11institutetext:  Institute of Astrophysics, FORTH, N. Plastira 100, 70013,
Heraklion, Greece22institutetext: Max-Planck-Institut für Radioastronomie, Auf
dem Hügel 69, 53121 Bonn, Germany33institutetext: Université Paris Cité, CNRS,
Astroparticule et Cosmologie, 75013 Paris, France44institutetext: Fakultät für
Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld,
Germany55institutetext: ASTRON, Netherlands Institute for Radio Astronomy,
Oude Hoogeveensedijk 4, 7991 PD, Dwingeloo, The Netherlands66institutetext:
Laboratoire de Physique et Chimie de l’Environnement et de l’Espace,
Université d’Orléans / CNRS, 45071 Orléans Cedex 02, France 77institutetext:
Observatoire Radioastronomique de Nançay, Observatoire de Paris, Université
PSL, Université d’Orléans, CNRS, 18330 Nançay, France88institutetext:
Dipartimento di Fisica “G. Occhialini”, Universitá degli Studi di Milano-
Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy99institutetext: INFN,
Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano,
Italy1010institutetext: INAF - Osservatorio Astronomico di Brera, via Brera
20, I-20121 Milano, Italy1111institutetext: Institute for Gravitational Wave
Astronomy and School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK1212institutetext: INAF - Osservatorio
Astronomico di Cagliari, via della Scienza 5, 09047 Selargius (CA),
Italy1313institutetext: Hellenic Open University, School of Science and
Technology, 26335 Patras, Greece1414institutetext: Kavli Institute for
Astronomy and Astrophysics, Peking University, Beijing 100871, P. R.
China1515institutetext: School of Physics, Faculty of Science, University of
East Anglia, Norwich NR4 7TJ, UK1616institutetext: Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), Am Muühlenberg 1, 14476
Potsdam, Germany1717institutetext: Gran Sasso Science Institute (GSSI),
I-67100 L’Aquila, Italy 1818institutetext: INFN, Laboratori Nazionali del Gran
Sasso, I-67100 Assergi, Italy 1919institutetext: Universitá di Cagliari,
Dipartimento di Fisica, S.P. Monserrato-Sestu Km 0,700 - 09042 Monserrato
(CA), Italy2020institutetext: Department of Astrophysics/IMAPP, Radboud
University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The
Netherlands2121institutetext: School of Physics, Trinity College Dublin,
College Green, Dublin 2, D02 PN40, Ireland2222institutetext: Jodrell Bank
Centre for Astrophysics, Department of Physics and Astronomy, University of
Manchester, Manchester M13 9PL, UK2323institutetext: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, P. R.
China2424institutetext: E.A. Milne Centre for Astrophysics, University of
Hull, Cottingham Road, Kingston-upon-Hull, HU6 7RX, UK2525institutetext:
Centre of Excellence for Data Science, Artificial Intelligence and Modelling
(DAIM), University of Hull, Cottingham Road, Kingston-upon-Hull, HU6 7RX,
UK2626institutetext: Arecibo Observatory, HC3 Box 53995, Arecibo, PR 00612,
USA2727institutetext: IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-
Yvette, France 2828institutetext: Institut für Physik und Astronomie,
Universität Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, 14476, Potsdam,
Germany2929institutetext: Ollscoil na Gaillimhe — University of Galway,
University Road, Galway, H91 TK33, Ireland3030institutetext: Laboratoire
Univers et Théories LUTh, Observatoire de Paris, Université PSL, CNRS,
Université de Paris, 92190 Meudon, France3131institutetext: Florida Space
Institute, University of Central Florida, 12354 Research Parkway, Partnership
1 Building, Suite 214, Orlando, 32826-0650, FL, USA3232institutetext: Ruhr
University Bochum, Faculty of Physics and Astronomy, Astronomical Institute
(AIRUB), 44780 Bochum, Germany 3333institutetext: Advanced Institute of
Natural Sciences, Beijing Normal University, Zhuhai 519087, China

# The second data release from the European Pulsar Timing Array

I. The dataset and timing analysis

J. Antoniadis  0000-0003-4453-776 The second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta S.
Babak  0000-0001-7469-4250 The second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta A.-S. Bak Nielsen  0000-0002-1298-9392 The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayThe second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta C. G. Bassa  0000-0002-1429-9010 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta A. Berthereau The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta M. Bonetti  0000-0001-7889-6810 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta E. Bortolas
The second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta
P. R. Brook  0000-0003-3053-6538 The second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta M. Burgay  0000-0002-8265-4344 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta R. N. Caballero
0000-0001-9084-9427 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta A. Chalumeau  0000-0003-2111-1001 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta D. J. Champion
0000-0003-1361-7723 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta S. Chanlaridis  0000-0002-9323-9728 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta S. Chen  0000-0002-3118-5963
The second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta I. Cognard
0000-0002-1775-9692 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta G. Desvignes
0000-0003-3922-4055 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta M. Falxa R. D. Ferdman The second data release from
the European Pulsar Timing ArrayThe second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:eptaThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta A. Franchini  0000-0002-8400-0969 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta J. R. Gair  0000-0002-1671-3668 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta B. Goncharov
0000-0003-3189-5807 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta E. Graikou The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta J.-M. Grießmeier
0000-0003-3362-7996 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta L. Guillemot
0000-0002-9049-8716 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta Y. J. Guo The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta H. Hu  0000-0002-3407-8071
The second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta F. Iraci The
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayLABEL:eptaLABEL:epta D. Izquierdo-Villalba  0000-0002-6143-1491
The second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayLABEL:eptaLABEL:epta J. Jang<EMAIL_ADDRESS>0000-0003-4454-0204 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta J. Jawor  0000-0003-3391-0011 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta G. H. Janssen
0000-0003-3068-3677 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta A. Jessner
0000-0001-6152-9504 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta R. Karuppusamy  0000-0002-5307-2919 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta E. F. Keane
0000-0002-4553-655X The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta M. J. Keith  0000-0001-5567-5492 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta M. Kramer The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta M. A. Krishnakumar  0000-0003-4528-2745 The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayThe second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:eptaLABEL:inptaLABEL:inpta K. Lackeos
0000-0002-6554-3722 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta K. J. Lee 3388The second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayLABEL:eptaLABEL:epta K. Liu<EMAIL_ADDRESS>second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta Y. Liu  0000-0001-9986-9360
The second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayLABEL:eptaLABEL:epta A. G. Lyne The second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayLABEL:eptaLABEL:epta J. W. McKee  0000-0002-2885-8485 The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayThe second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta R. A. Main The second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta M. B. Mickaliger  0000-0001-6798-5682 The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta I. C. Niţu
0000-0003-3611-3464 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta A. Parthasarathy  0000-0002-4140-5616 The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta B. B. P. Perera
0000-0002-8509-5947 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta D. Perrodin  0000-0002-1806-2483 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta A. Petiteau
0000-0002-7371-9695 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta N. K. Porayko The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayThe second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta A. Possenti The second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayLABEL:eptaLABEL:epta H. Quelquejay Leclere The second data release
from the European Pulsar Timing ArrayThe second data release from the European
Pulsar Timing ArrayLABEL:eptaLABEL:epta A. Samajdar  0000-0002-0857-6018 The
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta S. A.
Sanidas The second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta
A. Sesana The second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta G. Shaifullah<EMAIL_ADDRESS>0000-0002-8452-4834 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayThe second data release from the European
Pulsar Timing ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta L. Speri  0000-0002-5442-7267 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta R. Spiewak The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayLABEL:eptaLABEL:epta B. W. Stappers The second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta S. C. Susarla
0000-0003-4332-8201 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta G. Theureau  0000-0002-3649-276X The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta C. Tiburzi
The second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta E. van der
Wateren  0000-0003-0382-8463 The second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta A. Vecchio
0000-0002-6254-1617 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing
ArrayLABEL:eptaLABEL:epta V. Venkatraman Krishnan  0000-0001-9518-9819 The
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta J. P. W.
Verbiest  0000-0002-4088-896X The second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayLABEL:eptaLABEL:epta J. Wang  0000-0003-1933-6498 The second data
release from the European Pulsar Timing ArrayThe second data release from the
European Pulsar Timing ArrayThe second data release from the European Pulsar
Timing ArrayThe second data release from the European Pulsar Timing ArrayThe
second data release from the European Pulsar Timing ArrayThe second data
release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta L. Wang and
Z. Wu The second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayLABEL:eptaLABEL:epta
0000-0002-1381-7859 The second data release from the European Pulsar Timing
ArrayThe second data release from the European Pulsar Timing ArrayThe second
data release from the European Pulsar Timing ArrayThe second data release from
the European Pulsar Timing ArrayLABEL:eptaLABEL:epta

(Received ; accepted )

Pulsar timing arrays offer a probe of the low-frequency gravitational wave
spectrum ($1-100$ nanohertz), which is intimately connected to a number of
markers that can uniquely trace the formation and evolution of the Universe.
We present the dataset and the results of the timing analysis from the second
data release of the European Pulsar Timing Array (EPTA). The dataset contains
high-precision pulsar timing data from 25 millisecond pulsars collected with
the five largest radio telescopes in Europe, as well as the Large European
Array for Pulsars. The dataset forms the foundation for the search for
gravitational waves by the EPTA, presented in associated papers. We describe
the dataset and present the results of the frequentist and Bayesian pulsar
timing analysis for individual millisecond pulsars that have been observed
over the last $\sim$25 years. We discuss the improvements to the individual
pulsar parameter estimates, as well as new measurements of the physical
properties of these pulsars and their companions. This data release extends
the dataset from EPTA Data Release 1 up to the beginning of 2021, with
individual pulsar datasets with timespans ranging from 14 to 25 years. These
lead to improved constraints on annual parallaxes, secular variation of the
orbital period, and Shapiro delay for a number of sources. Based on these
results, we derived astrophysical parameters that include distances,
transverse velocities, binary pulsar masses, and annual orbital parallaxes.

###### Key Words.:

(Stars:) pulsars: general – (Stars:) pulsars – Gravitational waves –
Astronomical instrumentation, methods, and techniques

##  1 Introduction

Pulsar timing arrays (PTAs; Sazhin, 1978; Detweiler, 1979) search for
gravitational waves (GWs) at nanohertz (nHz) frequencies by observing a suite
of stable, rapidly rotating millisecond pulsars (MSPs Backer, 1993) over
decadal timescales (Foster & Backer, 1990; Desvignes et al., 2016; Verbiest et
al., 2016). These GWs may be produced by inspiralling supermassive black holes
binaries (SMBHBs; see e.g. Sesana et al., 2004; Sesana, 2013), cosmic strings
(Damour & Vilenkin, 2001; Grishchuk, 2005; Boyle & Buonanno, 2008), phase
transitions in the early Universe (Schwaller, 2015), quantum fluctuations in
the primordial gravitational field (Grishchuk, 2005; Lasky et al., 2016), or
from a primordial magnetic field (see Caprini & Figueroa, 2018, and references
therein).

Several PTA experiments are currently operational: the European Pulsar Timing
Array (EPTA; Desvignes et al., 2016), the North American Nanohertz Observatory
for Gravitational Waves (NANOGrav; McLaughlin, 2013), the Parkes Pulsar Timing
Array (PPTA; Manchester et al., 2013), the Indian Pulsar Timing Array (Joshi
et al., 2022, InPTA), the MeerKAT Pulsar Timing Array (MPTA) based in South
Africa (Miles et al., 2023), and the Chinese Pulsar Timing Array (CPTA; Lee,
2016). These regional PTAs are also organised under the International Pulsar
Timing Array (IPTA; Verbiest et al., 2016; Perera et al., 2019). Although
individual GW sources remain an exciting prospect for detection with PTAs (see
e.g. Babak et al., 2016; Falxa et al., 2023, and references therein) it is
likely that the first signal detected by PTAs will be a stochastic GW
background (GWB) arising from multiple overlapping sources (Rosado et al.,
2015).

The recent detection of a common red noise process by EPTA, NANOGrav, PPTA and
IPTA (Chen et al., 2021; Alam et al., 2021a; Goncharov et al., 2022;
Antoniadis et al., 2022, respectively) suggests that the characteristic
spatial quadrupole correlation of the GWB (Hellings & Downs, 1983) can become
detectable with a modest expansion of PTA datasets. In this article, we
present the latest EPTA data release, henceforth referred to as EPTA DR2. The
release contains high-precision time-of-arrival (TOA) data for 25 MSPs, as
well as the corresponding timing models. We also make available the full suite
of software libraries required to reproduce these timing models. This EPTA
data release paper is accompanied by two papers (the EPTA and InPTA
Collaborations, 2023a, b), reporting the modelling of stochastic noise
processes present in individual pulsar datasets and the search for GW signals,
respectively.

The article is organised as follows; in Section 2 we introduce the MSP
selection process and provide details on the EPTA observing systems and
observations. In Section 3 we describe the data curation and combination
process, and in Section 4 we present the timing solutions for the 25 pulsars
in our dataset. In Section 5 we discuss the implications of these results. We
conclude with a brief summary in Section 6.

##  2 Observations and data processing

The EPTA uses data from six European radio telescopes: the Effelsberg
$100\text{\,}\mathrm{m}$ radio telescope (EFF) in Germany, the
$76\text{\,}\mathrm{m}$ Lovell telescope at Jodrell Bank Observatory (JBO) in
the United Kingdom, the large radio telescope operated by the Nançay Radio
Observatory (NRT) in France, the $64\text{\,}\mathrm{m}$ Sardinia Radio
Telescope (SRT) operated by the Italian National Institute for Astrophysics
(INAF) through the Astronomical Observatory of Cagliari (OAC), and the
Westerbork Synthesis Radio Telescope (WSRT) operated by ASTRON, the
Netherlands Institute for Radio Astronomy. In addition, these telescopes
regularly operate as the Large European Array for Pulsars (LEAP), which offers
an equivalent diameter of up to $194\text{\,}\mathrm{m}$ (Bassa et al.,
2016b). In this data release, we also incorporated, for the first time, data
from the Mark II telescope at JBO.

The first EPTA data release (henceforth DR1; Desvignes et al., 2016) only
included data from legacy data recording systems. Most of these made use of
the incoherent dedispersion scheme implemented on custom-built hardware,
placing a fundamental limit on achievable timing precision. In EPTA DR2, we
added data from next-generation coherent dedispersion recording systems that
offer a significant increase in bandwidth and sensitivity at each telescope.
These new backends use hardware based on field-programmable gate arrays
(FPGAs) to carry out the conversion of the electrical signal into a digital
data stream and to apply polyphase filterbanks and, in some cases, some pre-
filtering of the recorded band to reject known radio frequency interference
(RFI). For most of the next-generation EPTA recording backends, this data
processing step is implemented on the second generation of the Reconfigurable
Open Architecture Computing Hardware (ROACH) platform, developed by the
Collaboration for Astronomy Signal Processing and Electronics Research
(CASPER; Hickish et al., 2016). In the following sub-sections, we describe the
pulsar recording systems and data processing schemes for each telescope in
more detail (see Table 1 for an overview).

Table 1: Summary of telescopes, backends, and their operating frequencies.
tempo2 flags are constructed from these values using the telescope
abbreviations, the backend names and the central observing frequency for
characterising system noise properties through the ‘group’ flag or the sub-
band centre frequencies for determining phase offsets or JUMPS with ‘system’
flags. For example, for a TOA from an observation made at the Nançay Radio
Telescope with the NUPPI backend, from the sub-band
$1420\text{\,}\mathrm{MHz}$ when the backend was acquiring its full bandwith
around a central frequency of $1854\text{\,}\mathrm{MHz}$, the system flag is
NRT.NUPPI.1420 and the group flag is NRT.NUPPI.1484. A dash-symbol indicates
that no sub-bands were created and the system and group flags are identical.

Telescope (Abbreviation) |  Receiver or Backend | Centre Frequency ($\mathrm{MHz}$) |  Sub-bands ($\mathrm{MHz}$) | Category | Polarisation
---|---|---|---|---|---
Effelsberg 100-m Radio Telescope (EFF) |  EBPP | $1360\text{\,}1410\text{\,}2639\text{\,}$ |  - | Legacy | Full Stokes
P200 | $1380\text{\,}$ |  $1365\text{\,}1425\text{\,}$ | Modern | Full Stokes
P217 | $1380\text{\,}$ |  $1365\text{\,}1425\text{\,}$ | Modern | Full Stokes
S110 | $2487\text{\,}$ |  - | Modern | Full Stokes
S60 | $4857\text{\,}$ |  - | Modern | Full Stokes
Jodrell Bank Observatory (JBO) |  |  |  |  |
\+ Lovell Telescope |  DFB | $1400\text{\,}1520\text{\,}$ |  - | Legacy | Full Stokes
ROACH | $1520\text{\,}$ |  $1420\text{\,}1620\text{\,}$ | Modern | Full Stokes
\+ Mark II |  MK2 | $1520\text{\,}$ |  - | Modern | Full Stokes
Nançay Radio Telescope (NRT) |  BON | $1400\text{\,}1600\text{\,}2000\text{\,}$ |  - | Legacy | Full Stokes
NUPPI | $1484\text{\,}1854\text{\,}2154\text{\,}2539\text{\,}$ |  $1292\text{\,}1420\text{\,}1548\text{\,}1676\text{\,}$; $1662\text{\,}1790\text{\,}1918\text{\,}2046\text{\,}$; $1962\text{\,}2090\text{\,}2218\text{\,}2346\text{\,}$; $2411\text{\,}2667\text{\,}$ | Modern | Full Stokes
Westerbork Synthesis Radio Telescope (WSRT) |  PuMaI | $323\text{\,}328\text{\,}367\text{\,}382\text{\,}840\text{\,}1380\text{\,}2273\text{\,}$ |  - | Legacy | Dual
PuMaII | $350\text{\,}1380\text{\,}2273\text{\,}$ |  - | Modern | Full Stokes
The Large European Array for Pulsars (LEAP) |  LEAP | $1396\text{\,}$ |  - | Modern |
Sardinia Radio Telescope (SRT)${\dagger}$ |  ROACH | $357\text{\,}1396\text{\,}$ |  - | Modern | -

  * ${\dagger}$

Data from SRT were only included as part of the LEAP mode of observations for
this data release.

###  2.1 Effelsberg Radio Telescope

The Effelsberg $100\text{\,}\mathrm{m}$ single-dish radio telescope located
near Bad Munstereifel is maintained and operated by the Max Planck Institut
für Radioastronomie. The Effelsberg Berkeley Pulsar Processor (EBPP; Backer et
al., 1997) was used to record observations up to 2009. A ROACH-based backend
became the main EFF pulsar backend in 2011 (Lazarus et al., 2016). This system
is capable of coherently dedispersing and folding dual-polarisation data
streams over a total bandwidth of up to $\sim$$500\text{\,}\mathrm{MHz}$.

The observations in the Effelsberg Radio Telescope were conducted in $L$, $S$,
and $C$ bands 111See https://eff100mwiki.mpifr-
bonn.mpg.de/doku.php?id=information_for_astronomers:rx:p200mm for a detailed
description of the observing systems with the Effelsberg Radio Telescope..
Depending on the central frequency, different receivers were used, as
summarised in Lazarus et al. (2016). $L-$band observations were performed with
the P200mm and P217mm receivers of the telescope. These offer bandwidths of
$140\text{\,}\mathrm{MHz}$ and $240\text{\,}\mathrm{MHz}$, respectively.

$S-$band observations around $2600\text{\,}\mathrm{MHz}$ were made with the
S110mm receiver. The receiver frequency coverage increased from
$80\text{\,}\mathrm{MHz}$ to $300\text{\,}\mathrm{MHz}$ after the second half
of 2018. $C-$band observations were made with the S60mm receiver, which offers
$500\text{\,}\mathrm{MHz}$ of bandwidth.

The typical integration length per source was $30\text{\,}\min$, which
included a $2\text{\,}\min$ scan of the noise diode for polarisation
calibration. Data reduction of the folded coherently dedispersed pulsar
archives, including flux and polarisation calibrations, was carried out via
the CoastGuard pipeline, which is part of the toaster software library (see
Lazarus et al., 2016, for a detailed description). The removal of RFI was
carried out with the pazi command in psrchive. For each pulsar, we produced a
profile template at each observing band by adding the three observations with
the highest signal-to-noise ratio (S/N). Data taken with the P200mm and P217mm
receivers were analysed with the same template. Each observation was
integrated over frequency, time, and polarisation to produce integrated pulse
profiles. For $L-$band data, observations were divided into two subbands,
except for PSRs J1738+0333, J1843$-$1113 and J2322+2057, which were typically
weak and required integration over the entire band. TOAs were then produced by
cross-correlating the pulse profiles with the respective pulse-profile
templates (see Lazarus et al., 2016, for details).

###  2.2 Lovell Telescope

The Lovell Telescope is located at the JBO, in Cheshire, UK. It is a
$76\text{\,}\mathrm{m}$ ($250\text{\,}\mathrm{f}\mathrm{t}$) parabolic dish on
an altitude-azimuth mount. The telescope is operated by the Jodrell Bank
Centre for Astrophysics at the University of Manchester. Since January 2009,
pulsar data have been processed using a clone of the Digital Filter Bank (DFB)
developed by the Australian National Telescope Facility. From April 2011
onwards, data were simultaneously processed using the DFB and a ROACH board.

Although the DFB initially operated over a bandwidth of
$128\text{\,}\mathrm{MHz}$ centred on $1400\text{\,}\mathrm{MHz}$, from
September 2009 onwards, the observing frequency coverage increased to
$512\text{\,}\mathrm{MHz}$ centred at $1520\text{\,}\mathrm{MHz}$, of which
the central $384\text{\,}\mathrm{MHz}$ were typically used. ROACH observations
covered a $512\text{\,}\mathrm{MHz}$ band centred on
$1532\text{\,}\mathrm{MHz}$. The edges of the band were masked, leaving a
total of $400\text{\,}\mathrm{MHz}$ of usable bandwidth.

The typical integration time per source was varied depending on the pulsar and
epoch, with the median observation times per pulsar ranging between
$\sim$$10\text{\,}\min 55\text{\,}\min$. These observations were time-stamped
using an on-site hydrogen maser clock and then corrected to Coordinated
Universal Time (UTC) using recorded offsets between local time kept by the
hydrogen maser and GPS time. At JBO, the observations were typically not flux
or polarisation calibrated.

For the ROACH backend, data streams from two orthogonal polarisations were
sampled at the Nyquist rate and digitised as 8-bit numbers. The
$512\text{\,}\mathrm{MHz}$ band was split into $16\text{\,}\mathrm{MHz}$ sub-
bands by a 32-channel polyphase filter. The signal in each
$16\text{\,}\mathrm{MHz}$ sub-band was then dedispersed and folded into 1024
bins at the pulse period in real-time. Each of the $16\text{\,}\mathrm{MHz}$
sub-bands was further divided into 64 channels, each
$0.25\text{\,}\mathrm{MHz}$ wide and recorded on disk in
$10\text{\,}\mathrm{s}$ long sub-integrations. The lowest and highest
$56\text{\,}\mathrm{MHz}$ of the band were discarded due to persistent and
known contamination by RFI sources at these frequencies. Therefore, the total
bandwidth recorded on disk was $400\text{\,}\mathrm{MHz}$. For the DFB, the
two orthogonal polarisation data streams, covering $512\text{\,}\mathrm{MHz}$,
were incoherently dedispersed and folded into 1024 bins at the pulse period
using $0.5\text{\,}\mathrm{MHz}$ wide channels and $10\text{\,}\mathrm{s}$
sub-integrations. In this case, the lowest and highest
$64\text{\,}\mathrm{MHz}$ of the total band were discarded due to
contamination by RFI.

The mitigation of RFI in both DFB and ROACH data was carried out using a
median filtering algorithm, which was followed by manual inspection. Since
November 2011, ROACH data have also been cleaned in real-time using a spectral
kurtosis method (Morello et al., 2022). For each pulsar observation, the
pipeline produced archive files with various frequency and time resolutions.
These were fully frequency-averaged archives with full time resolution
($10\text{\,}\mathrm{s}$ long sub-integrations); fully time-averaged profiles
with full frequency resolution ($0.25\text{\,}\mathrm{MHz}$ wide and
$0.50\text{\,}\mathrm{MHz}$ wide channels for the ROACH and DFB,
respectively); archives that were partially averaged in time and frequency
($1\text{\,}\mathrm{min}$ long sub-integrations and
$8\text{\,}\mathrm{MHz}$-wide and $12\text{\,}\mathrm{MHz}$-wide channels for
the ROACH and DFB, respectively) and finally; profiles that were fully
averaged in time and frequency.

TOAs were formed for each observation using the pat command in psrchive (van
Straten et al., 2012), which employs the FDM algorithm (Verbiest et al., 2016)
using the template profile for the appropriate frequency. In the work
presented here, for each epoch, we divided the partially (time-)averaged files
to produce two archives, one for each subband centred on
$1420\text{\,}\mathrm{MHz}1620\text{\,}\mathrm{MHz}$, respectively. These
files were then averaged in frequency and time individually, to finally
produce two TOAs spanning the full observation duration.

###  2.3 Mark II Telescope

The Mark II Telescope is also located at the JBO, in Cheshire, UK. It is an
altitude-azimuth mount instrument, with an elliptical dish with a major axis
of $38.1\text{\,}\mathrm{m}$ ($125\text{\,}\mathrm{f}\mathrm{t}$) and a minor
axis of $25.4\text{\,}\mathrm{m}$ ($83.3\text{\,}\mathrm{f}\mathrm{t}$). From
August 2017 onwards data have also been recorded for PSR J1713+0747 using this
telescope with the ROACH-board-based backend described above. The data were
processed in the same way as for the Lovell Telescope.

###  2.4 Nançay Radio Telescope

Regular EPTA timing observations were conducted with the NRT from late 2004.
These observations were made using the $L-$ and $S-$band receivers of the
telescope, with a frequency coverage of
$1.1\text{\,}\mathrm{GHz}1.8\text{\,}\mathrm{GHz}$, and
$1.7\text{\,}\mathrm{GHz}3.5\text{\,}\mathrm{GHz}$, respectively. From late
2004 until early 2014, the legacy Berkeley-Orléans-Nançay (BON) backend
(Cognard & Theureau, 2006) was used to record the pulsar timing data included
in the EPTA DR1 dataset (for a detailed description, see Desvignes et al.,
2016). Starting in August 2011, the Nançay Ultimate Pulsar Processing
Instrument (NUPPI) became the primary pulsar timing backend (Cognard et al.,
2013; Liu et al., 2020). The EPTA DR2 dataset includes data collected with the
NUPPI backend between August 2011 and early 2021. NUPPI observations have
durations ranging from $20\text{\,}\mathrm{min}80\text{\,}\mathrm{min}$, and
cover a frequency bandwidth of $512\text{\,}\mathrm{MHz}$ channelised into 128
channels that are coherently dedispersed in real-time. Observations with the
$L-$band receiver were made at a central frequency of
$1484\text{\,}\mathrm{MHz}$. Those with the $S-$band receiver were generally
centred on $2539\text{\,}\mathrm{MHz}$ and occasionally on $1854$ and
$2154\text{\,}\mathrm{MHz}$. Data collected before MJD 57924 were time-stamped
using a local rubidium clock, and later corrected to GPS standard time stamped
using recorded offsets between the clock and the Paris Observatory Universal
Time. Data collected after this epoch were directly stamped with the GPS time
standard, as the backend was locked to the GPS signal. Data were calibrated
for polarisation using a short scan on a reference noise diode, conducted
prior to each observation, with the
SingleAxis222http://psrchive.sourceforge.net/manuals/pac/ method of psrchive,
to correct for the differential phase and amplitude between the two
polarisations. Since late 2019, this simple calibration scheme was further
complemented by regular observations of bright polarised pulsars in a mode
where the horn rotated by $\sim$$180\text{\,}\mathrm{\SIUnitSymbolDegree}$
across the $1\text{\,}\mathrm{h}$ observation, enabling a better determination
of the polarimetric response of the NRT at the epoch of the observations.
These observations and the procedure followed for analysing them are described
in Guillemot et al. (2023). Automatic RFI mitigation was performed on
polarisation-averaged archives with the full frequency and time resolution
available (typically, $4\text{\,}\mathrm{MHz}$ and
$5\text{\,}\mathrm{s}60\text{\,}\mathrm{s}$ time resolution), using a Python
script based on the surgical RFI-cleaning algorithm of the CoastGuard software
package (Lazarus et al., 2016). Observations corrupted by strong RFI,
calibration issues, incidental backend faults, or those that contained no
visible pulsar signal, were discarded. We formed template profiles with the
four Stokes parameters and with four frequency sub-bands by integrating the
eight highest S/N observations, smoothing the average using a wavelet
smoothing method. Finally, we used these polarimetric profiles and the matrix
template matching (MTM) technique implemented in pat (van Straten, 2006) to
extract TOAs from the NUPPI observations. By modelling the transformation
between polarised light curves, the MTM method corrects potential polarisation
artefacts (caused by, e.g., calibration issues), yielding more accurate TOAs.
The procedure developed at Nançay to improve the polarisation calibration of
the NUPPI data will be presented in detail in Guillemot et al. (2023).

###  2.5 Sardinia Radio Telescope

The Sardinia Radio Telescope is a 64 m parabolic dish with an altitude-azimuth
mount. The receiver that is used primarily for EPTA observations is the dual-
band $L$/$P$ receiver (original $P-$band =
$305\text{\,}\mathrm{MHz}410\text{\,}\mathrm{MHz}$ and $L$-band = $1.3-1.8$
GHz). Data during the EPTA observations were mainly acquired with a DFB as
well as a ROACH backend, the latter of which is a copy of those installed at
JBO and Effelsberg. Both backends are capable of performing real-time folding
of the incoming data, each with a bandwidth of $500\text{\,}\mathrm{MHz}$.
Additionally, an 8-node CPU cluster installed with ROACH allows the baseband
recording of the full LEAP bandwidth ($128\text{\,}\mathrm{MHz}$). The data
were time-stamped using a local hydrogen maser. EPTA observations at the SRT
started in March 2014 and went on until July 2016, and were resumed from May
2018 onwards. The intermediate gap was due to repairs of the $L$/$P$ receiver,
refurbishment of the active surface of the dish, and relocation of the control
room along with the digital instruments to a permanent structure. Most of
these observations were conducted as part of the LEAP observing programme. For
the work in this paper, these data were directly integrated into the LEAP data
products, rather than being included as independent pulsar timing data (see
Section 2.7).

###  2.6 Westerbork Synthesis Radio Telescope

The WSRT was an equatorial mount linear array consisting of $14$ dishes of
$25\text{\,}\mathrm{m}$ diameter which were coherently added to form a
$93\text{\,}\mathrm{m}$ equivalent dish. Pulsar observations formed a key
science operation and were carried out using the Pulsar Machine (PumaI; Strom,
2002) backend from 1999 until mid-2010. From 2007 to mid-2015, pulsar
observations were performed using the second generation Pulsar Machine
(PuMaII; Karuppusamy et al., 2008) which was a coherent dedispersion backend
capable of simultaneously recording data across a total bandwidth of
$64\text{\,}\mathrm{MHz}128\text{\,}\mathrm{MHz}$. For both PuMa backends, the
WSRT was equipped with the turret-mounted Multi-Frequency FrontEnds (MFFEs;
Tan, 1991), which allowed for rapid changes in observing frequency. From 2013
onwards, the effective size of WSRT for pulsar observations gradually
decreased due to dishes being removed from the array as the infrastructure was
converted to adapt the telescopes for the new phased array feeds; the APERture
Tile In Focus (APERTIF; van Cappellen et al., 2022). To account for the
decrease in sensitivity, the observing time per pulsar was steadily increased
during this transition period, which ended on 15 June 2015 when the WSRT
officially ended its multiband operations.

The PumaI backend was used in conjunction with the MFFEs to observe pulsars at
$328\text{\,}\mathrm{MHz}382\text{\,}\mathrm{MHz}800\text{\,}\mathrm{MHz}1420\text{\,}\mathrm{MHz}2200\text{\,}\mathrm{MHz}$,
with the data recorded in a custom format. TOAs were created from the PuMaI
backend by first converting the observations to ASCII total intensity profiles
and then cross-correlating them with templates as described in Desvignes et
al. (2016). The PuMaII backend operated across a total bandwidth of
$80\text{\,}\mathrm{MHz}$ at $350\text{\,}\mathrm{MHz}$, and
$160\text{\,}\mathrm{MHz}$ at
$1380\text{\,}\mathrm{MHz}2273\text{\,}\mathrm{MHz}$. The total bandwidth was
subdivided into an overlapped polyphase filter bank scheme with individual
channels of $10\text{\,}\mathrm{MHz}20\text{\,}\mathrm{MHz}$ each with an
overlap of $1.25\text{\,}\mathrm{MHz}0\text{\,}\mathrm{MHz}$ at
$350\text{\,}\mathrm{MHz}1380\text{\,}\mathrm{MHz}$, respectively. Besides
multiband operation, PuMaII was also operated in LEAP mode (see Section 2.7)
centred at $1398\text{\,}\mathrm{MHz}$, $16.25\text{\,}\mathrm{MHz}$ of each
$20\text{\,}\mathrm{MHz}$ used to match other telescopes; with a total
bandwidth for timing observations set at $130\text{\,}\mathrm{MHz}$.

The duration of pulsar observations at the WSRT varied with pulsar and epoch
and lasted between $15\text{\,}\min 45\text{\,}\min$. However, at later
epochs, the duration per pulsar was extended to account for the loss of
sensitivity because fewer dishes were available. These observations were time-
stamped to UTC at the observatory using a GPS-referenced local hydrogen maser.
Daily averages of the local maser clock were also compared with other
observatory masers as part of regular, very large baseline interferometry
(VLBI) recordings. The local clock at WSRT was affected by a discrete offset
of $10\text{\,}\mathrm{s}$ between
$\sim$$56\,808\text{\,}\mathrm{M}\mathrm{J}\mathrm{D}56\,903\text{\,}\mathrm{M}\mathrm{J}\mathrm{D}$
due to a failure of the local Network Time Protocol (Mills, 1997) server.

To achieve phase coherence between the dishes, standard interferometric
coherent phasing of the array was performed at the start of each observing
run. By observing an astronomical calibrator, the signal amplitudes for each
polarisation were equalised for all individual antennas. This maximised the
vector sum of each polarised data stream, leading to very stable polarisation
behaviour, especially at $1380\text{\,}\mathrm{MHz}2273\text{\,}\mathrm{MHz}$
which was stable for at least a couple of days. No further calibration (flux
or polarisation) was performed, although the polarisation response was cross-
validated against calibrated data from the other telescopes when the WSRT was
operated in LEAP mode.

The PuMAII backend recorded raw data in
psrdada333http://psrdada.sourceforge.net/ format, which were dedispersed using
DM values stored in TEMPO-style ephemeris files. PuMaII produced psrchive
archives for each band with a final sub-integration length of
$10\text{\,}\mathrm{s}$ and 64 channels of
$10\text{\,}\mathrm{MHz}20\text{\,}\mathrm{MHz}$, depending on the observing
band. The data were Nyquist sampled for each channel at a resolution of
$25\text{\,}\mathrm{ns}$ leading to a variable number of bins for the pulsars;
thus, the fastest pulsars have 256 bins, while the slowest have 8192.

The excision of the RFI in WSRT was performed using a custom tool based on
psrchive, followed by manual inspection. To compensate for decreased amplitude
response at the band edges and obtain continuous frequency coverage, files per
frequency band were split and re-added, providing effective bandwidths of
$70\text{\,}\mathrm{MHz}$ at $350\text{\,}\mathrm{MHz}$,
$160\text{\,}\mathrm{MHz}$ at $1380\text{\,}\mathrm{MHz}$,
$130\text{\,}\mathrm{MHz}$ at $1396\text{\,}\mathrm{MHz}$ (LEAP mode) and
$150\text{\,}\mathrm{MHz}$ at $2200\text{\,}\mathrm{MHz}$.

For each of the band-averaged datasets, individual observations were further
reprocessed for RFI removal using
ITERATIVE_CLEANER444https://github.com/larskuenkel/iterative_cleaner, inspired
by algorithms from the CoastGuard (Lazarus et al., 2016) Pulsar Processing
Suite. Each observation was then averaged over frequency and time to produce
the pulse profiles. Templates for each pulsar were produced by averaging the
smallest number of profiles that contribute $\sim$90% of the S/N of the entire
dataset, and TOAs were generated following the recommendations in Appendix A
of (Verbiest et al., 2016), by using pat with the FDM algorithm.

###  2.7 The Large European Array for Pulsars (LEAP)

LEAP performs simultaneous monthly observations of more than twenty MSPs at
1.4 GHz with the five EPTA telescopes mentioned above (Bassa et al., 2016b).
During each observation, the telescopes switch between the pulsar and a nearby
phase calibrator. The typical exposure time per pair of pulsar and phase
calibrator is $45-60$ and $2-3$ min, respectively. All phase calibrators were
selected from the VLBA calibrator
catalogue555http://www.vlba.nrao.edu/astro/calib/vlbaCalib.txt. During the
observations, baseband data were recorded at the Nyquist rate with 8-bit
sampling and 8$\times$16 MHz subbands. The data were later assembled at JBO
where they were correlated, calibrated for polarisation, and coherently
combined with a dedicated software correlator (Smits et al., 2017), using
Effelsberg as a geometrical and time reference. Narrowband RFI from a
particular telescope was also removed before the data were combined. The
combined baseband data of each pulsar were then coherently dedispersed and
folded to form 10 s sub-integrations with a 1 MHz frequency resolution, using
the dspsr software package. Data were visually inspected to remove any
remaining impulse RFI. Next, for each pulsar, the data from each individual
observation were averaged in time and frequency and used to calculate the TOA
of the integrated profile with the MCMC implementation of the canonical
template matching scheme (Taylor, 1992; Verbiest et al., 2016). The template
was created using the observation with the highest S/N (which was eventually
not included in the timing dataset) using a wavelet smoothing scheme to remove
the radiometer noise on top of the pulsar signal. Most of the processing of
the dedispersed, folded data was carried out with the psrchive software
package.

##  3 Data preparation, combination, and timing analysis

The EPTA DR2 includes tempo2-compatible pulsar ephemerides and TOA files, with
the latter produced as described in the previous section and references
therein. Following the customised processing steps at individual telescopes,
described in Section 2, the final TOA sets were transferred to a central
repository, for which a continuous integration scheme was developed, providing
quick-look plots of the timing residuals for inspection. The data from the
central repository were then combined in parallel, using standard manual steps
as presented in Verbiest et al. (2016); Desvignes et al. (2016) as well as a
semi-automated combination scheme. In both cases, the results were manually
inspected and cross-verified for consistency, with the final data release
containing data from the manual process. In the following paragraphs, we
describe source selection, combination steps, and the resulting combined
dataset in more detail.

###  3.1 Source selection

Due to the large number of telescope and back-end combinations across EPTA
observatories, the heterogeneous recording and processing schemes, and the
complex RFI environment in most telescopes, curation and vetting of EPTA data
require long lead times. Furthermore, the noise modelling for individual
pulsars is computationally expensive, requiring a detailed and iterative
analysis of the possible noise models that may be applicable for each pulsar
dataset. For these reasons, we adopted a source selection scheme which
maximises the detectability of a stochastic GWB through the ${\rm
S/N_{A}^{2}}$ statistic of Rosado et al. (2015); Speri et al. (2023),

$\text{S/N}_{\text{A}}^{2}=2\sum_{a>b}\int\frac{\Gamma^{2}_{ab}\,S^{2}(f)\,T_{ab}}{P_{a}(f)P_{b}(f)}df\,,$
(1)

taking into account the fact that each pulsar contributes differently to the
PTA response, due to its inherently distinct noise properties, $P_{j}(f)$.
Here $\Gamma_{ab}$ represents the overlap reduction function that translates
the mean spectral density of an isotropic stochastic red-noise process,
$S(f)$, to cross-correlation power between pulsars $a$ and $b$ that have been
observed for a common duration of $T_{ab}$. The simple ranking produced
through this scheme was then improved by applying the coupling matrix
formalism introduced in Roebber (2019) as adapted by Speri et al. (2023) to
prioritise pulsars that maximise the response to HD-like correlations while
maintaining the ability to distinguish between competing dipolar and monopolar
signals. Using this methodology, we found that a subset of 25 pulsars out of
the 42 included in the DR1, were sufficient to recover at least
$90\text{\,}\%98\text{\,}\%$ of the full array sensitivity to a simulated
stochastic GW background with an amplitude of $3\times 10^{-15}$ at a
frequency of 1 yr-1, and a spectral index of $\gamma=13/3$. The same subset of
pulsars would also recover at least 95% of the total sensitivity to possible
individually resolvable, monochromatic gravitational wave sources across all
frequencies. These 25 pulsars comprise the EPTA DR2. Their distribution on the
sky can be seen in Figure 1.

Figure 1: Sky projection of the 25 pulsars included in the EPTA DR2 dataset.

###  3.2 Combination of the dataset

We followed the timing and combination steps described in Verbiest et al.
(2016) and Desvignes et al. (2016) to combine the data across telescopes. For
each pulsar, data from different telescopes were combined using tempo2 to form
the joint dataset, starting with parameters from Desvignes et al. (2016) and
using a summary TOA file, following Perera et al. (2019), Verbiest et al.
(2016) and Desvignes et al. (2016). To align the data from different observing
systems, we fitted for an arbitrary phase offset (commonly referred to as
jump) for each sub-band and backend combination, using the NUPPI sub-band
centred at $1420\text{\,}\mathrm{MHz}$ as our reference dataset. For a small
number of individual backend datasets, discrete time offsets were detected and
estimated using multi-pulsar analysis. These were also removed using the
tempo2 TIME keyword. During certain observing runs, data were collected using
both legacy and new backends, or in both single-telescope and LEAP modes. As
these observations represented the same signal and noise, we eliminated the
older backend and non-LEAP data. However, we kept the data outside the LEAP
bands to better constrain dispersion-measure (DM) variations. The overlapping
data were removed after the jump values were determined. With this final set
of curated TOAs, we then produced initial timing solutions for each pulsar
using tempo2. For these solutions, the timing parameters were fitted for
iteratively using tempo2, until the linearised timing solution converged. For
each of the pulsars, we then investigated the likelihood of introducing new
timing parameters that were not fitted for in DR1, using a 5$\sigma$ detection
threshold, as well as a number of $F-$statistic and information criteria based
tests.

All initial timing models include the spin frequency and its derivative, DM
and its first and second derivative, the astrometric parameters (position,
proper motions, and in several cases the annual parallax). For binary pulsars,
we included fits for five Keplerian parameters and a selection of post-
Keplerian (pK) parameters, depending on the pulsar. The full set of parameters
included in each timing model is listed in Tables 7 to 13. We note that we
used equatorial coordinates to fit for the positions of most pulsars, except
for PSRs J0030+0451, J1022+1001 and J1730$-$2304, for which we used ecliptic
coordinates, as their ecliptic latitude is less than $1\text{\,}\deg$. We used
the DE440 version of the JPL solar system ephemeris (Park et al., 2021) and
TT(BIPM2021) (Petit, 2009)666https://www.bipm.org/en/time-ftp/tt-bipm- as our
reference clock standard. We also applied the default spherical of the Solar
Wind electron density model implemented in tempo2 to correct for solar wind-
related DM variations, fixing the average density in the ecliptic plane at
$1\text{\,}\mathrm{ua}$ to $7.9\text{\,}{\mathrm{cm}}^{-3}$(Madison et al.,
2019) following Tiburzi et al. (2021), except for PSR J1022+1001, as described
in Section 3.4.

The combination scheme described above produced the full EPTA DR2 dataset, an
overview of which can be found in Figure 2. This dataset is used in the pulsar
timing analysis presented below, as well as in associated work, namely the
single-pulsar noise modelling in the EPTA and InPTA Collaborations (2023a),
and the search for GWs in the EPTA and InPTA Collaborations (2023b). Based on
the full DR2 dataset, we also produced additional dataset versions for GW
searches. Details for these versions can be found in Appendix A, as well as in
the EPTA and InPTA Collaborations (2023a) and the EPTA and InPTA
Collaborations (2023b).

###  3.3 Outlier analysis

The EPTA DR2 dataset was checked for outliers using the following procedures.
The first step to eliminate outliers was performed when compiling single
telescope data, either by custom automated data flagging or manual inspection.
After initial combination, outstanding outliers, such as TOAs with residuals
offset by more than 10 times the root mean square (rms) of the timing
solutions, were flagged and the observation archives reinspected. We found
that these were typically associated with low S/N and were therefore removed.
Additionally, we removed observations with known calibration issues as well as
those displaying corrupted polarisation profiles or systematic trends in
single-epoch timing residuals. A similar analysis was carried out using the
semi-automated analysis, including tests from expected correlations such as
excess contribution from the Solar Wind or epoch-wise offsets. Finally, we
inspected the whitened residuals using the results of the noise modelling
analysis (see the next subsection), which revealed that all remaining TOAs
were within 5$\sigma$ of the whitened timing residuals, indicating that no
additional outlier removal was needed.

Figure 2: Overview of the full EPTA DR2 dataset. Empty circles denote legacy
data, while filled squares show new EPTA backends. Vertical lines bound the
range of the ‘DR2new’ dataset. Readers can refer to Sec. 3 and A for details.

###  3.4 Bayesian timing analysis

To obtain the final timing solution for each pulsar, we performed a Bayesian
timing analysis on the combined dataset using the temponest toolkit (Lentati
et al., 2014). For this step, the timing measurements obtained with tempo2
were provided as initial guesses. temponest is based on the software packages
tempo2 (Hobbs et al., 2006) and Multinest (Feroz et al., 2009). It explores
the parameter space of a non-linear pulsar timing models using nested sampling
(Skilling, 2004) based on Bayesian inference (Lentati et al., 2014), to
provide robust estimates for the timing parameters.

For the analysis of each pulsar, in addition to the deterministic pulsar
timing parameters, we included a set of stochastic parameters to characterise
the noise components in the dataset. Within our adopted framework, the white
noise is described by a multiplicative factor $E_{\rm f}$ (EFAC) that accounts
for the possible underestimation of TOA uncertainties, and a factor $E_{\rm
q}$ that is added in quadrature (EQUAD) to model any other possible source of
noise, such as pulse phase jitter and systematics (e.g. Liu et al., 2011,
2012). These are related to the uncertainty of the TOA measurement,
$\sigma_{\rm r}$, as follows:

$\sigma=\sqrt{E^{2}_{\rm q}+E^{2}_{\rm f}\sigma^{2}_{\rm r}}.$ (2)

Such pairs of noise parameters were introduced for each back-end/frontent
combination. These combinations correspond to different -group flags in the
TOA files (see Table 1). The long-term red-noise processes in the data were
described by two types of models that account for both the chromatic and
achromatic noise components, respectively. In both cases, the noise components
were modelled as a stationary, stochastic process with a power-law spectrum
given by:

$S(f)=\frac{A^{2}}{C_{\rm 0}}\left(\frac{f}{f_{\rm r}}\right)^{-\gamma},$ (3)

where $S(f)$, $A$, $\gamma$ and $f_{\rm r}$ correspond to the spectral density
of the power as a function of frequency $f$, the spectral amplitude, the
spectral index, and the reference frequency (set at
$1\text{\,}{\mathrm{yr}}^{-1}$), respectively. The spectrum in both cases has
a low-frequency cutoff, determined by the inverse of the total time span of
the dataset. The constant $C_{\rm 0}$ was set to $1$ for the chromatic noise
component and to $12\pi^{2}$ for the achromatic term. The spectral amplitude
of the chromatic noise component is proportional to the observing radio
frequency as $A\propto\nu^{-\alpha}$, where the chromatic index $\alpha$ is
either $2$ for modelling the features induced by the DM variations, or $4$ for
capturing scattering variation in the data. For each pulsar, we used a
customised noise model obtained by following the procedures outlined in
Chalumeau et al. (2022), which employs a Bayesian model-selection framework to
determine the components in the noise model and the number of frequency bins
used to sample the spectrum of each component. For 21 pulsars, the noise model
includes a chromatic component to model DM variation, while for 10 pulsars it
includes an achromatic component. Only one pulsar (PSR J1600$-$3053) favours
the inclusion of a chromatic component for modelling scattering variation.
J2322+2057 is the only pulsar in our sample that does not show any evidence
for the presence of red noise. The noise components for each pulsar are
summarised in the tables of Appendix B. More details of our customised noise
models can be found in the accompanying paper on noise-modelling in this
series (the EPTA and InPTA Collaborations, 2023a).

During the analysis with temponest, both timing and noise parameters were
sampled simultaneously, while the time offsets were analytically marginalised.
All model parameters were sampled with uniform priors, except for EQUAD and
the amplitude of the red-noise processes, which used a log-uniform prior. For
PSR J1022+1001, whose line of sight lies very close to the ecliptic plane, we
also fitted the solar wind density at the orbit distance of the Earth, instead
of fixing its value to $7.9\text{\,}{\mathrm{cm}}^{-3}$ as was done for the
other pulsars. This helped to obtain an estimate of the timing parallax
consistent with other pulsar timing and VLBI measurements (see Section 4.3 for
more details).

##  4 Results

Table LABEL:tab:source provides a summary of the EPTA DR2 dataset, while the
full timing solutions are listed in Tables 7 through 13. Our timeseries have
time spans of 13.6 to 24.5 years and contain $\sim$800 to $\sim$6000 TOAs.
With the exception of PSR J1909$-$3744, all timing solutions use data from at
least two telescopes. In the following, we describe some new timing
measurements in DR2 and compare them with the results by Desvignes et al.
(2016); Perera et al. (2019); Alam et al. (2020); Reardon et al. (2021);
henceforth referred to as EPTA DR1, IPTA DR2, NG12 and PPTA DR2, respectively.
Unless otherwise noted, quoted values and uncertainties represent the median
and 68.3% confidence intervals (C.I.) of the one-dimensional marginalised
posterior distribution function.

Table 2: Overview of the EPTA DR2 25 pulsar data set. The columns represent the name of the pulsar, the telescopes that have collected the data, the number of TOAs, the timespan $T_{\rm span}$ of the data set, the median $\sigma_{\rm TOA}$ in each frequency band, the weighted rms (wrms) of the timing residuals, and the weighted rms after subtracting both red and DM noise (i.e. of the whitened residuals). The corresponding frequency coverage for the P, L, S, and C bands is 0.3–1.0, 1.0–2.0, 2.0–4.0, and 4.0–8.0 GHz, respectively. The asterisk next to the pulsar name denotes that this pulsar is also included in the InPTA data release (also see Appendix A).  Pulsar Jname | Telescopes | MJD range | $T_{\rm span}$ | $N_{\rm TOA}$ | Median $\sigma_{\rm TOA}$ ($\mu$s) | wrms | wrms,
---|---|---|---|---|---|---|---
(yr) | P | L | S | C | ($\mu$s) | whitened ($\mu$s)
J0030+0451 | EFF, LT, NRT, WSRT, LEAP | 51275–59294 | 22.0 | 4069 | — | 3.40 | 6.07 | — | 2.85 | 2.30
J0613$-$0200∗ | EFF, LT, NRT, WSRT, LEAP | 50931–59293 | 22.9 | 2909 | 7.40 | 1.43 | 6.16 | — | 2.47 | 1.06
J0751+1807∗ | EFF, LT, NRT, WSRT, LEAP | 50460–59294 | 24.2 | 3613 | — | 2.22 | 4.43 | — | 2.12 | 1.50
J0900$-$3144 | LT, NRT | 54286–59269 | 13.6 | 6064 | — | 2.95 | 8.92 | — | 4.28 | 2.60
J1012+5307∗ | EFF, LT, NRT, WSRT, LEAP | 50647–59295 | 23.7 | 5325 | 3.77 | 1.76 | 5.59 | 4.78 | 1.28 | 1.02
J1022+1001∗ | EFF, LT, NRT, WSRT, LEAP | 50361–59294 | 24.5 | 2445 | 11.3 | 2.19 | 4.42 | 2.99 | 1.78 | 1.56
J1024$-$0719 | EFF, LT, NRT, WSRT, LEAP | 50841–59294 | 23.1 | 2522 | — | 2.38 | 7.64 | — | 1.10 | 1.02
J1455$-$3330 | LT, NRT | 53375–59117 | 15.7 | 2815 | — | 7.23 | 17.8 | — | 2.52 | 2.46
J1600$-$3053∗ | EFF, LT, NRT, WSRT, LEAP | 53998–59230 | 14.3 | 2982 | — | 0.48 | 1.50 | — | 2.68 | 0.37
J1640+2224 | EFF, LT, NRT, WSRT, LEAP | 50459–59385 | 24.4 | 2006 | — | 3.57 | 7.65 | — | 1.13 | 1.10
J1713+0747∗ | EFF, LT, NRT, WSRT, LEAP | 50360–59295 | 24.5 | 5003 | 3.13 | 0.32 | 0.54 | 0.67 | 0.43 | 0.20
J1730$-$2304 | EFF, LT, NRT | 53397–59279 | 16.1 | 1315 | — | 1.57 | 7.26 | — | 1.00 | 0.83
J1738+0333 | EFF, LT, NRT, WSRT | 54103–59259 | 14.1 | 1019 | — | 4.31 | — | — | 2.90 | 2.33
J1744$-$1134∗ | EFF, LT, NRT, WSRT, LEAP | 50460–59230 | 24.0 | 1946 | 3.60 | 0.90 | 2.29 | 1.15 | 1.01 | 0.56
J1751$-$2857 | LT, NRT | 53746–59111 | 14.7 | 398 | — | 3.17 | — | — | 3.73 | 2.34
J1801$-$1417 | LT, NRT | 54206–59214 | 13.7 | 449 | — | 4.09 | — | — | 3.94 | 2.46
J1804$-$2717 | LT, NRT | 53766–59145 | 14.7 | 723 | — | 5.94 | — | — | 1.80 | 1.63
J1843$-$1113 | EFF, LT, NRT, WSRT | 53156–59293 | 16.8 | 893 | — | 1.37 | 2.64 | — | 3.47 | 0.81
J1857+0943∗ | EFF, LT, NRT, WSRT, LEAP | 50458–59258 | 24.1 | 1540 | — | 1.70 | 4.05 | 6.85 | 1.38 | 1.05
J1909$-$3744∗ | NRT | 53368–59115 | 15.7 | 2503 | — | 0.29 | 0.57 | — | 0.73 | 0.14
J1910+1256 | LT, NRT | 53725–59282 | 15.2 | 538 | — | 2.65 | — | — | 2.03 | 1.77
J1911+1347 | EFF, LT, NRT | 54095–59282 | 14.2 | 882 | — | 1.22 | 1.36 | — | 1.06 | 0.75
J1918$-$0642 | EFF, LT, NRT, WSRT, LEAP | 52095–59294 | 19.7 | 1361 | — | 2.04 | 4.14 | — | 1.78 | 1.31
J2124$-$3358∗ | LT, NRT | 53365–59213 | 16.0 | 2018 | — | 3.70 | 12.6 | — | 2.24 | 2.17
J2322+2057 | EFF, LT, NRT | 53905–59268 | 14.7 | 804 | — | 10.9 | — | — | 4.08 | 4.08

For 12 of the 25 pulsars in DR2, thanks to the extended data and increased
sensitivity of the next-generation receivers, we measure several new timing
parameters in comparison to DR1. Some of these measurements are also reported
here for the first time. These results are discussed in detail in the
following subsections. In addition, DR2 includes improved timing solutions for
the binary systems J0751$+$1807, J0900$-$3144, J1024$-$0719, J1455$-$3330,
J1713$+$0747, J1857$+$0943, J1909$-$3744 and J1910$+$1256, as well as for the
isolated pulsars J0030$+$0451, J1744$-$1134, J1843$-$1113, J2124$-$3358, and
J2322$+$2057, even though no additional timing parameters were measured
compared to EPTA DR1. For PSR J1024$-$0719, where we only fit the second spin
frequency derivative as in previous work (Bassa et al., 2016c), we performed
additional investigations to explore the measurability of higher order spin
frequency derivatives (see Section 5.5). For PSR J1857$+$0943, we omitted the
fit for secular variation of the orbital projected semi-major axis ($\dot{x}$)
which was poorly constrained from the timing analysis. Meanwhile, a rough
estimate of this parameter may still be inferred by searching for the annual
orbital parallax (Section 5.4).

###  4.1 PSR J0613$-$0200

PSR J0613$-$0200 is a binary pulsar with a white dwarf (WD) companion (Bassa
et al., 2016a). The new timing solution includes three pK parameters, namely
the secular change of the orbital period, $\dot{P}_{\rm b}$, the third
harmonic of the Shapiro delay (SD), $h_{3}$, and the ratio between the third
and fourth SD harmonics, $\zeta$ (Freire & Wex, 2010). The uncertainty of the
measurement on $\dot{P}_{\rm b}$ has improved by a factor of three compared to
EPTA DR1, while estimates for the SD parameters with EPTA data are reported
here for the first time. The astrometric solution includes a significant
measurement of the timing parallax. These measurements agree with the
constraints in NG12 and PPTA DR2, but are in slight tension with previous EPTA
estimates in DR1. The results are summarised in Table 7.

###  4.2 PSR J1012$+$5307

PSR J1012$+$5307 is a 1.8 M⊙ pulsar in a 14.4 h orbit around a low-mass WD
companion (van Kerkwijk et al., 1996; Lazaridis et al., 2009; Antoniadis et
al., 2016; Ding et al., 2020; Mata Sánchez et al., 2020). The DR2 dataset for
this pulsar now includes more than 5000 TOAs with an average precision of
$\sim$5 $\mathrm{\mu s}$. Previous mass estimates (Antoniadis et al., 2016)
for the system indicate that the expected SD amplitude is now larger than the
EPTA timing precision for the pulsar (5 $\mathrm{\mu s}\times
5000^{-1/2}\simeq 0.06$ $\mathrm{\mu s}$). This prompted us to include the
orthometric SD parameters in our fit, resulting in a significant detection of
$h_{3}$ and $h_{4}$ for the first time (see Table 8). The updated timing
solution also includes a highly significant measurement of $\dot{P}_{\rm b}$
and an improved estimate of the parallax, which is consistent with previous
timing estimates (Lazaridis et al., 2009; Desvignes et al., 2016; Perera et
al., 2019), but in tension with the estimates from VLBI and Gaia (Ding et al.,
2020; Antoniadis, 2020, 2021). Possible reasons for this discrepancy may be
related to systematics between dynamical and kinematic reference frames (Liu
et al., 2023), or covariance between the parallax, DM variability, and solar-
wind timing signatures. These will be investigated in a future publication.

###  4.3 PSR J1022$+$1001

In the EPTA DR1 dataset, the only measurable pK parameter for this system was
the advance of periastron, $\dot{\omega}$. The updated estimate for this
parameter is in agreement with the previous value, with an uncertainty
improved by five times. The new solution includes three additional pK
parameters, namely the two SD parameters and $\dot{P}_{\rm b}$, all
constrained with a significance greater than $3\sigma$ (see Table 8). In
particular, $\dot{P}_{\rm b}$ is measured for the first time in this system.
Both the SD and solar wind amplitude measurements are in agreement with those
reported in Reardon et al. (2021). The updated timing parallax is slightly
lower than the VLBI estimate for the system (Deller et al., 2019).

###  4.4 PSR J1455$-$3330

This MSP is in a 76 d binary system with a WD companion. The measured
$\dot{x}$ is $-1.98(6)\times 10^{-14}$ ls s-1 (see Table 8), which is in
agreement with the NG12 result. In addition, we report a tentative measurement
of $\dot{P}_{\rm b}$ for the first time. Further timing analysis may lead to a
clear detection of this parameter. The measured parallax agrees with IPTA DR2
and EPTA DR1.

###  4.5 PSR J1600$-$3053

PSR J1600$-$3053 is a binary MSP in orbit with a WD companion of an orbital
period of 14.3 day. Using the EPTA DR2 data, we obtain a measurement of the pK
parameter $\dot{P}_{\rm b}$ (see Table 9) for the first time. The pK
parameters measured in this system, including $\dot{P}_{\rm b}$, $\dot{x}$,
$\dot{\omega}$ and SD, are all consistent with those reported by NG12 and PPTA
DR2.

###  4.6 PSR J1640$+$2224

PSR J1640$+$2224 is an MSP with a WD companion in a 175-day orbit (Deng et
al., 2020). The new timing solution was derived using the DDH model in tempo2
and includes significant measurements of the timing parallax, SD parameters,
and the first derivative of the orbital period. The SD parameters suggest that
the pulsar has a low mass, although the measurement uncertainties are still
too large to provide a stringent constraint. The parallax estimate is
consistent with that derived with VLBI and the timing solutions by other PTAs
(see Section 5.1). The secular orbital period change can be attributed to
kinematic effects, although at face value our estimate is in slight tension
with expectations, given the position and proper motion of the pulsar (see
Section 5.3). The uncertainty for the $\dot{x}$ estimate is improved by three
(five) times compared to EPTA DR1 (IPTA DR2). The timing parameters are listed
in Table 9.

###  4.7 PSR J1730$-$2304

The proper motion along the ecliptic latitude was not measured in either EPTA
DR1, IPTA DR2 or PPTA DR2, due to the proximity of this isolated pulsar to the
ecliptic plane. Here, we report the first tentative measurement of this
parameter, $-4.4(1.8)$ mas yr-1. The revised timing solution is explained in
Table 9.

###  4.8 PSR J1738$+$0333

A pulsar timing analysis of PSR J1738$+$0333 based on data collected with the
Arecibo and EPTA telescopes resulted in a significant detection of the timing
parallax and first orbital period derivative (Freire et al., 2012). The latter
is consistent with the general relativity prediction given the masses of the
pulsar and its companion, which have been measured using optical spectroscopy
(Antoniadis et al., 2012). Here, we report for the first time a $4\sigma$
measurement of $\dot{P}_{\rm b}=-3.0(7)\times 10^{-14}$, using EPTA-only data.
This estimate is approximately $2\sigma$ higher than that of Freire et al.
(2012) ($-1.7(3)\times 10^{14}$), which could be due to systematics introduced
by the long integration times typically used in EPTA observations ($\sim
0.5-1$ hour). The latter correspond to a significant fraction of the 7.5 h
binary orbit and may result in smearing of the $\dot{P}_{\rm b}$ timing
signature.

###  4.9 PSR J1751$-$2857

This 110-day period binary pulsar (Stairs et al., 2005) is only monitored by
the EPTA. Compared to EPTA DR1, the new solution (Table 10) provides
significantly improved estimates for all timing parameters. We also report a
marginal detection of the timing parallax; $\varpi=1.1(4)$ mas.

###  4.10 PSR J1801$-$1417

PSR J1801$-$1417 is an isolated pulsar and only monitored by the EPTA. Here,
we report a marginal detection of the timing parallax signature at
approximately $3\sigma$ (see Tables 10 and 3).

###  4.11 PSR J1804$-$2717

PSR J1804$-$2717 is a binary MSP with a WD companion of an orbital period of
11.1 day. It is currently only monitored by the EPTA. The new timing solution
includes a significant measurement of the timing parallax, as well as proper
motion parameters that are seven times more precise compared to the EPTA DR1
solution (see Tables 3 and 11).

###  4.12 PSR J1911$+$1347

PSR J1911$+$1347 is a 4.63-ms isolated MSP. The astrometric solution for this
pulsar now includes a parallax measurement (see Table 3. This is also the
first parallax measurement reported in this pulsar. The new timing solution is
consistent with EPTA DR1 and IPTA DR2, with a significant improvement in the
measurement precision of all timing parameters.

###  4.13 PSR J1918$-$0642

We report an improved measurement of the SD signature, with
$h_{3}=8.3(2)\times 10^{-7}$ $\mu$s, and $\zeta=0.908(9)$, which agrees well
with previous estimates. Compared to EPTA DR1, the timing solution now also
includes a measured timing parallax (as seen in Table 12) that is consistent
with the value reported in NG12, as well as the VLBI parallax reported by Ding
et al. (2023), as can be seen in Table 3.

##  5 Discussion

###  5.1 Parallaxes and distances

Table 3: Summary of the timing parallax measurements in the EPTA DR2. The $\varpi_{\rm mes}$ column gives the median and 68.3 percentiles of the marginalised posteriod distribution. $S_{1400}$ gives the pulsar flux densities that were used to infer the bias-corrected distances, $D_{\rm corr}$ (all taken from the ATNF catalogue v1.69 Hobbs et al. (2004); see Section 5.1 for details on parallax inversion). The last two columns list the magnitude of the proper motion and the inferred transverse velocity ($V_{\rm trans}=4.74\mu D_{\rm corr}$) of the system, respectively. Pulsar name | $\varpi_{\rm mes}$ | $S_{1400}$ | $D_{\rm corr}$ | $\varpi_{\rm ref}$ | Reference | $\mu$ | $V_{\rm trans}$
---|---|---|---|---|---|---|---
| (mas) | (mJy) | (kpc) | (mas) |  | (mas yr-1) | (km s-1)
J0030$+$0451 | 3.09 $\pm$ 0.06 | 1.09 | 0.323${}_{-0.006}^{+0.006}$ | 3.02${}^{+0.07}_{-0.07}$ | Ding et al. (2023) | $6.33_{-0.08}^{+0.09}$ | 9.7${}_{-0.2}^{+0.2}$
J0613$-$0200 | 1.00 $\pm$ 0.05 | 2.25 | 0.99${}_{-0.05}^{+0.05}$ | 1.25${}^{+0.13}_{-0.13}$ | DR1 | $10.521_{-0.005}^{+0.005}$ | 50${}_{-3}^{+3}$
J0751$+$1807 | 0.85 $\pm$ 0.04 | 1.35 | 1.17${}_{-0.05}^{+0.06}$ | 0.82${}^{+0.17}_{-0.17}$ | DR1 | $13.543_{-0.071}^{+0.068}$ | 75.1${}_{-3.5}^{+3.6}$
J1012$+$5307 | 0.90 $\pm$ 0.08 | 3.8 | 1.07${}_{-0.08}^{+0.10}$ | 1.17${}^{+0.04}_{-0.05}$ | Ding et al. (2023) | $25.622_{-0.004}^{+0.004}$ | 131${}_{-11}^{+11}$
J1022$+$1001 | 1.16 $\pm$ 0.08 | 3.9 | 0.85${}_{-0.05}^{+0.06}$ | 1.39${}^{+0.04}_{-0.03}$ | Deller et al. (2019) | $23_{-3}^{+3}$ | 94${}_{-13}^{+15}$
J1024$-$0719 | 1.01 $\pm$ 0.04 | 1.5 | 0.98${}_{-0.04}^{+0.04}$ | 0.94${}^{+0.06}_{-0.06}$ | Ding et al. (2023) | $59.75_{-0.01}^{+0.01}$ | 279${}_{-11}^{+11}$
J1455$-$3330 | 1.3 $\pm$ 0.1 | 0.73 | 0.76${}_{-0.05}^{+0.06}$ | 1.04${}^{+0.35}_{-0.35}$ | DR1 | $8.097_{-0.010}^{+0.011}$ | 29${}_{-2}^{+2}$
J1600$-$3053 | 0.72 $\pm$ 0.02 | 2.44 | 1.39${}_{-0.04}^{+0.04}$ | 0.53${}^{+0.06}_{-0.06}$ | Reardon et al. (2021) | $6.984_{-0.010}^{+0.011}$ | 45.9${}_{-1.3}^{+1.3}$
J1640$+$2224 | 0.8 $\pm$ 0.2 | 0.46 | 1.08${}_{-0.19}^{+0.28}$ | 0.68${}^{+0.08}_{-0.08}$ | Ding et al. (2023) | $11.526_{-0.007}^{+0.006}$ | 62${}_{-12}^{+14}$
J1713$+$0747 | 0.88 $\pm$ 0.01 | 8.3 | 1.136${}_{-0.013}^{+0.013}$ | 0.90${}^{+0.03}_{-0.03}$ | DR1 | $6.292_{-0.001}^{+0.001}$ | 33.9${}_{-0.4}^{+0.4}$
J1730$-$2304 | 2.08 $\pm$ 0.06 | 4.0 | 0.48${}_{-0.01}^{+0.01}$ | 1.57${}^{+0.18}_{-0.18}$ | Ding et al. (2023) | $20.7_{-0.3}^{+0.5}$ | 47.3${}_{-1.6}^{+1.7}$
J1738+0333 | — | 0.34 | — | 0.68${}^{+0.05}_{-0.05}$ | Freire et al. (2012) | $8.713_{-0.023}^{+0.023}$ | 60.129${}_{-4.264}^{+4.494}$
J1744$-$1134 | 2.58 $\pm$ 0.03 | 2.6 | 0.388${}_{-0.004}^{+0.005}$ | 2.38${}^{+0.08}_{-0.08}$ | DR1 | $21.018_{-0.004}^{+0.004}$ | 38.6${}_{-0.4}^{+0.5}$
J1751$-$2857 | 1.1 $\pm$ 0.4 | 0.46 | 0.79${}_{-0.21}^{+0.43}$ | — | — | $8.67_{-0.19}^{+0.20}$ | 38${}_{-12}^{+15}$
J1801$-$1417 | 0.8 $\pm$ 0.3 | 1.54 | 1.00${}_{-0.25}^{+0.46}$ | — | — | $11.01_{-0.06}^{+0.06}$ | 59${}_{-17}^{+21}$
J1804$-$2717 | 1.1 $\pm$ 0.3 | 0.4 | 0.8${}_{-0.2}^{+0.3}$ | — | — | $17.1_{-0.4}^{+0.4}$ | 74${}_{-18}^{+23}$
J1857$+$0943 | 0.89 $\pm$ 0.06 | 5.0 | 1.11${}_{-0.07}^{+0.08}$ | 0.70${}^{+0.26}_{-0.26}$ | DR1 | $6.050_{-0.005}^{+0.005}$ | 32${}_{-2}^{+2}$
J1909$-$3744 | 0.94 $\pm$ 0.02 | 1.8 | 1.06${}_{-0.02}^{+0.02}$ | 0.86${}^{+0.01}_{-0.01}$ | Liu et al. (2020) | $37.026_{-0.004}^{+0.004}$ | 186.6${}_{-3.9}^{+4.0}$
J1911$+$1347 | 0.40 $\pm$ 0.09 | 0.9 | 2.2${}_{-0.4}^{+0.6}$ | — | — | $4.683_{-0.007}^{+0.007}$ | 52.2${}_{-9.9}^{+11.6}$
J1918$-$0642 | 0.75 $\pm$ 0.07 | 0.58 | 1.3${}_{-0.1}^{+0.1}$ | 0.6${}^{+0.1}_{-0.1}$ | Ding et al. (2023) | $9.29_{-0.01}^{+0.01}$ | 58.2${}_{-5.1}^{+5.6}$
J2124$-$3358 | 2.1 $\pm$ 0.1 | 4.5 | 0.47${}_{-0.02}^{+0.02}$ | 2.50${}^{+0.36}_{-0.36}$ | DR1 | $52.25_{-0.02}^{+0.02}$ | 117.6${}_{-5.5}^{+5.6}$

We report timing parallaxes for 20 of the systems contained in this data
release. For PSRs J1640$+$2224, J1751$-$2857, J1801$-$1417, J1804$-$2717,
J1911$+$1347, and J1918$-$0642 the parallax was not measured in EPTA DR1,
while for PSRs J0900$-$3144, J1738$+$0333, J1843$-$1113, J1910$+$1256,
J2322$+$2057, we still do not detect a parallax signature. In what follows we
briefly compare our new results with EPTA DR1 and the estimates obtained by
NANOGrav and VLBI (Vigeland et al., 2018; Deller et al., 2019; Alam et al.,
2021a; Ding et al., 2023).

For sources monitored by both EPTA and NANOGrav, we find that the parallax
estimates are in agreement, with the EPTA generally being more precise, due to
the improved sampling resulting from the larger number of observations. Unlike
NG12, we do not find a significant parallax signature for PSR J2322$+$2057.

To obtain distance estimates we invert the parallaxes using Lutz-Kelker-like
volume and luminosity priors (Lutz & Kelker, 1973; Binney & Merrifield, 1998),
as described in Verbiest et al. (2010, 2012); Igoshev et al. (2016)777The code
to correct for the Lutz-Kelker bias is available here:
http://psrpop.phys.wvu.edu/LKbias/. More specifically we use a volumetric
prior given by,

$P_{\rm
D}(\varpi)\propto\frac{R^{1.9}}{\varpi^{4}}\exp\left({-\frac{|\sin{b}|}{\varpi
E}-5\frac{R-R_{0}}{R_{0}}}\right),$ (4)

where $E$ is the scale height, assumed to be $500\text{\,}\mathrm{pc}$ for
MSPs, $b$ is the Galactic latitude of the source, $R_{0}$ is the distance of
the sun from the galactic centre ($8.5\text{\,}\mathrm{kpc}$), and $R$ is the
distance between the source and the galactic centre as,

$R=\sqrt{R_{0}^{2}+\frac{\cos{b}}{\varpi}-2R_{0}\frac{\cos{b}\cos{l}}{\varpi}},$
(5)

where $l$ is the Galactic longitude of the source. For the luminosity prior we
adopt,

$P_{\rm
L}(\varpi)\propto\frac{1}{\varpi}\exp\left(-0.5\left[\frac{\log{S_{1400}}+1.1-2\log{\varpi}}{0.9}\right]^{2}\right),$
(6)

where $S_{1400}$ is the mean flux density of the pulsar at 1400 MHz. The use
of these priors ensures that the posterior distribution function of the
distance is well behaved, so that statistical moments can be defined even in
the presence of large uncertainties.

A summary of the distance estimates obtained with the method described above
is given in Table 3. Four MSPs in our samples, PSRs J1640+2224, J1751$-$2857,
J1801$-$1417 and J1804$-$2717, have a poorly constrained parallax and thus the
corresponding distance estimates depend sensitively on the prior assumptions
and are highly asymmetric. Transverse velocities, calculated using the bias-
corrected distances, and the measured proper motions, are also listed in Table
3. The proper motions derived here are in most cases consistent with VLBI
estimates, but have approximately three times smaller uncertainties Deller et
al. (2019). For PSR J1022+1001, where numerous authors have reported
inconsistent parallax determinations from pulsar timing and VLBI campaigns, we
find a strong correlation between the parallax and the adopted model for the
solar wind electron density. This will be explored in detail by a future work
(Liu et al. in prep.).

###  5.2 Pulsar mass measurements

We detect the SD signature with a significance greater than $3\sigma$ in nine
systems. Figure 3 shows the marginalised posterior probability functions for
the masses of the corresponding pulsars. These estimates are based solely on
SD posteriors and do not take into account all available information, such as
additional pK parameters or independent measurements from optical
spectroscopy. A more robust analysis of the astrophysical parameters for EPTA
DR2 pulsars will be presented in a future publication.

We find that the mass estimates derived from the SD posteriors are generally
consistent with the measurements obtained with previous iterations of the EPTA
data, as well as those derived with independent datasets (e.g. see Desvignes
et al., 2016; Verbiest et al., 2016; Perera et al., 2018, 2019; Liu et al.,
2020; Reardon et al., 2021; Alam et al., 2021b). In most cases, the EPTA DR2
constraints are significantly more precise.

An SD detection for PSRs 1012+5307 and J1022+1001 is reported for the first
time. For the former system, the SD parameter posteriors are not yet
constrained with sufficient precision to provide informative constraints on
the pulsar mass. If one considers the mass ratio of the system, $q\equiv
m_{\rm p}/m_{\rm c}=10.44(11)$ (Mata Sánchez et al., 2020), and discards the
SD of posterior samples that correspond to physically meaningless values for
the companion mass (e.g. $m_{\rm c}/{\mathrm{M}_{\odot}}\lesssim 0.14$ or
$\gtrsim 0.3)$, the pulsar mass is constrained to be in the $1.7-2.0$ M⊙
range. As the precision of $h_{3}$ and $h_{4}$ improves with more data, the SD
signature will ultimately provide an independent measurement of the component
masses. Combined with the parallax, $\dot{P}_{\rm b}$, and WD spectroscopic
estimates, this will significantly improve the constraints on gravitational
dipole radiation, also providing a valuable test for low-mass WD cooling
models.

Figure 3: Posterior probability distributions for $m_{\rm p}$, for the 9
systems with significant SD measurements. Solid and dashed lines represent the
median and 68.3% C.I. of the marginalised posterior distribution function. In
each panel, the inset provides a zoomed view around the peak of the
distribution.

For PSR J1022+1001, the posterior distribution of the SD parameters suggests
that the pulsar mass is relatively low. For the corresponding constraints on
the component masses, the expected magnitude for the relativistic periastron
advance is $\dot{\omega}_{\rm
GR}\simeq$$0.01\text{\,}\mathrm{\SIUnitSymbolDegree}\text{\,}{\mathrm{yr}}^{-1}$,
which is $\sim$2.5$\sigma$ higher than the measured value for this parameter.
This can be explained either by an extremely low pulsar mass ($\sim$1 M⊙), or
by systematics and non-relativistic contributions to the measured
$\dot{\omega}$.

###  5.3 Secular variation of the orbital period

Table 4: Measured orbital period change and the expected contribution from extrinsic effects. Columns $25$ list the estimates for the Shklovskii effect, the contribution due to Galactic acceleration, the intrinsic change in the orbital period, and that due to GW damping. The final column shows the calculated kinematic distance. The superscripts indicate special cases where we assumed or cited measurements from literature: ${a}$ \- Mass and parallax taken from literature; ${*}$ \- Pulsar mass is assumed to be 1.4 M⊙; ${{\dagger}}$ \- Parallax value used to determine external orbital period variation effects taken from literature. See 5.3 for more details.  Pulsar name |  $\dot{P}_{\rm b,Obs}$ ($10^{-15}$) |  $\dot{P}_{\rm b,Shk}$ ($10^{-15}$) |  $\dot{P}_{\rm b,Gal}$ ($10^{-15}$) |  $\dot{P}_{\rm b,int}$ ($10^{-15}$) |  $\dot{P}_{\rm b,GW}$ ($10^{-15}$) |  $D_{\rm kin}$ (Kpc) |
---|---|---|---|---|---|---|---
J0613$-$0200 | $35^{+2}_{-2}$ | $28^{+1}_{-1}$ | $3.3^{+0.2}_{-0.2}$ | $3.6^{+2.8}_{-2.8}$ | $-3.7^{+1.2}_{-1.8}$ | $1.27^{+0.11}_{-0.10}$ |
J0751+1807∗ | $-35^{+0.5}_{-0.5}$ | $11.9^{+0.6}_{-0.5}$ | $0.66^{+0.06}_{-0.07}$ | $-47.6^{+0.7}_{-0.8}$ | $-34.2^{\ast}$ | — |
J1012+5307 | $54.6^{+0.6}_{-0.6}$ | $90.3^{+7.9}_{-7.4}$ | $-4.13^{+0.08}_{-0.06}$ | $-31.6^{+7.4}_{-7.9}$ | $-6^{+5}_{-32}$ | $0.78^{+0.38}_{-0.06}$ |
J1022+1001 | $218^{+9}_{-9}$ | $753^{+243}_{-186}$ | $-66^{+1}_{-1}$ | $-469^{+187}_{-243}$ | $-0.52^{+0.08}_{-0.08}$ | $0.32^{+0.10}_{-0.08}$ |
J1455$-$3330 | $4593^{+2202}_{-2235}$ | $805^{+62}_{-59}$ | $-209^{+13}_{-13}$ | $3997^{+2203}_{-2246}$ | — | $4.6^{+2.1}_{-2.1}$ |
J1600$-$3053† | $365^{+32}_{-31}$ | $287^{+39}_{-33}$ | $52.0^{+4.8}_{-4.9}$ | $25^{+46}_{-49}$ | $-0.11^{+0.01}_{-0.01}$ | $2.1^{+0.2}_{-0.2}$ |
J1640+2224 | $9501^{+1929}_{-1948}$ | $5556^{+1255}_{-1090}$ | $-2296^{+330}_{-382}$ | $6207^{+2285}_{-2334}$ | $-0.0006^{+0.0002}_{-0.0003}$ | — |
J1713+0747 | $264^{+73}_{-72}$ | $642^{+7}_{-7}$ | $-333^{+3}_{-3}$ | $-45^{+73}_{-72}$ | $-0.0067^{+0.0001}_{-0.0002}$ | $1.06^{+0.13}_{-0.13}$ |
J1738+0333a | $-30^{+7}_{-7}$ | $6.44^{+0.08}_{-0.08}$ | $-0.367^{+0.005}_{-0.005}$ | $-36^{+7}_{-7}$ | $-28^{+1}_{-1}$ | — |
J1909$-$3744† | $509.2^{+0.9}_{-0.9}$ | $513.3^{+8.8}_{-9.0}$ | $2.58^{+0.07}_{-0.07}$ | $-7^{+9}_{-9}$ | $-2.67^{+0.02}_{-0.02}$ | $1.154^{+0.002}_{-0.002}$ |

Using EPTA DR2, we measured binary orbital period derivatives for a total of
ten pulsars. The measurements in PSRs J1022$+$1001, J1600$-$3053,
J1640$+$2224, J1713+0747 (all with $\gtrsim 3\sigma$ significance), PSR
J1455$-$3330 (marginal), and J1738+0333 are new compared to those obtained
with the EPTA DR1 dataset. Overall, the measured values reported here are
consistent with EPTA DR1 (see Table 4), although with improved uncertainties.

GW emission causes a change in the orbital period in binary systems (Peters,
1964). The observed $\dot{P}_{\rm b}$ however, may also include extrinsic
contributions such as distance- and location-dependent kinematic effects. We
determined these extrinsic contributions to the observed $\dot{P}_{\rm b}$
using the bias-corrected distances given in Table 3. For PSRs J1600$-$3053,
J1909$-$3744 and J1738+0333, we used the reference parallax values from
Reardon et al. (2021), Liu et al. (2020) and Freire et al. (2012),
respectively (shown in Table 3). We consider two kinematic contributions. The
first is the so-called Shklovskii effect (Shklovskii, 1970), which scales with
the proper motion of a binary

$\frac{\dot{P}_{\rm b,Shk}}{P_{\rm b}}=\frac{\mu^{2}d}{c},$ (7)

where $d$ is the distance, $\mu$ is the proper measured motion, and $c$ is the
speed of light, respectively. The second is caused by differences in the
acceleration by the galactic gravitational potential at the solar system and
binary pulsar locations (Damour & Taylor, 1991). To correct for this, we used
the Milky Way potential model of McMillan (2017), which considers the gas disc
and the halo density profile, and is calibrated against Galactic maser
sources.

We then calculated the intrinsic orbital period change ($\dot{P}_{\rm b,int}$)
by subtracting the kinematic contributions from the observed value,

$\dot{P}_{\rm b,Int}=\dot{P}_{\rm b,Obs}-\dot{P}_{\rm b,Shk}-\dot{P}_{\rm
b,Gal}.$ (8)

From Table 4, it can be seen that for all pulsars except PSRs J0751$+$1807,
J1012+5307, J1640+2224 and J1738+0333, the derived $\dot{P}_{\rm b,int}$ is
consistent with zero within 2$\sigma$.

We also calculated the change in the orbital period by GW damping
($\dot{P}_{\rm b,GW}$) using the mass estimates presented in Section 5.2,
where available. We note that here we used the measurement of the SD parameter
$h_{3}$ and assumed a 1.4 M⊙ pulsar mass to estimate $\dot{P}_{\rm b,GW}$ for
PSR J0751+1807. The masses of PSR J1738$+$0333 were directly taken from
Antoniadis et al. (2012). For PSRs J0751+1807, J1012+5307 and J1738+0333, the
derived $\dot{P}_{\rm b,int}$ is consistent with the predicted $\dot{P}_{\rm
b,GW}$, suggesting that gravitational damping can account for the non-zero
intrinsic secular variation of orbital period.

For PSR J1640+2224, the timing parallax and corresponding distance estimate
agree well with those inferred using VLBI measurements (Ding et al., 2023).
Therefore, the marginal $\sim 2.7\sigma$ detection of $\dot{P}_{\rm b,int}$,
which cannot be accounted for by GW damping, is unlikely to be due to an
incorrect distance estimate. Although the most probable explanation for the
discrepancy is the low-significance measurement of $\dot{P}_{\rm b,Obs}$
($\sim 4.9\sigma$), another intriguing explanation could be that the system
experiences an additional acceleration due to a nearby object.

For pulsars with a $\dot{P}_{\rm b,int}$ consistent with either zero or
anticipated contribution from GW damping, we calculated the kinematic
distances $\dot{P}_{\rm b,obs}$, by assuming a net zero $\dot{P}_{\rm b}$
(Bell & Bailes, 1996). More specifically, we estimate the kinematic distances
as

$D_{\rm kin}=\frac{c\left(\dot{P}_{\rm b,Obs}-\dot{P}_{\rm b,Gal}-\dot{P}_{\rm
b,GW}\right)}{\mu^{2}P_{\rm b}}.$ (9)

For PSR J1455$-$3330, the masses were not measured with our data and the
kinetic distance is determined without considering the orbital period
derivative GW damping (which should nonetheless be negligible considering its
76-day orbital period). The results are summarised in Table 4. Comparison
between the kinematic distance and the bias-corrected parallax distance
indicates that they are, in general, consistent with each other.

###  5.4 Searches for annual orbital parallax signatures

For binary pulsars, the proper motion of the system changes the viewing
geometry of the orbit with respect to the Earth. This effect induces an
apparent secular variation in the projected semi-major axis of the binary
orbit (Kopeikin, 1996)888Here we used the astronomical convention as in
Edwards et al. (2006), where $\Omega$ and $i$ are different from those in
Kopeikin (1995, 1996) following: $\Omega=\pi/2-\Omega_{\rm K96}$, $i=\pi-
i_{\rm K96}$. This applies to Eq. 10–12.:

$x=x_{\rm int}\left[1+\cot
i(\mu_{\alpha}\cos\Omega-\mu_{\delta}\sin\Omega)(t-t_{0})\right],$ (10)

which gives

$\frac{\dot{x}}{x_{\rm int}}=\mu\cot i\cos(\theta_{\mu}+\Omega).$ (11)

Here $\mu\equiv\sqrt{\mu^{2}_{\alpha}+\mu^{2}_{\delta}}$999We note that here
$\mu_{\alpha}$, $\mu_{\delta}$ follows the notation as stated in Table 7.,
$\Omega$ is the longitude of the ascending node of the orbit and
$\theta_{\mu}$ is the position angle of the proper motion on the sky. It can
be seen that if the proper motion and orbital inclination angle are measured,
a measurement of $\dot{x}$ can then be used to determine $\Omega$, with an
ambiguity of $\cos(-\theta_{\mu}-\Omega)=\cos(\theta_{\mu}+\Omega)$.
Typically, the orbital inclination is obtained from the SD parameter $\sin i$,
with a $\pi-i$ ambiguity. Therefore, the determination of $\Omega$ can have
four possible pairs of solutions for the orbital inclination and ascending
node (e.g. Liu et al., 2020). Meanwhile, for binary pulsars, the annual motion
of the Earth around the Sun also introduces an apparently periodic variation
in the viewing geometry of the orbit, an effect known as the annual orbital
parallax. The variation in the projected semi-major axis as a result of this
effect can be written as (Kopeikin, 1995):

$x=x_{\rm int}\left[1-\frac{\cot
i}{d}(\Delta_{I_{0}}\cos\Omega-\Delta_{J_{0}}\sin\Omega)\right],$ (12)

where

$\displaystyle\Delta_{I_{0}}$ $\displaystyle=$
$\displaystyle-X\sin\alpha+Y\cos\alpha,$ (13) $\displaystyle\Delta_{J_{0}}$
$\displaystyle=$
$\displaystyle-X\sin\delta\cos\alpha-Y\sin\delta\sin\alpha+Z\cos\delta.$ (14)

Here, $\vec{r}=\\{X,Y,Z\\}$ is the position vector of the Earth with respect
to the barycentre of the solar system, and ($\alpha$, $\delta$) are the
spherical coordinates of the barycentre of the binary system. For nearby
binary pulsars in a wide orbit, the annual orbital parallax can be measurable,
giving a unique pair of solutions for the inclination and ascending node (e.g.
for PSR J1713+0747). We investigated sources with measured $\dot{x}$ and SD
parameters to search for this annual orbital parallax signatures. These
sources include PSRs J1600$-$3053, J1857+0943101010Although there is still no
measured $\dot{x}$ for J1857+0943, we include it here as Desvignes et al.
(2016) give tentative constraints on $\Omega$., J1640+2224, J1022+1001,
J1012+5307. We used the T2 binary model in tempo2, where the annual orbital
parallax effect is described with the KOM and KIN parameters, corresponding to
the longitude of the ascending node ($\Omega$) and inclination angle ($i$),
respectively. The SD parameter $s\equiv\sin i$ is then treated as a function
of KIN. We mapped the KOM-KIN space with the temponest toolkit, following a
scheme similar to described in Desvignes et al. (2016). In detail, we fixed
the set of white noise parameters to their maximum likelihood values and chose
to analytically marginalise over the spin and astrometric parameters. We set
customised uniform linear priors for KOM, KIN and $M_{\rm c}$. For KIN and
$M_{\rm c}$ the prior ranges are set to include all possible values of these
parameters that are allowed from the SD measurements (see tables in Appendix
B). For KOM the prior range was set from
$0\text{\,}\mathrm{\SIUnitSymbolDegree}360\text{\,}\mathrm{\SIUnitSymbolDegree}$.
The sampling was conducted with the constant efficiency option turned off, in
order to map the multimodal parameter space more effectively.

For PSR J1600$-$3053, four possible solutions were found in the mapping of KIN
and KOM using DR1 dataset (Desvignes et al., 2016), which was consistent with
a negligible annual orbital parallax parallax. With the EPTA DR2 data, we are
now able to single out one solution for these two parameters:
$\Omega=82^{+12}_{-8}$ deg, $i=110.4^{+0.9}_{-0.7}$ deg. Based on the number
of samples ($\sim 337$k) in the posterior distribution, the logarithmic
likelihood ratio of this solution to the others is $\gtrsim 5.5$. This means
that the signature of the annual orbital parallax is clearly detected in this
binary pulsar system.

(a) PSR J1600$-$3053

(b) PSR J1640$+$2224

(c) PSR J1022$+$1001

(d) PSR J1012$+$5307

(e) PSR J1857$+$0943

Figure 4: Posterior distributions of KOM and KIN from the search for annual-
orbital parallax effect, for PSRs J1600$-$3053 (top left), J1640+2224 (top
right), J1022+1001 (middle left), J1012+5307 (middle right), and J1857+0943
(bottom). The solid, dashed and dotted lines in the side panels stand for the
median, 1$\sigma$ confidence interval and 2$\sigma$ confident interval of the
distribution. The stars in the plots remark the maximum likelihood values of
the two parameters each obtained from the posterior distribution in the side
panel.

For PSR J1640+2224, two possible solutions for KOM and KIN have been found;
$\Omega=5^{+3}_{-2}$ deg, $i=111^{+4}_{-5}$ deg, and $\Omega=-332^{+3}_{-5}$
deg, $i=66^{+4}_{-3}$ deg, respectively. The logarithmic likelihood ratio of
these two islands is approximately 2.1, which favours the former solution.
Thus, there is weak evidence for the detection of the annual orbital parallax
signal.

In PSR J1012+5307, from the KOM-KIN mapping, the longitude of the ascending
node and the orbital inclination are restricted to two regions. The 2$\sigma$
range of $\Omega$ in these two regions, are 47–142 and 218–318 deg,
respectively. Although the most likely value of $\Omega$ is around 150 deg,
the preference for this area of the solution is unclear.

For PSR J1022+1001, four solutions in KOM and KOM have been found. These are
1). $\Omega=17^{+2}_{-3}$ deg, $i=56^{+4}_{-4}$ deg, 2). $\Omega=50^{+3}_{-2}$
deg, $i=56^{+4}_{-4}$ deg, 3). $\Omega=197^{+2}_{-3}$ deg, $i=125^{+4}_{-4}$
deg, 4). $\Omega=229^{+3}_{-2}$ deg, $i=125^{+4}_{-4}$ deg. Although two of
them are slightly preferred in probability, they cannot be distinguished with
high confidence, which means that the signal of annual orbital parallax is not
detected.

Desvignes et al. (2016) did not report measurable secular variation of the
projected semi-major axis in J1857+0943, and with the new data set we also do
not measure significant $\dot{x}$. Still, as in Desvignes et al. (2016), we
nonetheless attempted to search for an annual orbital parallax to see if the
constraint on the longtidue of the ascending node can be improved. In
Desvignes et al. (2016), there was only a tentative constraint on the
ascending node. With the new dataset, there is now a tentative preference for
$\Omega$ in the range of 100 to 200 deg. For the orbital inclination, the
posterior space around 88 deg is slightly preferred, compared to that peaked
around 92 deg. However, there is no clear evidence for neither of the two
solutions.

The analysis of PSR J1713+0747 yields $\Omega=91.1\pm 0.5$ deg and $i=71.3\pm
0.2$ deg as discussed in Seciont 4. These are consistent with previous
analyses (Zhu et al., 2019; Alam et al., 2021a; Reardon et al., 2021), as well
as with the results based on EPTA DR1 (Desvignes et al., 2016).

###  5.5 High order spin frequency derivatives in J1024$-$0719

There is a significant measurement of the second spin frequency derivative in
PSR J1024$-$0719 (Bassa et al., 2016c; Kaplan et al., 2016). Based on these
results, it is anticipated that this pulsar is in orbit with a main-sequence
star named 2MASS J10243869$-$0719190, of an orbital period longer than 200 yr
(Bassa et al., 2016c). Here with the new EPTA dataset, in addition to our
timing analysis presented in Table 8, we conducted another round of analysis
including modelling of the third, fourth and fifth spin frequency derivatives.
These derivatives, if measurable, can be used to derive the properties of the
Keplerian orbit of the pulsar (Bassa et al., 2016c). Table 5 shows the results
of this analysis, and a comparison with previous work. As can be seen, there
is a tentative change in the measured second spin frequency derivative
compared to previous results (even when the reference epoch of the spin
frequency and the number of modelled spin frequency derivatives are both the
same). This is expected given that the EPTA DR2 dataset covers a slightly
longer timeline / orbit phase of the binary system. It is also intriguing to
see that with the new and more sensitive data, there is now a tentative
detection of the fifth spin frequency derivative. This though needs to be
confirmed by future observations.

| Bassa et al. (2016c) | Kaplan et al. (2016) | EPTA DR2, F0-2 | EPTA DR2, F0-5
---|---|---|---|---
Epoch of spin frequency (MJD) | 55000 | 56236 | 55000 | 55000
Spin frequency, $\nu$ (s-1) | 193.7156834485468(7) | 193.7156863778085(8) | 193.715683448548(2) | 193.715683448549(3)
First derivative of $\nu$ (s-2) | $-6.95893(15)\times 10^{-16}$ | $-6.9638(4)\times 10^{-16}$ | $-6.9593(2)\times 10^{-16}$ | $-6.9598(2)\times 10^{-16}$
Second derivative of $\nu$ (s-3) | $-3.92(2)\times 10^{-27}$ | $-4.1(10)\times 10^{-27}$ | $-3.57(7)\times 10^{-27}$ | $-2.7(4)\times 10^{-27}$
Third derivative of $\nu$ (s-4) | $<2.7\times 10^{-36}$ | 1.1(7)$\times 10^{-34}$ | — | $-3(3)\times 10^{-36}$
Forth derivative of $\nu$ (s-5) | $<4.5\times 10^{-44}$ | — | — | $-1.7(7)\times 10^{-43}$
Fifth derivative of $\nu$ (s-6) | — | — | — | 2.2(9)$\times 10^{-51}$
Table 5: Spin frequency and its derivatives for PSR J1024$-$0719, measured
from previous work and analysis in this paper.

##  6 Conclusions

In this paper, we presented the EPTA DR2 dataset and results from a combined
frequentist and Bayesian timing analyses. This dataset contains high-precision
timing data for 25 MSPs, collected with the five largest radio telescopes in
Europe along with LEAP. The DR2 dataset combines data from EPTA DR1 (Desvignes
et al., 2016) with those recorded by a new generation of data acquisition
systems. The dataset extends to the beginning of 2021 and has baselines
ranging from 14 to 25 years.

We conducted timing analysis of the dataset based on a Bayesian framework to
measure the timing parameters of the pulsars. This has yielded a collection of
new measurements including annual parallaxes, secular variation of orbital
period, Shapiro delay and so forth in some pulsars. We also derived a group of
astrophysical parameters of these pulsars, including distances, transverse
velocities, binary masses, and annual orbital parallaxes. The DR2 dataset
builds the foundation for searching for GW signals. The results of this search
are reported in an accompanying publication (the EPTA and InPTA
Collaborations, 2023b).

###### Acknowledgements.

The European Pulsar Timing Array (EPTA) is a collaboration between European
and partner institutes, namely ASTRON (NL), INAF/Osservatorio di Cagliari
(IT), Max-Planck-Institut für Radioastronomie (GER), Nançay/Paris Observatory
(FRA), the University of Manchester (UK), the University of Birmingham (UK),
the University of East Anglia (UK), the University of Bielefeld (GER), the
University of Paris (FRA), the University of Milan-Bicocca (IT), the
Foundation for Research and Technology, Hellas (GR), and Peking University
(CHN), with the aim to provide high-precision pulsar timing to work towards
the direct detection of low-frequency gravitational waves. An Advanced Grant
of the European Research Council allowed to implement the Large European Array
for Pulsars (LEAP) under Grant Agreement Number 227947 (PI M. Kramer). The
EPTA is part of the International Pulsar Timing Array (IPTA); we thank our
IPTA colleagues for their support and help with this paper and the external
Detection Committee members for their work on the Detection Checklist. Part of
this work is based on observations with the 100-m telescope of the Max-Planck-
Institut für Radioastronomie (MPIfR) at Effelsberg in Germany. Pulsar research
at the Jodrell Bank Centre for Astrophysics and the observations using the
Lovell Telescope are supported by a Consolidated Grant (ST/T000414/1) from the
UK’s Science and Technology Facilities Council (STFC). ICN is also supported
by the STFC doctoral training grant ST/T506291/1. The Nançay radio Observatory
is operated by the Paris Observatory, associated with the French Centre
National de la Recherche Scientifique (CNRS), and partially supported by the
Region Centre in France. We acknowledge financial support from “Programme
National de Cosmologie and Galaxies” (PNCG), and “Programme National Hautes
Energies” (PNHE) funded by CNRS/INSU-IN2P3-INP, CEA and CNES, France. We
acknowledge financial support from Agence Nationale de la Recherche
(ANR-18-CE31-0015), France. The Westerbork Synthesis Radio Telescope is
operated by the Netherlands Institute for Radio Astronomy (ASTRON) with
support from the Netherlands Foundation for Scientific Research (NWO). The
Sardinia Radio Telescope (SRT) is funded by the Department of University and
Research (MIUR), the Italian Space Agency (ASI), and the Autonomous Region of
Sardinia (RAS) and is operated as a National Facility by the National
Institute for Astrophysics (INAF). The work is supported by the National SKA
programme of China (2020SKA0120100), Max-Planck Partner Group, NSFC 11690024,
CAS Cultivation Project for FAST Scientific. This work is also supported as
part of the “LEGACY” MPG-CAS collaboration on low-frequency gravitational wave
astronomy. JA acknowledges support from the European Commission (Grant
Agreement number: 101094354). JA and SCha were partially supported by the
Stavros Niarchos Foundation (SNF) and the Hellenic Foundation for Research and
Innovation (H.F.R.I.) under the 2nd Call of the “Science and Society – Action
Always strive for excellence – Theodoros Papazoglou” (Project Number: 01431).
AC acknowledges support from the Paris Île-de-France Region. AC, AF, ASe, ASa,
EB, DI, GMS, MBo acknowledge financial support provided under the European
Union’s H2020 ERC Consolidator Grant “Binary Massive Black Hole Astrophysics”
(B Massive, Grant Agreement: 818691). GD, KLi, RK and MK acknowledge support
from European Research Council (ERC) Synergy Grant “BlackHoleCam”, Grant
Agreement Number 610058. This work is supported by the ERC Advanced Grant
“LEAP”, Grant Agreement Number 227947 (PI M. Kramer). AV and PRB are supported
by the UK’s Science and Technology Facilities Council (STFC; grant
ST/W000946/1). AV also acknowledges the support of the Royal Society and
Wolfson Foundation. JPWV acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG) through thew Heisenberg programme (Project No.
433075039) and by the NSF through AccelNet award #2114721. NKP is funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
Projektnummer PO 2758/1–1, through the Walter–Benjamin programme. ASa thanks
the Alexander von Humboldt foundation in Germany for a Humboldt fellowship for
postdoctoral researchers. APo, DP and MBu acknowledge support from the
research grant “iPeska” (P.I. Andrea Possenti) funded under the INAF national
call Prin-SKA/CTA approved with the Presidential Decree 70/2016 (Italy). RNC
acknowledges financial support from the Special Account for Research Funds of
the Hellenic Open University (ELKE-HOU) under the research programme “GRAVPUL”
(grant agreement 319/10-10-2022). EvdW, CGB and GHJ acknowledge support from
the Dutch National Science Agenda, NWA Startimpuls – 400.17.608. BG is
supported by the Italian Ministry of Education, University and Research within
the PRIN 2017 Research Program Framework, n. 2017SYRTCN. LS acknowledges the
use of the HPC system Cobra at the Max Planck Computing and Data Facility. The
work presented here is a culmination of many years of data analysis as well as
software and instrument development. In particular, we thank Drs. N. D’Amico,
P. C. C. Freire, R. van Haasteren, C. Jordan, K. Lazaridis, P. Lazarus, L.
Lentati, O. Löhmer and R. Smits for their past contributions. We also thank
Dr. N. Wex for supporting the calculations of the galactic acceleration as
well as the related discussions. The EPTA is also grateful to staff at its
observatories and telescopes who have made the continued observations
possible. Author contributions. The EPTA is a multi-decade effort and all
authors have contributed through conceptualisation, funding acquisition, data-
curation, methodology, software and hardware developments as well as (aspects
of) the continued running of the observational campaigns, which includes
writing and proofreading observing proposals, evaluating observations and
observing systems, mentoring students, developing science cases. All authors
also helped in (aspects of) verification of the data, analysis and results as
well as in finalising the paper draft. Specific contributions from individual
EPTA members are listed in the CRediT111111https://credit.niso.org/ format
below. InPTA members contributed in uGMRT observations and data reduction to
create the InPTA data set which is employed while assembling the DR2full+ and
DR2new+ data sets. JJan, KLi, GMS equally share the correspondence of the
paper. CRediT statement:
Conceptualisation: APa, APo, AV, BWS, CGB, CT, GHJ, GMS, GT, IC, JA, JJan,
JPWV, JW, JWM, KJL, KLi, MK.
Methodology: APa, AV, DJC, GMS, IC, JA, JJan, JPWV, JWM, KJL, KLi, LG, MK.
Software: AC, AJ, APa, CGB, DJC, GMS, IC, JA, JJan, JJaw, JPWV, KJL, KLi, LG,
MJK, RK.
Validation: AC, APa, CGB, CT, GMS, GT, IC, JA, JJan, JPWV, JWM, KLi, LG.
Formal Analysis: APa, CGB, DJC, DP, EvdW, GHJ, GMS, JA, JJan, JPWV, JWM, KLi.
Investigation: APa, APo, BWS, CGB, DJC, DP, GMS, GT, IC, JA, JJan, JPWV, JWM,
KLi, LG, MBM, MBu, MJK, RK.
Resources: APa, APe, APo, BWS, GHJ, GMS, GT, HH, IC, JA, JJan, JPWV, JWM, KJL,
KLi, LG, MJK, MK, RK.
Data Curation: AC, AJ, APa, BWS, CGB, DJC, DP, EG, EvdW, GHJ, GMS, GT, HH, IC,
JA, JJan, JPWV, JWM, KLi, LG, MBM, MBu, MJK, MK, NKP, RK, SChe, YJG.
Writing – Original Draft: APa, GMS, JA, JJan, KLi, LG.
Writing – Review & Editing: AC, AF, APa, APo, DJC, EB, EFK, GHJ, GMS, GT, JA,
JJan, JPWV, JWM, KLi, MK, SChe, VVK.
Visualisation: APa, GMS, JA, JJan, KLi.
Supervision: APo, ASe, AV, BWS, CGB, DJC, EFK, GHJ, GMS, GT, IC, JA, JPWV,
KJL, KLi, LG, MJK, MK, VVK.
Project Administration: APo, ASe, AV, BWS, CGB, CT, GHJ, GMS, GT, IC, JJan,
JPWV, JWM, KLi, LG, MK.
Funding Acquisition: APe, APo, ASe, BWS, GHJ, GT, IC, JA, JJan, LG, MJK, MK.

Data Availability Files containing the TOAs and pulsar timing models can be
downloaded from https://epta.pages.in2p3.fr/epta-dr2/. The same site also
provides the software environments necessary to recreate the timing analysis
presented in this article.

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##  Appendix A Dataset versions

The full EPTA DR2 dataset, hereafter referred to as “DR2full”, is the primary
source of data for the pulsar timing analyses presented in this paper, as well
as the accompanying articles on the single-pulsar noise analysis (the EPTA and
InPTA Collaborations, 2023a) and the search for correlated signals in (the
EPTA and InPTA Collaborations, 2023b). Apart from the DR2full, additional
datasets were created, guided by iterative analyses carried out during the
detailed investigation of single pulsar noise models in the EPTA and InPTA
Collaborations (2023a), as well as multiple analyses carried out to search for
artefacts and spurious signals in the common signal searches carried out in
the EPTA and InPTA Collaborations (2023b).

The first of these, the ‘DR2full+’, consists of a combination of the DR2full
dataset with the first data release of the InPTA (henceforth InPTA-DR1
Tarafdar et al., 2022) for an overlapping set of 10 pulsars, marked with an
asterisk in Table 2. This dataset aims to improve the overall sensitivity of
the DR2full to DM-variation linked noise-processes. The combination was
performed using the ‘narrowband’ TOAs from the InPTA-DR1 for two observing
bands centred around $500\text{\,}\mathrm{MHz}1460\text{\,}\mathrm{MHz}$,
respectively. Unlike the EPTA data, the InPTA-DR1 data requires optimisation
based on the flux density of the pulsar being observed, as well as, observing
setup specific details leading to changes in the number of sub-bands
generated, ranging from a minimum of $4$ to a maximum of $16$. We refer
interested readers to Tarafdar et al. (2022) for the full details on the
InPTA-DR1 data processing. To combine the InPTA-DR1 data with the EPTA DR2
data, for each pulsar we used the ephemeris produced from the EPTA DR2 dataset
and followed the same steps as described in Section 3. Specifically, we fitted
for phase offset for each sub-band in the InPTA data individually with respect
to the reference system in the EPTA data (see Section 3).

The single-pulsar noise analysis of these additional sets of data are
presented in the EPTA and InPTA Collaborations (2023a). For the ‘DR2full+’
dataset, we performed single-pulsar Bayesian timing analysis following the
same description as in Section 3.4, using customised noise models determined
by the EPTA and InPTA Collaborations (2023a). The results of this analysis
were later used for the gravitational wave searches presented in the EPTA and
InPTA Collaborations (2023b).

Table 6: Overview of the name designations for the EPTA dataset Dataset | MJD range |  Notes
---|---|---
DR2full | 50360.76 – 59385.10 |  Full EPTA DR2 data
DR2new | 55611.40 – 59385.10 |  Data only from the new backends collected in the past 10 yr
DR2full+ | 50360.76 – 59644.16 |  DR2full + InPTA DR1 for 10 overlap pulsars
DR2new+ | 55611.40 – 59644.16 |  DR2new + InPTA DR1 for 10 overlap pulsars

A significant difference between the EPTA DR1 and DR2 datasets lies in the use
of the technique of coherent dedispersion. This allows for a far more accurate
modelling of the frequency-dependent DM delays, leading to the recovery of a
sharper profile and thus, improved timing performance. To test the
improvements derived from using only coherently dedispersed data, the ‘DR2new’
dataset was created. This dataset spans, approximately, the final $10.3$
years. Similar to the DR2full dataset, we also appended InPTA-DR1 data to the
DR2new to produce the ‘DR2new+’ dataset.

##  Appendix B Pulsar Ephemerides

The measurements of the pulsar timing parameters obtained from the Bayesian
timing analysis are presented in the following pages, in Tables 7–13 for all
25 EPTA DR2 pulsars. The measurement values are the medians of the posterior
distributions and the errors denote the 1-$\sigma$ confidence intervals for
the respective parameters.

Table 7: Measured timing model parameters for PSRs J0030$+$0451, J0613$-$0200,
J0751$+$1807 and J0900$-$3144\. Here the epoch of spin frequency, sky position
and DM are all set to MJD 55000. The definitions of the proper motion terms
are: $\mu_{\alpha}=\dot{\alpha}\cos\delta$, $\mu_{\delta}=\dot{\delta}$,
$\mu_{\lambda}=\dot{\lambda}\cos\beta$, $\mu_{\beta}=\dot{\beta}$. The
components in the red-noise model used in the Bayesian analysis to obtain the
timing solution are shown in the last row, where RN, DM, SV stand for
achromatic red noise, chromatic red noise for DM variation, and scattering
variation, respectively. All of the above apply to all tables in this section.
---
Pulsar Jname | J0030$+$0451 | J0613$-$0200 | J0751$+$1807 | J0900$-$3144
Right ascension, $\alpha$ (J2000) | — | 06:13:43.975688(1) | 07:51:09.155329(6) | 09:00:43.953105(8)
Declination, $\delta$ (J2000) | — |  $-$02:00:47.22541(4) | 18:07:38.4858(5) |  $-$31:44:30.8950(1)
Ecliptic longitude $\lambda$ (deg) | 8.910356334(10) | — | — | —
Ecliptic latitude $\beta$ (deg) | 1.4456958(4) | — | — | —
Spin frequency, $\nu$ (Hz) | 205.530695938456(2) | 326.6005620234831(4) | 287.457853995106(1) | 90.011841919354(1)
Spin frequency derivative, $\dot{\nu}$ (s-2) | $-4.2977(1)\times 10^{-16}$ | $-1.023017(8)\times 10^{-15}$ | $-6.43455(6)\times 10^{-16}$ | $-3.96012(7)\times 10^{-16}$
DM (cm-3 pc) | 4.331(1) | 38.7759(7) | 30.2457(8) | 75.691(2)
DM1 (cm-3 pc yr-1) | 0.0002(1) | $-8(2)\times 10^{-5}$ | $-0.00046(7)$ | 0.0009(5)
DM2 (cm-3 pc yr-2) | $-2(2)\times 10^{-5}$ | $-1.8(6)\times 10^{-5}$ | 10(20)$\times 10^{-6}$ | $-0.00019(9)$
Proper motion in $\alpha$, $\mu_{\rm\alpha}$ (mas yr-1) | — | 1.837(2) | $-2.70(1)$ | $-1.03(2)$
Proper motion in $\delta$, $\mu_{\rm\delta}$ (mas yr-1) | — | $-10.359(6)$ | $-13.27(7)$ | 1.99(2)
Proper motion in $\lambda$, $\mu_{\rm\lambda}$ (mas yr-1) | $-5.523(5)$ | — | — | —