∎

11institutetext: B. Charnay 22institutetext: LESIA, Observatoire de Paris,
Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules
Janssen, 92195 Meudon, France.
22email<EMAIL_ADDRESS>33institutetext: J. M. Mendonça
44institutetext: National Space Institute, Technical University of Denmark,
Denmark 55institutetext: L. Kreidberg 66institutetext: Max Planck Institute
for Astronomy, Heidelberg, Germany 77institutetext: N. B. Cowan
88institutetext: Department of Earth & Planetary Sciences, Department of
Physics, McGill University, Montréal, Canada 99institutetext: J. Taylor
1010institutetext: Department of Physics (Atmospheric, Oceanic and Planetary
Physics), University of Oxford, UK 1111institutetext: T. J. Bell
1212institutetext: Department of Physics, McGill University, 3600 rue
University, Montréal, QC H3A 2T8, Canada 1313institutetext: O. Demangeon
1414institutetext: Instituto de Astrofísica e Ciências do Espaço, Universidade
do Porto, Porto, Portugal 1515institutetext: B. Edwards 1616institutetext:
Department of Physics and Astronomy, University College London, London, UK
1717institutetext: Carole A. Haswell 1818institutetext: School of Physical
Sciences, The Open University, Walton Hall, Milton Keynes, MK17 8TT, UK
1919institutetext: G. Morello 2020institutetext: AIM, CEA, CNRS, Université
Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, Gif-sur-Yvette,
France 2121institutetext: L. V. Mugnai 2222institutetext: Dipartimento di
Fisica, La Sapienza Università di Roma, P.le A. Moro 2, 00185 Roma, Italy
2323institutetext: E. Pascale 2424institutetext: Dipartimento di Fisica, La
Sapienza Università di Roma, P.le A. Moro 2, 00185 Roma, Italy
2525institutetext: G. Tinetti 2626institutetext: Department of Physics and
Astronomy, University College London, London, UK 2727institutetext: P.
Tremblin 2828institutetext: Université Paris-Saclay, UVSQ, CNRS, CEA, Maison
de la Simulation, 91191, Gif-sur-Yvette, France 2929institutetext: R. Zellem
3030institutetext: Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, USA

# A survey of exoplanet phase curves with Ariel

Benjamin Charnay  João M. Mendonça  Laura Kreidberg  Nicolas B. Cowan  Jake
Taylor  Taylor J. Bell  Olivier Demangeon  Billy Edwards  Carole A. Haswell
Giuseppe Morello  Lorenzo V. Mugnai  Enzo Pascale  Giovanna Tinetti  Pascal
Tremblin  Robert T. Zellem

(Received: date / Accepted: date)

###### Abstract

The ESA-Ariel mission will include a tier dedicated to exoplanet phase curves
corresponding to $\sim$10$\%$ of the science time. We present here the current
observing strategy for studying exoplanet phase curves with Ariel. We define
science questions, requirements and a list of potential targets. We also
estimate the precision of phase curve reconstruction and atmospheric retrieval
using simulated phase curves. Based on this work, we found that full-orbit
phase variations for 35-40 exoplanets could be observed during the 3.5-yr
mission. This statistical sample would provide key constraints on atmospheric
dynamics, composition, thermal structure and clouds of warm exoplanets,
complementary to the scientific yield from spectroscopic transits/eclipses
measurements.

###### Keywords:

Exoplanets Ariel space mission Atmospheres Phase curves

##  1 Introduction

The ESA-Ariel mission tinetti18  due for launch in 2029, will conduct a survey
of $\sim$1000 exoplanets by transit and/or secondary eclipse spectroscopy.
Ariel’s main goals are to investigate:

  1. 1.

the physical processes shaping planetary atmospheres

  2. 2.

the chemical diversity of exoplanets

  3. 3.

the formation and evolution of planetary systems

Ariel is a 1-m telescope with simultaneous coverage from 0.5 to 7.8 $\mu$m.
Data will be collected by 3 instruments: FGS (3 photometric channels: VIS-
Phot, FGS-1 and FGS-2, from 0.5 to 1.1 $\mu$m), NIRSpec (1.1-1.95 $\mu$m and
R$\sim$10) and AIRS (two channels: AIRS-CH0 for 1.95-3.9 $\mu$m with
R=100-200; AIRS-CH1 for 3.9-7.8 $\mu$m with R=30-60). Ariel’s observational
strategy is based on a 4-Tier approach. Tier 1 (Reconnaissance survey) will
consist of a general transit survey at low spectral resolution (photometric
bands) of $\sim$1000 exoplanets to detect primary atmospheres and clouds. Tier
2 (Deep survey) aims at characterizing the chemical composition, clouds and
thermal structure of $\sim$500 planets (transits and eclipses at R$\sim$100
for AIRS-CH0 and R$\sim$30 for AIRS-CH1). Tier 3 (Benchmark planets) will
consist of a detailed analysis of $\sim$50 planets amongst the best targets
(transits and eclipses at R$\sim$200 for AIRS-CH0 and R$\sim$60 for AIRS-CH1).
Finally, Tier 4 will correspond mostly of phase curve observations. Over
Ariel’s 3.5-yr mission, roughly 10$\%$ of the science time will be dedicated
to phase curves.

The principle of phase curve observations is to measure the radiation
reflected or emitted by an exoplanet throughout its orbit around its star.
Planets are inherently three-dimensional objects, and only by observing their
phase variations can we obtain a complete picture of the global atmospheric
chemistry and climate. Ariel has several advantages compared to other space
missions for studying phase curves. Firstly, Ariel’s spectral bands cover most
of the thermal emission of warm/hot planets allowing accurate estimates of
atmospheric heat redistribution and Bond albedo. Secondly, simultaneous
observations with the different Ariel’s channels may enable us to disentangle
reflected light from thermal emission for favourable cases. Finally, phase
curves are very time consuming so the forthcoming Jame Webb Space Telescope
(JWST) will probably only observed one to three dozen exoplanet phase curves
as explained in the next section. Conversely, Ariel provides a unique
opportunity to perform a statistical survey of exoplanet spectroscopic phase
curves.

We present here the plan defined by the Ariel Phase Curve Working Group to
measure thermal phase curves of 35-40 exoplanets. We start in Section 2 with a
brief review of phase curves to highlight their interest, lessons from past
measurements and expectations for the coming decade with JWST. In Section 3,
we outline five science questions to be investigated with Ariel phase curves.
We describe the target list and the observational strategy in Section 4. We
then discuss the effect of systematic error and the expected precision for
atmospheric retrieval based on simulated phase curves from a 3D model (Section
5). We finish with a summary and conclusions in Section 6.

##  2 Brief Review of Phase Curves

Thermal phase curves of a short-period planet constrain its atmospheric
circulation—specifically the transport of energy from the day to night
hemisphere of a synchronously-rotating planet (for a recent review, see
parmentier2018 ). The combination of thermal eclipse depths and phase
amplitudes allows one to estimate a planet’s large-scale energy budget: how
much incident flux is absorbed, and where it moves before being re-radiated to
space. In other words, thermal phase curves enable the construction of simple
Trenberth diagrams for exoplanets read2016 . Phase variations can also be
inverted to produce a longitudinal map of the planet cowan2008 . There are now
excellent publicly-available codes to perform these inversions numerically
louden2018  and analytically luger2019 .

Planets on eccentric orbits present different challenges and opportunities
langton2008 . They exhibit eccentricity seasons and hence do not have a static
temperature map and cannot be mapped via phase variations. On the other hand,
the time-variable incident flux allows us to break the degeneracy between
advective and radiative timescales that plagues circular phase curves
cowan2011a ; kataria15 . This opportunity has been leveraged for most of the
nearby eccentric hot Jupiters laughlin2009 ; lewis2013 ; deWit2016 ; deWit2017
.

###  2.1 Review of Previous Phase Curve Measurements

Phase curves have so far been published for more than two dozen planets, and
observations are in hand, but not yet published, for about two dozen more. The
vast majority of these phase curves were obtained with the Spitzer Space
Telescope werner2004 , primarily as part of the post-cryogenic “warm” mission.
Multi-epoch phase curves and/or phase curves of non-transiting planets are
fraught with astrophysical and detector degeneracy and hence have been less
useful harrington2006 ; cowan2007 ; krick2016 . On the other hand, the out-of-
eclipse baseline can constrain the phase variations of short-period planets
wong2014 .

Due to its Earth-trailing orbit, Spitzer was capable of uniterrupted
monitoring of an entire planetary orbit. Indeed, continuous Spitzer phase
curves of transiting (and eclipsing) planets have been the most useful,
starting with the continuous, half-orbit observations knutson2007 . Full-orbit
continuous phase observations cowan2012  offer a look at all of the longitudes
on the planet, as well as providing two eclipses with which to remove first-
order detector and astrophysical noise (the in-eclipse system flux should be
the same at the start and the end of the observations).

More recently, a few near-infrared phase curves have been measured with the
Hubble Space Telescope (HST) stevenson14 ; kreidberg2018a ; arcangeli2019 .
HST phase curves have two advantages: 1) the NIR coincides with the peak
dayside emission of many hot Jupiters, and 2) HST enables spectroscopic phase
curves. HST has offered us a glimpse of what exoplanet characterization will
be like in the era of JWST, but HST’s low Earth orbit makes it impossible to
obtain continuous measurements throughout a planet’s orbit. Full-orbit phase
curves are therefore stitched together, which requires a model of the
telescope and detector systematics, which can lead to increased astrophysical
uncertainty stevenson14 ; keating2017 ; louden2018 ; mendoca2018 ; keating2019
; morello2019 .

Lastly, optical phase curves have been observed for a handful of planets with
Kepler and TESS. For the hottest planets, these phase curves still primarily
probe thermal emission from the planet, while for cooler planets they probe
reflected starlight parmentier2016 . In cases where the temperature of the
planet can be established with thermal measurements, it has been possible to
measure not only the geometric albedo of an exoplanet but also to map the
albedo across the dayside of the planet with an optical phase curve demory2013
. A planet’s reflected light can also be disentangled from its thermal
emission with polarimetric phase measurements, but these measurements have so
far proven exceedingly difficult and rare berdyugina2011 ; wiktorowicz2015 ;
bott2016 .

###  2.2 Trends from Past Measurements

The past decade of exoplanet phase curve measurements has revealed a few
intriguing trends and some notable exceptions. First of all, it has been
observed that the day-to-night temperature contrast increases with equilibrium
temperature cowan2011b ; schwartz2015 , likely due to the shorter radiative
relaxation timescale of hotter atmospheres showman2002 . However, the strong
temperature dependence of the radiative timescale has made it difficult to
tease out the variations in wind speed perez-becker2013 ; komacek2016 ;
tan2019 .

Phase curve measurements have also uncovered trends in albedo: the Bond
albedos inferred from Spitzer thermal phase measurements are significantly
greater than the geometric albedos inferred from Kepler optical eclipse
measurements schwartz2015 , a dichotomy known as the “Albedo Problem”
crossfield2015 . One hypothesis is that most hot Jupiters are dark at optical
wavelengths but reflective in the near-infrared schwartz2015 ; this can be
tested with NIR spectroscopic or —ideally— polarimetric observations. More
recently, it was shown that the Bond albedo inferred from thermal phase curves
is negatively correlated with planetary mass zhang2018 . If corroborated, this
trend could be due to metallicity or surface gravity, but the details remain
to be worked out.

The nightside flux of a synchronously rotating exoplanet is a direct measure
of heat transport: any power radiated from the nightside must have been
conveyed there from the dayside. Two recent studies have noted that the
nightside effective temperatures of hot Jupiters are remarkably uniform at
$\sim$1100 K, independent of their equilibrium temperature, $T_{\rm eq}$, or
dayside effective temperature, $T_{\rm day}$ keating2019 ; beatty2019 ,
possibly due to optically thick nightside clouds. The ultra-hot Jupiters, on
the other hand, exhibit an increase in nightside temperature (and larger phase
offsets zhang2018 ), likely due to hydrogen dissociation and recombination
bell2018 ; tan2019 ; mansfield2020 .

The broad trends observed among exoplanets have occasionally been bucked by
individual planets. Most thermal phase curves peak before the eclipse,
indicative of eastward winds, as predicted by global circulation models
showman2002 . The hotspot offset of HAT-P-7b as seen in the optical with
Kepler, however, seems to change direction over time armstrong2016 , while the
Spitzer phase curve of CoRoT-2b exhibits a significant westward offset
dang2018 . Likewise, while most hot Jupiters appear to have optical geometric
albedos of a few percent heng2013 , Kepler-7b is about an order of magnitude
more reflective demory2013 . Lastly, while nightside temperatures generally
seem to be a weak function of equilibrium temperature, the hot Jupiter
WASP-43b has been reported to have a very low nightside temperature
stevenson14 , at odds with other hot Jupiters with similar equilibrium
temperatures. If taken at face value, the above oddballs suggest that
planetary age (CoRoT-2b), gravity (Kepler-7b), and orbital period (WASP-43b)
significantly affect the large-scale absorption and transport of energy on hot
Jupiters. Indeed, comparing phase curves of exoplanets with similar bulk
properties is a promising approach to establish the fundamental parameters
governing planetary climates Keating2020 .

###  2.3 What to expect from JWST

We know of four phase curve observations planned for Early Release Science or
Guaranteed Time Observations: WASP-43b (MIRI/LRS ERS and NIRSpec/BOTS
GTO)111Backup ERS targets are WASP-103b and KELT-16b, WASP-121b and TOI-193.01
(NIRISS/SOSS GTO). It is safe to say that JWST spectroscopic phase curves will
be even more informative than the handful of HST phase curves due to the much
greater collecting area and spectral coverage.

It remains to be seen how many additional phase curve measurements will be
observed as part of JWST General Observer (GO) programs. One can obtain a
pessimistic estimate from the Hubble Space Telescope, which, like JWST, enjoys
enormous proposal pressure from a wide variety of astronomers. The first HST
phase curve was observed in 2013 and published the following year stevenson14
. Since that landmark paper, Hubble phase curves have been obtained and
published at a rate of about one per year. The allocation of JWST guaranteed
time observations, on the other hand, bodes well for phase curves: three
planets will benefit from full orbit phase measurements as part of GTO. So one
could optimistically expect that JWST will observe complete phase curves for
approximately three planets per year for the duration of the mission. Given
its expected duration of 5–10 years, we therefore estimate that between 7 and
30 full orbit phase curves will be acquired by JWST over the course of the
mission. We may reasonably expect the lion’s share of those to be for planets
with orbital periods shorter than one day, the cutoff between small and medium
proposals. Regarding planets with longer orbital periods, it would make sense
to point JWST at temperate and terrestrial planets since it is the only space
telescope able to meaningfully characterize their atmospheres and climate
cowan2015 .

##  3 Science questions for Ariel phase curves

We have defined five science questions to investigate with Ariel phase curves.
They are focused on the coupling between atmospheric dynamics, chemical
composition, thermal structure and clouds. Our objective is to answer these
questions using a statistical sample of phase curves, covering a large
parameter space.

SQ1: Which parameters control atmospheric heat redistribution?
Phase curve measurements will allow us to determine atmospheric heat
redistribution as a function of stellar irradiation, planetary radius,
metallicity and eccentricity. Measuring dayside/nightside emission and phase
offsets from the sub-stellar point of the phase curve peak for a large range
of planetary parameters would provide information about the circulation
regime, the atmospheric radiative timescales, wind speeds and the effect of
nightside clouds.

SQ2: How do atmospheric composition $\&$ thermal structure change from dayside
to nightside?
We expect significant variations in the chemical composition as a function of
longitude for strongly irradiated planets due to thermochemistry and molecular
dissociation. Atmospheric circulation should smooth out these variations,
producing chemical disequilibrium. In addition, longitudinal/vertical
temperature variations are expected, with stratospheric thermal inversion on
the dayside of some hot Jupiters due to absorption of incoming stellar flux at
low pressure.

SQ3: What is the atmospheric composition of low-mass planets?
A major question for Ariel is to understand the atmospheric composition or the
atmospheric metallicity of low-mass planets. 3D climate models predict that
the day-to-night heat redistribution decreases with atmospheric metallicity
due to a reduction of the atmospheric radiative timescale menou12 ; charnay15b
. Consequently, low-mass planets with high atmospheric metallicity should have
high-amplitude phase curves (see Fig. 1). Phase curves can thus be used to
measure independently of transit spectroscopy atmospheric metallicity.
Finally, they can reveal the presence of an atmosphere on a rocky planet
through the inferred heat redistribution (e.g. kreidberg19 ).

Figure 1: Simulated phase curve of GJ 1214b for one orbit with AIRS-CH1
(3.9-7.8 micron) and for different atmospheric compositions. The red line is
the GCM flux variations charnay15b , blue points are simulated Ariel
observations and the light blue curve is a sinusoidal fit. The corresponding
SNR for the amplitude is: 0.38, 1.63, 4.9 and 7.5 for the solar metallicity,
10$\times$solar metallicity, 100$\times$solar metallicity and pure water case
respectively. Without heat redistribution, the SNR would reach 11.1. The
thermal phase curve is easily detected for high-metallicity cases
($\geq$100$\times$solar).

SQ4: What is the albedo of exoplanets?
Most exoplanets are at least partially cloudy. A major questions is to
understand the nature of these aerosols. Are they condensate clouds or
photochemical hazes? The albedo of exoplanets is related to the optical
properties and cover of clouds/hazes. Transitions in the values of the albedo
are expected due to the disappearance of some clouds with temperature, as for
the L-T transition for brown dwarfs parmentier2016 . Measuring the albedo
would help to determine the nature of these aerosols. The wavelength-
integrated albedo is also a fundamental parameter for the thermal balance of
planets.

SQ5: How do thermal structure and aerosols vary in time?
Atmospheric variability could be caused by shear instability and shocks,
variation in cloud cover, or magnetic interactions fromang16 ; parmentier2016
; rogers17 . Given that brown dwarfs with the same size, composition, and
temperature as hot Jupiters have long been known to exhibit weather, it will
come as no surprise if hot Jupiters are variable, too. It would be useful to
measure the magnitude and timescale of such variability. 3D cloud-free
simulations suggest variations of the order of just 1$\%$ for the phase curve
amplitude over timescales of a few days to a few weeks komacek20 . The
presence of clouds may lead to a higher photometric variability (e.g. a few
percent for the phase curve amplitude), as suggested by some 3D simulations
charnay20  and brown dwarfs observations biller17 .

Science Questions | Observations | Required precision
---|---|---
SQ1 Atmospheric heat transport | photometric | 10$\%$ for amplitude and 5∘ for offset
SQ2 Variations of composition | spectroscopic | 0.5 dex for molecular abundances
SQ3 Low-mass planets | photometric | 0.5 dex for the metallicity
SQ4 Albedo | photometric | 0.1 for geometric and Bond albedo
SQ5 Time variability | photometric | 2$\%$ for amplitude and 3∘ for offset
Table 1: Science questions and requirements for Ariel phase curves. Figure 2:
Relative phase curve amplitude in AIRS-CH1 (3.9-7.8 micron) as a function of
metallicity and effective temperature from a grid of 2D ATMO models moses20 .
Simulations performed with the radius, rotation and stellar type of GJ436b.

General requirements for the science questions (SQs):
Table 1 shows the type of observations (photometric or spectroscopic phase
curves) and the required precision for each science question. For most of
them, the required precision is equivalent to 10$\%$ on the maximal amplitude
(i.e. the case with no heat redistribution) of the phase curve. According to
Fig. 2, a precision of 0.5 dex on the metallicity corresponds to 10$\%$ of
precision on the maximal amplitude. For SQ5, we chose a required precision of
2$\%$ for the amplitude (SNR$>$50). In conclusion, the general requirement to
address our science questions is to reach a signal-to-noise ratio (SNR) higher
than 10 (i.e. detection at 10 sigma) for the fitting of photometric phase
curves, assuming no heat redistribution and no offset. For spectroscopic phase
curves, we consider that the requirement is to reach SNR$>$10 for the emission
spectra at Tier 2 resolution (R$\sim$50) in each of 10 orbital phase bins.

##  4 Target list and observing strategy

###  4.1 Target list

We constructed a list of potential targets for phase curves from the whole
Ariel target list edwards19 . We divided planets into four categories:

  * •

rocky planets with planetary radii lower than 1.8 R⊕

  * •

sub-Neptunes with planetary radii between 1.8 and 3.5 R⊕

  * •

Neptunes with planetary radii between 3.5 and 7 R⊕

  * •

Giants with planetary radii larger than 7 R⊕

We limited the sample to planets with orbital periods of less than 5 days. We
classified them by the SNR of the maximal phase curve amplitude (no heat
redistribution). The SNR is computed accumulating a whole number of orbits up
to 10 days at Tier 1 resolution for rocky/sub-Neptunes, observing one orbit at
Tier 1 resolution for Neptunes and giant planets. For the latter, we also
computed the SNR for observations with a total integration time of 10$\%$ of
the orbital period at Tier 2 resolution for spectroscopic phase curves. The
SNR is evaluated assuming that the phase curve is a pure sine with no offset
and decomposed into observation bins of 1 hour. We first compute the SNR for a
1-hour observation at full phase using:

$SNR_{\rm 1h}=\frac{F_{\rm planck}(T_{\rm p})}{F_{\rm planck}(T_{\rm
s})}\left(\frac{R_{\rm p}}{R_{\rm s}}\right)^{2}\times\frac{1}{\sigma_{\rm
1h}}$ (1)

where $T_{p}$ is the planet equilibrium temperature with no heat
redistribution assuming a Bond albedo of $A_{B}$=0.3 ($T_{p}<$700K) or
$A_{B}$=0.1 ($T_{p}>$700K), $T_{s}$ is the star temperature, $R_{p}$ and
$R_{s}$ are the planet and star radii. These parameters are given in the Ariel
target list edwards19 . $\sigma_{1h}$ is the noise for a 1-hour observation
and obtained with the Ariel Radiometric Model (Ariel-RAD) mugnai20 . We used
the maximal value of SNR between the different Ariel’s spectral bands (Tier 1
resolution).

By computing the error on the estimation of the amplitude of the phase curve
assumed to be a pure sine, the SNR for a full orbit (with $P$ the orbital
period in hours and $P\gg$1h) can be expressed as (see Appendix):

$SNR_{\rm orbit}=0.5\times SNR_{\rm 1h}\times\sqrt{P/2}$ (2)

We validated this metric with comparison to Spitzer observations of LHS3844b
(see Appendix). We limit our sample to planets reaching SNR$>$10, which
corresponds to a 10-sigma detection of the phase curve, for the conditions
given above for each planet category. We also limit the number of giant
planets to 20. With all these conditions, we get 44 planets (see the bottom
panel in Fig. 3) including:

  * •

1 rocky planet

  * •

8 sub-Neptunes

  * •

15 Neptunes

  * •

20 giants

All planets in this sample reach SNR$>$10 for photometric (Tier 1 resolution)
thermal phase curves, which is the requirement for SQ1 (heat redistribution),
3 (the composition of low-mass planets and exo-Neptunes) and 4 (albedo). All
giant planets fulfill the conditions for spectroscopic phase curves (i.e.
SNR$>$10 in 10$\%$ of the orbital period at Tier 2 resolution, SQ2) and for
variability (SQ5). Six Neptunes reach SNR$>$5 for spectroscopic phase curves.

This target list covers a large parameter space with planetary mass from 0.01
to 4 MJup, equilibrium temperature from 600 to 2200 K, and expected
atmospheric metallicity from 1 to 300$\times$solar. 15 planets from our sample
are already known, including 1 rocky planet, 2 sub-Neptunes, 2 Neptunes, 10
giant planets (see Table 2). We found that a total of 136 giant planets from
the Ariel target list reach the conditions for spectroscopic phase curves (see
the top panel Fig. 3). They include 83 already known giant planets, most of
them coming from the WASP, HAT-P and KELT transit surveys. This implies a lot
of choice for the selection of giant planets. Our remaining targets targets
are simulated discoveries from the Transiting Exoplanet Survey Satellite
(TESS). TESS is surveying nearby stars for transiting planets with near-
complete sky-coverage ricker15 . TESS has delivered over 1000 planet
candidates in its first year (Guerrero et al., submitted). Some of these may
be exceptional targets for phase curve observations, including several new hot
Neptunes with short orbital periods and high equilibrium temperatures (i.e.
LTT 9779b, HD 219666b, TOI 132b, and TOI 824b). TESS is currently on track to
detect over 2000 planets around bright nearby stars, providing the large
statistical sample needed for Ariel.

Figure 3: Targets for phase curves as a function of planetary radius, planet period and temperature (color bar). Vertical dashed lines represent the radius limits of the 4 categories: rocky planets, sub-Neptunes, Neptunes and giants planets. The top panel shows all targets reaching the conditions for phase curves. The bottom panel is limited to the 20 best giant planets.  Planet | Period | Orbits | SNR | SNR | Thermal/Reflected
---|---|---|---|---|---
| (days) |  | (thermal emission) | (reflected light) | (FGS)
GJ 1214 b | 1.58 | 1 | 11.0 | 0.3 | 0.0
K2-266b | 0.66 | 4 | 11.1 | 1.5 | 0.0
55 Cnc e | 0.74 | 4 | 10.3 | 0.3 | 0.7
GJ 436b | 2.64 | 1 | 15.2 | 0.9 | 0.0
GJ 3470b | 3.34 | 1 | 10.4 | 0.6 | 0.0
HD 189733b | 2.22 | 1 or 3 | 205.2 | 3.0 | 0.04
HD 209458b | 3.52 | 1 or 3 | 168.1 | 2.3 | 0.2
XO-6b | 3.77 | 1 | 107.6 | 3.5 | 0.9
WASP-77Ab | 1.36 | 1 | 106.1 | 3.5 | 0.3
KELT-7b | 2.73 | 1 | 105.3 | 2.8 | 1.0
WASP-74b | 2.14 | 1 | 97.6 | 3.4 | 0.8
XO-3b | 3.19 | 1 | 95.1 | 3.5 | 1.0
WASP-82b | 2.71 | 1 | 78.8 | 3.0 | 1.3
WASP-14b | 2.24 | 1 | 75.1 | 2.3 | 0.7
KELT-14b | 1.71 | 1 | 71.1 | 3.9 | 0.8
Table 2: Known planets from the phase curve target list with the number of
orbits required to reach SNR$>$10 (3 orbits are indicated for multi-epoch
phase curves) and the SNR at Tier 1 resolution for thermal and reflected light
phase curves. For thermal emission, we give the highest value of SNR between
the different photometric channels, assuming a Bond albedo of $A_{B}$=0.3
($T_{p}<$700K) or $A_{B}$=0.1 ($T_{p}>$700K). For reflected light curves, we
give the SNR integrating the flux from 0.5 to 1.1 $\mu$m (the 3 FGS channels),
assuming a geometric albedo equal to the Bond albedo. The last column is the
ratio of the flux from thermal emission to the flux from reflected light
integrated from 0.5 to 1.1 $\mu$m. Horizontal lines separate sub-Neptunes,
Neptunes and giant planets. Note that with our limit at 1.8 REarth, 55 Cnc e
is listed as a sub-Neptune but it is likely a rocky planet.

###  4.2 Observational Strategy

The plan is to perform continuous phase curve observations, even when multiple
orbits are required. The observations will start and end with the secondary
eclipse, which represents the reference for the phase curve. We will keep a
margin of 7$\%$ of the period before and after so as not to miss the eclipse.
This corresponds to a maximal error of 0.1 on the eccentricity, $e$ (the
uncertainty on the eclipse time is $2eP/\pi$). The study of variability (SQ5)
with multi-epoch phase curves will be limited to one or two planets (e.g. HD
189733b) with 3 phase curves (i.e. 3 orbits). The observations of all planets
from our target list, including 2 planets with multi-epoch phase curves; would
take $\sim$175 days of observing time, or $\sim$13$\%$ of Ariel science time.
Removing 1 sub-Neptune, 5 giants and keeping only one planet with multi-epoch
phase curves, the observing time goes down to 10$\%$ of Ariel science time,
compatible with the current mission time dedicated to Tier 4. We can,
therefore, expect that the final target list for Ariel phase curves will
include 35-40 exoplanets and will represent a statistical sample for gas
planets. It is worth mentioning that the observation of phase curves also
provide transits and eclipses. In the current observing plan for Ariel, a
single transit or eclipse takes 2.5 times the transit duration. For our phase
curve observing plan, the equivalent of 28$\%$ of the observation time is
dedicated to transits or eclipses. This high fraction significantly reduces
the cost of phase curves. It comes from the fact that we are targeting short
period planets (P$<$ 5 days) with a mean period of $\sim$2 days.

###  4.3 Phase curves of reflected light and synergy with TESS, CHEOPS and
PLATO

A difficulty for Ariel phase curves will be to distinguish reflected light
from thermal emission. The last column in Table 2 shows the ratio of the
thermal flux by the reflected light for FGS (0.5-1.1 $\mu$m). Reflected light
dominates only for HD189733b and HD209458b as well as warm Neptunes and sub-
Neptunes. Figure 4 shows the fraction of the total flux (thermal
emission+reflected light) covered by the different ARIEL’s instruments (FGS,
NIRSpec and AIRS). Reflected light dominates for FGS only for effective
temperatures lower than 1700K for a G star and 1500K for an M star. Therefore,
the thermal emission will represent a non-negligible fraction of the flux
measured with FGS for most of hot Jupiters, even if we limit the measurements
to VIS-Phot (0.5-0.6 $\mu$m). However, the contribution of reflected light is
almost negligible in NIRSpec and AIRS channels for these planets. It would be
possible to extrapolate the thermal flux at short wavelength (i.g. in FGS
channels) to extract the reflected light. This method has been applied to
simulated Ariel eclipse observations zellem19 . According to this study, the
geometric albedo could be measured with a precision of $\pm$0.02 for $\sim$10
giant planets from Tier 3. In addition, inhomogeneous cloud and temperature
distributions lead to differences in the shape of phase curves (i.e. the
offset) between thermal emission and reflected light. This could also help to
distinguish thermal emission from reflected light. Additional modeling work
will be needed on this aspect.

Only giant planets from the phase curve target list are expected to reach a
SNR sufficient to detect reflected light and to measure the geometric albedo
(see Table2). For these planets and assuming that the reflected light can be
distinguished from the thermal emission (as explained above), the geometric
albedo would be measured with a precision of around $\pm$0.03 with one phase
curve. This is comparable to the precision for albedo measurements from
eclipses of Tier 3 planets zellem19 . Note that previous observations suggest
a geometric albedo around 0.1 for most of warm/hot giant exoplanets heng2013 ;
angerhausen2015 . For Neptunes and sub-Neptunes, a mean geometric albedo could
be derived from the cumulative signal of several planets. For comparison,
phase curves and eclipses from the Kepler telescope provided geometric albedo
measurements with also a precision of around $\pm$0.01-0.03 heng2013 ;
demory2013 ; angerhausen2015 ; parmentier2016 . The Kepler telescope had a
collecting area similar as Ariel and cumulated more than 1200 days of phase
curve observation for each target in the field of view of the primary mission.
Ariel phase curves will generally be limited to one planetary orbit, giving a
SNR typically 20-30 times lower than Kepler for the same target. However,
Ariel will target much brigther stars than in the Kepler survey. The precision
for the geometric albedo and the amplitude of phase curves could then be
relatively close to that of Kepler.

CHEOPS has reached a photometric precision of 10-17 ppm in 1 hour for
WASP-189, a magnitude 6.6 star lendl2020 . This would correspond to
$\sim$30-50 ppm for a magnitude 9 star. CHEOPS, as a mono-target instrument
like Ariel, will not be able to stare at numerous targets for an extended
period of time, but will still be able to make significant contributions for a
handful of golden targets. TESS stares at the same targets for at least 27
days and at best 1 years (without considering possible extensions of the
mission) and can thus achieve exquisite phase curves over time. TESS is
currently reaching a precision of around 85 ppm in similar conditions. In
contrast, Ariel will have a precision of around 50 ppm with VISPhot and FGS
for stars brighter than magnitude 11 in V. Taking into account the observation
durations, we can expect relatively similar precisions for phase curves of
reflected light between CHEOPS, TESS and Ariel. PLATO will have a photometric
precision of $\sim$30 ppm in 1 hour for bright solar-like stars (mag(V)$<$11).
In addition, the PLATO fields (around 11$\%$ of the whole sky for 2 PLATO
fields of 49∘, which is the current plan) will be observed for 1-3 years,
cumulating a few hundred orbits for short-period planets around bright stars.
PLATO phase curves of reflected light should achieve a much higher precision
than the other space telescopes mentioned above. Since there will be many
giant planets (136 with the current list) reaching the requirements for Ariel
phase curves, we plan to choose those which will benefit from a precise
visible phase curve observed by TESS, CHEOPS or PLATO. This would be a strong
synergy, combining visible and infrared phase curves from these telescopes,
together with accurate stellar characterization.

Figure 4: Fraction of the total flux (thermal emission+reflected light)
covered by the different ARIEL’s instruments (FGS, NIRSpec and AIRS). Solid
lines show the thermal emission and dashed lines show the reflected flux as a
function of the planetary effective temperature. The calculation is done
assuming a Bond albedo and a geometric albedo of 0.1, for a Sun-like star
(left panel) and for an M star as GJ1214b (right panel).

##  5 Simulations

We used theoretical models to evaluate Ariel’s potential performance for phase
curve observations of a hot Jupiter planet. We test our theoretical tools on
the planet WASP-43b and assume that the atmosphere is either cloud-free or
covered by clouds on the night side. We simulated Ariel phase curves based on
3D atmospheric model results and used simulated spectra to explore our
retrieval framework on Ariel observations.

###  5.1 Phase curves calculated from a global circulation model

Global Circulation Models (GCMs) are powerful tools that self-consistently
simulate 3D atmospheric temperature distributions and here can predict thermal
phase curves. Using a 3D GCM, THOR, we have simulated the phase curves of
WASP-43b hellier11  with a clear atmosphere. WASP-43b has twice the mass of
Jupiter, orbits its parent star in 19.2 hours with a semi-major axis of 0.0153
AU gillon12  and an estimated equilibrium temperature of 1440 K blecic14 . The
parent star is a K7 star of 0.73 MSun.

THOR is a flexible GCM that can calculate phase curves and is based on a
dynamical core that solves the non-hydrostatic compressible Euler equations on
an icosahedral grid mendonca16 . To represent the radiative processes in the
GCM simulations, we used a two-band formulation calibrated to reproduce the
results from more complex codes on WASP-43b mendonca18a .

The multi-wavelength phase curves are obtained from post-processing the 3D
simulated atmosphere with a more sophisticated radiative transfer model
mendonca18a . The spectra include cross-sections of the main absorbers in the
infrared - EXOMOL: H2O barber06 , CH4 yurchenko14 , NH3 yurchenko11 , HCN
harris06 , H2S azzam16 ; HITEMP rothman10 : CO2, CO; HITRAN rothman13 : C2H2.
The Na and K resonance lines are also added draine11  and H2-H2 H2-He CIA
richard12 . We assumed that the atmosphere has solar abundances. As shown in
kataria15 , the HST-WFC3 spectrum stevenson14  of the dayside in WASP-43b is
compatible with a clear atmosphere with solar abundances. Our THOR simulation
includes a chemical relaxation method fully coupled with the 3D dynamics for 4
different chemical species mendonca18b : H2O, CH4, CO and CO2. This
implementation allows us to study chemical disequilibrium in the atmosphere
due to atmospheric transport, which we use later to explore the potential of
our retrieval framework to find departures from chemical equilibrium in future
Ariel observations of WASP-43b. Other molecules in the atmosphere are assumed
to be in chemical equilibrium, and their concentrations are calculated with
the FastChem model stock18 . The stellar spectrum was obtained from the
PHOENIX model (allard95 , husser13 ).

The disc averaged planet spectrum is calculated at each orbital phase by
projecting the outgoing intensity for each geographical location of the
observed hemisphere. In Fig. 5, we show the theoretical phase curves simulated
with THOR. The different lines show the planet spectra for each orbital phase:
the red lines represent the hotter dayside of the atmosphere and blue the
colder nightside. The left panel in Fig. 5 corresponds to the WASP-43b
simulation with a clear atmosphere. We also show the impact of a grey cloud
deck on the night side of the planet in the right panel of Fig. 5. The cloud
distribution and optical properties are parameterized (see mendonca18a  for
further details). Clouds block most of the radiative flux from the deep hot
layers in the planet, which reduces the observed infrared flux for orbital
phases near the primary transit mendonca18a  and reduces the spectral
features. Therefore, clouds can limit atmospheric retrieval for spectroscopic
phase curves. In contrast, clouds generally make the detection of photometric
phase curves easier. As shown for WASP-43b (see Fig. 4), nightside clouds tend
to increase the amplitude of thermal phase curves (see parmentier2021 . In
addition, dayside clouds can increase the amplitude of phase curves of
reflected light and also of thermal phase curves for absorbing clouds. In this
last case, the effect of clouds may remain limited compared to that of the
atmospheric metallicity (see Fig. 1 and charnay15b ).

Figure 5: Emission plus reflected light at different orbital phases obtained
with THOR. The circular color map shows the color for different orbital
phases. The primary transit occurs at orbital phase 0∘ (blue) and the
secondary eclipse at 180∘ (red). The two panels show the phase curve values
for a simulation with clear sky (left panel) and a cloudy night side (right
panel).

###  5.2 Effect of systematics and retrieval of photometric phase curves

Phase curve observations are a technically challenging measurement, requiring
stable instrument performance spanning hours to days. The largest amplitude
phase curves observed to date have peak-to-trough variation of order 0.1$\%$,
but instrument systematic noise can be orders of magnitude larger than the
expected signal (e.g. knutson07 ).

We carried out several simulations to evaluate Ariel’s potential performance
for phase curve observations. Ariel is designed for greater stability than
either Spitzer or HST which have been the workhorse facilities for phase curve
observations so far. One caveat is that Ariel is required to meet this
stability requirement for timescales of just 10 hours, whereas nearly all
phase curves have longer duration than that (see tinetti18  and Ariel Yellow
Book). A work-around is to observe the phase curve in multiple overlapping
pieces. To assess the feasibility of this strategy, we simulated several test
light curves with varying assumptions about instrument performance.

We evaluated three potential cases:

  1. 1.

An optimistic case, with a continuous stare over the full phase curve with no
instrument systematic noise.

  2. 2.

A realistic case, with the phase curve split into three visits. Each visit is
separated by a constant offset drawn from a normal distribution with mean 0
and standard deviation 10-3. We considered two sub-cases: one with an overlap
between visits of 10$\%$ in orbital phase, and the other with a 25$\%$
overlap.

  3. 3.

A pessimistic case, same as case (2) above except with an additional linear
trend for each visit, also drawn from a normal distribution with mean 0 and
standard deviation 10-3.

The amplitude of the systematic noise was inspired by long-duration Spitzer
and Hubble observations, which show long-term systematic trends and offsets
that are still poorly explained (e.g. kreidberg18 ). Spitzer and Hubble’s data
also show additional noise from intrapixel sensitivity variations and charge
trapping (long15 , ingalls16 , zhou17 ), but Ariel observations are less
likely to be affected by these noise sources thanks to the stability
requirements and uninterrupted pointing.

For each case (optimistic, realistic, pessimistic), the simulated phase curves
for WASP-43b with clouds based on the THOR results are shown in Fig. 6. We
generated 500 instances of each phase curve with randomly generated photon
shot noise calculated with Ariel-Radmugnai20  and the assumed instrument
systematic noise model. We calculated the broadband phase curve from AIRS
Channel 0 (1.95 - 3.9 microns). The time resolution of the simulated light
curves was 0.01 times the orbital period of the planet.

We fit a single sinusoid as our nominal phase variation model. We also
performed a fit with a double sinusoid model that also included the first
harmonic; however, the inferred amplitude of the harmonic was consistent with
zero, so we focus here on results from the single sinusoid case.

We show the inferred sinusoid amplitudes and offsets for each case in Table 3.
For case (1), the optimistic phase curve with continuous phase coverage and no
instrument systematics, we detected the phase curve amplitude at high
confidence (52 sigma for WASP-43b) and measured the phase offset to an
accuracy of 2-3 degrees. This precision is more than sufficient to achieve the
science goals for phase curve observations. For example, the measured phase
curve will constrain the atmospheric metallicity to better than 0.5 dex and
enable a search for variability due to weather.

WASP-43b | Amplitude (ppm) | Hotspot offset (degrees)
---|---|---
1\. Optimistic | 1560 +/- 30 | 4.2 +/- 1.0
2a. Realistic (10$\%$ overlap) | 1560 +/- 30 | 4.3 +/- 2.1
2b. Realistic (25$\%$ overlap) | 1560 +/- 30 | 3.9 +/- 1.8
3a. Pessimistic (10$\%$ overlap) | 990 +/- 250 | 5.9 +/- 3.5
3b. Pessimistic (25$\%$ overlap) | 1520 +/- 150 | 4.6 +/- 3.2
Table 3: Accuracy of retrieved phase curve amplitude and hotspot offset for
different scenarios of systematic noise. Figure 6: Simulated phase curves for
WASP-43b for different assumptions about instrument performance. The
optimistic case is a continuous stare with no instrument systematics, the
realistic case is a phase curve split into 10-hour stares with 10$\%$ overlap
in phase coverage with randomly drawn offsets between each visit, and the
pessimistic case adds a randomly drawn linear trend to each 10-hour visit.

For case (2), the light curve with three visits with constant offsets between
them, we find that we can recover the phase curve amplitude almost as well as
for case (1). The estimated sinusoid amplitudes are consistent, and the
precision on the hotspot offset increases by a factor of two.

The more pessimistic case (3) gives consistent results with the other two
cases; however, the precision is noticeably worse. The phase curve amplitude
is detected at 10 sigma confidence, but only if the visits overlap by 25$\%$
in phase coverage. We conclude that depending on the phase curve amplitude and
overlap in phase coverage, the pessimistic noise scenario may not allow us to
achieve our science goals.

The ideal scenario is that Ariel observes phase curves continuously. If this
is not possible, multi-epoch phase curves are a viable alternative. The
optimal strategy depends on the instrument noise performance. If the detector
is stable and has minimal trends in time (change in flux less than 1e-3 over
24 hours), the phase curve can be split into visits with just 10$\%$ overlap
in phase coverage with no negative impact on science goals. By contrast, if
linear systematic trends are present, at least 25$\%$ overlap in phase
coverage is required to measure the phase curve amplitude and offset at
sufficient precision, and it may only be possible for larger amplitude phase
curves. The science will still be possible for the most pessimistic case, but
the number of observable phase curves will decrease due to the large overlap
in phase coverage that is required.

Ariel’s instrument systematics should be carefully assessed early on in the
mission to determine the optimal strategy for phase curve observations.

###  5.3 Retrieval of spectroscopic phase curves

To better understand the capability of Ariel to reveal the global climate
behaviour of planets, we performed 1D spectral retrievals on the disc average
flux of WASP-43b produced by THOR. We chose to perform retrievals with a phase
angle step of 60∘. The error on the observations was estimated using Ariel-Rad
for an integration time of 1 hour.

We used the radiative transfer and retrieval framework NEMESIS irwin08 ;
Taylor2020  to perform 1D spectral retrievals. We use the correlated-k
approach to model our spectra lacis91 ; Chubb2020 , which has been shown to be
effective and accurate when compared to using the line-by-line or cross-
section approaches garland19 . The line lists used in the retrieval are those
used to calculate THOR’s phase curves, however, NEMESIS uses a more recently
developed line list for H2O Polyansky18 .

We model the atmosphere with a uniformly mixed vertical chemical structure and
a temperature profile following the description of guillot10  and line13 .
Fig. 7 shows the retrieved temperature profiles for the cloud-free WASP-43b
simulation. The temperature retrieval shows that we can generally retrieve the
gradient of the profile within one sigma and the vertical shape of the profile
within two sigma, apart from at a phase angle of 300∘ which is slightly
outside of 2 sigma at the lowest pressures.

Figure 7: Retrieved temperature structures for the cloud free WASP-43b
numerical experiment. We present the median, 1-sigma and 2-sigma intervals in
decreasing opacity of purple, and the temperature structure from THOR is
presented in black.

In this retrieval exercise, we also test if we can distinguish between
atmospheres in equilibrium and disequilibrium. The disequilibrium profiles are
obtained from THOR simulations with solar metallicity and C/O. As seen in Fig.
8, we find that our retrieved chemical abundances are within one sigma of the
input value, with the tightest constraints being on the water abundance. The
water abundance in our simulation is 3.6$\times 10^{-4}$, compatible with HST
constraints of $0.32-16\times 10^{-4}$ kreidberg14b . We show that for H2O and
CO it is not possible to distinguish between the disequilibrium and
equilibrium case, but it is possible for CO2 and CH4, with the greatest
difference being for CH4. It may however only be possible to put an upper
limit on CH4 and CO2 due to their low abundance which results in less of an
impact on the observed spectrum. H2O abundance is retrieved with a precision
of 3 dex for 1 hour observations. For 3 hour observations (6 different phases
for WASP-43b), H2O abundance is retrieved with a precision of 3 dex. The
global mean H2O abundance is retrieved with a precision of around 60$\%$ over
the full orbit. This precision is sufficient to address the SQ2 for
spectroscopic phase curves. We see from the chemistry retrieval that the error
bars for CO are large for a phase angle of 180 degrees, this is due to the
isothermal nature of the temperature-pressure profile in the region of the
atmosphere to which we are sensitive to (i.e., around 0.1 bar), hence the
features would appear muted and have a greater uncertainty.

Figure 8: Retrieved abundances (VMR) and associated uncertainty for the cloud-
free WASP-43b simulations. They are shown by the vertical lines for each
molecule. The horizontal solid and dashed lines are the disequilibrium and
equilibrium abundances respectively, the values are taken to be around
$\sim$0.1 to 0.01 bar, the region in which our simulations are sensitive to.
Phase 0∘ and phase 180∘ correspond to the transit and the eclipse
respectively. Synthetic observations use disequilibrium chemistry.

We perform the same retrieval exercise for an atmosphere that is cloudy. The
3D cloudy atmosphere was built from the simulations in mendonca18a  and shown
in Fig. 5. To model the cloud in our retrieval framework we consider 3 extra
parameters: the opacity of the cloud, the fractional scale height, and the
cloud base pressure. We also assume that the cloud is non-scattering and grey
in nature. We recognise that considering the scattering properties of the
cloud could help to improve the retrieved atmospheric properties Taylor2020b .
The retrieved abundances in Fig. 9 demonstrate that we are able to detect and
constrain the abundance of H2O with comparable precision to the cloud free
scenario. We find that the other molecules are trickier to constrain due to
the muted features introduced by the cloud. The retrieved thermal profiles in
Fig. 10 demonstrate the inability to recover the shape of the thermal
structure for the cloudy atmosphere. The shape of the thermal structure
calculated using THOR is not able to be captured by a Guillot-style
parameterisation, given that the former is a state-of-the-art thermal
structure parameterisations used in exoplanet emission retrievals. In other
words, we demonstrate that observations from future instruments will require a
more complex vertical temperature parameterisation for cloudy atmospheres.

Figure 9: Retrieved abundances (VMR) and associated uncertainty for the cloudy
WASP-43b simulations. They are shown by the vertical lines for each molecule.
The horizontal solid and dashed lines are the disequilibrium and equilibrium
abundances respectively, the values are taken to be around $\sim$0.1 to 0.01
bar, the region in which our simulations are sensitive to. Phase 0∘ and phase
180∘ correspond to the transit and the eclipse respectively. Synthetic
observations use disequilibrium chemistry. Figure 10: Retrieved temperature
structures for the cloudy WASP-43b numerical experiment. We present the
median, 1-sigma and 2-sigma intervals in decreasing opacity of purple, and the
temperature structure from THOR is presented in black.

##  6 Summary and conclusions

Phase curves of thermal emission and reflected light are a powerful technic to
characterize exoplanetary atmospheres. They provide insights into the
atmospheric dynamics and longitudinal variations of the thermal structure, the
chemical composition and clouds. In the next decade, JWST will likely observe
a dozen planets with phase curves, preferentially smaller and cooler planets
than Ariel can efficiently observe.

We defined 5 science questions: 1) heat redistribution, 2) longitudinal
composition $\&$ temperature variations, 3) atmospheric composition of low-
mass planets, 4) albedo $\&$ clouds and 5) time variability, to investigate
with phase curves. They are focused on the coupling between atmospheric
dynamics, chemical composition, thermal structure, and clouds. For most of
these science questions, the requirement is to reach a SNR$>$10, i.e.
detection at 10 sigma of photometric phase curves (at Tier 1 resolution),
assuming no heat redistribution.

We constructed a list of 44 potential targets for phase curves from the Ariel
general target list. These planets fulfill the requirements for the science
questions and were divided into four size categories (rocky, sub-Neptunes,
Neptunes and giants). Based on this potential target list, we expect that
35-40 exoplanets could be observed if 10$\%$ of Ariel science time were
dedicated to phase curves. According to this study, Neptune-size and giant
planets are excellent targets for Ariel. They comprise $\sim$80$\%$ of planets
in the target list. However, our observing plan only partially answers SQ4
(albedo $\&$ clouds), mostly based on reflected light. For giant planets (for
which there is a multitude of good targets), an ideal strategy would be to
choose those which will benefit from a precise visible phase curve observed by
TESS, CHEOPS or PLATO. In particular, there would be a strong synergy by
combining the hundreds of PLATO visible phase curves with Ariel infrared phase
curves.

The study of the effect of systematics shows that multi-epoch observations
reduce the precision of phase-curve reconstruction. That performance
degradation is drastic when a linear trend is present in the instrumental
systematic effects.We therefore recommend performing continuous observations,
starting before an eclipse and ending after the following eclipse. We also
recommend to carefully assessing Ariel’s instrument systematic effects early
on in the mission.Depending on their nature, we could consider multi-epoch
phase curves, filling gaps in the observing scheduling.

The atmospheric retrieval of a simulated spectroscopic phase curves of cloud-
free WASP-43b allows to measure the abundance of water within 0.5 dex for a
3-h observation. Combining measurements over the whole orbit, the mean
abundance is retrieved with a 60$\%$ precision. In addition, the spectroscopic
phase curve of WASP-43b allows distinguishing chemical equilibrium and
disequilibrium thanks to CH4 and CO2 variations. It is important to note that
all the giant planets on our target list have a SNR for a full phase curve
similar to or higher than WASP-43b. This would imply a precision for
atmospheric retrieval generally better than our simulated case for WASP-43b.

To conclude, the Ariel mission is a unique opportunity to perform a
statistical survey of exoplanet phase curves, mostly warm/hot gaseous planets.
These observations will be complemented by Ariel transit/eclipse spectroscopy
and by measurements from other telescopes (e.g. TESS, CHEOPS, PLATO, JWST and
ELTs). Together, they will provide insights into the global climate of these
planets as well as a context for the interpretation of all Ariel observations.

###### Acknowledgements.

B.C. acknowledges financial support from CNES. J.M.M. work on Ariel is
supported by PRODEX grant (PEA: 4000127377). T.J.B. acknowledges support from
the McGill Space Institute Graduate Fellowship, the Natural Sciences and
Engineering Research Council of Canada’s Postgraduate Scholarships-Doctoral
Fellowship, and from the Fonds de recherche du Québec – Nature et technologies
through the Centre de recherche en astrophysique du Québec. L.V.M. and E.P.
were funded by the ASI grant n. 2018.22.HH.O. C.A. Haswell’s work on Ariel is
supported by STFC under grant ST/T00178X/1

##  7 Appendix: Expression of the SNR for phase curves

###  7.1 A simple metric for phase curves

We assume that the relative thermal flux variation with time (for an orbital
period $P$) is given by:

$F(t)=\frac{F_{\rm max}}{F_{\rm star}}\times\frac{\left(\cos(2\pi
t/P)+1\right)}{2}$ (3)

Phase curve data can be fitted with a cosine function $f(t)=A\times\cos(2\pi
t/P)+B$. For $P\gg$1h, the uncertainty on the amplitude $A$ is:

$\delta A=\sqrt{\frac{2}{P}}\sigma_{\rm 1h}$ (4)

where $\sigma_{\rm 1h}$ is the noise (in ppm) in the given spectral or
photometric band for a 1-hour observation of the host star and obtained with
Ariel-Rad, and $P$ is expressed in hours. With $A=\frac{F_{\rm max}}{2F_{\rm
star}}$, the SNR of the full phase curve is:

$SNR_{\rm orbit}=\frac{A}{\delta A}=0.5\times SNR_{\rm 1h}\times\sqrt{P/2}$
(5)

where $SNR_{\rm 1h}=\frac{F_{\rm max}}{F_{\rm star}}\frac{1}{\sigma_{\rm 1h}}$
is the SNR for a 1-hour observation at full phase given in equation (1).

###  7.2 Comparison to Spitzer data of LHS 3844b

We compared our expressions (4) and (5) to the analysis of the Spitzer phase
curves of LHS 3844b kreidberg19 . In this study, the planet-to-star flux
variation is binned over 25 equally spaced intervals over the orbital period
with 1-sigma uncertainties of $\sim$50 ppm. The peak-to-trough amplitude of
the phase variation is 350$\pm$40 ppm from MCMC fitting. This corresponds to a
SNR of $\sim$9\. Applying formula (4) and (5), we find an uncertainty of the
peak-to-trough amplitude of $\sim$30 ppm and a SNR of 12. We note that our
basic fitting tends to underestimate the uncertainty on the amplitude of the
phase curve since it assumes a perfect sine wave and does not take into
account the transit and the eclipse. In addition, our formula gives the
statistical uncertainty, so there would naturally be some deviation compared
to a given dataset. Finally, the uncertainty on the peak-to-trough sine
amplitude for real LHS 3844b data is higher than predicted by the idealized
model because it was simultaneously fit with an instrument systematic noise
model. The instrument systematics for Ariel are expected to be much less
severe than for Spitzer, so the Ariel uncertainties are expected to match
those calculated with Equation (4). Using noise estimations from Ariel-Rad, we
predict a SNR of $\sim$13 with Ariel for the same observing duration
($\sim$3.8 days) and the same spectral range (4-5 $\mu$m) as Spitzer. The SNR
is consistent with our previous estimation and it would rises up to 17 with
AIRS-CH1. We conclude that our metric compares favourably with previous
Spitzer observations and analysis of LHS 3844b.

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