11institutetext: 1 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o
Dipartimento di Fisica, Edificio G. Marconi, I-00185 Roma, Italy
2 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 Monteporzio
Catone, Italy
3 AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, F-91191 Gif-
sur-Yvette, France
4 CEA Saclay, DFR/IRFU, Service d’Astrophysique, Bat. 709, 91191 Gif-sur-
Yvette, France
5 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, F-75014, Paris,
France
6 INFN-Bologna, Via Irnerio 46, I-40126 Bologna, Italy
7 Dipartimento di Fisica e Scienze della Terra, Universitá degli Studi di
Ferrara, Via Giuseppe Saragat 1, I-44122 Ferrara, Italy
8 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, I-40129 Bologna,
Italy
9 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Tapada da Ajuda, PT-1349-018 Lisboa, Portugal
10 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa,
Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal
11 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, PT-1749-016 Lisboa, Portugal
12 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via
Piero Gobetti 93/3, I-40129 Bologna, Italy
13 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2,
34151 Trieste, Italy
14 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34131
Trieste, Italy
15 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The
Netherlands
16 ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo,
28692 Villanueva de la Cañada, Madrid, Spain
17 FRACTAL S.L.N.E., calle Tulipán 2, Portal 13 1A, 28231, Las Rozas de
Madrid, Spain
18 European Space Agency/ESOC, Robert-Bosch-Str. 5, D-64293 Darmstadt, Germany
19 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany
20 Mullard Space Science Laboratory, University College London, Holmbury St
Mary, Dorking, Surrey RH5 6NT, UK
21 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822,
F-69622, Villeurbanne, France
22 INAF-IASF Milano, Via Alfonso Corti 12, I-20133 Milano, Italy
23 Aix-Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
24 INAF-Osservatorio Astronomico di Brera, Via Brera 28, I-20122 Milano, Italy
25 Dipartimento di Fisica ”Aldo Pontremoli”, Universitá degli Studi di Milano,
Via Celoria 16, I-20133 Milano, Italy
26 INFN-Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
27 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The
Netherlands
28 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
29 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
30 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71,
53121 Bonn, Germany
31 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5,
Canada
32 Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
33 Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada
34 Carnegie Observatories, Pasadena, CA 91101, USA
35 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405,
Orsay, France
36 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK
37 INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, I-35122
Padova, Italy
38 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-
Universität München, Scheinerstrasse 1, 81679 München, Germany
39 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1,
D-85748 Garching, Germany
40 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino
Torinese (TO), Italy
41 INFN-Sezione di Roma Tre, Via della Vasca Navale 84, I-00146, Roma, Italy
42 Department of Mathematics and Physics, Roma Tre University, Via della Vasca
Navale 84, I-00146 Rome, Italy
43 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université
de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, F-31400 Toulouse, France
44 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131
Napoli, Italy
45 Department of Astronomy & Astrophysics, University of Toronto, 50 St George
Street, Toronto, Ontario M5S 3H4, Canada
46 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
47 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
48 Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 2-4,
08034 Barcelona, Spain
49 Department of Physics ”E. Pancini”, University Federico II, Via Cinthia 6,
I-80126, Napoli, Italy
50 INFN section of Naples, Via Cinthia 6, I-80126, Napoli, Italy
51 Dipartimento di Fisica e Astronomia “Augusto Righi” - Alma Mater Studiorum
Università di Bologna, via Piero Gobetti 93/2, I-40129 Bologna, Italy
52 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125,
Firenze, Italy
53 Institut national de physique nucléaire et de physique des particules, 3
rue Michel-Ange, 75794 Paris Cédex 16, France
54 Centre National d’Etudes Spatiales, Toulouse, France
55 Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
56 Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK
57 European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma,
Italy
58 Aix-Marseille Univ, CNRS, CNES, LAM, Marseille, France
59 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix,
Switzerland
60 Department of Astronomy, University of Geneva, ch. dÉcogia 16, CH-1290
Versoix, Switzerland
61 INFN-Padova, Via Marzolo 8, I-35131 Padova, Italy
62 Istituto Nazionale di Astrofisica (INAF) - Osservatorio di Astrofisica e
Scienza dello Spazio (OAS), Via Gobetti 93/3, I-40127 Bologna, Italy
63 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46,
I-40126 Bologna, Italy
64 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, N-0315 Oslo, Norway
65 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Drive, Pasadena, CA, 91109, USA
66 von Hoerner & Sulger GmbH, SchloßPlatz 8, D-68723 Schwetzingen, Germany
67 University of Lyon, UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, France
68 Université de Genève, Département de Physique Théorique and Centre for
Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
69 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin
katu 2, 00014 University of Helsinki, Finland
70 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk
4, 7991PD, Dwingeloo, The Netherlands
71 Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham, DH1 3LE, UK
72 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
73 University of Applied Sciences and Arts of Northwestern Switzerland, School
of Engineering, 5210 Windisch, Switzerland
74 Observatoire de Sauverny, Ecole Polytechnique Fédérale de Lau- sanne,
CH-1290 Versoix, Switzerland
75 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade
120, DK–8000 Aarhus C, Denmark
76 Institute of Space Science, Bucharest, Ro-077125, Romania
77 Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna,
Tenerife, Spain
78 Instituto de Astrofísica de Canarias, Calle Vía Làctea s/n, 38204, San
Cristòbal de la Laguna, Tenerife, Spain
79 Dipartimento di Fisica e Astronomia “G.Galilei”, Universitá di Padova, Via
Marzolo 8, I-35131 Padova, Italy
80 Centre for Electronic Imaging, Open University, Walton Hall, Milton Keynes,
MK7 6AA, UK
81 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
82 Infrared Processing and Analysis Center, California Institute of
Technology, Pasadena, CA 91125, USA
83 Universidad Politécnica de Cartagena, Departamento de Electrónica y
Tecnología de Computadoras, 30202 Cartagena, Spain
84 INAF-IASF Bologna, Via Piero Gobetti 101, I-40129 Bologna, Italy
85 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700
AV Groningen, The Netherlands
86 Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France
87 INFN-Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
88 Dipartimento di Fisica, Universitá degli Studi di Torino, Via P. Giuria 1,
I-10125 Torino, Italy
89 School of Physics and Astronomy, Cardiff University, The Parade, Cardiff,
CF24 3AA, UK
90 Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH,
UK
91 Space Science Data Center, Italian Space Agency, via del Politecnico snc,
00133 Roma, Italy
92 SISSA, International School for Advanced Studies, Via Bonomea 265, I-34136
Trieste TS, Italy
93 INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste TS, Italy
94 Universidad de la Laguna, E-38206, San Cristóbal de La Laguna, Tenerife,
Spain
95 Dipartimento di Fisica - Sezione di Astronomia, Universitá di Trieste, Via
Tiepolo 11, I-34131 Trieste, Italy
96 Dept. of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
97 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del
Cavaliere, 100, I-00100 Roma, Italy
98 School of Physics, HH Wills Physics Laboratory, University of Bristol,
Tyndall Avenue, Bristol, BS8 1TL, UK
99 Research Program in Systems Oncology, Faculty of Medicine, University of
Helsinki, Helsinki, Finland
100 Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland
101 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
102 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of
Helsinki, Helsinki, Finland
103 Centre de Calcul de l’IN2P3, 21 avenue Pierre de Coubertin F-69627
Villeurbanne Cedex, France
104 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, E-28049
Madrid, Spain
105 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg
16, 69120 Heidelberg, Germany
106 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, D-
69120 Heidelberg, Germany
107 Sorbonne Universités, UPMC Univ Paris 6 et CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
108 ICC&CEA, Department of Physics, Durham University, South Road, DH1 3LE, UK
109 INFN, Sezione di Lecce, Via per Arnesano, CP-193, I-73100, Lecce, Italy
110 Department of Mathematics and Physics E. De Giorgi, University of Salento,
Via per Arnesano, CP-I93, I-73100, Lecce, Italy
111 Institute for Computational Science, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland
112 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008,
Santiago, Chile
113 Department of Physics, P.O.Box 35 (YFL), 40014 University of Jyväskylä,
Finland
114 Department of Physics and Astronomy, University College London, Gower
Street, London WC1E 6BT, UK

#  Euclid preparation: I. The Euclid Wide Survey

Euclid Collaboration: R. Scaramella1,2<EMAIL_ADDRESS>J. Amiaux3
Y. Mellier4,5 C. Burigana6,7,8 C.S. Carvalho9 J.-C. Cuillandre3 A. Da
Silva10,11 A. Derosa12 J. Dinis10,11 E. Maiorano12 M. Maris13,14 I. Tereno9,10
R. Laureijs15 T. Boenke15 G. Buenadicha16 X. Dupac16 L.M. Gaspar Venancio15 P.
Gómez-Álvarez16,17 J. Hoar16 J. Lorenzo Alvarez15 G.D. Racca15 G. Saavedra-
Criado15 J. Schwartz18 R. Vavrek16 M. Schirmer19 H. Aussel3,5 R. Azzollini20
V.F. Cardone1,2 M. Cropper20 A. Ealet21 B. Garilli22 W. Gillard23 B.R.
Granett24 L. Guzzo24,25,26 H. Hoekstra27 K. Jahnke19 T. Kitching20 M.
Meneghetti12,28 L. Miller29 R. Nakajima30 S.M. Niemi15 F. Pasian14 W.J.
Percival31,32,33 M. Sauvage3 M. Scodeggio22 S. Wachter34 A. Zacchei14 N.
Aghanim35 A. Amara36 T. Auphan23 N. Auricchio12 S. Awan20 A. Balestra37 R.
Bender38,39 C. Bodendorf39 D. Bonino40 E. Branchini41,42 S. Brau-Nogue43 M.
Brescia44 G.P. Candini20 V. Capobianco40 C. Carbone22 R.G. Carlberg45 J.
Carretero46 R. Casas47,48 F.J. Castander47,48 M. Castellano2 S.
Cavuoti44,49,50 A. Cimatti51,52 R. Cledassou53,54 G. Congedo55 C.J.
Conselice56 L. Conversi16,57 Y. Copin21 L. Corcione40 A. Costille58 F.
Courbin59 H. Degaudenzi60 M. Douspis35 F. Dubath60 C.A.J. Duncan29 S. Dusini61
S. Farrens3 S. Ferriol21 P. Fosalba47,48 N. Fourmanoit21 M. Frailis14 E.
Franceschi12 P. Franzetti22 M. Fumana22 B. Gillis55 C. Giocoli62,63 A.
Grazian37 F. Grupp38,39 S.V.H. Haugan64 W. Holmes65 F. Hormuth19,66 P.
Hudelot5 S. Kermiche23 A. Kiessling65 M. Kilbinger3 R. Kohley16 B. Kubik67 M.
Kümmel38 M. Kunz68 H. Kurki-Suonio69 S. Ligori40 P.B. Lilje64 I. Lloro70 O.
Mansutti14 O. Marggraf30 K. Markovic65 F. Marulli12,28,51 R. Massey71 S.
Maurogordato72 M. Melchior73 E. Merlin2 G. Meylan74 J.J. Mohr38,39 M.
Moresco12,51 B. Morin3 L. Moscardini6,12,51 E. Munari14 R.C. Nichol36 C.
Padilla46 S. Paltani60 J. Peacock55 K. Pedersen75 V. Pettorino3 S. Pires3 M.
Poncet54 L. Popa76 L. Pozzetti12 F. Raison39 R. Rebolo77,78 J. Rhodes65 H.-W.
Rix19 M. Roncarelli12,51 E. Rossetti51 R. Saglia38,39 P. Schneider30 T.
Schrabback30 A. Secroun23 G. Seidel19 S. Serrano47,48 C. Sirignano61,79 G.
Sirri28 J. Skottfelt80 L. Stanco61 J.L. Starck3 P. Tallada-Crespí81 D.
Tavagnacco14 A.N. Taylor55 H.I. Teplitz82 R. Toledo-Moreo83 F.
Torradeflot46,81 M. Trifoglio84 E.A. Valentijn85 L. Valenziano12,28 G.A.
Verdoes Kleijn85 Y. Wang82 N. Welikala55 J. Weller38,39 M. Wetzstein39 G.
Zamorani12 J. Zoubian23 S. Andreon24 M. Baldi12,28,51 S. Bardelli12 A.
Boucaud86 S. Camera40,87,88 G. Fabbian89,90 R. Farinelli84 J. Graciá-Carpio39
D. Maino22,25,26 E. Medinaceli62 S. Mei86 C. Neissner46 G. Polenta91 A.
Renzi61,79 E. Romelli14 C. Rosset86 F. Sureau3 M. Tenti28 T. Vassallo38 E.
Zucca12 C. Baccigalupi13,14,92,93 A. Balaguera-Antolínez78,94 P. Battaglia84
A. Biviano13,14 S. Borgani13,14,93,95 E. Bozzo60 R. Cabanac43 A. Cappi12,72 S.
Casas3 G. Castignani51 C. Colodro-Conde78 J. Coupon60 H.M. Courtois67 J.
Cuby58 S. de la Torre58 S. Desai96 D. Di Ferdinando6 H. Dole35 M.
Fabricius38,39 M. Farina97 P.G. Ferreira29 F. Finelli6,84 P. Flose-Reimberg5
S. Fotopoulou98 S. Galeotta14 K. Ganga86 G. Gozaliasl99,100 I.M. Hook101 E.
Keihanen100 C.C. Kirkpatrick69 P. Liebing20 V. Lindholm100,102 G. Mainetti103
M. Martinelli104 N. Martinet58 M. Maturi105,106 H.J. McCracken107 R. B.
Metcalf51,84 G. Morgante12 J. Nightingale108 A. Nucita109,110 L. Patrizii28 D.
Potter111 G. Riccio44 A.G. Sánchez39 D. Sapone112 J.A. Schewtschenko36 M.
Schultheis72 V. Scottez5 R. Teyssier111 I. Tutusaus47,48 J. Valiviita102,113
M. Viel13,14,92,93 W. Vriend85 L. Whittaker56,114

(Received August XXXX, 2021; accepted XXXX, 2021)

Euclid is a mission of the European Space Agency, designed to constrain the
properties of dark energy and gravity via weak gravitational lensing and
galaxy clustering. It will carry out a wide area imaging and spectroscopy
survey (EWS) in visible and near infrared bands, covering approximately
$15\,000\,\mathrm{deg}^{2}$ of extragalactic sky on six years. The wide-field
telescope and instruments are optimized for pristine PSF and reduced
straylight, producing very crisp images.

This paper presents the building of the Euclid reference survey: the sequence
of pointings of EWS, Deep fields and Auxiliary fields for calibrations, and
spacecraft movements followed by Euclid as it operates in a step-and-stare
mode from its orbit around the Lagrange point L2.

Each EWS pointing has four dithered frames; we simulate the dither pattern at
pixel level to analyse the effective coverage. We use up-to-date models for
the sky background to define the Euclid region-of-interest (RoI). The building
of the reference survey is highly constrained from calibration cadences,
spacecraft constraints, and background levels; synergies with ground-based
coverage are also considered. Via purposely-built software, we first generate
a schedule for the Auxiliary and Deep fields observations. On a second stage,
the RoI is tiled and scheduled with EWS transit observations, with an
algorithm optimized to prioritize best sky areas, produce a compact coverage,
and ensure thermal stability.

The reference survey RSD_2021A is the optimized result of a modern survey
design. It fulfills all constraints and is a good proxy for the final
solution. The wide survey covers $\approx 14\,500\,\mathrm{deg}^{2}$. The
limiting AB magnitudes ($5\sigma$ point-like source) achieved in its footprint
are estimated to be 26.2 (visible) and 24.5 (near infrared); for spectroscopy,
the Hα line flux limit is $2\times 10^{-16}$ erg cm-2 s-1 at
$1600\,\mathrm{nm}$; and for diffuse emission the surface brightness limits
are 29.8 (visible) and 28.4 (near infrared) mag arcsec-2.

###### Key Words.:

cosmology – space vehicles – dark matter – dark energy – survey – all sky

###### Key Words.:

cosmology – space vehicles – dark matter – dark energy – survey – all sky

##  1 Introduction

Observations of distant type Ia supernovae (e.g. Riess et al., 1998;
Perlmutter et al., 1999) together with those of the cosmic microwave
background (CMB; e.g. de Bernardis et al., 2000, 2002; Hanany et al., 2000;
Pryke et al., 2002; Hinshaw et al., 2013; Planck Collaboration et al., 2020a,
b) suggest that the spatial curvature of the Universe is close to zero.
Despite the large contribution of dark matter (DM), however, the total matter
density is still much lower than the critical matter density. As a
consequence, a non-zero value for the cosmological constant $\Lambda$ is
usually introduced to complete the cosmological model. Although a cosmological
constant can describe the data, it is not generally appealing (Weinberg,
1989), and alternative solutions have been investigated, such as an evolving
quantum field (dark energy, or DE), and a modification to general relativity
on cosmological scales. We refer to Amendola et al. (2018) for an extensive
review of theoretical models.

To learn more about the nature of DE and DM, we need to quantify their impact
on cosmological observations. In particular, we need to determine $H(z)$, the
expansion history of the Universe as a function of redshift $z$, using
geometrical tests, and measure the growth of large-scale structures through
gravitational instability. The latter can be captured using the time
derivative of the matter density contrast $\delta\equiv\delta\rho/\rho$,

$\frac{{\rm d}\ln(\delta)}{{\rm d}\ln(a)}\,\simeq\,\Omega_{\rm m}^{\gamma},$
(1)

where $a\equiv 1/(1+z)$ is the cosmic scale factor, and $\Omega_{\rm m}$ is
the mean density divided by the critical density. For the canonical $\Lambda$
cold dark matter ($\Lambda$CDM) model in linear theory, $\gamma\simeq 0.55$,
whereas it differs for other models of DE (Amendola et al., 2018). We note
that the growth of cosmic structures is also influenced by the DM
characteristics.

The exact nature of dark energy can be tested via its equation-of-state,
$w=p/(\rho c^{2})$, which directly influences the expansion history. In the
general case, $w$ is a function of the scale factor, $w=w(a)$, and simplifies
to $w=-1$ in case of a cosmological constant. Using a truncated Taylor
expansion, one can write $w=w_{0}+w_{a}\,(1-a)$ and seek constraints on the
possible values in the $w_{0}-w_{a}$ plane. Ideally, we should have a model
that describes the redshift dependence of $w(a)$, and predicts how the growth
of structure is affected by a modification of gravity. Nonetheless, $\gamma$,
$w_{0}$ and $w_{a}$ provide convenient generic parameterizations that can be
used to compare the expected performance of various cosmological probes. When
Fisher matrix techniques are used (Euclid Collaboration: Blanchard et al.,
2020), the confidence areas in two-dimensional parameter spaces are ellipses.
The inverse of the area of the 2-$\sigma$ ellipse in the $w_{0}-w_{a}$ plane,
after marginalisation over all other cosmological and nuisance parameters,
defines the DE figure of merit (FoM; Albrecht et al., 2006; Laureijs et al.,
2011). Hence, the larger is the FoM, the better (more informative) is the
experiment.

Two of the best cosmological probes are galaxy clustering (GC) and weak
gravitational lensing (WL), especially once combined in the so-called $3\times
2$pt statistics. These are the two-point correlation of galaxies positions (GC
uses galaxies as test particles in the expanding space-time to map the mass
density contrast $\delta$ over time), the shear two-point correlations (WL
exploits the cumulative distortion effect of the tidal gravitational fields
along the line of sight on the shapes of the galaxy images; see e.g.
Kilbinger, 2015, for a review), and the cross-correlation of the lens
positions with the shear of the source galaxies, known as galaxy-galaxy
lensing. Despite tremendous progress in GC and WL experiments in recent years
(e.g. Alam et al., 2017; DES Collaboration et al., 2021; Gil-Marín et al.,
2020; Hildebrandt et al., 2020), much larger cosmological volumes need to be
surveyed.

Even though a clear “target precision” is lacking, Laureijs et al. (2011)
argues that a ${\rm FoM}\geq 400$ provides constraints on $w(a)$ and $\gamma$
that can test key aspects of our current cosmological model. This target ${\rm
FoM}\geq 400$ has driven the design of Euclid, a medium class mission of the
European Space Agency (ESA) that combines GC and WL.

To meet its primary science goal, Euclid (Laureijs et al., 2011; Racca et al.,
2016) has to observe a large fraction of the extra-galactic sky both with
multiband imaging and slitless spectroscopy. The sky area and mean number
density of galaxies are specified by the scientific requirements of the GC and
WL experiments (Laureijs et al., 2011; Rassat et al., 2008; Cropper et al.,
2013; Massey et al., 2013):

  * •

a $15\,000\,\deg^{2}$ survey of the extra-galactic sky, jointly for WL and GC
to be completed in six years with all the necessary calibrations;

  * •

an average galaxy number density of $30\,{\rm arcmin}^{-2}$ that are useful
for WL in the optical imaging data;

  * •

an average galaxy number density of $1700\,\deg^{-2}$ with reliable redshifts
from the $\mathrm{H}_{\alpha}$ emission line spectroscopic data, useful for
GC.111This is a revision of the Laureijs et al. (2011) value after the removal
in 2014 of blue-grism exposures in the wide survey.

Moreover, we want to minimise systematic residuals in the error budget by
maximising the uniformity in the coverage of the observed sky (Laureijs et
al., 2011; Scaramella et al., 2014).

The resulting survey is the Euclid Wide Survey (EWS) that will cover
$15\,000\,\deg^{2}$ to a minimum depth of $m_{\rm AB}=24.5$ mag in the visible
band with a signal-to-noise ratio (SNR) of $10$ for sources extended as the
$z\sim 1$ galaxies (details in Sect. 5.2.2 and Laureijs et al., 2011; Cropper
et al., 2016). In the near-infrared _Y, J_ and _H_ bands, a depth of $m_{\rm
AB}=24.0$ mag will be reached with a minimum SNR of $5$ for point sources
(Laureijs et al., 2011). This is sufficient to complement ground-based multi-
band observations that will be used to determine photometric redshifts
(photo-$z$s) for the WL sources (Euclid Collaboration: Desprez, G. et al.,
2020). Using slitless spectroscopy, Euclid will detect line emission with a
sensitivity of $f_{{\rm H}\alpha}\geq 2\times
10^{-16}\,\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}$; and a SNR of 3.5
for a typical source of size $0\aas@@fstack{\prime\prime}5$ (Maciaszek et al.,
2014, 2016). Space-based observations provide excellent and consistent image
quality at visible wavelengths for WL, and sufficient depth in the NIR bands,
both unattainable from the ground for this type of survey.

Figure 1: The Euclid timeline. The EWS has to be carried out within the six
year mission baseline and will start 3 months after the launch, following a
commissioning (1 month) and a Performance Verification period (2 months). An
extension of operations beyond the six years is possible and will be decided
in due time. A faster replanning is allowed during the Early Operation Phase
of six months. The three main Data Releases (DR#) are shown. The plan is to
have $2500\,\deg^{2}$ made public in DR1, to grow to $7500$ in DR2 and be
complete at DR3 for $15\,000\,\deg^{2}$. In addition, four Quick Data Releases
(Q#) are foreseen, each of $\sim 50\,\deg^{2}$.

With this design, Fisher matrix analyses forecast that EWS $3\times 2$pt
datasets will obtain a DE FoM of 500 for a non-flat $w_{0}-w_{a}$ CDM model in
an optimistic setting (defined by the range of scales used; see Euclid
Collaboration: Blanchard et al., 2020). It is interesting to compare Euclid
$3\times 2$pt forecasts with the constraints obtained by DES, which can be
considered as representative of ongoing Stage III surveys. DES Year 1
constraints for a $w_{0}-w_{a}$ CDM model (Abbott et al., 2019) are
intermediate results, where the uncertainties obtained for $w_{0}$ and $w_{a}$
are a factor of 10 larger than the Euclid forecasts in the pessimistic setting
(roughly translating to a factor of 100 in the FoM). The DES DE FoM from the
DES Year 3 analysis of a more complete dataset is not yet available, but we
can use instead the results on $S_{8}=\sigma_{8}(\Omega_{\rm m/0.3})^{1/2}$ to
compare Stage-III constraints with Stage-IV Euclid forecasts.

Using a $3\times 2$pt dataset and marginalizing over 25 nuisance parameters
and 7 cosmological parameters of a flat $w$CDM model, DES Year 3 gets $3.2\%$
errors on $S_{8}$ (DES Collaboration et al., 2021). Using the Fisher matrix
for $3\times 2$pt obtained in Euclid Collaboration: Blanchard et al. (2020)
for a $w_{0}-w_{a}$ CDM model, and adding additional 10 nuisance parameters to
represent the uncertainty on the shear multiplicative bias of each redshift
bin (in order to increase the number of nuisance parameters to have a fair
comparison with DES), we marginalize over a total of 25 nuisance and
cosmological parameters, getting $1.25\%$ errors on $S_{8}$ in a pessimistic
setting and $0.68\%$ errors in an optimistic setting. Although the comparison
is not completely fair, because of the different intrinsic alignments and bias
models, less nuisance parameters considered in the Euclid forecast and a
slightly different set of cosmological parameters, these numbers are roughly
in agreement with what one would obtain by simply scaling the DES Year 3
results for the Euclid increase in area and source number density.

The EWS has to be carried out within the six year mission baseline and will
start 3 months after the launch, following a commissioning (1 month) and a
Performance Verification period (2 months). Figure 1 shows the Euclid timeline
with the data release planning. The first major ‘data release’ (DR1),
corresponding to 2500 deg2 of the EWS is planned to take place one year after
$T_{1}$ ($T_{1}=14$ months after launch), the second data release (DR2), is
expected to release 7500 deg2 three years after $T_{1}$, and the final one
(DR3) will release the full survey (15 000 deg2) six years after $T_{1}$. In
between there will be other ‘quick data releases’: Q1 of $50$ deg2 is planned
at $T_{1}$, and Q2, Q3 and Q4, will take place two, four, and five years after
Q1, respectively.

In addition to the main survey, a significant fraction of time will be spent
to calibrate the instruments and to characterise the target galaxies. This
results in some fields to be observed to greater depth than the wide survey
(typically 2 magnitudes deeper). These deep fields have great legacy value
beyond the cosmological core science. While aspects of non-core science did
not influence the design of the spacecraft and instruments, they are taken
into account in the design of the EWS to maximise Euclid scientific return. In
fact, it must be noticed that the large decrease in the background with the
wavelength dramatically increases the SNR in the NIR bands when compared to
Earth-based observations affected by airglow, which instead increases with
wavelength. This makes even a small space telescope competitive with a large
ground telescope which suffers from a background dominated by atmospheric
emission in the NIR bands. The relative gain is such that, in order to cover
the same areas planned for Euclid and at the same depths, a ground-based NIR
survey on existing facilities would need to observe for several centuries.
Regarding other space-based facilities, we notice that the James Webb Space
Telescope (JWST) will be in orbit as well and with its diameter of
$6.5\,\mathrm{m}$ will go much deeper and faster than Euclid although only on
very small areas (JWST field of view is 75 time smaller than the Euclid one).
Hence the two facilities are complementary and, moreover, JWST will likely
benefit from targets selected from the Euclid Surveys.

The challenge is to fit all these observations into a finite time allocation
set by the limitation of the mission, which is six years, whilst fulfilling a
wide range of constraints, which are reviewed in detail in this paper. Part of
the survey optimisation involves selecting the best areas of sky to use, which
in turn relies on a good model of the properties of the observable sky, such
as Galactic extinction and the zodiacal background. We also need to model the
distribution of (bright) stars, as their stray light lowers the observed
galaxy number density.

This paper focuses on the design of the EWS, while the deep fields are
described in a companion paper (Scaramella _et al._ , in prep., hereafter
[Sc21]). The EWS design takes into account the main backgrounds which impact
any large area survey, the sequence of operations, the many limitations to the
pointing of the telescope. The EWS is at an advanced stage, fulfilling the key
survey requirements over the full mission. Survey scenarios at this stage
therefore show the detailed feasibility of the mission but are subject to
further optimisation. Nevertheless, the results we present and their
discussion are instructive and useful for any future large area survey from
space or ground which aims to combine imaging and spectroscopy.

The paper is organised as follows. The spacecraft is described in Sect. 2,
followed by a summary of Euclid’s instruments in Sect. 3. In Sect. 4 the
reference observation sequence (ROS) is introduced, including a study of
dithering scenarios. Models of zodiacal light, stray light effects, and other
environmental properties, define the ‘region of interest’ (RoI) used as input
for the implementation of the Euclid Reference Survey Definition (RSD). These
effects and the properties of the resulting RoI are presented in Sect. 5,
where we also discuss complementary ground-based observations. Section. 6
describes the implementation of the calibration program. Observations of
sample characterisation fields and the EDS are briefly mentioned in this
context. The construction of the EWS is presented in Sect. 7. We present the
most recent outcome of the survey optimisation (mid 2021) in Sect. 8. This
solution is a good proxy for the actual survey. We conclude in Sect. 9.

In the Appendix we provide a list of acronyms used in this paper.

##  2 The spacecraft and telescope

###  2.1 The spacecraft

The spacecraft comprises a service module (SVM) and a payload module (PLM),
connected by an interface structure designed to maximise thermal decoupling.
The PLM includes the main instruments, the folded beam optical components of
the telescope, the radiators, and the fine guidance system (FGS). The SVM
provides the main Spacecraft services: Power Generation, conditioning and
distribution, Sun shield and Solar Array, telecommunication with ground (Low
and High Gain antenna), Attitude and Orbit Control System (including FGS) and
support the Instruments Warm Electronics. Details are given in Laureijs et al.
(2011) and Racca et al. (2016).

Figure 2: _Left panel:_ Euclid Spacecraft Reference Frame. $X_{\rm SC}$ points
toward the Sun disk center. The edge and center of the joint FoV are offset by
$None$ and $None$, respectively. The longer side of the FoV is typically
aligned with ecliptic meridians during observations. _Right panel:_ Euclid
Optical Reference Frame. Notice that for the latter the $X_{\rm opt}$ $Y_{\rm
opt}$ plane is defined as looking onto the sky with $\hat{z}$ towards the
spacecraft. There are four additional chips used as fine guidance sensors
(FGS) placed on each side of the VIS FoV.

Euclid has severe constraints in pointing to ensure maximal thermal stability,
which are described in this paper and limit the standard operations. Therefore
it is important to describe in detail the attitude of the spacecraft. The
Euclid Spacecraft Reference Frame $(O_{\rm SC},X_{\rm SC},Y_{\rm SC},Z_{\rm
SC})$ is defined as follows (see Fig. 2 for a graphical representation):

  * •

$O_{\rm SC}$: origin is at the point of intersection of the longitudinal
launcher axis with the launcher adapter interface plane (the plane of
separation of the spacecraft from the launcher);

  * •

$+Z_{\rm SC}$ is in the direction perpendicular to the launcher interface
plane, positive in the direction of the launch;

  * •

$+X_{\rm SC}$ is in the launcher interface plane, directed to a physical mark
on the interface ring nominally aligned with the solar array such that the
$+X_{\rm SC}$ vector is perpendicular to the solar array and pointing towards
the sun;

  * •

$+Y_{\rm SC}$ is in the remaining direction of the right-handed orthogonal
triad.

The orientation of the telescope optical reference frame, projected onto the
sky, is also specified in Fig. 2. The field of view (FoV) Reference Frame is
centred on the centre of the FoV itself and is such that

$X_{\rm FoV}=-X_{\rm OPT}$ and $Y_{\rm FoV}=Y_{\rm
OPT}-0\aas@@fstack{\circ}82$, taking into account the shift of the edge of the
FoV of $0\aas@@fstack{\circ}47$, and its half size of of
$0\aas@@fstack{\circ}35$ (see Fig. 2 and Table 1).

###  2.2 The telescope

Euclid’s PLM (Racca et al., 2016) is designed around a three-mirror anastigmat
Korsch design telescope with silicon carbide (SiC) mirrors and truss (Korsch,
1972; Pamplona et al., 2016). The sizes of the telescope components are:
primary pupil $R_{1}=0.6$ m, primary mirror (M1) stopper $R_{2}$ = 0.1975 m,
spider arm mean length $R_{3}$ = 0.44 m, spider arm thickness $L$ = 0.012 m.
This provides a total collecting area of
$A=\pi\left(R_{1}^{2}-R_{2}^{2}\right)-3R_{3}L=0.99\,\mathrm{m}^{2}$. Euclid
has two instruments onboard, the visible imager (VIS; Sect. 3.1) and the near-
infrared spectrometer and photometer (NISP; Sect. 3.2). The wavelength
separation at $\sim 920$ nm between the two instruments is performed by a
dichroic plate located at the exit pupil of the telescope. The two focal
planes image the same part of the sky, allowing multiple data acquisition with
a single telescope pointing (see Sect. 4.1). The coordinates of the focal
plane as projected on the sky are shown in Fig. 2.

###  2.3 Pointing angles

The main reference frames are shown in Fig. 2. The following angles (see Fig.
3) have operational ranges that constrain a pointing and therefore target
visibility:

  * •

Solar aspect angle (SAA): the angle between the spacecraft’s $+Z_{\rm SC}$
axis (telescope pointing direction) and the direction to the centre of the
Solar disk;

  * •

alpha angle (AA): the angle between the Sun vector projected onto the $X_{\rm
SC}$–$Y_{\rm SC}$ plane and the $+X_{\rm SC}$ axis. It increases as the
spacecraft rotates clockwise about its $+Z_{\rm SC}$ axis;

  * •

Solar panel Solar aspect angle (SPSAA): The angle between the spacecraft
$+X_{\rm SC}$ axis and the direction to the centre of the Solar disk.

Figure 3: Illustration of the relevant pointing angles defined in the text. In
the top right image the minimal SAA (pointing towards the Sun) is indicated,
while the maximum SAA (pointing away from the Sun) adopted in the survey is
$110\degr$ (see also Sect. 2.5). In the lower right image AA is shown; the
maximum range allowed for the survey is $\pm 5\degr$, corresponding to a large
margin with respect to the spacecraft capabilities.

###  2.4 Orbit and operation mode

Euclid will operate at the Sun-Earth Lagrangian point L2, following a yearly
orbit with a libration within $\pm 0\aas@@fstack{\circ}41$ across the ecliptic
plane (Fig. 4). The Lissajous orbit is dynamically unstable and requires
regular orbital maintenance, currently planned to last one day every four
weeks, i.e. $\sim 3\%$ of the total mission time. This orbit offers a very
stable thermal environment and maximises the visible sky at any time.

Figure 4: _Left panel:_ Euclid’s transfer orbit following launch at time
$T_{0}$, and the subsequent Lissajous orbit at the Sun-Earth Lagrangian point
L2 (SEL2). _Right panel:_ Euclid’s main step-and-stare operating mode, showing
the steps as rotations around $X_{\rm SC}$.

Euclid employs a step-and-stare mode, acquiring data on a fixed sky field and
then slewing to the next pointing. Slews come in two types, depending on the
value of the eigenslew $\epsilon$ defined as the angle between one field
quaternion222In 3-dimensional space, according to Euler’s rotation theorem,
any rotation or sequence of rotations of a rigid body or coordinate system
about a fixed point is equivalent to a single rotation by a given angle
$\theta$ about a fixed axis (called the Euler axis) that runs through the
fixed point. Therefore the quaternion fully describes the spacecraft attitude
and a single rotation quaternion relates one pointing to another. and the
next, which can be decomposed in an arc connecting two separate pointings on
the sky plus a rotation around $Z_{\rm SC})$. Eigenslews $\epsilon\leq$$$ are
considered ‘small slews’, and $\epsilon>$$$ are ‘large slews’. Euclid has
adopted a specific hybrid on-board Attitude and Orbit Control System (AOCS)
architecture where slews are performed using four reaction wheels in stop and
go mode, and pointing stabilisation is achieved using low-noise cold gas
micro-thrusters. With this solution, slew time is reduced thanks to the high
torque provided by the reaction wheels. Slews only consume cold gas for the
tranquilization transient phase, large slews requiring longer tranquilization
periods. Therefore, Euclid’s lifetime slew budget is limited to $950$ large
and $2.5\times 10^{5}$ small slews. The latter are weighted in the budget: for
$\epsilon\leq$$$ they count as a single slew, and for
${1\aas@@fstack{\circ}2}<\epsilon\leq$$$ a penalty occurs proportional to
$\epsilon$. The slew constraints imply that the EWS must be implemented mostly
with small slews (preferably with $\epsilon\leq$$$), and that fields observed
consecutively in time must be spatially adjacent.

###  2.5 Transits and visibility

The SAA and AA ranges define how much the spacecraft can deviate from
observing at ‘transit’ meridian, which are the two ecliptic meridians defined
by the perpendicular to the spacecraft’s $X_{\rm SC}$ axis (for a transit
${\rm SAA}=90\degr$). The size and geometry of the Sun shield limit the SAA
and AA ranges that can be used for observations. The ranges (and variations
of) SAA and AA are constrained further by the fact that Euclid needs great
thermal stability to minimise temporal point spread function (PSF) variations.
The SAA limits allow the telescope to ‘depoint’ (i.e. to rotate around $Y_{\rm
SC}$) from transit to a maximum of $3\degr$ towards the Sun (${\rm
SAA}=87\degr$), and up to $20\degr$ away from the Sun (${\rm SAA}=110\degr$),
while the AA limits allow the telescope to rotate around $Z_{\rm SC}$ up to
$|{\rm AA}|\leq 5\degr$.

In addition, the orbit libration mentioned in Sec. (2.4) imposes an additional
$0\aas@@fstack{\circ}41$ buffer for the spacecraft orientation angles with
respect to the Sun. This decreases the allowed ranges of both SAA and AA by
$0\aas@@fstack{\circ}41$ on each side of their range intervals.

The allowed ranges of SAA and AA define the instantaneous sky visibility,
shaped along the full circle defined by the two meridian transits (see Fig.
5). The orbit progresses with Euclid’s revolution around the Sun, continuously
changing the visible sky enabling a full sky survey. Given the symmetry of the
transit meridians, the spacecraft has access to the same region of the sky
every six months by pointing in the ‘leading’ direction (towards the direction
of motion around the Sun) or, flipping the telescope, six months later
pointing in the ‘trailing’ direction.

Figure 5: Strip of sky observable by Euclid at a given epoch (in light red).
The dashed blue line is the great circle orthogonal to the Sun with the
satellite observing orthogonal to the sun for a transit at $0\deg$. The
asymmetry of the red region with respect to the transit blue lines is due to
the asymmetry of the usable SAA range with respect to $90\degr$ (wider when
pointing away from the Sun).

Figure 5 shows an example of the instantaneous sky visibility for a generic
transit. It is evident that two small regions located at the ecliptic poles
have perennial (continuous) visibility, whereas the lowest ecliptic latitudes
can be observed only when crossed by a transit meridian. In practice, at any
given time (or from a given position in the orbit) Euclid can scan an annulus
on the sky, and consequently most of the sky must be observed at or close to
transit. Observations that require long and regular visibility (such as for
the EDS) can only be fulfilled in a very limited area on the sky at high
ecliptic latitudes.

In general, a depointing (SAA $\neq 90\degr$) induces a rotation of the focal
plane with respect to the transit ecliptic meridian. To counterbalance, the
spacecraft must rotate around the $+Z_{\rm SC}$ axis to keep the alignment
with the transit meridian. The amplitude of this rotation must stay within
$|{\rm AA}|\leq 4\aas@@fstack{\circ}59$ to fulfil the thermal and orbit
libration constraints. This effect becomes larger with increasing ecliptic
latitudes and requires ad hoc solutions for the scheduling (see Sect. 7.4.2).

##  3 The instruments

Figure 6: _Left panel_ : The VIS FPA, illustrating the detector placement. The
dashed line shows the joint FoV of both instruments. Two narrow strips at the
extremes of the Y axis are outside the joint FoV. _Right panel_ : The NISP
FPA, with two narrow strips at the extremes of the X axis outside the FoV (the
Reference Frame is $X_{\rm FoV}-Y_{\rm FoV}$ see also Fig. 2 and the
definition at the end of Sect. 2.1).

###  3.1 VIS instrument

The visible imager instrument (VIS) contains a focal plane array (FPA)
consisting of $6\times 6$ Teledyne e2v CCDs ($4\,{\rm k}\times 4\,\rm{k}$
pixels each) with a Nyquist driven plate scale of
$0\aas@@fstack{\prime\prime}1$ pixel-1, yielding a field-of-view (FoV) of 0.56
deg2 including detector gaps. For details, see Table 1 and the left panel of
Fig. 6. In addition to the gaps, the central four rows in each detector serve
as charge injection lines (which cannot be read out).

VIS is optimised to detect spatially resolved images of galaxies in the
550–900 nm passband (hereafter referred to as the ‘VIS band’; Cropper et al.,
2014, 2016). The VIS nominal survey images (dithered; see Sect. 4) will have
at a minimum SNR of $10\,\sigma$ (average $15.8\,\sigma$) for extended sources
(full width at half maximum, FWHM, such that
${\rm{FWHM}_{gal}}>1.25\,{\rm{FWHM}_{PSF}}\simeq
0\aas@@fstack{\prime\prime}225$) at a detection limit of $m_{\rm AB}=24.5$ mag
(see Sect. 5.2.2). These will enable accurate galaxy shape measurements for an
average of 30 arcmin-2 galaxies over the survey area (Laureijs et al., 2011;
Massey et al., 2013). Besides WL shape measurement, VIS data are also used to
improve photo-$z$ estimation, by enabling optimal photometric extraction of
the less resolved, complementary ground observations thanks to its
diffraction-limited image quality. To maximise the SNR for the shape
measurements, the VIS band is rather broad (see Fig. 7), encompassing the
Sloan Digital Sky Survey (SDSS) $r$ and $i$ bands, and the bluer half of the
$z$-band.

The VIS central data processing unit constructs the images from the pixel data
and compresses them in a lossless manner in approximately $250\,\rm{s}.$ No
additional image processing will be done on board to maintain full control
over systematic errors. The data will be transferred to the ground with a rate
of approximately $520\,\rm{Gbit/day}$ (Racca et al., 2016).

###  3.2 NISP instrument

The near-infrared spectrometer and photometer (NISP; Maciaszek et al., 2014,
2016) contains an array of $4\times 4$ HAWAII-2RGs detectors ($2\,{\rm
k}\times 2\,{\rm k}$ pixels each) with a plate scale of
$0\aas@@fstack{\prime\prime}3$ pixel-1, undersampling its diffraction limited
PSF (Fig. 6, right panel). Table 1 shows the size of the FPA, FoV, and gaps
between the detectors. Note that in the $Y$ direction of the focal plane, the
central gap ($86\aas@@fstack{\prime\prime}1$ wide) is narrower than the two
outer gaps ($101\aas@@fstack{\prime\prime}4$ wide).

NISP is designed to carry out slitless spectroscopy (NISP-S) and imaging
photometry (NISP-P) at near-infrared (NIR) wavelengths (see Fig. 7). By using
its grism and filter wheel assemblies (GWA and FWA, respectively), NISP can
switch between slitless spectroscopy and imaging modes, which are detailed in
the following. NISP will transfer data to the ground with a rate of
approximately $290\,\rm{Gbit/day}$, for a total of $810\,\rm{Gbit/day}$,
smaller than the spacecraft allocation of $850\,\rm{Gbit/day}$.

Table 1: Sizes of the VIS and NISP focal planes and their corresponding FoVs,
as well as angular sizes of detectors, detector gaps, and width of the VIS
charge injection lines (see Figs. 2 and 6).

| VIS | NISP
---|---|---
| X Size | Y Size | X Size | Y Size
Focal Plane [mm] | 302.71 | 336.59 | 155.85 | 164.48
Plate scale [arcsec/mm] | 8.33 | 16.70
FoV [deg] | 0.700 | 0.778 | 0.723 | 0.763
Detectors [′] | 6.82 | 6.89 | 10.21 | 10.21
Detector gaps [′′] | 12.7 | 64.4 | 50.6 | 101.4 / 86.1
Charge injection gaps [′′] | N/A | 0.4 | N/A | N/A

Figure 7: Total sensitivity of Euclid’s photometric and spectroscopic bands
(solid and dashed lines, respectively). The sensitivity (in electrons per
photon) includes all optical surfaces as well as the detectors’ average
quantum efficiency, all considered at their expected end of life (EOL)
performance after six years at L2 (degraded by radiation damage and
contamination, see e.g. Gaspar Venancio et al., 2020).

Figure 8: The (simplified) ROS executed for each field, observing four
dithered pointings. Each pointing results in a VIS, $Y$, $J$ and $H$ image,
and one spectral exposure. Filter and grism wheel movements are shown,
together with stabilisation times (“Stab”). Not shown are wheel movements
during the dither slew, readout and processing done simultaneously with other
operations and slews, overheads, and various inline calibrations.

####  3.2.1 Slitless spectroscopy

Euclid has a “blue” grism (BGS000) covering the 0.92–1.25
$\mathrm{\SIUnitSymbolMicro m}$ wavelength range, and three “red” grisms
(RGS000, RGS180 and RGS270) covering 1.25–1.85 $\mathrm{\SIUnitSymbolMicro m}$
(Costille et al., 2016). The blue grism is not used for the EWS observations,
and only employed for part of the EDS. The numeric labels indicate the
dispersion directions, offset by $90\degr$ for the red grisms. Different
dispersion directions are required to disentangle the spectra of various
objects in the slitless spectroscopic exposures of the EWS. Due to a non-
conformity discovered in 2020 (Laureijs et al., 2020), the RGS270 will not be
used in the survey observations. Instead, the RGS000 and RGS180 will be
rotated in the ROS by $-4\degr$ and $+4\degr$, respectively (see Sect. 4.1 for
details).

The red grisms disperse the light with nearly constant spectral resolution of
$1.354\,{\rm nm/px}$ which gives ${\cal R}=\lambda/\Delta\lambda\sim 450$ for
an object of diameter $0\aas@@fstack{\prime\prime}5$. This is larger than the
minimum required value of 380 to achieve an error on the measured redshift of
$\sigma(z)<0.001(1+z)$. The spectroscopic observations support the GC probe
and are optimised to detect the redshifted H$\alpha$ emission of galaxies at
$z=0.9$–$1.8$. With a detection limit of $2\times
10^{-16}\,\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}$ $(3.5\,\sigma)$
for a typical source of size $0\aas@@fstack{\prime\prime}5$ at 1600 nm (see
Sect. 5.2.2), NISP should be able to determine spectroscopic redshifts for at
least 1700 galaxies $\deg^{-2}$ on average in the corresponding wavelength
range 1250–1850 nm. This estimate, however, strongly depends on the intrinsic
luminosity function of H$\alpha$ emitters, which is still uncertain in the
redshift range observed by Euclid (Pozzetti et al., 2016). Because the
redshift is based on an emission line, passive galaxies will be
underrepresented in the spectroscopic sample, with a bias against dense
enviroments.

####  3.2.2 Photometry

Photometry will be measured for objects down to a minimum of $m_{\rm AB}=24.0$
mag for $5\,\sigma$ point-like source in the $Y$, $J$ and $H$ passbands. The
photometric data support the GC experiment by providing the reference images
needed to extract the spectra in the (slitless) dispersed images. The NIR
photometric data, however, critically complement the ground-based observations
(Sect. 5.5) in getting accurate photometric redshift estimates, at the primary
probe level mainly needed for the WL experiment and essential for many other
astronomical science aspects.

###  3.3 The Euclid joint field of view

The intersection of the VIS and NISP FoVs defines the Euclid joint FoV, with
the $X$ and $Y$ dimensions defined by VIS and NISP, respectively. The Euclid
FoV is $0.53$ deg2. Its borders are shown by the dashed lines in Fig. 6,
resulting from the overlap of the VIS and NISP FoVs aligned on an edge. The
left and right edges of the NISP FoV and the top and bottom edges of the VIS
FoV are outside the joint Euclid FoV.

##  4 Observing and dithering sequences

###  4.1 The reference observation sequence (ROS)

Euclid executes a highly optimised reference observation sequence (ROS; see
Fig. 8) at every survey field, exploiting the instruments’ inter-operability.
The ROS visits four nearby pointings at every field, covering an area of
$0.53\,\deg^{2}$ (see Sect. 3.3) common to both instruments and fulfilling the
galaxy number density and SNR requirements detailed in Laureijs et al. (2011).
Small ‘dither slews’ are performed between the pointings, taking 66 s.

At each pointing, VIS takes an image and NISP a simultaneous spectral exposure
with the red grism, both lasting about 570 s (note that these times are not
frozen yet). Once the VIS shutter is closed, the GWA and FWA move for the
three NISP images of 112 s each. A 2 s margin is allocated between the end of
a NISP exposure and the wheel actuation, ensuring the NISP exposure is
completed before the wheel is moving to avoid compromising the last frame.
Moreover, a stabilisation time of 10 s is considered between a wheel movement
and the following exposure. During the NISP imaging, VIS takes biases, flats
and other calibration frames. In addition, VIS also takes a shorter science
exposure of 108 s during the $H$-band exposure in the first pointing, in order
to help with the PSF dynamic range on relatively bright stars that saturate
during the standard, longer exposures. Details of these sequences are given
e.g. in Cropper et al. (2016) and Maciaszek et al. (2016).

After each pointing a dither step is applied and a new grism position is
selected. The ROS uses the RGS000 and RGS180 at two angles each, offset by
four degrees, to allow for sufficient decontamination of the overlapping
slitless spectra.

The total duration of the ROS, including dither slews and overheads is 4214 s.
At the end of the ROS a slew towards the next field is performed. Most of
these slews are small ($\epsilon\leq$$$) and referred to as ‘field slews’. The
slew duration is a function of the (eigen-) angular rotation. On average it is
$182\,{\rm s}$, implying a total length of $4396\,{\rm s}$ for the ROS (slews
included). This is less than the upper limit of $4400\,{\rm s}$ as defined at
mission system level during budget allocation. On occasion, the ERS requires
larger slews that are limited to a maximum number of 950 over the full
mission. In the most recent EWS solution (see Sect. 8), the ‘large slews’
comprise 1% of all non-dither slews applied.

###  4.2 The dithering strategy

For each field, the ROS obtains multiple exposures with dithered pointings to
mitigate detector defects and cosmic rays, and to meet the required depth. The
depth will vary across the field, not only because of masked defects, but
predominantly because the NISP and VIS focal planes have different detector
and gap sizes (see Fig. 6, and Table 1). The dithering strategy between
pointings, used for the ERS, must meet the following requirements:

  1. 1.

95% of the survey area shall be covered with at least three exposures in VIS.
90% of the survey area shall be covered by at least three exposures in each of
the NISP $Y$, $J$ and $H$ bands.

  2. 2.

90% of each survey field shall be covered by three or more spectroscopic
exposures, and 50% by four or more spectroscopic exposures (using different
grism orientations).

  3. 3.

The NISP imaging of the fields covered by the NISP spectroscopic channel in
the EWS shall be acquired over the whole image with depth on average fainter
than $m_{\rm AB}=24.0$ mag for $5\,\sigma$ point-like source.

To analyse the sky coverage of a given dither pattern, we have simulated the
ROS observations at the pixel level for a nearly square sky region. The latter
covers several FoVs in each of the two dimensions ($N\times N$ joint FoVs,
where $3\leq N\leq 8$) to avoid boundary effects. The pixel count statistics
(number of exposures per sky area) are then computed with an integration time
map calculator.

In the various dither patterns, the minimum dither step in each of the two
directions is sized such as to ensure that the slew is larger than the largest
of the detector gaps in that direction (which, according to Table 1 are the
NISP gaps). The maximum dither step is sized to prevent gaps of one line or
row of detectors to overlap with the next gaps. In our simulations, we used
slightly larger values for the detector gaps than the ones present in the as-
built instruments (see Table 1). For VIS, we used
$13\aas@@fstack{\prime\prime}6\,$ in the $X$ direction,
$67\aas@@fstack{\prime\prime}6\,$ in the $Y$ direction and
$0\aas@@fstack{\prime\prime}4\,$ for the width of the charge injection lines,
while for NISP we used 50″ in $X$ and 100″ in $Y$. Subsequent fields are
shifted by about 0.7 degrees, setting in the simulation an overlapping of 1%
(in area) between contiguous FoVs. This constitutes the basic set-up of our
simulations.

The dither step size is affected by errors, namely after a dither slew there
is a pointing error of 11″at 3 $\sigma$. The overlap between contiguous FoVs
is also affected by an uncertainty related to the absolute pointing error
(APE), the baseline input being the requirement applicable to industry and its
translation into APE. The line-of-sight has an uncertainty off-set introduced
by the rotation of the filter and grism wheels (that is compensated by the
spacecraft AOCS).

We produce coverage maps in two ways. The first method is deterministic
(method D). In this case, we consider the basic set-up and add a deterministic
small shift (of 11″) to each step of the dithering pattern. The implementation
of this further displacement is necessary to be safe in filling the gaps when
considering the estimate of the of the uncertainties in the dithering step. No
random errors are considered in this method. In the second method (MC), we
consider uncertainties on dithering steps and directions (a pointing error of
11″at 3 $\sigma$), on off-set repeatability, and other relevant parameters. In
this method, the actual overlapping between contiguous FoVs is affected by the
APE uncertainty; we consider an error on APE with a Gaussian random amplitude
of 4.5″at $1\,\sigma$ and uniform random orientation. Monte Carlo iterations
are then run on a representative patch to extract the coverage maps and the
statistics of pixel numbers for the desired area inside the simulated patch
for each of the dither patterns presented in Sects.4.2.1 and 4.2.2.

Figure 9: The seven dither patterns that we have analysed are shown in the
spacecraft reference frame (see Fig. 2). For clarity, a shift of $5\arcsec$
(or of $-5\arcsec$) in the $X{\rm sc}$ direction is applied to the “J” (or to
the “N”) pattern in the figure so to distinguish it from the “S” pattern. Only
the “J” and the “S” patterns, which have identical first two steps, are fully
compliant with the survey stringent requirements.

####  4.2.1 Minimal pattern

The dither pattern defined in Laureijs et al. (2011) was minimal in the sense
that it only respected the stringent constraints on the size of a dither step.
The latter is constrained to a minimum of $100\arcsec$ by the reaction wheels
to prevent mechanical damage to wheel ball bearings in small rotation regime,
and to a maximum of $396\arcsec$ by the size of the star catalogue available
to the AOCS. The pattern (referred to as “J” pattern given its shape, see Fig.
9) is defined as follows, starting from the pointing of the first exposure:

  * •

Dither step 1: $\Delta X_{\rm SC}=50\arcsec$, $\Delta Y_{\rm SC}=100\arcsec$;

  * •

Dither step 2: $\Delta X_{\rm SC}=\leavevmode\nobreak\ \leavevmode\nobreak\
0\arcsec$, $\Delta Y_{\rm SC}=100\arcsec$;

  * •

Dither step 3: $\Delta X_{\rm SC}=\leavevmode\nobreak\ \leavevmode\nobreak\
0\arcsec$, $\Delta Y_{\rm SC}=100\arcsec$.

These are the minimum values that need to be ensured by the AOCS. As discussed
earlier, in order to account for the error in the dithering step, the dither
is commanded to be $11\arcsec$ larger than the minimum step.

In method D, uncertainties in the telescope pointing were not considered. In
total, we simulated $3\times 3$ joint FoVs (i.e. nine adjacent executions of
the ROS). For a simple visualisation of the integration time map, we simulated
the sky coverage with the resolution of the VIS and NISP FPAs degraded to 1″.
The width of the VIS charge injection lines were increased to 1″ to be
included in the simulation. The top panels in Fig. 10 show, for VIS and NISP,
the central $\simeq 50\aas@@fstack{\prime}0\,\times\,56\aas@@fstack{\prime}7$
of the “J” integration time map, i.e. a single survey field and its boundaries
(containing $\simeq 10^{7}$ simulated pixels).

Figure 10: Illustration of the 4 stacked integration time maps for “J” (top)
and “S” (bottom) dithering patterns. The left half displays the maps for VIS,
and the right half those for NISP. The colour scaling gives the number of
stacked exposures per map pixel. Details are given in the text.

In method MC, the statistics on the number of exposures per map pixel were
evaluated with the same tool, but now considering random errors on some
pointing parameters. We simulated a larger sky area ($7\times 7$ joint FoVs)
to allow a corresponding increase in the number of realisations and a better
characterisation of the overlapping regions between the FoVs. Here, we adopted
a resolution of $0\aas@@fstack{\prime\prime}2$ to accurately quantify the
effect of the finer geometric characteristics of the VIS and NISP FoVs.
Statistics extracted for areas of different sizes (for example of
$4\aas@@fstack{\circ}0\times 4\aas@@fstack{\circ}0$) around the central joint
FoV show that the coverage requirements are fulfilled at a global level. On a
field-to-field basis, the coverage can vary within the associated standard
deviations.

####  4.2.2 The baseline “S” dither pattern

Looking for a potentially better dither pattern, we analysed six other
patterns with four vertices (Fig. 9). Two cases were proposed by Arendt,
Kashlinski and Mosley based on previous experience (private communication;
labelled as AKM2 and AKM3), and four cases (“S”, “R”, “N”, “X”) were taken
from Markovič et al. (2017), who focused on optimising the performance of the
spectroscopic survey only, without taking into account the concurrent
constraints required by the imaging part. The two AKM patterns exceed the
maximum step size, with only AKM3 satisfying the constraint on the joint
visibility.

Table 2: The statistics (mean and standard deviation) on the number, $X$, of exposures per pixel for the “S” dither pattern: percentages of covering for individual ($X=0,1,2,3,4$) and cumulative ($X>4$, $X\geq 3$, $X\geq 4$) bins. The percentage of pixels with $X=0$ or 1 is negligible or very small for both VIS and NISP. Boldface values refer to the requirements 1 and 2 of Sect. 4.2.  | VIS | NISP
---|---|---
Covering | mean $\pm$ st. dev. (%) | mean $\pm$ st. dev. (%)
$X=0$ | $(3.57\pm 5.71)\times 10^{-6}$ | $0.00\pm 0.00$
$X=1$ | $0.23\pm 0.05$ | $0.99\pm 0.04$
$X=2$ | $4.34\pm 0.03$ | $8.13\pm 0.02$
$X=3$ | $47.51\pm 0.09$ | $40.46\pm 0.10$
$X=4$ | $36.52\pm 0.32$ | $42.12\pm 0.16$
$X>4$ | $11.40\pm 0.29$ | $8.30\pm 0.17$
$X\geq 3$ | ${\bf 95.43\pm 0.03}$ | ${\bf 90.88\pm 0.05}$
$X\geq 4$ | $47.92\pm 0.08$ | ${\bf 50.42\pm 0.06}$

Among the four possibilities suggested in Markovič et al. (2017), only the “S”
pattern, which is the one closest to the reference “J”, meets all constraints.
Its statistics on the number of exposures per pixel derived from simulations
performed considering the most relevant sources of uncertainty are given in
Table 2. The “S” pattern improves, particularly for NISP, upon the “J”
pattern, decreasing the fraction of pixels with a single exposure (from
$0.59$% to $0.23$% in the case of VIS and from $3.47$% to $0.99$% in the case
of NISP) while increasing the fraction of pixels with two exposures (from
$3.91$% to $4.34$% in the case of VIS and from $4.79$% to $8.13$% in the case
of NISP). For the “S” pattern, the percentages of $X\geq 3$ covering are
$95.43\pm 0.03$ and $90.88\pm 0.05$ for VIS and NISP, respectively, while the
percentage of $X\geq 4$ covering for NISP is $50.42\pm 0.06$ (the statistics
reported in Table 2 are extracted from 500 realisations). Thus, at least under
the considered uncertainty in specifications, the requirement 1 of Sect. 4.2
for VIS and NISP imaging is satisfied at $\sim 14.3\,\sigma$ level and at
$\sim 17.6\,\sigma$ level, respectively, while the requirement 2 of Sect. 4.2
for NISP spectroscopy is satisfied at $\sim 17.6\,\sigma$ level and at $\sim
7\,\sigma$ level for the $X\geq 3$ covering and $X\geq 4$ covering,
respectively. We note that the standard deviation of the cumulative case
($X\geq 3$ or $X\geq 4$) covering cannot be derived by a simple analytical
propagation of the standard deviations of simpler cases because of the
presence of mutual correlations. These, however, are taken into account in the
numerical simulation.

The “S” pattern was therefore chosen as the baseline dither pattern for the
ERS (see also Markovič et al., 2017). The coverage map is shown in the bottom
panels of Fig. 10, and in Fig. 11 we display the “S” pattern as it appears on
sky. It is defined as follows.

  * •

Dither step 1: $\Delta X_{\rm sc}=50\arcsec$, $\Delta Y_{\rm sc}=100\arcsec$.

  * •

Dither step 2: $\Delta X_{\rm sc}=\leavevmode\nobreak\ \leavevmode\nobreak\
0\arcsec$, $\Delta Y_{\rm sc}=100\arcsec$.

  * •

Dither step 3: $\Delta X_{\rm sc}=50\arcsec$, $\Delta Y_{\rm sc}=100\arcsec$.

We note that by increasing the number of dithers and their step size, a more
uniform coverage can be achieved (see e.g. Rowe et al., 2011). However, this
needs to be balanced against the total survey area, the mission duration,
readout overheads etc. Other patterns can still be studied and implemented in
case of a revision of the current survey and hardware limitations.

Figure 11: The adopted “S” dither pattern projected on the sky for a leading
(left) and a trailing (right) pointing. See Fig. 2 for reference frames. The
red frame is the first to be observed, and the dots mark the centre of each
frame. Note how the pattern on the sky is invariant if the telescope is
flipped.

##  5 The EWS region of interest

The constraints that drive the implementation of the Euclid Survey areas fall
into three main categories (see Sect. 1):

  * •

environmental;

  * •

calibration needs;

  * •

spacecraft constraints.

Environmental properties are external physical constraints of the observable
sky, namely zodiacal background, Galactic extinction, foreground stellar
density and blinding stars. The zodiacal background and extinction affect the
SNR, as does the stray light from bright stars and Galactic plane. The star
density also affects the number of useful objects, because their light is
dispersed and overlaps with the galaxy spectra of interest, or affects the
ability of measure reliable shapes. Moreover, an increased star density also
increase the multiplicative bias in weak lensing shear estimates, if
unaccounted for (Hoekstra et al., 2017).

Together, these constraints lead to the definition of the EWS RoI, a
collection of four contours enclosing two larger ‘mainlands’ and two smaller
‘islands’. Each contour is defined by a series of joint segments derived from
either the zodiacal light (ecliptic latitude segment), dust extinction
(Galactic caps segment), and the stellar density (Galactic latitude segment).

###  5.1 Environment models

The cosmological measurements are strongly mediated by the nature of the areas
of the sky selected for the survey. In the following, we discuss the models of
the environment used in the survey planning.

####  5.1.1 Zodiacal background

A large amount of literature is available on this subject and several models
and estimates have been proposed over the years. Updating previous work done
by the DUNE consortium (Refregier & Douspis, 2008), we have used a lean and
conservative model of the zodiacal background obtained by combining the
spectral dependence proposed by Aldering (2001) for the proposal for the SNAP
satellite (Aldering et al., 2002) and the angular dependence found in Leinert
et al. (1998). This ‘basic’ model assumes a cylindrical symmetry with respect
to the Sun. In this time-invariant and symmetrical model, the zodiacal
background flux density $\zeta$ depends solely on wavelength $\lambda$ and
ecliptic latitude $\beta$,

$\zeta(\lambda,\hskip 0.55496pt\beta)=f_{\lambda}(\lambda)\,g(|\hskip
0.55496pt\beta|).$ (2)

For the spectral dependence, Aldering (2001) suggests

$f_{\lambda}(\lambda)=\begin{cases}\kappa&,\,\lambda\in[0.41\,$\mathrm{\SIUnitSymbolMicro
m}$;\lambda_{\rm ref}]\\\ \kappa\,10^{-0.730\,(\lambda-\lambda_{\rm
ref})/$\mathrm{\SIUnitSymbolMicro m}$}&,\,\lambda\in[\lambda_{\rm
ref},2.2\,$\mathrm{\SIUnitSymbolMicro m}$]\end{cases}$ (3)

with $\lambda_{\rm ref}=0.61$ $\mathrm{\SIUnitSymbolMicro m}$. The
normalisation constant $\kappa$ fixes the flux density to the North Ecliptic
Pole (NEP; $\beta=$$$), such that $\kappa=1.76\times
10^{-18}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,\text{\AA}^{-1}\,\mathrm{arcsec}^{-2}$,
corresponding to $m_{\mathrm{AB}}=23.05$ mag arcsec-2. For comparison, this is
just 3% lower than the value given by Leinert et al. (1998) for the NEP at
$0.50\,$\mathrm{\SIUnitSymbolMicro m}$$.

For the dependence on ecliptic latitude $\beta$, we have $g(|\hskip
0.55496pt\beta|)$ as a dimensionless, monotonically decreasing function over
the interval $\beta\in[$$,$$]$, normalised to $g($$)=1$. Leinert et al. (1998)
report values for $\zeta$ (their table 17) as a function of $\beta$ and
elongation from the Sun, for a wavelength of
$0.50\,$\mathrm{\SIUnitSymbolMicro m}$$. We reproduce their values for an
elongation of $None$ (applicable to Euclid) in Table 3, showing that at
$\beta=$$$ ($None$) the zodiacal background is 2 times (3 times) higher than
at the NEP. This dependence on latitude is in good agreement with values
measured by the SMEI satellite (Buffington et al., 2016). Euclid uses fixed
integration times and thus does not compensate for increased background. We
therefore limit the EWS to $|\beta|\geq$$$, corresponding to a reduction of
20/30% in VIS/J SNR compared to the NEP when taking into account also the
stray light (see Sect. 5.1.3 and Fig. 19). This still allows for a suitable
number density of detected galaxies for WL and GC averaged over the survey.

Table 3: The ecliptic latitude dependence of the simple zodiacal background
model of Leinert et al. (1998). Here, $\zeta$ is in units of
$10^{-8}\,\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,$\mathrm{\SIUnitSymbolMicro
m}$^{-1}\,\mathrm{sr}^{-1}$.

$|\hskip 0.55496pt\beta|$ | $None$ | $None$ | $None$ | $None$ | $None$ | $None$ | $None$ | $None$ | $None$ | $None$ | $None$
---|---|---|---|---|---|---|---|---|---|---|---
$\zeta(0.50\,$\mathrm{\SIUnitSymbolMicro m}$,\hskip 0.55496pt\beta)$ | 259 | 251 | 225 | 193 | 166 | 147 | 132 | 104 | 86 | 79 | 77
$g(\hskip 0.55496pt\beta)$ | 3.36 | 3.26 | 2.92 | 2.51 | 2.16 | 1.91 | 1.71 | 1.35 | 1.12 | 1.03 | 1.0

Table 4: The normalisation of the basic zodiacal model at the average wavelength of Euclid bands (see Fig. 7).  band | $\lambda_{*}$ | $f_{\lambda}(\lambda_{*})$ | $m_{\rm AB}$ | ${\rm F}(\lambda_{*})=\lambda_{*}f_{\lambda}(\lambda_{*})$ | ${\rm F}(\nu_{*})=\nu_{*}f_{\nu}(\nu_{*})$ | ${\rm F}(\nu_{*})=\nu_{*}f_{\nu}(\nu_{*})$
---|---|---|---|---|---|---
|  [$\mathrm{\SIUnitSymbolMicro m}$] |  [$\mathrm{erg}\;\mathrm{s}^{-1}\,\mathrm{cm}^{-2}\,\text{\AA}^{-1}\,\mathrm{arcsec}^{-2}$] | $[\mathrm{mag\,arcsec}^{-2}]$ | $[{\rm nW}\,{\rm m}^{-2}\,{\rm s}^{-1}\,{\rm sr}^{-1}]$ |  [MJy sr-1] |  [${\rm\mu}$Jy arcsec-2]
VIS | 0.716 |  1.47$\times 10^{-18}$ | 22.90 | 448 | 0.107 | 2.515
$Y$ | 1.080 |  8.62$\times 10^{-19}$ | 22.67 | 367 | 0.132 | 3.103
$J$ | 1.367 |  4.92$\times 10^{-19}$ | 22.68 | 286 | 0.131 | 3.079
$H$ | 1.770 |  2.50$\times 10^{-19}$ | 22.85 | 188 | 0.111 | 2.609
blue grism | 1.145 |  7.15$\times 10^{-19}$ | 22.66 | 348 | 0.133 | 3.126
red grism | 1.550 |  3.62$\times 10^{-19}$ | 22.74 | 238 | 0.123 | 2.891
Normalisation | 0.610 |  1.76$\times 10^{-18}$ | 23.05 | 456 | 0.093 | 2.186

This basic model results in a background that is constant in time, depends on
the ecliptic latitude as in Leinert et al. (1998), and follows an exponential
decay for $\lambda\geq 0.61$ $\mathrm{\SIUnitSymbolMicro m}$. In Table 4 we
report corresponding numeric values and magnitudes for the various Euclid
bands. Here we used the simple average wavelength of a band, defined as

$\lambda_{*}=\int\lambda\,T(\lambda)\,{\rm d}\lambda\;/\int T(\lambda)\,{\rm
d}\lambda\,,$ (4)

where $T(\lambda)$ is the end-of-life (EOL) throughput (see Fig. 7). For
background-limited observations, we could instead weight the integrals in Eq.
(4) by the background flux density; this would slightly change the values of
$\lambda_{*}$, as we will discuss in a separate paper.

Better fits to the dust emission detected by the COBE Differential Microwave
Radiometer, can be obtained by considering more sophisticated models, such as
the ones of Kelsall et al. (1998). These models include a slab model that is
not centered on the ecliptic plane and add inhomogeneous clouds. These result
in a zodiacal background that depends not only on the wavelength and direction
of observation, but also on the observation’s epoch. In such models the
minimum background no longer coincides with the ecliptic poles. Instead, it
circles the poles with a yearly period, and the value at the NEP has a
corresponding sinusoidal variation with a 20% peak-to-peak variation (see also
Pyo et al., 2012).

Figure 12:  A comparison of the zodiacal emission at
$1.25\,$\mathrm{\SIUnitSymbolMicro m}$$. For a random subset of the pointings
in the reference survey we show results from the conservative basic model (red
triangles) and the estimates by the more complex model (blue diamonds). The
latter (Maris et al., 2019) depends on the observation time and angular
direction, and it no longer has cylindrical symmetry. This causes the scatter
in the blue points. The Euclid $YJH$-bands show a similar behaviour.

We have implemented such a model (Maris et al., 2006, 2019), which includes
time dependence, differences in trailing and leading directions and possible
deviations of the pointing directions from the orthogonal direction to the
Sun. The model is evaluated for $10^{3}$ random fields of the reference survey
binned in ecliptic latitude, and shows scattering due to different observation
epochs. It predicts for all bands on average $15-20$% lower background values
than the basic model, and 30% less flux at the NEP (see Fig. 12). We use this
more complex model for more precise estimates done a posteriori once a
reference pointing solution is obtained using the basic model as an input.

We have adopted the basic model as our baseline to define the ecliptic
latitude exclusion zone presented in Sect. 5.2, because it is more
conservative, whilst providing reasonable margins. This model is also the
reference model adopted for Mission Performance Evaluations, and hence Euclid
might detect a slightly larger number density of usable galaxies than our
current predictions. We note that among the different models in the literature
some could yield background values larger than the adopted basic model,
because of a different normalisation (see e.g. Wright, 1998).

####  5.1.2 Galactic extinction

Extinction by interstellar dust is estimated from the $E(B-V)$ reddening
maps333In particular Ebv_xgal_ns2048_REL5.fits, found at
http://hyperstars.lmpa.eu/mamd/planck_dust_model.html. produced by the Planck
Collaboration et al. (2014). The map’s resolution of $5^{\prime}$ is high
compared to the linear size of Euclid’s FoV of $\sim$$$ $\sim$$$. We smoothed
the map with a 2 deg wide Gaussian kernel, such that the segments of the RoI
boundary that are determined by $E(B-V)$ have a comparable smoothness as the
segments determined by Galactic and ecliptic latitude.

To apply rigidly the original $E(B-V)=0.08$ mag limit of Laureijs et al.
(2011) would cause highly convoluted Region of Interest (RoI, see Sect. 5.2)
contours and holes inside the contiguous survey areas. To achieve a larger RoI
with compact regions, a slightly adjusted upper limit of $E(B-V)=0.09$ mag was
chosen, while allowing local excursions up to 0.17 mag to simplify contours
and avoid local holes. These settings define a first version of the Galactic
exclusion zone. The introduction of the ecliptic exclusion zone (Sect. 5.1.1)
in the two resulting Galactic caps, divides them into two larger mainlands and
two smaller islands, as discussed in Sect. 5.2. The median value of $E(B-V)$
over the RoI is $0.037$ mag (more statistics are given in Sect. 5.2.2). This
approach meets the performance requirement on mean galaxy number density,
while preserving a connected survey that optimally complements ground-based
data (Sect. 5.5).

We also use the smoothed $E(B-V)$ to estimate the SNR in the RoI. To this end
we must compute the total extinction in magnitude for an Euclid band of
central wavelength $\lambda$:

$-2.5\operatorname{log_{10}}{(F_{\rm obs}/F_{0})}=A_{\lambda}\;3.1\;E(B-V).$
(5)

Here, $F_{{\rm obs}}$ is the observed flux and $F_{0}$ is the flux in the
absence of extinction. The total extinction in the $V$-band is quantified by
$A_{V}=R_{V}\;E(B-V)$ where $R_{V}=3.1$ parameterises the dust extinction in
our Galaxy. We infer the extinction scaling coefficient $A_{\lambda}$ with
respect to the $V$-band from the dust extinction curves of Gordon et al.
(2003) for the Euclid channels (VIS band, $Y$, $J$, $H$, red grism band) based
on their central wavelength. Results are given in Table 5 for the NEP with
$E(B-V)=0.07$ mag.

Table 5: Extinction scaling coefficients $A_{\lambda}$ for the Euclid passband
central wavelengths, and total extinction $A$ at the NEP.

band | $\lambda\,[$\mathrm{\SIUnitSymbolMicro m}$]$ | $A_{\lambda}$ |  $A$ [mag]
---|---|---|---
VIS | 0.72 | 0.68 | 0.148
$Y$ | 1.10 | 0.34 | 0.073
$J$ | 1.40 | 0.23 | 0.050
$H$ | 1.80 | 0.16 | 0.034
red grism | 1.60 | 0.18 | 0.039

Figure 13: The 1034 blinding stars present in Euclid’s RoI. There are 1003
stars over the RoI which have $Y$, or $J$, or $H\,<\,$4 AB mag (in red), and
31 stars with $i\,<\,$4 AB mag and all three bands $Y,J,H\,>\,$4 AB mag (in
blue). The size of the dots scales down with the magnitude from $-2$ to $+4$.
Those nearby bright stars are evenly distributed over the RoI, uncorrelated to
the Galactic latitude. The three Euclid Deep Fields (see Sect. 6.2), free of
blinding stars, are shown in green.

####  5.1.3 Bright sources and stray light

The EWS allows for $15\%$ of its area to be lost due to various effects such
as dead pixels, cosmic rays, etc. This masking budget allocates $2\%$ for
bright stars and an additional $2\%$ for their ghost images. Stars brighter
than $m_{\rm AB}\sim 17.5$ mag for VIS and $16.0$ mag for NISP will saturate
the detectors for the baseline integration times, and we refer to them as
‘bright’ stars.

Due to image persistence constraints for NISP, stars with $m_{\rm AB}\leq 4$
mag must never be present within the FoV for all three photometric bands. We
also apply this rule to the VIS instrument due to stray light considerations
as discussed below. In the following, we refer to both cases as ‘blinding’
stars. We selected three catalogs in the literature that include all the
brightest stars in the sky and sample the wavelength domain of the two
instruments: $i$-band for VIS from the ATLAS All-Sky Stellar Reference Catalog
(Tonry et al., 2018), $Y$-band for NISP from the spectrally matched magnitudes
of stars from the Tycho2/2MASS catalogs (Pickles & Depagne, 2010), and $J$ and
$H$-band for NISP from the 2MASS Point Source Catalog (Skrutskie et al.,
2006). In total, there are 1034 blinding stars within the RoI (Fig. 13). 1003
of those stars were first selected through the $Y$ ¡ 4 or $J$ ¡ 4 or $H$ ¡ 4
AB mag NISP criteria, and 31 stars with $Y,J,H$ ¿ 4 mag were added through the
$i$ ¡ 4 mag VIS criteria (there are 275 $i$ ¡ 4 mag stars within the RoI, the
majority already selected through the NISP criteria). Survey fields with
blinding stars will be skipped on all instruments, amounting to $3\%$ of the
RoI area. Bright (non-blinding) stars will be observed but locally masked
during data reduction. The observed area lost in this way is accounted for in
the pre-allocated masking budget, which does not incorporate areas skipped on
purpose because of the presence of blinding stars.

In the following we show how stray light from stellar sources inside and
outside the FoV affects the observations and dictates the RoI Galactic
latitude threshold. Stray light is generated in various ways, such as
reflections on mechanical structures, scattering on contaminated optical
surfaces and their intrinsic surface roughness, by diffraction on the edges of
mechanical parts such as baffles, spider arms, or by multiple internal
reflections between optical elements. Stray light creates an additional
background contribution that can be diffuse or structured (ghosts).

Up to 2014, the stray light was expected to be a minor fraction of the total
diffuse background (this was initially specified to be less than 20% of the
zodiacal background at the NEP). However, with lessons learned from the Gaia
mission, a study on stray light contamination was carried out by ESA and the
industry for Euclid. Gaspar Venancio et al. (2020) have studied the stray
light aspects extending the analysis from pure in-field (dominated by the
mirrors particulate contamination) to the far out-of-field domain (dominated
by the internal structural multiple reflections). Both can contribute
significantly to the stray light level, as out-of-field stray light, though
largely attenuated by diffuse scattering process, integrates over the full sky
and becomes dominant when getting close to the Galactic plane. They found that
particulate contamination on the mirrors will be the main contributor: stray
light in some sky areas can become comparable and even exceed the local
zodiacal background. Consequently, Euclid must stay clear of the Galactic
plane, since sources both inside and outside the FoV (in-field and out-field
stray light, respectively) contribute.

Figure 14: Impact from bright stars on the SNR achievable by a target galaxy
$m_{\rm AB}=24.5$ mag (extended source) in VIS when placed over a full Euclid
FoV ($\sim$$\,\times$$$, with a scheme of the 6$\times$6 CCD mosaic layout
superimposed) in a typical part of the Euclid RoI. The colour scale is the SNR
of a target source placed in the field at that position. The degradation in
SNR when the target galaxy is placed closer and closer to a bright star is
evident. _Left panel_ : a $m_{\scriptstyle\rm VIS}=4$ AB mag star has effects
on almost 50% of the FoV, however science is impacted on less than 10% of the
FoV, the solid yellow area where the target galaxy is at ${\rm SNR}<14$ (see
text). _Right panel_ : common stars of magnitude $m_{\scriptstyle\rm VIS}=8$,
$9$, and $10$ have a very limited impact overall with respect to the FoV size:
less than 0.1% of the FoV lost on average across the RoI.

Figure 15: The RoI outline (17 354 deg2) with the accepted ranges of the
stellar density, dust extinction, and zodiacal light.

The normalised diffusion irradiance profile (NDI, Gaspar Venancio et al.,
2016) describes the profile of the scattered light in the telescope focal
plane for a point source at a given position either within the FoV or up to 20
degrees away from the optical axis. Euclid’s Korsch optical design (Korsch,
1972) effectively suppresses scattered light. For Euclid and its enhanced
baffling (Gaspar Venancio et al., 2020), in-field stray light will impact the
SNR of faint galaxies from our science goal defined limit of
$m_{\mathrm{V}IS}=24.5$, causing them to fall below the minimal value of ${\rm
SNR}=10$. To ensure such SNR level is realized through the mission science
pipeline involving all the steps of data processing and signal extraction, the
system team in charge of scaling the mission design built margins by adopting
a conservative goal of ${\rm SNR}=14$ using their own internal radiometric SNR
metric. The SNR of a $m_{\scriptscriptstyle\rm VIS}=24.5$ galaxy degrades from
the mission median value of 17.5 to 14 if the galaxy is at an angular
separation of $6.8,4.1,2.5,1.5$, and $1.0$ arcminutes from a bright star of
$m_{\scriptstyle\rm VIS}=4$, $5$, $6$, $7$ and $8$ AB mag, respectively. The
left panel in Fig. 14 illustrates the case for a $m_{\mathrm{V}IS}=4$ star,
the impacted area representing just 9% of the entire field-of-view, a truly
remarkable performance made possible by the Korsch optical design. For fainter
stars, of magnitude 9 and 10, the radius of SNR degradation to a level of 14
is reached at a $8$ and $1.5$ arcseconds radius, respectively. As shown in the
right panel of Fig. 14 the impact of these stars is negligible beyond the core
of the PSF. The average density of the 8, 9, and 10 magnitude stars (Zakharov
et al., 2013) over the RoI amounts to $0.6$, $1.6$, and $3.8$ in-field stars
per FoV for the VIS: their collected impact will be limited to a tenth of a
percent of area loss of the Euclid FoV on average across the RoI. Accounting
for the sparse 4 to 7 AB mag stars does not alter these statistics. A
comparable performance is expected from the NISP as the mission design drove
an NDI dominated by the telescope, not by the instruments.

In summary, for individual bright stars Euclid will skip observing tiles in
which at least one of the four exposures will contain a $m_{AB}\leq
4\,\mathrm{mag}$ star. For a handful of extremely bright stars also nearby
tiles will be skipped, according to an avoidance radius for the tile center
set at a level of stray light yielding a 15% degradation in SNR (NDI model).
Areas affected by in-field stars fainter than $m_{AB}=4\,\mathrm{mag}$ will be
masked during the data reduction phase, as well as ghosts originating from the
dichroic.

All the stars outside the telescope field of view will also contribute
globally to the background level of stray light. Their combined effect is to
add a diffuse cumulative component that depends on the pointing direction of
the telescope. In consequence, this effects scales with the Galactic latitude,
the NDI defining an intensity ratio of the collected brightness of the Galaxy.
The out-field stray light 2D map adopted in this paper is a Besançon model of
the Galaxy flux (Robin et al., 2012, 2014). The model is scaled at the
relevant wavelength to match at the one percent level the out-field stray
light level computed at 12 selected points across the whole sky. The levels
were estimated by the system team for the 2018 Mission Critical Design Review
(MCDR) effort.

Figure 16: Distribution of reddening values (_left_), stellar density
(_middle_) and zodiacal background (_right_) across the RoI. Below each graph,
the box-and-whiskers plot marks the mean (red line), median (black line),
interquartile range (empty box), plus the minimum and maximum values.

A spacecraft stray light model based on the estimation of the NDI was then
used by the system team to describe the stray light due to the diffusion in
the telescope and gauge its effects on the PSF and local background level on
the focal plane (Gaspar Venancio et al., 2016, 2020). Assuming the entrance of
the telescope is illuminated by a distant point source (collimated light),
then the NDI is defined as the ratio of light irradiance (power per unit area)
on the image plane to the source irradiance in object space at the entrance of
the telescope. The NDI is computed for both VIS and NISP using the ASAP
optical software Breault Research Organization (2014), a ray-tracing program
that uses a statistical Monte Carlo approach. The computation is done with
telescope and instrument optical and mechanical models and associated
contamination assumptions. Then the NDI is applied on the sky for a mesh of
pointing directions on the sky (with sampling equal to the Euclid FoV) over
the full sky in order to estimate for each possible pointing direction the
cumulative out-of-field stray light maps.

Conservative estimates of these contaminants established by the mission system
team are adopted for the background and associated noise computations
presented in this paper. This drives in particular the RoI definition with
respect to Galactic latitude (from a minimum of $|\hskip
0.55496pt\beta|\geq$$$ to nominal $|\hskip 0.55496pt\beta|\geq$$$), and how
close the EWS can get to the Galactic bulge, refining the Galactic exclusion
zone.

Finally, an additional concern is that due to stray light: Euclid must avoid
pointing within a circle centered on the position of Solar system planets,
Mars and Jupiter having the largest radius, of 13 deg.

Figure 17: The sky constraints as experienced by Euclid from L2, and the RoI
avoiding the worst regions: the four quadrants of the RoI are in blue. The
solid yellow lines trace the ecliptic latitude threshold of $|\beta|=$$$. The
red solid lines trace the galactic latitude threshold of $|b|=$$$. For
declination $\delta\geq+$$$ the threshold is $|b|=$$$ with respect to the
northern ground surveys (Sect. 5.2.1).

###  5.2 The Euclid Region of Interest

####  5.2.1 The RoI four main areas

The dominant factors that determine the RoI are the zodiacal background, the
Galactic extinction, and stray light due to the Galactic stellar density.
Minor contributors such as emission from Galactic cirrus were ignored, being
at least five magnitudes fainter than the total background over most of the
RoI (Sect. 5.3). The EWS will also skip fields containing blinding stars
(Sect. 5.1.3), leaving only their faint effect imprinted on the out-field
stray light.

The main outline of the RoI is defined by the extinction limits, an ecliptic
latitude threshold of $|\beta|\leq$$$, and a Galactic latitude threshold of
$|b|\leq$$$. Note that for declination $\delta\geq+$$$ we set $|b|\leq$$$,
since the Euclid complementary ground surveys were designed and started in
2017 with this value from Laureijs et al. (2011). Section 5.5 describes how
the RoI is affected by these ground-based surveys.

The RoI fragments into four quadrants delimited by the yellow contours in Fig.
15. and detailed in Table 6. The RoI is best presented on this equirectangular
projection. We use elliptical projections when highlighting aspects of area
conservation. Our sky projections were produced with IPAC’s Montage
package444http://montage.ioac.caltech and the University of Groningen’s
Kapteyn package for Python555https://www.astro.rug.nl/software/kapteyn/.

The environment limits and their impact on the RoI are shown in Fig. 15, while
Fig. 16 shows the distribution of extinction, stellar counts from Gaia
(limited at ${\rm mag}=20$) and zodiacal background within the RoI. In total,
the present RoI encompasses 17 354 deg2 that are compliant for Euclid’s core
cosmology science. The EWS can be constructed from any 15 000 deg2 within.
Some parts of the EWS will inevitably be of lower quality for cosmology, yet
their legacy value is high. For example, the Small Magellanic Cloud (SMC) is
inside the RoI, although clearly at odds with the survey constraints. Pushing
further into the Galactic plane, though, would rapidly reach extinction levels
unacceptable for Euclid’s core science, as is evident from Fig. 17 that shows
the combination of all constraints, highlighting the best parts of the Euclid
sky.

Table 6: The four separate quadrants of the RoI (see Fig. 17). The total area is 17 354 deg2. Quadrant | Name | Area [deg2]
---|---|---
S-.5pt |  |
| Northern “mainland” | 7142
S-.5pt |  |
| Northern “island” | 1575
S-.5pt |  |
| Southern “island” | 1700
S-.5pt |  |
| Southern “mainland” | 6937

From Fig. 17 it is clear that the area of the RoI changes with ecliptic
longitude. Figure 18 shows the area of the RoI as a function of ecliptic
longitude (in bins of $1\degr$ of ecliptic longitude). The plot contains two
global maxima, which coincide with the longitudes that cross both a mainland
and an island, and two global minima, which coincide with the intersection of
the ecliptic and Galactic planes.

Figure 18:  The available RoI area as a function of the ecliptic longitude.
The troughs are produced by the intersection of the Galactic and ecliptic
planes and lead to unallocated time.

Given that one degree of longitude is scanned by one day of orbit and that in
one day ${\sim}10\,\text{deg}^{2}$ of EWS are observed, in a six year mission
it is only possible to observe a maximum of $60\,\text{deg}^{2}$ of EWS sky,
per degree of longitude. In practice, this time must be shared with
calibrations and EDF observations that collectively take ${\sim}20\text{\%}$
of the total time. This lowers the EWS allocation to a maximum average value
of $48\,\text{deg}^{2}$ per degree of longitude. From the analysis of Figs. 17
and 18, it is clear that the sky in the RoI is not uniform enough to fill this
quota. Given the limited pointing range of the telescope, with observations at
or close to transit, this inevitably leads to the depletion of the available
(i.e., yet unobserved) sky in some ecliptic longitudes. This reveals an
intrinsic limitation to the maximum efficiency attainable by the EWS, in which
in some parts of the year there will be unallocated time periods that increase
in duration towards the end of the mission. This is an important feature of
EWS solutions, as discussed in Sect. 8.2.

####  5.2.2 SNR and survey depth in the RoI

Figure 17 highlights the best parts of the Euclid sky. In the following, we
compute the corresponding SNR maps, which provide the quantitative context.

For our SNR computations we take into account the following aspects at the
hardware level: telescope and instruments’ internal backgrounds, photometric
zero points (encoding the total throughput), read noise and dark current.
These are independent of sky position and were taken from the latest available
ground characterisation measurements. At the environmental level, we include
all-sky maps for the zodiacal background, extinction, and stray light from the
Galaxy as detailed in Sect. 5.1.

At the operational level, we allow for three exposures (VIS and NISP imaging)
and four exposures for NISP spectroscopy, the FPA geometries, integration
times, and the size of the measurement apertures. This is motivated by the
fact that 90% (50%) of the survey area is covered with at least three (four)
imaging exposures (see Table 1).. The SNR measurement metrics are evaluated as
follows: for VIS, we consider an extended source with a total magnitude of
$m_{\rm AB}=24.5$ mag in a $1\aas@@fstack{\prime\prime}3$ diameter aperture,
capturing $94\%$ of the flux. For NISP photometry, we consider a point-like
source with a total magnitude of $m_{\rm AB}=24.0$ mag in the $Y$, $J$, $H$
bands in a
$0\aas@@fstack{\prime\prime}9\,\times\,0\aas@@fstack{\prime\prime}9$
($3\,\times\,3$ pixel) aperture, capturing $\sim 80$% of the flux for $Y$ and
$J$, 70% for $H$. For NISP spectroscopy, we consider an emission line with a
flux of $2\times 10^{-16}\,\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-2}$
at an observed wavelength of 1600 nm, measured in a 4$\times$4 pixel wide
aperture in the dispersed images.

In this way we verify that the scientific requirements of the Euclid project
are met. Global statistics of the SNR are summarised in Table 7. The median
survey depths converted and scaled to a $5\,\sigma$ point like source
($5\,\sigma$ point-like source) performance metric for imaging are listed.

Figure 19: Top: VIS band SNR in an equirectangular celestial projection (same
referential as Fig. 17); the entire RoI is within specifications (${\rm
SNR}\geq 10$, the first level of light green). Middle: NISP SNR for the
$J$-band ($Y$\- and $H$-bands are similar, see Table 7 for overall
statistics). Bottom: the NISP red grism band SNR. The more convoluted
isocontours of the VIS band SNR are caused by the stronger dust extinction
versus the NISP. Greyed regions outside the RoI are excluded due to extreme
reddening and/or high stellar density and/or high zodiacal background (c.f.
RoI definition and Fig. 17.) Table 7: SNR statistics for the RoI, for each
channel: VIS band (boldface values refers to extended objects), NIR bands
($Y$, $J$, $H$, values refer to point like objects) and red grism band, $S$
(italic values in the last column refer to $None$ diameter sources). The
median depth here is evaluated for $5\,\sigma$ point-like source.

| VIS | $Y$ | $J$ | $H$ | $S$
---|---|---|---|---|---
Minimum SNR | 10.0 | 5.0 | 5.7 | 5.7 | 3.2
Median SNR | 15.9 | 6.5 | 7.8 | 7.2 | 4.5
Maximum SNR | 19.8 | 7.8 | 9.0 | 8.5 | 6.6
Median depth [AB mag] | 26.2 | 24.3 | 24.5 | 24.4 | $-$

The resulting SNR maps for VIS and NISP are shown on Fig. 19. All four
quadrants are fully green, within specifications, for all channels for their
respective depth metrics (VIS, NISP-P, and NISP-S).

We note that the SNR computations do not consider the contamination of galaxy
samples by stars; to this end we have introduced the thresholds to Galactic
latitude. The greyed areas in Fig. 19 illustrate where a certain component
(such as extinction) is out of range. These may appear inside the RoI (e.g. at
the location of the SMC). Non greyed areas outside the RoI reflect an
evolution of the criteria that led to the RoI definition, for example by
tightening the Galactic latitude threshold from $|b|\geq$$$ to $|b|\geq$$$
after the northern ground-based surveys had been defined for $|b|\geq$$$.

Our more complex zodiacal model (Section 5.1.1) predicts a lower background
that varies with time and position along the orbit. This modulation happens at
a level far below the typical range of zodiacal background within the RoI
(Fig. 12), and hence we do not expect the median performance to change with
this model.

In summary, Fig. 19 shows that the SNR in the VIS band exceeds the requirement
of ${\rm SNR}\geq 10$ over the whole RoI, with a median value of nearly 16.
This gain is mostly related to longer than required integration times, driven
by the needs of the spectroscopic channel. Likewise, the $YJH$ photometric
data are well above the ${\rm SNR}\geq 5$ requirement. A negligible area (less
than 50 deg2) of the NISP spectroscopy is below the ${\rm SNR}\geq 3.5$
requirement (Fig. 19. bottom). The median SNR for spectroscopy is $4.5$, a
comfortable margin. Hence all specifications are exceeded, and on average
Euclid will go deeper than initially planned.

Figure 20: Best SNR areas in each hemisphere / Galactic cap. In yellow the
best 1300 deg2 in each Galactic cap, in red the best 2600 deg2 (including the
yellow area). These are similar to the areas planned for the first and second
data release, $2500$ deg2 and $5000$ deg2 respectively (cf. Fig. 46).

####  5.2.3 Best SNR areas of the Euclid sky

The areas of sky where the largest SNR can be achieved on average are offset
from the ecliptic poles due to the out of field stray light from the Galaxy
and the Large Magellanic Cloud (LMC). The SNR in these areas is close to the
maximum values listed in Table 7. Figure 20 shows yellow and orange filled
areas that were derived from an average of the VIS, $Y$, $J$ and $H$ SNR maps.
The boundaries have been smoothed in this representation. The areas shown in
yellow represent the best 1300 deg2 in each Galactic cap (or celestial
hemisphere), and the orange area the best 2600 deg2 (including the yellow
area). The EWS seeks to cover these best areas first.

Figure 21: Euclid VIS and $J$ band $1\,\sigma$ asinh limiting surface
brightness at the $10^{\prime\prime}\times 10^{\prime\prime}$ scale (LSB
performance) and the nearby Universe: Local Group galaxies are represented by
large cyan dots and galaxies up to $z=0.03$ by red dots, while the purple
ellipses indicate the actual sky area covered by the nearest clusters of
galaxies (Virgo, Coma, Fornax, Hydra are within the RoI). The RoI exhibits a
0.4 magnitude range in sensitivity for the four Euclid bands from the best
area down to the ecliptic plane.

Figure 22: $E(B-V)$ over the RoI at the native $5^{\prime}$ resolution of the
Planck/IRAS map, and surface brightness of the Diffuse Galactic Light (cirrus)
in the VIS and $J$ bands. The two islands (smaller parts of the RoI) collect
the worst areas of the Euclid sky due to the combined presence of cirrus and
the proximity of the Galactic plane. The small green ellipses show the three
Euclid Deep Fields.

###  5.3 Limiting surface brightness

As described above, we have a complete knowledge of the background reaching
Euclid’s focal planes. Similarly to what we did for faint compact sources, we
can derive the detection performance for diffuse emission such as non-resolved
stellar populations in tidal streams of galaxies, intracluster light, and the
cosmic infrared background (CIB). In the following, we will do this
considering the same noise properties as before, assuming a stack of three
exposures for both VIS and NISP, as this applies to the 90% level coverage of
the imaging survey (see Sect. 4).

We use the limiting surface brightness metric adopted by Mihos et al. (2013),
which is based on the asinh magnitude introduced by Lupton et al. (1999). This
conservative metric is a good description of the actual signal properties at
very low $\sigma$-levels relevant for such science. It has the merit of
reflecting an actual science performance: the determination of the light
profile of Messier 101 down to a surface brightness of $\mu_{B}=29.5$ mag
arcsec-2 corresponds to the $1\,\sigma$ limit in the Mihos et al. (2013)
study. At $1\,\sigma$, the asinh magnitude is 0.5 magnitudes smaller than the
corresponding standard magnitude. This limiting surface brightness is computed
directly from the photometric zero point of the system and the background
noise property from our SNR study. Because of the very limited contamination
of bright stars (see Sect. 5.1.3) here we only need to consider the diffuse
background. The limiting surface brightness expressed at the pixel scale is:

$\mu\;=\;{\rm
ZP}-2.5\operatorname{log_{10}}(b)-a\,\operatorname{asinh}(f/2b),$ (6)

where, at the native resolution, ZP is the photometric zero point in electrons
per second; $b$ is the noise per pixel in the image, assumed to be the Poisson
standard deviation of the background counts per pixel, B, i.e. $b=\sqrt{B}$;
$a=2.5\operatorname{log_{10}}({\rm e})=1.08574$ and $f$ is the level in
electrons per pixel of the extended astronomical source.

Given Euclid’s small plate scales ($0\aas@@fstack{\prime\prime}1$ and
$0\aas@@fstack{\prime\prime}3$ pixel-1 for VIS and NISP, respectively), the
depth metric relation must be brought to the physical scale of common features
encountered in the near-field (galaxies, streams, shells, dwarfs, etc): we
adopt a generic $10^{\prime\prime}\,\times\,10^{\prime\prime}$ scale while
conforming to the standard performance unit for extended emissions in
magnitude per square arcsecond. We shift from the pixel scale to our scale of
interest considering a square area of $n$ native pixels on the side. By
averaging over the larger area, the estimate for the noise is scaled down by a
factor $\sqrt{n\times\,n}$ while the zero point gets shifted by
$-2.5\operatorname{log_{10}}(n\times\,n)$ for flux conservation. Scaling to
the magnitude per square arcsecond unit adds $-2.5\operatorname{log_{10}}(p)$,
with $p$ the area of the native pixel in square arcsecond (0.01 for VIS, 0.09
for NISP). For the adopted $10^{\prime\prime}\,\times\,10^{\prime\prime}$
scale ($n$=100 for VIS, 33.33 for NISP) the combined effects on the limiting
surface brightness amount to $-2.5\operatorname{log_{10}}(1/(0.01\times
100))=0$ for VIS, and $-2.5\operatorname{log_{10}}(1/(0.09\times
33.33))=+1.193$ for NISP (the larger the physical scale, the lower the noise
and the greater the performance).

We can now explore the Euclid RoI at the asinh $1\,\sigma$ level $(f=b)$. The
result is a map of limiting surface brightness for each band. The VIS and NISP
maps show essentially the same structures (shape, amplitude, location), and in
Fig. 22 they are combined in a single map (see the color bar for the
amplitudes in VIS and $J$-band). The maximum range from the best area to the
worst, at the ecliptic plane limit, is $\sim 0.4\,{\rm mag}\,{\rm
arcsec}^{-2}$, corresponding to the total background level ratio of 2.25
between these best and worst areas. The median limiting surface brightness
across the four bands over the RoI is: $m_{\scriptscriptstyle\rm VIS}=29.8$,
$Y=28.2$, $J=28.4$, $H=28.4$ AB mag arcsec-2, $1\,\sigma$ asinh magnitude at
the $10^{\prime\prime}\,\times\,10^{\prime\prime}$ scale ($-0.25$ mag for the
minimum performance, $+0.15$ mag for the maximum over the RoI). We note that
our various background components are conservative estimates and these levels
can be considered a safe performance. Our more complex zodiacal background
model presented in Sect. 5.1.1 indicates how intensity varies with time and
position along the orbit. This will in consequence modulate the total
background, hence the depth, although at a level lower than the 0.4 magnitude
range depth seen here over the RoI (Fig. 12). The median performance is not
expected to change. We also note that a depth metric based on the integrated
Sersic radial profile over a whole galaxy, digging deep into the noise,
typically adds at least two magnitudes with respect to this contrast-oriented
metric. This is taking particularly into account Euclid’s pristine image
quality that will enable an effective masking of the foreground and background
compact sources.

For an illustration of the scientific potential, the map in Fig. 22 features
the nearby extra-galactic Universe up to a redshift of $z=0.03$: more than
$10\,000$ bright ($K$-band magnitude $<12$) galaxies (2MRS catalog Huchra et
al., 2012), including several members from the Local Group, and four nearby
clusters of galaxies all falling within the Euclid RoI (some additional
targets that are located outside the RoI might be observed during the
unallocated time, see Sect. 8.2). The three Euclid Deep Fields are shown in
yellow on the map of Fig. 22. They are designed to be $+2$ mag deeper than the
EWS for compact sources, but the gain in depth is comparable for the diffuse
emission (for specific details, see our companion paper on the Euclid Deep
Fields, [Sc21]).

Such capacity at detecting faint nebulous objects will make the Diffuse
Galactic Light (DGL, cirrus) an ubiquitous component of the background over
the entire RoI as it averages to a level of 27.1 mag arcsec-2 in the VIS
imaging (derived from multi-band dedicated CFHT-MegaCam observations to help
characterise the RoI). Based on the scaling of the DGL’s albedo (Gordon,
2004), we derive an average of 27.9 mag arcsec-2 over the RoI in the NISP
$J$-band. This extra background is shown in the map in Fig. 22. Note that it
is included in our derivation of the limiting surface brightness, but it has
no impact since it is more than five magnitudes fainter than the combination
of zodiacal background and out-field stray light. We know that structures
exist in the DGL down to the arcsecond scale even at high Galactic latitude
(Miville-Deschênes et al., 2016). The DGL is, however, truly diffuse over the
great majority of the RoI, although as shown in Fig. 22 some parts of the
Euclid sky are inevitably worse than others. The two islands of the RoI
(regions S-.5pt
II
and S-.5pt
III
, see Sect. 5.2.1) correspond to the worst parts of the Euclid sky due to
their proximity to the Galactic plane and a considerable presence of cirrus.

Figure 23: Expected average number density of VIS galaxies satisfying the WL
requirements (notice that the local impact of foreground galaxies or clusters
is not considered here). _Upper panel:_ counts as a function of the local
background (in mag arcsec-2) for different levels of $E(B-V)$ reddening;
_Lower panel:_ counts as a function of reddening for different levels of the
background. Lines are linear fits to the simulations.

Figure 24: Expected counts of WL galaxies in arcmin-2 (see text; note that the
local impact of foreground galaxies is not considered here) in ecliptic
coordinates with the Galactic plane removed (black). The decrease in the
number counts is evident going towards the ecliptic plane, where the zodiacal
background is higher. See also Scaramella et al. (2014).

Figure 25: Expected counts of GC galaxies (with reliable spectroscopic
redshifts) in deg-2 (see text) in ecliptic coordinates. Here the number counts
decrease faster with latitude than for the WL counts (cf. Fig. 24) since the
impact of stars (spectra) from our Galaxy on the GC 1D spectra is more severe
than the impact of stars (pointlike) on the 2D WL data.

###  5.4 Expected number of sources

The expected counts of galaxies that satisfy the WL requirements in terms of
${\rm SNR}>10$ (as measured by SourceExtractor, Bertin & Arnouts, 1996) and
size (${\rm FWHM_{gal}}>1.25\,{\rm FWHM_{PSF}}$) can be inferred using
realistic image simulations. In our analysis, we used sky simulations produced
with SkyLens666 http://metcalf1.difa.unibo.it/blf-portal/skylens.html, in
which the input galaxies magnitude, spectro-morphological, and redshift
distributions were drawn from the HST Ultra Deep Field (HUDF). Since the HUDF
is a very small field (11 square arcmin), the galaxies therein may not be
representative of the mean properties of galaxies on the whole sky. Given the
larger size of the COSMOS field, its counts are more robust. We have thus
corrected the UDF counts such as to reproduce the magnitude distribution in
the COSMOS field. We have also estimated the SNR based on the specified
throughput of the VIS instrument and including Galactic extinction and the
zodiacal background. Note this estimates are conservative because the latest
throughput estimates for the VIS SNR are larger than the initially specified
ones that we used.

By simulating images under various conditions, and extracting the sources from
them, the dependencies of the galaxy density with extinction and background
were determined, as shown in Fig. 23. The results were interpolated from each
pointing to derive an estimate of the spatial map of number counts, shown in
Fig. 24. These estimates are currently being updated to include stray light,
which is quite dependent on the local environment of bright stars and their
spectral energy distribution (cf. Sect. 5.1.3), and to include the latest
instrument parameters and data reduction methods.

Figure 26: Weight (GC expected number of good redshifts deg-2) of observed
fields along a typical EWS progression (see Sect. 7) with a superimposed
moving median taken on the previous year (red line). The approximately regular
cusps every 6 months are due to the paucity of observable fields close to the
Galactic and ecliptic planes and their increased background and extinction.
The decrease of the median and the upper envelope in the second half of the
mission is due to a relatively lower quality of the sky areas still available
for observations in those periods. In fact, the best areas are observed in the
first half of the mission, see Fig. 20 and Fig. 46

A similar approach was carried out for GC, where spectra were simulated, then
extracted and finally measured (Zoubian et al., 2014), yielding a similar sky
map of expected number of reliable redhifts over the sky (see Jamal et al.,
2018, other methods are being developed). Figure 25 shows an updated version
from the ‘Science Performance Verification’ #2 (SPV2) GC simulation (SPV is an
end-to-end simulated exercise of the whole data reduction chain777A summary of
the SPV2 exercise can be found in pages 4 and 5 of the EC newsletter at:
https://www.euclid-ec.org/Documents/Newsletter/EC-Newsletter_issue08.pdf .).
As for the WL case, the GC simulations are currently being updated to include
the latest estimated effects of stray light, instrument characteristics and
data reduction procedures. We expect that the overall fraction of recovered
redshifts in simulations will increase with updated models and throughput,
coupled with better simulations and data reduction, since the simple but up to
date estimates for spectra now yield ${\rm SNR}>3.5$ almost everywhere (see
Fig. 19). The indicative expected number of reliable redshifts of each field
(used as a relative weight) as a function of time for a typical EWS (see Sect.
7) is shown in Fig. 26. Only 2% of the fields have a weight below 1000, the
consequence of an excellent survey efficiency and a good RoI selection. The
overall normalisation of these numbers is based on model estimates of the
intrinsic number of Hα emitters plus current estimates of contamination rate
and a preliminary data reduction success rate. The latter is expected to be
improved. At present, the number of H-alpha emitters observable with Euclid
remains uncertain, but recently Bagley et al. (2020) forecasted this number to
be $\sim 3300\deg^{-2}$ in the redshift range measurable by Euclid. According
to this forecast, to meet the initial Euclid specification of
$1700\,\deg^{-2}$ would require to reliably measure the redshifts of half of
the underlying population.

Figure 27: A simulated spectrum of a typical (mass, star formation rate)
galaxy from the Euclid weak lensing population at $z=1$ (D. Masters, private
communication), overlapping the Euclid VIS and NISP $Y,J,H$ bands. The Vera C.
Rubin Observatory photometric bands ($u,g,r,i,z,y$) are also shown,
highlighting the importance of the ground-based data to the photometric
redshift derivation as the $4000\,$Å break falls within the $i$-band. The
total sensitivity for the ground is given at 1.2 airmass.

Figure 28: Current plan (2021) of the ground-based coverage of the Euclid sky
based on the six most powerful ground-based wide-field imagers, and the Euclid
RoI (in solid yellow). Vera C. Rubin Observatory (LSST and suggested northern
and southern extensions: purple, green and pink rectangles) and Blanco (DES,
red) in the South. Subaru (WISHES), JST (JEDIS-$g$), Pan-STARRS, and CFHT
(CFIS) in the North (orange). The best SNR 2600 deg2 areas for Euclid in each
Galactic cap (black dotted contours) are key to steer the ground efforts with
respect to the mission’s first data release (DR1). The Galactic referential is
shown in thin light green.

###  5.5 The EWS coverage by ground-based telescopes

Weak lensing tomography and the need to account for the contamination by
galaxy intrinsic alignments require solid estimates of the redshifts of the
galaxies used as sources in the weak lensing analysis. Euclid will exploit
galaxies up to a redshift of $z\sim 2$ with the majority of galaxies, at the
lensing goal of $m_{\scriptstyle\rm VIS}=24.5$, lying at a redshift $z\sim 1$.
A typical spectral energy distribution of such galaxy is shown on the
simulated spectrum of Fig. 27. The NISP depth goals ($Y,J,H$ = 24.0, see Sect.
3.2.2) were scaled to capture the flux of this galaxy population at the
required SNR for proper photometric redshift derivation. However, since the
weak lensing imaging band through the broad VIS band ($r$+$i$+$z$) does not
sample key features of a $z=1$ galaxy energy distribution, in particular the
$4000\,$Å break which falls within the $i$-band (Fig. 27), complementary bands
are needed to reach the required redshift precision. The $g,r,i,z$ bands are
critical in particular, as introduced by Laureijs et al. (2011).

Large projects aiming at obtaining photometry through the Sloan bands over
large parts of the sky were on the rise at the time of the Euclid mission
definition. Since photometry in those bands does not require the observing
conditions of a space observatory, this critical part of the mission was left
for external up-and-coming photometric surveys by ground-based facilities
located across the two celestial hemispheres in order to reach the entire
Euclid sky. The minimal depths needed to derive photometric redshifts for the
WL probe are 25.7, 25.1, 24.8, 24.6 AB mag in the $g,r,i,z$ bands,
respectively, for $5\,\sigma$ point-like source. These levels were first
introduced in Laureijs et al. (2011) and later on fine tuned to optimally
match the spectral energy distribution of the $z$=1 $m_{\scriptstyle\rm
VIS}$=24.5 galaxy populations anchored on the $Y,J,H$ depths of 24.0 that will
be achieved by Euclid.

At the time of the mission selection in 2012, Laureijs et al. (2011) commented
on ground surveys that were still speculative: only the Dark Energy Survey
(DES, Dark Energy Survey Collaboration et al., 2016) was about to start on its
broad-band imaging effort, with nearly 4500 deg2 of its of 5000 deg2 goal
overlapping the EWS. As of the end of 2020, the now completed DES has secured
in the $g,r,i,z$ bands coverage of nearly a third of the EWS area over the
south Galactic cap while the rest of the Euclid RoI is an on-going effort.
Together, the following six most powerful ground-based wide-field telescopes
will eventually deliver the photometry needed by Euclid across the $u,g,r,i,z$
bands from the northern and southern hemispheres (Fig. 28), the $u$-band being
a solid bonus for photometric redshifts at any depth.

Extensive community-based actions led to the Canada-France Imaging Survey
(CFIS, Ibata et al., 2017), which ought to cover by 2025 the northernmost 4800
deg2 of the Euclid RoI in the $u$\- and $r$-band, using 314 MegaCam nights on
the 3.6 m Canada-France-Hawaii Telescope (CFHT). Spain’s 2.6 m Javalambre
Survey Telescope (Cenarro et al., 2018) should start in 2021 covering that
area with the Javalambre-Euclid Deep Imaging Survey in $g$-band (JEDIS-$g$,
100 nights). Pan-STARRS (USA, 2$\,\times\,$1.8 m telescopes, Chambers et al.,
2016) joined in 2018 to provide the $i$-band by 2025 as a result of their on-
going Near Earth Object (NEO) search. Finally, a group of Japanese scientists
joined the Euclid Consortium in 2020 through the contribution of Subaru Hyper
Suprime-Cam (HSC, Miyazaki et al., 2018) time (40 nights). WISHES (Wide
Imaging with Subaru HSC of the Euclid Sky) will cover the northern area in the
$z$-band, the most demanding band in terms of depth, hence requiring an 8 m
class telescope. The telescopes actively collecting data are now working in
concert as part of the Ultraviolet Near-Infrared Optical Northern Survey
(UNIONS), an independent consortium motivated by the shared effort for Euclid,
to cover the northernmost sky over the complete set of photometric bands, with
a completion date around 2025. Canadian and University of Hawaii UNIONS
members launched an effort in 2019 to gather $g$-band data with Subaru-HSC to
complement the Spanish effort. Note that since CFHT and Subaru cannot
effectively observe from Hawai’i at declinations $\delta\geq+80^{\circ}$, the
EWS RoI has been trimmed around the equatorial pole by a few tens of degrees,
a minor hit since the area was already mostly rejected due to high dust
extinction.

Meanwhile the Vera C. Rubin Observatory (USA, Ivezić et al., 2019) is
approaching first light and the start of the Legacy Survey of Space and Time
(LSST) should be in phase with Euclid. When on the sky, Rubin will be the most
powerful wide-field imager ever built and the Euclid minimal depths in the
$ugriz$ bands will be reached within one year of normal Rubin LSST operations
over the 8000 deg2 overlapping the Euclid RoI in the southern sky; this will
supersede the DES dataset. The Rubin Observatory being such a powerful
machine, the Euclid Consortium is investigating with the Rubin community a
northern survey extension serving various strategic Rubin scientific niches
(Rhodes et al., 2017). Such an extension (3000 deg2 of Euclid RoI area) would
fill the $+$2 to $+$30 deg declination gap between the main component of the
LSST and the on-going Euclid northernmost sky effort (Fig. 28).

##  6 Calibration and Deep Field observations

Euclid’s three major modes of observation (VIS, NISP-P and NISP-S) and their
tight scientific requirements imply a thorough and extensive calibration
program throughout the mission, which serves two main purposes, namely the
calibration of the flight hardware (instrument calibrations, see Sect. 6.1)
and the characterisation of the target galaxies and quantification of any
biases that may arise in the WL and GC experiments (sample characterisation,
see Sect. 6.2).

###  6.1 Telescope and instrument calibrations

The hardware calibration focuses on the properties of the optics, detectors,
electronics, and the opto-mechanical aspects of structural components. The
associated performance regarding PSF, throughput, quantum efficiency, noise
and bias, is subject to change due either to variations in operational
conditions (orbit, depointing, on-board power dissipation) or to long term
ageing (micro meteorite pitting, particles and UV radiation damage). The
optical performance is furthermore affected by contamination from material
outgassing. Monitoring and accurate correction of these effects is paramount
to the scientific success of Euclid and requires repeated execution of
calibration observations with varying cadences.

The hardware calibrations can be divided into on-sky (e.g. transmission) and
off-sky calibrations (e.g. flat fields). The latter have the least constraints
as they can be executed independently of the spacecraft’s pointing; the
scheduling must merely respect the required cadence within some tolerance. On-
sky calibrations have the additional constraint that they must minimally
disturb the thermal equilibrium of the spacecraft. To this end, a selection of
targets is available, from which we choose those that optimally merge with the
scientific observations.

In the following we provide a summary of the main aspects of the calibrations
that impact the building of the EWS (excluding the additional calibration data
taken during the performance verification phase, and shorter instrumental
calibrations that are integrated in the ROS; see Sect. 4.1).

####  6.1.1 Self-calibration and VIS nonlinearity

Quite important for hardware calibrations are the self-calibration
observations, a block of about 18 h observing a field near the NEP with
perennial visibility (for details about this field see [Sc21]). Besides
monitoring the total system transmission, these observations provide the data
for a large range of additional Calibration Products. The self-calibration
observations are scheduled eleven times per year, approximately on a monthly
basis, and back-to-back with a VIS non-linearity sequence of about 9 h
duration.

####  6.1.2 VIS PSF calibration

Another large Calibration Block is a sequence of VIS PSF observations lasting
$20.5$ h, targeting one of about a dozen stellar fields featuring a suitable
range of magnitudes and spectral energy distributions (SEDs), while minimising
Galactic extinction and polarisation effects on the PSF ellipticity. The
Euclid VIS PSF has in fact a weak dependence on polarisation of the incident
light. Most fields in the EWS and EDS have low levels of Galactic polarisation
at VIS wavelengths, but we must ensure that the PSF calibration fields are
also selected to have low polarisation. Additional dedicated observations of
polarised regions are separately planned to measure the PSF polarsation
dependence, in orbit. These observations are used to routinely update and
validate the VIS PSF model, and must be taken with the spacecraft in thermal
equilibrium. This condition is met after about one week without large changes
in the SAA and AA attitude angles, i.e. as close as possible to the values
used in the preceding days of EWS observations. The VIS PSF calibration data
hence need to be embedded in the currently executed EWS patch. In the global
schedule (Fig. 29), the longitudes of the PSF calibration fields are marked on
the top row, and the related observations are shown by the red strips within
the EWS observation blocks.

####  6.1.3 NISP calibrations

Two types of NISP calibrations are relevant in the description of the EWS.
First, the NISP nonlinearity calibration requires a data-intensive special
readout mode and must be done one detector at a time, requiring a total of 49
h. In the current implementation of the EWS, these calibrations are scheduled
approximately every six months. In a future version of the EWS, these
observations might be partially executed in parallel with one of the VIS on-
sky calibrations, pending a confirmation of the instrument inter-operability.

The second set of NISP calibrations is a one-time check of the NISP-S
wavelength dispersion solution (NISP-S-PN-1 in the 2nd year of Fig. 29). This
is initially obtained during the performance verification phase prior to the
beginning of the survey, and repeated once, about one year after the start of
the EWS. The dispersion solution is obtained from compact planetary nebula
(PN) with strong emission lines, being stepped across a larger number of
positions in the NISP focal plane. The dispersion solution is transferred to
the self-cal field (Sect. 6.1.1) to establish a set of secondary standards to
monitor the stability of the dispersion. Currently, the PN can be chosen from
a list of 24 suitably compact PNe, which will be down-selected to some extent
pending further ground-based spectroscopy.

###  6.2 Euclid Deep Fields and Euclid Auxiliary Fields

The Euclid schedule devotes much time to deep observations for galaxy sample
characterisation. For GC, one needs to quantify biases in redshift
measurements due to contamination and emission line misclassification
(completeness-purity calibrations). For WL one needs to quantify biases in
shear estimation due to noise (Viola et al., 2014), colour gradients
(Semboloni et al., 2013; Er et al., 2018), and the calibration of photo-$z$s.
To this end, Euclid will observe three types of fields:

  1. 1.

Deep observations of six well-known fields that have extensive ground- and
space-based multi-wavelength photometric and spectroscopic coverage. These are
hereafter known as the Euclid Auxiliary Fields (EAFs): CDFS, COSMOS-Wide,
SXDS, VVDS-Deep, CANDELS/AEGIS, and CANDELS/GOODS-N. They are covered with 1–4
Euclid FoVs, i.e. spanning 0.5–2.0 deg2. The six EAFs are observed for
photometric redshift calibration and colour gradient calibration purposes.

  2. 2.

Repeated observations of two 20 deg2 fields at different times to obtain
different dispersion angles to calibrate spectral confusion. These are called
the Completeness-Purity-Calibration fields (CPC).

  3. 3.

Deep observations of large (10–20 deg2) fields, two magnitudes deeper than the
EWS, for calibration of the noise bias. These are the three Euclid Deep Fields
(EDFs: EDF-North, EDF-South, and EDF-Fornax).

Defining the exact locations and footprints of the three EDFs required
substantial effort.888The properties of the three EDFs can be found at
https://www.cosmos.esa.int/web/euclid/euclid-survey . EDF-North, located at
the NEP, is visited 40 times with the ROS to reach a depth two magnitudes
deeper than the EWS, while the other two EDFs, at lower latitudes and larger
local background, need more visits to reach the required depth. To maximise
synergy, EDF-North and EDF-South are chosen to have the same centers of CPC-
North and CPC-South, respectively.

The EDFs and EAFs will have great scientific and legacy value due to the depth
of the observations. A detailed description of their planned observations is
presented in [Sc21].

###  6.3 Computing the stage-1 schedule

ECTile is the software purposely developed to generate the scheduling of the
EWS. It has two major stages. In the preparatory stage-1, ECTile computes the
schedule of the various calibrations, EDFs and EAFs. We review stage-1 in this
section. The core of ECTile is stage-2, which computes the EWS; it is
described in Sect. 7.

The observations of calibration fields, EDFs, EAFs, and also ecliptic ‘polar
caps’ (see below), hereafter called ‘targets’, share the same traits: most are
only visible during a short interval at a given time (apart from the poles
themselves), must observe a specific region (some with a specific
orientation), and some must be re-observed with a fixed cadence. Given these
restrictive properties, their schedule is computed in stage-1, followed by the
scheduling of the EWS in stage-2 in the remaining time available. When
scheduling the targets we need to make sure to leave enough time for EWS in
each year, in order to fulfil the planned public delivery of reduced data to
the community (Sect. 1) .

We note that for the reasons explained in Sect. 7.4.2, the polar caps at high
ecliptic latitudes ($|\hskip 0.55496pt\beta|\geq$$$) are also part of the
stage-1 schedule. Each of the two polar caps is covered with a fixed number of
patches (thirteen in the northern polar cap, six in the southern polar cap),