# Wideband timing of the Parkes Pulsar Timing Array UWL data

Małgorzata Curyło  Astronomical Observatory, University of Warsaw, Aleje
Ujazdowskie 4, 00-478 Warsaw, Poland  Timothy T. Pennucci  Institute of
Physics, Eötvös Loránd University, Pázmány P.s. 1/A, 1117 Budapest, Hungary
Matthew Bailes  Centre for Astrophysics and Supercomputing, Swinburne
University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia  N.
D. Ramesh Bhat  International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA 6102, Australia  Andrew D. Cameron  Centre for
Astrophysics and Supercomputing, Swinburne University of Technology, P.O. Box
218, Hawthorn, Victoria 3122, Australia  Australian Research Council Centre of
Excellence for Gravitational Wave Discovery (OzGrav), Australia  Shi Dai
School of Science, Western Sydney University, Locked Bag 1797, Penrith, NSW
2751, Australia  George Hobbs  CSIRO Astronomy and Space Science, Australia
Telescope National Facility, PO Box 76, Epping NSW 1710, Australia  Agastya
Kapur  CSIRO Astronomy and Space Science, Australia Telescope National
Facility, PO Box 76, Epping NSW 1710, Australia  Department of Physics and
Astronomy and MQ Research Centre in Astronomy, Astrophysics and
Astrophotonics, Macquarie University, NSW 2109, Australia  Richard N.
Manchester  CSIRO Astronomy and Space Science, Australia Telescope National
Facility, PO Box 76, Epping NSW 1710, Australia  Rami Mandow  CSIRO Astronomy
and Space Science, Australia Telescope National Facility, PO Box 76, Epping
NSW 1710, Australia  Department of Physics and Astronomy and MQ Research
Centre in Astronomy, Astrophysics and Astrophotonics, Macquarie University,
NSW 2109, Australia  Matthew T. Miles  Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn,
Victoria 3122, Australia  Australian Research Council Centre of Excellence for
Gravitational Wave Discovery (OzGrav), Australia  Christopher J. Russell
CSIRO Astronomy and Space Science, Australia Telescope National Facility, PO
Box 76, Epping NSW 1710, Australia  Daniel J. Reardon  Centre for Astrophysics
and Supercomputing, Swinburne University of Technology, P.O. Box 218,
Hawthorn, Victoria 3122, Australia  Australian Research Council Centre of
Excellence for Gravitational Wave Discovery (OzGrav), Australia  Ryan M.
Shannon  Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia  Australian
Research Council Centre of Excellence for Gravitational Wave Discovery
(OzGrav), Australia  Renée Spiewak  Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn,
Victoria 3122, Australia  Jodrell Bank Centre for Astrophysics, School of
Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, UK
Andrew Zic  CSIRO Astronomy and Space Science, Australia Telescope National
Facility, PO Box 76, Epping NSW 1710, Australia  Department of Physics and
Astronomy and MQ Research Centre in Astronomy, Astrophysics and
Astrophotonics, Macquarie University, NSW 2109, Australia  Xing-Jiang Zhu
Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai
519087, China

###### Abstract

In 2018 an ultra-wide-bandwidth low-frequency (UWL) receiver was installed on
the 64-m Parkes Radio Telescope enabling observations with an instantaneous
frequency coverage from 704 to 4032 MHz. Here, we present the analysis of a
three-year data set of 35 millisecond pulsars observed with the UWL by the
Parkes Pulsar Timing Array (PPTA), using wideband timing methods. The two key
differences compared to typical narrow-band methods are, firstly, generation
of two-dimensional templates accounting for pulse shape evolution with
frequency and, secondly, simultaneous measurements of the pulse time-of-
arrival (ToA) and dispersion measure (DM). This is the first time that
wideband timing has been applied to a uniform data set collected with a single
large-fractional bandwidth receiver, for which such techniques were originally
developed. As a result of our study, we present a set of profile evolution
models and new timing solutions including initial noise analysis. Precision of
our ToA and DM measurements is in the range of 0.005 $-$ 2.08 $\mu$s and
(0.043$-$14.24)$\times 10^{-4}$ cm-3 pc, respectively, with 94% of the pulsars
achieving a median ToA uncertainty of less than 1 $\mu$s.

pulsar timing, data analysis

††software: PulsePortraiture (Pennucci et al., 2014), Enterprise (Ellis et
al., 2019), PSRCHIVE (Hotan et al., 2004), Tempo (Nice et al., 2015), Tempo2
(Hobbs et al., 2006).

##  1 Introduction

Pulsar timing array (PTA) experiments provide extraordinary means to study a
wide range of physical phenomena across nearly all branches of physics and
astronomy. These include characteristics of neutron stars themselves but can
also relate to solar system dynamics, general relativity or nHz gravitational
waves (GWs) generated by various processes such as supermassive black hole
inspirals or cosmic strings (e.g. Burke-Spolaor et al., 2019; Vallisneri et
al., 2020). However, as all of the above processes may have a very subtle
effect on timing measurements (of the order of several ns), an increase of the
precision and accuracy is a vital element of current PTA efforts. The three
major pillars of PTA, working under a joint venture as the International
Pulsar Timing Array: IPTA; Manchester & IPTA, 2013, are the European Pulsar
Timing Array (EPTA; Desvignes et al., 2016), the North American Nanoherthz
Observatory for Gravitational waves (NANOGrav; McLaughlin, 2013) and the
Parkes Pulsar Timing Array (PPTA; Hobbs, 2013). They have also been recently
joined by the Indian Pulsar Timing Array (InPTA; Paul et al., 2019; Tarafdar
et al., 2022) and are supported by the Chinese Pulsar Timing Array (CPTA; Lee,
2016) and MeerKAT Pulsar Timing Array (MPTA; Miles et al. 2022, submitted).

There are a number of possible improvements that can be applied to
observational strategies, instrumentation and analysis techniques, such as
increasing the number of pulsars in the array and the cadence of observations,
or enlarging telescope apertures. In particular, several recent projects and
facilities such as the Five Hundred Meter Aperture Spherical Telescope (FAST;
Hobbs et al., 2019) and the Canadian Hydrogen Intensity Mapping Experiment
(CHIME/Pulsar Project; Amiri et al., 2021; CHIME Collaboration et al., 2022)
will soon join PTA efforts and significantly increase available observing time
and collecting area.

Another approach is to utilize wideband receivers. In the first instance, the
uncertainty of a timing measurements $\sigma$ depends on the observing system
and is described by the radiometer equation (Lorimer & Kramer, 2004):

$\sigma\propto\frac{T_{\rm sys}}{A_{\rm eff}}(\tau\Delta f)^{1/2}$ (1)

where $T_{\rm sys}$ is the system’s temperature, $A_{\rm eff}$ effective
aperture, $\tau$ integration time and $\Delta f$ is the bandwidth. Apart from
improvements based on Eq. 1, wideband receivers will also significantly
broaden our capabilities of studying processes related to the interstellar
medium (ISM), such as scintillation and DM variability. Currently, there are
two telescopes with large instantaneous frequency coverage used for PTA
observations: Effelsberg 100-m Telescope with 600$-$3000 MHz ultra-broadband
(UBB) receiver111https://www3.mpifr-bonn.mpg.de/staff/pfreire/BEACON.html and
Parkes Radio Telescope (Murriyang) with ultra-wide-bandwidth low-frequency
(UWL) receiver covering the largest frequency range of 704$-$4032 MHz (Hobbs
et al., 2020). Installation of a new wideband receiver is also planned for
MeerKAT (Kramer et al., 2016) and ultra-wideband (UWB) feed at the Green Bank
Telescope used by NANOGrav has recently entered a commissioning phase (Bulatek
& White, 2020).

In order to fully benefit from wideband observations it is important to adjust
current analysis techniques. Growing apertures, observation time and frequency
coverage will significantly increase computational requirements making
processing of long PTA data sets arduous. Moreover, as many millisecond
pulsars exhibit an intrinsic profile evolution with frequency, pulse shape can
change drastically between the extreme ends of the band. These two effects,
known collectively as the $large~{}bandwidth~{}problem$ are the main factors
driving the development of new wideband timing methods.

In general, the procedure of timing analysis consists of measuring the
observed pulse times-of-arrival (ToAs) via template-matching technique which
are then compared with the timing model. The model is supposed to accurately
predict the periodicity of the pulsar’s radio emission, fitting for parameters
such as spin period and its derivative, position and proper motion, or binary
parameters if applicable. Differences between the observations and the model
(the residuals), may manifest as either white or red noise (systematic and
time correlated, respectively). In the former case, the main contributors are
radiometer noise and jitter (Helfand et al., 1975; Shannon et al., 2014),
while for the latter these could be pulsar intrinsic spin noise (Shannon &
Cordes, 2010), dispersion measure (DM) variations or gravitational waves. In
other words, the sensitivity of the PTA depends on the accuracy of both the
ToA measurement (and so on the template used) and timing solutions, including
a proper characterization of noise sources.

Typically, template-matching is performed using a frequency-averaged profile
template which will deviate from the true pulse shape at different
frequencies. In order to mitigate this problem, it is common to divide the
band and measure the so called sub-banded ToAs with one average template per
sub-band and/or use additional parameters in the timing model (FD parameters).
The wideband timing techniques (presented by Pennucci et al., 2014 and Liu et
al., 2014) offer an elegant and more direct solution by generation of a two-
dimensional profile template maintaining frequency resolution (model of the
profile evolution) and a simultaneous measurement of one ToA and DM at a
reference frequency for the whole band. In particular, wide-band timing
methods from Pennucci et al. (2014) were applied to the 12.5 year data set
from NANOGrav (Alam et al., 2021a, hereafter NG12.5) and directly compared
with a corresponding narrow-band release (Alam et al., 2021b). It was shown
that the number of ToAs was reduced by a factor of 33, while maintaining a
similar level of precision in terms of timing model and noise parameters (at
least $2\sigma$ agreement). NG12.5 have also reported improvement of timing
results by 10-15% for pulsars which were impaired by stronger environmental
effects such as high DM and scintillation. Similarly, Nobleson et al. (2022)
presented a wideband analysis of five pulsars observed with the upgraded Giant
Metrewave Radio Telescope (uGMRT) at low frequencies between 300 and 500 MHz.
Apart from showing the aforementioned consistency between results obtained
with wide- and narrow-band methods, Nobleson et al. (2022) also emphasize an
increased precision of low-frequency DM measurements and prove wideband timing
methods to be beneficial even for small fractional bandwidths. Other examples
of implementations of methods presented by Pennucci et al. (2014) can be found
in Fonseca et al. (2021) and Sharma et al. (2022), where the analyses focus on
detailed binary parameter estimation and study of new PTA-candidate pulsars,
respectively.

Here, we present wideband timing of 35 millisecond pulsars observed with the
UWL by the PPTA using PulsePortraiture (Pennucci et al., 2016, 2014). This is
the first time such analysis is performed on observations gathered by a single
ultra-wide-bandwidth receiver with instantaneous fractional bandwidth of
approximately 6:1. It is also the first study of the new PPTA UWL data set in
general (note, however, that UWL observations were used to estimate the DM for
the timing analysis of the previous data release; Reardon et al., 2021).
Continuous frequency coverage of an ultra-wide band and utilization of
PulsePortraiture allowed us to describe the evolution of pulse profiles and
subsequently measure ToAs and DMs with raw uncertainties at least two times
smaller then those obtained for the previous data set (Kerr et al., 2020,
hereafter PPTA DR2).

The paper is organized as follows. In Section 2 we describe the UWL receiver,
observation strategies and the new data set. In Section 3 we present data
preparation procedures and the principles of wideband timing and noise
analysis. Section 4 contains our results and their discussion, including
profile evolution and timing models, noise analysis and notes on a few
individual pulsars. Summary and conclusions are presented in Section 5.

##  2 Observations

Observations analysed in this paper were collected between November 2018 and
March 2022 with the 64-m Parkes Radio Telescope (Murriyang) located in New
South Wales, Australia. The starting date marks the transition from the
previous observing systems to the UWL which has been continuously carrying out
all observations ever since. Full description and technical details of the new
receiver and associated systems can be found in Hobbs et al. (2020) and below
we introduce it only briefly.

Both the feed and low-noise amplifiers are cryogenically cooled and maintain
low, 22 K temperature for the majority of the band. Pre-processed data
consists of 26 critically sampled sub-bands, each 128 MHz wide, continuously
covering 704$-$4032 MHz frequency range. For timing purposes, we use data
coherently de-dispersed and folded into 1024 phase bins by the DSPSR suite
(van Straten & Bailes, 2011) on the Medusa GPU cluster. Noteworthy, Medusa is
the only processing system presently used as opposed to seven employed in the
previous data release PPTA DR2.

For flux calibration purposes we observe two bright sources (PKS B1934$-$638
and PKS B0407$-$658) approximately once per session. Additionally, each pulsar
observation is preceded (and sometimes also followed) by a two-minute
injection of a noise diode signal.

The list of all 35 pulsars observed with the UWL for PPTA project (P456) is
presented in Table 1. Observations are carried out with a standard cadence of
approximately 1$-$3 weeks. 25 of the sources were included in the previous
data release, while the remaining 10 were added to the array between 2018 and
2020 and are currently being reviewed as potential candidate pulsars for PTA.

##  3 Methods

###  3.1 Wideband timing

We refer the reader to Pennucci et al. (2014); Pennucci (2019) for a detailed
explanation of the wideband timing procedures, however, a brief summary is
presented below.

We first make a smooth, noise-free, average (in time and frequency) profile
from one, highest signal to noise ratio (S/N) observation, which is then used
iteratively to align several tens of epochs comprising our initial frequency
resolved template. This step is similar for both narrow- and wide-band
techniques, however, the standard phase shift between the profiles and the
noise-free template is additionally a function of frequency, i.e. is described
by a dispersion law:

$\phi_{\rm n}(\nu_{\rm n})=\phi_{0}+\frac{\rm K\cdot DM}{\rm P_{\rm
S}}(\nu_{\rm n}^{-2}-\nu_{\phi_{0}}^{-2})$ (2)

where $\phi_{0}$ is the phase offset at reference frequency222The reference
frequency is chosen such that there is zero covariance between DM and
$\phi_{0}$ (see Appendix in Pennucci et al. (2014) $\nu_{\phi_{0}}$),
$\rm{K=4149.37759336~{}MHz^{2}~{}cm^{3}~{}pc^{-1}~{}s}$ is dispersion constant
and $P_{\rm S}$ is the spin period of the pulsar. A collection of these
aligned profiles (the portrait) is then decomposed into eigenvectors via
principal component analysis (PCA) and the ones with the highest S/N (along
with the mean profile) are used to model the frequency evolution of the
profile. At the final step, a spline function is fitted to the projection of
each mean substracted portrait profile onto the significant eigenvectors. In
short, we can reconstruct the profile shape at any desired frequency by
summing up the product of $n_{\rm eig}$ spline functions $S_{\rm i}(\nu)$ and
significant eigenvectors $\hat{e_{\rm i}}$ and adding it to the mean profile
$\tilde{p}$:

$\mathrm{T(\nu)=\sum_{i=1}^{n_{eig}}S_{i}(\nu)\hat{e_{i}}+\tilde{p}}$ (3)

These procedures then allow for a simultaneous measurement of one ToA and DM
per whole band (at a reference frequency $\nu_{\phi_{0}}$) for each
observation by minimizing the statistic:

$\chi^{2}=\sum_{\rm n,k}\frac{|d_{\rm nk}-a_{\rm n}t_{\rm nk}e^{\rm-2\pi
ik\phi_{\rm n}}|^{2}}{\sigma_{\rm n}^{2}}$ (4)

where indices $k$ and $n$ run over Fourier frequencies and channels,
respectively, $d_{\rm nk}$ is a one-dimensional (along phase axis) discrete
Fourier transform (DFT) of the profile and $t_{\rm nk}$ is the DFT of our
template, $a_{\rm n}$ is the amplitude scaling factor and $\sigma_{\rm n}^{2}$
is the Fourier-domain noise level.

###  3.2 Data preparation

In order to prepare data for our wideband analysis, we first cleaned it from
radio-frequency-interference (RFI), calibrated and frequency averaged down to
416 channels (each 8 MHz wide) by an automatic pipeline (also used in Kerr et
al., 2020). Afterwards, we prepared our portraits by aligning and averaging
nearly all available UWL observations as explained in Sec. 3.1. The two
exceptions here were the brightest or faintest sources. In the former case,
10$-$30 observations were sufficient enough to produce high S/N portraits and
at the same time allowed for more accurate RFI excision. In case of the
faintest sources in the data set, some of the observations were heavily
corrupted by RFI or instrumental errors, and due to a much smaller number of
observing epochs relative to brighter pulsars they significantly affected the
portrait and so were removed. We also note, that each portrait was manually
checked for RFI and channels with $\rm S/N\sim 0$ before modeling procedures
because: i) summing up the profiles may bring up previously missed
contaminated channels, and ii) any spurious signal or impaired channels can
significantly affect the model of profile evolution. Finally, once the
templates were obtained, we derived spline models and subsequently measured
one ToA and one DM per band for each observation as explained in Sec 3.1. All
of the above procedures were performed using an open source
code333https://github.com/pennucci/PulsePortraiture/ \- PulsePortraiture
(Pennucci et al., 2014).

The last step before proceeding to timing analysis was to filter out all bad
epochs in our data sets. For that purpose, we have defined four conditions
which had to be met in order to include particular observation in the further
analysis. These were:

  1. 1.

Observation time: $t_{obs}>300$ s.
Nominal length of each observation is 1.1 hour. There is however, a subset of
shorter observations due to technical limitations, e.g. changing weather
conditions, finite length of observing session, RFI, etc. We do include those
partial observations in our analysis, only if their length exceeds 300 s. Such
a low threshold allows to include more observations of the brightest pulsars,
whereas short ones usually didn’t fulfill also the remaining conditions.

  2. 2.

ToA S/N: $S>25$.
The quality of low S/N observations results in poor estimation of measured
ToAs and DMs (high uncertainties). Note however, that the threshold applies to
S/N of the wideband ToAs not observations, and is defined as:
$S\equiv\sqrt{\sum_{\rm n}S_{\rm n}^{2}}$, where $S_{n}$ $\equiv a_{\rm
n}\sqrt{\sum_{\rm k}|t_{\rm nk}|^{2}}/\sigma_{\rm n}$ (for a more detailed
description see Appendix A in Alam et al., 2021a).

  3. 3.

Goodness-of-fit: $\chi^{2}_{\rm reduced}<1.25$.
The $\chi^{2}$ statistics is calculated for each observation, taking into
account the model of profile evolution. Large values of this parameter can
imply non-curated RFI or low quality profiles.

  4. 4.

Highest/lowest frequency ratio: $f_{\rm ratio}>1.1$.
Observations filtered out based on this condition are either heavily
contaminated by RFI or affected by serious instrumental issues, where the
signal throughout majority of the band is lost. All observations which we
included in our analysis had $f_{\rm ratio}>2.9$ which corresponds to the
effective bandwidth of approximately 2600 MHz (most of observations with
$f_{\rm ratio}\sim 2.9$ were cut below 1400 MHz due to strong RFI). For full
bandwidth $f_{\rm ratio}=5.7$.

This procedure allowed us to remove most of the bad epochs automatically and
only a small fraction of individual observations were later flagged manually
due to large residuals or DM/ToA uncertainties, which, as expected, occured
mainly for low quality profiles. Each epoch which was commented out was
additionally checked by eye to determine whether it could be curated and added
back to the data set. This procedure was possible only because of a still
relatively small number of observations, however we acknowledge the need for
more accurate RFI zapping algorithms for future analyses. In general, we used
between 80% to 100% of available UWL observations for each pulsar.

Finally, we note that all of the pulsars in the data set were analysed using
total intensity profiles (Stokes I), apart from J0437$-$4715 for which we used
the polarimetric invariant profile, as was also done in previous analyses (van
Straten et al., 2001; Kerr et al., 2020). The invariant interval can be used
to avoid additional red noise from polarization calibration errors and to
reduce dependence of the observation on the parallactic angle (Hotan et al.,
2006). It is given by (Hotan et al., 2006):

$\rm S_{\rm inv}=I^{2}-Q^{2}-U^{2}-V^{2}$ (5)

where $I,Q,U,V$ are Stokes parameters. It can be used in the case of the least
polarized sources or when large part of the emission is unpolarized, which is
why we applied it only to J0437-4715. Our final data set consists of frequency
dependent profile templates (portraits), spline models of profile evolution
and a list of measured ToAs and DMs with their uncertainties for each pulsar.

###  3.3 Timing solutions and noise analysis

As explained in Section 3.1, each wideband measurement consists of one ToA and
a corresponding DM at the time of the observation along with their
uncertainties. It is possible to obtain a wideband ToA without fitting for the
DM, however that would still necessitate providing an external and precise
measurement of the DM at the time of the observation in order to properly
align the data with the template. In either case, further analysis of the
timing and noise models also requires using both measurements per observation,
as opposed to sub-banded ToAs, single wideband ToA does not contain the full
information of the dispersive delay within the observing band. In
consideration of that, in our work we used two packages, i.e. Tempo (Nice et
al., 2015) for pulsar timing and ENTERPRISE (Ellis et al., 2019) for noise
modeling, where the new wideband likelihood has been already been implemented
(alternatively, wideband analysis can also be performed in PINT; Luo et al.,
2021). The mathematical description of these implementations can be found in
Appendix B in Alam et al., 2021a. Below, we present all details of obtaining
our wideband timing solutions and corresponding noise models along with a
general overview of the augmented likelihood.

We started with initial timing solutions obtained for the PPTA DR2 (Goncharov
et al., 2021) applying the following changes. First, we removed all noise
parameters, as they have to be derived again using the new wideband
likelihood. We have also updated the clock standard and solar system ephemeris
to TT(BIPM2019) and DE438, respectively. Another change was applied to the
choice of binary models. The majority of pulsars with binary companions from
PPTA DR2 were fitted with the T2 model which is available only in the Tempo2
package (Hobbs et al., 2006). Here instead, for most pulsars we used the ELL1
model (adequate for low eccentricity orbits; Lange et al., 2001), for one
pulsar (J1643-1224) we used DD model (to include measurement of eccentricity;
Damour & Taylor, 1992), for two pulsars (J0437$-$4715, J1713$+$0747) we used
the DDK model (allowing for measurements of annual-orbital parallax; Kopeikin,
1995, 1996), and four pulsars (J1017$-$7156, J1022$+$1001, J1545$-$4550,
J1600$-$3053) were fitted with ELL1H model (which includes modelling of
Shapiro delay; Freire & Wex, 2010).

The next element of our timing analysis is modeling of the DM variability. We
used the DMX model which assumes that the DM is constant within chosen time
intervals (e.g. a fraction of a day or several days), and models the
subsequent changes in a piecewise-constant manner. The choice of the DMX epoch
length depends on various factors, e.g. observing strategy, desired precision
or expected ISM/solar wind variability. In this work we have applied DMX
epochs between 1 to 60 days, depending on the number of measurements in order
to avoid overfitting. Each DM from our wideband measurements was then used as
a prior on the DMX value in the corresponding epoch. In the case there were
more ToAs for a given epoch, the prior was calculated from a weighted average.

Finally, noise present in the timing residuals was modelled using Bayesian
inference implemented within ENTERPRISE.

To account for the white noise, ToA uncertainties for each measurement
$\sigma_{\rm j}^{\rm ToA}$ were modified by 2 Gaussian-noise components:

$\rm\sigma_{\rm j}^{2}=(EFAC~{}\sigma_{\rm j}^{ToA})^{2}+EQUAD^{2}$ (6)

where EFAC encapsulates unknown systematic errors associated with observing
systems and analysis, and EQUAD added in quadrature characterizes any
additional, system-independent white noise.

Within the wideband likelihood, two new white noise components have been
introduced: DMEFAC and DMEQUAD which are analogous to these described above
but are applied to DM uncertainties. In addition to that, we also used a new
DMJUMP parameter, which can be referred to the standard JUMP parameter (used
to account for phase offsets of unmodeled profile evolution), but here it
represents the offset between the mean wideband DM and individual wideband DM
measurements. In other words, it accounts for the ambiguity of determining the
absolute DM. Finally, a standard ECORR parameter can be omitted in the
wideband analysis, as it is describing the correlation between sub-banded ToAs
naturally not present here (noise sources contributing to ECORR are absorbed
by EQUAD in the wideband likelihood).

We set narrow Gaussian EFAC and DMEFAC priors on 1.00 $\pm$ 0.25. Priors for
the other parameters are drawn from uniform distributions given by:
$\mathrm{log_{10}{(EQUAD)}\in[-8.5,-5.0]}$,
$\mathrm{log_{10}{(DMEQUAD)}\in[-7.0,0.0}]$,
$\mathrm{log_{10}{(DMJUMP)}\in[-0.01,0.01]}$.

Table 1: Summary of timing analysis for 35 pulsars. 25 of the listed pulsars are high priority sources observed as part of the PPTA project, while the remaining 10 (marked with a star) have been added to the array after the installation of UWL, between 2018 and 2020. The 10th column shows the S/N of the pulse portraits. Pulsar | Span | DM | P | ToAs | Pars | RMS | Med $\sigma_{\rm ToA}$ | Med $\sigma_{\rm DM}$ | S/N | $\mathrm{n_{eig}}$ | Figure
---|---|---|---|---|---|---|---|---|---|---|---
| [yr] | [ cm-3 pc] | [ms] |  |  | $\rm[\mu s]$ | $\rm[\mu s]$ | [$\times 10^{-4}$ cm-3 pc] |  |  |
J0030$+$0451* | 3.1 | 4.33 | 4.87 | 34 | 5 | 0.784 | 0.732 | 4.423 | 427 | 1 | 7
J0125$-$2327* | 3.1 | 9.60 | 3.68 | 105 | 13 | 0.529 | 0.129 | 0.974 | 5503 | 3 | 7
J0348$+$0432* | 2.7 | 40.47 | 39.12 | 22 | 10 | 2.511 | 2.080 | 14.24 | 211 | 0 | 7
J0437$-$4715 | 3.3 | 2.64 | 5.76 | 222 | 13 | 0.195 | 0.005 | 0.043 | 18191 | 6 | 7
J0613$-$0200 | 3.4 | 38.78 | 3.06 | 76 | 13 | 0.269 | 0.158 | 0.807 | 1633 | 2 | 7
J0614$-$3329* | 3.1 | 37.05 | 3.15 | 90 | 10 | 0.813 | 0.741 | 4.171 | 490 | 1 | 7
J0711$-$6830 | 3.3 | 18.41 | 5.49 | 157 | 7 | 0.435 | 0.550 | 3.211 | 3927 | 2 | 8
J0900$-$3144* | 2.7 | 75.61 | 11.11 | 65 | 12 | 0.931 | 0.612 | 3.421 | 2376 | 2 | 8
J1017$-$7156 | 3.3 | 94.22 | 2.34 | 173 | 16 | 0.133 | 0.120 | 0.684 | 2090 | 2 | 8
J1022$+$1001 | 3.4 | 10.25 | 16.45 | 59 | 12 | 0.566 | 0.254 | 1.517 | 6189 | 3 | 8
J1024$-$0719 | 3.3 | 6.48 | 5.16 | 50 | 9 | 0.771 | 0.619 | 4.105 | 1516 | 2 | 8
J1045$-$4509 | 3.3 | 58.15 | 7.47 | 54 | 12 | 1.316 | 0.781 | 4.173 | 2408 | 2 | 8
J1125$-$6014 | 3.3 | 52.93 | 2.63 | 97 | 15 | 0.203 | 0.119 | 0.744 | 970 | 2 | 9
J1446$-$4701 | 3.3 | 55.83 | 2.19 | 73 | 13 | 0.665 | 0.425 | 2.741 | 431 | 0 | 9
J1545$-$4550 | 3.3 | 68.39 | 3.58 | 100 | 15 | 0.327 | 0.212 | 2.144 | 876 | 2 | 9
J1600$-$3053 | 3.3 | 52.33 | 3.60 | 53 | 16 | 0.263 | 0.113 | 0.832 | 2694 | 2 | 9
J1603$-$7202 | 3.3 | 38.04 | 14.84 | 105 | 12 | 0.706 | 0.361 | 2.129 | 6643 | 2 | 9
J1643$-$1224 | 3.3 | 62.41 | 4.62 | 42 | 14 | 1.25 | 0.286 | 1.459 | 2870 | 2 | 9
J1713$+$0747 | 2.3 | 15.92 | 4.57 | 43 | 13 | 0.241 | 0.048 | 0.384 | 9893 | 2 | 10
J1730$-$2304 | 3.3 | 9.62 | 8.12 | 39 | 6 | 1.244 | 0.287 | 1.677 | 3660 | 2 | 10
J1741$+$1351* | 2.6 | 24.20 | 3.74 | 16 | 10 | 0.460 | 0.345 | 2.640 | 145 | 0 | 10
J1744$-$1134 | 3.3 | 3.14 | 4.07 | 88 | 8 | 0.327 | 0.080 | 0.524 | 8670 | 2 | 10
J1824$-$2452A | 2.8 | 119.90 | 3.05 | 11 | 5 | 4.481 | 0.123 | 0.710 | 388 | 1 | 10
J1832$-$0836 | 2.7 | 28.20 | 2.72 | 21 | 8 | 0.246 | 0.227 | 2.107 | 290 | 1 | 10
J1857$+$0943 | 3.1 | 13.30 | 5.36 | 25 | 10 | 0.391 | 0.257 | 2.189 | 1366 | 2 | 11
J1902$-$5105* | 2.7 | 36.25 | 17.40 | 31 | 7 | 2.884 | 0.329 | 1.910 | 121 | 0 | 11
J1909$-$3744 | 3.4 | 10.39 | 2.95 | 220 | 15 | 0.231 | 0.027 | 0.176 | 15208 | 1 | 11
J1933$-$6211* | 3.1 | 11.50 | 3.54 | 84 | 12 | 0.752 | 0.764 | 4.842 | 1381 | 1 | 11
J1939$+$2134 | 2.7 | 71.01 | 1.56 | 12 | 5 | 0.856 | 0.010 | 0.050 | 2049 | 2 | 11
J2051$-$0827* | 2.6 | 20.73 | 4.51 | 34 | 8 | 8.643 | 0.680 | 5.657 | 1462 | 2 | 11
J2124$-$3358 | 3.3 | 4.59 | 4.93 | 67 | 8 | 1.698 | 0.985 | 5.760 | 410 | 1 | 12
J2129$-$5721 | 3.3 | 31.85 | 3.73 | 109 | 12 | 0.679 | 0.509 | 3.512 | 1895 | 1 | 12
J2145$-$0750 | 3.3 | 9.00 | 16.05 | 51 | 13 | 0.823 | 0.186 | 1.179 | 7117 | 2 | 12
J2150$-$0326* | 2.6 | 20.67 | 3.50 | 26 | 10 | 2.011 | 1.115 | 6.827 | 128 | 0 | 12
J2241$-$5236 | 3.3 | 11.41 | 2.19 | 151 | 13 | 0.283 | 0.070 | 0.465 | 8651 | 2 | 12

##  4 Results and discussion

Table 1 contains basic pulsar parameters and summarizes the results of our
ToA/DM measurements and timing analysis. Apart from nominal DM, orbital
period, median ToA/DM uncertainties and RMS of the timing residuals, we also
listed the number of ToAs and fitted timing model parameters, S/N of the
average portraits, and the number of eigenprofiles as a general
characteristics of the profile evolution models. In Appendix A we show all the
residual and DM variability plots (referenced in the last column of Table 1)
and in Appendix B we present tables of our timing model parameters.

###  4.1 Average portraits

As described in Sec. 3.1, an average portrait is composed of several tens of
aligned and averaged observations. As an example, we show the portrait of
J0125$-$2327 in Fig. 1.

Each gap in frequency coverage corresponds to zero-weighted channels due to
strong RFI. The overall contamination of the whole band is relatively low and
the two most significant sources of spurious signal are mobile ($<1000$ MHz)
and WiFi/Bluetooth ($\sim 2400$ MHz) transmissions.

Any residual intensity variations preserved despite time averaging of the
portrait are eliminated by the normalization of each frequency channel with
respect to the mean profile, and so the amplitude in Fig. 1 is in arbitrary
units. This procedure is meant to ensure that the model correctly describes
the intrinsic profile shape changes irrespective to the effects of spectral
index or ISM variability (specifically the diffractive scintillation).

Figure 1: Data portrait of J0125$-$2327 with the mean profile shown in the
upper panel.

###  4.2 Profile evolution models

Profile evolution with frequency is encoded in the spline curve model, and the
number of significant eigenvectors (eigenprofiles) can be treated as a proxy
for the level of profile evolution complexity. Noteworthy, the later depends
not only on the intrinsic profile shape changes, but also on any distortions
of the signal (e.g. due to ISM or RFI) and observing system/analysis failures.

When there are no eigenprofiles detected, all measurements are referred to the
mean profile. In such a case, there is either none/little profile evolution or
S/N for a given pulsar is too low. The first eigenprofile corresponds to a
gradient of the mean profile changes (provided it is the only eigenprofile
detected), while the second and third may encompass more extreme profile
evolution, possible ISM effects such as unmodeled scattering, or issues
related to data reduction and analysis. These could be, e.g. misalignment of
the profiles comprising the average portrait and inaccuracies in polarization
calibration. More than three eigenprofiles can be detected for very high S/N
pulsars, but usually they arise from systematic errors or unmitigated RFI.

The number of eigenprofiles for each pulsar in our data set is listed in Table
1. The majority of pulsars required two eigenprofiles (19 out of 35), one was
detected for eight pulsars and none in case of five sources. The remaining
three pulsars were described by three (J0125$-$2327, J1022$+$1001) and six
eigenprofiles (J0437$-$4715). There is an evident relation between the
complexity of the spline model and S/N of the average portrait, indicating
that the quality of observations is one of the leading factors affecting the
precision of pulse shape modeling. We discuss this further in Sec. 4.3.

Below, as an example, we present the profile evolution model for
J0125$-$2327\. This is a new pulsar to the PPTA discovered in the Green Bank
Northern Celestial Cap Pulsar Survey (Stovall et al., 2014), characterized by
a profile with two leading components exhibiting strong evolution with
frequency. Fig. 2 shows the observed profile shapes at four frequencies
throughout the UWL band extracted from the average portrait (teal) and modeled
pulse profiles (black). Despite obvious complexity of the pulse shape and its
evolution, the model is closely tracing all visible details with high
accuracy.

J0125$-$2327 is one of only three pulsars for which we detected three
significant eigenprofiles (shown in Fig. 3). Each eigenprofile is smooth and
well resolved with high S/N. Additionally, coordinate curves comprising the
spline model shown in Fig. 4 are clear and do not exhibit any signs of RFI
contamination. In fact, the presented model can be considered as a flagship
example given the complexity of the pulse shape and the level of detail
captured.

Figure 2: Observed (teal) and reconstructed by the spline model (black) pulse
profiles at four frequencies throughout the UWL band for J0125$-$2327 (upper
left to lower right: 894, 1380, 2699 and 3180 MHz). Figure 3: Mean profile and
three eigenprofiles for J0125$-$2327\. Units at the y-axis are arbitrary.
Figure 4: Spline model of profile shape evolution with frequency for
J012$5-$2327\. Purple points are mean substracted profiles comprising the
average portrait projected onto the eigenprofiles. Color scale and size of the
points correspond to frequency and S/N of the profile, respectively. The green
line represents the spline model of profile evolution.

In the case of the majority of our pulsars which are described by two, high
S/N eigenprofiles, their spline models do not indicate any substantial issues
and fit observed profiles at various frequencies with a comparable level of
accuracy as shown above. Five out of 35 pulsars with no significant
eigenvectors have portrait $\rm S/N<430$, usually with low flux measured above
2500 MHz. Similarly, subtle profile shape characteristics and its changes can
be lost for models with on only one eigenprofile, especially in the upper part
of the band where the flux of most of pulsars is usually lower. Two extreme
examples of such sources are J0030$+$0451 and J2129$-$5721, both with large
spectral indices $\alpha_{\rm J0030}=-2.4$ and $\alpha_{\rm J2129}=-3.9$
(Spiewak et al., 2022). In order to capture any profile evolution, and more
accurately predict the pulse shapes at lower frequencies, we decided to remove
the upper part of the band above 3000-3500 MHz. This resulted in a detection
of a previously not present eigenprofile for J2129$-$5721 and a much better
fit to the observed profile shapes in case of both pulsars.

Finally, we would like to stress, that the fact that all pulsars we studied
required only zero to three eigenprofiles indicates excellent quality of UWL
observations maintained throughout the whole band (free of substantial
instrumental, calibration or profile alignment errors). In addition to that,
our study demonstrates a notable efficiency of PulsePortraiture, where even
complex pulse shape changes can be captured and described by a reasonably
simple model.

###  4.3 Sub-banded ToAs

Figure 5: Sub-banded residuals and DM measurements for J0125$-$2327\. The
bottom panel shows residuals as a function of frequency. Each color
corresponds to one of the frequency bands.

Although wideband timing methods offer a number of improvements such as
simplification of timing models and lowering data volumes, a parallel analysis
of sub-banded ToAs may help reveal frequency-dependent effects related either
to physical phenomena, such as DM variations, or technical issues. At the same
time, wideband profile evolution modeling is also sensitive to instrumental
problems (for instance, resulting in spurious eigenprofiles) and was shown to
improve timing of scintillating and weak pulsars (Alam et al., 2021a).
Moreover, comparison of noise models from narrow- and wideband data sets can
help discriminate various sources of noise, because of a different set of
parameters. Therefore, synergy between the two approaches would be highly
beneficial for high precision pulsar timing, allowing for better
characterizations of individual pulsars, ISM and their models.

Following the above arguments, we divided all UWL observations into eight sub-
bands and calculated wideband ToAs separately in each band, using the model
derived for the whole band. The first six bands have an equal width of 208 MHz
and cover frequency range from 704 MHz to 1952 MHz, while bands seven and
eight are broader (416 MHz and 1664 MHz, respectively) providing a more even
distribution of S/N.

In Fig. 5 we present sub-banded residuals as a function of time and frequency
(top and bottom panels, respectively) and DM measurements for pulsar
J0125$-$2327 as an example. This is one of the best timed sources in our
sample, with $\sigma_{\rm ToA}=129$ ns, a portrait with high $\rm S/N=5503$
and a detailed model containing three, well defined eigenprofiles presented in
Sec. 4.2. Nevertheless, the bottom panel of Fig. 5 shows an evident frequency
drift of the residuals with an amplitude of approximately $3.79~{}\mu$s. After
a thorough inspection we found such drift in all of our pulsars, however with
varying significance. Median amplitude of the drift is about $1.65~{}\mu$s and
in most cases it lies below the phase bin resolution.

This leads to the first hypothesis: if pulse shape changes occur at such small
scales, then the spline models may not be able to capture their frequency
evolution in full detail resulting in frequency dependence of the residuals.
However, we find that scenario unlikely, as power spectra of the average
portraits show an expected exponential drop of the signal with harmonic
number, while any unresolved features would remain above the white noise at
all harmonics.

More importantly, phase resolution can be a limiting factor in the process of
profile alignment for the average portraits. As noted in Pennucci, 2019,
intrinsic profile evolution is entangled with the absolute DM or its
variations and the alignment of the portraits may be accurate up to one phase
bin level. Furthermore, due to the dispersive law, any inaccuracies in the DM
measurements will induce profile smearing on timescales increasing with
bandwidth, thus the effect will be more prominent for ultra-wide-band
receivers, such as the UWL. The rotational period of J0125-2327 is 3.68 ms,
which means that the amplitude of the frequency drift seen in Fig. 5 is equal
to approximately one phase bin, further supporting this scenario. The
corresponding $\delta$DM that would induce this level of drift across the UWL
band is $\rm\sim 5\times 10^{-4}\,cm^{-3}$ pc, which is comparable with the
median DM uncertainty of $\rm\sim 1\times 10^{-4}\,cm^{-3}$ pc.

Another possibility is that these drifts are induced or enhanced by timing and
noise modeling systematics or during the decomposision of the average
portraits with PCA, which can be supported by the form of their frequency
dependence. Of course, all of the above hypotheses do not have to be mutually
exclusive and the observed frequency drift may very well be a function of
multiple factors.

Four pulsars, where the drift is larger than the bin resolution have either
zero or one eigenprofile (J1446$-$4701, J2124$-$3358 and J2129$-$5721,
respectively), have a broad profile possibly affected by scattering
(J1045$-$4509) and a weak upper-most part of the band which was cut out during
modeling (J2129$-$5721). In these cases we expect some of the information on
the profile evolution to be missing or obscured, and therefore a few possible
improvements would require observations with higher S/N and detailed modeling
of the ISM effects.

This work is the first to report and delineate the unmodeled frequency
dependence in the timing residuals from wideband timing, although
PulsePortraiture and wideband techniques have been already used for multiple
and diverse studies as was mentioned in Sec. 1. The phenomenon is naturally
present in narrow-band analyses which use frequency-averaged templates and is
usually corrected by applying additional FD parameters to the timing model in
order to account for profile evolution with frequency. The main goal of the
wideband technique is to capture and model these frequency-dependent effects
and thus detection of the drift in our wideband data is unsettling. We note
however, that a similar trend has been recently detected also in other data
sets including wideband analysis from NANOGrav (via private communication).
Additionally, timing models from the latest MPTA release included FD
parameters despite using frequency evolving templates (Miles et al. 2022,
submitted).

The fact that the drift was observed only now may have multiple reasons.
Frequency dependent phenomena will be less pronounced or even absolutely
undetectable for narrow-bandwidth observations, and wideband timing methods
have, in fact, never been used on data with more than 2 GHz instantaneous
frequency coverage and to such extend (nearly all new timing analyses now
include wideband methodology). For instance, wideband analysis presented in
NG12.5 did not report the need for FD parameters, because the ToAs were
effectively calculated only for two sub-bands (as opposed to eight sub-bands
shown in Fig. 5), which obscured any such detectable drifts. Similarly,
Tarafdar et al. (2022) also presented both narrow- and wideband timing of 14
pulsars as the first data release from the InPTA and explicitly showed no need
for additional corrections for frequency-dependent effects. However, InPTA
analysis was also perfomed on only two sub-bands and produced narrow-band,
frequency-resolved templates by iteratively fitting for DM to the multi-band
observations. Noteworthy, Tarafdar et al. (2022) pointed out that differences
in denoising the templates lead to a discrepancy in DM estimates obtained by
narrow- and wideband techniques, which might be worth further investigation.

Interestingly, parallel narrow- and wideband analyses presented in e.g. NG12.5
and Nobleson et al. (2022) yielded remarkable agreement between the results
from the two methods. This indicates that frequency-evolving templates and
wideband measurements provide timing precision at least as good as standard
procedures despite the observed frequency drift, however its actual
significance will be a matter of debate. This argumentation also applies to
this work as ToA/DM uncertainties and RMS of the residuals we obtained reach
low, sub-$\mu$s precision which is an improvement when compared to PPTA DR2. A
complete and fully restrictive comparison will be possible after obtaining
narrow- and wideband results for the next PPTA data release which is already
in the process of preparation.

Nevertheless, observed frequency dependence of the residuals is a noteworthy
complication, because wideband timing aims to model all frequency dependent
effects in its extraction of a single ToA per observation, thus rendering ad
hoc frequency-dependent modeling of the timing residuals obsolete. The
solution to this problem is beyond the scope of this work, but is under
investigation in several groups.

###  4.4 Timing models

The nominal set of fitted parameters in our timing models included spin period
P and spin down rate P1, and five astrometric parameters: right ascension RA
and declination DEC, proper motion in both directions PMRA and PMDEC and
parallax PX. Binary pulsars were additionally fitted for orbital period PB,
projected semi-major axis A1, epoch of ascending node TASC, first and second
Laplace parameters (EPS1 and EPS2, respectively). For a few pulsars, proper
motions and parallax were poorly constrained so we excluded them from the
fitting. In particular, this is the case of pulsars close to the ecliptic such
as J1022$+$1001 or J1730-2304. Additionally, first time derivative of orbital
period PBDOT, rate of change of periastron and projected semi-major axis
(OMDOT and XDOT, respectively) also could not be measured for most pulsars and
were excluded (except from models for: J0437$-$4715, J1017$-$7156,
J2145$-$0750). For all four pulsars with ELL1H binary model we find both
orthometric Shapiro delay parameters H3 and H4. Finally, in case of two
pulsars with DDK binary model (J0437$-$3715 and J1713$+$0747) instead of
fitting for the aforementioned binary parameters we used epoch and longitude
of periastrion (T0 and OM, respectively) and eccentricity of the orbit (E).
Tables containing the timing results are presented in Appendix B. Timing model
parameters we obtained are consistent with those presented in PPTA DR2 within
error limits. The most significant discrepancies were measured for PMRA, PMDEC
and PX parameters which is expected because of the much shorter length of our
data set (for instance, PX for J2145$-$0750 obtained by us and in PPTA DR2 is
3.4(7) and 1.40(8), respectively).

###  4.5 Noise analysis, timing residuals and DM variations

For the majority of our pulsars EFAC and DMEFAC parameters have values between
1.0 and 1.3 which indicates that the ToA and DM measurements are free of
substantial systematic errors (note however, that this is also a function of
narrow prior distributions).

There are two pulsars with excess white noise characterized by a slightly
increased DMEFAC (J2124$-$3358 and J2241$-$5236) up to 1.58 and 1.41,
respectively. The absolute value of the DMJUMP parameter in most cases is less
than 0.0007 cm-3 pc indicating a proper alignment of profiles comprising the
average portrait and a consistent fit for DM from modeling and noise analysis.
There are, however, two pulsars (J1824$-$2452A and J2051$-$0827), where DMJUMP
goes up to 0.009 and there might be several reasons for that including
significant scattering, large DM variations or simply the fact that they have
much less data than other pulsars in the data set. Additionally, J2051$-$0827
is a black widow (eclipsing binary) and together with aforementioned ISM
effects this can impede obtaining a correct profile alignment.

Our initial noise models for most pulsars are dominated by large EQUAD values
of order of a few hundred nanoseconds. This is due to the fact that EQUADs
absorb all of the unmodeled sources of noise including all kinds of red noise
but also jitter and issues with modeling the DM variability (usually covered
by ECORR parameter which is omitted in wideband methodology). We have obtained
sub-$\mu$s RMS in 26 pulsars out of which 20 belong to the main PPTA array.
Nearly all pulsars (94%) have median ToA uncertainties lower than $1~{}\mu$s
except two low priority sources which are observed by the PPTA only since
recently (J0348$+$0432 and J2150$-$0326). In case of DM uncertainties, our
measurements are in the range of (0.043 - 14.24)$\times 10^{-4}$ cm-3 pc. We
have obtained DM precision of $10^{-5}$ cm-3 pc for 10 pulsars and down to
$10^{-6}$ cm-3 pc in the case of two (J0437$-$4715 and J1939$+$2134). This is
the level of precision achieved by Nobleson et al. (2022) for corresponding
pulsars despite the fact that their observations covered 200-500 MHz band
where DM measurements can be measured with higher precision because of the
larger delays and stronger signal in the low-frequency regime.

Finally, we would also like to note an observed dependence of the DM on usable
bandwidth (cleaned of all spurious channels). For instance, there is a subset
of observations, where $f_{\rm ratio}<3$ because the lower part of the band
(below 1400 MHz) was cut out by automatic pipelines due strong RFI. Usually,
this resulted in lower or higher estimated DMs when compared to the ones
obtained for the whole band even by $\sim 0.005$ cm-3 pc (see e.g.
J0125$-$2327 in Fig. 7), which is at least one order of magnitude larger than
the typical DM uncertainty obtained in the data set. This effect might be
correlated with the observed frequency drift of the residuals reported in Sec.
4.3 or to the frequency-dependent DM (explained further in Sec. 4.7.2) and
will be investigated together in the future work.

###  4.6 Pulsars not included in the main PPTA array

Apart from the top priority pulsars observed for nearly two decades, PPTA has
also been monitoring 10 new and/or lower priority sources since the start of
UWL operation. Their timing potential is currently being investigated and this
work may serve as an additional point for their evaluation (Mandow et al. in
preparation).

Most of these pulsars (6 out of 10) have low $\rm S/N<500$, which makes
modeling of their profile evolution difficult, however their ToA uncertanties
reach precise timing requirements with values between $\sim 350$ and $~{}700$
ns (apart from J0348$+$0432 and J2150$-$0326 already mentioned in the previous
section). The remaining four sources have high S/N and low ToA uncertanties in
the range of 129 - 764 ns. Their models are characterized with good resolution
and up to three eigenprofiles, including a fine example of J0125$-$2327
discussed in detail in Sec. 4.1 and Sec. 4.3. Pulsar J0900$-$3144 is already a
part of EPTA with timing baseline of approximately 7 years and ToA precision
of $4.27~{}\mu$s (Desvignes et al., 2016), while our work gives $\sigma_{\rm
ToA}=0.612~{}\mu$s, which indicates that this is a source worth further
monitoring. J1933$-$6211 has a profile shape with multipeak leading component
strongly evolving with frequency, however we detected only one eigeprofile for
it. Further monitoring and collecting more observations would be highly
beneficial for a better characterization of this pulsar, especially given that
it is a typical binary pulsar with a white dwarf companion (Graikou et al.,
2017) which makes it potentially valuable addition to the PPTA project.
J2051$-$0827 is relatively bright ($\rm S/N=1462$), however it is one of the
most difficult sources in our sample to describe. Its pulse shape consists of
one peak component strongly affected by scattering and additionally it is an
eclipsing binary so modeling its profile evolution and noise analysis are
particularly challenging. J0348$+$0432 and J2150$-$0326 are the worst two
pulsars in our sample in terms of median ToA uncertainties so directly they
might not be of much value to the PPTA (although further observations and
individual analysis may improve their solutions). Nevertheless, monitoring
such sources can also lead to a better understanding of pulsars emission or
ISM effects which in turn may help constructing better timing and noise models
in general (Kerr et al., 2020).

Finally, we also report the measurements of the binary orbital periods for two
pulsars, which were not previously published: J0125-2327 and J2051-0827 (Tab.
2 and Tab. 7 in Appendix B).

###  4.7 Notes on individuals pulsars

####  4.7.1 J1022$+$1001

J1022$+$1001 is an interesting source because of a long-standing controversy
regarding its pulse shape instability over timescales ranging from several to
several tens of minutes (see Padmanabh et al., 2021 and references therein).
There have been various attempts to explain this peculiar behavior, which
include suggested polarization calibration errors (van Straten, 2013), strong
scintillation coupled with intrinsic profile evolution (Shao & You, 2016) or
effects directly related to the pulsar magnetosphere (Ramachandran & Kramer,
2003). Unfortunately, there is still no consensus as these results are often
contradicting.

In our wideband model for this pulsar, a complex shape evolution is evident as
the two-peak pulse components change their relative height. Notwithstanding,
the reconstructed profiles trace the shape evolution with a good precision
(similarly to the model of J0125$-$2327 shown in Fig. 2).

The spline model for J1022$+$1001 profile evolution required 3 eigenprofiles.
They can either correspond to subtle corrections to the intrinsic profile
shape changes or to the absorbed ISM/instrumental effects as was mentioned
earlier in Sec. 4.2. It might be tantalising to discern temporal pulse shape
variability in the second and/or third eigenprofile, although spline models
are supposed to trace only stationary profile changes with frequency. The
initial noise model returns a very large EQUAD of 1.4 $\mu$s, which is
significantly exceeding recently estimated jitter of $\sim 120$ ns for one
hour integration (Parthasarathy et al., 2021) and this might indeed reflect
the scatter induced by pulse instability in time (and other noise sources as
explained in Sec. 4.5).

####  4.7.2 J2241$-$5236

Frequency-dependent DM (Cordes et al., 2016) is thought to be detectable
mostly for bright, high-DM pulsars with little to no profile evolution, which
would make J2241$-$5236 a nearly perfect target. If a true nature of phase
offsets deviates from $\nu^{-2}$ given by the dispersion law (Eq. 2), the DM
measured at different parts of the band will vary. In fact, detection of this
phenomenon for J2241$-$5236 was recently reported by (Kaur et al., 2022) based
on three days of observations obtained by uGMRT, Murchison Widefield Array
(MWA) and Parkes UWL in November 2019. The reported DM changes scale as
$\rm\delta DM\sim\nu^{2.5\pm 0.1}$ indicating that DM measured at lower
frequencies is notably higher then the one measured in the upper part of the
band.

Figure 6: Sub-banded DM measurements as a function of frequency for
J2241$-$5236\. Grey circles represent each individual measurement, while red
points show median values and their uncertainties for each sub-band. The
y-axis on the plot was narrowed down to $\rm\delta DM$ in the range of ($-$
0.1,0.1) cm-3 pc for a better visibility omitting a few outlying measurements,
however they were included in the calculation of the median value. Nominal DM
is shown in the legend of the plot and it corresponds to the black line
centered at $\rm\delta DM=0$.

In our wideband analysis, we have initially obtained a similar trend when
studying 20-min integrations and data portrait consisting of approximately ten
highest S/N observations. The number of detected eigenprofiles strongly
depended on the number of used channels (it was either zero or one) despite
the overall S/N exceeding 3500. In the final analysis we have used full 1 hour
integrations and all available UWL observations which resulted in doubling the
S/N of average portraits for most pulsars. Our new model for J2241$-$5236 has
two, well resolved eigenprofiles and the sub-banded residuals show a small
frequency drift of 0.67 $\mu$s relative to a bin width of 2.14 $\mu$s.

In Fig. 6 we show sub-banded DM measurements as a function of frequency. All
of our DM measurements now have $\rm\delta DM\sim 0$ with respect to the
nominal DM and so any chromatic trend of the DM detected previously was
reduced significantly.

These results indicate that detection of frequency-dependent ISM phenomena may
be more difficult than previously expected. Even best candidate pulsars with
sharp integrated profile shapes and high stability may exhibit non-negligible
profile evolution at small sub-$\mu$s scales, which is challenging both to
detect and model. Additionally, we would like to emphasize again that the
measured results will be a function of the somewhat arbitrarily determined
profile evolution model and disentangling this from actual ISM or other
frequency-dependent effects is a non-trivial problem (Hassall et al., 2012).

##  5 Summary and conclusions

We have presented wideband timing analysis of the new UWL observations
collected under the Parkes Pulsar Timing Array project between November 2018
and March 2022. The main output of our work is a presentation of precise
models of profile shape evolution with frequency for 35 pulsars and wideband
timing measurements (simultaneously estimated ToAs and DMs). We have also
performed an initial noise analysis extended with wideband likelihood
including white noise components. Results presented here are the very first
demonstration of using methods presented in Pennucci et al. (2014) on
observations with instantaneous bandwidth larger than 2 GHz (fractional
bandwidth of UWL is 1.41), which is soon to become a standard in high
precision pulsar timing.

Precision of our ToA and DM measurements is in the range of 0.005 $-$ 2.08
$\mu$s and (0.043 $-$ 14.24)$\times 10^{-4}$ cm-3 pc, respectively. Comparison
of raw ToAs from our work and the previous data release PPTA DR2 yields an
increase of the precision by a factor of two, however due to utilization of
different receivers, backends and analysis methods this gives only a rough yet
promising estimate. A proper comparative analysis will be possible after
finalization of two, parallel releases (narrow- and wideband) comprising all
available PPTA data extending back to 2004.

In case of two pulsars, J0030$+$0451 and J2129$-$5721, very low S/N above 3000
or 3500 MHz affected the accuracy of the modeled profile evolution. We decided
to exclude the uppermost frequencies from our analysis which improved both
models, however wideband timing procedures should ultimately be able to
resolve such issues in a less brute-force manner. For instance, this could be
achieved by adjusting the normalization algorithm so that it does not
artificially elevate or overestimate information from low S/N channels.

We have detected a frequency drift of the sub-banded residuals obtained with
wideband methodology with an average amplitude of $\sim 1~{}\mu$s which is
below the phase bin resolution of our observations. This might indicate
profile alignment issues, however we cannot yet rule out other possible
explanations, including timing/noise model systematics, nuances of PCA
procedures and the effect of non- or poorly modeled ISM effects, such as
scattering. The frequency dependence of the residuals will be investigated
further in detail and the results will be presented elsewhere.

Wideband models presented here will be used for timing analysis of the next
full data release (DR3) from PPTA which is now under development. The nominal
DR3 data set, despite using frequency evolving templates, will produce sub-
banded ToAs of UWL observations in order to combine them with the previous
releases in a consistent way. However, we also do intend to produce a wideband
DR3 in parallel, where we will reproduce 14-year long data sets with methods
presented here and combine it with UWL observations.

The Parkes radio telescope (Murriyang) is part of the Australia Telescope
National Facility which is funded by the Australian Government for operation
as a National Facility managed by CSIRO. MC is supported by the Polish
National Science Center through research grant NR 2021/41/N/ST9/01512. SD is
the recipient of an Australian Research Council Discovery Early Career Award
(DE210101738) funded by the Australian Government.

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##  Appendix A Timing residuals and DM measurements

Below, we present plots with timing residuals (top panels) and DM variability
(bottom panels). For each pulsar, ToA error bars are corrected by the obtained
white noise components, namely EFAC and EQUAD, while DM measurements include
DMEFAC and DMJUMP. Red and black points in DM plots show the wideband
measurements and DMX model, respectively. If DMX model points are missing in
any pulsars plot it means that there is only one DMX bin encompassing the
whole timing baseline. This is the case for the least frequently observed
pulsars.

Figure 7: Timing residuals and DM variations for J0030$+$0451, J0125$-$2327,
J0348$+$0432, J0437$-$4715, J0613$-$0200 and J0614$-$3329. Figure 8: Timing
residuals and DM variations for J0711$-$6830, J0900$-$3144, J1017$-$7156,
J1022$+$1001,J1024$-$0719 and J1045$-$4509. Figure 9: Timing residuals and DM
variations for J1125$-$6014, J1446$-$4701,J1545$-$4550, J1600$-$3053,
J1603$-$7202 and J1643$-$1224. Figure 10: Timing residuals and DM variations
for J1713$+$0747, J1730$-$2304, J1741$+$1351, J1744$-$1134, J1824$-$2452A and
J1832$-$0836. Figure 11: Timing residuals and DM variations for J1857$+$0943,
J1902$-$5103, J1909$-$3744, J1933$-$6211, J1939$+$2134 and J2051$-$0827.
Figure 12: Timing residuals and DM variations for J2124$-$3358, J2129$-$5721,
J2145$-$0750, J2150$-$0326 and J2241$-$5236.

##  Appendix B Timing parameters

Below we present tables with timing model parameters. The parameters which
were used in the fitting are marked in bold.

=65mm

Table 2: Timing parameters for J0030$+$0451, J0125$-$2327, J0348$+$0432,
J0437$-$4715 and J0613$-$0200.

|  J0030$+$0451 |  J0125$-$2327 |  J0348$+$0432 |  J0437$-$4715 |  J0613$-$0200
---|---|---|---|---|---
MJD range | 58486 - 59607 | 58480 - 59607 | 58486 - 59470 | 58441 - 59648 | 58441 - 59665
Right ascension (RA) [hh:mm:ss] | 00:30:27.4234(3) | 01:25:1.074328(5) | 03:48:43.64127(7) | 04:37:16.0445973(9) | 06:13:43.977048(3)
Declination (DEC) [dd:mm:ss] | 04:51:39.726(10) | -23:27:8.1349(1) | 04:32:11.462(2) | -47:15:9.99983(1) | -02:00:47.3405(1)
Proper motion in RA [mas $yr^{-1}$] | -6.2100 | 37.14(8) | – | 121.378(8) | 1.84(4)
Proper motion in DEC [mas $yr^{-1}$] | 0.5000 | 10.5(2) | – | -71.42(1) | -10.6(1)
Spin frequency [$s^{-1}$] | 205.530698975007(4) | 272.081088484952(1) | 25.560636549613(1) | 173.6879478266896(2) | 326.6005667292508(8)
Spin frequency derivative [$s^{-2}$] | -4.296(3)$\cdot 10^{-16}$ | -1.36332(10)$\cdot 10^{-15}$ | -1.5764(7)$\cdot 10^{-16}$ | -1.728459(8)$\cdot 10^{-15}$ | -1.02301(6)$\cdot 10^{-15}$
DM [$cm^{-3}pc$] | 4.332798 | 9.598062 | 40.466112 | 2.640302 | 38.779608
Parallax [mas] | 2.7900 | 1.4(3) | – | 6.3(3) | 0.7(2)
Binary model | – | ELL1 | ELL1 | DDK | ELL1
Orbital period [d] | – | 7.2771996382(8) | 1.0242406060(5)$\cdot 10^{-1}$ | 5.74104584368(9)$\cdot 10^{-0}$ | 1.19851255685(5)$\cdot 10^{-0}$
Projected semi-major axis [lt-s] | – | 4.7297995(1) | 0.140988(1) | 3.36674563(2) | 1.09144391(9)
Epoch of periastron [MJD] | – | – | – | 59048.3754(6) | –
Longitude of periastron [deg] | – | – | – | 1.54(4) | –
Eccentricity of orbit | – | – | – | 0.00001920(1) | –
Epoch of ascending node [MJD] | – | 59039.36296968(4) | 58978.0641405(2) | – | 59052.42591812(1)
First Laplace parameter | – | 0.00000009(6) | -0.00009(3) | – | 0.0000038(2)
Second Laplace parameter | – | 0.00000053(5) | -0.00003(2) | – | 0.0000032(2)
Time derivative of orbital period | – | – | – | – | –
Rate of advance of periastron | – | – | – | 0.0137870 | –
Rate of change of projected semi-major axis | – | – | – | – | –
Sine of inclination | – | – | – | – | –
Companion mass [$M_{\odot}$] | – | – | – | 0.223972 | –
Shapiro delay parameter h3 [$\mu s$] | – | – | – | – | –
Shapiro delay parameter h4 [$\mu s$] | – | – | – | – | –

=65mm

Table 3: Timing parameters for J0614$-$3329, J0711$-$6830, J0900$-$3144,
J1017$-$7156 and J1022$+$1001.

|  J0614$-$3329 |  J0711$-$6830 |  J0900$-$3144 |  J1017$-$7156 |  J1022$+$1001
---|---|---|---|---|---
MJD range | 58481 - 59596 | 58441 - 59664 | 58607 - 59595 | 58441 - 59664 | 58425 - 59665
Right ascension (RA) [hh:mm:ss] | 06:14:10.34823(2) | 07:11:54.15385(1) | 09:00:43.95220(2) | 10:17:51.312124(8) | 10:22:57.988(7)
Declination (DEC) [dd:mm:ss] | -33:29:54.1312(2) | -68:30:47.23742(8) | -31:44:30.8721(2) | -71:56:41.57562(3) | 10:01:52.8(3)
Proper motion in RA [mas $yr^{-1}$] | 0.5800 | -15.45(7) | -1.2(3) | -7.36(3) | -16.4260
Proper motion in DEC [mas $yr^{-1}$] | -1.9200 | 14.08(7) | 1.8(3) | 6.83(3) | 1.6741
Spin frequency [$s^{-1}$] | 317.594454701573(4) | 182.1172372978466(7) | 90.011843174693(1) | 427.6219116054646(7) | 60.7794488431391(8)
Spin frequency derivative [$s^{-2}$] | -1.7570(4)$\cdot 10^{-15}$ | -4.9461(6)$\cdot 10^{-16}$ | -3.958(1)$\cdot 10^{-16}$ | -4.0508(6)$\cdot 10^{-16}$ | -1.6017(8)$\cdot 10^{-16}$
DM [$cm^{-3}pc$] | 37.049000 | 18.406857 | 75.689026 | 94.223226 | 10.252000
Parallax [mas] | 1.1000 | – | 1.8620 | 0.9(9) | 1.5506
Binary model | ELL1 | – | ELL1 | ELL1H | ELL1H
Orbital period [d] | 53.58461255(3) | – | 18.737635770(5) | 6.5118987118(2) | 7.8051301640(10)
Projected semi-major axis [lt-s] | 27.6387917(6) | – | 17.2488093(5) | 4.83004758(6) | 16.7654220(6)
Epoch of periastron [MJD] | – | – | – | – | –
Longitude of periastron [deg] | – | – | – | – | –
Eccentricity of orbit | – | – | – | – | –
Epoch of ascending node [MJD] | 59056.1281453(2) | – | 59105.63920900(7) | 59053.90652703(1) | 59048.78549539(4)
First Laplace parameter | 0.00004976(4) | – | 0.00000986(4) | -0.00007136(2) | 0.00009625(7)
Second Laplace parameter | 0.00017382(4) | – | 0.00000350(4) | 0.00012288(2) | -0.00001333(8)
Time derivative of orbital period | – | – | – | – | –
Rate of advance of periastron | – | – | – | 0.016(8) | 0.0109124
Rate of change of projected semi-major axis | – | – | – | – | –
Sine of inclination | – | – | – | – | –
Companion mass [$M_{\odot}$] | 0.240000 | – | – | – | –
Shapiro delay parameter h3 [$\mu s$] | – | – | – | 0.00000011(4) | 0.0000011(4)
Shapiro delay parameter h4 [$\mu s$] | – | – | – | 0.00000010(5) | 0.0000004(5)

=65mm

Table 4: Timing parameters for J1024$-$0719, J1045$-$4509, J1125$-$6014,
J1446$-$4701 and J1545$-$4550.

|  J1024$-$0719 |  J1045$-$4509 |  J1125$-$6014 |  J1446$-$4701 |  J1545$-$4550
---|---|---|---|---|---
MJD range | 58443 - 59665 | 58454 - 59645 | 58442 - 59665 | 58443 - 59665 | 58442 - 59665
Right ascension (RA) [hh:mm:ss] | 10:24:38.64907(2) | 10:45:50.17978(3) | 11:25:55.242909(3) | 14:46:35.70981(1) | 15:45:55.945423(4)
Declination (DEC) [dd:mm:ss] | -07:19:19.9693(6) | -45:09:54.0569(3) | -60:14:6.80611(3) | -47:01:26.7964(2) | -45:50:37.50900(9)
Proper motion in RA [mas $yr^{-1}$] | -35.3(3) | -5.9(3) | 11.13(3) | -4.20(10) | -0.35(4)
Proper motion in DEC [mas $yr^{-1}$] | -48.1(7) | 4.4(3) | -13.04(3) | -2.8(2) | 2.39(8)
Spin frequency [$s^{-1}$] | 193.715686208170(5) | 133.793151505030(2) | 380.1730997159584(5) | 455.644022907970(2) | 279.6977022449402(6)
Spin frequency derivative [$s^{-2}$] | -6.971(1)$\cdot 10^{-16}$ | -3.165(2)$\cdot 10^{-16}$ | -5.3938(3)$\cdot 10^{-16}$ | -2.0362(2)$\cdot 10^{-15}$ | -4.10347(4)$\cdot 10^{-15}$
DM [$cm^{-3}pc$] | 6.480135 | 58.150964 | 52.934539 | 55.830112 | 68.392666
Parallax [mas] | 1.1(8) | 3.3416 | 1.7(3) | 0.4(5) | 0.8(2)
Binary model | – | ELL1 | ELL1 | ELL1 | ELL1H
Orbital period [d] | – | 4.083529193(1) | 8.7526035149(2) | 2.7766607312(9)$\cdot 10^{-1}$ | 6.2030648297(3)
Projected semi-major axis [lt-s] | – | 3.0151308(5) | 8.3391936(5) | 0.0640121(2) | 3.84690438(9)
Epoch of periastron [MJD] | – | – | – | – | –
Longitude of periastron [deg] | – | – | – | – | –
Eccentricity of orbit | – | – | – | – | –
Epoch of ascending node [MJD] | – | 59049.0111262(1) | 59053.335201025(8) | 59053.9339604(1) | 59056.49624717(2)
First Laplace parameter | – | -0.0000205(4) | -0.00000002(3) | 0.000003(6) | -0.00000871(5)
Second Laplace parameter | – | -0.0000104(4) | -0.00000064(1) | 0.000010(6) | -0.00000968(4)
Time derivative of orbital period | – | – | – | – | –
Rate of advance of periastron | – | – | – | – | –
Rate of change of projected semi-major axis | – | – | – | – | –
Sine of inclination | – | – | 0.977(7) | – | –
Companion mass [$M_{\odot}$] | – | – | 0.33(5) | – | –
Shapiro delay parameter h3 [$\mu s$] | – | – | – | – | 0.00000007(6)
Shapiro delay parameter h4 [$\mu s$] | – | – | – | – | -0.00000010(9)

=65mm

Table 5: Parameter files.

| J1600-3053 | J1603-7202 | J1643-1224 | J1713+0747 | J1730-2304
---|---|---|---|---|---
MJD range | 58443 - 59665 | 58443 - 59664 | 58454 - 59664 | 58426 - 59265 | 58455 - 59664
Right ascension (RA) [hh:mm:ss] | 16:00:51.902531(5) | 16:03:35.67088(3) | 16:43:38.16602(2) | 17:13:49.536688(5) | 17:30:21.6849(3)
Declination (DEC) [dd:mm:ss] | -30:53:49.4528(2) | -72:02:32.8216(2) | -12:24:58.633(1) | 07:47:37.4514(1) | -23:04:31.12(7)
Proper motion in RA [mas $yr^{-1}$] | -0.77(6) | -2.4(1) | 5.9(2) | 4.7(1) | 20.0246
Proper motion in DEC [mas $yr^{-1}$] | -6.8(2) | -7.2(2) | 1.9(10) | -3.9(4) | -4.9776
Spin frequency [$s^{-1}$] | 277.9377110420382(7) | 67.3765821486238(5) | 216.373340194411(2) | 218.811843674173(2) | 123.110288948911(1)
Spin frequency derivative [$s^{-2}$] | -7.3383(5)$\cdot 10^{-16}$ | -7.092(4)$\cdot 10^{-17}$ | -8.645(2)$\cdot 10^{-16}$ | -4.086(3)$\cdot 10^{-16}$ | -3.060(1)$\cdot 10^{-16}$
DM [$cm^{-3}pc$] | 52.328259 | 38.042070 | 62.301899 | 15.917131 | 9.622926
Parallax [mas] | 1.0(2) | 0.7019 | 2.6(8) | 0.0(4) | 1.7(8)
Binary model | ELL1H | ELL1 | DD | DDK | –
Orbital period [d] | 14.3484575522(8) | 6.3086295734(7) | 147.0173956(1) | 67.82512992(2) | –
Projected semi-major axis [lt-s] | 8.8016553(1) | 6.8806683(3) | 25.0725735(5) | 32.3424220(1) | –
Epoch of periastron [MJD] | – | – | 58987.082(2) | 58847.926(1) | –
Longitude of periastron [deg] | – | – | 321.849(4) | 176.240(6) | –
Eccentricity of orbit | – | – | 0.00050576(3) | 0.000074955(9) | –
Epoch of ascending node [MJD] | 59056.37152772(2) | 59050.18353655(4) | – | – | –
First Laplace parameter | -0.00000546(2) | 0.00000160(8) | – | – | –
Second Laplace parameter | -0.00017365(2) | -0.00000925(7) | – | – | –
Time derivative of orbital period | – | – | – | – | –
Rate of advance of periastron | -0.007(8) | – | – | – | –
Rate of change of projected semi-major axis | – | – | -0.04(1) | – | –
Sine of inclination | – | – | – | – | –
Companion mass [$M_{\odot}$] | – | – | – | 0.538032 | –
Shapiro delay parameter h3 [$\mu s$] | 0.00000032(7) | – | – | – | –
Shapiro delay parameter h4 [$\mu s$] | 0.00000011(10) | – | – | – | –

=65mm

Table 6: Timing parameters for J1741$+$1351, J1744$-$1134, J1824$-$2452A,
J1832$-$0836 and J1857$+$0943.

|  J1741$+$1351 |  J1744$-$1134 |  J1824$-$2452A |  J1832$-$0836 |  J1857$+$0943
---|---|---|---|---|---
MJD range | 58627 - 59571 | 58442 - 59658 | 58627 - 59664 | 58627 - 59607 | 58486 - 59624
Right ascension (RA) [hh:mm:ss] | 17:41:31.13983(1) | 17:44:29.421749(3) | 18:24:32.0076(2) | 18:32:27.589563(5) | 18:57:36.388636(7)
Declination (DEC) [dd:mm:ss] | 13:51:44.0624(3) | -11:34:54.7976(2) | -24:52:10.94(5) | -08:36:55.1437(3) | 09:43:17.1470(2)
Proper motion in RA [mas $yr^{-1}$] | -8.7609 | 18.86(4) | -0.2339 | -8.13(8) | -2.6(1)
Proper motion in DEC [mas $yr^{-1}$] | -6.5447 | -9.5(2) | -8.9459 | -21.2(4) | -5.9(2)
Spin frequency [$s^{-1}$] | 266.869166434737(3) | 245.4261233068814(5) | 327.40552589863(2) | 367.767120938621(2) | 186.494081052219(1)
Spin frequency derivative [$s^{-2}$] | -2.1524(2)$\cdot 10^{-15}$ | -5.3818(4)$\cdot 10^{-16}$ | -1.73531(2)$\cdot 10^{-13}$ | -1.1194(2)$\cdot 10^{-15}$ | -6.2038(8)$\cdot 10^{-16}$
DM [$cm^{-3}pc$] | 24.198581 | 3.139444 | 119.902567 | 28.195970 | 13.298525
Parallax [mas] | 0.4586 | 3.1(2) | -0.2430 | 1.1(3) | 1.6(5)
Binary model | ELL1 | – | – | – | ELL1
Orbital period [d] | 16.335347843(6) | – | – | – | 12.32717119165031
Projected semi-major axis [lt-s] | 11.0033142(4) | – | – | – | 9.230780241
Epoch of periastron [MJD] | – | – | – | – | –
Longitude of periastron [deg] | – | – | – | – | –
Eccentricity of orbit | – | – | – | – | –
Epoch of ascending node [MJD] | 59102.65959876(9) | – | – | – | 59058.66451258(5)
First Laplace parameter | -0.00000399(8) | – | – | – | -0.0000215720
Second Laplace parameter | -0.00000896(7) | – | – | – | 0.0000024568
Time derivative of orbital period | – | – | – | – | –
Rate of advance of periastron | – | – | – | – | –
Rate of change of projected semi-major axis | – | – | – | – | –
Sine of inclination | 0.941114 | – | – | – | 0.9990(4)
Companion mass [$M_{\odot}$] | 0.254703 | – | – | – | 0.253681
Shapiro delay parameter h3 [$\mu s$] | – | – | – | – | –
Shapiro delay parameter h4 [$\mu s$] | – | – | – | – | –

=65mm

Table 7: Timing parameters for J1902$-$5105, J1909$-$3744, J1933$-$6211,
J1939$+$2134 and J2051$-$0827.

| J1902-5105 | J1909-3744 | J1933-6211 | J1939+2134 | J2051-0827
---|---|---|---|---|---
MJD range | 58604 - 59595 | 58427 - 59658 | 58485 - 59607 | 58566 - 59567 | 58659 - 59596
Right ascension (RA) [hh:mm:ss] | 19:02:2.84424(4) | 19:09:47.424713(1) | 19:33:32.41353(2) | 19:39:38.56131(2) | 20:51:7.5244(4)
Declination (DEC) [dd:mm:ss] | -51:05:57.0563(9) | -37:44:14.91222(5) | -62:11:46.6962(2) | 21:34:59.1205(3) | -08:27:37.76(2)
Proper motion in RA [mas $yr^{-1}$] | -3.0375 | -9.49(1) | -5.4(2) | 0.0610 | 6.4920
Proper motion in DEC [mas $yr^{-1}$] | -11.1896 | -35.72(5) | 11.2(2) | -0.4044 | 0.8409
Spin frequency [$s^{-1}$] | 573.92104402934(2) | 339.3156919155664(2) | 282.212317830028(2) | 641.92821931762(1) | 221.79628690412(4)
Spin frequency derivative [$s^{-2}$] | -3.035(1)$\cdot 10^{-15}$ | -1.61469(2)$\cdot 10^{-15}$ | -3.078(2)$\cdot 10^{-16}$ | -4.33064(7)$\cdot 10^{-14}$ | -6.36(4)$\cdot 10^{-16}$
DM [$cm^{-3}pc$] | 36.250000 | 10.391734 | 11.499000 | 71.014000 | 20.729900
Parallax [mas] | – | 1.26(5) | – | 0.2368 | –
Binary model | ELL1 | ELL1 | ELL1 | – | ELL1
Orbital period [d] | 2.011803739(1) | 1.53344945354(2)$\cdot 10^{-0}$ | 12.819406522(2) | – | 0.0991102480(6)
Projected semi-major axis [lt-s] | 1.901956478 | 1.89799100(10) | 12.2815798(3) | – | 0.045075(4)
Epoch of periastron [MJD] | – | – | – | – | –
Longitude of periastron [deg] | – | – | – | – | –
Eccentricity of orbit | – | – | – | – | –
Epoch of ascending node [MJD] | 59099.3814765(2) | 59042.266169044(4) | 59051.25497668(4) | – | 59127.024438(2)
First Laplace parameter | 0.0000000000 | 0.00000007(5) | 0.00000135(4) | – | 0.0000000000
Second Laplace parameter | 0.0000000000 | -0.00000007(3) | -0.00000029(4) | – | 0.0000000000
Time derivative of orbital period | – | – | – | – | –
Rate of advance of periastron | – | – | – | – | –
Rate of change of projected semi-major axis | – | – | – | – | –
Sine of inclination | – | 0.9983(3) | – | – | –
Companion mass [$M_{\odot}$] | – | 0.202(6) | – | – | –
Shapiro delay parameter h3 [$\mu s$] | – | – | – | – | –
Shapiro delay parameter h4 [$\mu s$] | – | – | – | – | –

=65mm

Table 8: Timing parameters J2124$-$3358, J2129$-$5721, J2145$-$0750,
J2150$-$0326 and J2241$-$5236.

| J2124-3358 | J2129-5721 | J2145-0750 | J2150-0326 | J2241-5236
---|---|---|---|---|---
MJD range | 58441 - 59645 | 58441 - 59645 | 58441 - 59645 | 58659 - 59595 | 58441 - 59644
Right ascension (RA) [hh:mm:ss] | 21:24:43.83534(4) | 21:29:22.78125(3) | 21:45:50.45355(5) | 21:50:27.23531(10) | 22:41:42.039596(4)
Declination (DEC) [dd:mm:ss] | -33:58:45.4778(9) | -57:21:14.3313(3) | -07:50:18.589(2) | -03:26:32.832(4) | -52:36:36.27108(4)
Proper motion in RA [mas $yr^{-1}$] | -13.9(4) | 9.3(2) | -10.3(8) | – | 18.75(3)
Proper motion in DEC [mas $yr^{-1}$] | -49.8(8) | -9.7(3) | -7(2) | – | -5.17(5)
Spin frequency [$s^{-1}$] | 202.793896594889(3) | 268.359230930218(3) | 62.2958887629109(7) | 284.84324670257(1) | 457.3101563491688(8)
Spin frequency derivative [$s^{-2}$] | -8.463(2)$\cdot 10^{-16}$ | -1.5019(2)$\cdot 10^{-15}$ | -1.1563(5)$\cdot 10^{-16}$ | -6.622(9)$\cdot 10^{-16}$ | -1.44219(6)$\cdot 10^{-15}$
DM [$cm^{-3}pc$] | 4.592633 | 31.846624 | 9.002578 | 20.670000 | 11.410342
Parallax [mas] | 3(1) | 0.4232 | 3.4(7) | – | 1.2(2)
Binary model | – | ELL1 | ELL1 | ELL1 | ELL1
Orbital period [d] | – | 6.625492997(2) | 6.8389025093(9) | 4.044550594(3) | 1.4567223817(2)$\cdot 10^{-1}$
Projected semi-major axis [lt-s] | – | 3.5005689(4) | 10.1641111(4) | 3.320713(1) | 0.02579534(6)
Epoch of periastron [MJD] | – | – | – | – | –
Longitude of periastron [deg] | – | – | – | – | –
Eccentricity of orbit | – | – | – | – | –
Epoch of ascending node [MJD] | – | 59042.5329168(1) | 59043.17563464(5) | 59128.4793005(3) | 59041.98680351(6)
First Laplace parameter | – | -0.0000043(2) | -0.00000691(8) | 0.0000033(6) | -0.000004(5)
Second Laplace parameter | – | -0.0000118(2) | -0.00001808(7) | 0.0000112(7) | -0.000003(5)
Time derivative of orbital period | – | – | – | – | –
Rate of advance of periastron | – | – | 0.0242437 | – | –
Rate of change of projected semi-major axis | – | – | – | – | –
Sine of inclination | – | – | – | – | –
Companion mass [$M_{\odot}$] | – | – | – | – | –
Shapiro delay parameter h3 [$\mu s$] | – | – | – | – | –
Shapiro delay parameter h4 [$\mu s$] | – | – | – | – | –