11institutetext: Faculty of Aerospace Engineering, Delft University of
Technology, Delft, The Netherlands
11email<EMAIL_ADDRESS>22institutetext: CNRS/INSU, LATMOS-IPSL,
Guyancourt, France

# From exo-Earths to exo-Venuses

Flux and Polarization Signatures of Reflected Light

G. Mahapatra  , Now at SRON Netherlands Institute for Space Research, Leiden,
The Netherlands11 F. Abiad 11 L. Rossi 22 D.M.Stam 11

(Accepted 11 January, 2023)

###### Abstract

Context. Terrestrial-type exoplanets in or near stellar habitable zones appear
to be ubiquitous. It is, however, unknown which of these planets have
temperate, Earth-like climates or e.g. extreme, Venus-like climates.

Aims. Technical tools to distinguish different types of terrestrial-type
planets are crucial for determining whether a planet could be habitable or
incompatible with life as we know it. We investigate the potential of
spectropolarimetry for distinguishing exo-Earths from exo-Venuses.

Methods. We present numerically computed fluxes and degrees of linear
polarization of starlight that is reflected by exoplanets with atmospheres in
evolutionary states ranging from similar to the current Earth to similar to
the current Venus, with cloud compositions ranging from pure water to 75%
sulfuric acid solution, for wavelengths between 0.3 and 2.5 $\mu$m. We also
present flux and polarization signals of such planets in stable but spatially
unresolved orbits around the star Alpha Centauri A.

Results. The degree of polarization of the reflected starlight shows larger
variations with the planetary phase angle and wavelength than the total flux.
Across the visible, the largest degree of polarization is reached for an
Earth-like atmosphere with water clouds, due to Rayleigh scattering above the
clouds and the rainbow feature at phase angles near 40∘. At near-infrared
wavelengths, the planet with a Venus-like CO2 atmosphere and thin water clouds
shows the most prominent polarization features due to Rayleigh-like scattering
by the small cloud droplets. A planet in a stable orbit around Alpha Centauri
A would leave temporal variations on the order of 10-13 W/m3 in the total
reflected flux and 10-11 in the total degree of polarization as the planet
orbits the star and assuming a spatially unresolved star-planet system. Star-
planet contrasts are in the order of 10-10 and vary proportionally with
planetary flux.

Conclusions. Current polarimeters appear to be incapable to distinguish
between the possible evolutionary phases of spatially unresolved terrestrial
exo-planets, as a sensitivity close to 10-10 would be required to discern the
planetary signal given the background of unpolarized starlight. A
telescope/instrument capable of achieving planet-star contrasts lower than
10-9 should be able to observe the large variation of the planet’s resolved
degree of polarization as a function of its phase angle and thus be able to
discern an exo-Earth from an exo-Venus based on its clouds’ unique
polarization signatures.

###### Key Words.:

Exo-planets – Venus – Radiative transfer – Polarimetry

##  1 Introduction

Despite having similar sizes, being formed around the same time and from
similar materials, it is clear that the Earth and Venus have evolved into
dramatically different worlds. While it is generally acknowledged that Venus
once had much larger amounts of water than today, it is still debated whether
Venus was once more Earth-like with oceans of water before the runaway-
greenhouse-effect took off (Donahue et al. 1982), or whether the atmospheric
water vapour never actually condensed on the surface (Turbet et al. 2021).
Bullock & Grinspoon (2001) conducted a detailed study of the possible
evolution of Venus’s climate over long time periods starting with a water
vapour enriched atmosphere. Terrestrial-type exoplanets are also expected to
harbour a wide variety of atmospheric compositions with maybe only a few
planets hospitable to life as we know it. Various climate models suggest that
the likelihood of a planetary atmosphere exhibiting a Venus-like runaway-
greenhouse-effect is higher than that of an atmosphere in an Earth-like,
N2-dominated state (Lincowski et al. 2018, and references therein). A study by
Kane et al. (2020) even shows that Jupiter’s migration might have stimulated
the runaway-greenhouse-effect on Venus, suggesting that there could be more
Venus-analogs than Earth-analogs in planetary systems with Jupiter-like
planets.

As planned powerful telescopes and dedicated, sensitive detection techniques
will allow us to characterize smaller exoplanets in the near-future, it will
become possible to probe terrestrial-type planets in and near the habitable
zones of solar-type stars and to find out whether they resemble Earth or
Venus, or something else all together. The high-altitude cloud deck on an exo-
Venus would make it difficult to use a technique like transit spectroscopy for
the characterization of the planet as the clouds themselves would block the
transmission of the starlight and apart from a spectral dependence of the
cloud optical thickness which could leave a wavelength dependent transmission
through the cloud tops, the microphysical properties of the cloud particles,
such as their composition, shape and size distribution would remain a mystery.
Also, the clouds would inhibit measuring trace gas column densities as they
would block the planet’s lower atmosphere and only allow transit spectroscopy
of the highest regions of the atmosphere (see, e.g. Lustig-Yaeger et al.
2019a, and references therein). Indeed, a Venus-like ubiquitous cloud deck
could possibly be mistaken for the planet’s surface, as one would only measure
the transmittance through the gaseous atmosphere above the clouds, possibly
inferring the atmosphere to be thin and eroding (Lustig-Yaeger et al. 2019b).

Jordan et al. (2021) modelled the photochemistry of some of the primary
sulphuric chemical species that should be responsible for the formation and
sustenance of Venus’s sulfuric acid solution clouds, such as SO2, OCS and H2S,
and found that the abundances of such species above the cloud deck would
depend heavily on the effective temperature and distance to the parent star,
with their abundances decreasing with increasing temperature and being
depleted, as we see on Venus, in the presence of a star like the Sun. Thus it
would be challenging to rule out the possibility of an exoplanet being a Venus
analog solely on the basis of the detection of such chemical species in
transmission spectra (Jordan et al. 2021). Indeed, the full characterization
of rocky exoplanets and their classification appears to require the direct
imaging of starlight that is reflected by such planets and/or the thermal
radiation that is emitted by them. While telescopes able to perform such
measurements are not yet available, plans are underway for their development
and deployment (Keller et al. 2010; The Luvoir Team 2019).

While most of telescopes and instruments are designed for only measuring total
fluxes of exoplanets, including (spectro)polarimetry is also being considered.
The main reason to include (spectro)polarimetry (see e.g. Rossi et al. 2021,
and references therein) is that it increases the contrast between star and
planet, as the stellar flux will be mostly unpolarized when integrated over
the disk (Kemp et al. 1987), while the flux of the reflected starlight will
usually be (linearly) polarized. And in addition, (spectro)polarimetry can be
used for the characterization of planetary atmospheres and surfaces. As a
classic example of the latter, Hansen & Hovenier (1974) used Earth-based
measurements of the disk-integrated degree of polarization of sunlight that
was reflected by Venus in three spectral bands and across a broad phase angle
range, to deduce that the particles forming Venus’s main cloud deck consist of
75$\%$ sulfuric acid solution, that the effective radius of their size
distribution is 1.05 $\mu$m, and that the effective width of the distribution
is 0.07. They also derived the cloud top altitude (at 50 mbars) by determining
the amount of Rayleigh scattering in the gas above the cloud tops at a
wavelength of 0.365 $\mu$m. This was later confirmed by the Pioneer Venus
mission which performed in-situ measurements using a nephelometer on a probe
that descended through the clouds (Knollenberg & Hunten 1980).

Polarimetry proved to be an effective technique for disentangling Venus’s
cloud properties because the scattering particles leave a unique angular
polarization pattern in the reflected sunlight depending on the particles’
micro- and macro-physical properties (for an extensive explanation of the
application of polarimetry for the characterization of planetary atmospheres,
see Hansen & Travis 1974). While multiple scattered light usually has a low
degree of polarization, and thus dilutes the angular polarization patterns of
the singly scattered light, the angles where the absolute degree of
polarization reaches a local maximum and/or where it is zero (the so-called
’neutral points’) are preserved and thus still allow for the characterization
of the particles.

Another factor in the successful application of (spectro)polarimetry for the
characterization of Venus’s clouds and hazes is that with Earth-based
telescopes, inner planet Venus can be observed at a wide range of phase
angles, thus allowing observations of the angular variation of the degree of
polarization due to the light that has been singly scattered by the
atmospheric constituents. In our solar system, only Venus, Mercury, and the
Moon can be observed at a large phase angle range with Earth-based telescopes
(ignoring the proximity of Mercury to the Sun). To effectively apply
polarimetry to the outer planets in the solar system, a polarimeter onboard a
space mission would be needed. An example of such an instrument was OCCP
onboard NASA’s Galileo mission (Russell et al. 1992) that orbited Jupiter.
Regarding exoplanets, however, the range of observable phase angles depends on
the inclination angle of the planetary orbits: for a face-on orbit, the
planet’s phase angle will be 90∘ everywhere along the orbit, while for an
edge-on orbit, the phase angle will range from close to 0∘ (when the planetary
disk is fully illuminated) to 180∘ (when the night-side of the planet is in
view). The precise range of accessible phase angles would of course depend on
the observational technique and e.g. the use of a coronagraph or star-shade.

Here we investigate the total flux and degree of polarization of starlight
that is reflected by terrestrial-type exoplanets, focusing on the possible
evolutionary stages of Venus as described by Bullock & Grinspoon (2001). Our
goal is to identify characteristic signatures that could help to identify the
properties of exo-Venuses, thus to guide the design of future telescope
instruments. We compute the disk-integrated total and polarized fluxes of
light reflected between wavelengths of 0.3 to 2.5 $\mu$m. First, we study the
single scattering properties of spherical cloud droplets of pure water (H2O)
or 75$\%$ sulphuric acid (H2SO4) in order to identify potentially distinct
signatures for each particle type as a function of wavelength and planetary
phase angle. Second, we compute the multiple scattered flux and polarization
signals that are integrated over the planet’s illuminated disk as functions of
the planet’s phase angle. Third, we compute the signals of the planets in the
four evolutionary phases in stable orbits around the nearby solar-type Alpha
Centauri A, simulating the observations of such planets if they are spatially
unresolved from their parent star.

The outline of this paper is as follows. In Sect. 2, we define the fluxes and
polarization of planets, and we describe our numerical algorithm and the four
model planets in the evolutionary phases as described by Bullock & Grinspoon
(2001). In Sect. 3, we present the total and polarized fluxes as computed for
planets that are spatially resolved from their star and for planets that are
spatially unresolved. In the latter case, the planet’s signal is thus combined
with the stellar light. We specifically assume that our model planet orbits
the solar-type star Alpha Centauri A. In Sect. 4, we summarize our results and
present our conclusions.

##  2 Numerical method

###  2.1 Flux and polarization definitions

In this paper, we present the flux and polarization signals of starlight that
is reflected by potentially habitable exoplanets that orbit solar-type stars,
and in particular, Alpha Centauri A. Because these planets will be very close
in angular distance to their parent star, they will usually be spatially
unresolved, i.e. it will not be possible to spatially separate the planet’s
signal from that of its parent star. The flux vector ${\bf F}_{\rm u}$ (u=
’unresolved’) that describes the light of the star and its spatially
unresolved planet, and that arrives at a distant observer is then written as

${\bf F}_{\rm u}(\lambda,\alpha)={\bf F}_{\rm s}(\lambda)+{\bf F}_{\rm
p}(\lambda,\alpha),$ (1)

with ${\bf F}_{\rm s}$ the star’s flux vector and ${\bf F}_{\rm p}$ that of
the planet. Furthermore, $\lambda$ is the wavelength (or wavelength band), and
$\alpha$ is the planetary phase angle, i.e. the angle between the star and the
observer as measured from the center of the planet. We assume that the light
of the star is captured together with the starlight that is reflected by the
planet. A telescope with a coronagraph or star-shade would of course limit the
amount of captured direct starlight, depending on its design and the angular
distance between the star and the planet.

A flux (column) vector is given by (Hansen & Travis 1974)

${\bf F}=[F,Q,U,V],$ (2)

with $F$ the total flux, $Q$ and $U$ the linearly polarized fluxes, and $V$
the circularly polarized flux. The dimensions of $F$, $Q$, $U$, and $V$ are W
m-2, or W m-3 when defined per wavelength.

Measurements of FGK-stars, such as the Sun and Alpha Centauri A, indicate that
their (disk-integrated) polarized fluxes are virtually negligible (Kemp et al.
1987; Cotton et al. 2017), thus we describe the star’s flux (column) vector
that arrives at the observer located at a distance $D$ as

${\bf F}_{\rm s}(\lambda)=F_{\rm s}(\lambda)\hskip 2.84544pt{\bf
1}=\frac{R_{\rm s}^{2}}{D^{2}}\hskip 2.84544pt\pi B(\lambda,T_{\rm s})\hskip
2.84544pt{\bf 1},$ (3)

with $\pi B$ the stellar surface flux, $T_{\rm s}$ the star’s effective
temperature, $R_{\rm s}$ the stellar radius, and ${\bf 1}$ the unit (column)
vector. The parameter values that we adopt for the Alpha Centauri A system are
listed in Table 1.

Because of the huge distances to stars and their planets, flux vector ${\bf
F}_{\rm p}$ of the starlight that is reflected by an exoplanet pertains to the
planet as a whole, thus integrated across the illuminated and visible part of
the planetary disk. It is given by (see e.g. Rossi et al. 2018)

$\displaystyle{\bf F}_{\rm p}(\lambda,\alpha)$ $\displaystyle=$ $\displaystyle
A_{\rm G}(\lambda)\hskip 2.84544pt{\bf R}_{\rm p}(\lambda,\alpha)\hskip
2.84544pt\frac{r_{\rm p}^{2}}{D^{2}}\frac{R_{\rm s}^{2}}{d^{2}}\hskip
2.84544pt\pi B(\lambda,T_{\rm s})\hskip 2.84544pt{\bf 1}$ (4) $\displaystyle=$
$\displaystyle A_{\rm G}(\lambda)\hskip 2.84544pt{\bf R}_{\rm
1p}(\lambda,\alpha)\hskip 2.84544pt\frac{r_{\rm p}^{2}}{D^{2}}\frac{R_{\rm
s}^{2}}{d^{2}}\hskip 2.84544pt\pi B(\lambda,T_{\rm s}).$ (5)

Here, $A_{\rm G}$ is the planet’s geometric albedo, ${\bf R}_{\rm p}$ the
matrix describing the reflection by the planet and ${\bf R}_{\rm 1p}$ its
first column, $r_{\rm p}$ is the planet’s radius, $d$ the distance between the
star and the planet, and $D$ the distance to the observer. The planet’s
reflection is normalized such that planetary phase function $R_{\rm 1p}$,
which is the first element of ${\bf R}_{\rm 1p}$, equals 1.0 at
$\alpha=0^{\circ}$.

The contrast $C$ between the total flux of the planet and the total flux of
the star is then given by

$C(\lambda,\alpha)=\frac{F_{\rm p}(\lambda,\alpha)}{F_{\rm s}(\lambda)}=A_{\rm
G}(\lambda)\hskip 2.84544ptR_{\rm 1p}(\lambda,\alpha)\frac{r_{\rm
p}^{2}}{d^{2}},$ (6)

with $F_{\rm p}$ the first element of the planetary flux vector ${\bf F}_{\rm
p}$. Using the parameters from Table 1, the contrast $C$ between a planet with
the radius of Venus at a Venus-like distance from Alpha Centauri A equals
about $2\cdot 10^{-9}A_{\rm G}$ at $\alpha=0^{\circ}$ (at this phase angle,
the planet would actually be precisely behind the star with respect to the
observer and thus out of sight).

The degree of polarization of the spatially resolved planet (without including
any direct light of the star) is defined as

$P_{\rm p}=\frac{\sqrt{Q_{\rm p}^{2}+U_{\rm p}^{2}}}{F_{\rm p}},$ (7)

where we ignore the planet’s circularly polarized flux $V_{\rm p}$ as it is
expected to be very small compared to the linearly polarized fluxes (Rossi &
Stam 2018). We also ignore the circularly polarized fluxes in our radiative
transfer computations, as this saves significant amounts of computing time
without introducing significant errors in the computed total and linearly
polarized fluxes (see Stam & Hovenier 2005).

Fluxes $Q_{\rm p}$ and $U_{\rm p}$ are defined with respect to the planetary
scattering plane, which is the plane through the planet, the star and the
observer. In case the planet is mirror-symmetric with respect to the planetary
scattering plane, linearly polarized flux $U_{\rm p}$ equals zero and we can
use an alternative definition of the degree of polarization that includes the
polarization direction as follows

$P_{\rm p}=-\frac{Q_{\rm p}}{F_{\rm p}}.$ (8)

If $P_{\rm p}>0$ ($P_{\rm p}<0$), the light is polarized perpendicular
(parallel) to the reference plane.

In case a planet is not completely spatially resolved from its parent star,
and the background of the planet on the sky is thus filled with (unpolarized)
starlight, the observable degree of polarization $P_{\rm u}$ can be written as
(cf. Eqs. 6-7)

$P_{\rm u}=\frac{\sqrt{Q_{\rm p}^{2}+U_{\rm p}^{2}}}{F_{\rm p}+xF_{\rm
s}}=\frac{F_{\rm p}}{F_{\rm p}+xF_{\rm s}}\hskip 2.84544ptP_{\rm
p}=\frac{C}{C+x}P_{\rm p},$ (9)

with $x$ the fraction of the stellar flux that is in the background, which
will depend on the angular distance between the star and the planet, on the
starlight suppressing techniques that are employed, such as a coronagraph or
star-shade, and on the spatial resolution of the telescope at the wavelength
under consideration. This equation also holds for the signed degree of
polarization as given by Eq. 8. If $x=1$, the planetary and the stellar flux
are measured together. In that case,

$P_{\rm u}=\frac{C}{C+1}P_{\rm p}\approx CP_{\rm p}.$ (10)

Here we used the fact that the contrast $C$ will usually be very small (on the
order of 10-9 as shown earlier).

The planet’s degree of polarization $P_{\rm p}$ and the contrast $C$ both
depend on $\lambda$ and $\alpha$, but generally in a different way. The
dependence of $P_{\rm u}$ on $\lambda$ and $\alpha$ will thus generally differ
from that of either $P_{\rm p}$ or $C$.

Table 1: The values of the parameters describing the planetary system of Alpha Centauri A used in our numerical modelling222The orbital distance $d$ of the planet has been chosen such that it receives the same stellar flux as Venus receives from the Sun, and in accordance with the orbit stability requirements for a planet around Alpha Centauri A predicted by Quarles & Lissauer (2016). For the radius of the Sun, $R_{\rm Sun}$, we use 695,700 km. . Parameter (unit) | Symbol | Value
---|---|---
Stellar radius ($R_{\rm Sun}$) | $R_{\rm s}$ | 1.2234
Stellar effective temperature (K) | $T_{\rm s}$ | 5790
Planet radius (km) | $r_{\rm p}$ | 6052
Planet orbital distance (AU) | $d$ | 0.86
Planet orbital period (yr) | $P$ | 0.76
Distance to the system (ly) | $D$ | 4.2
Angular separation (arcsecs) | $S$ | 0.67

###  2.2 Our radiative transfer algorithm

Our procedure to compute the flux vector ${\bf F}_{\rm p}$ (Eq. 5) of the
starlight that is reflected by the planet, is described in Rossi et al.
(2018). The radiative transfer algorithm is based on an efficient adding-
doubling algorithm (de Haan et al. 1987) and fully includes polarization for
all orders of scattering. With this algorithm, through the use of a Fourier-
series expansion of the planetary reflection matrix ${\bf R}_{\rm p}$, the
reflected flux vector can be computed for any planetary phase angle $\alpha$.

Our model planetary atmospheres consist of horizontally homogeneous layers.
For each layer, we prescribe the total optical thickness $b$, the single-
scattering albedo $a$, and the single-scattering matrix ${\bf P}$. Our layered
model atmosphere is bounded below by a Lambertian reflecting surface (i.e. the
light is reflected isotropically and unpolarized) with an albedo $a_{\rm
surf}$.

A layer’s optical thickness $b$ at a wavelength $\lambda$ is the sum of the
optical thicknesses of the gas molecules, $b^{\rm m}$, and, if present, the
cloud particles, $b^{\rm c}$. We ignore other atmospheric particles, such as
haze particles. The single-scattering matrix P of a mixture of gas molecules
and cloud particles in a layer is given by

${\bf P}(\Theta,\lambda)=\frac{b_{\rm sca}^{\rm
m}(\lambda)\leavevmode\nobreak\ {\bf P}^{\rm m}(\Theta,\lambda)+b_{\rm
sca}^{\rm c}(\lambda)\leavevmode\nobreak\ {\bf P}^{\rm
c}(\Theta,\lambda)}{b_{\rm sca}^{\rm m}(\lambda)+b_{\rm sca}^{\rm
c}(\lambda)},$ (11)

with subscript ‘${\rm sca}$’ referring to ‘scattering’, thus $b_{\rm sca}=ab$,
with $a$ the single scattering albedo. Furthermore, ${P^{\rm m}}$ is the
single-scattering matrix of the gas molecules, and ${P^{\rm c}}$ that of the
cloud particles. $\Theta$ is the single scattering angle:
$\Theta=180^{\circ}-\alpha$.

We use two types of model atmospheres to study the influence of an exoplanet’s
atmospheric evolution on the reflected light signals: an Earth-like and a
Venus-like atmosphere. For our Earth-like atmosphere, we define the pressure
and temperature across 17 layers, representing a mid-latitude summer profile
(following Stam 2008). For our Venus-like atmosphere, we use 71 layers with
pressure and temperature profiles from the Venus International Reference
Atmosphere (VIRA) (Kliore et al. 1985), representing a mid-latitude afternoon
profile. With these vertical profiles, and assuming anisotropic Rayleigh
scattering (Hansen & Travis 1974), we compute each layer’s single scattering
matrix ${\bf P}^{\rm m}$ and the scattering optical thickness $b_{\rm
sca}^{\rm m}$. We neglect absorption, thus $b^{\rm m}=b^{\rm m}_{\rm sca}$.
The depolarization factor for computing ${\bf P}^{\rm m}$ and $b_{\rm
sca}^{\rm m}$ for anisotropic Rayleigh scattering depends on the atmospheric
composition. For the Earth-like atmosphere, we use a (wavelength independent)
depolarization factor of 0.03, which is representative for dry air, and for
the Venus-like atmosphere, we use 0.09, which is representative for a pure CO2
atmosphere (Hansen & Travis 1974). We use wavelength-independent refractive
indices of 1.00044 and 1.00027 for the Venus-like and the Earth-like model
atmospheres, respectively; note that this assumption has a negligible effect
on the reflected total and polarized fluxes.

The cloud particles in our model atmospheres are spherical and distributed in
size according to a two-parameter gamma size distribution (see Hansen & Travis
1974) that is described by an effective radius $r_{\rm eff}$ and an effective
variance $v_{\rm eff}$. The terrestrial clouds are located between 1 and 3 km
altitude, and the Venusian clouds, depending on their evolutionary phase,
between 47 and 80 km. The cloud optical thickness has a uniform vertical
distribution through the altitude range (see Fig. 1).

The single-scattering properties of the cloud particles are computed using
Mie-theory (De Rooij & Van der Stap 1984), as these particles are expected to
be spherical. For these computations we specify the wavelength $\lambda$ and
$n_{\rm r}$, the refractive index of the cloud particles. The cloud particles
are composed of either pure water or a sulphuric acid solution with varying
concentration. We use the refractive index of water from Hale & Querry (1973)
and that of sulphuric acid with 75 $\%$ acid concentration from Palmer &
Williams (1975). We use a negligible value for the imaginary part of the
particles’ refractive indices, $n_{\rm i}$ = 10-8.

###  2.3 Cloud properties through the planet’s evolution

Figure 1: The four evolutionary phases of the model planets (Bullock &
Grinspoon 2001). In Phases 1 and 2, the clouds consist of liquid water
droplets, and in Phases 3 and 4, of liquid sulfuric acid solution droplets.
The cloud optical thickness is indicated by $b^{\rm c}$ and the cloud particle
effective radius by $r_{\rm eff}$. For the effective variance $v_{\rm eff}$ of
the size distributions in Phases 1-3, we use 0.1, and for Phase 4, $v_{\rm
eff}=0.07$.

It is suspected that early Venus had a thin, Earth-like atmosphere and
(possibly) an Earth-like ocean that was later lost due to the runaway
greenhouse effect (Donahue et al. 1982; Kasting 1988; Way & Del Genio 2020).
As the planet’s surface heated up, the water would have evaporated and
enriched the atmosphere with water vapor. The macroscopic cloud properties for
the 4 evolutionary phases that we will use are illustrated in Fig. 1.

We start our evolutionary model of Venus assuming Earth-like conditions (Phase
1), i.e. an atmosphere consisting of 78$\%$ N2 and 22$\%$ O2. Model
simulations showed that the actual dependence of the total and polarized flux
signals on the percentage of oxygen appeared to be negligible. Hence we used
the present day Earth atmosphere as the Earth-like atmosphere model while the
actual percentage of oxygen on an exo-planet could be different. The cloud
particles have an effective radius $r_{\rm eff}$ of 10 $\mu$m in agreement
with ISCCP (Tselioudis 2001) and an effective variance $v_{\rm eff}$ of 0.1.
The total cloud optical thickness $b^{\rm c}$ is 10.0 at
$\lambda=0.55\leavevmode\nobreak\ \mu$m and the cloud layer extends from 2 to
4 km.

The next evolutionary phases are also inspired by the Venus climate model of
Bullock & Grinspoon (2001). In Phase 2, the atmosphere is Venus-like as it
consists of pure CO2 gas, and has relatively thin liquid water clouds with
$b^{\rm c}=4$, and with the cloud tops at 80 km. For this phase, we use
$r_{\rm eff}$ of 0.5 $\mu$m, which is smaller than the present day value,
because the atmosphere is expected to be too hot for strong condensation to
take place thus preventing the particles to grow larger. In Phase 3, the
clouds are thick sulphuric-acid solution clouds, with $b^{\rm c}=120$ and the
cloud tops at 65 km, because the atmosphere is cool enough to allow
condensation and/or coalescence of saturated vapour over a large altitude
range. Since the region of condensation covers a large altitude range, the
particles can grow large until they evaporate. In this phase, $r_{\rm
eff}=2\leavevmode\nobreak\ \mu$m, which is twice the effective radius of the
present day Venus cloud particles. For both phases 2 and 3, we use $v_{\rm
eff}=0.1$. In Phase 4, the clouds have the present-day properties of Venus’s
clouds with $b^{\rm c}=30$ and the cloud tops at 65 km (Rossi et al. 2015;
Ragent et al. 1985). For the cloud particle sizes in this phase, we use
$r_{\rm eff}=1.05$ $\mu$m and $v_{\rm eff}=0.07$ following the values derived
by Hansen & Hovenier (1974). We ignore the absorption by cloud particles in
the UV in all of our Venus-like clouds to avoid adding complexity and because
the exact nature and location of the UV-absorption is still under debate
(Titov et al. 2018).

Figure 2: The total flux and degree of polarization of incident unpolarized
light that has been singly scattered by four different types of cloud
particles, as functions of the phase angle $\alpha$ and the wavelength
$\lambda$. Left column: the total flux or phase function (single scattering
matrix element $P_{\rm 11}$). Right column: the degree of polarization
($-P_{\rm 21}/P_{\rm 11}$). First row: H2O particles with $r_{\rm
eff}=10\leavevmode\nobreak\ \mu$m and $v_{\rm eff}=0.1$ (belonging to the
Phase 1 model planet: ’Current Earth’); Second row: H2O particles with $r_{\rm
eff}=0.5\leavevmode\nobreak\ \mu$m and $v_{\rm eff}=0.1$ (Phase 2: ’Thin
clouds Venus’); Third row: 75% H2SO4 particles with $r_{\rm
eff}=2.0\leavevmode\nobreak\ \mu$m and $v_{\rm eff}=0.1$ (Phase 3: ’Thick
clouds Venus’); Fourth row: 75% H2SO4 particles with $r_{\rm
eff}=1.05\leavevmode\nobreak\ \mu$m and $v_{\rm eff}=0.07$ (Phase 4: ’Current
Venus’).

Figure 2 shows the phase function (i.e. single scattering matrix element
$P_{\rm 11}$) and the degree of linear polarization for unpolarized incident
light (the ratio of single scattering matrix elements $-P_{\rm 21}$/$P_{\rm
11}$) that has been singly scattered by the four different types of cloud
particles as functions of $\alpha$ (i.e. 180∘ \- $\Theta$), for a range of
wavelengths $\lambda$.

As can be seen in Fig. 2, the phase functions show strong forward scattering
peaks (near $\alpha=180^{\circ}$, thus when the night-side of the planet would
be turned towards the observer) that decrease with increasing $\lambda$, thus
with decreasing effective particle size parameter $x_{\rm eff}=2\pi r_{\rm
eff}/\lambda$ (for the large H2O cloud particles, with $r_{\rm eff}=10$
$\mu$m, this decrease is not readily apparent from the figure). The H2O
particles with $r_{\rm eff}=10\leavevmode\nobreak\ \mu$m show a moderate local
maximum in the phase function around $\alpha=40^{\circ}$, which is usually
referred to as the primary rainbow (see e.g. Hansen & Travis 1974). The large
H2O and the H2SO4 particles also produce higher fluxes towards
$\alpha=0^{\circ}$ that are referred to as the glory (Laven 2008; García-Muñoz
et al. 2014; Markiewicz et al. 2014; Rossi et al. 2015; Markiewicz et al.
2018). For the small H2O particles and the H2SO4 particles at larger
wavelengths, the phase functions become more isotropic and the glory and other
angular features disappear.

Figure 2 also shows the degree of linear polarization of the singly scattered
light. This degree of polarization appears to be more sensitive to the
particle composition than the scattered flux, especially for $\lambda$ between
0.5 and 2 $\mu$m, where H2O particles yield relatively high positive degrees
of polarization (perpendicular to the scattering plane) between phase angles
of about 20∘ and 100∘, whereas the H2SO4 particles impart a mostly negative
degree of polarization through a broad range of phase angles, except for
narrow regions around $\alpha=20^{\circ}$ and $80^{\circ}$. The tiny, $r_{\rm
eff}=0.5$ $\mu$m, water droplets have a strong, broad positive polarization
region for $\lambda\geq 1\leavevmode\nobreak\ \mu$m, where they are so small
with respect to the wavelength that they scatter like Rayleigh scatterers.

As mentioned before (see e.g. Hansen & Travis 1974; Hansen & Hovenier 1974),
patterns in the single scattering degree of polarization are generally
preserved when multiple scattered light is added, as the latter usually has a
low degree of polarization, and thus adds mostly total flux, which subdues
angular features, but does not change the angular pattern (local maxima,
minima, neutral points) itself. The single scattering angular features in the
polarization will thus also show up in the polarization signature of a planet
as a whole, and can be used for characterisation of the cloud particle
properties and thus possibly of various phases in the evolution of a Venus-
like exoplanet. This will be investigated in the next section.

##  3 Results

Here we present the disk-integrated total flux and degree of polarization of
incident unpolarized starlight that is reflected by the model planets at
different wavelengths $\lambda$ and for phase angles $\alpha$ ranging from 0∘
to 180∘. The actual range of phase angles at which an exoplanet can be
observed depends on the inclination angle $i$ of the planet’s orbit (the angle
between the normal on the orbital plane and the direction towards the
observer): $\alpha$ ranges between $90^{\circ}-i$ to $90^{\circ}+i$.
Obviously, at $\alpha=0^{\circ}$, the planet would be precisely behind its
star, and at 180∘ it would be precisely in front of its star (in transit).
Other phase angles might be inaccessible due to restrictions of inner working
angles of telescopes and/or instruments. For completeness, we include all
phase angles in our computations.

Section 3.1 shows results for spatially resolved planets and Sect. 3.2 for
planets that are spatially unresolved from their star. In particular, we show
these latter results for a model planet orbiting the star Alpha Centauri A at
a distance where the incident stellar flux is similar to the solar flux that
reaches Venus. Because our model planets are all mirror-symmetric with respect
to the reference plane, their linearly polarized flux $U_{\rm p}$ equals zero
and will not be discussed further.

###  3.1 Flux and polarization of spatially resolved planets

Figure 3: Left column: The total flux (or phase function); and right column:
The degree of polarization of incident unpolarized starlight that is reflected
by the model planets in the 4 evolutionary phases as functions of $\alpha$ and
$\lambda$. First row: Phase 1 (’Current Earth’); Second row: Phase 2 (’Thin
clouds Venus’); Third row: Phase 3 (’Thick clouds Venus’); Fourth row: Phase 4
(’Current Venus’). The phase functions are normalised such that at
$\alpha=0^{\circ}$, they equal the planet’s geometric albedo $A_{\rm G}$.

Figure 3 shows the total flux $F_{\rm p}$ (the planetary phase function) and
degree of polarization $P_{\rm p}$ as functions of $\alpha$ and $\lambda$ for
the four evolutionary phases illustrated in Fig. 1. The total fluxes are
normalized such that at $\alpha=0^{\circ}$, they equal the planet’s geometric
albedo $A_{\rm G}$ (see Eq. 5). Figure 4 shows $A_{\rm G}$ of the planets in
the four evolutionary phases as functions of the wavelength $\lambda$. Table 2
lists the geometric albedo’s at 0.5, 1.0, 1.5 and 2.0 $\mu$m. The ’Current
Earth’ (Phase 1) shows very little variation in $A_{\rm G}$, and the ’Thin
clouds Venus’ (Phase 2) has the lowest albedo because of the small cloud
particles and the small cloud optical thickness. The geometric albedo’s of the
’Thick clouds Venus’ (Phase 3) and the ’Current Venus’ (Phase 4) are very
similar. Thus across the wavelength region investigated in this paper, the
’Current Earth’ (Phase 1) has the highest geometric albedo.

Table 2: The model planets’ geometric albedo’s $A_{\rm G}$ for the four evolutionary phases at four wavelengths. $\lambda$ ($\mu$m) | 0.5 | 1.0 | 1.5 | 2.0
---|---|---|---|---
Current Earth | 0.757 | 0.752 | 0.740 | 0.739
Thin clouds Venus | 0.186 | 0.184 | 0.240 | 0.310
Thick clouds Venus | 0.727 | 0.627 | 0.580 | 0.564
Current Venus | 0.726 | 0.573 | 0.500 | 0.484
Figure 4: The planets’ geometric albedo’s $A_{\rm G}$ as functions of the
wavelength $\lambda$ for the four evolutionary phases.

For each model planet, the total flux $F_{\rm p}$ decreases with increasing
$\alpha$ mostly because less of the planet’s observable disc is illuminated.
The planet with thin H2O clouds (Phase 2) is very dark over all $\alpha$’s
because the cloud optical thickness $b^{\rm c}$ is small and the surface is
black. The total fluxes show vague similarities with the single scattering
phase functions of the cloud particles (Fig. 2). In particular, for the
’Current Earth’ (Phase 1) with large H2O particles, $F_{\rm p}$ increases
slightly around $\alpha=40^{\circ}$, the rainbow angle. Also, the decrease of
$F_{\rm p}$ with $\lambda$ is stronger for the Venus-type planets with H2SO4
clouds (Phases 3 and 4) than for the planet with the large H2O particles
(’Current Earth’, Phase 1), because the single scattering phase function of
the sulfuric acid particles decreases stronger with $\lambda$ than that of the
water droplets (see Fig. 2).

Unlike $F_{\rm p}$, the degree of polarization $P_{\rm p}$ of each of the
model planets, shows angular and spectral features that depend strongly on the
cloud properties and should thus allow distinguishing between the different
evolutionary phases. In Phase 1 (’Current Earth’), $P_{\rm p}$ is high and
positive up till $\lambda=0.5\leavevmode\nobreak\ \mu$m and around
$\alpha=90^{\circ}$, which is due to Rayleigh scattering by the gas above the
clouds. Starting at the shortest $\lambda$, $P_{\rm p}$ increases slightly
with $\lambda$ before decreasing. This is due to the slightly larger
contribution of multiple scattered light, with a lower degree of polarization,
at the shortest wavelengths. A Rayleigh-scattering peak is also seen for Phase
4 (’Current Venus’), except there, the peak decreases more rapidly with
$\lambda$ because the clouds are higher in the atmosphere and there is thus
less gas above them. In Phase 3 (’Thick clouds Venus’), the Rayleigh-
scattering peak is suppressed by the contribution of low polarized light that
is reflected by the thicker clouds below the gas. In Phase 2 (’Thin clouds
Venus’), the relatively thin clouds are higher in the atmosphere than in Phase
1 (’Current Earth’), which is why the Rayleigh-scattering peak only occurs at
the very shortest wavelengths (the peak is hardly visible in Fig. 3). Because
the Phase 2 cloud particles are small ($r_{\rm eff}=0.5\leavevmode\nobreak\
\mu$m), they themselves give rise to a Rayleigh-scattering peak at
$\lambda\geq 1.0\leavevmode\nobreak\ \mu$m.

The two model planets with the H2O cloud particles (Phases 1 and 2) show a
narrow region of positive polarization between 30∘ and 40∘, which is the
rainbow peak (see Fig. 2). On exoplanets, this local maximum in $P_{\rm p}$
could be used to detect liquid water clouds on exoplanets (Karalidi et al.
2011, 2012; Bailey 2007). In Phase 1 (’Current Earth’), the rainbow region
starts near the Rayleigh scattering peak of the gas and extends towards the
largest wavelengths. In Phase 2 (’Thin clouds Venus’), with the small water
droplets, the rainbow only occurs at the shortest wavelengths. With increasing
wavelength, it broadens and disappears into the cloud particles’ Rayleigh
scattering peak.

The H2SO4 cloud particles (Phases 3 and 4) have their own specific
polarization patterns, such as the broad negative polarization region at
$\alpha\gtrapprox 80^{\circ}$, which can be traced back to their single
scattering patterns (Fig. 2). In Phase 3 (’Thick clouds Venus’), the cloud
particles give rise to a sharp positive polarization peak at the shortest
wavelengths and for 20${}^{\circ}\leq\alpha\leq 30^{\circ}$. In Phase 4
(’Current Venus’), there is a broader, lower, positive polarization branch
across this phase angle range, which resembles the positive polarization
branch of the tiny H2O droplets in Phase 2 (’Thin clouds Venus’). However, at
the longer wavelengths, the phase angle dependence of the polarization of the
latter planet is very different which should help to distinguish between such
planets. This emphasizes the need for measurements at a wide range of
wavelengths and especially phase angles (if the planet’s orbital inclination
angle allows this).

###  3.2 Flux and polarization of spatially unresolved planets

Figure 5: A sketch of the geometries within the Alpha Centauri system: the
orbital plane of the stars Alpha Centauri A and B is inclined by about 80∘
with respect to the observer on Earth. Our model planet (the blue dot) orbits
Alpha Centauri A. In this sketch, the line of nodes of the planet’s orbit was
chosen to coincide with that of the stellar orbits. The inclination angle
$i_{\rm m}$ of the planet’s orbit with respect to the stellar orbital plane is
45∘, and the inclination angle of the planet’s orbit with respect to the
observer is 80${}^{\circ}-45^{\circ}=35^{\circ}$. The phase angles of the
planet in this sketch would range from $90^{\circ}-35^{\circ}=55^{\circ}$ to
$90^{\circ}+35^{\circ}=125^{\circ}$.

In the previous section, we showed the signals of spatially resolved planets,
thus without background starlight. When observing an exoplanet in the
habitable zone of a solar-type star, it will be difficult to avoid the
starlight. Here, we show the total flux of the planet $F_{\rm p}$, the star-
planet contrast $C$ (see Eq. 6), the spatially resolved degree of polarization
of the planet $P_{\rm p}$ (thus without the starlight) and the spatially
unresolved degree of polarization of the combined star-planet signal $P_{\rm
u}$ (thus including the starlight). While the total planet fluxes shown in
Fig. 3 were normalized at $\alpha=0^{\circ}$ to the planets’ geometric
albedo’s $A_{\rm G}$, here they are computed according to Eq. 5, and thus
depend on the parameters of the planet-star system. We assume our model
planets orbit Alpha Centauri A.

The solar-type star Alpha Centauri A is part of a double star system, and the
orbital parameters of ourplanets are chosen based on the stable planet orbital
distances and orbital inclination angles around this star as predicted by
Quarles & Lissauer (2016). Figure 5 shows a sketch of the system. We use a
planetary orbital distance $d$ of 0.86 AU, such that each model planet
receives a stellar flux similar to the solar flux received by Venus.
Additional system parameter values are listed in Table 1. According to Quarles
& Lissauer (2016), stable orbits around Alpha Centauri A can be found for a
range of angles between the planetary orbital plane and the plane in which the
two stars orbit, and thus for a range of inclination angles $i$ of the
planetary orbit.

Figure 6: Variation of the planet’s phase angle $\alpha$ along the planet’s
orbit around Alpha Centauri A for different mutual inclination angles $i_{\rm
m}$ of the planetary orbit with respect to the orbital plane of the two stars.
Top: $\Omega$, the longitude of the ascending node of the planet’s orbit, is
0∘. For $i_{\rm m}=-10^{\circ}$, the planet is then in a face-on orbit
($i=0^{\circ}$), while for $i_{\rm m}=80^{\circ}$, it is in an edge-on orbit
($i=90^{\circ}$). Bottom: $\Omega=205^{\circ}$, and the planetary orbit is
aligned with the node of the stellar orbital plane.

Figure 6 shows the variation of the planetary phase angle $\alpha$ along a
planetary orbit for two values of the longitude of the orbit’s ascending node
$\Omega$: for $\Omega=0^{\circ}$ (the line connecting the planet’s ascending
and descending nodes is perpendicular to the line to the observer) and for
$\Omega=205^{\circ}$ which represents the configuration of Earth with Alpha
Centauri A. The orbital phase of the planet is defined such that at an orbital
phase angle of 0∘, $\alpha=180^{\circ}$. The inclination angle $i_{\rm m}$ is
the angle between the plane in which the stars move and the planetary orbital
plane. For $\Omega=0^{\circ}$, $i_{\rm m}=-10^{\circ}$ would yield a face-on
planetary orbit ($i=0^{\circ}$) with $\alpha=90^{\circ}$ everywhere along the
orbit. For $i_{\rm m}=80^{\circ}$, the orbit is edge-on ($i=90^{\circ}$) and
$\alpha$ varies between 0∘ and 180∘. Figure 6 also shows the range of $\alpha$
for $i_{\rm m}=0^{\circ}$ and 35∘. According to Quarles & Lissauer (2016), the
latter is the most probable orientation of a stable planetary orbit. For these
two cases, the accessible phase angles range from 80∘ to 100∘, and from 45∘ to
135∘, respectively. For $\Omega=205^{\circ}$, the maximum range of $\alpha$
would be from 20∘ to 160∘, depending on $i_{\rm m}$.

Figure 7: The total planetary flux $F_{\rm p}$ (in W m-3) and the star-planet
contrast $C$, the degree of polarization $P_{\rm p}$ of the spatially resolved
planet, and $P_{\rm u}$, the degree of polarization of the star and the
spatially unresolved planet, all for a ’Current Venus’ model planet (Phase 4)
as functions of the planet’s orbital phase for four mutual inclination angles
$i_{\rm m}$ and four wavelengths $\lambda$: $0.5\leavevmode\nobreak\ \mu$m
(row 1); $1.0\leavevmode\nobreak\ \mu$m (row 2); $1.5\leavevmode\nobreak\
\mu$m (row 3); and $2.0\leavevmode\nobreak\ \mu$m (row 4). The longitude of
the ascending node of the planetary orbit, $\Omega$, is 205∘.

Figure 7 shows $F_{\rm p}$, $P_{\rm p}$ and $P_{\rm u}$ (the spatially
unresolved planet, thus with starlight included) for Phase 4 (’Current Venus’)
as functions of the planet’s orbital phase for $\Omega=205^{\circ}$, four
values of $i_{\rm m}$ (-10∘, 80∘, 0∘, and 35∘), and four wavelengths (0.5,
1.0, 1.5, and 2.0 $\mu$m). The plots for $F_{\rm p}$ also show the contrast
$C$. Because $C$ is the ratio of the planetary flux $F_{\rm p}$ to the stellar
flux $F_{\rm s}$ (Eq. 6), its variation with the orbital phase is proportional
to that of $F_{\rm p}$.

At the two orbital phases in each plot where all the lines cross, the
planetary phase angles $\alpha$ are the same (see Fig. 6) and thus all $F_{\rm
p}$ and $P_{\rm p}$ are the same. The plots for $F_{\rm p}$ appear to be very
similar for the different wavelengths, apart from a difference in magnitude
which is mainly due to the decrease of the stellar flux that is incident on
the planet with increasing wavelength, although the planetary albedo $A_{\rm
G}$ and phase function $R_{\rm 1p}$ also decrease with increasing $\lambda$ as
can be seen in Fig. 3. This wavelength dependence of $F_{\rm p}$ also causes
the decrease of the contrast $C$ with increasing wavelength (i.e. the planet
darkens with increasing $\lambda$), as $C$ is independent of the wavelength
dependence of the stellar flux (see Eq. 6). The shape differences between the
$F_{\rm p}$ (and $C$) curves are due to the wavelength dependence of the
planetary flux that can also be seen in Fig. 3.

At each wavelength $\lambda$, the largest variation in $F_{\rm p}$ with the
orbital phase is seen for $i_{\rm m}=80^{\circ}$, because for that
configuration the variation of $\alpha$ along the orbit is largest (see Fig.
6). The degree of polarization $P_{\rm p}$ of the planet shows significant
variation with the orbital phase at all wavelengths. A particularly striking
feature for the geometry with $i_{\rm m}=80^{\circ}$ is the double peak close
to the orbital phases of 150∘ and 200∘. As can be seen in Fig. 6 for
$\Omega=205^{\circ}$ and $i_{\rm m}=80^{\circ}$, $\alpha$ decreases from about
160∘ at an orbital phase of 0∘, to 20∘ at an orbital phase around 175∘, to
then increase again with increasing orbital phase. Tracing this path of
$\alpha$ through the $P_{\rm p}$ panel in the bottom row of Fig. 3 explains
the double peaked behaviour of $P_{\rm p}$ and its wavelength dependence as
shown in Fig. 7. For the other values of $i_{\rm m}$, the phase angle range
that is covered along the orbit is smaller, and therefore the variation in
$P_{\rm p}$ is also smaller.

The degree of polarization of the spatially unresolved planet, $P_{\rm u}$,
shows similar variations along the orbital phase as $P_{\rm p}$, except that
most features are flattened out because of the addition of the unpolarized
stellar flux, which is independent of the orbital phase angle. The double
peaked feature for $i_{\rm m}=80^{\circ}$ remains strong, however, as at those
orbital phase angles, the contrast $C$ is relatively large and thus the
influence of the added stellar flux relatively small. The variation in the
polarisation of the unresolved system due to the orbiting planet is on the
order of 10-11.

Figure 8: Similar to Fig. 7 except for the model planets in the four
evolutionary phases and $\Omega=0^{\circ}$ and $i_{\rm m}=80^{\circ}$ (thus
for an edge–on orbit, $i=90^{\circ}$): Phase 1 (’Current Earth’), Phase 2
(’Thin clouds Venus’), Phase 3 (’Thick clouds Venus’), Phase 4 (’Current
Venus’). The wavelengths $\lambda$ are like before: $0.5\leavevmode\nobreak\
\mu$m (row 1); $1.0\leavevmode\nobreak\ \mu$m (row 2);
$1.5\leavevmode\nobreak\ \mu$m (row 3); and $2.0\leavevmode\nobreak\ \mu$m
(row 4).

Figure 9: Similar to Fig. 8 except for the most probable, stable orbit around
Alpha Centauri A, i.e. for $\Omega=205^{\circ}$ and $i_{\rm m}$= 35∘ (Quarles
& Lissauer 2016). The wavelengths are like before: $0.5\leavevmode\nobreak\
\mu$m (row 1); $1.0\leavevmode\nobreak\ \mu$m (row 2);
$1.5\leavevmode\nobreak\ \mu$m (row 3); and $2.0\leavevmode\nobreak\ \mu$m
(row 4).

Figure 8 is similar to Fig. 7, except for the four model planets in the
different evolutionary phases and all for $i_{\rm m}=80^{\circ}$ and
$\Omega=0^{\circ}$. Because here the planetary orbits are seen edge-on
($i=90^{\circ}$), the full range of phase angles is covered, which makes it
possible to explore the full extent of variation of flux and polarization
signals. Because of this large phase angle range, $F_{\rm p}$ varies strongly
with the orbital phase. The wavelength dependence of the total flux can be
traced back to Fig. 3, where in particular the Phase 2 planet (’Thin cloud
Venus’) is dark at all wavelengths, but relatively bright at the longest
wavelengths and small phase angles. As was the case in Fig. 7, the variation
of $C$ is the same as that of $F_{\rm p}$, except for the off-set due to the
stellar flux. The largest values of $C$ (about 1.6$\times 10^{-9}$), are found
for $\lambda=\leavevmode\nobreak\ 0.5\leavevmode\nobreak\ \mu$m and around the
orbital phase of 180∘ (at 180∘, the planets would actually be behind the
star).

Furthermore in Fig. 8, $P_{\rm p}$ depends strongly on $\lambda$ and the
planet’s evolutionary phase. At 0.5 $\mu$m, the Phase 1 planet (’Current
Earth’) shows the largest values of $P_{\rm p}$ due to the Rayleigh scattering
gas above the low altitude clouds. At the longer wavelengths, where the
Rayleigh scattering is less prominent, the curves for the Phase 1 planet
clearly show the positive polarization of the rainbow around the orbital
phases of 140∘ and 220∘ (see Fig. 6). For the Phase 2 planet (’Thin clouds
Venus’) and 0.5 $\mu$m, the small cloud particles cause positive polarization
around 150∘ and 210∘, which connect the rainbow and the Rayleigh scattering
maximum in Fig. 3. At longer wavelengths, the broad positive polarization
signature of Rayleigh scattering by the cloud particles dominates the curves,
while the curves for the Phase 3 and 4 planets show mostly negative
polarization apart from the orbital phases around 180∘. When adding the
starlight, the angular features of the polarization $P_{\rm u}$ are suppressed
along those parts of the orbits where $C$ is smallest, thus away from the
orbital phase angle of 180∘. In particular within 180∘ $\pm 40^{\circ}$,
$P_{\rm u}$ still shows distinguishing features, although they are very small
in absolute sense (smaller than 10-10).

Figure 9 is similar to Fig. 8 except here the model planets are in the most
probable stable orbit around Alpha Centauri A as predicted by Quarles &
Lissauer (2016), namely with $\Omega=\leavevmode\nobreak\ 205^{\circ}$ and
$i_{\rm m}=\leavevmode\nobreak\ 35^{\circ}$. As can be seen in Fig. 6, for
this geometry $\alpha$ varies between about 60∘ and 120∘. In Fig. 9, $F_{\rm
p}$ shows a similar variation as the curves in Fig. 8, although less
prominent, as the planets do not reach a ’full’ phase (where
$\alpha=0^{\circ}$) nor the full night phase ($\alpha=180^{\circ}$) along
their orbit. The flux curves in Fig. 9 also miss small angular features that
appear in the single scattering phase functions of the cloud particles (see
Fig. 2), such as the glory, again because the planets do not go through the
related phase angles.

In this particular orbital geometry, $P_{\rm p}$ shows less pronounced angular
features than for the same model planets in edge-on orbits (Fig. 8) because of
the more limited phase angle range. For example, the ’Current Earth’ (Phase 1)
shows no rainbow despite the H2O clouds, because the phase angle of about 40∘
is not reached. In the visible ($\lambda=0.5\leavevmode\nobreak\ \mu$m),
$P_{\rm p}$ reaches the largest values for the ’Current Earth’ (Phase 1). At
longer wavelengths, $P_{\rm p}$ of the ’Thin clouds Venus’ (Phase 2) strongly
dominates because of the Rayleigh scattering by the small cloud particles. The
’Thick clouds Venus’ (Phase 3) shows predominant negative polarization at all
wavelengths and across the whole orbital phase angle range except at
$\lambda=0.5\leavevmode\nobreak\ \mu$m around an orbital phase angle of 20∘.

The polarization of the spatially unresolved planets, $P_{\rm u}$, clearly
shows the suppression of the polarization features due to the added starlight
towards the smaller and larger orbital phase angles, where the planets are
darker. While the Phase 2 planet (’Thin clouds Venus’) is relatively dark ($C$
is very small), its Rayleigh scattering polarization signal is so strong that
its unresolved polarization signal is larger than that of the other planets,
except the Phase 1 planet (’Current Earth’) at 1.0 $\mu$m and orbital phase
angles close to 180∘.

###  3.3 Evolutionary phases across $\alpha$ and $\lambda$

Figure 10: Top: The planet models that yield the largest absolute degree of
polarization $|P_{\rm p}|$ over all phase angles $\alpha$ and wavelengths
$\lambda$: Phase 1 - ’Current Earth’ (blue); Phase 2 - ’Thin clouds Venus’
(light orange); Phase 3 - ’Thick clouds Venus’ (dark orange); and Phase 4 -
’Current Venus’ (brown). Bottom: The maximum values of $|P_{\rm p}|$ of the
four model planets as functions of $\alpha$ and $\lambda$.

In Fig. 10, we show which evolutionary phase has the highest values of
$|P_{\rm p}|$ across all phase angles $\alpha$ and wavelengths $\lambda$. We
find that $|P_{\rm p}|$ of the Phase 1 planet (’Current Earth’) dominates
between 30∘ and 150∘, and mostly for $\lambda<1.0\leavevmode\nobreak\ \mu$m.
In particular, around $\alpha=40^{\circ}$ and up to $\lambda=2.0$ $\mu$m, the
polarization signal of the rainbow produced by the large water cloud particles
is about 0.1 (see the bottom plot of Fig. 10). For
$\lambda>1.0\leavevmode\nobreak\ \mu$m, the Phase 2 planet (’Thin clouds
Venus’) shows the strongest polarization due to the Rayleigh scattering by the
small H2O cloud particles, as can clearly be seen in the bottom plot. The
small patches where the strongest polarization signal is from the ’Thick
clouds Venus’ (Phase 3), for example near $\alpha=20^{\circ}$ and
$\lambda<1.0\leavevmode\nobreak\ \mu$m, or from the ’Current Venus’ (Phase 4)
are due to the single scattering polarization features of the H2SO4 cloud
particles, as can be seen in Fig. 2.

The accessible phase angle range for direct observations of such exoplanets
obviously depends on the actual orientation of the planetary orbits and cannot
be optimized by the observer. Precisely because of that, Fig. 10 clearly
indicates that measurements should be performed across a broad wavelength
range, including wavelengths below 1 $\mu$m to allow distinguishing between
Earth-like and Venus-like planets in various evolutionary phases.

##  4 Summary and conclusions

We presented the total flux and linear polarization of starlight that is
reflected by model planets of various atmospheric types to investigate whether
different phases in the evolution of planets like the Earth and Venus can be
distinguished from each other. We have used four planet models to represent
possible evolutionary phases. Phase 1 (’Current Earth’) has an Earth-like
atmosphere and liquid water clouds; Phase 2 (’Thin clouds Venus’) has a Venus-
like CO2 atmosphere and thin water clouds; Phase 3 (’Thick clouds Venus’) has
a Venus-like CO2 atmosphere and thick sulphuric acid solution clouds; and
Phase 4 (’Current Venus’) has a CO2 atmosphere and thin sulphuric acid
solution clouds. We have computed the total flux and polarization signals
specifically for model planets orbiting our neighbouring solar-type star Alpha
Centauri A using predicted stable orbits (Quarles & Lissauer 2016) in its
habitable zone.

We have computed the reflected starlight for wavelengths $\lambda$ ranging
from 0.3 $\mu$m to 2.5 $\mu$m and for planetary phase angles $\alpha$ from 0∘
to 180∘. We not only present the fluxes and polarization of spatially resolved
model planets (thus without background starlight) but also of spatially
unresolved planets, thus the combined signal of the planet and the star. For
the latter cases, we also computed the planet-star contrast $C$ as a function
of $\alpha$ and $\lambda$ to determine what would technically be required to
detect the planetary signals upon the background starlight.

The range of planetary phase angles $\alpha$ at which a planet can be observed
(spatially resolved or unresolved) depends on the inclination of the planetary
orbit with respect to the observer. We have specifically studied the reflected
light signals of planets orbiting solar-type star Alpha Centauri A. This star
is part of a double star system with solar-type star Alpha Centauri B. The
distance between the two stars varies from 35.6 to 11.2 AU (M-dwarf Alpha
Centauri C or Proxima Centauri orbits the pair at a distance of about 13,000
AU). Dynamical computations (Quarles & Lissauer 2016) predict stable planetary
orbits around Alpha Centauri A in a narrow range of mutual inclination angles
$i_{\rm m}$ between the orbital planes of the two stars and that of the
planet. In particular, the most stable orbit has $i_{\rm m}=35^{\circ}$, which
provides an $\alpha$ range from 60∘ to 120∘. We find that with this orbital
geometry the degree of polarization of the planet would be largest for the
’Current Earth’ (Phase 1) across the visible ($\lambda<1.0\leavevmode\nobreak\
\mu$m) due to Rayleigh scattering by the gas above the clouds. At near
infrared wavelengths (1.0 $\mu$m $<\lambda<2.5\leavevmode\nobreak\ \mu$m), the
polarization of the ’Thin clouds Venus’ (Phase 2) is highest, because this
planet has small cloud droplets that scatter like Rayleigh scatterers at the
longer wavelengths.

The well-known advantage of measuring the degree of polarization for the
characterization of (exo)planets is that the angular features in the signal of
the planet as a whole are similar to the angular features in the light that
has been singly scattered by the gas molecules and cloud particles, which are
very sensitive to the microphysical properties (such as the size distribution,
composition, shape) of the scattering molecules and cloud particles and to the
atmosphere’s macrophysical properties (such as cloud altitude and thickness).
The reflected total flux is much less sensitive to the atmospheric properties
than the degree and direction of polarization (see e.g. Hansen & Travis 1974,
for several examples). Indeed, the variations of the planetary flux $F_{\rm
p}$ along a planet’s orbit appear to be mostly due to the change of the
fraction of the planetary disk that is illuminated and visible to the
observer. They provide limited information on the planet’s atmospheric
characteristics, especially if one takes into account that with real
observations, the planet radius will be unknown unless the planet happens to
transit its star. The variation of the planetary flux $F_{\rm p}$ with phase
angle $\alpha$ and wavelength $\lambda$ is similar that of planet-star
contrast $C$, the ratio of $F_{\rm p}$ to the stellar flux $F_{\rm s}$. For a
Venus-like planet orbiting Alpha Centauri A, $C$ is on the order of 10-9.

Our numerical simulations show that variations in $P_{\rm p}$, the degree of
polarization of the spatially resolved planets (thus without any starlight),
with $\alpha$ combined with variations with $\lambda$ could be used to
distinguish the planetary evolutionary phases explored in this paper:

  * •

Phase 1 planets (’Current Earth’) show strong positive (perpendicular to the
reference plane through the star, planet, and observer) polarization around
$\alpha=40^{\circ}$ due to scattering by large water cloud droplets (the
rainbow) and also higher polarization for $\lambda<0.5\leavevmode\nobreak\
\mu$m and $\alpha\approx 90^{\circ}$ due to Rayleigh scattering by the gas
above the clouds.

  * •

Phase 2 planets (’Thin clouds Venus’) polarize light negatively (parallel to
the reference plane) across most phase angles and at visible wavelengths. At
near-infrared wavelengths, they have strong positive polarization around
$\alpha=90^{\circ}$ due to Rayleigh scattering by the small cloud droplets,
and a ’bridge’ of higher polarization from the Rayleigh maximum to the rainbow
angle ($\alpha\approx 40^{\circ}$) with decreasing $\lambda$.

  * •

Phase 3 planets (’Thick clouds Venus’) have predominantly negative
polarization from the visible to the near-infrared, with small regions of
positive polarization for $\lambda<1.0\leavevmode\nobreak\ \mu$m and for
$20^{\circ}\leq\alpha\leq 30^{\circ}$, that are characteristic for the 75%
H2SO4 cloud particles with $r_{\rm eff}=2.0\leavevmode\nobreak\ \mu$m.

  * •

Phase 4 planets (’Current Venus’) yield similar polarization patterns as the
Phase 3 planets, except with more prominent negative polarization for
$10^{\circ}\leq\alpha\leq 30^{\circ}$ and for 0.5 $\mu$m $\leq\lambda\leq
2\leavevmode\nobreak\ \mu$m. Rayleigh scattering by the small cloud particles
produces a maximum of positive polarization around $\alpha=90^{\circ}$ and for
$\lambda>2\leavevmode\nobreak\ \mu$m.

Our simulations of the planetary polarisation $P_{\rm p}$ do not include any
background starlight, and can thus reach several percent to even 20$\%$ for
the ’Current Earth’ (Phase 1) planet in an edge-on orbit (Fig. 8). Whether or
not such polarization variations could be measured depends strongly on the
techniques used to suppress the light of the parent star. If the background of
the planet signal contains a fraction $x$ of the flux of the star, the degree
of polarization of the light gathered by the detector pixel that contains the
planet will equal $(C/(C+x))P_{\rm p}\approx(C/x)P_{\rm p}$ with $C$ the
contrast between the total fluxes of the planet and the star (Eq. 9). For a
Venus-like planet orbiting Alpha Centauri A, $C$ is on the order of 10-9, thus
$x$ should be as small as 10-4 to get a polarisation signal on the order of
10-6, assuming $P_{\rm p}$ is about 0.1. Not only excellent direct starlight
suppression techniques, but also a very high spatial resolution would help to
decrease $x$.

Our simulations further show that temporal variations in the total flux
$F_{\rm u}$ of Alpha Centauri A with a spatially unresolved terrestrial-type
planet orbiting in its habitable zone would be less than 10-12 W/m3. The
degree of polarization $P_{\rm u}$ of the combined star and planet signals,
would show variations smaller than 0.05 ppb. To identify this planetary flux
on top of the stellar flux, a very high-sensitivity instrument would be
required and an even higher sensitivity would be required to subsequently
characterize the planetary atmosphere. Recall that the orbital period of such
a planet, and with that the period of the signal variation and presumably the
stability requirements of an instrument, would be about 0.76 years. Bailey et
al. (2018) computed the polarization signal of spatially unresolved, hot,
cloudy, Jupiter-like planet HD 189733b to be on the order of $\sim$20 ppm.
Because of their large size, Jupiter-like planets, and in particular those in
close-in orbits that receive large stellar fluxes, would clearly be less
challenging observing targets than terrestrial-type planets.

The HARPS instrument on ESO’s 3.6 m telescope includes polarimetric
observations with a polarimetric sensitivity of $10^{-5}$ (snik2010harps).
Planetpol on the 4.2 m William Herschel Telescope (WHT) on La Palma achieved a
sensitivity to fractional linear polarization (the ratio of linearly polarized
flux to the total flux) of $10^{-6}$. While PlanetPol did not succeed in
detecting exoplanets, it did provide upper limits on the albedo’s of a number
of exoplanets (Lucas et al. 2009). The Extreme Polarimeter (ExPo), that was
also mounted on the WHT, was designed to target young stars embedded in
protoplanetary disks and evolved stars surrounded by dusty envelopes, with a
polarimetric sensitivity better than $10^{-4}$ (Rodenhuis et al. 2012). The
HIPPI-2 instrument uses repeated observations of bright stars in the SDSS g’
band for achieving better than 3.5 ppm accuracy on the 3.9-m Anglo-Australian
Telescope and better than 11 ppm on the 60-cm Western Sydney University’s
telescope (Bailey et al. 2020). The POLLUX-instrument on the LUVOIR space
telescope concept aims at high-resolution (R$\sim$120,000) spectropolarimetric
observations across ultra-violet and visible wavelengths (100-400 nm) to
characterize atmospheres of terrestrial-type exoplanets (The Luvoir Team 2019;
Rossi et al. 2021). The EPICS instrument planned to be mounted on the ELT
telescope, is designed to achieve a contrast of 10-10 depending the angular
separation of the objects (Kasper et al. 2010).

In our simulations, we have neglected absorption by atmospheric gases.
Including such absorption would yield lower total fluxes in specific spectral
regions, depending on the type and amount of absorbing gas, its vertical
distribution, and on the altitude and microphysical properties of the clouds
and hazes. Including absorption by atmospheric gases could increase or
decrease the degree of polarization, depending on the amount and vertical
distribution of the absorbing gas and on the microphysical properties of the
scattering particles at various altitudes (see e.g. Trees & Stam 2022; Stam
2008, for examples of polarization spectra of Earth-like planets). While
measuring total and polarized fluxes of reflected starlight across gaseous
absorption bands is of obvious interest for the characterization of planets
and their atmospheres, the small numbers of photons inside gaseous absorption
bands would make such observations extremely challenging.

We have also neglected any intrinsic polarization of Alpha Centauri A.
Measurements of the degree of linear polarization of FKG stars indicate that
active stars like Alpha Centauri A have a typical mean polarization of 28.5
$\pm$ 2.2 ppm (Cotton et al. 2017). This could add to the challenges in
distinguishing the degree of polarization of the planet from that of the star
if the planet is spatially unresolved, although the phase angle variation of
the planetary signal and the direction of polarization of the planet signal
(i.e. usually either perpendicular or parallel to the plane through the star,
the planet, and the observer) could be helpful provided of course that the
instrument that is used for the observations has the capability to measure the
extremely small variations in the signal as the planet orbits its star (on the
order of 10-9).

State-of-the-art instruments with sensitivity to polarization signals down to
$10^{-6}$ (i.e. 1000 ppb) are still a few orders of magnitude away from
detecting variations in polarization signals from spatially unresolved exo-
Earths or exo-Venuses around nearby solar-type stars such as Alpha Centauri A.
To be able to distinguish between the different planetary evolutionary phases
explored in this paper, e.g. between water clouds or sulphuric acid clouds,
variations on the order of 10-9 and hence significant improvements in
sensitivity would be needed if the planets are spatially unresolved. The
variation in the degree of polarization of spatially resolved planets along
their orbital phase should be detectable by instruments capable of achieving
star-planet contrasts of 10-9 and that would allow to distinguish between
water clouds or sulphuric acid clouds. Current high-contrast imaging
instruments manage to directly image self-luminous objects such as young
exoplanets and brown dwarfs in NIR total fluxes at contrasts of 10-2-10-6
(Bowler 2016; Nielsen et al. 2019; Langlois et al. 2021; van Holstein 2021).
Further, instruments such as EPICS on ELT and concepts for instruments on
future space observatories such as HabEx (Gaudi et al. 2020) and LUVOIR (The
Luvoir Team 2019) hold the promise for attaining contrasts of $\sim$10-10.
Reaching such extreme contrasts would make it possible to directly detect
terrestrial-type planets and to use polarimetry to differentiate between exo-
Earths and exo-Venuses.

###### Acknowledgements.

This work was supported by the _Netherlands Organisation for Scientific
Research (NWO)_ through the User Support Programme Space Research, project
number ALW GO 15-37. We thank the referee for the valuable feedback that
improved this paper.

## References

  * Bailey (2007) Bailey, J. 2007, Astrobiology, 7, 320
  * Bailey et al. (2020) Bailey, J., Cotton, D. V., Kedziora-Chudczer, L., De Horta, A., & Maybour, D. 2020, Publications of the Astronomical Society of Australia, 37
  * Bailey et al. (2018) Bailey, J., Kedziora-Chudczer, L., & Bott, K. 2018, Monthly Notices of the Royal Astronomical Society, 480, 1613
  * Bowler (2016) Bowler, B. P. 2016, Publications of the Astronomical Society of the Pacific, 128, 102001
  * Bullock & Grinspoon (2001) Bullock, M. A. & Grinspoon, D. H. 2001, Icarus, 150, 19
  * Cotton et al. (2017) Cotton, D. V., Marshall, J. P., Bailey, J., et al. 2017, Monthly Notices of the Royal Astronomical Society, 467, 873
  * de Haan et al. (1987) de Haan, J. F., Bosma, P., & Hovenier, J. 1987, Astronomy and Astrophysics, 183, 371
  * De Rooij & Van der Stap (1984) De Rooij, W. & Van der Stap, C. 1984, Astronomy and Astrophysics, 131, 237
  * Donahue et al. (1982) Donahue, T., Hoffman, J., Hodges, R., & Watson, A. 1982, Science, 216, 630
  * García-Muñoz et al. (2014) García-Muñoz, A., Pérez-Hoyos, S., & Sánchez-Lavega, A. 2014, Astronomy & Astrophysics, 566, L1
  * Gaudi et al. (2020) Gaudi, B. S., Seager, S., Mennesson, B., et al. 2020, arXiv preprint arXiv:2001.06683
  * Hale & Querry (1973) Hale, G. M. & Querry, M. R. 1973, Applied optics, 12, 555
  * Hansen & Hovenier (1974) Hansen, J. E. & Hovenier, J. 1974, Journal of the Atmospheric Sciences, 31, 1137
  * Hansen & Travis (1974) Hansen, J. E. & Travis, L. D. 1974, Space science reviews, 16, 527
  * Jordan et al. (2021) Jordan, S., Rimmer, P. B., Shorttle, O., et al. 2021, ApJ, 922, 44. doi:10.3847/1538-4357/ac1d46
  * Kane et al. (2020) Kane, S. R., Vervoort, P., Horner, J., & Pozuelos, F. J. 2020, The Planetary Science Journal, 1, 42
  * Karalidi et al. (2011) Karalidi, T., Stam, D., & Hovenier, J. 2011, Astronomy & Astrophysics, 530, A69
  * Karalidi et al. (2012) Karalidi, T., Stam, D., & Hovenier, J. 2012, Astronomy & Astrophysics, 548, A90
  * Kasper et al. (2010) Kasper, M., Beuzit, J.-L., Verinaud, C., et al. 2010, in Ground-based and Airborne Instrumentation for Astronomy III, Vol. 7735, SPIE, 948–956
  * Kasting (1988) Kasting, J. F. 1988, Icarus, 74, 472
  * Keller et al. (2010) Keller, C. U., Schmid, H. M., Venema, L. B., et al. 2010, in Ground-based and airborne instrumentation for astronomy III, Vol. 7735, International Society for Optics and Photonics, 77356G
  * Kemp et al. (1987) Kemp, J. C., Henson, G., Steiner, C., & Powell, E. 1987, Nature, 326, 270
  * Kliore et al. (1985) Kliore, A. J., Moroz, V. I., & Keating, G. M. 1985, Advances in Space Research, 5
  * Knollenberg & Hunten (1980) Knollenberg, R. G. & Hunten, D. M. 1980, Journal of Geophysical Research: Space Physics, 85, 8039
  * Langlois et al. (2021) Langlois, M., Gratton, R., Lagrange, A.-M., et al. 2021, Astronomy & Astrophysics, 651, A71
  * Laven (2008) Laven, P. 2008, Applied optics, 47, H133
  * Lincowski et al. (2018) Lincowski, A. P., Meadows, V. S., Crisp, D., et al. 2018, The Astrophysical Journal, 867, 76
  * Lucas et al. (2009) Lucas, P. W., Hough, J. H., Bailey, J. A., et al. 2009, Monthly Notices of the Royal Astronomical Society, 393, 229
  * Lustig-Yaeger et al. (2019a) Lustig-Yaeger, J., Meadows, V. S., & Lincowski, A. P. 2019a, The Astronomical Journal, 158, 27
  * Lustig-Yaeger et al. (2019b) Lustig-Yaeger, J., Meadows, V. S., & Lincowski, A. P. 2019b, The Astrophysical Journal Letters, 887, L11
  * Markiewicz et al. (2014) Markiewicz, W. J., Petrova, E., Shalygina, O., et al. 2014, Icarus, 234, 200
  * Markiewicz et al. (2018) Markiewicz, W. J., Petrova, E. V., & Shalygina, O. S. 2018, Icarus, 299, 272
  * Nielsen et al. (2019) Nielsen, E. L., De Rosa, R. J., Macintosh, B., et al. 2019, The Astronomical Journal, 158, 13
  * Palmer & Williams (1975) Palmer, K. F. & Williams, D. 1975, Applied Optics, 14, 208
  * Quarles & Lissauer (2016) Quarles, B. & Lissauer, J. J. 2016, AJ, 151, 111
  * Ragent et al. (1985) Ragent, B., Esposito, L., Tomasko, M., et al. 1985, Advances in Space Research, 5, 85
  * Rodenhuis et al. (2012) Rodenhuis, M., Canovas, H., Jeffers, S. V., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 84469I
  * Rossi et al. (2021) Rossi, L., Berzosa-Molina, J., Desert, J.-M., et al. 2021, Experimental Astronomy, 1
  * Rossi et al. (2018) Rossi, L., Berzosa-Molina, J., & Stam, D. M. 2018, Astronomy & Astrophysics, 616, A147
  * Rossi et al. (2015) Rossi, L., Marcq, E., Montmessin, F., et al. 2015, Planetary and Space Science, 113, 159
  * Rossi & Stam (2018) Rossi, L. & Stam, D. M. 2018, Astronomy & Astrophysics, 616, A117
  * Russell et al. (1992) Russell, E. E., Brown, F. G., Chandos, R. A., et al. 1992, Space Sci. Rev., 60, 531
  * Snik et al. (2011) Snik, F., Kochukhov, O., Piskunov, N., et al. 2011, Solar Polarization 6, 437, 237. doi:10.48550/arXiv.1010.0397
  * Stam (2008) Stam, D. 2008, Astronomy & Astrophysics, 482, 989
  * Stam & Hovenier (2005) Stam, D. & Hovenier, J. 2005, Astronomy & Astrophysics, 444, 275
  * The Luvoir Team (2019) Team, L. et al. 2019, arXiv preprint arXiv:1912.06219
  * Titov et al. (2018) Titov, D. V., Ignatiev, N. I., McGouldrick, K., Wilquet, V., & Wilson, C. F. 2018, Space Science Reviews, 214, 1
  * Trees & Stam (2022) Trees, V. J. H. & Stam, D. M. 2022, A&A, 664, A172
  * Tselioudis (2001) Tselioudis, G. 2001, ISCCP Definition of Cloud Types
  * Turbet et al. (2021) Turbet, M., Bolmont, E., Chaverot, G., et al. 2021, Nature, 598, 276
  * van Holstein (2021) van Holstein, R. 2021, PhD thesis, Leiden University
  * Way & Del Genio (2020) Way, M. J. & Del Genio, A. D. 2020, Journal of Geophysical Research: Planets, 125, e2019JE006276