11institutetext:  ETH Zurich, Institute for Particle Physics & Astrophysics,
Wolfgang-Pauli-Str. 27, 8093 Zurich, Switzerland 22institutetext: National
Center of Competence in Research PlanetS (www.nccr-planets.ch)
33institutetext: European Southern Observatory, Karl-Schwarzschild-Str. 2,
85748 Garching, Germany 44institutetext: Research School of Astronomy &
Astrophysics, Australian National University, ACT 2611, Australia
55institutetext: STAR Institute, University of Liège, 19C allée du Six Août,
4000 Liège, Belgium 66institutetext: NASA Goddard Space Flight Center, 8800
Greenbelt Rd, Greenbelt, MD, 20771, USA 77institutetext: Department of
Physics, and Institute for Research on Exoplanets, Université de Montréal,
Montréal, H3T 1J4, Canada 88institutetext: University of Oxford, Department of
Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, Sherrington
Road, Oxford OX1 3PU, United Kingdom 99institutetext: DTU Space, National
Space Institute, Technical University of Denmark, Elektrovej 328, DK-2800 Kgs.
Lyngby, Denmark 1010institutetext: Department of Extrasolar Planets and
Atmospheres, Institute for Planetary Research, German Aerospace Centre,
Rutherfordstr. 2, 12489 Berlin 1111institutetext: Zentrum für Astronomie und
Astrophysik, Technische Universität Berlin, Hardenbergstrasse 36, D-10623
Berlin, Germany 1212institutetext: Univ. Grenoble Alpes, CNRS, IPAG, F-38000
Grenoble, France 1313institutetext: Centre Spatial de Liège, Université de
Liège, Avenue Pré-Aily, 4031 Angleur, Belgium 1414institutetext: Institute of
Astronomy, KU Leuven, Celestijnenlaan 200D, 3001, Leuven, Belgium
1515institutetext: University of Zurich, Institute of Computational Sciences,
Winterthurerstrasse 190, 8057 Zurich, Switzerland 1616institutetext:
Observatoire astronomique de l’Université de Genève, chemin Pegasi 51b, 1290
Versoix, Switzerland 1717institutetext: Large Binocular Telescope Observatory,
933 North Cherry Avenue, Tucson, AZ 85721, USA 1818institutetext: Steward
Observatory, Department of Astronomy, University of Arizona, 993 N. Cherry
Ave, Tucson, AZ, 85721, USA 1919institutetext: Leiden Observatory, Leiden
University, 2333CA Leiden, The Netherlands 2020institutetext: Department of
Space, Earth & Environment, Chalmers University of Technology, Onsala Space
Observatory, 439 92 Onsala, Sweden 2121institutetext: Institut de Ciències de
l’Espai (ICE, CSIC), Campus UAB, C/Can Magrans s/n, 08193 Bellaterra, Spain
2222institutetext: Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA 2323institutetext: Department of Astronomy, Stockholm
University, Alba Nova University Center, 10691 Stockholm, Sweden
2424institutetext: Department of Space, Earth and Environment, Astronomy and
Plasma Physics, Chalmers University of Technology, SE-412 96 Gothenburg,
Sweden 2525institutetext: University of Exeter, School of Physics and
Astronomy, Stocker Road, Exeter, EX4 4QL, UK 2626institutetext: IAS, CNRS (UMR
8617), bat 121, Univ. Paris-Sud, F-91405 Orsay, France 2727institutetext:
University of Tartu, Tartu Observatory, 1 Observatooriumi Str., 61602
Tõravere, Tartumaa, Estonia 2828institutetext: Centro de Astrobiología (CAB,
CSIC-INTA), Depto. de Astrofísica, ESAC campus 28692 Villanueva de la Cañada
(Madrid), Spain 2929institutetext: Max-Planck-Institut für Astronomie,
Königstuhl 17, 69117 Heidelberg, Germany 3030institutetext: SRM Institute of
Science and Technology, Chennai, India 3131institutetext: Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena,
CA 91109, USA 3232institutetext: Department of Astronomy, University of
Michigan, Ann Arbor, MI 48109, USA 3333institutetext: Physikalisches Institut,
Universität Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
3434institutetext: Landessternwarte, Zentrum für Astronomie der Universität
Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany 3535institutetext: Freie
Universität Berlin, Department of Earth Sciences, Malteserstr. 74-100, 12249
Berlin, Germany

3636institutetext: Instituto de Astrofísica de Canarias (IAC), E-38200 La
Laguna, Tenerife, Spain 3737institutetext: Dept. Astrofísica, Universidad de
La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain 3838institutetext:
Institut d’Estudis Espacials de Catalunya (IEEC), C/Gran Capità 2-4, 08034
Barcelona, Spain 3939institutetext: Department of Astronomy, Yale University,
New Haven, CT 06511, USA 4040institutetext: Nicolaus Copernicus Astronomical
Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland
4141institutetext: Department of Earth and Planetary Sciences, University of
California, Riverside, 900 University Ave. Riverside, CA, USA 92521
4242institutetext: Deshbandhu College, University of Delhi 110019 Delhi, India
4343institutetext: Vanderbilt University, Department of Physics & Astronomy,
6301 Stevenson Center Ln., Nashville, TN 37235, USA 4444institutetext:
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3
0HA, UK 4545institutetext: www.life-space-mission.com

# Large Interferometer For Exoplanets (_LIFE_):

I. Improved exoplanet detection yield estimates for a large mid-infrared
space-interferometer mission

S.P. Quanz, Correspondence<EMAIL_ADDRESS>Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): M. Ottiger Large Interferometer For Exoplanets (_LIFE_):Large
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(_LIFE_): G. Chauvin Large Interferometer For Exoplanets (_LIFE_):Large
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(_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
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Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): A. García Muñoz Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): C. Gascón Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): J. H. Girard Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): A. Glauser
Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): J.L. Grenfell Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): G. Guidi Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): J. Hagelberg Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): R. Helled Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): M.J. Ireland Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): M. Janson Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): R.K.
Kopparapu Large Interferometer For Exoplanets (_LIFE_):Large Interferometer
For Exoplanets (_LIFE_): J. Korth Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): T. Kozakis Large
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(_LIFE_): S. Kraus Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): A. Léger Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): L. Leedjärv
Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): T. Lichtenberg Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): J. Lillo-Box Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): H. Linz Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): R. Liseau Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): J. Loicq
Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): V. Mahendra Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): F. Malbet Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): J. Mathew Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): B. Mennesson Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): M.R. Meyer
Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): L. Mishra Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): K. Molaverdikhani Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): L. Noack Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): A.V. Oza Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): E. Pallé Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): H. Parviainen Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): A.
Quirrenbach Large Interferometer For Exoplanets (_LIFE_):Large Interferometer
For Exoplanets (_LIFE_): H. Rauer Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): I. Ribas Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): M. Rice Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): A. Romagnolo Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): S. Rugheimer Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): E.W. Schwieterman Large Interferometer
For Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): E.
Serabyn Large Interferometer For Exoplanets (_LIFE_):Large Interferometer For
Exoplanets (_LIFE_): S. Sharma Large Interferometer For Exoplanets
(_LIFE_):Large Interferometer For Exoplanets (_LIFE_): K.G. Stassun Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): J. Szulágyi Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): H.S. Wang Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_):Large Interferometer For Exoplanets
(_LIFE_): F. Wunderlich Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_): M.C. Wyatt Large Interferometer For
Exoplanets (_LIFE_):Large Interferometer For Exoplanets (_LIFE_): the _LIFE_
Collaboration Large Interferometer For Exoplanets (_LIFE_):Large
Interferometer For Exoplanets (_LIFE_):

(Received: ¡date¿ / Accepted: ¡date¿)

###### Abstract

Context. One of the long-term goals of exoplanet science is the atmospheric
characterization of dozens of small exoplanets in order to understand their
diversity and search for habitable worlds and potential biosignatures.
Achieving this goal requires a space mission of sufficient scale that can
spatially separate the signals from exoplanets and their host stars and thus
directly scrutinize the exoplanets and their atmospheres.

Aims. We seek to quantify the exoplanet detection performance of a space-based
mid-infrared (MIR) nulling interferometer that measures the thermal emission
of exoplanets. We study the impact of various parameters and compare the
performance with that of large single-aperture mission concepts that detect
exoplanets in reflected light.

Methods. We have developed an instrument simulator that considers all major
astrophysical noise sources and coupled it with Monte Carlo simulations of a
synthetic exoplanet population around main-sequence stars within 20 pc of the
Sun. This allows us to quantify the number (and types) of exoplanets that our
mission concept could detect. Considering single visits only, we discuss two
different scenarios for distributing 2.5 years of an initial search phase
among the stellar targets. Different apertures sizes and wavelength ranges are
investigated.

Results. An interferometer consisting of four 2 m apertures working in the
4–18.5 $\mu$m wavelength range with a total instrument throughput of 5% could
detect up to $\approx$550 exoplanets with radii between 0.5 and 6 R⊕ with an
integrated S/N$\geq$7\. At least $\approx$160 of the detected exoplanets have
radii $\leq$1.5 R⊕. Depending on the observing scenario, $\approx$25–45 rocky
exoplanets (objects with radii between 0.5 and 1.5 ⊕) orbiting within the
empirical habitable zone (eHZ) of their host stars are among the detections.
With four 3.5 m apertures, the total number of detections can increase to up
to $\approx$770, including $\approx$60–80 rocky eHZ planets. With four times 1
m apertures, the maximum detection yield is $\approx$315 exoplanets, including
$\leq$20 rocky eHZ planets. The vast majority of small, temperate exoplanets
are detected around M dwarfs. The impact of changing the wavelength range to
3–20 $\mu$m or 6–17 $\mu$m on the detection yield is negligible.

Conclusions. A large space-based MIR nulling interferometer will be able to
directly detect hundreds of small, nearby exoplanets, tens of which would be
habitable world candidates. This shows that such a mission can compete with
large single-aperture reflected light missions. Further increasing the number
of habitable world candidates, in particular around solar-type stars, appears
possible via the implementation of a multi-visit strategy during the search
phase. The high median S/N of most of the detected planets will allow for
first estimates of their radii and effective temperatures and will help
prioritize the targets for a second mission phase to obtain high-S/N thermal
emission spectra, leveraging the superior diagnostic power of the MIR regime
compared to shorter wavelengths.

###### Key Words.:

Telescopes – Techniques: interferometric – Infrared: planetary systems –
Techniques: high angular resolution – Methods: numerical – Planets and
satellites: detection – Planets and satellites: terrestrial planets

##  1 Introduction

One of the major objectives of exoplanet science is the atmospheric
characterization of a statistically relevant sample of small exoplanets.
Specific emphasis will be on temperate terrestrial planets to investigate
whether there are other worlds similar to Earth that may harbor life. While
occurrence rates of Earth-like planets around solar-type stars are still
somewhat uncertain (e.g., Bryson et al., 2021), thanks to transiting exoplanet
discovery missions such as _Kepler_ (Borucki et al., 2010) and the _Transiting
Exoplanet Survey Satellite_ (TESS; Ricker et al., 2015) and ongoing long-term
radial velocity (RV) surveys, we know that, statistically, planets with radii
and masses comparable to or slightly larger than those of Earth and with
shorter orbital periods are very abundant (e.g., Mayor et al., 2011; Tuomi et
al., 2019; Kunimoto & Matthews, 2020; Bryson et al., 2021). Some major
detections were even made within 20 pc of the Sun with both transit searches
(e.g., Berta-Thompson et al., 2015; Vanderspek et al., 2019; Gillon et al.,
2017, 2017) and RV surveys (e.g., Anglada-Escudé et al., 2016; Ribas et al.,
2016; Jeffers et al., 2020; Astudillo-Defru et al., 2017; Díaz et al., 2019;
Zechmeister et al., 2019), with RV planets typically being closer to the Sun
because of the geometric bias of transiting planets.

Going forward, and focusing on the atmospheric characterization of small
planets, the _James Webb Space Telescope_ (_JWST_) might reveal whether some
of these objects transiting nearby M dwarfs have managed to retain their
atmospheres (e.g., Koll et al., 2019) despite the high level of activity of
their host stars, in particular at younger ages (e.g., Ribas et al., 2016;
MacGregor et al., 2018; Johnstone et al., 2019); an in-depth investigation of
atmospheric constituents with _JWST_ seems, however, rather challenging (e.g.,
Kreidberg & Loeb, 2016; Morley et al., 2017; Krissansen-Totton et al., 2018).
The _Atmospheric Remote-sensing Infrared Exoplanet Large-survey_ mission
(Ariel; Tinetti et al., 2018) of the European Space Agency (ESA) will provide
transmission and emission spectra of hundreds of exoplanets, but the focus
will be on objects with hot or warm hydrogen-dominated atmospheres; only a few
small, relatively hot exoplanets will be studied. Upgraded or new fully
optimized instruments at existing 8-meter-class ground-based telescopes may
have a chance of directly detecting the nearest small exoplanet, Proxima Cen b
(e.g., Lovis et al., 2017). Due to their unprecedented spatial resolution and
sensitivity, the upcoming 30–40 m ground-based extremely large telescopes
(ELTs) will be powerful enough to directly detect small planets around the
nearest stars. Instruments working at mid-infrared (MIR) wavelengths, such as
the _Mid-infrared ELT Imager and Spectrograph_ (METIS; Brandl et al., 2018,
2021), will detect the thermal emission of the planets (Quanz et al., 2015;
Bowens et al., 2021). Instruments working at optical or near-infrared (NIR)
wavelengths and featuring high-resolution spectrographs coupled with adaptive
optics systems, such as the _Planetary Camera and Spectrograph_ (PCS; Kasper
et al., 2021) and the _High Resolution Spectrograph_ (HIRES; Marconi, 2020),
aim at detection in reflected light.

Unfortunately, none of the currently planned ground-based instrument projects
and approved space missions is capable of investigating in detail the
atmospheres of several dozen small exoplanets, including a sizable subsample
residing in or close to the so-called habitable zone (HZ) of their host stars
(Kasting et al., 1993; Kopparapu et al., 2013). This is one of the reasons
why, in the context of the Astrophysics Decadal Survey in the United States,
new flagship missions, the _Habitable Exoplanet Observatory_ (HabEx; Gaudi et
al., 2020) and the _Large UV/Optical/IR Surveyor_ (LUVOIR; The LUVOIR Team,
2019), are currently under assessment; one of their main science drivers is
the direct detection and characterization of temperate terrestrial exoplanets
in reflected light111We note that during the refereeing process of this paper,
the Consensus Study Report “Pathways to Discovery in Astronomy and
Astrophysics for the 2020s” was published by the National Academies of
Sciences, Engineering, and Medicine recommending a large ($\sim$6 m aperture)
infrared/optical/ultraviolet (IR/O/UV) space telescope as a future flagship
mission (National Academies of Sciences & Medicine, 2021)..

Here, we focus on a different observational approach and a new initiative that
aims at developing a space-based MIR nulling interferometer capable of
detecting and characterizing the thermal emission of (temperate) rocky
exoplanets. The characterization of temperate exoplanets in the MIR was
recently announced to be a potential science theme for a future science
mission within ESA’s Voyage 2050
program222https://www.cosmos.esa.int/web/voyage-2050. The idea to employ
interferometric nulling for exoplanet science was originally proposed by
Bracewell (1978) and later followed up in Leger et al. (1995) and Angel &
Woolf (1997); in the late 1990s to mid 2000s, concept studies were carried out
by both ESA and NASA: the _Darwin_ mission and the _Terrestrial Planet Finder
- Interferometer_ (_TPF-I_) mission, respectively. In the end, these concepts
did not go forward for implementation because of technical challenges, but
also because our understanding of the exoplanet population was significantly
more limited. This has, however, changed. Given the enormous scientific
progress in exoplanet research since the mid 2000s and significant advances
related to key technologies, it is time to reassess such a mission concept and
quantify its potential scientific return. In 2018, a first such study was
published (Kammerer & Quanz, 2018), which investigated the exoplanet detection
yield of a space-based MIR nulling interferometer based on exoplanet
occurrence rates from NASA’s _Kepler_ mission; it claimed that a few hundred
small exoplanets (radii between 0.5 and $6\leavevmode\nobreak\
\mathrm{R}_{\oplus}$) could be within reach for such an instrument. These
promising results, in combination with ongoing lab activities related to
nulling interferometry, resulted in the creation of the _Large Interferometer
For Exoplanets (LIFE)_ initiative333www.life-space-mission.com.

The present paper is the first in a series of papers currently in preparation.
It is assumed that a mission such as _LIFE_ would consist of two main phases:
(1) a search phase to directly detect a large sample of exoplanets orbiting
nearby stars and (2) a characterization phase to reobserve a subsample of
these exoplanets and investigate their properties and atmospheres in detail.
In the following, we focus on the search phase and quantify how many
exoplanets _LIFE_ would be able to detect depending on different mission
parameters. Future work will focus more on the characterization phase of the
mission, including questions related to atmospheric diversity and evolution,
habitability, and the search for indications of biological activity. In this
context, the MIR wavelength regime offers complementary information and even
several advantages compared to studies at optical or NIR wavelengths. These
include more direct constraints on the temperature and size of the objects and
a large set of molecular absorption lines of main atmospheric species,
including biosignatures (e.g., Schwieterman et al., 2018; Catling et al.,
2018), some of which, for example CH4, might be easier to detect in the MIR
than at shorter wavelengths.

In comparison to previous studies that quantify the detection yield of an MIR
nulling interferometer (e.g., Kammerer & Quanz, 2018; Quanz et al., 2018,
2021), we have significantly updated and improved our simulation approach as
further described below. Similar detection yield analyses were carried out for
the reflected light missions mentioned above, enabling a direct comparison of
the different mission concepts.

We note that in the ideal case, exoplanet surveys with existing and future
high-precision RV instruments (e.g., CARMENES, NIRPS, ESPRESSO, MAROON-X,
HARPS3, and EXPRES) will continue to uncover a significant fraction of the
exoplanet population within 20 pc, including rocky, potentially habitable,
planets444As an alternative to ground-based RV searches, space-based high-
precision astrometry missions could be envisioned (e.g., Malbet & Sozzetti,
2018; Janson et al., 2018).. In this case, the search phase of future direct
detection exoplanet missions would be shortened and more of the limited
observing time could be allotted to the characterization of the objects.
Whether in the end the low-amplitude RV signals from Earth-like planets can be
separated from astrophysical noise sources, such as stellar jitter (e.g.,
Oshagh et al., 2017), and to what extent, consequently, those RV surveys will
be complete, remains to be seen.

In Sect. 2 we describe in detail the setup of our yield simulations. The
results are presented in Sect. 3, and we discuss them in Sect. 4. We conclude
and summarize our main findings in Sect. 5.

##  2 Setup of yield simulations

The general approach of our Monte Carlo-based yield simulations is described
in Kammerer & Quanz (2018), but we have implemented several updates as
detailed in the following.

###  2.1 Stellar target catalog

We used a new _LIFE_ target star catalog that includes single main-sequence
stars and wide separation binaries of spectral types FGKM out to 20 pc of the
Sun. While the catalog includes a total of 1732 objects, only a subset was
considered for the simulations of the search phase depending on the
optimization strategy (see Sect. 2.6 for details). The catalog and its
creation are further explained in Appendix A.

###  2.2 Exoplanet population

In our simulations we generated an artificial population of exoplanets around
the stars in our target catalog and did not consider any known exoplanets. The
underlying exoplanet occurrence rates as a function of the radius and orbital
period follow the results from NASA’s ExoPaG SAG13 (Kopparapu et al., 2018)
for single FGK stars and Dressing & Charbonneau (2015) for single M stars.
This allows for a comparison with the results obtained in the context of the
_HabEx_ and _LUVOIR_ studies mentioned above, which used a very similar
underlying exoplanet population. Binary stars with measured (apparent)
separations greater than 50 AU were treated as single stars. For binary
systems with smaller separations, the occurrence rates were scaled down by a
factor of 0.3 over the entire period and radius range (cf. Kraus et al.,
2016). We focused our analysis on planets with radii, $R_{\textrm{p}}$, in the
range $0.5\,R_{\oplus}\leq R_{\textrm{p}}\leq 6\,R_{\oplus}$ for FGK stars and
$0.5\,R_{\oplus}\leq R_{\textrm{p}}\leq 4\,R_{\oplus}$ for M stars. Orbital
periods, $P_{\textrm{P}}$, in the range $0.5\,d\leq P_{\textrm{P}}\leq 500\,d$
and $0.5\,d\leq P_{\textrm{P}}\leq 200\,d$ were considered for FGK and M
stars, respectively. Cold, Neptune-, or Jupiter-like objects with separations
$\geq$3 au, for which the SAG13 statistics cannot be applied (cf. Dulz et al.,
2020) and which were included in the _HabEx_ and _LUVOIR_ studies, were not
considered as they are typically too cold for a detection within a reasonable
amount of integration time. All planets were assumed to have circular orbits
and were assigned a random Bond albedo, $A_{\textrm{B}}\in[0,0.8)$555For
reference, the Bond albedos of Venus and Mercury are $\approx$0.8 and
$\approx$0.1, respectively, bracketing the values for the Solar System
planets., and a geometric albedo, $A_{\textrm{g}}\in[0,0.1)$666These low
values are motivated by the MIR wavelength range we are considering., that is
constant over the wavelength range we consider (cf. Kammerer & Quanz, 2018).
Both albedos were uniformly distributed in the considered intervals. The
planets were treated as black bodies with their whole surface area radiating
with an equilibrium temperature ($T_{\textrm{eq}}$) determined by the
luminosity of their host star, their Bond albedo, and their orbital
separation.

In Table 1 we define two types of exoplanets that are important throughout the
paper: rocky planets orbiting within the empirical habitable zone (eHZ) and
exo-Earth candidates (EECs). We also provide the respective occurrence rates
as provided by our exoplanet population.

Table 1: Types of exoplanets that are of particular importance throughout the
paper. The first row shows our definition of a rocky planet orbiting within
the empirical habitable zone (eHZ). The second row defines exo-Earth
candidates (EECs) as used in the yield estimates for _HabEx_ and _LUVOIR_. The
last columns summarize the occurrence rates for these objects as provided by
the assumed exoplanet population.

Planet type |  $R_{\textrm{P}}$ [R⊕] | Stellar flux range [S⊕] | Occurrence ratesaa$a$Occurrence rates are values for single stars averaged over our input catalog for the given range of spectral types. Because the stellar flux range is spectral type dependent, the number of objects falling within this range, and hence the occurrence rates, varies with spectral type as well.
---|---|---|---
|  |  | M stars | FGK stars
Rocky eHZ | 0.5 – 1.5 | 1.776 – 0.32bb$b$The flux range is given by the “recent Venus” and “early Mars” limits and is spectral type dependent (Kopparapu et al., 2014). The values given here correspond to 1 M⊕ planet orbiting a solar twin. We note that both limits take into account the luminosity evolution of the Sun, which was fainter during the epochs when Venus and Mars provided habitable conditions. For present-day solar luminosity, these insolation limits correspond to separations of 0.75 and 1.77 au, respectively, excluding Venus from, but including Mars in, the eHZ. | 0.558 | 0.370
Exo-Earth Candidates (EECs) | 0.82 / 0.62cc$c$For EECs, the lower limit of the radius range depends on the separation from the star; closer to the star, planets are required to have a larger radius. This can be described by $R_{\textrm{P}}^{\textrm{min}}=0.8\cdot S^{0.25}$, where $S$ is the insolation. The _HabEx_ and _LUVOIR_ studies focused primarily on solar-type stars and used a corresponding expression based on the semimajor axis, $a$, i.e., $R_{\textrm{P}}^{\textrm{min}}=0.8\cdot a^{-0.5}$. As we are also interested in M stars, we had to convert this into an expression for $S$. – 1.4 | 1.107 – 0.356dd$d$The flux range is given by the “runaway greenhouse” and “maximum greenhouse” limits and is spectral type dependent (Kopparapu et al., 2014). The values given here correspond to a 1 M⊕ planet orbiting a solar twin. | 0.312 | 0.164

777

###  2.3 Simulating spacecraft, instrument, and noise sources with LIFEsim

In order to estimate the signal-to-noise ratio (S/N) of our simulated
exoplanets we have developed the new simulation tool LIFEsim (Dannert et al.,
2022). This tool enables us to simulate the temporally modulated signal that a
planetary system would leave in an observing sequence with a space-based
nulling interferometer (cf. Lay, 2005) and further includes the most relevant
– and wavelength-dependent – astrophysical noise sources. This is an important
difference from our earlier exoplanet detection yield estimates, where,
instead of explicitly simulating the interferometer transmission and signal
modulation for every simulated planet, constraints on the inner working angle
of the instrument and sensitivity were used to assess the discovery potential
(Kammerer & Quanz, 2018; Quanz et al., 2018).

Figure 1: Artist’s impression of the _LIFE_ nulling-interferometry mission,
consisting of four collector spacecraft in a rectangular array configuration
sending light to a beam combiner spacecraft in the center. The present
analysis assumes an X-array configuration with a baseline ratio of 6:1.

In the following, we considered an interferometer consisting of four collector
spacecraft in a so-called X-array configuration that feed their beams into a
fifth beam combiner spacecraft located in their center (see Fig. 1).

The ratio between the long and the short baseline of the X-array was assumed
to be 6:1 for the time being and a $\pi/2$ phase shift was applied between two
conjugated transmission maps (cf. Defrère et al., 2010). The short baselines
of the array are referred to as “nulling baselines” and are responsible for
creating the central dark fringe that nulls the host star. The long baselines
are referred to as “imaging baselines” and are responsible for a higher-
frequency modulation of the transmission map perpendicular to the fringe
pattern created by the nulling baselines. The resulting modulation map, the
difference between the two conjugate transmission maps, effectively suppresses
the signal coming from any centrally symmetric source (such as emission from
optically thin and smooth exozodiacal dusk disks with random inclination) so
that only the shot noise of the source contributes to the S/N of the
observations. The 6:1 baseline ratio has been shown to be more robust against
instability noise compared to a 2:1 baseline ratio (Lay, 2006), but additional
trade-off studies are needed to further validate this choice. A detailed
description of LIFEsim is provided in Dannert et al. (2022), but we refer the
reader to Defrère et al. (2010) for a description of the general nulling and
beam combination scheme and the resulting final modulation map for an X-array
interferometer.

In our simulations we assumed that for each new target star the array is
reconfigured so that the center of the projected eHZ (cf. Table 1; Sect. 2.2)
falls within the first transmission maximum of the nulling baselines at a
reference wavelength of 15 $\mu$m. However, we imposed a minimum separation
between two adjacent collector spacecraft of at least 10 m and allow for a
maximum separation of 600 m. Initial tests had shown that having the reference
wavelength between 15 and 20 $\mu$m resulted in comparable detection yields,
but that shorter (e.g., 10 $\mu$m) or longer (e.g., 25 $\mu$m) reference
wavelengths provided lower yield numbers. Keeping the baseline lengths of the
configuration in mind (from a technical perspective), we decided to use 15
$\mu$m as reference for all analyses presented in the following. The aperture
diameter $D$ of the collector spacecraft is a free parameter in our instrument
model, and we discuss the impact of aperture size on the results in Sect. 3.2.
For the computation of the total photon flux received by the collector
spacecraft we ignored any possible obscuration from a secondary mirror. To be
conservative, we assumed 5%888This 5% throughput is applied to the modulation
maps, which already contain only 50% of the incoming light. for the optical
throughput of the instrument, but will update this number when the concept for
the optical layout is maturing. In earlier studies in the context of the
_Darwin/TPF-I_ missions a throughput of 10% was assumed (e.g., Lay et al.,
2007; Defrère et al., 2010) and also at the _Large Binocular Telescope
Interferometer (LBTI)_ the most recent estimate for the optical throughout
around $\sim$11$\leavevmode\nobreak\ \mu$m is $\approx$0.11 (S. Ertel, private
communication). For the detector quantum efficiency, we assumed 70% over the
full wavelength range, which is identical to the _Darwin_ studies mentioned
above. Recent experiments with 15-micron-cutoff HgCdTe detector arrays have
yielded quantum efficiencies of $\gtrsim$0.8 between 6 and 12 $\mu$m
wavelengths (Cabrera et al., 2020) and the Si:As IBC detectors of the
JWST/MIRI instrument achieve $\gtrsim$0.7 between 12 and 20 $\mu$m (Rieke et
al., 2015).

At the moment, our S/N calculations are photon-based and include all major
astrophysical noise terms. We implicitly assumed that our measurements would
not be limited by instrumental effects (see Sect. 2.7 below for the definition
of our detection criterion). The impact of phase and/or amplitude variations
as major systematic noise sources is currently being assessed. Also, thermal
background from the aperture mirrors and the instrument optics, and detector-
related noise sources will be included in subsequent work. For the mirrors of
the collector spacecraft and the instrument optics not to contribute
significantly to the measurement implies a required temperature of
$\lesssim$40 K (Defrère et al., 2010). Noise terms that were explicitly
included are:

Photon noise from the simulated planets: Given the distance, radius, and
equilibrium temperature of our simulated planets, their photon flux (assuming
black-body emission) and related noise are fully described.

Photon noise from stellar leakage: Depending on the distance to the star, its
radius, and the length of the nulling baseline, a small fraction of stellar
photons may “leak” through the central dark fringe and hence contribute to the
photon noise.

Photon noise from exozodi disks: For each simulated planetary system we
randomly assigned a level of emission from a dusty exozodi disk following the
observed (nominal) distribution from the Hunt for Observable Signatures of
Terrestrial Systems (HOSTS) survey (Ertel et al., 2018, 2020). To compute the
spectral energy distribution (SED) of the exozodi disk we used the publicly
available code from Kennedy et al. (2015). All exozodi disks were assumed to
be optically thin, smooth (i.e., without any substructure) and seen face-on.
We refer the reader to Sect. 4.3.2 for a discussion about these assumptions.

Photon noise from local zodiacal light: The optically thin zodiacal dust in
our Solar System is a source of significant MIR emission. In LIFEsim the
surface brightness is described by a pointing-dependent 2D model originally
developed for the _Darwin_ simulator and based on data from _Cosmic Background
Explorer_ (COBE; Kelsall et al., 1998). Compared to the original _COBE_ data,
the model slightly over-predicts the flux in the 6-20 $\mu$m range and for a
pointing direction with a relative latitude of more than 90∘ from the Sun by
10-20%. For wavelengths shorter than 6 $\mu$m the difference increases to a
factor of 2-3 at 3 $\mu$m. However, at these shorter wavelengths the total
photon noise is strongly dominated (up to several orders of magnitude) by the
contribution from stellar leakage. In our simulations we assumed that we
always point in anti-sunward direction (_LIFE_ will be launched to the Earth-
Sun L2 point) but considered the true latitude of the target star.

In our S/N calculations we implicitly assumed that the combined beams are fed
through single-mode fibers before the signal is spectrally dispersed. The
effective field-of-view (FoV) of the fibers, and hence of each collector
spacecraft, is wavelength dependent and given by $\lambda/D$.

In Fig. 2 we show an example of how the various noise terms compare to the
incoming photon flux from a 1 R⊕ exoplanet with an effective temperature of
276 K orbiting at 1 au from a Sun-like star at 10 pc distance. The system is
assumed to be located within the ecliptic and contains an exozodi disk with
the same brightness as the zodiacal light in the Solar System.

Figure 2: Example illustrating the photon flux and noise contributions from
the various astrophysical sources in our nulling-interferometry simulations:
exoplanet flux (1 R⊕ and 276 K effective temperature located at 10 pc; dashed
red line), flux from a Sun-like star (black line with stars), local zodiacal
light (green line with ticks), and exozodi (1 zodi; solid blue line). The
corresponding photon noise contributions (1-$\sigma$) are shown with the same
color code, but as dotted lines.

###  2.4 Mission parameters

Assuming a total mission lifetime of 5-6 years, we assigned an available on-
source observing time of 2 years to the initial search phase. This translates
into 2.5 years of mission time considering 25% of general mission overhead;
the remaining time of the mission is dedicated to detailed follow-up
observations of a subset of the detected exoplanets and possibly an ancillary
science program. The slew time from one target to the next was fixed to 10
hours, which is part of the 2 year observing time. For the moment we only
considered single visits of target stars during the survey.

###  2.5 Setup of Monte Carlo simulations

To create the exoplanet population we used the freely available P-Pop Monte
Carlo tool999https://github.com/kammerje/P-pop, which for each star of the
target catalog randomly draws exoplanets from the distributions described
above (cf. Kammerer & Quanz, 2018) and puts them at random positions along
their orbits. The orbital inclination was also randomly chosen for each
system, but planets in multi-planet systems were assumed to be co-planar. To
ensure that multi-planet systems were dynamically stable we applied a
stability criterion following the approach by He et al. (2019) that is based
on the mutual Hill radius of neighboring planets. Specifically, for circular
orbits as assumed here, a system was considered stable if for all planet pairs
within the system

$\Delta=\frac{a_{out}-a_{in}}{R_{H}}>8\quad,$

where $a_{out}$ and $a_{in}$ are the semimajor axes of the outer and inner
planet, respectively, and $R_{H}$ is the mutual Hill radius given by

$R_{H}=\frac{a_{in}+a_{out}}{2}\Bigg{[}\frac{m_{in}+m_{out}}{3M_{*}}\Bigg{]}^{1/3}\quad,$

with $m_{in}$ and $m_{out}$ being the mass of the inner and outer planet,
respectively, and $M_{*}$ being the mass of the host star. If a system or pair
of planets was unstable, we re-drew the system, which happened in less than 2%
of the cases. In total, we generated 500 planetary systems per target star.
All planets were then run through LIFEsim in order to compute their photon
fluxes as well as the photon noise from the various sources listed above.

Figure 3: Total exoplanet detection yield from our reference case scenario
simulations ($D=2$ m; $\lambda=4-18.5\,\mu m$) in the radius vs. stellar
insolation plane. The plots show the number of expected planet detections per
grid cell, including the statistical 1-$\sigma$ uncertainty from the Monte
Carlo approach but excluding uncertainties in the exoplanet occurrence rates.
Left panel: Scenario 1 (search phase optimized for maximizing the total number
of exoplanets). Right panel: Scenario 2 (search phase optimized for maximizing
the number of rocky eHZ exoplanets.)

###  2.6 Distribution of observing time: Two scenarios

We considered two scenarios that determine the distribution of the available
on-source observing time of 2 years: maximizing the total number of detected
exoplanets (scenario 1) or the number of rocky exoplanets orbiting within the
eHZ of their host star (scenario 2; see Table 1 for definition). We note that
the eHZ includes a much larger range of insolations than the “classical” HZ
(Kasting et al., 1993; Kopparapu et al., 2013), but a much smaller range than
the “extended hydrogen” HZ (Pierrehumbert & Gaidos, 2011), which is estimated
to reach $\approx$10 au ($\approx 0.01\;S_{\oplus}$) for a G-type star.
Depending on the main science goals of _LIFE_ , one may prefer either of the
two scenarios, but, as we will see below, maximizing the number of temperate,
rocky exoplanets (scenario 2) leads to a decrease in the total number of
detectable planets (scenario 1).

The algorithm to distribute the observing time was similar to the one
discussed in Lay et al. (2007) and considered that for each star in a given
Monte Carlo run one can compute the detection efficiency (defined as number of
detected planets per time interval $\delta t$). The number of detected planets
depends on the threshold one puts on the S/N of the planets, which in turn
depends on the assumed aperture size of the collector spacecraft and the
assumed length of $\delta t$ (in our analysis we assumed $\delta t$=1h). Also,
one can decide which subset of planets to focus on (i.e., scenario 1 or
scenario 2). By computing the number of detectable planets for all stars and
over a sufficiently large range of time intervals, one can identify the star
that offers the maximum possible detection yield for the smallest time
interval. This star and the corresponding planet(s) as well as the length of
the required time interval were saved, and the star offering the second best
detection efficiency was searched. We repeated this process until the
available observing time (including the 10 h slew time from one star to the
next) was used up. This yielded the total number of detectable planets per
star as well as a rank-ordered list of target stars based on their expected
contribution to the detection yield. We then calculated the gain (i.e., planet
yield per time) as an average over all 500 Monte Carlo realizations for each
star. This allowed us to construct an observing sequence, which yielded the
final average numbers we are quoting below. For completeness we note that in
this analysis we implicitly assumed that the X-array of the collector
spacecraft did an integer number of full rotations around its center
irrespective of the assumed integration time. This allowed for an easier
computation of the exoplanets’ signals passing through the interferometer’s
modulation map (Dannert et al., 2022).

###  2.7 Detection criterion

In the following, we required a S/N$\geq$7 spectrally integrated over the full
wavelength range for a planet to be considered a detection. This choice
compensates for the lack of an instrumental noise model in the current
simulations. Under the assumptions that the instrumental noise contribution is
equal to or lower than the astrophysical noise and that the total noise can be
written as the square root of the sum of the instrumental and astrophysical
noise (i.e., $\sigma_{tot}=\sqrt{(\sigma_{inst})^{2}+(\sigma_{astro})^{2}}$),
a total S/N$\geq$7 corresponds to an astrophysical signal of
S/N${}_{astro}\geq$5.

As we were only considering photon noise, we verified that (slightly modified
versions of) published signal extraction algorithms for nulling-interferometry
data (e.g., Thiébaut & Mugnier, 2006; Mugnier et al., 2006) actually achieve a
performance close to the ideal photon-noise limited case and can also be
applied to multi-planet systems (Dannert et al., 2022).

##  3 Results

For the two scenarios outlined in Sect. 2.6, we chose an aperture size of
$D=2$ m as our reference case, but we also investigated cases with $D=1$ m and
$D=3.5$ m (the latter corresponding to the aperture of ESA’s _Herschel_
spacecraft, the largest monolithic infrared space telescope ever launched).
Besides using 4–18.5 $\mu$m as wavelength range in the reference case, we also
computed detection yields for 3–20 $\mu$m and 6–17 $\mu$m. We note that for
determining the final wavelength range not only the expected detection yield
during a 2.5-year search phase should be considered, but also the scientific
importance of molecular bands for atmospheric characterization at the short
and long wavelength end (Konrad et al., 2021) and technical aspects. We remind
the reader that in all cases the assumed instrument throughput is 5% (see
Sect. 2.3).

Figure 4: Total exoplanet detection yield from our reference case scenario
simulations ($D=2$ m; $\lambda=4-18.5\,\mu m$) using the planet classification
scheme introduced by Kopparapu et al. (2018) (see Table 2). For comparison, we
also plot the number of terrestrial exoplanets within the eHZ as defined in
Sect. 2.6 as the leftmost bar labeled “Rocky eHZ.” The bars show the number of
expected planet detections, including the statistical 1-$\sigma$ uncertainty
from the Monte Carlo approach but excluding uncertainties in the exoplanet
occurrence rates. Left panel: Scenario 1, i.e., the search phase is optimized
for maximizing the total number of exoplanets. Right panel: Scenario 2, i.e.,
the search phase is optimized for maximizing the number of rocky eHZ
exoplanets.

Figure 5: Total exoplanet detection yield from our reference case scenario
simulations ($D=2$ m; $\lambda=4-18.5\,\mu m$) shown as a function of spectral
type of the host star (left panel: Scenario 1; right panel: Scenario 2). The
bars show the number of expected planet detections, including the statistical
1-$\sigma$ uncertainty from the Monte Carlo approach but excluding
uncertainties in the exoplanet occurrence rates. In both panels, the green
bars (labeled “Rocky HZ”) show the number of rocky eHZ exoplanets, which are a
subset of the planets in the 0.5–1.5 R⊕ bin in Fig. 3.

###  3.1 Exoplanet yield of reference case scenarios

In Fig. 3 we show the expected number of detectable exoplanets and the
standard deviation resulting from our 500 Monte Carlo runs in the radius
versus stellar insolation plane for the reference case setup and the two
scenarios described in Sect. 2.6. Figure 4 is based on the same information,
but this time we follow the exoplanet classification scheme introduced by
Kopparapu et al. (2018) (see Table 2). This scheme was also used in the final
study reports by the _LUVOIR_ and _HabEx_ teams (The LUVOIR Team, 2019; Gaudi
et al., 2020), which allows for an easier comparison between the different
mission concepts. A current short-coming is that the scheme assumes a Sun-like
host star. Variations in the host star SED could potentially alter the stellar
flux condensation boundaries of the considered chemical species by a few
percent. A more robust analysis with different host stellar spectral types,
including M dwarfs, is needed to correct this. We note that in Fig. 4, we also
show the number of terrestrial exoplanets located within the eHZ as defined in
Sect. 2.6. for comparison with the other classes of exoplanets. It is
important to keep in mind that in all cases the number of detectable
exoplanets as a function of their radius and received insolation is influenced
by both the assumed underlying exoplanet population and our technical
assumptions.

Table 2: Exoplanet classification scheme introduced by Kopparapu et al. (2018). Together with the planet types defined in Table 1, we apply this scheme in Figs. 4, 8, 9, and 14, as well as in Figs. 18 and 21 in Appendix C. For reference, Venus would be classified as a “hot, rocky” planet and Earth and Mars as “warm, rocky” planets. Planet type |  $R_{\textrm{P}}$ [R⊕] | Stellar flux range [S⊕]
---|---|---
|  | Hot | Warm | Cold
Rocky | 0.5 – 1 | 182 – 1.0 | 1.0 – 0.28 | 0.28 – 0.0035
Super-Earths | 1 – 1.75 | 187 – 1.12 | 1.12 – 0.30 | 0.30 – 0.0030
Sub-Neptunes | 1.75 – 3.5 | 188 – 1.15 | 1.15 – 0.32 | 0.32 – 0.0030
Sub-Jovians | 3.5 – 6 | 220 – 1.65 | 1.65 – 0.45 | 0.45 – 0.0030

These plots show that within the present simulation framework, _LIFE_ would
discover hundreds of nearby exoplanets, the vast majority of which have radii
between 0.5 and 3 R⊕. It also shows that the choice of observing scenario has
a significant impact on the planet yield: on the one hand the number of rocky
exoplanets orbiting within the eHZ can indeed be significantly increased by a
factor of $\approx$1.6 if one optimizes the observing strategy accordingly
(scenario 2). For EECs (see Table 1) it is even a factor of $\approx$2\.
However, this comes at a price as the resulting total number of detectable
exoplanets is significantly smaller compared to scenario 1 ($\approx$350 vs.
$\approx$550). Because of their moderate temperatures, rocky exoplanets in the
eHZ are fainter than objects orbiting closer to the star and require larger
amounts of integration time to be detected. While for scenario 1 the typical
observing time per target is between 15 and 35 hours, it is between 50 and 130
hours for scenario 2.

In Fig. 5 we show the distribution of detectable exoplanets as a function of
the spectral type of their host star. For the rocky eHZ exoplanets there is a
strong preference for M dwarfs. This is because M dwarfs are, on average, much
more numerous and hence there is a larger number of M dwarfs close to the Sun.
In addition, they have a higher occurrence rate of terrestrial exoplanets (cf.
Dressing & Charbonneau, 2015). In Fig. 6 we show the distance distribution of
the detected exoplanets for both scenarios. By maximizing the number of rocky
eHZ exoplanets one exclusively observes stars within $\sim$10 pc of the Sun.

Figure 6: Distance distribution of the detected planet populations shown in
Fig. 3. The bars show the number of expected planet detections, including the
statistical 1-$\sigma$ uncertainty from the Monte Carlo approach but excluding
uncertainties in the exoplanet occurrence rates.

Another important parameter to look at is the detection efficiency, that is,
the number of detectable rocky eHZ exoplanets relative to the total number of
such exoplanets that were generated in our simulations. This is illustrated in
Fig. 7, which is based on the results for scenario 2. One can see that,
depending on the received stellar insolation (or, to a first approximation,
the resulting equilibrium temperature), only a certain fraction of the
exoplanets is detected. As indicated above, the main reason is the required
sensitivity rather than the spatial resolution. Still, some of the simulated
exoplanets are indeed at an orbital phase where they escape a detection with
the interferometer. However, it is reassuring that $\geq$50% detection
efficiency is achieved for exoplanets with $T_{\textrm{eq}}\geq 225$ K, or
insolations within $0.8\;S_{\oplus}\leq S_{\textrm{p}}\leq 1.5\;S_{\oplus}$.
This number could be further increased by implementing a multi-visit search
phase. Work on quantifying the gain in detection efficiency (and survey
completeness) as a function of number of visits in currently ongoing.

Figure 7: Detection efficiency for rocky eHZ exoplanets for our scenario 2.
Top panel: Equilibrium temperature distribution of all exoplanets present in
the surveyed sample (gray) and all detected exoplanets (blue). Bottom panel:
Same as above, but as a function of stellar insolation. In both panels the
detection efficiency (y axis on the right-hand side) is shown with the dashed
red line.

Figure 8: Same as Fig. 4, but now for $D=1.0$ m.

Figure 9: Same as Fig. 4, but now for $D=3.5$ m.

###  3.2 Impact of aperture size and wavelength range

In Figs. 8 and 9 we show the expected detection yield for apertures with $D=1$
m and $D=3.5$ m, respectively. The format is the same as for the reference
case shown in Fig. 4; the plots corresponding to Fig. 3 are available in
Appendix C, where we also show in Fig. 18 the relative changes in yield
compared to the reference case. Figure 10 provides a summary of the impact of
the aperture size on the total _LIFE_ exoplanet detection yield during the
2.5-year search phase. It shows that, as expected, aperture size strongly
affects the number of detectable exoplanets and it is important to point out
that the gain (loss) when going to larger (smaller) apertures is most
significant for small exoplanets of all temperatures and cool exoplanets of
all sizes. Specifically, Figs. 8 and 9 show that in the case of $D=3.5$ m, the
number of rocky eHZ exoplanets and EECs would increase to $\approx$63 (+132%)
and $\approx$28 (+161%), respectively, in scenario 1. The corresponding
numbers in scenario 2 are $\approx$78 (+78%) and $\approx$39 (+88%). The
relative smaller gain in scenario 2 compared to scenario 1 is explained by the
higher number of exoplanets already detected with the reference aperture size
of $D=2$ m. In case of $D=1$ m, the number of rocky eHZ exoplanets and EECs
would decrease to $\approx$6 (-76%) and $\approx$2 (-81%), respectively, in
scenario 1. In scenario 2, the numbers would go down to $\approx$17 (-61%) and
$\approx$7 (-64%) for rocky eHZ exoplanets and EECs, respectively.

Figure 10: Total expected exoplanet detection yield as a function of aperture
size and for each of the two scenarios. The numbers shown here are the sum of
the mean numbers shown in the grid cells in Figs. 3, 16, and 17. Figure 11:
Detection yield comparison between _LUVOIR A/B_ , _HabEx_ , and _LIFE_. For
_LIFE_ we show the numbers for the $D=2$ m reference case and for _HabEx_ the
numbers from the baseline 4-meter concept. Jovian planets are not shown,
because they were not included in the _LIFE_ simulations (cf. Sect. 2.2; see
text for important details).

The effect of changing the wavelength range is much weaker by comparison, and
generally the number of detectable planets only increases or decreases by a
few percent. Figures 19 and 20 in Appendix C show the results in the same
format as Fig. 3, and the changes relative to the reference case with
$\lambda=4-18.5\,\mu m$ are shown in Fig. 21.

##  4 Discussion

###  4.1 Total exoplanet yields

Figure 12: Median S/N of the detected exoplanets in our reference case
scenario simulations in the radius vs. stellar insolation plane (left panel:
Scenario 1; right panel: Scenario 2). We note that the 1D distributions on top
and to the right of the grids (as well as colored area) represent the numbers
of detected exoplanets, including the 1-$\sigma$ uncertainties shown in Fig.
3, and not the marginalized distributions of the S/N.

Looking at the total number of detectable exoplanets and their properties
reveals how diverse the expected _LIFE_ exoplanet yield will be. This sample
spans about four orders of magnitudes in planet insolation and about a factor
of 10 in planet radius. In addition to investigations concerning the
habitability of a subset of the sample, _LIFE_ has the potential to address a
number of scientific questions related to the formation and evolution of
exoplanets and their atmospheres.

In Fig. 11 we provide a comparison with the detection yields published in the
_HabEx_ and _LUVOIR_ study reports (Gaudi et al., 2020; The LUVOIR Team,
2019)101010See Stark et al. (2019) for details on the yield calculations for
the reflected light missions.. This plot suggests that for exoplanets with
radii up to 6 R⊕, _LIFE_ , with four times $D=2$ m apertures, can achieve
overall detection yields comparable to those of the _LUVOIR-A_ (15-meter
aperture) and _LUVOIR-B_ (8-meter aperture) concepts; _HabEx_ , with a 4-meter
primary mirror, is predicted to yield fewer detections. It needs to be noted,
though, that while in our simulations planets with radii $>$6 R⊕ were not
included, _LUVOIR A_ and _B_ and _HabEx_ are predicted to detect $\approx$117,
$\approx$102 and $\approx$31 of these Jovian planets, respectively. It is also
important to mention that the numbers for _LIFE_ are the sum of numbers for
the various planet types shown in Fig. 4. The resulting overall numbers of
detectable planets differ slightly from those shown in Fig. 10 because some
detectable planets fall outside the insolation ranges defined in Table 2.

Overall, these results show that in principle both approaches, large, single-
aperture reflected light missions and interferometric MIR missions (under the
assumptions laid out in Sect. 2), offer unprecedented opportunities for the
direct detection and detailed investigations of hundreds of nearby exoplanets.
Going forward, it will be important to investigate possible scientific
synergies between missions such as _HabEx_ or _LUVOIR_ and _LIFE_ because at
least a subset of the exoplanets detected by one approach is likely also
detectable by the other.

Figure 13: Comparison of exoplanet detections with _LIFE_ to known Solar
System planets and exoplanets. The _LIFE_ yield for the reference case
(scenario 2) is shown in red contours using a kernel density estimate of the
detected sample. Every shaded contour level corresponds to 50 exoplanets
detected in the respective parameter space. Blue points represent 60 out of
the 79 known exoplanets within 10 pc of the Sun for which we could estimate
the radius and insolation level. Gray points represent the four rocky planets
in the Solar System (E=Earth, V=Venus, Ma=Mars, and Me=Mercury).

The single most important parameter related to the number of detectable
exoplanets is, unsurprisingly, the aperture size of the collector spacecraft.
While here we focus on a 2-D array architecture for the interferometer with
four collector spacecraft with aperture sizes between $D=1$ m and $D=3.5$ m,
Dandumont et al. (2020) recently presented a similar yield analysis based on 4
different implementations of a two-aperture Bracewell interferometer: 2
CubeSat options (with a 0.5 or 1 m baseline and 0.08 m apertures), a Proba
mission option (with a 5 m baseline and 0.25 m apertures), and the _Fourier
Kelvin Stellar Interferometer_ concept presented in Danchi et al. (2008) and
Danchi & Barry (2010) (with a 12 m baseline and 0.5 m apertures). The trend
shown here continues down to CubeSat apertures and the detection of at least
$\approx$10–15 rocky eHZ exoplanets requires an aperture size of at least
$D=1$ m. The strong dependence of the detection yield on the aperture size
results from the fact that, in the vast majority of cases, the local zodiacal
dust emission is an important noise term and the effective FoV of the
collector spacecraft scales with $\lambda/D$ (cf. Sect. 2.3). In Appendix D we
provide an overview of the relative contributions of the various noise terms
to the total noise for planets detected in the reference case scenarios (Fig.
23).

Another key result from our analyses is that, depending on how the observing
time is distributed amongst the stellar targets, both the number of detected
exoplanets and the type of detected exoplanets can vary significantly. This
illustrates a strong need for the community to clearly define and prioritize
the scientific objectives of such a mission in order to derive the appropriate
observing strategy. An additional parameter that needs to be considered in
this context is the completeness of the survey, that is, how important it is
to have detected, with a certain level of confidence, all (or at least most)
exoplanets from a specific subset of the exoplanet population in the solar
neighborhood. Higher completeness requires more observing time per target star
(including multiple visits) and hence a lower number of detectable exoplanets
overall.

The results from the search phase, and also its duration, have an immediate
impact on the follow-up strategy during the characterization phase of the
mission. It is hence important to understand how well the fundamental
properties of the detected exoplanets (such as radius and temperature, but
also their orbital position) can be constrained from single-epoch data. In
Fig. 12 we present some first indications by showing the median S/N of the
detected exoplanets as a function of their radius and insolation. Because of
the much longer average integration time per star in scenario 2 (cf. Sect.
3.1), the median S/N is, in many cases, significantly higher than in scenario
1. Interestingly, the exoplanets in most grid cells (and certainly warm and
hot exoplanets with radii $>1.5\leavevmode\nobreak\ R_{\oplus}$) are detected
with sufficiently high S/N that some first-order estimate of their radius and
effective temperature and maybe even a rough analysis of their SED based on
(very) low-resolution spectroscopy appears feasible. This aspect needs to be
investigated further as the possibility to obtain spectral information already
from single-epoch data allowing for a first characterization and
classification of the exoplanets has an impact on the follow-up strategy
during the characterization phase. For completeness we note that the largest,
hottest planets receiving the highest levels of insolation do not show the
highest median S/N. This is because the S/N is related to the location of the
exoplanets in the transmission map of the interferometer, which maximizes the
throughput for exoplanets located in the eHZ and not for close-in exoplanets
(cf. Sect. 2.3).

Figure 14: Same as the right panel in Fig. 4, but ignoring all M-type dwarfs
in the target catalog and spending the full search phase on FGK stars only.
The panels show the results for $D=1.0$ m, $D=2.0$ m, and $D=3.5$ m from left
to right, respectively.

Finally, whether or not future exoplanet imaging space missions will have to
carry out a somewhat extended search phase, will also depend on the progress
and results of ongoing and future ground-based RV surveys. In Fig. 13 we show
a comparison between the expected _LIFE_ detection yield (reference case;
scenario 2) and currently known exoplanets within 10 pc of the Sun drawn from
the NASA Exoplanet Archive111111https://exoplanetarchive.ipac.caltech.edu. If
the insolation is not provided for planets in the archive, it is calculated
via $L_{\star}[\mathrm{L_{\odot}}]/a[\mathrm{AU}]^{2}$, with $L_{\star}$ the
host star luminosity, $L_{\odot}$ the solar luminosity, and $a$ the semimajor
axis of the exoplanet orbit. A missing radius measurement is estimated using
forecaster121212https://github.com/chenjj2/forecaster (Chen & Kipping, 2016).
This leads to 60 out of the 79 confirmed exoplanets within 10 pc for which we
can assign both radius and insolation. Fig. 13 shows on the one hand that
there is an interesting sample of planets already known within 10 pc from
which a preliminary target list could be compiled. On the other it
demonstrates that one expect a factor of 5 more planets to be found within 10
pc with _LIFE_. New (or the continuation of) systematic RV exoplanet searches
in the solar neighborhood will be fundamentally important to either provide
future imaging missions with a predefined exoplanet target list or at least
provide them with stringent constraints on the existence of nearby planetary
systems. The same is true for systematic searches of exozodi disks around
nearby stars. As mentioned above, already during the search phase typical
integration times are easily on the order of days. This means that S/Ns $>$5
per spectral channel131313First science requirements for the spectral
resolution and wavelength coverage of _LIFE_ are presented im (Konrad et al.,
2021). Previous works in this direction and in the context of atmospheric
characterization of terrestrial exoplanets at MIR wavelengths suggested
spectral resolutions of up to $R\approx 40$ (e.g., Des Marais et al., 2002;
Léger et al., 2019). are costly and knowing interesting or promising targets
beforehand saves valuable observing time.

###  4.2 Rocky HZ exoplanets: M-star preference and detection efficiency

As shown in Fig. 4, _LIFE_ could detect $\approx$25–45 of rocky exoplanets
located within the eHZ of their host stars and $\approx$10–20 EECs following
the definition of Kopparapu et al. (2018). However, these numbers are strongly
affected by the aperture size of the collector spacecraft. As shown in Figs. 8
and 9, decreasing (increasing) the aperture size yields a significant decrease
(increase) in the number of rocky temperate exoplanets. These findings will be
of great importance during upcoming trade-offs between mission cost, where
aperture size will be an important parameter, and science return. We stress
that this is not only important for the search phase, but it is even more
relevant for the characterization phase that aims at investigating a subsample
of the detected exoplanets in greater detail with high-S/N spectra (Konrad et
al., 2021).

Following similar arguments presented in Stark et al. (2014, 2015), Quanz et
al. (2021) argued that in order to obtain statistically robust results on the
fraction of rocky HZ exoplanets that are indeed habitable, at least 30 (better
50) exoplanets in that part of parameter space need to be studied. According
to the numbers presented above, this appears to be achievable with _LIFE_.
However, the vast majority of these planets is found around M stars and at
this point in time it is unknown whether exoplanets orbiting within the HZ
around M stars are able to retain (secondary) atmospheres because of the high
activity of M-type stars, in particular at young ages (e.g., Tian & Ida, 2015;
Luger & Barnes, 2015; Lingam & Loeb, 2018; Godolt et al., 2019; Atri & Mogan,
2021). It has been shown that under certain circumstances such exoplanets,
which are very likely tidally locked, may still provide habitable conditions
(e.g., Leconte et al., 2015; Ribas et al., 2016; Turbet et al., 2016; Boutle
et al., 2017), but empirical data are still lacking. There is, however, hope
that _JWST_ will be able to address this fundamentally important question and,
for a few cases, investigate the existence of atmospheres of rocky exoplanets
transiting M stars (e.g., Koll et al., 2019). Also, a deep characterization
effort of the potential M-star targets should be carried out, including the
high-energy radiation budget and its past history, in order to understand
which stars may have provided a more quiescent environment for their expected
planets. If rocky exoplanets orbiting M stars can retain atmospheres, then
_LIFE_ is in an excellent position to robustly characterize a significantly
larger sample. If not, then one may want to reconsider the observing strategy
and possibly de-prioritize M stars in the stellar input catalog. Figure 14
shows the results for the most extreme case, where all M stars are ignored and
the full search phase is spent on FGK stars. In this case, $\approx$25 rocky
eHZ exoplanets can be expected assuming an aperture size of $D=3.5$ m; with
$D=2.0$ m this number would drop to $\approx$11, limiting the statistical
power of the analysis. In this context, two points are important to mention:
(a) in order to further increase the number of detected rocky eHZ planets
around FGK stars, it will be important to investigate how a search phase with
a multi-visit strategy would affect the results. We remind the reader that a
detection efficiency $\geq$50% is currently achieved for exoplanets with
$T_{\textrm{eq}}\geq 225$ K or insolations within $0.8\;S_{\oplus}\leq
S_{\textrm{p}}\leq 1.5\;S_{\oplus}$. Ignoring the M stars and repeating the
analysis shown in Fig. 7 for FGK stars only reveals that the overall detection
efficiency is indeed lower (see Fig. 22 in Appendix C). Hence, we can expect
to gain additional detections when multiple visits per star are applied; (b)
As we discuss in Sect. 4.3.3, our underlying occurrence rates for rocky,
temperate planets around FGK stars may be on the rather conservative side.

Similar to the expected total detection yield, also the number of predicted
EECs can be compared to those published in the _HabEx_ and _LUVOIR_ study
reports (Gaudi et al., 2020; The LUVOIR Team, 2019). _HabEx_ , with its
4-meter baseline concept, is expected to detect $\approx$8 EECs, while
_LUVOIR-A_ and _LUVOIR-B_ are predicted to directly image $\approx$54 and
$\approx$28 EECs, respectively. Considering the $\approx$20 EECs that _LIFE_
is expected detect (assuming $D=2$ m and scenario 2), one has to keep in mind
its preference for planets around M stars, while _HabEx_ and _LUVOIR_ have a
strong detection bias for planets around solar-type stars. This suggests that
there is only limited overlap in primary discovery space for EECs between the
missions if they were to carry out independent search phases. Hence, it will
be important to check the potential overlap assuming that one mission is
following-up after the other (e.g., _LIFE_ following after _LUVOIR/HabEx_). In
addition, looking more carefully at the underlying EEC occurrence rates, it
shows that the numbers cannot be directly compared (see Sect. 4.3.3 below).

###  4.3 Remaining limitations and uncertainties in the simulations

####  4.3.1 Treatment of noise sources

Compared to previous works the yield simulations presented here are based on a
more realistic treatment of the observing technique and include all major
astrophysical noise terms. One of the next crucial steps is to continue the
development of an instrument concept including a noise breakdown structure so
that quantitative instrumental noise estimates can be included in the
simulations. The calculations from Lay (2004) indicate that the noise
contributions from photon noise and instrumental noise may indeed not be very
different. However, these calculations were done for a specific example and
how the relative contributions scale with stellar and planet properties and
exozodi brightness needs to be investigated. Also, recent work by Dandumont et
al. (2020) and other previous analyses in the context of _TPF-I_ (e.g., Lay,
2006) or _Darwin_ (e.g., Defrère et al., 2010) can serve as starting points.
In addition to the noise budget, important instrument parameters such as
overall throughput and detector quantum efficiency need to be further
validated. For the interested reader we provide a summary of the status of
some key technologies relevant for _LIFE_ in Appendix B.

####  4.3.2 Treatment of exozodiacal and zodiacal dust

While we do take into account emission from potential exozodiacal dust disks
using the nominal distribution from the HOSTS survey (Ertel et al., 2020), it
needs to be acknowledged that there is still considerable uncertainty in the
median exozodi level: while in the nominal distribution the median zodi level
is $\bar{z}$$\approx$3.2, Ertel et al. (2020) show that one can only be
confident at the 1$\sigma$ level that the median is below 9 zodis and at the
2$\sigma$ level that it is below 27 zodis. In order to quantify the impact of
these uncertainties on the detection yield, we did the following experiment
for the reference case scenarios: the exozodi level distribution was shifted
by adding multiples of the median absolute deviation (MAD) of the distribution
(MAD($z$)$\approx$2.7) to each individual exozodi level in the sample. In the
most extreme case we analyzed the distribution was shifted by 9$\cdot$MAD,
resulting in a median $\bar{z}$$\approx$27 corresponding to the 2$\sigma$
level mentioned above. In this case, the total number of detectable planets
decreased by $\lesssim$6% and the number of rocky, HZ planets changed even
less. One reason for this somewhat limited impact is that, compared to noise
from stellar leakage and local zodiacal dust emission, noise from exozodiacal
dust disks contributes only little to the total noise budget of detected
planets (see Fig. 23). Hence, in a statistical sense, the current
uncertainties may not have a strong impact on the overall results. Still,
additional observational efforts determining exozodi levels would further
improve upon the current statistical uncertainties and, maybe even more
importantly, would also help prioritize the most promising individual targets
for future space missions. In addition, at the moment, the HOSTS survey does
not show a correlation between spectral type and the level of exozodi emission
(Ertel et al., 2020), and hence we apply the same underlying distribution of
exozodi levels to all target stars irrespective of spectral type. A larger
data set would be required to further confirm this current finding.

Furthermore, the inclination of exozodiacal dust disks and possible spatial
offsets and substructures (e.g., “clumps,” such as those seen in the zodiacal
light; Reach, 2010; Krick et al., 2012) are not considered in our simulations.
As long as exozodiacal dust disks are centrally symmetric and optically thin,
their contribution to the photon noise in a _LIFE_ measurement is to first
order independent from their inclination; hence, changing the inclination of
the disks has no measurable impact on the results. Spatial offsets and disk
substructures would, however, have an impact on the S/N calculations. Defrère
et al. (2010) looked at the specific case of an Earth-Sun twin located at 15
pc. They modeled planet induced resonant structures in exozodi disks with
varying dust density and inclination and investigated up to what exozodi level
the planet would still be detectable. They concluded that around 10 $\mu$m
wavelength, up to $\sim$15 and $\sim$7 zodis are acceptable for disks with
inclinations between 0-30∘ and up to 60 ∘, respectively. For edge-on systems
this limit drops to $\sim$1.4 zodis. In order to further refine the results
presented here, analyses as the ones presented in Defrère et al. (2010) could
to be carried out, possibly enlarging the covered parameter space. However, as
already noted by the authors, advanced signal processing approaches may
further relax the constraints in terms of acceptable zodi levels mentioned
above. Also, if one considers the nominal distribution of exozodi levels
published by the HOSTS team, the above mentioned zodi level limits appear to
be on the high-end side: about two-thirds of our simulated systems with
randomly assigned zodi levels based on the nominal distribution have disks
with $\leq$7 zodis. Hence, we do not expect exozodi dust disks to be a show-
stopper for _LIFE_. However, systems that are seen (close to) edge-on may pose
a real challenge and have to be investigated in more detail, and a coordinated
effort to reduce the existing uncertainties in the median exozodi level of
nearby stars remains certainly important.

Defrère et al. (2010) also addressed the question of disk offsets, where the
geometric center of the exozodi disk is shifted away from the center of the
star, which would lead to additional flux through the modulation map of the
interferometer. They focused on a Sun-like star at 15 pc and considered only
face-on systems. When looking at the modulated signal covering the full
wavelength range, systems with up to $\sim$50 zodis and offsets as large as
$\sim$0.5 au were considered acceptable. As the offset increases, the level of
acceptable zodis decreases, but for offsets as large as 1 au, $\sim$5 zodis
could still be tolerated. It hence seems that, apart from potentially extreme
cases, disk offsets are not a major concern for _LIFE_. For reference: in the
Solar System, the center of the zodiacal cloud is shifted by only about 0.013
AU from the Sun (Landgraf & Jehn, 2001).

Similarly to the exozodi disks, also our zodiacal dust model does not contain
any substructures. As mentioned above, and shown in Fig. 23, the MIR emission
from the zodiacal dust is an important astrophysical noise source in a typical
_LIFE_ observation and the model should hence be further refined to correct
for the current overestimation of the emission shortward of 6 $\mu$m.

####  4.3.3 Occurrence rates of small, temperate exoplanets

An additional uncertainty in our results is related to the underlying
exoplanet population. While in some parts of the parameter space the
occurrence rate of exoplanets was robustly measured by the Kepler mission,
there remains significant uncertainty related to the completeness and
reliability for the occurrence rates of rocky, temperate exoplanets around
Sun-like stars (e.g., Bryson et al., 2020). We note that recent estimates for
$\eta_{\oplus}$, which is the fraction of stars with terrestrial exoplanets
within their HZ, are higher than the ones resulting from our underlying
distributions: Bryson et al. (2021) provided two values, $\eta_{\oplus}^{\rm
o}=0.58^{+0.73}_{-0.33}$ and $\eta_{\oplus}^{\rm o}=0.88^{+1.28}_{-0.51}$, for
the occurrence rate of planets with radii between 0.5 and 1.5 $R_{\oplus}$
orbiting in the eHZ of stars with effective temperatures between 4800 and 6300
K141414Our definition of the eHZ is identical to their “optimistic” HZ case;
the superscript “o” in $\eta_{\oplus}^{\rm o}$ refers to the word
“optimistic”.. These bounds represent two extreme assumptions about the
extrapolation of completeness beyond orbital periods where the Kepler DR25
completeness data are available. For EECs around solar-type stars, Bryson et
al. (2021) found a lower bound of $\eta_{\oplus}^{\rm
EEC}=0.18^{+0.16}_{-0.28}$ and an upper bound of $\eta_{\oplus}^{\rm
EEC}=0.28^{+0.30}_{-0.09}$. In our simulations, $\eta_{\oplus}^{\rm o}\approx
0.37$ and $\eta_{\oplus}^{\rm EEC}\approx 0.16$ for FGK dwarfs (and
$\approx$0.56 and $\approx$0.31 for M dwarfs, respectively; see Table 1).
Hence, in particular for solar-type stars, we might be underestimating the
number of EECs and rocky, eHZ planets. Also, our value for $\eta_{\oplus}^{\rm
EEC}$ is lower than the one used in the _HabEx_ and _LUVOIR_ concept studies,
which was $\eta_{\oplus}^{\rm EEC}=0.24^{+0.46}_{-0.16}$. One reason for this
difference is that we did not keep this parameter constant throughout our
stellar sample, but we let it vary between various spectral types (see notes
in Table 1). Hence, at least for this specific subset of planets, a direct
quantitative comparison between our results and the other mission studies is
not immediately straightforward. In a future study, we will further
investigate the impact of the various values of $\eta_{\oplus}$ and their
statistical and systematic uncertainties on the resulting detection yield (cf.
Léger et al., 2015).

Overall, it is clear that reanalyses of the _Kepler_ data, in combination with
additional results from _K2_ , _TESS_ , and the upcoming _PLAnetary Transits
and Oscillations of stars (PLATO)_ mission (Rauer et al., 2014) are extremely
important to provide a more robust empirical basis for future updates of the
analyses presented here. In particular _PLATO_ is designed to detect Earth-
like planets in the HZ of solar-like stars and will improve our knowledge of
the occurrence rate and formation mechanism of these targets.

##  5 Summary and conclusions

We have presented new and more realistic results for the exoplanet detection
yield of a space-based MIR nulling interferometer based on the exoplanet
statistics observed by the _Kepler_ mission and targeting main-sequence FGKM
stars within 20 pc of the Sun. Taking into account all major astrophysical
noise terms and adding some margin for the not yet included instrumental noise
effects (we require a S/N$\geq$7 for a detection), we find that an
interferometer array consisting of four 2 m apertures and covering the 4–18.5
$\mu$m wavelength range with a total throughput of 5% will yield, depending on
the observing strategy, between $\approx$350 and $\approx$550 directly
detected exoplanets with radii between 0.5 and 6 R⊕ within a 2.5-year search
phase. Between $\approx$160 and $\approx$190 of these exoplanets have radii
between 0.5 and 1.5 R⊕. As there is some freedom in how to assign observing
time to the stellar targets, one can attempt to maximize the number of
detected planets in certain areas of parameter space. We demonstrated this
with two scenarios where either the total number of exoplanets or the number
of rocky planets within the empirical HZ is maximized. The observing strategy
must be adapted to the overall scientific objectives of the _LIFE_ mission
since it influences the number (and types) of detected exoplanets.

Keeping the instrument throughput fixed at 5%, we find the number of
detectable exoplanets to be a strong function of aperture size. In our current
analysis, the wavelength range has a negligible impact on the exoplanet yield.
We have shown that $\approx$25–45 rocky exoplanets within the empirical HZ of
their host stars are expected to be detectable with four 2 m apertures, but
this number could go up to $\approx$60–80 if the aperture size were increased
to 3.5 m. In this case, the total number of detectable planets could go up to
$\approx$770\. With four 1 m apertures, the number of rocky exoplanets within
the empirical HZ would be $\leq$20 and the total detection yield $<$320\.
Irrespective of aperture size, the vast majority of rocky exoplanets orbiting
within the empirical HZ are detected around M dwarfs. It will be important to
further investigate if these planets could in principle possess atmospheres
despite the high-energy UV flux and flaring activity these stars display. To
further increase the number of detected rocky HZ planets around FGK stars,
multiple visits per star during the search phase need to be considered in
future work.

All numbers presented here (i.e., the total number of detected planets and the
number of rocky planets within the HZ) are competitive with those predicted
for current mission concepts searching for exoplanets in reflected light.
Further studies investigating potential scientific and operational synergies
between a reflected light and a thermal emission mission should be considered.
Such efforts are particularly important for small temperate planets, such as
EECs, because reflected light missions have a strong bias for detecting these
objects primarily around solar-type stars, while _LIFE_ has a strong bias for
planets around M stars. We note, however, that when comparing the numbers of
detectable EECs with those predicted for the _LUVOIR_ and _HabEx_ missions,
the underlying occurrence rates are not identical. The simulations presented
here use lower values for EECs around FGK stars. This shows that care must be
taken when comparing predicted detection yields of future missions, and
additional efforts, such as obtaining new data and investigating new data
analysis approaches, are needed to further refine the statistical occurrence
rates that form the basis for all yield calculations.

Comparing the predicted primary discovery space of _LIFE_ with known
exoplanets within 10 pc of the Sun shows that there are $>$40 objects,
$\approx$15 of which have predicted radii $<$1.5 R⊕, that could be added to a
target list today. To minimize the time that future exoplanet imaging space
missions have to devote to an initial search phase, continuing ground- and
space-based detection surveys is crucial.

We note that both the overall exoplanet detection yield and the observing time
required to robustly characterize the atmospheric properties of rocky,
temperate exoplanets with an MIR interferometer are strong functions of
apertures size, which must be considered in future trade-off studies. The MIR
regime is particularly rich in molecular absorption bands of the main
constituents of terrestrial exoplanet atmospheres, including major
biosignatures (e.g., Schwieterman et al., 2018; Catling et al., 2018). Also,
thermal emission spectra provide more information about the atmospheric
structure and allow for a more direct measurement of the planetary radius than
reflected light data (e.g., Line et al., 2019). The relatively high S/N
(integrated over the full wavelength range) that most detectable planets in
our simulations have suggests that decent estimates for their radii and
effective temperatures – and in some cases even rough SEDs – seem possible. In
this case, the data from a single-epoch observation obtained during the search
phase will already provide crucial information for categorizing and
prioritizing the planets for the follow-up characterization phase (for further
details, see Dannert et al., 2022).

Our results show that when investigating and selecting future large exoplanet
imaging space missions, for instance in the context of ESA’s Voyage 2050
program, a concept such as _LIFE_ should be considered a serious contender and
may be required to ultimately assess the habitability of exoplanets151515See,
e.g., ”Exoplanet Strategy Report 2018” from the National Academies of
Sciences, Engineering, and Medicine available at
https://www.nap.edu/catalog/25187/exoplanet-science-strategy.. Taking into
account the heritage from the _Darwin_ and _TPF-I_ studies and more recent
progress based on various local activities, new coordinated efforts to further
understand and increase the technological readiness level of key components
have started as part of the _LIFE_ initiative (e.g., Gheorghe et al., 2020).

###### Acknowledgements.

We thank the anonymous referee for a critical and constructive review of the
original manuscript which helped improve the quality of the paper. This work
has been carried out within the framework of the National Centre of Competence
in Research PlanetS supported by the Swiss National Science Foundation. SPQ,
EA and HSW. acknowledge the financial support of the SNSF. SK acknowledges
funding from an ERC Starting Grant (grant agreement No. 639889). DE
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (project Four
Aces; grant agreement No 724427). JL-B acknowledges financial support received
from “la Caixa” Foundation (ID 100010434) and the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 847648, with fellowship code LCF/BQ/PI20/11760023. RA is a
Trottier Postdoctoral Fellow and acknowledges support from the Trottier Family
Foundation. This work was supported in part through a grant from FRQNT. TL
acknowledges funding from the Simons Foundation (SCOL award No. 611576). Part
of this work was conducted at the Jet Propulsion Laboratory, California
Institute of Technology, under contract with NASA. This research has made use
of the SIMBAD database, operated at CDS, Strasbourg, France, and of the
Washington Double Star Catalog maintained at the U.S. Naval Observatory. This
research has made use of the NASA Exoplanet Archive, which is operated by the
California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.
This research has made use of the following Python packages: astropy (Astropy
Collaboration et al., 2013, 2018), matplotlib (Hunter, 2007), numpy (Van Der
Walt et al., 2011), and scipy (Virtanen et al., 2020).

_Author contributions:_ SPQ initiated the project, devised the analyses and
wrote the manuscript. MO and FD wrote the LIFESim tool and created the
figures. AGh and EF contributed to the LIFESim tool. JK simulated the
exoplanet populations. FM created the _LIFE_ target star catalog. All authors
discussed the results and commented on the manuscript.

## References

  * Angel & Woolf (1997) Angel, J. R. P. & Woolf, N. J. 1997, ApJ, 475, 373
  * Anglada-Escudé et al. (2016) Anglada-Escudé, G., Amado, P. J., Barnes, J., et al. 2016, Nature, 536, 1
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Astudillo-Defru et al. (2017) Astudillo-Defru, N., Díaz, R. F., Bonfils, X., et al. 2017, Astronomy & Astrophysics, 605, L11
  * Atri & Mogan (2021) Atri, D. & Mogan, S. R. C. 2021, MNRAS, 500, L1
  * Berta-Thompson et al. (2015) Berta-Thompson, Z. K., Irwin, J., Charbonneau, D., et al. 2015, Nature, 527, 204
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
  * Boutle et al. (2017) Boutle, I. A., Mayne, N. J., Drummond, B., et al. 2017, A&A, 601, A120
  * Bowens et al. (2021) Bowens, R., Meyer, M. R., Delacroix, C., et al. 2021, A&A, 653, A8
  * Bracewell (1978) Bracewell, R. N. 1978, Nature, 274, 780
  * Brandl et al. (2021) Brandl, B., Bettonvil, F., van Boekel, R., et al. 2021, The Messenger, 182, 22
  * Brandl et al. (2018) Brandl, B. R., Absil, O., Agócs, T., et al. 2018, in Proceedings of the SPIE, ed. H. Takami, C. J. Evans, & L. Simard, Leiden Univ. (Netherlands) (SPIE), 107021U
  * Bryson et al. (2020) Bryson, S., Coughlin, J., Batalha, N. M., et al. 2020, AJ, 159, 279
  * Bryson et al. (2021) Bryson, S., Kunimoto, M., Kopparapu, R. K., et al. 2021, AJ, 161, 36
  * Cabrera et al. (2020) Cabrera, M. S., McMurtry, C. W., Forrest, W. J., et al. 2020, Journal of Astronomical Telescopes, Instruments, and Systems, 6, 011004
  * Catling et al. (2018) Catling, D. C., Krissansen-Totton, J., Kiang, N. Y., et al. 2018, Astrobiology, 18, 709
  * Chen & Kipping (2016) Chen, J. & Kipping, D. 2016, The Astrophysical Journal, 834, 17
  * Colavita et al. (2009) Colavita, M. M., Serabyn, E., Millan-Gabet, R., et al. 2009, PASP, 121, 1120
  * Danchi & Barry (2010) Danchi, W. C. & Barry, R. K. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7734, Optical and Infrared Interferometry II, ed. W. C. Danchi, F. Delplancke, & J. K. Rajagopal, 77340M
  * Danchi et al. (2008) Danchi, W. C., Barry, R. K., Lawson, P. R., Traub, W. A., & Unwin, S. 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7013, Optical and Infrared Interferometry, 70132Q
  * Dandumont et al. (2020) Dandumont, C., Defrère, D., Kammerer, J., et al. 2020, Journal of Astronomical Telescopes, Instruments, and Systems, 6, 035004
  * Dannert et al. (2022) Dannert, F., Ottiger, M., Quanz, S. P., et al. 2022, arXiv e-prints, arXiv:2203.00471
  * Defrère et al. (2018a) Defrère, D., Absil, O., Berger, J. P., et al. 2018a, Experimental Astronomy, 46, 475
  * Defrère et al. (2010) Defrère, D., Absil, O., den Hartog, R., Hanot, C., & Stark, C. 2010, A&A, 509, A9
  * Defrère et al. (2018b) Defrère, D., Léger, A., Absil, O., et al. 2018b, Experimental Astronomy, 46, 543
  * Des Marais et al. (2002) Des Marais, D. J., Harwit, M. O., Jucks, K. W., et al. 2002, Astrobiology, 2, 153
  * Díaz et al. (2019) Díaz, R. F., Delfosse, X., Hobson, M. J., et al. 2019, A&A, 625, A17
  * Dressing & Charbonneau (2015) Dressing, C. D. & Charbonneau, D. 2015, ApJ, 807, 45
  * Dulz et al. (2020) Dulz, S. D., Plavchan, P., Crepp, J. R., et al. 2020, ApJ, 893, 122
  * Ertel et al. (2020) Ertel, S., Defrère, D., Hinz, P., et al. 2020, AJ, 159, 177
  * Ertel et al. (2018) Ertel, S., Defrère, D., Hinz, P., et al. 2018, AJ, 155, 194
  * Gaudi et al. (2020) Gaudi, B. S., Seager, S., Mennesson, B., et al. 2020, arXiv e-prints, arXiv:2001.06683
  * Gheorghe et al. (2020) Gheorghe, A. A., Glauser, A. M., Ergenzinger, K., et al. 2020, in Optical and Infrared Interferometry and Imaging VII, ed. P. G. Tuthill, A. Merand, & S. Sallum, Vol. 11446, International Society for Optics and Photonics (SPIE), 497 – 505
  * Gillon et al. (2017) Gillon, M., Demory, B.-O., Van Grootel, V., et al. 2017, Nature Astronomy, 1, 0056
  * Gillon et al. (2017) Gillon, M., Triaud, A. H. M. J., Demory, B.-O., et al. 2017, Nature, 542, 456
  * Godolt et al. (2019) Godolt, M., Tosi, N., Stracke, B., et al. 2019, A&A, 625, A12
  * He et al. (2019) He, M. Y., Ford, E. B., & Ragozzine, D. 2019, MNRAS, 490, 4575
  * Hinz et al. (2014) Hinz, P., Bailey, V. P., Defrère, D., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9146, Optical and Infrared Interferometry IV, ed. J. K. Rajagopal, M. J. Creech-Eakman, & F. Malbet, 91460T
  * Holman & Wiegert (1999) Holman, M. J. & Wiegert, P. A. 1999, AJ, 117, 621
  * Hunter (2007) Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90
  * Janson et al. (2018) Janson, M., Brandeker, A., Boehm, C., & Martins, A. K. 2018, Future Astrometric Space Missions for Exoplanet Science, ed. H. J. Deeg & J. A. Belmonte, 87
  * Jeffers et al. (2020) Jeffers, S. V., Dreizler, S., Barnes, J. R., et al. 2020, Science, 368, 1477
  * Johnstone et al. (2019) Johnstone, C. P., Khodachenko, M. L., Lüftinger, T., et al. 2019, Astronomy & Astrophysics, 624, L10
  * Kammerer & Quanz (2018) Kammerer, J. & Quanz, S. P. 2018, A&A, 609, A4
  * Kasper et al. (2021) Kasper, M., Cerpa Urra, N., Pathak, P., et al. 2021, The Messenger, 182, 38
  * Kasting et al. (1993) Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108
  * Kelsall et al. (1998) Kelsall, T., Weiland, J. L., Franz, B. A., et al. 1998, ApJ, 508, 44
  * Kennedy et al. (2015) Kennedy, G. M., Wyatt, M. C., Bailey, V., et al. 2015, ApJS, 216, 23
  * Koll et al. (2019) Koll, D. D. B., Malik, M., Mansfield, M., et al. 2019, The Astrophysical Journal, 886, 140
  * Konrad et al. (2021) Konrad, B. S., Alei, E., Angerhausen, D., et al. 2021, arXiv e-prints, arXiv:2112.02054
  * Kopparapu et al. (2018) Kopparapu, R. K., Hébrard, E., Belikov, R., et al. 2018, ApJ, 856, 122
  * Kopparapu et al. (2013) Kopparapu, R. k., Ramirez, R., Kasting, J. F., et al. 2013, The Astrophysical Journal, 765, 131
  * Kopparapu et al. (2014) Kopparapu, R. K., Ramirez, R. M., SchottelKotte, J., et al. 2014, ApJ, 787, L29
  * Kraus et al. (2016) Kraus, A. L., Ireland, M. J., Huber, D., Mann, A. W., & Dupuy, T. J. 2016, AJ, 152, 8
  * Kreidberg & Loeb (2016) Kreidberg, L. & Loeb, A. 2016, The Astrophysical Journal Letters, 832, L12
  * Krick et al. (2012) Krick, J. E., Glaccum, W. J., Carey, S. J., et al. 2012, ApJ, 754, 53
  * Krissansen-Totton et al. (2018) Krissansen-Totton, J., Garland, R., Irwin, P., & Catling, D. C. 2018, The Astronomical Journal, 156, 114
  * Kunimoto & Matthews (2020) Kunimoto, M. & Matthews, J. M. 2020, AJ, 159, 248
  * Landgraf & Jehn (2001) Landgraf, M. & Jehn, R. 2001, Ap&SS, 278, 357
  * Lay (2004) Lay, O. P. 2004, Appl. Opt., 43, 6100
  * Lay (2005) Lay, O. P. 2005, Applied Optics IP, 44, 5859
  * Lay (2006) Lay, O. P. 2006, in SPIE Astronomical Telescopes + Instrumentation, ed. J. D. Monnier, M. Schöller, & W. C. Danchi (SPIE), 62681A–14
  * Lay et al. (2007) Lay, O. P., Martin, S. R., & Hunyadi, S. L. 2007, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6693, Techniques and Instrumentation for Detection of Exoplanets III, ed. D. R. Coulter, 66930A
  * Leconte et al. (2015) Leconte, J., Wu, H., Menou, K., & Murray, N. 2015, Science, 347, 632
  * Léger et al. (2019) Léger, A., Defrère, D., García Muñoz, A., et al. 2019, Astrobiology, 19, 797
  * Léger et al. (2015) Léger, A., Defrère, D., Malbet, F., Labadie, L., & Absil, O. 2015, ApJ, 808, 194
  * Leger et al. (1995) Leger, A., Puget, J. L., Mariotti, J. M., Rouan, D., & Schneider, J. 1995, Ap&SS, 223, 172
  * Line et al. (2019) Line, M., Quanz, S. P., Schwieterman, E. W., et al. 2019, BAAS, 51, 271
  * Lingam & Loeb (2018) Lingam, M. & Loeb, A. 2018, J. Cosmology Astropart. Phys., 2018, 020
  * Lovis et al. (2017) Lovis, C., Snellen, I., Mouillet, D., et al. 2017, Astronomy & Astrophysics, 599, A16
  * Luger & Barnes (2015) Luger, R. & Barnes, R. 2015, Astrobiology, 15, 119
  * MacGregor et al. (2018) MacGregor, M. A., Weinberger, A. J., Wilner, D. J., Kowalski, A. F., & Cranmer, S. R. 2018, The Astrophysical Journal Letters, 855, L2
  * Malbet & Sozzetti (2018) Malbet, F. & Sozzetti, A. 2018, Astrometry as an Exoplanet Discovery Method, ed. H. J. Deeg & J. A. Belmonte, 196
  * Marconi (2020) Marconi, A. 2020, in Ground-based and Airborne Instrumentation for Astronomy VIII, ed. C. J. Evans, J. J. Bryant, & K. Motohara, Vol. 11447, International Society for Optics and Photonics (SPIE), 414 – 425
  * Martin et al. (2012) Martin, S., Booth, A., Liewer, K., et al. 2012, Appl. Opt., 51, 3907
  * Mason et al. (2001) Mason, B. D., Wycoff, G. L., Hartkopf, W. I., Douglass, G. G., & Worley, C. E. 2001, AJ, 122, 3466
  * Mayor et al. (2011) Mayor, M., Marmier, M., Lovis, C., et al. 2011, arXiv e-prints, arXiv:1109.2497
  * Meixner et al. (2019) Meixner, M., Cooray, A., Leisawitz, D., et al. 2019, arXiv e-prints, arXiv:1912.06213
  * Morley et al. (2017) Morley, C. V., Kreidberg, L., Rustamkulov, Z., Robinson, T., & Fortney, J. J. 2017, The Astrophysical Journal, 850, 121
  * Mugnier et al. (2006) Mugnier, L., Thiébaut, E., & Belu, A. 2006, in EAS Publications Series, Vol. 22, EAS Publications Series, ed. M. Carbillet, A. Ferrari, & C. Aime, 69–83
  * National Academies of Sciences & Medicine (2021) National Academies of Sciences, E. & Medicine. 2021, Pathways to Discovery in Astronomy and Astrophysics for the 2020s (Washington, DC: The National Academies Press)
  * Oshagh et al. (2017) Oshagh, M., Santos, N. C., Figueira, P., et al. 2017, A&A, 606, A107
  * Pecaut & Mamajek (2013) Pecaut, M. J. & Mamajek, E. E. 2013, ApJS, 208, 9
  * Peñín et al. (2020) Peñín, L. F., Scoarnec, Y., Fernández-Ibarz, J. M., et al. 2020, in Proceedings of the Small Satellite Conference, Technical Session I: Space Mission Architectures, Paper 02
  * Pierrehumbert & Gaidos (2011) Pierrehumbert, R. & Gaidos, E. 2011, ApJ, 734, L13
  * Quanz et al. (2021) Quanz, S. P., Absil, O., Benz, W., et al. 2021, Experimental Astronomy
  * Quanz et al. (2015) Quanz, S. P., Crossfield, I., Meyer, M. R., Schmalzl, E., & Held, J. 2015, International Journal of Astrobiology, 14, 279
  * Quanz et al. (2018) Quanz, S. P., Kammerer, J., Defrère, D., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10701, Optical and Infrared Interferometry and Imaging VI, 107011I
  * Rauer et al. (2014) Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249
  * Reach (2010) Reach, W. T. 2010, Icarus, 209, 848
  * Ribas et al. (2016) Ribas, I., Bolmont, E., Selsis, F., et al. 2016, Astronomy & Astrophysics, 596, A111
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Rieke et al. (2015) Rieke, G. H., Ressler, M. E., Morrison, J. E., et al. 2015, PASP, 127, 665
  * Sakon et al. (2018) Sakon, I., Roellig, T. L., Ennico-Smith, K., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10698, Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave, ed. M. Lystrup, H. A. MacEwen, G. G. Fazio, N. Batalha, N. Siegler, & E. C. Tong, 1069817
  * Schwieterman et al. (2018) Schwieterman, E. W., Kiang, N. Y., Parenteau, M. N., et al. 2018, Astrobiology, 18, 663
  * Stark et al. (2019) Stark, C., Belikov, R., Bolcar, M., et al. 2019, in American Astronomical Society Meeting Abstracts, Vol. 233, American Astronomical Society Meeting Abstracts #233, 402.05
  * Stark et al. (2015) Stark, C. C., Roberge, A., Mandell, A., et al. 2015, ApJ, 808, 149
  * Stark et al. (2014) Stark, C. C., Roberge, A., Mandell, A., & Robinson, T. D. 2014, ApJ, 795, 122
  * The LUVOIR Team (2019) The LUVOIR Team. 2019, arXiv e-prints, arXiv:1912.06219
  * Thiébaut & Mugnier (2006) Thiébaut, E. & Mugnier, L. 2006, in IAU Colloq. 200: Direct Imaging of Exoplanets: Science & Techniques, ed. C. Aime & F. Vakili, 547–552
  * Tian & Ida (2015) Tian, F. & Ida, S. 2015, Nature Geoscience, 8, 177
  * Tinetti et al. (2018) Tinetti, G., Drossart, P., Eccleston, P., et al. 2018, Experimental Astronomy, 46, 135
  * Tuomi et al. (2019) Tuomi, M., Jones, H. R. A., Butler, R. P., et al. 2019, arXiv e-prints, arXiv:1906.04644
  * Turbet et al. (2016) Turbet, M., Leconte, J., Selsis, F., et al. 2016, Astronomy & Astrophysics, 596, A112
  * Van Der Walt et al. (2011) Van Der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science & Engineering, 13, 22
  * Vanderspek et al. (2019) Vanderspek, R., Huang, C. X., Vanderburg, A., et al. 2019, The Astrophysical Journal Letters, 871, L24
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9
  * Zechmeister et al. (2019) Zechmeister, M., Dreizler, S., Ribas, I., et al. 2019, Astronomy & Astrophysics, 627, A49

##  Appendix A Stellar sample

The stellar sample was compiled from querying the SIMBAD
database161616http://simbad.cds.unistra.fr/simbad/ (Wenger et al. 2000) for
objects within 20 pc. We removed the substellar objects (planets and brown
dwarfs) using the object type parameter. For the remaining stellar objects we
focused on main-sequence stars as indicated by the luminosity class of the
objects (in case no luminosity class was given we assumed the objects were
main-sequence objects). Based on the spectral type of the object we then
assigned effective temperature, radius and mass using the relation published
in Table 5 of Pecaut & Mamajek (2013), which is based on empirical data. In
order to assess which objects are members of binary or multiple systems we
used SIMBAD’s hierarchical link. This feature connects an object with its
parent and child objects. To keep the complexity of our multiple star sample
low we decided to only use wide binaries where planetary orbits are possible
around both components and thus are most similar to orbits around single
stars. We therefore excluded all systems with more than two components and
also those that had binary subtypes as SIMBAD object type. This was necessary
because not all binary components had their own SIMBAD entry (for example, if
the objects cannot be observed individually due to too small separations). In
that case the system has no children and we cannot distinguish it from a
single star. We also removed objects with incomplete information for the
stellar parameters (this step also included binaries, where one component was
not listed as a main-sequence star). To obtain an estimate for the separation
between the remaining binary systems, we cross-matched the system position
with the Washington Visual Double Star Catalog (WDS; Mason et al. 2001) by
drawing a circle with a radius of 1 arcsec around the system position. If the
position of a WDS object was close enough to fall within the circle we assumed
that the two objects were the same physical system. In the case the WDS
catalog had more components listed per system than SIMBAD (e.g., because of
background stars) we took the separation between the two main components. The
separation is normally given for two different observations. We kept the
smaller one as the assumed physical separation of the binary system. In order
to ensure that all stellar components of the remaining binary systems could
harbor stable planetary systems between 0.5 and 10 AU we used the stability
criterion from Holman & Wiegert (1999) that takes into account the stellar
masses, their separation and the eccentricity of the binary orbit (which we
assumed to be zero). Systems that did not fulfill this stability criterion
were removed.

The catalog that was the basis for our simulations consists of 1732 stars in
total (123 wide binary components and 1609 single stars). The distance and
spectral type distributions are shown in Fig. 15. The catalog is available
upon request. We are continuously improving the catalog and plan for an
updated version to include the results from _Gaia_ data release 3.

Figure 15: Properties of target stars considered in our study. Top panel:
Distance distribution of stars from the stellar input catalog in bins of 2.5
pc. Single stars are shown in orange, wide binaries (see text) in blue, and
the sum in green. Bottom panel: Spectral type distribution of stars from the
stellar input catalog. Colors are the same as above.

##  Appendix B Technological readiness

The readiness of key technologies relevant for a space mission such as _LIFE_
are summarized in Defrère et al. (2018b) and Quanz et al. (2021), but given
their importance for the mission success, we discuss three aspects in the
following:

Nulling interferometry: It is important to understand that the _LIFE_
measurement principle has been demonstrated successfully in the lab at ambient
temperatures (Martin et al. 2012). One of the next steps is to build a
corresponding experiment, but fully under cryogenic conditions, for a broad
wavelength coverage and with sensitivity as one of the key drivers (as needed
for a space mission). This is underway in the form of the _Nulling
Interferometry Cryogenic Experiment_ at ETH Zurich (Gheorghe et al. 2020). In
addition, new nulling instruments have been proposed for the _Very Large
Telescope Interferometer_ , including one working in the 3–5 $\mu$m range
(Defrère et al. 2018a). These efforts join previous successful projects
related to N-band ground-based nulling interferometry, with the Keck Nuller
(Colavita et al. 2009) and the LBTI (Hinz et al. 2014). For _LIFE_ it will be
important to leverage the experience from these projects and realize possible
synergies.

Autonomous formation flying: To meet the assumptions made for our simulations,
a high level of autonomous formation flying of all spacecraft will be
required. Specifically, the baselines of the array should be rearranged for
every new target star in order to maximize the transmission of photons from a
specific distance from the star, and, during an observation, the array should
rotate in order to modulate the signal from potential exoplanets. ESA’s
Proba-3 mission (current launch date: 2023) will demonstrate many critical
aspects of formation flying for _LIFE_. Proba-3 will feature two cube-sized
spacecraft with lengths of $\sim$1.5 m and masses of 200-300 kg and will
maintain a virtual “rigid structure” with millimeter and arcsecond relative
precision. In addition, Proba-3 aims specifically at demonstrating manoeuvres
relevant for _LIFE_ , such as station-keeping at distances from 25 m up to 250
m, approaching and separating in formation without losing millimeter
precision, repointing the formation, and the combination of station-keeping,
resizing, and re-targeting manoeuvres. Details can be found in Peñín et al.
(2020). We note that some relevant technology related to formation flying was
developed in both France and Germany in the context of the _Darwin_ mission
and has been flying on the Swedish _Prototype Research Instruments and Space
Mission technology Advancement_ space
experiment171717https://earth.esa.int/web/eoportal/satellite-
missions/p/prisma-prototype.

High quantum-efficiency, low-noise MIR detectors: The low photon rate of
exoplanets (cf. Fig. 2) and the need to integrate for many hours, if not days,
will put strong requirements on the detector technology in terms of quantum
efficiency (in our simulations we assumed 70%; see Sect. 2.3), low read-out
noise and dark current, and high stability. In addition, a wavelength coverage
of at least $\sim$4–18.5 $\mu$m is required based on atmospheric retrieval
analyses quantifying the characterization potential of _LIFE_ for Earth-like
atmospheres (cf. Konrad et al. 2021). In this overall context, the technology
development plan for the 5.9-meter _Origins Space Telescope_
181818https://asd.gsfc.nasa.gov/firs/docs/OriginsVolume2TechDevelopmentPlanREDACTED.pdf,
another mission concept proposed in the context of NASA’s 2020 astrophysics
decadal survey (Meixner et al. 2019), is of great importance as its proposed
MISC instrument (Sakon et al. 2018) would cover the same wavelength range as
_LIFE_. While a detailed noise budget and requirements breakdown for _LIFE_
are still being worked on, it is clear that two types of detector technologies
can be considered (and eventually traded): HgCdTe detectors (as, for instance,
used in NEOCam) and Si:As detectors (as used in _JWST/MIRI_). Currently, the
3-11 $\mu$m range is better covered by HgCdTe than Si:As since the latter is
basically transparent below 10 $\mu$m wavelengths. First efforts to go up to
15 $\mu$m with HgCdTe detectors were already reported (Cabrera et al. 2020)
and new arrays with cutoff wavelength $>$16 $\mu$m have been grown,
hybridized, and packaged and are undergoing testing (cf. _OST_ Technology
Development Plan). A key parameter for HgCdTe detectors will be the achievable
dark current. For Si:As detectors, in addition to potential challenges related
to dark current and 1/f noise, a general problem is the question of
availability. Industrial fabricators that have built these detectors in the
past (e.g., for _Spitzer_ , _WISE_ , and _JWST_) have stopped their production
of low background detectors in the relevant wavelength range and it is unclear
under what conditions and on what timescales restarting the production would
be an option.

##  Appendix C Additional figures for non-reference cases

Figures 16, 17, 18, 19, 20 , and 21 show the expected detection yield for the
non-reference cases. Figure 22 shows the detection efficiency for FGK-dwarf
host stars (i.e., ignoring M dwarfs).

Figure 16: Same as Fig. 3, but now for $D=1.0$ m.

Figure 17: Same as Fig. 3, but now for $D=3.5$ m.

Figure 18: Impact of the aperture size on the exoplanet detection yield. The
numbers are relative to those shown for the reference scenarios with $D=2$ m
in Fig. 4. Left: Scenario 1, with $D=3.5$ m in the top panel and $D=1.0$ m in
the bottom panel. Right: Scenario 2, with $D=3.5$ m in the top panel and
$D=1.0$ m in the bottom panel.

Figure 19: Same as Fig. 3, but now for a wavelength range of 3–20 $\mu$m.

Figure 20: Same as Fig. 3, but now for a wavelength range of 6–17 $\mu$m.

Figure 21: Impact of the wavelength coverage on the exoplanet detection yield.
The numbers are relative to those shown for the reference scenarios with
$\lambda=4-18.5\,\mu m$ in Fig. 4. Left: Scenario 1, with $\lambda=3-20\,\mu
m$ in the top panel and $\lambda=6-17\,\mu m$ in the bottom panel. Right:
Scenario 2, with $\lambda=3-20\,\mu m$ in the top panel and $\lambda=6-17\,\mu
m$ in the bottom panel.

Figure 22: Same as Fig. 7, but ignoring M stars and only considering FGK
stars.

##  Appendix D Distribution of noise contributions for detected planets in the
reference case scenarios

Figure 23 summarizes the contribution of the main noise terms to the overall
noise budget for detected planets orbiting FGK stars or M stars.

Figure 23: Donut charts illustrating how the main noise terms considered in
our simulations (stellar leakage and noise from the thermal emission of the
local and exozodiacal dust disks) contribute to the overall noise budget for
the detected planets. The left column shows the results for planets detected
around FGK stars and the right column for planets detected around M stars. The
top row is for reference case scenario 1 and the bottom row for reference case
scenario 2. The numbers correspond to the mean relative contribution of the
various noise terms to the total noise per detected planet averaged over all
planets. The quoted uncertainties are the corresponding standard deviations
(graphically indicated by the colored arcs inside and outside of the donuts).
For M stars, noise from exozodiacal dust disks is basically negligible, and
stellar leakage and noise from the zodiacal dust disk contribute equally to
the total noise in both scenarios. This trend is generally the same for FGK
stars in scenario 1, even though the relative share of exozodi noise is
larger. For scenario 2, however, stellar leakage clearly dominates the noise
budget of planets around FGK stars.