distance $R$ from the GC defines the rotation curve of the
Galaxy. In the last decade, there have been a number of studies to improve the
measured rotation curve of the Galaxy using a variety of different methods
(e.g., Bovy et al., 2012; Bhattacharjee et al., 2014; Reid et al., 2014,
2019). In this work, we used the results and rotation curve model from Reid et
al. (2019).

For every MSP, we transformed the coordinates of the pulsars from their
ecliptic coordinates to the cartesian Galactocentric coordinates ($X$, $Y$,
$Z$) using
Astropy333https://docs.astropy.org/en/stable/api/astropy.coordinates.Galactocentric.html.
The distance from the Sun to the Galactic mid-plane was taken from Reid et al.
(2019) ($Z_{\odot}=5.5(58)$ pc), and the distance from the Sun to the GC was
taken from GRAVITY Collaboration et al. (2018) ($R_{\odot}=8.12(3)$ kpc). We
then derived the Galactocentric velocity components ($V_{\rm X}$, $V_{\rm Y}$,
$V_{\rm Z}$). For subtracting the Sun’s peculiar velocity from these
components, we used the values of ($U$, $V$, $W$)⊙ =
($11.10(74),12.24(47),7.25(37)$) km s-1 from Schönrich et al. (2010).

To correct for differential rotation of the Galaxy, we calculated the
distances of the pulsars from the Galactic centre, and used the universal
rotation curve of the Galaxy from Reid et al. (2019, Tab. 4) to calculate the
$X$ and $Y$ components of the Galactic rotation at the position of MSPs.
Finally, we subtracted these from the velocities of MSPs and derived their
transverse velocities with respect to their own LSR. All velocities reported
in this work are thus corrected for the peculiar velocity of the Sun and the
rotation curve of the Galaxy.

##  4 Results: astrometry

In this section we report our astrometric results, including positions, proper
motions, parallaxes for our sample pulsars, and compare with MSPs for which
these have been previously measured in the literature.

###  4.1 Positions and proper motions

In Table 2, positions from timing analyses in equatorial and ecliptic
coordinates for $77$ MSPs are given. These were not sampled using temponest
and only fitted using tempo2. In addition, the positions in Galactic
coordinates are derived from ecliptic coordinates using Astropy. All
uncertainties shown are computed via the median with $68.3\%$ equally-tailed
credible intervals.

In Table 4, we show proper motions from timing analyses in both equatorial and
ecliptic coordinates. In this table, the proper motions in Galactic
coordinates are derived from the ecliptic proper motions using Astropy. The
reported values and uncertainties are the lower bound ($16\%$), median
($50\%$), and upper bound ($84\%$) of marginalized posterior probability
distributions. The timing baseline of observations for each pulsar, as well as
the best previous measurements of proper motions in equatorial coordinates and
their corresponding references, are listed. In our study, our limited timing
baseline means that our proper motion results are not superior to all previous
results, mainly because the relative error is proportional to the observing
span $T^{-3/2}$, and some of the pulsars have been timed for over a decade
with other instruments.

Out of $77$ pulsars, $69$ had proper motions with statistical significance
greater than $3\sigma$, and are shown in the upper portion of Table 4. The
remaining $8$ pulsars (i.e. those listed in the lower portion of the Table)
consist of an isolated MSP (J1721$-$2457) with relatively poor rms timing
residuals of $3.7\mu$s and less than two deg from the ecliptic, and seven
binary MSPs (J0955$-$6150, J1022$+$1001, J1327$-$0755, J1653$-$2054,
J1705$-$1903, J1802$-$2124, J1811$-$2405) that have either poor timing
residuals or small proper motions (as low as $0.21(15)$ mas yr-1 for
J0955$-$6150). PSR J1022$+$1001 is near the ecliptic plane, and as expected
had a very accurate proper motion in ecliptic longitude, but a poor constraint
in ecliptic latitude. For this pulsar, we set the proper motion and orbital
parameters from the high accuracy VLBI study of Deller et al. (2016) and
derived a parallax of $1.2(4)$ mas.

We typically require at least two years of data in order to separate the
annual parallax from the proper motion of a pulsar, due to the motion of the
pulsar against the background of the sky and the orbit of the Earth around the
Sun. We note that it is not always simple to break this degeneracy for the
pulsars with high rms timing residuals, and worse for pulsars in (wide) binary
systems. A solution to this is to implement VLBI techniques, which can often
measure pulsar parallaxes significantly better, e.g. Deller et al. (2008)
measured the parallax for PSR J0437$-$4715 to be $6.396(54)$ mas with the
Australian Long Baseline Array.

All but two of the proper motions listed in Table 4 are consistent with
previous results within $3\sigma$. One component of the proper motion of PSRs
J1446$-$4701 is just over $3\sigma$ inconsistent with the values in Reardon et
al. (2021), but the total proper motion is the same to within 8%. The origin
of this discrepancy is unclear. We checked the use of the Solar System
ephemeris that Reardon et al. (2021) used for their timing analysis, and our
parallax remained the same to within errors. This inconsistency might be due
to the different noise models that were adopted or be due to different noise
properties at different epochs.

Figure 1 presents the trajectories of the $69$ MSPs since $5$ Myr ago until
the present in Galactic longitude ($l$) and Galactic latitude ($b$), assuming
zero radial velocity for the MSPs. Both effects of the peculiar velocity of
the Sun and the Galactic differential rotation are subtracted from the
observed proper motions of MSPs. In addition, the Galactic gravitational
potential is assumed to be zero, because the oscillation period in the
galactic potential ($>100$Myr) is much larger than 5 Myr, and so the MSPs are
simply moving along their corrected proper motion vectors. The area in gray
indicates the part of the sky that the MeerKAT radio telescope is not able to
reach. From the diagram it is obvious that the MSPs are both moving towards
and away from the galactic plane, consistent with a relaxed ancient
population.

Figure 1: Celestial distribution of the 69 MSPs in this study, shown in Galactic longitude, $l$ and Galactic latitude, $b$. The curves are the paths of these MSPs from $5\,\rm Myr$ up until the present, and the current positions of these MSPs are shown with black stars. The effects of the peculiar motion of the Sun and the Galactic rotation curve have been subtracted from the observed proper motions. The black unit vectors are showing the proper motions with at least $3\sigma$ significance, and the length of the brown vectors are showing the magnitudes of velocities with at least $3\sigma$ significance in parallax detections. See the text for more details.  Table 2: Pulsar positions in Equatorial ($\alpha$,$\delta$), Ecliptic ($\lambda$,$\beta$), and Galactic ($l$,$b$) coordinates to 3 significant figures. The digits in parentheses are showing $1\sigma$ uncertainty in the last significant digits. The last column shows the reference epochs (in MJD) at which the positions are referred to. | Equatorial coordinates |  | Ecliptic coordinates |  | Galactic coordinates |
---|---|---|---|---|---|---
Pulsar | $\alpha$ | $\delta$ |  | $\lambda$ | $\beta$ |  | $l$ | $b$ | Epoch
| (hh:mm:ss) | (dd:mm:ss) |  | (∘) | (∘) |  | (∘) | (∘) | (MJD)
J0030$+$0451 |  $00$:$30$:$27.42348(5)$ |  $+04$:$51$:$39.7144(17)$ |  | $8.910338462(13)$ | $1.4457050(5)$ |  | $113.141$ | $-57.611$ | $59251$
J0125$-$2327 |  $01$:$25$:$1.075151(3)$ |  $-23$:$27$:$8.13167(5)$ |  | $9.743403184(9)$ | $-29.830147598(15)$ |  | $188.932$ | $-81.569$ | $59155$
J0437$-$4715 |  $04$:$37$:$16.0492624(13)$ |  $-47$:$15$:$10.027930(15)$ |  | $50.469059178(11)$ | $-67.873319464(5)$ |  | $253.395$ | $-41.963$ | $59187$
J0610$-$2100 |  $06$:$10$:$13.602953(6)$ |  $-21$:$00$:$27.74435(13)$ |  | $93.34234093(3)$ | $-44.41732764(4)$ |  | $227.746$ | $-18.184$ | $59152$
J0613$-$0200 |  $06$:$13$:$43.977070(3)$ |  $-02$:$00$:$47.34294(13)$ |  | $93.799014159(15)$ | $-25.40716602(4)$ |  | $210.413$ | $-9.305$ | $59152$
J0614$-$3329 |  $06$:$14$:$10.348230(3)$ |  $-33$:$29$:$54.13156(4)$ |  | $95.410779861(16)$ | $-56.871120173(12)$ |  | $240.501$ | $-21.827$ | $59155$
J0636$-$3044 |  $06$:$36$:$34.398358(5)$ |  $-30$:$44$:$1.10761(7)$ |  | $103.35534034(3)$ | $-53.74983738(2)$ |  | $239.526$ | $-16.416$ | $59163$
J0711$-$6830 |  $07$:$11$:$54.153093(12)$ |  $-68$:$30$:$47.23387(10)$ |  | $204.06089121(19)$ | $-82.88867768(2)$ |  | $279.531$ | $-23.280$ | $59149$
J0900$-$3144 |  $09$:$00$:$43.95220(3)$ |  $-31$:$44$:$30.8722(4)$ |  | $150.54478942(11)$ | $-46.14670208(11)$ |  | $256.162$ | $9.486$ | $59149$
J0931$-$1902 |  $09$:$31$:$19.116246(6)$ |  $-19$:$02$:$55.05445(13)$ |  | $152.37696462(2)$ | $-31.77673163(4)$ |  | $250.999$ | $23.054$ | $59149$
J0955$-$6150 |  $09$:$55$:$20.847467(18)$ |  $-61$:$50$:$16.89532(13)$ |  | $197.40608543(8)$ | $-64.96064654(4)$ |  | $283.685$ | $-5.737$ | $59138$
J1012$-$4235 |  $10$:$12$:$12.938801(10)$ |  $-42$:$35$:$53.40253(12)$ |  | $176.78918824(4)$ | $-48.91085447(4)$ |  | $274.218$ | $11.225$ | $59152$
J1017$-$7156 |  $10$:$17$:$51.311783(15)$ |  $-71$:$56$:$41.57398(6)$ |  | $222.42774900(4)$ | $-67.736237859(16)$ |  | $291.558$ | $-12.553$ | $59132$
J1022$+$1001 |  $10$:$22$:$57.990(2)$ |  $+10$:$01$:$52.92(8)$ |  | $153.86581653(3)$ | $-0.06391(2)$ |  | $231.794$ | $51.101$ | $59166$
J1024$-$0719 |  $10$:$24$:$38.648440(3)$ |  $-07$:$19$:$19.98201(9)$ |  | $160.734318047(8)$ | $-16.04485601(3)$ |  | $251.702$ | $40.515$ | $59152$
J1036$-$8317 |  $10$:$36$:$40.59408(3)$ |  $-83$:$17$:$56.14254(4)$ |  | $253.01178441(3)$ | $-68.082219349(15)$ |  | $298.943$ | $-21.497$ | $59187$
J1045$-$4509 |  $10$:$45$:$50.179671(15)$ |  $-45$:$09$:$54.05686(17)$ |  | $186.51850583(6)$ | $-47.71477807(5)$ |  | $280.851$ | $12.254$ | $59142$
J1101$-$6424 |  $11$:$01$:$37.19154(3)$ |  $-64$:$24$:$39.33316(16)$ |  | $211.77519692(9)$ | $-60.54978942(5)$ |  | $291.417$ | $-4.023$ | $59142$
J1103$-$5403 |  $11$:$03$:$33.27734(4)$ |  $-54$:$03$:$43.1296(4)$ |  | $198.52939001(16)$ | $-53.10724535(11)$ |  | $287.421$ | $5.529$ | $59149$
J1125$-$5825 |  $11$:$25$:$44.352943(14)$ |  $-58$:$25$:$16.85965(11)$ |  | $207.30413917(5)$ | $-54.35535501(3)$ |  | $291.893$ | $2.602$ | $59132$
J1125$-$6014 |  $11$:$25$:$55.243299(4)$ |  $-60$:$14$:$6.80942(3)$ |  | $209.504093810(12)$ | $-55.659051349(9)$ |  | $292.504$ | $0.894$ | $59149$
J1216$-$6410 |  $12$:$16$:$7.321183(7)$ |  $-64$:$10$:$9.16656(4)$ |  | $221.617852389(19)$ | $-54.452089781(13)$ |  | $299.096$ | $-1.561$ | $59132$
J1327$-$0755 |  $13$:$27$:$57.58052(9)$ |  $-07$:$55$:$29.918(3)$ |  | $203.27816789(2)$ | $1.2046411(10)$ |  | $318.383$ | $53.848$ | $59281$
J1421$-$4409 |  $14$:$21$:$20.960491(9)$ |  $-44$:$09$:$4.57445(14)$ |  | $228.33837111(2)$ | $-28.29432179(4)$ |  | $319.497$ | $15.809$ | $59142$
J1431$-$5740 |  $14$:$31$:$3.49603(4)$ |  $-57$:$40$:$11.6601(4)$ |  | $236.41161578(9)$ | $-40.16383928(11)$ |  | $315.962$ | $2.660$ | $59129$
J1435$-$6100 |  $14$:$35$:$20.26018(2)$ |  $-61$:$00$:$58.0019(2)$ |  | $238.94141530(6)$ | $-42.97710038(6)$ |  | $315.186$ | $-0.641$ | $59132$
J1446$-$4701 |  $14$:$46$:$35.709721(8)$ |  $-47$:$01$:$26.79678(14)$ |  | $234.21533611(2)$ | $-29.41052857(4)$ |  | $322.500$ | $11.425$ | $59129$
J1455$-$3330 |  $14$:$55$:$47.976991(8)$ |  $-33$:$30$:$46.40280(19)$ |  | $231.347559978(19)$ | $-16.04479743(6)$ |  | $330.722$ | $22.562$ | $59149$
J1525$-$5545 |  $15$:$25$:$28.12719(4)$ |  $-55$:$45$:$49.8854(6)$ |  | $244.36826552(9)$ | $-35.71040433(14)$ |  | $323.439$ | $0.851$ | $59149$
J1543$-$5149 |  $15$:$43$:$44.145289(15)$ |  $-51$:$49$:$54.7135(2)$ |  | $246.13439909(4)$ | $-31.17770143(6)$ |  | $327.921$ | $2.479$ | $59132$
J1545$-$4550 |  $15$:$45$:$55.945403(5)$ |  $-45$:$50$:$37.50813(11)$ |  | $244.821544044(13)$ | $-25.29111285(3)$ |  | $331.892$ | $6.988$ | $59142$
J1547$-$5709 |  $15$:$47$:$24.121775(16)$ |  $-57$:$09$:$17.5931(2)$ |  | $248.44532730(4)$ | $-36.17005240(6)$ |  | $325.076$ | $-2.052$ | $59162$
J1600$-$3053 |  $16$:$00$:$51.902514(4)$ |  $-30$:$53$:$49.45343(17)$ |  | $244.347678961(9)$ | $-10.07185627(5)$ |  | $344.090$ | $16.451$ | $59138$
J1603$-$7202 |  $16$:$03$:$35.670781(16)$ |  $-72$:$02$:$32.82328(9)$ |  | $256.52028909(3)$ | $-49.96303799(2)$ |  | $316.630$ | $-14.496$ | $59127$
J1614$-$2230 |  $16$:$14$:$36.50974(3)$ |  $-22$:$30$:$31.542(2)$ |  | $245.788315625(16)$ | $-1.2568780(7)$ |  | $352.636$ | $20.192$ | $59146$
J1629$-$6902 |  $16$:$29$:$8.744211(5)$ |  $-69$:$02$:$45.46922(3)$ |  | $258.424015436(10)$ | $-46.517655909(9)$ |  | $320.371$ | $-13.926$ | $59127$
J1643$-$1224 |  $16$:$43$:$38.16612(2)$ |  $-12$:$24$:$58.6334(14)$ |  | $251.08723617(7)$ | $9.7783440(4)$ |  | $5.670$ | $21.218$ | $59146$
J1652$-$4838 |  $16$:$52$:$37.623911(18)$ |  $-48$:$38$:$51.7486(4)$ |  | $257.71020795(6)$ | $-25.92450458(11)$ |  | $338.005$ | $-2.920$ | $59021$
J1653$-$2054 |  $16$:$53$:$31.01268(7)$ |  $-20$:$54$:$54.977(9)$ |  | $254.49648386(6)$ | $1.633926(2)$ |  | $359.969$ | $14.261$ | $59149$
J1658$-$5324 |  $16$:$58$:$39.343556(7)$ |  $-53$:$24$:$6.97315(11)$ |  | $259.45266028(2)$ | $-30.52273666(3)$ |  | $334.869$ | $-6.625$ | $59188$
J1705$-$1903 |  $17$:$05$:$43.84864(4)$ |  $-19$:$03$:$41.414(5)$ |  | $257.16132332(7)$ | $3.7751398(10)$ |  | $3.248$ | $13.030$ | $59198$
J1708$-$3506 |  $17$:$08$:$17.616678(11)$ |  $-35$:$06$:$22.6595(5)$ |  | $259.21049750(4)$ | $-12.14991994(12)$ |  | $350.470$ | $3.124$ | $59215$
J1713$+$0747 |  $17$:$13$:$49.5367813(12)$ |  $+07$:$47$:$37.45024(4)$ |  | $256.668711830(6)$ | $30.700351152(10)$ |  | $28.751$ | $25.223$ | $58957$
J1719$-$1438 |  $17$:$19$:$10.076221(4)$ |  $-14$:$38$:$1.0020(4)$ |  | $260.016965101(16)$ | $8.45261155(11)$ |  | $8.859$ | $12.838$ | $59150$
J1721$-$2457 |  $17$:$21$:$5.50008(3)$ |  $-24$:$57$:$6.226(6)$ |  | $261.18398611(6)$ | $-1.8096426(17)$ |  | $0.387$ | $6.751$ | $59205$
J1730$-$2304 |  $17$:$30$:$21.68471(6)$ |  $-23$:$04$:$31.259(16)$ |  | $263.186095246(13)$ | $0.188838(4)$ |  | $3.137$ | $6.023$ | $59150$
J1732$-$5049 |  $17$:$32$:$47.766068(7)$ |  $-50$:$49$:$0.31465(13)$ |  | $265.16176864(2)$ | $-27.49163468(4)$ |  | $340.029$ | $-9.454$ | $59115$
J1737$-$0811 |  $17$:$37$:$47.109064(19)$ |  $-08$:$11$:$8.9798(11)$ |  | $264.30464763(7)$ | $15.1437208(2)$ |  | $16.932$ | $12.318$ | $59150$
J1744$-$1134 |  $17$:$44$:$29.4220844(14)$ |  $-11$:$34$:$54.80132(10)$ |  | $266.119458498(6)$ | $11.80517508(3)$ |  | $14.794$ | $9.180$ | $59150$
J1747$-$4036 |  $17$:$47$:$48.715690(13)$ |  $-40$:$36$:$54.7995(5)$ |  | $267.57913114(4)$ | $-17.20154481(13)$ |  | $350.208$ | $-6.412$ | $59130$
J1751$-$2857 |  $17$:$51$:$32.686848(5)$ |  $-28$:$57$:$46.5713(7)$ |  | $268.141987317(19)$ | $-5.5372894(2)$ |  | $0.646$ | $-1.124$ | $59150$
J1757$-$5322 |  $17$:$57$:$15.154960(7)$ |  $-53$:$22$:$26.59604(12)$ |  | $269.52715024(2)$ | $-29.93590243(4)$ |  | $339.637$ | $-13.980$ | $59163$
J1801$-$1417 |  $18$:$01$:$51.064902(5)$ |  $-14$:$17$:$34.5498(6)$ |  | $270.45421998(2)$ | $9.14561928(15)$ |  | $14.546$ | $4.162$ | $59150$
J1802$-$2124 |  $18$:$02$:$5.334393(11)$ |  $-21$:$24$:$3.676(5)$ |  | $270.48652542(4)$ | $2.0373763(13)$ |  | $8.382$ | $0.611$ | $59127$
J1811$-$2405 |  $18$:$11$:$19.854352(6)$ |  $-24$:$05$:$18.428(4)$ |  | $272.586046336(10)$ | $-0.6745983(11)$ |  | $7.073$ | $-2.559$ | $59145$
J1825$-$0319 |  $18$:$25$:$55.954707(11)$ |  $-03$:$19$:$57.5837(5)$ |  | $276.88733392(5)$ | $19.95175171(14)$ |  | $27.045$ | $4.138$ | $59155$
J1832$-$0836 |  $18$:$32$:$27.589497(5)$ |  $-08$:$36$:$55.1454(3)$ |  | $278.29199042(2)$ | $14.59068359(9)$ |  | $23.109$ | $0.256$ | $59147$
J1843$-$1113 |  $18$:$43$:$41.260405(4)$ |  $-11$:$13$:$31.1031(3)$ |  | $280.944529539(16)$ | $11.79998448(8)$ |  | $22.055$ | $-3.397$ | $59150$
J1902$-$5105 |  $19$:$02$:$2.844154(10)$ |  $-51$:$05$:$57.0563(2)$ |  | $280.99087040(3)$ | $-28.25008697(5)$ |  | $345.650$ | $-22.379$ | $59152$
J1903$-$7051 |  $19$:$03$:$38.783931(19)$ |  $-70$:$51$:$43.58175(12)$ |  | $277.69661567(4)$ | $-47.84766764(3)$ |  | $324.391$ | $-26.508$ | $59136$
Table 3: continued | Equatorial coordinates |  | Ecliptic coordinates |  | Galactic coordinates |
---|---|---|---|---|---|---
Pulsar | $\alpha$ | $\delta$ |  | $\lambda$ | $\beta$ |  | $l$ | $b$ | Epoch
| (hh:mm:ss) | (dd:mm:ss) |  | (∘) | (∘) |  | (∘) | (∘) | (MJD)
J1909$-$3744 |  $19$:$09$:$47.4245244(7)$ |  $-37$:$44$:$14.91930(3)$ |  | $284.220813110(2)$ | $-15.155611509(9)$ |  | $359.731$ | $-19.596$ | $59120$
J1918$-$0642 |  $19$:$18$:$48.027685(2)$ |  $-06$:$42$:$34.95735(12)$ |  | $290.314613645(9)$ | $15.35104745(3)$ |  | $30.027$ | $-9.123$ | $59155$
J1933$-$6211 |  $19$:$33$:$32.413330(4)$ |  $-62$:$11$:$46.69324(5)$ |  | $283.961757663(10)$ | $-39.885817735(13)$ |  | $334.431$ | $-28.631$ | $59132$
J1946$-$5403 |  $19$:$46$:$34.4933990(19)$ |  $-54$:$03$:$42.55867(3)$ |  | $288.139717466(5)$ | $-32.287682135(10)$ |  | $343.885$ | $-29.580$ | $59166$
J2010$-$1323 |  $20$:$10$:$45.922635(4)$ |  $-13$:$23$:$56.1327(3)$ |  | $301.924490472(7)$ | $6.49093303(7)$ |  | $29.446$ | $-23.540$ | $59155$
J2039$-$3616 |  $20$:$39$:$16.580827(4)$ |  $-36$:$16$:$17.23958(11)$ |  | $302.723254857(9)$ | $-17.24602148(3)$ |  | $6.331$ | $-36.517$ | $59166$
J2124$-$3358 |  $21$:$24$:$43.834930(4)$ |  $-33$:$58$:$45.49195(11)$ |  | $312.738766685(11)$ | $-17.81894426(3)$ |  | $10.925$ | $-45.438$ | $59155$
J2129$-$5721 |  $21$:$29$:$22.781548(3)$ |  $-57$:$21$:$14.33356(4)$ |  | $303.827979841(7)$ | $-39.900000443(10)$ |  | $338.005$ | $-43.570$ | $59145$
J2145$-$0750 |  $21$:$45$:$50.453394(17)$ |  $-07$:$50$:$18.5938(7)$ |  | $326.02458223(2)$ | $5.3130387(2)$ |  | $47.777$ | $-42.084$ | $59155$
J2150$-$0326 |  $21$:$50$:$27.235330(10)$ |  $-03$:$26$:$32.8318(4)$ |  | $328.604922236(17)$ | $9.06674597(12)$ |  | $53.573$ | $-40.757$ | $59199$
J2222$-$0137 |  $22$:$22$:$5.996923(12)$ |  $-01$:$37$:$15.7776(5)$ |  | $336.73199735(2)$ | $7.97708619(14)$ |  | $62.019$ | $-46.075$ | $59155$
J2229$+$2643 |  $22$:$29$:$50.883815(5)$ |  $+26$:$43$:$57.61393(13)$ |  | $350.695629791(19)$ | $33.29016684(4)$ |  | $87.693$ | $-26.284$ | $59251$
J2234$+$0944 |  $22$:$34$:$46.856913(9)$ |  $+09$:$44$:$30.0415(3)$ |  | $344.11900953(3)$ | $17.31853087(9)$ |  | $76.280$ | $-40.438$ | $59251$
J2241$-$5236 |  $22$:$41$:$42.0400897(10)$ |  $-52$:$36$:$36.272400(11)$ |  | $318.696416363(3)$ | $-40.393437599(3)$ |  | $337.457$ | $-54.927$ | $59132$
J2317$+$1439 |  $23$:$17$:$9.235606(4)$ |  $+14$:$39$:$31.29559(12)$ |  | $356.129406224(12)$ | $17.68024246(4)$ |  | $91.361$ | $-42.360$ | $59251$
J2322$+$2057 |  $23$:$22$:$22.320041(6)$ |  $+20$:$57$:$2.50080(14)$ |  | $0.135924003(19)$ | $22.87835838(4)$ |  | $96.515$ | $-37.310$ | $59251$
J2322$-$2650 |  $23$:$22$:$34.638615(5)$ |  $-26$:$50$:$58.38398(10)$ |  | $340.443069768(14)$ | $-20.89690479(3)$ |  | $28.637$ | $-70.228$ | $59155$
Table 4: Pulsar Proper Motions in Equatorial, Ecliptic, and approximate Galactic Coordinates. The Galactic proper motions are derived from the Ecliptic ones. The time span (column 2) of observations for each pulsar is provided in yr. The previous best measurements of proper motions in Equatorial coordinates and the corresponding references are shown in the final 3 columns. Proper motions with $>3\sigma$ and $<3\sigma$ detections are listed in the upper and lower portions of the table, respectively. The list of the references are provided at the end of the table. |  | Equatorial coordinates |  | Ecliptic coordinates |  | Galactic coordinates |  | Best Previous Measurement
---|---|---|---|---|---|---|---|---
Pulsar | Span | $\mu_{\alpha}\equiv\dot{\alpha}\cos\delta$ | $\mu_{\delta}$ |  | $\mu_{\lambda}\equiv\dot{\lambda}\cos\beta$ | $\mu_{\beta}$ |  | $\mu_{l}\equiv\dot{l}\cos b$ | $\mu_{b}$ |  | $\mu_{\alpha}$ | $\mu_{\delta}$ | Ref
| (yr) | (mas yr-1) | (mas yr-1) |  | (mas yr-1) | (mas yr-1) |  | (mas yr-1) | (mas yr-1) |  | (mas yr-1) | (mas yr-1) |
Total Proper Motion Detections $>3\sigma$
J0030$+$0451 | $2.54$ | $-6.4(11)$ | $1(2)$ |  | $-5.51(7)$ | $3(3)$ |  | $-6.2(6)$ | $2(3)$ |  | $-6.2(1)$ | $0.5(3)$ | (1)
J0125$-$2327 | $3.06$ | $37.15(5)$ | $10.75(6)$ |  | $38.18(3)$ | $-6.17(6)$ |  | $7.70(6)$ | $37.90(4)$ |  |  |  |
J0437$-$4715 | $2.86$ | $121.394(19)$ | $-71.489(19)$ |  | $85.96(2)$ | $-111.60(2)$ |  | $62.89(2)$ | $126.06(2)$ |  | $121.4385(20)$ | $-71.4754(20)$ | (2)
J0610$-$2100 | $3.05$ | $9.29(11)$ | $16.74(17)$ |  | $8.86(11)$ | $16.97(17)$ |  | $-11.81(17)$ | $15.08(15)$ |  | $9.11(3)$ | $16.45(3)$ | (3)
J0613$-$0200 | $3.05$ | $1.81(6)$ | $-10.31(17)$ |  | $2.08(6)$ | $-10.28(17)$ |  | $10.00(15)$ | $-3.11(11)$ |  | $1.836(5)$ | $-10.349(13)$ | (6)
J0614$-$3329 | $3.06$ | $0.60(4)$ | $-1.84(5)$ |  | $0.68(4)$ | $-1.81(5)$ |  | $1.93(4)$ | $-0.02(4)$ |  | $0.61(3)$ | $-1.74(4)$ | (4)
J0636$-$3044 | $2.99$ | $26.01(9)$ | $-19.26(10)$ |  | $27.92(9)$ | $-16.37(10)$ |  | $27.66(8)$ | $16.79(11)$ |  |  |  |
J0711$-$6830 | $3.06$ | $-15.54(5)$ | $14.24(6)$ |  | $-12.11(5)$ | $-17.25(5)$ |  | $-17.79(6)$ | $-11.21(5)$ |  | $-15.568(10)$ | $14.175(11)$ | (6)
J0900$-$3144 | $3.06$ | $-1.4(4)$ | $2.4(5)$ |  | $-2.3(3)$ | $1.6(5)$ |  | $-2.7(4)$ | $0.5(6)$ |  | $-1.01(5)$ | $2.02(7)$ | (5)
J0931$-$1902 | $3.06$ | $-2.44(11)$ | $-3.92(19)$ |  | $-0.80(9)$ | $-4.5(2)$ |  | $1.29(12)$ | $-4.47(19)$ |  | $-2.4(2)$ | $-4.4(4)$ | (1)
J1012$-$4235 | $3.05$ | $-4.15(15)$ | $5.20(16)$ |  | $-6.30(14)$ | $2.14(17)$ |  | $-6.40(15)$ | $1.82(17)$ |  |  |  |
J1017$-$7156 | $3.15$ | $-7.43(8)$ | $6.91(7)$ |  | $-8.93(6)$ | $-4.84(6)$ |  | $-10.03(8)$ | $1.49(9)$ |  | $-7.411(12)$ | $6.870(11)$ | (6)
J1024$-$0719 | $3.05$ | $-35.32(5)$ | $-48.39(12)$ |  | $-14.37(4)$ | $-58.16(13)$ |  | $8.62(7)$ | $-59.28(12)$ |  | $-35.270(17)$ | $-48.22(3)$ | (6)
J1036$-$8317 | $2.86$ | $-11.35(6)$ | $2.86(6)$ |  | $-1.83(6)$ | $-11.56(6)$ |  | $-11.15(6)$ | $-3.57(5)$ |  |  |  |
J1045$-$4509 | $3.10$ | $-6.25(19)$ | $4.9(2)$ |  | $-7.9(2)$ | $0.6(2)$ |  | $-7.8(2)$ | $1.4(2)$ |  | $-6.07(3)$ | $5.19(4)$ | (6)
J1101$-$6424 | $3.10$ | $-1.0(2)$ | $0.4(2)$ |  | $-1.0(2)$ | $-0.5(3)$ |  | $-1.1(3)$ | $0.0(3)$ |  |  |  |
J1103$-$5403 | $3.06$ | $-7.1(4)$ | $1.1(4)$ |  | $-6.2(4)$ | $-3.8(4)$ |  | $-7.1(4)$ | $-1.9(5)$ |  |  |  |
J1125$-$5825 | $3.15$ | $-8.95(13)$ | $1.79(13)$ |  | $-7.81(13)$ | $-4.72(14)$ |  | $-9.04(14)$ | $-1.21(15)$ |  | $-10.0(3)$ | $2.4(3)$ | (7)
J1125$-$6014 | $3.06$ | $11.09(3)$ | $-13.00(3)$ |  | $17.01(3)$ | $-1.57(4)$ |  | $14.71(4)$ | $-8.69(4)$ |  | $11.106(13)$ | $-13.037(14)$ | (6)
J1216$-$6410 | $3.15$ | $-7.80(5)$ | $2.69(5)$ |  | $-7.54(5)$ | $-3.36(5)$ |  | $-8.10(5)$ | $1.60(5)$ |  |  |  |
J1421$-$4409 | $3.10$ | $-10.27(13)$ | $-6.15(14)$ |  | $-7.28(10)$ | $-9.50(15)$ |  | $-11.78(15)$ | $-2.16(11)$ |  | $-11.6(4)$ | $-7.9(8)$ | (8)
J1431$-$5740 | $3.17$ | $0.0(3)$ | $2.5(4)$ |  | $-1.0(3)$ | $2.3(5)$ |  | $1.0(5)$ | $2.3(4)$ |  |  |  |
J1435$-$6100 | $3.15$ | $-5.6(2)$ | $-2.4(3)$ |  | $-4.00(18)$ | $-4.5(3)$ |  | $-6.1(3)$ | $0.0(3)$ |  |  |  |
J1446$-$4701 | $3.17$ | $-4.24(9)$ | $-2.45(16)$ |  | $-3.15(8)$ | $-3.73(18)$ |  | $-4.88(14)$ | $-0.32(14)$ |  | $-4.36(4)$ | $-3.00(7)$ | (6)
J1455$-$3330 | $3.06$ | $7.66(13)$ | $-2.3(2)$ |  | $8.01(9)$ | $0.1(2)$ |  | $5.5(2)$ | $-5.82(17)$ |  | $7.98(8)$ | $-2.0(2)$ | (1)
J1525$-$5545 | $3.06$ | $-6.8(5)$ | $-3.3(8)$ |  | $-5.5(5)$ | $-5.5(7)$ |  | $-7.6(8)$ | $1.1(7)$ |  |  |  |
J1543$-$5149 | $3.15$ | $-4.33(18)$ | $-2.5(2)$ |  | $-3.54(16)$ | $-3.5(3)$ |  | $-4.9(3)$ | $0.68(18)$ |  | $-4.3(14)$ | $-4(2)$ | (7)
J1545$-$4550 | $3.10$ | $-0.51(7)$ | $2.50(13)$ |  | $-1.09(6)$ | $2.29(12)$ |  | $1.12(12)$ | $2.23(11)$ |  | $-0.48(2)$ | $2.37(4)$ | (6)
J1547$-$5709 | $2.99$ | $-6.59(18)$ | $-5.9(2)$ |  | $-4.77(17)$ | $-7.4(3)$ |  | $-8.8(3)$ | $-0.5(2)$ |  |  |  |
J1600$-$3053 | $3.12$ | $-1.04(6)$ | $-7.5(2)$ |  | $0.48(4)$ | $-7.5(2)$ |  | $-5.91(18)$ | $-4.79(12)$ |  | $-0.960(7)$ | $-6.96(3)$ | (6)
J1603$-$7202 | $3.16$ | $-2.55(10)$ | $-7.40(11)$ |  | $-0.21(8)$ | $-7.83(11)$ |  | $-6.92(12)$ | $-3.66(9)$ |  | $-2.447(11)$ | $-7.356(13)$ | (6)
J1614$-$2230 | $3.05$ | $4.4(5)$ | $-29(3)$ |  | $9.52(7)$ | $-28(3)$ |  | $-18(3)$ | $-23.1(17)$ |  | $3.8(1)$ | $-32.5(7)$ | (1)
J1629$-$6902 | $3.16$ | $-6.50(3)$ | $-8.44(4)$ |  | $-4.45(3)$ | $-9.68(4)$ |  | $-10.62(4)$ | $-0.78(3)$ |  |  |  |
J1643$-$1224 | $3.05$ | $6.1(4)$ | $3.4(17)$ |  | $5.6(3)$ | $4.0(19)$ |  | $6^{+2}_{-3}$ | $-3.2^{+1.1}_{-1.3}$ |  | $5.970(18)$ | $3.77(8)$ | (6)
J1652$-$4838 | $2.33$ | $-3.8(3)$ | $-9.7(7)$ |  | $-2.5(3)$ | $-10.1(8)$ |  | $-9.8(7)$ | $-3.2(5)$ |  |  |  |
J1658$-$5324 | $2.82$ | $0.02(10)$ | $2.86(17)$ |  | $-0.33(11)$ | $2.84(17)$ |  | $2.27(13)$ | $1.74(15)$ |  | $0.02(10)$ | $4.90(23)$ | (9)
J1708$-$3506 | $2.67$ | $-5.8(3)$ | $-0.4(8)$ |  | $-5.8(2)$ | $-0.9(8)$ |  | $-3.8(7)$ | $4.5(5)$ |  | $-5.3(8)$ | $-2(3)$ | (7)
J1713+0747 | $1.98$ | $4.94(4)$ | $-4.07(9)$ |  | $5.30(4)$ | $-3.59(9)$ |  | $-1.45(8)$ | $-6.24(7)$ |  | $4.9254(14)$ | $-3.917(3)$ | (6)
J1719$-$1438 | $3.03$ | $3.76(10)$ | $-3.3(5)$ |  | $3.99(9)$ | $-3.0(5)$ |  | $-0.8(6)$ | $-5.0(3)$ |  | $1.9(4)$ | $-11(2)$ | (7)
J1730$-$2304 | $3.03$ | $20.3(11)$ | $0(21)$ |  | $20.29(6)$ | $1(21)$ |  | $11(19)$ | $-17(11)$ |  | $20.06(12)$ | $-4(2)$ | (6)
J1732$-$5049 | $3.22$ | $-0.60(9)$ | $-10.0(2)$ |  | $-0.07(9)$ | $-10.0(2)$ |  | $-8.8(2)$ | $-4.78(13)$ |  | $-0.41(9)$ | $-9.87(19)$ | (2)
J1737$-$0811 | $3.03$ | $-3.9(3)$ | $-11.8(13)$ |  | $-3.4(3)$ | $-11.4(10)$ |  | $-12.5^{+1.5}_{-1.8}$ | $-2.6^{+0.8}_{-0.9}$ |  |  |  |
J1744$-$1134 | $3.03$ | $18.85(2)$ | $-9.73(13)$ |  | $19.11(2)$ | $-9.21(13)$ |  | $1.11(12)$ | $-21.18(7)$ |  | $18.803(4)$ | $-9.390(18)$ | (6)
J1747$-$4036 | $3.14$ | $-1.3(2)$ | $0.0(7)$ |  | $-1.3(2)$ | $-0.1(6)$ |  | $-0.8(10)$ | $1.2(7)$ |  | $-1.30(13)$ | $-2.7(4)$ | (15)
J1751$-$2857 | $3.03$ | $-7.37(10)$ | $-4.8(10)$ |  | $-7.30(10)$ | $-5.0(10)$ |  | $-7.8(9)$ | $3.9(5)$ |  | $-7.4(1)$ | $-4.3(12)$ | (5)
J1757$-$5322 | $2.96$ | $-2.48(8)$ | $-10.02(16)$ |  | $-2.42(8)$ | $-10.04(16)$ |  | $-10.07(17)$ | $-2.33(9)$ |  |  |  |
J1801$-$1417 | $3.03$ | $-10.67(10)$ | $-1.8(6)$ |  | $-10.68(10)$ | $-1.9(6)$ |  | $-6.8^{+0.7}_{-0.6}$ | $8.4^{+0.4}_{-0.3}$ |  | $-10.89(12)$ | $-3.0(10)$ | (5)
J1825$-$0319 | $3.06$ | $3.1(2)$ | $-1.6(6)$ |  | $3.1(2)$ | $-1.8(6)$ |  | $0.0(5)$ | $-3.5(4)$ |  |  |  |
J1832$-$0836 | $3.05$ | $-8.06(8)$ | $-21.5(4)$ |  | $-9.30(8)$ | $-21.0(4)$ |  | $-22.8(4)$ | $-2.8(2)$ |  | $-8.06(5)$ | $-21.01(19)$ | (6)
J1843$-$1113 | $3.03$ | $-2.02(7)$ | $-2.9(4)$ |  | $-2.23(7)$ | $-2.7(4)$ |  | $-3.5(3)$ | $0.5(2)$ |  | $-1.91(7)$ | $-3.2(3)$ | (5)
J1902$-$5105 | $3.08$ | $-3.90(11)$ | $-8.5(3)$ |  | $-4.91(11)$ | $-8.0(3)$ |  | $-9.2(3)$ | $1.43(14)$ |  | $-4.8(13)$ | $4.4(16)$ | (9)
J1903$-$7051 | $3.17$ | $-6.56(13)$ | $-17.68(18)$ |  | $-9.35(13)$ | $-16.37(18)$ |  | $-18.31^{+0.20}_{-0.19}$ | $4.45^{+0.14}_{-0.13}$ |  | $-8.8(16)$ | $-16(2)$ | (9)
J1909$-$3744 | $3.26$ | $-9.513(12)$ | $-35.70(4)$ |  | $-13.852(11)$ | $-34.25(4)$ |  | $-36.83(4)$ | $-3.06(2)$ |  | $-9.5146(8)$ | $-35.776(3)$ | (6)
J1918$-$0642 | $3.06$ | $-7.02(5)$ | $-6.37(17)$ |  | $-7.84(4)$ | $-5.33(17)$ |  | $-8.83(14)$ | $3.44(11)$ |  | $-7.15(2)$ | $-5.97(5)$ | (1)
J1933$-$6211 | $3.19$ | $-5.58(4)$ | $11.05(7)$ |  | $-3.19(3)$ | $11.96(7)$ |  | $10.58(7)$ | $6.43(4)$ |  | $-5.54(7)$ | $10.7(2)$ | (10)
Table 5: continued

continued Equatorial coordinates Ecliptic coordinates Galactic coordinates
Best Previous Measurement Pulsar Span
$\mu_{\alpha}\equiv\dot{\alpha}\cos\delta$ $\mu_{\delta}$
$\mu_{\lambda}\equiv\dot{\lambda}\cos\beta$ $\mu_{\beta}$
$\mu_{l}\equiv\dot{l}\cos b$ $\mu_{b}$ $\mu_{\alpha}$ $\mu_{\delta}$ Ref (yr)
(mas yr-1) (mas yr-1) (mas yr-1) (mas yr-1) (mas yr-1) (mas yr-1) (mas yr-1)
(mas yr-1) Total Proper Motion Detections $>3\sigma$ J1946$-$5403 $3.01$
$-1.08(3)$ $-4.75(5)$ $-2.06(2)$ $-4.41(5)$ $-4.84(5)$ $0.55(3)$ J2010$-$1323
$3.07$ $2.63(8)$ $-6.1(4)$ $1.25(3)$ $-6.5(4)$ $-4.5(3)$ $-4.9(2)$ $2.4(3)$
$-5.6(3)$ (11) J2039$-$3616 $3.01$ $-10.11(5)$ $-8.66(15)$ $-12.06(4)$
$-5.64(16)$ $-10.17^{+0.15}_{-0.14}$ $8.60(8)$ J2124$-$3358 $3.07$ $-14.17(6)$
$-50.28(14)$ $-29.76(4)$ $-42.93(15)$ $-51.44(15)$ $9.11(8)$ $-14.109(19)$
$-50.36(4)$ (6) J2129$-$5721 $3.12$ $9.30(4)$ $-9.61(5)$ $4.54(3)$ $-12.59(5)$
$-12.06(6)$ $-5.76(3)$ $9.30(1)$ $-9.576(13)$ (6) J2145$-$0750 $3.07$
$-9.7(3)$ $-8.2(10)$ $-11.87(8)$ $-4.5(10)$ $-11.9(7)$ $4.4(8)$ $-9.48(2)$
$-9.11(7)$ (6) J2150$-$0326 $2.82$ $6.5(2)$ $-11.1(6)$ $2.28(9)$ $-12.7(6)$
$-5.5(5)$ $-11.3(7)$ J2222$-$0137 $3.07$ $44.8(2)$ $-5.7(6)$ $39.63(9)$
$-21.7(7)$ $23.7(4)$ $-38.5(6)$ $44.70(4)$ $-5.69(8)$ (12) J2229$+$2643 $2.54$
$-1.70(9)$ $-5.75(14)$ $-4.05(8)$ $-4.42(15)$ $-4.70(9)$ $-3.74(15)$ $-2.1(6)$
$-5.7(5)$ (1) J2234$+$0944 $2.54$ $6.9(2)$ $-33.2(5)$ $-6.48(13)$ $-33.3(5)$
$-16.6(3)$ $-29.6(5)$ $6.9(2)$ $-32.0(4)$ (1) J2241$-$5236 $3.19$ $18.875(12)$
$-5.286(13)$ $13.830(10)$ $-13.888(15)$ $-14.904(17)$ $-12.736(10)$
$18.881(4)$ $-5.294(5)$ (6) J2317$+$1439 $2.54$ $-1.45(9)$ $3.7(2)$ $0.20(6)$
$4.0(2)$ $0.51(6)$ $4.0(2)$ $-1.3(3)$ $3.6(5)$ (11) J2322$+$2057 $2.54$
$-18.06(10)$ $-15.24(15)$ $-22.83(7)$ $-6.09(17)$ $-22.82(8)$ $-6.14(17)$
$-18.4(4)$ $-15.4(5)$ (5) J2322$-$2650 $3.06$ $-2.37(9)$ $-8.20(17)$
$-5.59(6)$ $-6.44(19)$ $-8.40(16)$ $1.52(11)$ $-2.4(2)$ $-8.3(4)$ (13) Total
Proper Motion Detections $<3\sigma$ J0955$-$6150 $3.12$
$-0.08^{+0.15}_{-0.14}$ $-0.21^{+0.16}_{-0.15}$ $0.12^{+0.16}_{-0.17}$
$-0.19^{+0.15}_{-0.14}$ $0.06^{+0.17}_{-0.16}$ $-0.21(15)$ J1022$+$1001 $2.97$
$-27^{+50}_{-53}$ $-27^{+129}_{-136}$ $-15.70(13)$ $-36^{+140}_{-137}$
$9^{+84}_{-85}$ $-38^{+111}_{-109}$ $-14.92^{+0.05}_{-0.03}$ $-5.61(3)$ (11)
J1327$-$0755 $2.34$ $-4(2)$ $5(6)$ $-5.06^{+0.10}_{-0.11}$ $3(6)$ $-2(4)$
$6(5)$ J1653$-$2054 $3.04$ $-10.5(13)$ $-15(11)$ $-8.7^{+0.4}_{-0.3}$
$-16^{+11}_{-12}$ $-19(10)$ $-1(6)$ J1705$-$1903 $2.77$ $-1.7(9)$
$-1^{+7}_{-6}$ $-1.5(5)$ $-1(7)$ $-2(6)$ $1^{+4}_{-3}$ J1721$-$2457 $2.73$
$3.4(6)$ $5(8)$ $3.1(3)$ $5(8)$ $6(7)$ $0(4)$ J1802$-$2124 $3.16$ $-1.6(3)$
$-14^{+9}_{-8}$ $-1.7(3)$ $-14(8)$ $-13(7)$ $-6(4)$ $-1.13(12)$ $-3(4)$ (5)
J1811$-$2405 $3.12$ $0.53(9)$ $3(5)$ $0.59(4)$ $3(4)$ $3(4)$ $1(2)$ $0.53(6)$
$0$ (14)

  * •

References: (1) Arzoumanian et al. (2018), (2) Reardon et al. (2016), (3) van
der Wateren et al. (2022), (4) Guillot et al. (2019), (5) Desvignes et al.
(2016), (6) Reardon et al. (2021), (7) Ng et al. (2014), (8) Spiewak et al.
(2020), (9) Camilo et al. (2015), (10) Graikou et al. (2017), (11) Deller et
al. (2019), (12) Guo et al. (2021), (13) Spiewak et al. (2018), (14) Ng et al.
(2020), (15) Alam et al. (2021b)

###  4.2 Parallaxes

The parallax measurements made in this study, as well as the best previous
measurements of those pulsars in the literature and their corresponding
measurement techniques (timing or VLBI), are listed in Table 6 and shown in
Figure 2 ($>3\sigma$ detections). The parallax values and their uncertainties
are derived from the marginalized posterior probability distributions from
temponest. There are $35$ of our MSPs that had parallax measurements with
$>3\sigma$ significance. $12$ of these $35$ are new (PSRs J0125$-$2327,
J0614$-$3329, J0636$-$3044, J1421$-$4409, J1629$-$6902, J1652$-$4838,
J1658$-$5324, J1757$-$5322, J1801$-$1417, J1811$-$2405, J1933$-$6211 and
J1946$-$5403). We improved the precision of $4$ other parallaxes (of the $35$
MSPs) by a factor of 1.25 to 6 including PSRs J1446$-$4701, J1455$-$3330,
J2124$-$3358 and J2322$-$2650\. For the MSPs with less than $3\sigma$
significance in parallax, we calculated the $68\%$ confidence upper level and
listed them in the lower portion of Table 6.

To gain confidence in our parallax measurements, we compared them with their
values derived from other authors in Figure 2 as $22$ of our significant
($>3\sigma$) parallaxes have previous measurements. For PSR J1713$+$0747, we
measured the parallax of $0.88(10)$ mas using only $1.98$ yr of data as its
subsequent timing was compromised by a sudden step-change in its pulse profile
and gradual decay back to the original profile. In addition to the previously
measured parallax value of $0.763(21)$ mas by Reardon et al. (2021) using
timing analysis of the Parkes pulsar timing array second data release, a
parallax of $0.82(3)$ mas was obtained by Arzoumanian et al. (2018) using
timing analysis of the NANOGrav 11-year data set. Using the VLBI technique,
Chatterjee et al. (2009) measured a value of $0.95^{+0.06}_{-0.05}$ mas.
Desvignes et al. (2016) obtained a value of $0.90(3)$ mas, very close to our
value. All measurements are in agreement with our value to within about
$1\sigma$.

Of the $35$ MSPs with significant measurements, only our parallaxes for PSRs
J1643$-$1224 and J2322$-$2650 (shown in Figure 2) are markedly inconsistent
with the best previous measurements. We discuss each of these two pulsars in
turn below. Overall the standard deviation of the differences in relative
parallaxes for the $23$ pulsars with previous measurements in literature is
only $10$ percent, and if we exclude the two outliers discussed above, only
$7$ percent.

####  4.2.1 PSR J1643$-$1224

For PSR J1643$-$1224, we obtained the parallax value of
$\varpi=2.6^{+0.5}_{-0.6}$ mas which is quite inconsistent with the VLBI value
of $\varpi=1.1(1)$ mas obtained by Ding et al. (2023) and also has the lower
formal error of the measurements.

Matthews et al. (2016) measured a value of $0.7(6)$ mas using the NANOGrav
nine-year data set and a detailed model for DM variation that is known to be
significant. They compared their value with the measurement of Verbiest et al.
(2009), who obtained a value of $2.2(4)$ mas which did not include a complex
model for DM variation. Matthews et al. (2016) hence argued that the limited
DM variation modelling might be the possible source of the inconsistency
between their measurements.

Two more measurements come from Reardon et al. (2021), who obtain the parallax
to be $0.82(17)$ mas using the $24$ yr observations of the Parkes Pulsar
Timing Array data set and Desvignes et al. (2016), who obtain a parallax of
$1.17(26)$ mas. Clearly, the timing of this pulsar presents challenges.

PSR J1643$-$1224 has been shown to be located behind the HII region Sh 2$-$27
by Harvey-Smith et al. (2011) and undergoes annual DM variations. Its timing
with MeerKAT also exhibits strong evidence for chromatic variations (Miles et
al. in prep), probably due to time-dependent scattering in the HII region
which are not modelled by our techniques. Ocker et al. (2020) reported that
there is a significant excess DM for PSR J1643$-$1224 consistent with its
distance beyond the HII region.

We examined whether our parallax might be radio-frequency dependent, by
measuring the parallax using just the top half of our band ($1284$–$1712$
MHz), and we obtained a smaller value of $1.7(4)$ mas, which is in better
agreement ($1.5\sigma$) with the measurement of Ding et al. (2023). In our
noise analysis for this pulsar, using the entire band, the noise model with
the white, red, and DM noise parameters had the highest Bayesian evidence;
however, using only the top part of the band the noise model with the white
and red noise parameters had the highest Bayesian evidence. We know that this
pulsar is significantly affected by interstellar scattering, but this is not
currently included in our noise model. Weaknesses in our noise model such as
this may lead to small biases in the parallax measurement. This would be good
topic for a new study. For the rest of this paper, we adopt the VLBI parallax.

####  4.2.2 PSR J2322$-$2650

For PSR J2322$-$2650, we have measured a parallax of $1.3(2)$ mas, which is
very different from the previously measured value of $4.4(12)$ mas by Spiewak
et al. (2018) but since it has $6$ times lower uncertainty we favour our
value. Our parallax distance for PSR J2322$-$2650 is thus
$0.80^{+0.18}_{-0.12}$ kpc which is much nearer the YMW16 distance of $0.76$
kpc than the $0.23$ kpc suggested by Spiewak et al. (2018). We speculate that
the inconsistency between our values is due to the absence of any DM noise
model in the earlier analysis of Spiewak et al. (2018) and their poorer timing
residuals.

Table 6: Pulsar parallaxes and distances. MSPs for which we have measured the parallax distances with statistical significance of $>3\sigma$ and $<3\sigma$ are listed in the upper and lower portions of the table, respectively. The DM distances, with two decimal digits, are derived from the two DM models: NE2001 and YMW16. $68\%$ upper limits on parallaxes and lower limits on parallax distances are provided for parallaxes with statistical significance of less than $3\sigma$ (lower portion). |  |  |  | Best Previous Measurement |  | DM distances |
---|---|---|---|---|---|---|---
Pulsar |  | Parallax $\varpi$ | Distance $d$ |  | Parallax $\varpi$ | Technique | Ref. |  | NE2001 | YMW16 |
|  | (mas) | (kpc) |  | (mas) |  |  |  | (kpc) | (kpc) |
Significant Parallaxes Detections ($>3\sigma$) |
J0030$+$0451 |  | $2.91(18)$ | $0.34(2)$ |  | $3.04(5)$ | VLBI | (1) |  | $0.32$ | $0.34$ |
J0125$-$2327 |  | $0.84(11)$ | $1.19^{+0.17}_{-0.13}$ |  |  |  |  |  | $0.44$ | $0.87$ |
J0437$-$4715 |  | $6.8(4)$ | $0.148^{+0.009}_{-0.008}$ |  | $6.396(54)$ | VLBI | (2) |  | $0.14$ | $0.16$ |
J0613$-$0200 |  | $0.86(17)$ | $1.16^{+0.28}_{-0.19}$ |  | $1.01(9)$ | Timing | (3) |  | $1.71$ | $1.02$ |
J0614$-$3329 |  | $1.5(4)$ | $0.67^{+0.25}_{-0.14}$ |  | $<2.2$ | Timing | (4) |  | $1.90$ | $2.69$ |
J0636$-$3044 |  | $4.3(7)$ | $0.23^{+0.04}_{-0.03}$ |  |  |  |  |  | $1.00$ | $0.68$ |
J1024$-$0719 |  | $0.97^{+0.12}_{-0.13}$ | $1.03^{+0.15}_{-0.11}$ |  | $0.93(5)$ | VLBI | (1) |  | $0.39$ | $0.38$ |
J1125$-$6014 |  | $1.2^{+0.3}_{-0.2}$ | $0.86^{+0.22}_{-0.16}$ |  | $0.6(3)$ | Timing | (3) |  | $1.49$ | $0.99$ |
J1421$-$4409 |  | $1.2(4)$ | $0.8^{+0.5}_{-0.2}$ |  |  |  |  |  | $1.57$ | $2.09$ |
J1446$-$4701 |  | $0.7(2)$ | $1.4^{+0.7}_{-0.3}$ |  | $0.6(3)$ | Timing | (3) |  | $1.46$ | $1.57$ |
J1455$-$3330 |  | $1.1(3)$ | $0.94^{+0.33}_{-0.19}$ |  | $0.99(22)$ | Timing | (4) |  | $0.53$ | $0.68$ |
J1600$-$3053 |  | $0.51(11)$ | $2.0^{+0.5}_{-0.3}$ |  | $0.53(6)$ | Timing | (3) |  | $1.63$ | $2.53$ |
J1614$-$2230 |  | $1.3(2)$ | $0.77^{+0.14}_{-0.11}$ |  | $1.54(10)$ | Timing | (5) |  | $1.27$ | $1.39$ |
J1629$-$6902 |  | $0.9(2)$ | $1.1^{+0.3}_{-0.2}$ |  |  |  |  |  | $0.96$ | $0.96$ |
J1643$-$1224 |  | $2.6^{+0.5}_{-0.6}$ | $0.39^{+0.11}_{-0.07}$ |  | $1.1(1)$ | VLBI | (1) |  | $2.40$ | $0.79$ |
J1652$-$4838 |  | $2.4^{+0.8}_{-0.7}$ | $0.41^{+0.19}_{-0.10}$ |  |  |  |  |  | $3.38$ | $4.33$ |
J1658$-$5324 |  | $1.3(3)$ | $0.75^{+0.17}_{-0.12}$ |  |  |  |  |  | $0.93$ | $0.88$ |
J1713$+$0747 |  | $0.88(10)$ | $1.14^{+0.15}_{-0.11}$ |  | $0.763(21)$ | Timing | (3) |  | $0.89$ | $0.92$ |
J1730$-$2304 |  | $2.3(2)$ | $0.44^{+0.05}_{-0.04}$ |  | $2.0(1)$ | VLBI | (1) |  | $0.53$ | $0.51$ |
J1744$-$1134 |  | $2.61(9)$ | $0.384^{+0.014}_{-0.013}$ |  | $2.44(5)$ | Timing | (3) |  | $0.41$ | $0.15$ |
J1757$-$5322 |  | $1.2(3)$ | $0.81^{+0.25}_{-0.15}$ |  |  |  |  |  | $0.96$ | $0.94$ |
J1801$-$1417 |  | $0.6(2)$ | $1.7^{+1.1}_{-0.5}$ |  |  |  |  |  | $1.52$ | $1.10$ |
J1811$-$2405 |  | $0.72(15)$ | $1.4^{+0.4}_{-0.2}$ |  | $<0.4$ | Timing | (4) |  | $1.77$ | $1.83$ |
J1832$-$0836 |  | $0.7(2)$ | $1.4^{+0.6}_{-0.3}$ |  | $0.48(14)$ | Timing | (5) |  | $1.11$ | $0.81$ |
J1909$-$3744 |  | $0.92(3)$ | $1.09(4)$ |  | $0.86(1)$ | Timing | (3) |  | $0.46$ | $0.56$ |
J1918$-$0642 |  | $0.84(15)$ | $1.20^{+0.26}_{-0.18}$ |  | $0.71(7)$ | VLBI | (1) |  | $1.24$ | $1.03$ |
J1933$-$6211 |  | $0.7(2)$ | $1.5^{+0.6}_{-0.3}$ |  |  |  |  |  | $0.52$ | $0.65$ |
J1946$-$5403 |  | $0.81^{+0.10}_{-0.11}$ | $1.23^{+0.18}_{-0.14}$ |  |  |  |  |  | $0.87$ | $1.15$ |
J2010$-$1323 |  | $0.32(11)$ | $3.1^{+1.7}_{-0.8}$ |  | $0.41(12)$ | Timing | (10) |  | $1.02$ | $1.16$ |
J2124$-$3358 |  | $2.06^{+0.16}_{-0.15}$ | $0.48(4)$ |  | $2.3(2)$ | Timing | (3) |  | $0.27$ | $0.36$ |
J2145$-$0750 |  | $1.6(3)$ | $0.61^{+0.14}_{-0.09}$ |  | $1.603^{+0.063}_{-0.009}$ | VLBI | (9) |  | $0.57$ | $0.69$ |
J2222$-$0137 |  | $3.7(3)$ | $0.27(2)$ |  | $3.730(16)$ | VLBI | (6) |  | $0.31$ | $0.27$ |
J2241$-$5236 |  | $0.91(6)$ | $1.10^{+0.07}_{-0.06}$ |  | $0.96(4)$ | Timing | (3) |  | $0.51$ | $0.96$ |
J2322$+$2057 |  | $1.2(3)$ | $0.80^{+0.21}_{-0.15}$ |  | $0.98(26)$ | Timing | (5) |  | $0.80$ | $1.01$ |
J2322$-$2650 |  | $1.3(2)$ | $0.80^{+0.18}_{-0.12}$ |  | $4.4(12)$ | Timing | (7) |  | $0.32$ | $0.76$ |
Table 7: continued

| Parallax $\varpi$ | Distance $d$ |  | Best Previous Measurement |  | DM distances |
---|---|---|---|---|---|---|---
Pulsar | Measurement |  $1\sigma$ limit |  $1\sigma$ limit |  | Parallax $\varpi$ | Technique | Ref. |  | NE2001 | YMW16 |
| (mas) | (mas) | (kpc) |  | (mas) |  |  |  | (kpc) | (kpc) |
Weak Parallaxes Detections ($<3\sigma$) |
J0610$-$2100 | $0.5^{+0.5}_{-0.4}$ | $<1.0$ | $>1.0$ |  | $0.72(11)$ | VLBI | (1) |  | $3.54$ | $3.26$ |
J0711$-$6830 | $4^{+6}_{-3}$ | $<10.0$ | $>0.1$ |  |  |  |  |  | $0.86$ | $0.11$ |
J0900$-$3144 | $1.1^{+1.3}_{-0.8}$ | $<2.4$ | $>0.4$ |  | $0.77(44)$ | Timing | (8) |  | $0.54$ | $0.38$ |
J0931$-$1902 | $0.5^{+0.4}_{-0.3}$ | $<0.9$ | $>1.2$ |  | $0.7(7)$ | Timing | (5) |  | $1.88$ | $3.72$ |
J0955$-$6150 | $0.6^{+0.8}_{-0.4}$ | $<1.4$ | $>0.6$ |  |  |  |  |  | $4.04$ | $2.17$ |
J1012$-$4235 | $1.3^{+0.9}_{-0.8}$ | $<2.2$ | $>0.5$ |  |  |  |  |  | $2.51$ | $0.37$ |
J1017$-$7156 | $1.5^{+0.9}_{-0.8}$ | $<2.4$ | $>0.5$ |  | $0.6(6)$ | Timing | (3) |  | $2.98$ | $1.81$ |
J1022$+$1001 | $1.1(4)$ | $<1.5$ | $>0.7$ |  | $1.38^{+0.04}_{-0.03}$ | VLBI | (9) |  | $0.45$ | $0.83$ |
J1036$-$8317 | $0.6^{+0.7}_{-0.4}$ | $<1.3$ | $>0.7$ |  |  |  |  |  | $1.03$ | $0.93$ |
J1045$-$4509 | $1.0^{+1.0}_{-0.7}$ | $<2.0$ | $>0.5$ |  | $1.7(7)$ | Timing | (3) |  | $1.96$ | $0.33$ |
J1101$-$6424 | $1.0^{+1.3}_{-0.7}$ | $<2.3$ | $>0.4$ |  |  |  |  |  | $4.48$ | $2.18$ |
J1103$-$5403 | $1.4^{+1.9}_{-1.0}$ | $<3.3$ | $>0.3$ |  |  |  |  |  | $2.55$ | $1.68$ |
J1125$-$5825 | $1.3^{+0.8}_{-0.7}$ | $<2.1$ | $>0.5$ |  |  |  |  |  | $2.62$ | $1.74$ |
J1216$-$6410 | $0.9(4)$ | $<1.3$ | $>0.8$ |  | $0.6(3)$ | Timing | (3) |  | $1.33$ | $1.10$ |
J1327$-$0755 | $0.13^{+0.17}_{-0.09}$ | $<0.3$ | $>3.4$ |  |  |  |  |  | $1.73$ | $25.00$ |
J1431$-$5740 | $2.4^{+1.4}_{-1.3}$ | $<3.8$ | $>0.3$ |  |  |  |  |  | $2.55$ | $3.55$ |
J1435$-$6100 | $0.4^{+0.5}_{-0.3}$ | $<0.9$ | $>1.1$ |  |  |  |  |  | $2.16$ | $2.82$ |
J1525$-$5545 | $1.5^{+1.3}_{-1.0}$ | $<2.8$ | $>0.3$ |  |  |  |  |  | $2.37$ | $3.14$ |
J1543$-$5149 | $0.6^{+0.5}_{-0.4}$ | $<1.1$ | $>0.9$ |  |  |  |  |  | $2.43$ | $1.15$ |
J1545$-$4550 | $0.38(18)$ | $<0.6$ | $>1.8$ |  | $0.45(14)$ | Timing | (3) |  | $2.13$ | $2.25$ |
J1547$-$5709 | $0.6^{+0.6}_{-0.4}$ | $<1.2$ | $>0.8$ |  |  |  |  |  | $1.87$ | $2.70$ |
J1603$-$7202 | $1.1^{+0.7}_{-0.6}$ | $<1.8$ | $>0.6$ |  | $0.3(3)$ | Timing | (3) |  | $1.17$ | $1.13$ |
J1653$-$2054 | $0.24^{+0.34}_{-0.17}$ | $<0.6$ | $>1.7$ |  |  |  |  |  | $1.67$ | $2.63$ |
J1705$-$1903 | $0.6^{+0.6}_{-0.4}$ | $<1.2$ | $>0.8$ |  |  |  |  |  | $1.65$ | $2.34$ |
J1708$-$3506 | $0.5^{+0.4}_{-0.3}$ | $<0.9$ | $>1.1$ |  |  |  |  |  | $2.80$ | $3.32$ |
J1719$-$1438 | $0.30^{+0.23}_{-0.18}$ | $<0.6$ | $>1.9$ |  |  |  |  |  | $1.21$ | $0.34$ |
J1721$-$2457 | $0.24^{+0.29}_{-0.16}$ | $<0.6$ | $>2.0$ |  |  |  |  |  | $1.30$ | $1.39$ |
J1732$-$5049 | $0.7(4)$ | $<1.1$ | $>0.9$ |  |  |  |  |  | $1.41$ | $1.87$ |
J1737$-$0811 | $0.4^{+0.4}_{-0.3}$ | $<0.8$ | $>1.2$ |  |  |  |  |  | $1.71$ | $0.21$ |
J1747$-$4036 | $0.17^{+0.25}_{-0.12}$ | $<0.5$ | $>2.3$ |  | $0.1(7)$ | Timing | (5) |  | $3.39$ | $7.15$ |
J1751$-$2857 | $0.6(3)$ | $<0.9$ | $>1.2$ |  |  |  |  |  | $1.10$ | $1.09$ |
J1802$-$2124 | $0.4^{+0.5}_{-0.3}$ | $<0.9$ | $>1.2$ |  | $1.24(57)$ | Timing | (8) |  | $2.94$ | $3.03$ |
J1825$-$0319 | $0.5^{+0.4}_{-0.3}$ | $<0.9$ | $>1.1$ |  |  |  |  |  | $3.07$ | $3.86$ |
J1843$-$1113 | $0.32^{+0.17}_{-0.16}$ | $<0.5$ | $>2.0$ |  | $0.69(33)$ | Timing | (8) |  | $1.70$ | $1.71$ |
J1902$-$5105 | $0.20^{+0.25}_{-0.14}$ | $<0.5$ | $>2.3$ |  |  |  |  |  | $1.18$ | $1.65$ |
J1903$-$7051 | $0.8^{+0.8}_{-0.5}$ | $<1.6$ | $>0.6$ |  |  |  |  |  | $0.76$ | $0.93$ |
J2039$-$3616 | $0.25^{+0.14}_{-0.13}$ | $<0.4$ | $>2.5$ |  |  |  |  |  | $0.91$ | $1.70$ |
J2129$-$5721 | $0.12^{+0.13}_{-0.08}$ | $<0.3$ | $>4.1$ |  | $0.26(17)$ | Timing | (3) |  | $1.36$ | $6.16$ |
J2150$-$0326 | $0.30^{+0.28}_{-0.19}$ | $<0.6$ | $>1.7$ |  |  |  |  |  | $1.05$ | $1.98$ |
J2229$+$2643 | $0.6(3)$ | $<0.9$ | $>1.1$ |  | $0.2(3)$ | Timing | (5) |  | $1.43$ | $1.80$ |
J2234$+$0944 | $1.0(4)$ | $<1.4$ | $>0.7$ |  | $0.7(3)$ | Timing | (5) |  | $1.00$ | $1.58$ |
J2317$+$1439 | $0.29^{+0.16}_{-0.15}$ | $<0.5$ | $>2.2$ |  | $0.60(8)$ | Timing | (5) |  | $0.83$ | $2.16$ |

  * •

References: (1) Ding et al. (2023), (2) Deller et al. (2008), (3) Reardon et
al. (2021), (4) Guillemot et al. (2016), (5) Alam et al. (2021b), (6) Guo et
al. (2021), (7) Spiewak et al. (2018), (8) Desvignes et al. (2016), (9) Deller
et al. (2019), (10) Alam et al. (2021a)

Figure 2: Comparison of our $23$ parallax measurements with the measurements
from literature. All of our measurements are consistent with previous work
with only two exceptions, namely PSRs J1643$-$1224 and J2322$-$2650 that we
discuss in Section 4.2. Table 8: Pulsar transverse velocities. The second
column shows the velocities that are corrected for the peculiar velocity of
the Sun and the Galactic differential rotation ($v_{\rm\perp,LSR}$). The third
column shows the observed transverse velocities without any corrections
($v_{\rm\perp,obs}$). The fourth column shows the best previous reported
velocities, some of which did not correct for the local standard of rest. The
size of the error bars is showing $68$% confidence interval from posterior
probability distributions of velocities.

|  |  |  |  | Best Previous Reports
---|---|---|---|---|---
Pulsar | $v_{\rm\perp,LSR}$ |  | $v_{\rm\perp,obs}$ |  | $v_{\perp}$ | Ref.
| (km s-1) |  | (km s-1) |  | (km s-1) |
J0030$+$0451 | $25^{+4}_{-3}$ |  | $11^{+3}_{-2}$ |  | $15.4(2)$ | (1)
J0125$-$2327 | $204^{+31}_{-24}$ |  | $218^{+31}_{-24}$ |  |  |
J0437$-$4715 | $99(6)$ |  | $99(6)$ |  | $104.72(14)$ | (2)
J0610$-$2100† | $113^{+23}_{-17}$ |  | $126^{+23}_{-17}$ |  | $120^{+25}_{-17}$ | (1)
J0613$-$0200 | $58^{+14}_{-9}$ |  | $58^{+14}_{-9}$ |  | $49(5)$ | (3)
J0614$-$3329 | $22.4^{+2.3}_{-1.3}$ |  | $6.1^{+2.3}_{-1.3}$ |  |  |
J0636$-$3044 | $43^{+6}_{-5}$ |  | $36^{+6}_{-5}$ |  |  |
J1024$-$0719 | $278^{+44}_{-32}$ |  | $293^{+44}_{-33}$ |  | $300(20)$ | (1)
J1045$-$4509† | $16^{+4}_{-2}$ |  | $22^{+15}_{-6}$ |  | $22^{+16}_{-7}$ | (3)
J1125$-$6014 | $101^{+17}_{-12}$ |  | $70^{+18}_{-13}$ |  | $124^{+90}_{-40}$ | (3)
J1421$-$4409 | $21^{+27}_{-13}$ |  | $48^{+27}_{-13}$ |  |  |
J1446$-$4701 | $7^{+8}_{-5}$ |  | $33^{+15}_{-8}$ |  | $39^{+40}_{-14}$ | (3)
J1455$-$3330 | $53^{+12}_{-7}$ |  | $36^{+13}_{-7}$ |  |  |
J1545$-$4550† | $56^{+10}_{-5}$ |  | $27^{+12}_{-7}$ |  | $26^{+12}_{-7}$ | (3)
J1600$-$3053 | $57^{+20}_{-12}$ |  | $71^{+20}_{-13}$ |  | $62^{+8}_{-7}$ | (3)
J1614$-$2230 | $98^{+21}_{-16}$ |  | $108^{+22}_{-17}$ |  |  |
J1629$-$6902 | $26^{+16}_{-10}$ |  | $56^{+17}_{-11}$ |  |  |
J1643$-$1224† | $39^{+6}_{-5}$ |  | $30^{+5}_{-4}$ |  | $41^{+5}_{-4}$ | (1)
J1652$-$4838 | $5^{+7}_{-2}$ |  | $20^{+9}_{-5}$ |  |  |
J1658$-$5324 | $31.3^{+2.7}_{-1.8}$ |  | $10.1^{+2.7}_{-1.8}$ |  |  |
J1713$+$0747 | $28^{+4}_{-3}$ |  | $35^{+4}_{-3}$ |  | $39.1(11)$ | (3)
J1730$-$2304 | $54^{+26}_{-11}$ |  | $53^{+25}_{-10}$ |  | $54^{+3}_{-2}$ | (1)
J1744$-$1134 | $35.0(13)$ |  | $38.6(14)$ |  | $40.9^{+0.9}_{-0.8}$ | (3)
J1757$-$5322 | $21^{+12}_{-7}$ |  | $39^{+12}_{-7}$ |  |  |
J1801$-$1417 | $84^{+57}_{-24}$ |  | $85^{+57}_{-24}$ |  |  |
J1832$-$0836 | $134^{+61}_{-33}$ |  | $149^{+61}_{-33}$ |  | $170^{+80}_{-40}$ | (3)
J1843$-$1113† | $14^{+18}_{-5}$ |  | $24^{+20}_{-8}$ |  |  |
J1909$-$3744 | $179(7)$ |  | $191(7)$ |  | $203(3)$ | (3)
J1918$-$0642 | $44^{+11}_{-8}$ |  | $54^{+11}_{-8}$ |  | $44^{+6}_{-5}$ | (1)
J1933$-$6211 | $117^{+37}_{-20}$ |  | $88^{+37}_{-20}$ |  |  |
J1946$-$5403 | $21^{+3}_{-2}$ |  | $28^{+4}_{-3}$ |  |  |
J2010$-$1323 | $83^{+47}_{-22}$ |  | $99^{+49}_{-25}$ |  | $58.7^{+27}_{-16}$ | (4)
J2124$-$3358 | $114^{+10}_{-9}$ |  | $120^{+10}_{-9}$ |  | $110^{+11}_{-9}$ | (3)
J2145$-$0750 | $37^{+8}_{-5}$ |  | $37^{+8}_{-6}$ |  | $38.9^{+0.5}_{-1.9}$ | (4)
J2222$-$0137 | $47^{+5}_{-4}$ |  | $58^{+5}_{-4}$ |  |  |
J2234$+$0944† | $207^{+162}_{-68}$ |  | $227^{+162}_{-69}$ |  |  |
J2241$-$5236 | $79^{+7}_{-6}$ |  | $103^{+7}_{-6}$ |  | $97(5)$ | (3)
J2322$+$2057 | $82^{+23}_{-17}$ |  | $90^{+23}_{-17}$ |  |  |
J2322$-$2650 | $32^{+7}_{-4}$ |  | $32^{+8}_{-5}$ |  |  |

  * $\bullet$

We used our proper motions and other author’s (usually parallax) distances
(listed in Table 6) to calculate the velocities of the MSPs that are marked by
a “†”.

  * $\bullet$

References: (1) Ding et al. (2023), (2) Reardon et al. (2016), (3) Reardon et
al. (2021), (4) Deller et al. (2019),

##  5 Kinematics of the MSP sample

In this section we study the kinematics of our MSP sample, using only those
with parallax-based distances and velocities. While models of the ISM can be
used to estimate distances to pulsars, it is well known that this can lead to
large uncertainties. We choose here to base our kinematic analysis on MSPs
with high quality distances ($>3\sigma$ significance) only.

For a detailed analysis of the uncertainties involved in the use of DM-based
distances via ISM models, we refer the reader to Appendix A.

###  5.1 Distances

We used the posterior samples of the parallax measurements to derive the
posterior distribution of the distances. The distances of the $35$ MSPs with
$>3\sigma$ significance are listed in the upper portion of Table 6. For the
MSPs with $<3\sigma$ significance in parallax, we calculated $68\%$ confidence
lower limit on distances and listed them in the lower portion of Table 6.

Lutz & Kelker (1973) pointed out a bias in the parallax of stars and showed
that the bias depends on the significance of the parallax with low-
significance parallaxes more likely to be over-estimated due to volume
effects. This has the effect of pushing low-significance pulsar parallax
objects further from the Sun. Verbiest et al. (2010, 2012) confirmed the
existence of the Lutz-Kelker (LK) bias in observations, and they provided a
method in order to correct the values of parallaxes and distances. For our
discussions concerning pulsar distances and velocities, we will not adopt the
LK corrections for individual objects to ease comparison with other recent
authors (e.g., Matthews et al., 2016; Arzoumanian et al., 2018), but will
discuss what effects it would have on the median velocities of the population.
One problem with the LK correction is that the magnitude of the correction
depends upon the assumed model for the spatial (i.e. Galactic) distribution of
the pulsars, and also the luminosity function of MSPs. For high-precision
parallaxes, the effects are negligible.

###  5.2 Tangential Velocities

From the proper motions and distances of the MSPs, we calculated the
barycentric transverse velocities, $v_{\perp}$, and then using the method
explained in Section 3.4.1 we corrected them for the peculiar velocity of the
Sun and the rotation curve of the Galaxy, assuming zero radial velocity with
respect to our line of sight. In Table 8, a summary of $39$ velocities for two
subsets of our MSPs is given. The first subset is those MSPs with $>3\sigma$
significance for their parallax and proper motion measurements. The second
subset is those MSPs with $>3\sigma$ significance for their proper motions
measurements but $<3\sigma$ significance for their parallax measurements. For
MSPs in the second subset namely: PSRs J0610$-$2100, J1045$-$4509,
J1545$-$4550, J1643$-$1224, J1843$-$1113, and J2234$+$0944, we used parallaxes
measured by other authors listed in the fifth column of Table 6 to calculate
their transverse velocities (these velocities are denoted by a ${\dagger}$
after the pulsar name). We used the positions, proper motions, and parallaxes
of other MSPs in the pulsar catalogue (Manchester et al., 2005) and selected
those with significant parallaxes ($>3\sigma$) and calculated their transverse
velocities using our method described in Section 3 in order to increase our
MSP velocity sample.

####  5.2.1 Isolated versus Binary Millisecond Pulsars

We divided the MSPs into two subsets depending upon whether they are isolated
or binary MSPs. The velocity distributions for the isolated and binary MSPs
are presented in Figure 3. We obtained a mean velocity of $67(12)\,{\rm
km\,s^{-1}}$ for $16$ isolated MSPs, and a mean velocity of $72(8)\,{\rm
km\,s^{-1}}$ for the $49$ binary MSPs. Between these MSPs, PSR J1024$-$0719 is
a special case with an extremely wide binary with the orbital period of
$2$–$20$ kyr (Kaplan et al., 2016) (presented with a black bin in Figure 3).
Also, PSR J1300$+$1240 is a planetary system (Wolszczan, 1990) (presented with
an olive bin in Figure 3). We did not include them for calculating the mean
velocity of the binary MSPs as their evolutionary history is probably quite
different from the other systems. The two-sample Kolmogorov-Smirnov test was
performed for comparing the distributions of velocity samples. The maximum
absolute difference between the cumulative distribution functions of the two
samples was $0.17$ and the $p$-value was $0.87$. This means with our limited
sample that there is no statistical evidence that the two samples are drawn
from different distributions. Whatever causes isolated MSPs to be created does
not have an appreciably different effect on their velocities from that of the
binaries.

Figure 3: Histograms of transverse velocities for the isolated MSPs (red) and
the binary MSPs (blue). PSR J1024$-$0719 is shown in black due to its
extremely long orbital period ($2$–$20$ kyr (Kaplan et al., 2016)). PSR
J1300$+$1240 is shown in olive as it is a planetary system (Wolszczan, 1990).
These two pulsars are not included in obtaining the velocity dispersion of
binary MSPs as the evolutionary history is quite unlike that of the others in
the sample.

###  5.3 Dispersion of the Velocity Components

We obtained cylindrical Galactocentric velocity components of $v_{\rm\rho}$
(radial), $v_{\rm\phi}$ (rotational), and $v_{\rm z}$ (perpendicular) and then
calculated the components that are at least $70^{\circ}$ from the line of
sight (i.e. for the components that are nearly perpendicular to the line of
sight direction). This condition satisfies the $68\%$ confidence level
condition for the measures of the velocity components. The angles with respect
to the line of sight are:

$\cos(a_{\rm\rho})=\frac{d^{2}+\rho^{2}-\sqrt{(z-z_{\odot})^{2}+R^{2}_{\odot}}}{2\,\rho\,d},$
(3) $\cos(a_{\rm z})=\frac{z-z_{\odot}}{d},$ (4)

and

$\cos(a_{\rm\phi})=\sqrt{1-\cos(a_{\rm\rho})^{2}-\cos(a_{\rm z})^{2}},$ (5)

where $a_{\rm\rho}$, $a_{\rm\phi}$, and $a_{\rm z}$ are the angles between the
line of sight vector and the $\rho$, $\phi$, and $z$ components, respectively.
When the condition is satisfied, we have $\left|\cos
a_{\rm\rho,\phi,z}\right|\leq 0.34$. We implemented the direction cosines of
the line of sight vector for calculating $\cos(a_{\rm\phi})$. We have plotted
them in Figures 4, 5, and 6.

After identifying the desired components for every pulsar, we aimed to find
the dispersion (the standard deviation) for each velocity component. We are
not able to measure 3D space velocities of MSPs due to the lack of line-of-
sight radial velocities for many of them. We found the dispersion of the
components as follows:

$\sigma_{\rho}=101(17)\,{\rm km\,s^{-1}}~{}~{}~{}~{}\sigma_{\phi}=56(9)\,{\rm
km\,s^{-1}}~{}~{}~{}~{}\sigma_{z}=37(5)\,{\rm km\,s^{-1}}.$ (6)

There are multiple outliers in the data points of the velocity components in
the Figures. To remove the outliers to see any underlying trends, we used the
median absolute deviation (MAD) as a robust measure of dispersion, assuming a
Gaussian velocity distribution, and followed the criteria discussed by Leys et
al. (2013). Similar to Matthews et al. (2016), for every MSP with the velocity
component of $v_{\rm i}$ we obtained $\left|v_{\rm i}-M\right|/\rm MAD$, where
$M$ is the global median of the i-th velocity component. To be consistent with
Matthews et al. (2016) analysis, we chose $\left|v_{\rm i}-M\right|/\rm MAD>3$
to find outliers. After excluding the outliers, the dispersions of the
components changed to:

$\sigma_{\rho}=63(11)\,{\rm km\,s^{-1}}~{}~{}~{}~{}\sigma_{\phi}=48(8)\,{\rm
km\,s^{-1}}~{}~{}~{}~{}\sigma_{z}=19(3)\,{\rm km\,s^{-1}}.$ (7)

Before interpreting the velocities we should point out that there are multiple
selection effects at play in Figure 6. This figure tends to imply that
$\sigma_{z}$ of nearby MSPs is much lower compared to that of further MSPs.
This means it is more likely to find MSPs with low $v_{z}$ near the Sun. MSPs
that are near us are either moving very slowly and will always be near us, or
they are moving fast and just happen to be passing near us. MSPs rising out of
the Galactic plane will slow down eventually (e.g. a pulsar with a vertical
velocity of $\approx 80$ km s-1 will rise $\approx 2$ kpc) and then return
back towards the plane. The amount of time such pulsars spend around $2$ kpc
is much more than the time they spend nearby the Sun when they are passing
quickly through the plane. Therefore, for a short amount of time they are
expected to be observed with high luminosity, and so it is easier to measure
their parallaxes and proper motions: this leads to a strong selection effect
in the sample. Another selection effect is that the more distant pulsars in
the sample are likely to be drawn from the luminous end of the pulsar
luminosity function and could be seen at greater $z$-distances. Finally, the
proper motions of fast moving pulsars are easier to determine than those of
slow moving pulsars, which will likely boost the mean tangential velocity of
the sample at larger distances. All these selection effects are at play, and
simple comparisons of the observed MSP velocities as a function of $z$ or
distance needs to be performed with caution and best done by simulations as
was attempted for the slow pulsars by Lorimer et al. (1997).

Figure 4: Radial component of the cylindrical Galactocentric velocity
($v_{\rho}$) versus distance from the Galactic centre ($\rho$) for MSPs whose
radial velocity vectors are $>70^{\circ}$ from the line of sight direction.
The dotted line indicates the assumed position of the Sun. Velocities
calculated using MSPs in this work are in blue and from the pulsar catalogue
are in red. The velocity dispersion of these MSPs is $63(11)$ km s-1, if we
exclude PSR J0125$-$2327 and J1300$+$1240 as outliers. Figure 5: Rotational
component of the cylindrical Galactocentric velocity ($\Delta v_{\phi}$)
versus distance from the Galactic centre ($\rho$) for MSPs whose rotational
velocity vector are $>70^{\circ}$ from the line of sight direction. The dotted
line indicates the assumed position of the Sun. Velocities calculated using
MSPs in this work are in blue and from the pulsar catalogue are in red. The
velocity dispersion of these MSPs is this component is $46(8)$ km s-1,
excluding PSR J1909$-$3744 as an outlier. The signature of asymmetric drift is
visible with the mean drift velocity of $39(11)$ km s-1. Figure 6: $z$
component of the cylindrical Galactocentric velocity ($v_{z}$) versus distance
from the Galactic centre ($\rho$) for MSPs whose $z$-velocity vector are
nearly perpendicular to the line of sight direction ($>70^{\circ}$). The
dotted line indicates the assumed position of the Sun. Velocities calculated
using MSPs in this work are shown in blue and those taken from the pulsar
catalogue (psrcat) are shown in red. The velocity dispersion along the $z$
direction is $19(3)$ km s-1, excluding $6$ outliers namely PSRs J1801$-$1417,
J0610$-$2100, J1911$-$1114, J1227$-$4853, J1417$-$4402 and J1431$-$4715\. It
is clear that near the Sun (at $\rho\approx 8.12$ kpc) $\sigma_{z}$ is small,
and it becomes larger for MSPs further away from the Sun. As mentioned in the
text multiple selection effects are at play.

###  5.4 Asymmetric drift in the Galaxy

An asymmetric drift is predicted to be observed in normal pulsars with
characteristic ages greater than $10^{7}$ yr (Hansen & Phinney, 1997). MSPs
have characteristic ages greater than $10^{8}$ yr and are expected to be old
enough to have reached a virialized state; so they are also expected to
demonstrate such a drift. Cordes & Chernoff (1997) predicted an asymmetric
drift of $13\,{\rm km\,s^{-1}}$ for the case of a uniform surface density
model, and $25\,{\rm km\,s^{-1}}$ for the case of an exponential surface
density model, both with the MSP velocity dispersion of $60\,{\rm
km\,s^{-1}}$. In Figure 5, we can clearly see most of the MSPs moving opposite
to the direction of Galactic rotation, which is the signature of the
asymmetric drift. Utilising the rotational velocities, $V_{\phi}$, we obtained
a median rotational velocity of $38(16)\,{\rm km\,s^{-1}}$ and the mean
rotational velocity of $45(13)\,{\rm km\,s^{-1}}$. By removing the outliers
(as defined in 5.3), we obtained the median value of $33(14)\,{\rm
km\,s^{-1}}$ and the mean value of $38(11)\,{\rm km\,s^{-1}}$. Our values are
higher than the $25(12)\,{\rm km\,s^{-1}}$ obtained by Toscano et al. (1999).
Our median value is closer to $25\,{\rm km\,s^{-1}}$, the predicted value
corresponding to the exponential surface density model of Cordes & Chernoff
(1997) although the MSPs in our sample have higher mean velocities than
theirs.

##  6 Comparison with previous work

###  6.1 Millisecond Pulsars vs. Normal Pulsars

Soon after their discovery, a population study by Gunn & Ostriker (1970),
based on data from just $41$ young pulsars, argued that pulsars are probably
born in a disk distribution with an initial scale height of $\sim 80$ pc
(similar to the normal OB stars), and move with average velocities of $\sim
100\,{\rm km\,s^{-1}}$. In the first large scale study of pulsar proper
motions, Lyne et al. (1982) studied a sample of $26$ and showed that the
pulsars with Galactic scale height greater than a few tens of parsecs tend to
be moving away from the Galactic plane, probably due to a velocity kick at
birth. They found pulsars had an rms velocity of about $210$ km s-1. Later,
Lyne & Lorimer (1994) studied the youngest pulsars with proper motions, and
found they had a mean space velocity at birth of $450(90)$ km s-1.

By modelling the kinematics of the spatial distribution of MSPs, Cordes &
Chernoff (1997) estimated that such pulsars on average receive a $z$-velocity
kick at birth of just $52^{+17}_{-11}\,{\rm km\,s^{-1}}$, much less that than
of the normal pulsars. They suggested that the rms speed of young pulsars is
$\sim 5$ time larger than that of MSPs, and a significant contribution to the
observed $z$-velocity (the velocity component along Galactocentric $z$) of
MSPs originates from the diffusive processes that affect all old stars in the
disk.

We plotted the histograms of the transverse velocities of the normal pulsars
from the pulsar catalogue and the MSPs in Figure 7, to highlight how different
the two distributions are. In this Figure, the top, middle, and bottom panels
show the distributions of MSPs from this work, MSPs from the combination of
this work and the pulsar catalogue, and normal pulsars from the pulsar
catalogue, respectively. For the $87$ normal pulsars in the pulsar catalogue
with significant parallaxes and proper motions ($>3\sigma$), we found a mean
velocity of $246(21)\,{\rm km\,s^{-1}}$. Comparing the mean velocities showed
that the normal pulsars seem to be faster than MSPs by a factor of $\sim 3.2$.
The characteristic age of MSPs is generally higher than that of normal
pulsars, and accordingly, the old pulsars seem to be slower than young
pulsars. If we restrict our attention to the pulsars with characteristic ages
less than $3$ Myr, we find a mean transverse velocity of $283(44)$ km s-1,
$\sim 3.7$ times faster than the MSPs.

The velocities of the MSPs and normal pulsars as a function of Galactocentric
$z$ are shown in Figure 8. This figure demonstrates that pulsars with
velocities $<100$ km s-1 are more concentrated in the Galactic plane and are
less scattered compared to the pulsars with velocities $>100$ km s-1. In
addition, on average, normal pulsars seem to have a wider $z$-distribution
around the Galactic plane (i.e. larger $z$ scale height) compared to the MSPs
consistent with their higher velocities. This figure provides additional
evidence for the results from the numerical studies of Bhattacharya & van den
Heuvel (1991, and references therein); Tauris & Bailes (1996); Cordes &
Chernoff (1997) that suggested MSPs have lower velocities compared to the
young, long-period pulsars. The higher velocities of normal pulsars reinforce
the idea that the high recoil velocity that pulsars receive during their birth
in supernova explosions (Burrows, 2013; Verbunt et al., 2017; Deller et al.,
2019), and are in agreement with some of the core-collapse supernovae
simulations performed by Wongwathanarat et al. (2012); Müller (2020). On the
other hand, some of the low velocities of normal pulsars might be due to the
weak supernova kicks for a sub-population of the pulsars (Willcox et al.,
2021). Some caution in interpreting Figure 8 is required, as many selection
effects are at play.

Figure 7: Histograms of transverse velocities for our MSP sample (top panel
and in blue), our MSPs and MSPs in the pulsar catalogue (middle panel and in
gray), and the normal pulsars in pulsar catalogue that have $>3\sigma$
significance in parallaxes (bottom panel and in red). This plot indicates how
different the distributions of MSP velocities and the normal pulsar velocities
are. Figure 8: Transverse velocities versus Galactocentric $z$ coordinate of
pulsars. This plots shows the difference between the velocities of MSPs and
normal pulsars. Velocities of MSPs from this work, MSPs from the pulsar
catalogue, and normal pulsars from pulsar catalogue (Manchester et al., 2005)
are shown in blue, red, and olive, respectively. Pulsars from the pulsar
catalogue are shown if they have at least $3\sigma$ significance in parallax.
The mean velocity of the normal pulsars is $246(21)$ km s-1, $\sim 3.2$ times
the mean MSP transverse velocity of $77(8)$ km s-1. Also, the velocity
dispersion of the normal pulsars is $194(15)$ km s-1, $\sim 3$ times the MSP
velocity dispersion of $65(6)$ km s-1. Pulsars with velocities $<100$ km s-1
are often closer to the Galactic plane.

###  6.2 Velocity dispersions for millisecond pulsars

The dispersions of the $3$ velocity components that Matthews et al. (2016)
obtained from the analysis of NANOGrav nine-year MSP timing data set, after
excluding outliers, are $\sigma_{\rm\rho}=46\,{\rm km\,s^{-1}}$,
$\sigma_{\rm\phi}=40\,{\rm km\,s^{-1}}$, and $\sigma_{\rm z}=24\,{\rm
km\,s^{-1}}$. We obtained $\sigma_{\rm\rho}=63(11)\,{\rm km\,s^{-1}}$,
$\sigma_{\rm\phi}=48(8)\,{\rm km\,s^{-1}}$, and $\sigma_{\rm z}=19(3)\,{\rm
km\,s^{-1}}$. Our $\sigma_{\rm\phi}$ and $\sigma_{\rm z}$ are thus comparable
with their values. Also, they calculated the dispersion of velocity
components, using the fit equations to the local stellar data provided by
Aumer & Binney (2009), to be $\sigma_{\rm\rho}=34\,{\rm km\,s^{-1}}$,
$\sigma_{\rm\phi}=22\,{\rm km\,s^{-1}}$, and $\sigma_{\rm z}=18\,{\rm
km\,s^{-1}}$ for the characteristic age of $\tau\sim 5$ Gyr, and
$\sigma_{\rm\rho}=42\,{\rm km\,s^{-1}}$, $\sigma_{\rm\phi}=28\,{\rm
km\,s^{-1}}$, and $\sigma_{\rm z}=24\,{\rm km\,s^{-1}}$ for $\tau\sim 10$ Gyr.
This is showing that our $\sigma_{\rm z}$ is consistent with the model fit
corresponding to the characteristic age of $\tau\sim 5$ Gyr, but our
$\sigma_{\rm\rho}$ and $\sigma_{\rm\phi}$ are closer to the model fit
corresponding to the characteristic age of $\tau\sim 10$ Gyr. Both comparisons
show that the MSPs are consistent with having been drawn from the old disk
stellar population, and that they are subject to kick velocities at birth, as
discussed in the next section.

The fact that the $z$-components are the lowest is perhaps easiest to
understand. MSPs with low $z$-velocities spend more time near that galactic
disk (and hence the Sun) and are preferentially detected by MSP surveys, plus
as mentioned before, the $z$-velocity is on average lower than the birth
$z$-velocity as it exhibits simple harmonic motion in the Galactic potential.

Figure 9: Transverse velocity versus orbital period for binary MSPs.
Velocities calculated using MSPs in this work are in blue and from the pulsar
catalogue are in red. The correlation coefficient between velocity and binary
period (excluding PSRs J1024$-$0719, J1300$+$1240, and J2234$+$0611 for
reasons discussed in the text) was found to be $-0.27$, indicating a weak
anti-correlation, similar to that predicted.

###  6.3 Velocity distributions for millisecond pulsars

Hobbs et al. (2005) studied the kinematics of 233 pulsars and found the mean
two-dimensional speeds of $211(18)\,{\rm km\,s^{-1}}$ for all pulsars in their
sample, $307(47)\,{\rm km\,s^{-1}}$ for pulsars with characteristic ages less
than $3$ Myr, $87(13)\,{\rm km\,s^{-1}}$ for recycled pulsars, and
$209(19)\,{\rm km\,s^{-1}}$ for the normal pulsars with characteristic ages
greater than $3$ Myr. Our mean velocity of $78(8)\,{\rm km\,s^{-1}}$ for MSPs
is consistent with their mean two-dimensional speed of $87(13)\,{\rm
km\,s^{-1}}$ for recycled pulsars. Also, Hobbs et al. (2005) obtained the mean
2D speeds of $77(16)\,{\rm km\,s^{-1}}$ for seven isolated MSPs and
$89(15)\,{\rm km\,s^{-1}}$ for 28 binary MSPs. These are comparable with the
mean velocities of $67(12)\,{\rm km\,s^{-1}}$ and $77(9)\,{\rm km\,s^{-1}}$
for the $16$ solitary and $49$ binary MSPs in our sample, respectively,
excluding PSR J1024$-$0719\. Johnston et al. (1998), by studying scintillation
parameters for a sample of 49 pulsars, suggested that binary MSPs have higher
velocities compared with the isolated MSPs. Toscano et al. (1999) calculated
velocities for a sample of $23$ MSPs and presented that, on average, the
binary MSPs are one-third faster than isolated MSPs. However, Hobbs et al.
(2005) reported that there is no significant difference between the mean
velocities of both. Our results are also showing that the mean velocities of
both are not significantly different.

Figure 10: Transverse velocity versus minimum companion mass assuming orbital
inclination of $90^{\circ}$ and the pulsar mass of $1.35$ M⊙ for binary MSPs.
Velocities calculated using MSPs in this work are in blue and from the pulsar
catalogue are in red. The correlation coefficient between velocity and minimum
companion mass (excluding PSRs J1024$-$0719, J1300$+$1240, and J2234$+$0611
for reasons discussed in the text) was found to be $-0.10$, indicating almost
no correlation.

Using simulations, Tauris & Bailes (1996) predicted a mild inverse correlation
between the recoil velocity (the post-explosion velocity that a pulsar
receives at birth) and the orbital period of binary MSPs, depending upon the
birth component masses of the binary, the separation between the two objects,
and the evolution of the system during the common-envelope and mass transfer
stages. Toscano et al. (1999) did not find any significant correlation with
orbital period using a sample of $23$ binary MSPs. In addition, Hobbs et al.
(2005) looked for this correlation using the binary MSPs in their pulsar
sample, but they did not find any either. We also investigated the
relationship between transverse velocity and orbital period of binary MSPs in
Figure 9, and measured the correlation coefficient to be $-0.24$ (excluding
the very long orbital period pulsar PSRs J1024$-$0719, the planet pulsar
J1300$+$1240, and the eccentric MSP J2234$+$0611 - which probably has a
complex evolutionary history) showing a weak anti-correlation. We need to keep
in mind that many selection effects are influencing our observed MSP
distribution. For instance, binary MSPs with short orbital periods and heavy
companions might be less likely to appear in pulsar surveys unless
acceleration searches have been undertaken. Nevertheless, Tauris & Bailes
(1996) predicted an approximate velocity range of $25$–$270$ km s-1 for binary
MSPs and that those above $300$ km s-1 should be very rare. This is borne out
by our observations of our sample’s transverse velocities. In the standard
model MSPs obtain their velocities as the vector sum of three distinct
mechanisms. First, MSP progenitors, the binaries containing OB stars, are
formed in large stellar nurseries where they obtain velocities of 10s of km
s-1 due to stellar interactions. Second, when the neutron star is produced
there is a Blaauw momentum kick to the binary as a result of the mass loss of
the supernova (Blaauw, 1961), and possibly an asymmetric kick imparted to the
neutron star. To get very high velocities requires very compact binaries,
large mass loss and asymmetric retrograde kicks that do not disrupt the orbit.
These must be rare in MSP formation. As we only measure the 2D velocities the
presence of some very low velocities is not unexpected and could be a
projection effect, but MSP velocities can help constrain models for neutron
star formation in population synthesis codes such as COMPAS (Riley et al.,
2022) and STARTRACK (Belczynski et al., 2002).

Any correlation between the minimum masses of companions of binary MSPs and
their transverse velocities of binary MSPs was explored. We used the pulsar
catalogue to find the minimum mass of the companions and plotted them as a
function of transverse velocity in Figure 10. The minimum masses were
calculated assuming the orbital inclination angle to be $90^{\circ}$ and the
mass of the pulsar to be $1.35$ M⊙. We measured the correlation coefficient
between the minimum mass and the transverse velocity (excluding PSRs
J1024$-$0719, J1300$+$1240, and J2234$+$0611) to be only $-0.10$, showing at
best a very weak anti-correlation.

##  7 Conclusions

We have undertaken a study on the parallaxes and proper motions of $77$ MSPs,
observed with the MeerKAT radio telescope, over a time span of $\sim 3$ years.
Out of the $77$ MSPs, $35$ and $69$ had significant parallaxes and proper
motions (above $3\sigma$), respectively. We calculated the transverse
velocities for MSPs that had both significant parallaxes and proper motions.
For the MSPs for which we did not have significant parallaxes, we used
parallaxes taken from the literature. Pulsars with significant parallaxes and
proper motions from the pulsar catalogue psrcat were also added to our MSP
sample to produce the largest MSP velocity sample to date. We found that the
transverse velocity of MSPs has a mean of $78(8)$ km s-1 and an rms scatter of
$101$ km s-1.

The Lutz-Kelker effect on the derived transverse velocities was investigated
for a sample of $35$ pulsars with parallaxes. The median transverse velocities
increased by $\sim 2\%$ and $\sim 13\%$ for parallaxes with $>10\sigma$ and
$<10\sigma$ significance, respectively. The median of the whole population
increased by $\sim 7\%$.

Out of our $69$ MSPs with significant ($>3\sigma$) proper motions, the long
orbital period pulsar J1024$-$0719 had the highest transverse velocity of
$278^{+44}_{-32}$ km s-1, and the $12.4$ d orbital period binary pulsar PSR
J1652$-$4838 had the lowest transverse velocity of just $5^{+7}_{-2}$ km s-1.

Although the mean velocity of $72(8)$ km s-1 for $49$ binary MSPs is slightly
faster than that of $67(12)$ km s-1 for $16$ isolated ones, there was no
evidence that their distributions are statistically significantly different.
In comparison to the normal pulsars in the pulsar catalogue with the mean
transverse velocity of $246(21)$ km s-1, MSPs had a much lower mean transverse
velocity of $78(8)$ km s-1.

The velocity dispersions of the pulsars in the (cylindrical) Galactocentric
velocity components (radial, rotational, and perpendicular) were measured to
be $\sigma_{\rm\rho}=63(11)\,{\rm km\,s^{-1}}$, $\sigma_{\rm\phi}=48(8)\,{\rm
km\,s^{-1}}$, and $\sigma_{\rm z}=19(3)\,{\rm km\,s^{-1}}$ after removal of a
small number of outliers. The lower $z$-velocity component is likely a
consequence of the fact that high velocity MSPs will spend less time near the
Sun suppressing their representation in pulsar surveys.

The expected asymmetric drift was clearly seen in the rotational component of
the velocities, and the mean value was found to be $38(11)\,\rm km~{}s^{-1}$
after removing outliers. This is consistent with the number predicted by
Cordes & Chernoff (1997), who model the formation of MSPs in our galaxy. They
predicted an asymmetric drift of approximately $25$ km s-1 for their more
realistic (exponential surface density) Milky Way model.

The substantially increased sample of MSPs with good parallaxes and proper
motions presented in this study bolsters the case that MSPs arise out of the
old disk of the Milky Way and, are subject to kick velocities at birth
significantly smaller than those seen for young pulsars (Hobbs et al., 2005).
We explored the distribution of MSP velocities versus orbital period and found
a weak anti-correlation, in agreement with the predictions of Tauris & Bailes
(1996), based on simulations of the recoil velocities of MSPs in various
binary stellar evolutionary scenarios.

## Acknowledgements

We thank Dr. Simon Stevenson and Dr. Hao Ding for their suggestions and
comments that improved the manuscript. The MeerKAT telescope is operated by
the South African Radio Astronomy Observatory, which is a facility of the
National Research Foundation, an agency of the Department of Science and
Innovation. MS, MB, CF, MTM, DJR, and RMS acknowledge support through the
Australian Research Council Centre of Excellence for Gravitational Wave
Discovery (OzGrav), through project number CE17010004. RMS acknowledges
support through Australian Research Council Future Fellowship FT190100155. MK
acknowledges significant support from the Max-Planck Society (MPG) and the
MPIfR contribution to the PTUSE hardware. This work used the OzSTAR national
facility at Swinburne University of Technology. OzSTAR is funded by Swinburne
University of Technology and the National Collaborative Research
Infrastructure Strategy (NCRIS) that also supports the pulsars.org.au data
portal that was used extensively for this work. This research has made use of
NASA’s Astrophysics Data System and software such as: psrchive (van Straten et
al., 2012), tempo2 (Hobbs et al., 2006; Edwards et al., 2006), temponest
(Lentati et al., 2014), psrcat (Manchester et al., 2005), pulseportraiture
(Pennucci et al., 2014; Pennucci, 2019), pygedm (Price et al., 2021), numpy
(van der Walt et al., 2011), scipy (Virtanen et al., 2019), matplotlib
(Hunter, 2007), ipython (Pérez & Granger, 2007), astropy (Astropy
Collaboration et al., 2013, 2018).

## Data Availability

Data is available from the Swinburne pulsar portal: https://pulsars.org.au

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##  Appendix A On the use of Galactic electron density models for MSP
distances and velocities

The integrated electron column density ($n_{\rm e}$) along the line of sight
from the Earth to a pulsar at distance $d$ relates to the DM of the pulsar as

${\mathrm{DM}}=\int_{0}^{d}n_{\mathrm{e}}(l)dl.$ (8)

The distance to a pulsar can be derived from an observational DM by having a
model which describes the distribution of electrons throughout the Galaxy. An
inaccurate electron-density model might result in overestimating or
underestimating distances. By measuring new pulsar distances via pulsar
parallaxes, as in this work, we are able to provide an improved basis for
refining the ISM models for different lines of sight. There are two widely
used DM models, namely NE2001 (Cordes & Lazio, 2002, 2003) and YMW16 (Yao et
al., 2017). The comparison of the two has been carried out by many authors
(e.g., Yao et al., 2017; Deller et al., 2019; Ocker et al., 2020; Price et
al., 2021). They found that the different models have performance that is
dependent on where one is looking on the sky, and thus one model works better
than others depending on which part of the Galactic ISM is being probed by any
given pulsar.

###  A.1 Distances

Using the two Galactic electron-density models: NE2001 and YMW16, implemented
in PyGEDM (Price et al., 2021), we have calculated DM-based distances of the
$35$ MSPs with $>3\sigma$ significance in their parallaxes. In addition, for
MSPs with $<3\sigma$ significance in their parallaxes, we added those MSPs
whose DM distances were less than the lower distance limits obtained from the
probability distributions (the third column in Table 6). We also added other
MSPs from the pulsar catalogue (Manchester et al., 2005) with $>3\sigma$
significance in parallax into our MSP sample and calculated their DM-based
distances. We excluded all MSPs in globular clusters as they have different
origins and their velocities are dominated by their host clusters.

The top and bottom panels of Figure 11 show the comparison of parallax
distances with DM distances obtained from the NE2001 and YMW16 models,
respectively. In this Figure, points in red show a DM distance overestimation
(i.e. MSPs are closer than what the models predict) and points in blue show a
DM distance underestimation (i.e. MSPs are further than what the models
predict). The ratios of $d_{\rm DM}/d_{\rm\varpi}$ and $d_{\rm\varpi}/d_{\rm
DM}$ are represented by the area of the circles in red and blue for the cases
of overestimation and underestimation, respectively. The black circles in the
legend of Figure 11 provide the standard circle sizes for the ratios of $10$,
$5$, $2$, and $1$. We found the maximum underestimation based on the YMW16
model for PSR J1737$-$0811 with the distance ratio of $d_{\varpi}/d_{\rm
DM}\approx 5.9$ (considering the lower limit of its parallax distance), and
the maximum overestimation was for PSR J1652$-$4838 with the distance ratio of
$d_{\rm DM}/d_{\varpi}\approx 10.5$.

The $16\%$, $50\%$, and $84\%$ of the underestimation and overestimation of
the YMW16 distances ratios were obtained to be $1.4^{+1.1}_{-0.3}$ and
$1.2^{+1.1}_{-0.1}$, respectively. In addition, the underestimation and
overestimation of the NE2001 distances were obtained to be $1.6^{+1.1}_{-0.5}$
and $1.3^{+1.8}_{-0.2}$, respectively. Also, the distance ratios of
$d_{\varpi}/d_{\rm DM}$ were obtained for YMW16 model to be
$1.1^{+0.7}_{-0.3}$ and for NE2001 to be $1.2^{+1.3}_{-0.4}$. The comparison
of distance ratios highlighted that there are obviously patches in Figure 11
that the DM-based distances are systematically overestimated or underestimated
by the two models.

Figure 11: The ratios of the parallax derived distances and DM derived
distances. Top panel: distance ratios for the NE2001 model. Bottom panel:
Distance ratios for the YMW16 model. The blue points indicate DM model
distance underestimation ($d_{\varpi}/d_{\rm DM}$), and the red points
indicate distance overestimation ($d_{\rm DM}/d_{\varpi}$). The sizes of the
black circles in the legend indicate the distance ratios. The black crosses in
bottom panel show the parallaxes used for establishing the YMW16 model that we
might expect to fit well.

###  A.2 Velocities

The top panel of Figure 12 presents the histogram of transverse velocities for
the total sample that possesses parallax measurements. The mean transverse
velocity is $78(8)\,{\rm km\,s^{-1}}$. The fastest moving MSP for these
parallax-based velocities is the planet pulsar PSR J1300$+$1240 (Wolszczan,
1990), which has a velocity of $330^{+20}_{-18}\,{\rm km\,s^{-1}}$, and the
slowest-moving, PSR J1652$-$4838, has a velocity of just $5^{+7}_{-2}\,{\rm
km\,s^{-1}}$. Using the code
444https://github.com/jantoniadis/GAIA_pulsars_eDR3/blob/main/GAIA_pulsars_eDR3.ipynb
provided by Antoniadis (2021), we explored whether the LK corrections might
affect our velocities by breaking our sample into two groups, those with
$>10\sigma$ parallaxes and those with lower significance. The median
velocities of the first group before and after the LK correction were $67$ and
$68$ km s-1 respectively and the second group $43$ and $49$ km s-1. Clearly,
the LK correction raises the velocities of the second group and brings their
median closer to that of the (more accurate) group, as might be predicted by
the LK selection effect. Overall, the median velocities of the entire
population before and after the LK correction were $46$ and $50$ km s-1, an
increase of $\sim 7$ %.

To illustrate the effect of using DM-based distances rather than our preferred
parallax-based distances, we have derived transverse velocities using DM-based
distances obtained from NE2001 and YMW16 models. The middle and the bottom
panels of Figure 12 present the histograms of transverse velocities for the
MSPs that their velocities are calculated from YMW16 and NE2001 models,
respectively.

Using the YMW16 model, the mean transverse velocity is $71(8)\,{\rm
km\,s^{-1}}$. The fastest moving MSP, PSR J1300$+$1240, would have a velocity
of $407(100)\,{\rm km\,s^{-1}}$ in the DM model, and the slowest-moving, PSR
J1446$-$4701, has a velocity of $6^{+6}_{-4}\,{\rm km\,s^{-1}}$, assuming
$25\%$ distance errors. Using the NE2001 model, the mean transverse velocity
is $65(6)\,{\rm km\,s^{-1}}$. The fastest moving MSP in this model, PSR
J0610$-$2100, has a velocity of $292(80)\,{\rm km\,s^{-1}}$, and the slowest-
moving, PSR J1446$-$4701, has a velocity of $6^{+6}_{-4}\,{\rm km\,s^{-1}}$,
again assuming $25\%$ distance errors.

It is curious to note that distance errors in YMW16 are responsible for a
marked increase in the number of very low velocity MSPs (see Figure 12, lower
two panels). In reality there are much fewer low velocity MSPs in the sample
based on parallax distances only (Figure 12, upper panel), and this is merely
an artifact of moving the MSPs too close to the Sun, reducing their apparent
speeds.

We have performed a two-sample Kolmogorov-Smirnov test for comparing the
continuous distributions of the parallax-based velocities and either of the
two DM-based velocities to see if their macroscopic properties would differ.
For the YMW16 model, the maximum absolute differences between the cumulative
distribution functions were $0.11$ and the p-value was $0.86$. For the NE2001
model, the maximum absolute differences between the cumulative distribution
functions were $0.17$ and the p-values was $0.32$. There is thus no
statistical evidence that the parallax-based distances and DM-based distances
differ that markedly in $v_{\perp}$.

Figure 12: Histograms of parallax-based velocities compared to the DM-based
velocities. The transverse velocities are calculated using distances derived
from the parallax measurements in the top panel, YMW16 DM model in the middle
panel, and NE2001 DM model in the bottom panel. As discussed in Section A.2,
there is an excess of pulsars with very low velocities (lower two panels) for
which distances have been derived from an ISM model, compared to the sample
compiled from pulsars with parallaxes only (upper panel). This is an artifact
of these models placing pulsars too close to the Sun. The velocity standard
deviation of $\sigma_{\perp}$ for the YMW16 model compared to the one for
NE2001 model shows the number of low velocities in NE2001 model are higher.

For the $35$ pulsars with significant parallaxes ($>3\sigma$), $\sim 2/3$ of
their YMW16 DM-based distances are more than $1\sigma$ different from the
parallax distances. This affects any velocities calculated from DM-based
distances and makes the distribution of the velocities broader. It is beyond
the scope of this current work to examine each DM model and search for reasons
why they predict the wrong distances but using either the NE2001 or YMW16
models for calculating distances for finding velocities of individual objects
is subject to large uncertainties and for the rest of the paper we only
consider the transverse velocities of pulsars that have reliable parallax
measurements. It is very difficult to calibrate distance models at high
galactic latitudes in the absence of pulsars with parallaxes.