# Effect of multiple scattering on the Transmission spectra and the
Polarization phase curves for Earth-like Exoplanets

Manika Singla Indian Institute of Astrophysics, Koramangala 2nd block,
Bengaluru 560034  Department of Physics, Pondicherry University, Puducherry
605014  Aritra Chakrabarty Data Observatory Foundation, DO; Diagonal Las
Torres N∘2640, Building E, Peñalolén, Santiago, Chile  Facultad de Ingeniería
y Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal las Torres 2640,
Peñalolén, Santiago, Chile  Millennium Institute for Astrophysics, Chile
Sujan Sengupta Indian Institute of Astrophysics, Koramangala 2nd block,
Bengaluru 560034

###### Abstract

It is the most appropriate time to characterize the Earth-like exoplanets in
order to detect biosignature beyond the Earth because such exoplanets will be
the prime targets of big-budget missions like JWST, Roman Space Telescope,
HabEx, LUVOIR, TMT, ELT, etc. We provide models for the transmission spectra
of the Earth-like exoplanets by incorporating effects of multiple scattering.
For this purpose we numerically solve the full multiple-scattering radiative
transfer equations instead of using Beer-Bouguer-Lambert’s law that doesn’t
include the diffuse radiation due to scattering. Our models demonstrate that
the effect of this diffuse transmission radiation can be observationally
significant, especially in the presence of clouds. We also calculate the
reflection spectra and polarization phase curves of Earth-like exoplanets by
considering both cloud-free and cloudy atmospheres. We solve the 3D vector
radiative transfer equations numerically and calculate the phase curves of
albedo and disk-integrated polarization by using appropriate scattering phase
matrices and integrating the local Stokes vectors over the illuminated part of
the disks along the line of sight. We present the effects of the globally
averaged surface albedo on the reflection spectra and phase curves as the
surface features of such planets are known to significantly dictate the nature
of these observational quantities. Synergic observations of the spectra and
phase curves will certainly prove to be useful in extracting more information
and reducing the degeneracy among the estimated parameters of terrestrial
exoplanets. Thus, our models will play a pivotal role in driving future
observations.

planets and satellites: atmospheres — atmospheric effects — transmission
spectroscopy — reflection spectroscopy — radiative transfer — polarization —
scattering

##  1 Introduction

Over 5000 extra-solar planets have been detected till date and many techniques
are being developed to study their atmospheres in detail. Techniques such as
reflection, transmission and emission photometry and spectroscopy(Tinetti,
2006; Seager, 2010) help in characterizing the planetary atmospheres.
Characterizing the terrestrial exoplanets is extremely challenging because of
their very small size and low planet to star flux ratio (Selsis et al., 2008;
Rice, 2014). We will only be able to detect bio-signature on extra-terrestrial
planets unambiguously if we can precisely characterize the Earth-sized planets
which are in the circumstellar habitable zone of their host stars (Huang,
1959, 1960; Whitmire et al., 1991; Kasting et al., 1993; Kopparapu et al.,
2013; Torres et al., 2015; Kane et al., 2016; Fujii et al., 2018; Covone et
al., 2021). Presence of the biosignatures like oxygen, water, methane, etc.
signals the high chances of the presence of life on these planet. The presence
of oxygen and ozone is the result of an extended biomass production through
oxygenic photosynthesis (Owen, 1980; Sagan et al., 1993; Selsis et al., 2002;
Selsis, 2004; Segura et al., 2007; Seager, 2008; Selsis et al., 2008; Scharf,
2009; Grenfell et al., 2014; Fujii et al., 2018; Claudi & Alei, 2019).

When an exoplanet transits the host star, a fraction of the starlight passes
through the planetary atmosphere. The radiation interacts with the atmosphere
through scattering and absorption and provides the spectral fingerprints on
the transmitted flux. Model transmission spectra for terrestrial exoplanets
have previously been presented by Ehrenreich et al. (2006); Kaltenegger &
Traub (2009); Pallé et al. (2009, 2010); Yan et al. (2015); Wunderlich et al.
(2019, 2020); Lin et al. (2021); Gialluca et al. (2021); Madden & Kaltenegger
(2020), etc. In these models, only the total extinction of the incident
stellar flux is considered by using the Beer-Bouguer-Lambert’s law (Tinetti et
al., 2013). These models, albeit include the scattering opacity to the true
absorption opacity, do not incorporates the angular distribution of the
transmitted photons due to scattering in the planetary atmosphere. Sengupta et
al. (2020) have considered the in and out scattering for the hot Jupiter like
exoplanets while modeling the transmission spectra. In the present work, we
have calculated the transmission spectra of the Earth-like planets by solving
the multiple-scattering radiative transfer equations by using discrete space
theory (Peraiah & Grant, 1973), following Sengupta et al. (2020). We
demonstrate that diffuse transmission radiation due to scattering can affect
the overall broad natures of the transmission spectra, especially when the
single-scattering albedo increases in the presence of clouds.

When incoming stellar flux hits the solid planetary surface, some fraction of
it gets reflected, absorbed or transmitted depending on the wavelength and the
angle of incidence of the incident stellar radiation (Seager, 2010; Selsis et
al., 2008). Study of the reflection spectra and phase curves can add to the
information that is obtained from the transmission spectroscopy. Moreover,
these techniques can be used to characterize the planets with arbitrary
orbital alignment with respect to the line of sight. Previously, Sagan et al.
(1993) have obtained the reflection spectra of the Earth using the
observations by Galileo satellite. Also, Kawashima & Rugheimer (2019); Batalha
et al. (2019); Segura et al. (2005); Kaltenegger et al. (2007); Kitzmann et
al. (2010a, b); Rugheimer et al. (2013); Rugheimer & Kaltenegger (2018), among
many have calculated the reflected spectra for Earth-like exoplanets and also
studied the effects of clouds on the spectra. In this paper, we present new
model reflection spectra for the Earth-like exoplanets for the sake of
completeness of our investigation on the atmospheres of these planets. These
model spectra have been calculated by using the same numerical method
mentioned above.

Several polarimetric techniques are also increasingly being used for the study
of exoplanetary atmospheres. Polarimetric studies for planets were initiated
with the observation of the Solar system objects and are still being continued
(Coffeen, 1969; Coffeen & Gehrels, 1969; Hall & Riley, 1974; Michalsky &
Stokes, 1977; West et al., 1983; Joos & Schmid, 2007, etc.). Mallama (2009)
characterized the terrestrial exoplanets based on the phase curves of some
solar system planets. Stam et al. (2003); Stam (2003); Rossi et al. (2018),
etc. studied the polarization spectra for the extra-solar planets. By studying
the polarization profiles, we can extract information about the atmospheric as
well as physical properties such as cloud distribution, mean size of cloud
particulate as well as rotation-induced oblateness etc. as demonstrated by
Sengupta & Krishan (2001); Sengupta & Maiti (2006); Sengupta (2008); Sengupta
& Marley (2009, 2010, 2011, 2016); Sengupta (2016) for the case of brown
dwarfs and self-luminous exoplanets. In addition, phase dependent polarization
of reflected planetary radiation can help understanding more atmospheric
composition including biosignatures, surface constitutents like ocean, ice,
forest etc and thus the evidence of a habitable environment in exoplanets
(Zubko et al., 2008; Kedziora-Chudczer & Bailey, 2010; Rossi et al., 2017).
The traces of exo-moons are also being searched by means of polarization
(Sengupta & Marley, 2016; Molina et al., 2017, 2018).

The reflected light can be polarized because of the various scattering
processes which depend on the types of scatterers and the scattering mechanism
(Seager, 2010). Linear polarization signals from the starlight reflected from
the horizontally inhomogenous Earth-like planets is presented in Karalidi &
Stam (2012). Groot et al. (2020); Rossi & Stam (2018) studied the linearly or
circularly polarized signals from the sunlight reflected from the model Earth.
Polarization signals from starlight reflected by the Earth-like exoplanets
have been studied by Stam (2008); Fauchez et al. (2017); Wei & Zhong-quan
(2017); Muñoz (2018); Sterzik et al. (2019); Patty et al. (2021); Gordon et
al. (2022), among others. Wang et al. (2019) have used PARASOL data to
calculate the variation of the disk-integrated polarization. Karalidi et al.
(2011, 2012a); Michael F. Sterzik & Manev (2020) have modeled the polarized
signal from the clouds on exoplanets and Zugger et al. (2010, 2011) from the
exoplanetary oceans and atmospheres. Stam & Hovenier (2005) have estimated the
errors in the calculated phase functions and albedos of planets if
polarization is neglected.

Detecting the polarization signals of the reflected radiation is, however,
extremely difficult because of the very low signal to noise (S/N) ratio as
compared to that of the Solar-system planets. Some of the upcoming telescopes
will unravel the polarimetric properties of the Earth and the extra-solar
planets. LOUPE (Lunar Observatory for Unresolved Polarimetry of the Earth), a
small spectropolarimeter is being developed to observe the Earth from the Moon
as an exoplanet (Klindžić et al., 2021; Karalidi et al., 2012b) and also ELF
(Exo-Life finder) Telescope (Berdyugina et al., 2018) will be used for the
direct detection of exoplanet biosignatures. Other big-budget missions like
HabEx, LUVOIR, Roman Space Telescope, etc. will also have imaging polarimetric
facility. These missions will additionally have coronographic instruments
onboard which will allow us to detect polarimetric signals from the exoplanets
in the habitable zones directly.

Most of the above mentioned polarization models either use Monte Carlo method
or solve 1-D vector radiative transfer equations and invoke generalized
spherical harmonic expansion to integrate the scattering polarization over the
visible disk. In the present work we calculate the azimuth-dependent intensity
vectors by solving the 3-D vector radiative transfer equations. The disk
integrated flux and polarization are estimated by integrating the intensity
vector at each local point over the illuminated disk. The 1-D version of the
same numerical code has also been used to solve the radiative transfer
equations in their vector form for the calculation of polarized spectra of
rotation-induced oblate self-luminous exoplanets and cloudy brown dwarfs
(Sengupta & Marley, 2009, 2010; Marley & Sengupta, 2011; Sengupta, 2016;
Sengupta & Marley, 2016; Sengupta, 2018). However, in order to calculate the
polarization over the rotation-induced oblate disk of the object, spherical
harmonic expansion method was used in those work. This scalar version of the
same code has also been used to calculate the transmission spectra for the hot
Jupiters (Sengupta et al., 2020; Chakrabarty & Sengupta, 2020). Chakrabarty &
Sengupta (2021) have presented the polarization models for hot-Jupiters by
solving 3D vector radiative transfer equations. In the present work we employ
the same methodology to calculate the polarization for the Earth-like
exoplanets.

In the next section, we discuss about the necessary inputs used to calculate
the transmission spectra and the reflected spectra for Earth-like exoplanets.
In sections 3.1 and 3.2, we present the results of the transmission and the
reflection spectra. Vector phase curve models are presented in section 3.3. In
section 4, we analyze and discuss the results and finally, in the last
section, we present the conclusions of this work.

##  2 Atmospheric models for Earth-like exoplanets

We present models for transmission spectra, reflection spectra and the phase
curves for geometric albedo and linear polarization for the Earth-like
exoplanets orbiting around Sun-like stars. For calculating the reflection and
the transmission spectra as well as the scattering polarization, we take the
atmospheric chemical composition for the modern Earth-like exoplanets from
Kawashima & Rugheimer (2019) and opacity data , i.e. absorption and scattering
cross-sections for all the molecules that have been taken in the atmospheric
composition of the planet, from the database for PICASO (Batalha et al.,
2020). The observed temperature-pressure profile of the Earth’s atmosphere is
considered for the calculations. We consider two types of atmosphere in all of
our model calculations: cloudy and cloud-free. In the case of cloudy
atmospheres, we consider very thin clouds or haze and we have used an
approximate Rayleigh model to express the effect of these clouds/haze
following Sing et al. (2016); Kempton et al. (2017), etc.

In case of transmission spectra, we have included thin clouds with scattering
cross-sections ($\sigma$) equal to 100, 200 and 400 times the scattering
cross-section ($\sigma_{R}$) of the dominant atmospheric constituent i.e.
nitrogen in this case. The cloud deck and base have been fixed at 5x102 Pa and
5x103 Pa. For the case of reflection spectra, we have considered the cloud
position between the pressure levels of 1x103 Pa and 5x104 Pa with a
scattering cross-section equal to 400 times the scattering cross-section of
nitrogen gas. The cloud position (considering 100 % coverage) for the case of
reflected spectra is kept at deeper layers of the atmosphere while for the
case of the transmission spectra, the clouds are considered at the upper
layers of the atmosphere. It is because we can probe only the outer atmosphere
by transmission spectra. Also, we will probe complementary portions of the
atmosphere in terms of the altitude. A terrestrial exoplanet usually should
have water cloud in the upper atmosphere. For large cloud particulates, Mie
scattering phase matrix should be appropriate to describe the angular
distribution of photon before and after scattering. But for small size of
cloud particles, Rayleigh phase matrix serves the purpose reasonably well. In
the present work we have used Rayleigh phase matrix for water droplets
(Sengupta & Maiti, 2006).

##  3  Results

###  3.1  The Transmission Spectra

Using the atmospheric models described in Section 2, we have presented models
of transmission spectra of the Earth-like exoplanets. Studying the absorption
lines on the transmission spectra can directly tell us about the volatiles and
biosignatures present in their atmospheres. But such interpretation requires
accurate models of the broad continua of the spectra, especially in the
visible wavelengths and in the presence of clouds. Also, an accurate model can
help us understand how the presence of clouds can suppress the absorption
lines since detecting these absorption lines of the Earth-like planets is
already extremely challenging. Following Sengupta et al. (2020), we solve the
multiple-scattering radiative transfer equation for diffuse reflection and
transmission which is given as,

$\mu\frac{dI(\tau,\mu,\lambda)}{d\tau}=I(\tau,\mu,\lambda)-\frac{\omega}{2}\int_{-1}^{1}{p(\mu,\mu^{\prime})I(\tau,\mu^{\prime},\lambda)\text{d}\mu^{\prime}}-\frac{\omega}{4}Fe^{-\tau/\mu_{0}}p(\mu,\mu_{0}).$
(1)

Here, I($\tau$,$\mu$,$\lambda$) is the specific intensity of the transmitted
radiation along our line of sight. $\omega$ is the single scattering albedo, F
is the incident stellar flux along our line of sight (along the direction
-$\mu_{0}$), p($\mu$,$\mu^{\prime}$) is the scattering phase function and
$\tau$ is the optical depth along the line of sight. The detail formalism and
numerical technique are described in (Sengupta et al., 2020).

Surface albedo of the planet is not considered in this case as transmission
spectra predominantly convey the information of the upper atmosphere. Figure 1
presents the transmission depth with and without diffusion of radiation due to
scattering. When the diffusion due to scattering is not considered, especially
in the longer wavelengths where the values of $\omega$ are extremely low
($\omega$ $\approx$ 0), we can use the Beer-Bouguer-Lambert’s law instead of
solving the radiative transfer equations. Note that even when the diffuse
radiation due to scattering is not incorporated, total atmospheric optical
depth is determined by both the absorption and the scattering opacities.

Figure 1: Transmission depth of the Earth-like exoplanets with (solid) and
without (dashed) diffuse scattering for cloudy and cloud-free atmospheres (see
Section 2). Scattering opacity is however, included even in the case for
cloud-free atmosphere.

###  3.2  The Reflection Spectra

While orbiting the host star, an exoplanet reflects a part of the incident
starlight depending on its orbital phase, orbital inclination, and of course,
the atmospheric and surface constituents. The study of the reflected light
from the planets helps us to probe deeper into their atmospheres, detect their
surface as well as cloud properties. It also reduces the degeneracy among the
estimated parameters when combined with the results from the study of
transmission spectra.

We solve the scalar one-dimensional multiple-scattering radiative transfer
equation to model the geometric albedo (see, for example, Batalha et al.
(2019)) and the full-phase (fully illuminated disk) reflection spectra of the
Earth-like planets using the atmospheric model explained in Section 2. The
equation is somewhat similar to Equation 1 but follows a different geometry
(radial geometry) to calculate the layerwise optical depths and single-
scattering albedos. Surface albedo, which depends on the surface composition,
contributes to the overall reflectivity of the rocky planets. For example, if
the whole surface is covered with snow, the surface albedo is the highest,
i.e. around 0.9 and hence it contributes heavily to the geometric albedo but
if the whole surface is covered with ocean, the surface albedo is much less,
around 0.06, thus contributing much less to the geometric albedo (Kaltenegger
et al., 2007). For the present Earth-like exoplanets, we take the surface
albedo to be 0.14 at all wavelengths, where the surface components are 70%
ocean, 2% coast, and 28% land, which in turn, is divided into 30% grass, 30%
trees, 9% granite, 9% basalt, 15% snow, and 7% sand (Kaltenegger et al.,
2007). The surface reflection is assumed to be Lambertian i.e. isotropic in
nature. We calculate the surface albedo by summing all the components’ albedo
multiplied by their respective fraction of the planetary surface coverage.
Figure 2 shows the reflection spectra and the geometric albedo of cloudy and
cloud-free Earth-like planets with the annotations of oxygen and water
absorption lines.

Figure 2: Reflected spectra (top panel) and Geometric albedo (bottom panel)
for the Earth-like exoplanets orbiting around the Sun-like stars at a
resolution of 300. The blue plot is for the clear sky while the orange plot is
for the cloudy atmosphere. The absorption lines of O2 and H2O are shown.

###  3.3  The Phase Curve Models

The reflected light observable from the planets depends on their orbital phase
and the study of these phase curves conveys valuable information about the
atmospheres and surfaces of the Earth-like exoplanets. The orbital phase
($\alpha_{\rm orb}$) is 0o when the maximum area of the illuminated disk is
viewed and 180o when the minimum or no illuminated part of the planetary disk
is viewed. However, modeling these phase curves is cumbersome and requires us
to invoke three-dimensional radiative transfer models as explained by
(Chakrabarty & Sengupta, 2021). Here, we solve the 3-D vector radiative
transfer equation to calculate both the albedo (total reflectivity of the
disk) phase curves and the disk-integrated polarization phase curves.

The partial illumination of a planetary disk yields net non-zero disk-
integrated scattering polarization of the reflected light. A study of this
polarization can provide us information about the atmospheric clouds in
detail, surface composition, and also the light absorbers present in the
atmospheres (Chakrabarty & Sengupta, 2021). We assume the incident starlight
to be unpolarized and the polarization of the planet’s reflected light is
solely caused by the scattering process. We ignore polarization due to strong
magnetic field if any.

The state of polarization of each beam of ray after scattering is determined
by the scattering phase matrices which depend on the scattering mechanism. We
follow the methods prescribed by Chakrabarty & Sengupta (2021) to solve the
vector radiative transfer equations and calculate the phase dependent
reflected flux and the polarization ($P$) averaged over the illuminated
planetary disk. The corresponding atmospheric model is explained in Section 2.

Figure 3: Effect of surface albedo on the phase curves of albedo (or flux
ratio, F($\alpha_{\rm orb}$)/F0) and net polarization ($P$) integrated over
the illuminated planetary disk at a wavelength of 0.6 $\mu$m and at an orbital
inclination = 90o. We have used surface albedo 0.14 for our calculations. And
0.9 surface albedo is for the case of snowball (fully covered with snow)
planet.

We studied the effect of surface albedo on the overall disk albedo and
polarization as depicted in Figure 3. For the rest of the calculations, we
considered the value of surface albedo to be 0.14 as explained in Section 3.2.

Figure 4-5 show the the total albedo and the disk polarization ($P$) at
$\lambda=0.6~{}\mu$m and $\lambda=1~{}\mu$m for both the cloud-free and cloudy
atmospheres , by considering multiple scattering of the incident radiation.
Figure 6 shows the same for visible wavelength ($\lambda=0.6~{}\mu$m),
considering only single scattering. These phase curves can be detected with
the next-generation polarimetric missions which will use their coronagraphic
instruments to resolve the Earth-like exoplanets from their host stars. The
observable flux ratio i.e. the ratio of the observable reflected flux from the
planet to the observable starlight is shown in the figures. This indicates the
contrast required by those instruments to directly detect the reflection
spectra from such planets. Since the flux ratios are in the order of parts per
billion (ppb), detecting the polarization of the planets without resolving
them separately amidst the stellar glare will be impossible with the current
technology and hence not shown in the figures.

Figure 4: The phase curves of albedo (or flux ratio, F($\alpha$)/F0) and
polarization ($P$) integrated over the illuminated disk at a wavelength of 0.6
$\mu$m (visible) and at orbital inclinations angle 90o (solid) and 45o
(dashed) for both cloud-free and cloudy atmospheres.  Figure 5: Same as figure
4 but at a wavelength of 1.0 $\mu$m (near infrared).  Figure 6: Same as figure
4 but with single scattering of the incident radiation.

##  4  Analysis and Discussion

The transmission depth for modern Earth-like exoplanets orbiting around Sun-
like stars at wavelengths up to 2.0 $\mu$m is shown in Figure 1. We can see
that the transit depth reduces with the inclusion of diffuse radiation due to
scattering as explained by Sengupta et al. (2020). The transmission depth
increases with the inclusion of clouds as the cloud particles block the
transmitted flux through the atmosphere. Clouds also suppress the absorption
features of the molecules in shorter wavelengths. The effect of diffusion due
to scattering on the broadband continuum can be significant with respect to
the levels of the individual absorption features, especially in the presence
of atmospheric clouds. This calls for more accurate modeling of the
transmission spectra by solving the complete radiative transfer equation.
Otherwise detecting the features of the biosignatures of the Earth-like
planets can be confusing and may be erroneous. Of course, at longer wavelength
regions, all the plots are found to merge because the effect of scattering is
negligible.

The relatively stronger absorption lines such as O2, H2O are easily detectable
in the reflection spectra (see Figure 2). Clearly, the geometric albedo
increases with the decrease in wavelength because of the dominance of Rayleigh
scattering ($\propto\frac{1}{\lambda^{4}}$) at shorter wavelength. Also,
because of increased back scattering of the incident stellar radiation, the
presence of clouds significantly increases the geometric albedo and hence the
reflected flux at the visible wavelength region. .

Figure 3 shows the variation of the albedo and the disk-averaged polarization
($P$) for different orbital phase at a fixed wavelength ($\sim$ 0.6$\mu$m) and
an orbital inclination angle of 90o (edge-on view) for different surface
albedos. The value of the surface albedo depends on the surface composition of
the planet i.e. the amount of ocean cover, land cover, trees, ice, etc. The
surface albedo of 0.9 corresponds to the case where the whole surface of the
planet is covered with snow. The intermediate surface albedo of 0.05 in the
figure corresponds to the case where almost the whole surface is covered with
ocean and 0.1 corresponds to the case where half of the surface is covered
with ocean and the remaining half with trees and grass. As the surface
reflection is assumed to be Lambertian, it completely depolarizes the light
that is reflected in the upward direction from the surface of the planet,
i.e., from the bottom of the atmosphere (BOA) (Rossi et al., 2018). As a
result, with an increase in the surface albedo, the overall albedo increases
but the polarization ($P$) decreases. $P$ is evidently found to peak at around
90o orbital phase. The phase dependent polarization profile presented here is
consistent with that presented by (Sengupta & Maiti, 2006; Stam et al., 2003).

Figures 4 and 5 show the phase dependent light curves at wavelengths of 0.6
$\mu$m and 1.0 $\mu$m respectively for a planet with two orbital inclinations,
e.g., 45o and 90o Evidently, the peak-to-peak fluctuations of the light curves
decrease with a decrease in the orbital inclination. Moreover, these figures
also demonstrate the effects of clouds. The presence of clouds and non-zero
surface albedo increase the total albedo of the disk as expected. Figure 6
show the phase dependent light curve at a wavelength of 0.6 $\mu$m for the
same orbital inclinations by considering only single scattering at each
atmospheric layer. The effect of clouds on the albedo is the same as the case
for multiple scattering. But clouds do not affect the disk averaged
polarization because angle of scattering at each layer is the same. We note
that the peak polarization (i.e. at 90o phase angle) is 1 for clear sky as
well as cloudy atmosphere. This happens because we have approximated the
effects of clouds with the Rayleigh phase matrix and for the case of Rayleigh
scattering, the degree of single-scattering polarization is 1 at 90o phase
angle (see Fig. 3 of Chakrabarty & Sengupta (2021)). Basically, the single-
scattering approximation overestimates the observable polarization and
underestimates the total albedo.

Although, we see opposite behaviour of disk averaged polarization with the
clouds in case of hot-Jupiters (see Fig. 15 of Chakrabarty & Sengupta (2021)).
For hot-Jupiters, the polarization depends on single scattering albedo, but
for the Earth-like exoplanets, it depends on single scattering albedo as well
as the surface albedo as explained in the next paragraph.

However, to understand the total degree of polarization for a planet with a
Lambertian surface with non-zero surface albedo, we divide the total upward
(towards us) radiation at the top of atmosphere (TOA) into two streams: (i)
the downward incident radiations that get scattered back to the upward
direction and get polarized, especially at disk locations away from the
substellar point (e.g., Stam et al., 2006; Chakrabarty & Sengupta, 2021), and
(ii) the upward radiations from BOA that get transmitted in the same direction
which are predominantly unpolarized. For a cloud-free atmosphere, as the
wavelength increases, the intensity of stream-i decreases since the single-
scattering albedo of the atmosphere decreases, whereas the intensity of
stream-ii remains almost constant as we have assumed the same surface albedo
at both the wavelengths. Hence, the relative dominance of stream-ii increases
at higher wavelengths, and the polarization ($P$) drops significantly at
$\lambda=$1 $\mu$m compared to that at $\lambda=$0.6 $\mu$m. For the same
reason, the total disk albedo at 0.6 $\mu$m is only slightly higher than that
at 1 $\mu$m for the cloud-free case which is also suggested by Figure 2.

The effect of clouds is twofold. For a very low value of surface albedo, as in
the case of the gaseous planets, the presence of clouds increases the
depolarization of radiations due to multiple scattering as the single-
scattering albedo of the atmosphere increases (Chakrabarty & Sengupta, 2021).
This causes the total disk polarization to drop with the presence of clouds
while the disk albedo rises (see Figures 15-17 of Chakrabarty & Sengupta
(2021)). On the other hand, for a rocky planet with relatively high surface
albedo, we find that stream-ii dominates over stream-i at the TOA in the
absence of any cloud particle which causes a low value of disk polarization.
However, the presence of a cloud layer tends to strengthen stream-i by
reflecting back more of the downward radiations to the upward direction and
weaken stream-ii by reflecting them back in the downward direction. As a
result, the presence of clouds increases the overall disk-integrated degree of
polarization and also increases the albedo of the disk. Thus, polarization
serves as an indicator of the presence of clouds and helps us understand the
thickness and the properties of the cloud layers when combined with the scalar
spectrum of the planet.

##  5 Conclusions

This paper focuses on the various existing techniques that can be used
synergically to characterize Earth-like exoplanets. We have demonstrated how
the inclusion of diffuse radiation due to scattering can improve the model of
transmission spectra over the traditional approach of invoking the Beer-
Bouguer-lambert’s law. The difference is significant with respect to the
molecular absorption features that can serve as biosignatures. We have also
presented the reflection spectra including non-zero globally averaged surface
albedo and these spectra also carry information about the biosignatures and
the volatiles. However, obtaining the transmission or reflection spectra from
such small-sized planets with thin atmospheres is extremely challenging at
present but will be possible in the era of the upcoming big-budget missions
like HabEx, LUVOIR, TMT, ELT, etc. Our models will play a significant role in
the habitability study of the Earth-like planets using transmission,
reflection spectra and phase dependent linear polarization.

In this paper we demonstrate that atmospheric cloud can significantly affect
both the transmission and reflection spectra. The use of polarimetry can allow
us to study the properties of the clouds in great detail and reduce the
overshadowing effects of clouds. The coronagraphic instruments of those
upcoming missions in the upcoming decades will be able to directly image the
Earth-like planets in the habitable zones around their host stars. Leveraging
the polarimetric instruments in conjunction with these coronagraphic
instruments, we will be able to conduct a phase curve study of such planets.
Our vector phase curve models show the contrast required to resolve these
planets from their host stars and also predict the maximum observable
reflected flux and degree of polarization.

Evidently, the surface albedo and the clouds significantly dictate the nature
of the phase dependent light curves. Our approximate globally averaged
Lambertian representation of the surface albedo has allowed us to simplify the
calculations to some extent and develop an understanding of the effect of
surface albedo on the reflection spectra and phase dependent light curves.
However, in our upcoming work, we will consider individual surface components
and their wavelength-dependent reflection matrices to calculate the spectra
and the light curves more accurately.

Finally, our models should be useful in designing the instruments onboard the
upcoming missions, selecting the science targets, as well as extracting the
planetary properties from the spectra and phase curves, once obtained.

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