New models of reflection spectra for terrestrial exoplanets: Present and prebiotic Earth orbiting around stars of different spectral types

TL;DR

This paper develops new reflection spectra and albedo models for Earth-like exoplanets, including prebiotic conditions, across different star types, aiding future habitability assessments.

Contribution

It introduces comprehensive models of reflected spectra for modern and prebiotic Earth-like exoplanets considering various atmospheric and surface compositions across multiple star types.

Findings

Prebiotic Earth-like exoplanets scatter more starlight in optical wavelengths.

Cloud presence affects transmission spectra significantly.

Models validated against existing published spectra.

Abstract

In order to recognize a habitable exoplanet from future observed spectra, we present new model reflected spectra and geometric albedo for modern and prebiotic (3.9 Ga) Earth-like exoplanets orbiting within the habitable zone of stars of spectral types F, G, K and M. We compute this for various atmospheric and surface compositions of the planets. Molecules that are potential biosignatures and act as greenhouse agents are incorporated in our model atmosphere. Various combinations of solid and liquid materials such as ocean, coast, land consisting of trees, grass, sand or rocks determine the surface albedo of the planet. Geometric albedo and model reflected spectra for a set of nine potential habitable planets, including Proxima Centauri b, TRAPPIST-1d, Kepler-1649c and Teegarden's Star-b, are also presented. We employ the opacity data derived by using the open-source package Exo-Transmit and adopt different atmospheric Temperature-Pressure profiles depending on the properties of the terrestrial exoplanets. The model reflected spectra are constructed by numerically solving the multiple scattering radiative transfer equations. We verified our model reflected spectra for a few specific cases by comparing with those published by other researchers. We demonstrate that prebiotic Earth-like exoplanets and present Earth-like exoplanets with increased amount of greenhouse gases in their atmospheres scatter more starlight in the optical. We also present the transmission spectra for modern and prebiotic Earth-like exoplanets with cloudy and cloudless atmospheres.

Tables

Table 1. Table 1: Molecular abundance for the Earth’s atmosphere (Sagan et al., 1993 ) .

Molecule	Abundance (volume mixing ratio)
N2	0.78
O2	0.21
H2O	0.03 - 0.001
Ar	9$\times$10-3
CO2	3.5$\times$10-4
CH4	1.6$\times$10-6
N2O	3$\times$10-7
O3	10-7–10-8

Table 2. Table 2: Surface Bond albedo for various surface compositions considered in our calculations for the modeled reflected spectra.

S.No.	Surface composition	Surface albedo
1	Ocean cover (100$\%$)	0.06
2	Ocean (50$\%$), Trees and grass (50$\%$)	0.1
3	Present Earth-like	0.14
4	Prebiotic Earth-like	0.16
5	Ocean (83$\%$) and snow (17$\%$)	0.2
6	No solid or liquid surface	0

Table 3. Table 3: T eq,max is the maximum temperature at the top of the atmosphere of the planet and T surf,max is the maximum temperature at the bottom of the atmosphere after considering greenhouse effect. A max is the maximum possible Bond albedo, ESI is the Earth similarity index, R p and M p are the radius and mass of the planet and a is the orbital separation. [1] https://exoplanets.nasa.gov/exoplanet-catalog/ [2] https://phl.upr.edu/projects/earth-similarity-index-esi *estimate value

Planet	Teq,max	Tsurf,max	ABond,max	R${}_{}{}^{\left[1\right]}{}_p{}^{}$	M${}_{}{}^{\left[1\right]}{}_p{}^{}$	a[1]	ESI[2]
	(K)	(K)		(R⊕)	(M⊕)	(au)
Kepler-442b	257	290	0.216	1.34	2.36	0.409	0.84
Kepler-62e	291	328	0.52	1.61	36	0.427	–
Kepler-22b	286	323	0.486	2.33	36	0.849	–
TOI-700d	269	304	0.347	1.144	1.57	0.1633	0.93
Kepler-1649c	296	334	0.55	1.06	1.2	0.0649	0.90
Teegarden b	289	326	0.51	1.02∗	1.05	0.0252	0.95
Proxima b	258	292	0.2289	1.08∗	1.27	0.0485	0.87
TRAPPIST-1d	286	323	0.49	0.788	0.388	0.0223	0.90
TRAPPIST-1e	250	282	0.12	0.92	0.692	0.0292	0.85

Equations

$$\begin{array}{r}\mu \frac{dI(\tau ,\mu ,\lambda )}{d\tau }=I(\tau ,\mu ,\lambda )-\frac{\omega }{2}{\int }_{-1}^{1}p(\mu ,{\mu }^{\prime })I(\tau ,{\mu }^{\prime },\lambda )\text{d}{\mu }^{\prime }\\ -\frac{\omega }{4}F(\lambda )e^{-\tau (\lambda )/{\mu }_0}p(\mu ,{\mu }_0)\end{array}$$

$$p(\mu ,{\mu }^{\prime })=\frac{3}{4}[1+{\mu }^2{\mu }^{\prime 2}+\frac{1}{2}(1-{\mu }^2)(1-{\mu }^{\prime 2})],$$

$${\mu }_{j+1/2}=\sum _{k=1}^j{c}_k,j=1,2,...,\mathrm{J}$$

$${\int }_{-1}^{+1}f(\mu )d\mu =\sum _{j=1}^m{a}_{j}f({\mu }_j)$$

$$a_j=\frac{1}{P_m^{{}^{\prime }}({\mu }_j)}{\int }_{-1}^{+1}\frac{P_m(\mu )}{\mu -{\mu }_j}d\mu$$

$$T_{\mathrm{eq}}^4=(1-A){\frac{R_s}{2a}}^2{T}_{\mathrm{eff}}^4$$

$$T_{\mathrm{surf}}^4=T_{\mathrm{eq}}^4(1+\frac{3}{4}{\tau }_{\mathrm{g}})$$

$$\begin{gathered} T_1=-2.83h+273; \\ {T}_2=-3h+290.1 \end{gathered}$$

$$D=1-\frac{F_{trans}}{F_{star}}$$

$$F_{trans}=[1-(\frac{R_{pl,atm}}{R_{star}}{)}^2]F_{star}+F_{atm}$$

$$I(\lambda )=I_0(\lambda )e^{-{\tau }_{\lambda }/{\mu }_0}$$

$$\tau (\lambda ,z)=2{\int }_0^{l(z)}\chi (\lambda ,z)\rho (z)dl$$

$$l(z)=\int dl=\sqrt{(R_p+z_{max}{)}^2-(R_p+z{)}^2}$$

Full text

††journal: New Astronomy

\affiliation

[inst1]Indian Institute of Astrophysics, Koramangala, Bengaluru, 560034, India

\affiliation

[inst2]Department of Physics, Pondicherry University, Kalapet, Puducherry, 605014, India

1 Introduction

Since the discovery of 51 Pegasi b (Mayor and Queloz, 1995), more than 5000 exoplanets candidates have been discovered using various methods, yet a little has been investigated about their atmospheres. According to Bryson et al. (2020), around half of the Solar-type stars in our Galaxy might host rocky and potentially habitable planets within their habitable zones. But still we are far from finding any exoplanet that may have an ambient environment similar to that of the Earth. We will be a step closer to finding out such planets if we can characterize the atmospheres of terrestrial exoplanets (Selsis, 2004; Morley et al., 2015; Kaltenegger, 2017; Alonso, 2018; Kopparapu et al., 2020; Quanz et al., 2021).

The classical circumstellar habitable zone is defined as the region around a star where the surface temperature of a planet is appropriate for water to exist in liquid state (Huang, 1959, 1960; Whitmire et al., 1991; Kasting et al., 1993; Kopparapu et al., 2013). A few of the planets discovered by the NASA’s Kepler space mission, possibly located in the habitable zone of their host stars (Covone et al., 2021), are also of great interest. The recent TESS (Transiting Exoplanet Survey Satellite) discoveries include Super-Earth and Sub-Neptunes orbiting around HD 108236 (G3V), GJ 3929 b, which is a hot Earth sized planet orbiting around M3.5V star (Daylan et al., 2021; Kemmer et al., 2022). Many Earth-like planets, including TOI-700d, which lie in their host star’s habitable zone, discovered by TESS are also important (Kaltenegger et al., 2021). Many potentially habitable exoplanets have also been discovered by RV Spectrographs (Jurgenson et al., 2016; Wildi et al., 2017, and many more).

When stellar radiation is incident on the surface of a planet, parts of it get reflected, absorbed and transmitted depending on the wavelength of the radiation and the angle of incidence of the stellar flux (Seager, 2010; Perryman, 2018). The planetary reflected spectra are generated by the fraction of the incident stellar radiation reflected along our line of sight (Selsis et al., 2008). The interaction of the incident stellar radiation with the matter in the upper atmosphere of the planet introduces signatures of the atmospheric chemical composition in the reflected spectra. However, when an exoplanet transits across the host star, a tiny portion of the stellar disk is blocked yielding into a reduction in the stellar flux. At the same time, a fraction of star-light passes through the planetary atmosphere, if any, and brings the information on the chemical composition of the planetary atmosphere. This is known as the transmission spectrum (Pallé et al., 2009; Wunderlich et al., 2019). The signatures of the molecules present in the planetary atmosphere are revealed in the absorption features of reflected and transmitted spectra (Tinetti et al., 2013). If a combination of biosignatures, such as oxygen, ozone, water and methane, were detected in the atmosphere of rocky exoplanets in habitable zone, there would be a high possibility that the planet harbours life (Owen, 1980; Sagan et al., 1993; Selsis, 2004; Scharf, 2009; Grenfell et al., 2014; Fujii et al., 2018; Claudi and Alei, 2019).

Previous studies suggested that the potentially habitable planets can orbit stars of F, G, K and M spectral types (Selsis, 2000). According to Kasting et al. (1993), the most potentially habitable planets orbit around late F, G and early K-type main-sequence stars. Stars whose spectral type is earlier than F0 have less than 2 Gyr main sequence lifetimes and hence are very less likely to have planets that can harbour life (Segura et al., 2003). On the other hand, M dwarfs have much less probability of having life supporting planets orbiting around them because their habitable zones are much nearer and narrower and so the planets in the habitable zone are exposed to strong UV radiation and strong flares (Huang, 1959, 1960; Hart, 1979). Also most planets in the inner habitable zone of M dwarfs are tidally locked (Kasting et al., 1993; Segura et al., 2003; Martinez-Rodriguez et al., 2019) and the permanent day or night side of the planet may have hostile environment for life. Nearly 70$\%$ of all stars in our Galaxy are M dwarfs and rocky planets orbiting M dwarfs may be the most common in the universe (Henry et al., 2006; Shields et al., 2016; Meadows et al., 2018; Lin and Kaltenegger, 2020; Reylé et al., 2021; Sabotta et al., 2021). Therefore, it is important to include the exoplanets orbiting M dwarfs as well in any investigation and probe.

Exoplanets similar to the prebiotic Earth ($\sim$3.9 Ga) can also be the potential candidates for supporting life on them. Prebiotic Earth contained no free molecular oxygen but carbon dioxide and nitrogen as the most dominant gases in their atmospheres (Rugheimer et al., 2015). Prevalent oxygenation of the Earth’s atmosphere took place somewhere between 2.45 Ga and 2.32 Ga, which is known as the Great Oxidation Event GOE (Holland, 2002; Bekker et al., 2004; Guo et al., 2009). Discovery of the biomarkers in sedimentary rocks (banded iron formation) with age 2.7 Ga to 2.8 Ga, which are characteristic of photosynthetic cyanobacteria, indicates the appearance of O2 in the Earth’s atmosphere (Brocks et al., 1999). Before this period, life survived through anoxygenic photosynthesis process. The second oxygenation event took place around 0.8 Ga to 0.5 Ga, which is known as Neoproterozoic Oxygenation Event (NOE). During that period, oxygen probably accumulated to the levels that are required for the animal life (Shields-Zhou and Och, 2011; Och and Shields-Zhou, 2012; Hiatt et al., 2020).

About three decades ago, the Galileo space mission obtained the reflected spectra of the Earth over a relatively clear sky region of the Pacific Ocean, north of Borneo, which was analysed by Sagan et al. (1993). Previously, many groups have characterized the atmospheres of modern and prebiotic Earth-like exoplanets by calculating reflection and transmission spectra (Ehrenreich et al., 2006; Kaltenegger and Traub, 2009; Kitzmann et al., 2010b; Domagal-Goldman et al., 2014; Wunderlich et al., 2019; Kaltenegger et al., 2020; Lin et al., 2021). Studies have also been done for the Earth-like planets orbiting F, G, K and M stars (Segura et al., 2005; Grenfell et al., 2007; Rugheimer et al., 2013; Rugheimer and Kaltenegger, 2018). An open source radiative transfer model PICASO to calculate the reflected spectra of exoplanets was presented by Batalha et al. (2019). Earth’s transmission spectra through lunar eclipse observations were calculated by Pallé et al. (2009, 2010) and Yan et al. (2015).

Previously, Kreidberg and Loeb (2016), Meadows et al. (2016), Dong et al. (2017); Lovis et al. (2017); Luger et al. (2017); Meadows et al. (2018); Lin and Kaltenegger (2020); Scheucher et al. (2020) have characterized the atmosphere for Proxima Centauri b and De Wit et al. (2018); Krissansen-Totton et al. (2018); Moran et al. (2018); Zhang et al. (2018); Lustig-Yaeger et al. (2019); Hori and Ogihara (2020); Lin and Kaltenegger (2020); Turbet et al. (2020); Wunderlich et al. (2020); May et al. (2021) have extensively discussed about TRAPPIST-1 system and in particular the planets TRAPPIST-1d and e. Kaltenegger et al. (2013), on the other hand, modeled the transmission spectra for the planet Kepler-62e. Clouds also play a crucial role in determining reflection and transmission spectra (Kitzmann et al., 2010a, b, 2011a, 2011b; Kawashima and Rugheimer, 2019). Fauchez et al. (2019) demonstrated the effect of clouds and hazes on the transmission spectra of the planets in the habitable zone of TRAPPIST-1, and Pidhorodetska et al. (2020) worked on detectability of molecules through transmission spectroscopy.

In this paper, we present the new synthetic reflected spectra of exoplanets similar to the modern and prebiotic Earth orbiting around stars of F, G, K and M spectral types. If the atmosphere is optically thick at pressure level much smaller than 103 mbar, most of the incident stellar radiation in the optical wavelength region will get absorbed and reflected only by the planetary atmosphere. However, the reflecting properties of the surface play a crucial role in the re-emission of thermal radiation at the infrared wavelength region. The surface albedo of solid or liquid (ocean) surface is also considered, which provides better and realistic model spectra. We also calculate the spectra for nine Earth-like planets that lie in the habitable zone of their host stars.

We also present model transmission spectra for simulated terrestrial exoplanets with atmospheric composition similar to that of the modern as well as prebiotic Earth. For this purpose, we use the publicly available software package Exo-Transmit111https://github.com/elizakempton/Exo_Transmit (Kempton et al., 2017). A comparative study of the transmission spectra calculated using the Exo-Transmit code and the TauREx software package (Waldmann et al., 2015) was presented in Sengupta et al. (2020).

In the next Section, we discuss the methodologies adopted to calculate the reflected spectra and the validation of our results. In particular, numerical methodologies are discussed in Section 2.1 and the absorption and scattering opacities that are employed are described in Section 2.2 and we discuss about Temperature-Pressure profile in Section 2.3 . Results are presented in Section 3. The model reflected spectra for both cases - modern and prebiotic Earth-like exoplanets orbiting around stars of F, G, K and M spectral types are presented in Section 3.1 and the model reflected spectra of specific and interesting habitable terrestrial planets are shown in Section 3.2. The model transmission spectra are presented in Section 3.3. Finally we discuss our results with specific conclusions in Section 4.

2 Methodology

2.1 Numerical Methodology

To calculate the reflected spectra, we solved the multiple-scattering radiative transfer equation for diffused reflection and transmission, which for a plane-parallel geometry and azimuthal symmetry, is given by (Chandrasekhar, 1960; Sengupta et al., 2020):

[TABLE]

where $I(\tau,\mu,\lambda)$ is the specific intensity of the diffused radiation field along the direction $\mu=\cos\theta$, $\theta$ being the angle between the axis of symmetry and the ray path, $\omega$ is the albedo for single scattering, $F(\lambda)$ is the incident stellar flux in the direction $-\mu_{0}$ and $\tau$ is the optical depth such that $d\tau=-\chi dz$, where $\chi$ is the total absorption coefficient or extinction coefficient, i.e., the sum of true absorption and absorption due to scattering. In the above equation, $p(\mu,\mu^{\prime})$ is the scattering phase function that describes the angular distribution of the photons before and after scattering. The scattering phase function depends on the nature of the scatterers. For scattering by atoms and molecules, the angular distribution is described by Rayleigh scattering phase function and is given by Chandrasekhar (1960),

[TABLE]

where $\mu$ and $\mu^{\prime}$ are the cosine of the angle before and after scattering with respect to the normal.

As pointed out by Sengupta et al. (2020), in a scattering medium, the radiation field has two components: the reflected and transmitted intensities, which suffer one or more scattering process, and the directly transmitted flux, which is known as the reduced incident flux (Chandrasekhar, 1960), $\pi F(\lambda)e^{-\tau(\lambda)/\mu_{0}}$ in the direction $-\mu_{0}$. So, the reflected and the transmitted intensities that are incorporated through the second term in the right hand side of the above equation, do not include the reduced incident flux, which is described by the third term.

We solved the multiple scattering radiative transfer equations by using the discrete space theory that was developed by Peraiah and Grant (1973) and Peraiah (2002). The numerical code has extensively been used to solve the vector radiative transfer equations to incorporate scattering polarized spectra of brown dwarfs and self-luminous exoplanets (Sengupta and Marley, 2009, 2010, 2016; Marley and Sengupta, 2011; Sengupta, 2016a, 2018). For the present work, we used the scalar version of the same numerical code by using the following steps:

As the vertical atmosphere is heterogeneous with respect to temperature, pressure and optical depth, we divided it into many “shells” of small optical depths. The thickness of each shell is less than or equal to a critical thickness $\rm{\tau_{critical}}$, which is calculated on the basis of the physical characteristics of the medium. If $\tau\leq\rm{\tau_{critical}}$, the reflection and the transmission operators have non-negative elements (Peraiah, 2001). We assume a constant temperature and pressure over each shell, and then integrate the radiation over all the shells. 2. 2.

The integration of the transfer equation is performed on the shells, which is a two-dimensional grid bounded by [rn, rn+1]$\times$[$\mu$j-1/2, $\mu$j+1/2], where, rn is the radial grid and $\mu_{j+1/2}$ is the angular grid:

[TABLE]

Here, ck are the weights of Gauss-Legendre quadrature formula. We used the plane-parallel approximation by making the shell curvature equal to zero. 3. 3.

The Gauss’ quadrature formula is given as (Chandrasekhar, 1960):

[TABLE]

where $\mu_{1}$,….., $\mu_{m}$ are the zeros of P${}_{m}(\mu)$ and

[TABLE]

where, P${}_{m}(\mu)$ is known as the Legendre Polynomial of order $m$. We used the 8-point Gauss’ Quadrature Formula. 4. 4.

We obtained the transmission and reflection operators of the shell by comparing these discrete equations with the canonical equations of the interaction principle, which relates the incident and emergent radiation from a medium of given optical depth. 5. 5.

Combining all the shells by star algorithm (Peraiah, 2002), we obtained the total radiation field. Star algorithm combines the radiation for two consecutive shells by putting them together and calculating the radiation field as a whole.

The numerical method has been described in detail in Peraiah and Grant (1973), Peraiah (2002), Sengupta and Marley (2009) and Sengupta et al. (2020).

In order to validate our numerical calculations, we compared the model reflected spectra of a terrestrial exoplanet orbiting around a solar-type star with the observed reflected spectra of the Earth (Sagan et al., 1993) obtained by the Galileo spacecraft. This is presented in Figure 1. We found an overall good match of the observed low-resolution spectrum with our model spectrum, in particular the dominant water and oxygen bands. The intensity decreases with wavelength in the infrared region because of the nature of input solar spectra and Rayleigh scattering (which is proportional to $\lambda^{-4}$).

Figure 2 shows the comparison between the reflected spectra for TRAPPIST-1e as calculated by our model with that calculated by Lin and Kaltenegger (2020). We used the same chemical abundance as in Lin and Kaltenegger (2020) for verification purposes. Temperature-Pressure ($\rm{T}$-$\rm{P}$) profiles employed for this case were the same as considered by O’Malley-James and Kaltenegger (2019). Here also, the overall nature is the same and the slight variations are due to different opacities used and also they used vertically variable atmospheric abundance, but we used vertically homogenous atmospheric abundance. We also compared our model reflected spectrum for prebiotic Earth-like exoplanets orbiting around Sun-like stars (atmosphere composed of only N2 and CO2; Rugheimer et al. 2015) with the model spectrum calculated by S. Ranjan (priv. comm.). Figure 3a demonstrates that the reflection spectrum of prebiotic terrestrial exoplanets calculated by us matches very well with that derived by S. Ranjan. The slight variation is again due to the differences in opacities used in both models. Figure 3b shows the comparison of the geometric albedo, which also matches very well. Here there are no absorption lines because the considered molecules show absorption beyond the limit of the wavelength considered in this work (2.49 $\mu$m).

2.2 Absorption and Scattering Opacity

In order to calculate the reflection and the transmission spectra, we calculated the absorption and scattering coefficients of the atmosphere by using the Exo-Transmit software package (Kempton et al., 2017). In this package, the opacities for 30 molecular and atomic species on a fixed temperature-pressure-wavelength grid are tabulated. The wavelength grid ranges from 0.3 to 30 $\mu$m at low spectral resolution of $\mathcal{R}\approx$ 1000. The temperature and pressure range at which the absorption and scattering coefficients were calculated for each wavelength were 100–3000 K and $10^{-6}$–$10^{6}$ mbar respectively. The opacities were derived by using the line lists given by Lupu et al. (2014). The gas opacities were adopted from the widely used database of Freedman et al. (2008, 2014). The individual opacity sources are the atomic and molecular opacity weighted by their abundances and the total Rayleigh scattering opacity. Since the Earth’s atmosphere is sufficiently cool, we neglected the collision induced absorption of hydrogen. We adopted the molecular abundances of the present Earth (Sagan et al., 1993) as described in Table 1.

For the prebiotic Earth-like exoplanets, we considered a carbon dioxide dominated atmosphere with the molecular abundance as 10$\%$ CO2, trace amounts of CH4 and the remaining N2 as considered by Kaltenegger et al. (2007). The atmospheric abundances of Proxima Centauri b, Kepler-442b, Kepler-62e, Kepler-22b, Kepler-1649c, TOI-400d, Teegarden’s Star b, Trappist-1d and TRAPPIST-1e were considered to be the same as that of the present Earth. For the verification purpose for TRAPPIST-1e, the atmospheric abundance and $\rm{{T}}$-$\rm{{P}}$ profile were adopted following O’Malley-James and Kaltenegger (2019). The molecular abundances were incorporated in the equation of states file in the Exo-Transmit package, in which pressure and temperature were also included for the atmospheric layers.

2.3 Temperature-Pressure profile

For the terrestrial exoplanets, the internal temperature is negligible as compared to the irradiated temperature. Thus the incident stellar flux at the top-most layer of the atmosphere and the molecules present in the atmosphere determine the Temperature-Pressure ($T$-$P$) profile of the terrestrial exoplanets.

Most of the stellar radiation gets reflected from the upper layers of the planetary atmosphere. So, the temperature structure of the outer layers of the atmosphere mostly determines the reflected spectra. In the case of transmission spectra also, the lower atmosphere cannot be probed for most of the wavelengths (Kaltenegger and Traub, 2009). Hence, for the present Earth-like exoplanets orbiting around stars of F, G, K and M spectral types, we adopted the Earth’s atmospheric $T$-$P$ profile (NOAA NASA US Air Force, 1976) , which is shown in Figure 4. As we go upwards from the solid surface of the Earth, the temperature decreases continuously with the decrease in pressure, thus following the ideal gas law in the tropospheric region. This region extends roughly about 9 km at the poles and 17 km at the equator (Caballero et al., 2022). We will roughly consider the height of the tropopause equal to 11 km for our calculations. In the stratospheric region , which extends about 35 km above the tropopause, the temperature increases with the decrease in pressure due to the presence of ozone gas , which absorbs the ultraviolet radiation. This is known as the temperature inversion (NOAA NASA US Air Force, 1976). The $T$-$P$ profile for the atmospheres of early Earth-like exoplanets is taken the same as considered in Kaltenegger et al. (2007) for Epoch 0 (3.9 Ga).

3 Results and Analysis

3.1 The Reflection Spectra

While orbiting its host star, an exoplanet reflects part of the starlight along our line of sight. We observe the maximum reflected radiation when the planet is almost at full phase or at zero degree phase angle, i.e, just before or after the secondary eclipse position.

In the present investigation, we considered terrestrial planets around stars of three sub-classes 0, 2 and 5 of F, G, K and M spectral types so that late to early stages of each spectral type are included. All the stars were considered to be of main sequence dwarfs of luminosity class V. The input stellar fluxes at the surface of the planets orbiting within the habitable zone of their host stars are shown in Figure 5. The fluxes at the stellar surface for F, G and K spectral types were obtained from ESO library (Pickles, 1998). For the M spectral type, the stellar fluxes were obtained from PHOENIX model (Husser et al., 2013) generated through publicly available code petitRADTRANS (Mollière et al., 2019). The equilibrium temperatures of the planets were assumed to be same as that of the present Earth, i.e. 288 K.

3.1.1 Reflected spectra for present Earth-like exoplanets

We calculated the reflected spectra by solving the multiple-scattering radiative transfer equation for plane-parallel stratification (equation (1)). To estimate the surface Bond albedo of the rocky planets, we considered few different types of surfaces with the compositions given in Table 2.

The surface composition of the present Earth is 70$\%$ ocean, 2$\%$ coast and 28$\%$ land, which is divided into 30$\%$ grass, 30$\%$ trees, 9$\%$ granite, 9$\%$ basalt, 15$\%$ snow and 7$\%$ sand (Kaltenegger et al., 2007). And the surface composition for prebiotic Earth is 70$\%$ ocean, 2$\%$ coast and 28$\%$ land. The land surface consists of 35$\%$ basalt, 40$\%$ granite, 15$\%$ snow and 10$\%$ sand with no land vegetation (Kaltenegger et al., 2007; Rugheimer and Kaltenegger, 2018). In the sixth scenario, no solid or liquid surface exists, which means that the atmosphere of the planet is so optically thick, that the incoming stellar radiation gets reflected only from the atmosphere and it does not reach up to the surface. Hence, in this case, surface albedo does not affect the reflected spectra or the geometric albedo. Zero surface albedo may also mean the gaseous planets, which is beyond the scope of this work.

We calculated the surface Bond albedo by weighted sum of all the components’ albedo. And the weight factors are the respective fractions of the planetary surface coverage. The reflected spectra for present Earth-like exoplanets orbiting around solar type of star for different surface albedo are shown in Figure 6. Reflected flux increases with the increase in the surface albedo and it is steeper than the input stellar flux because of Rayleigh scattering. The effect of surface albedo on the geometric albedo for the present Earth-like exoplanets is shown in Figure 6b. We can see that geometric albedo increases with the increase in surface albedo. However, it decreases with the wavelength because Rayleigh scattering is not significant at longer wavelength region.

The reflected spectra for the present Earth-like exoplanets orbiting around stars of F, G, K and M spectral types are shown in Figure 7. The absorption lines of H2O (0.72 $\mu$m, 0.82 $\mu$m, 0.94 $\mu$m, 1.10 $\mu$m and 1.87 $\mu$m), O2 (0.63 $\mu$m, 0.69 $\mu$m, 0.76 $\mu$m) and CH4 (1.60 $\mu$m) are also shown in this figure. The flux decreases with the increase in the wavelength in the infrared region. This is because of two reasons: firstly, the input stellar flux also decreases with the increase in wavelength in infrared and secondly Rayleigh scattering dominates in the shorter wavelength region (Ityaksov et al., 2008). The reflected spectra has the planetary atmospheric features as well as the stellar atmospheric features.

In the present study, we ignored the effect of strong stellar ultra-violet irradiation that may alter the planetary environment by dissociating water molecules and energy limited hydrogen loss (Sanz-Forcada et al., 2011; Sengupta, 2016b). Presence of sufficient initial water content at the planetary surface may still avoid the planet to become parched under such situation. However, since most of the planets in the habitable zone of M dwarfs are tidally locked, the presence of an Earth-like planet is rare (Martinez-Rodriguez et al., 2019).

In order to investigate the effect of various greenhouse gases on the geometric albedo, we increased the abundance of CO2 by two orders in magnitude, CH4 by four orders in magnitude and H2O by one order in magnitude. This increase is compensated by altering the abundance of N2. The geometric albedo of the present Earth-like exoplanets with increased abundances of atmospheric greenhouse gases is presented in Figure 8. We found that the geometric albedo increases slightly in the shorter wavelength region because of the increase in Rayleigh scattering. However, the scattering could have drastic effect in the thermal re-emission at the near and far infrared wavelength region and hence in determining the surface temperature of the planet by an increased greenhouse effect.

3.1.2 Early Earth-like exoplanets

The reflected spectra for the prebiotic Earth orbiting around stars of F, G, K and M spectral types are presented in the Figure 9. And the geometric albedo (for surface albedo 0.16) is presented in Figure 10. We see very less absorption lines because only N2, CO2 and CH4 were considered in the atmospheric composition for the prebiotic Earth. The absorption lines of CO2 (1.4 $\mu$m, 1.6 $\mu$m and 2 $\mu$m) and CH4 (1.66 $\mu$m) can be seen. The overall nature of spectra remains the same as that for the modern Earth case.

A comparison between the geometric albedo for present and prebiotic Earth with zero surface albedo is shown in the Figure 11. Prebiotic Earth-like exoplanets scatter more starlight as compared to the present Earth-like exoplanets because of greater abundances of greenhouse gases (mainly CO2). The absorption lines for the present Earth are also shown in this figure.

3.2 Reflected spectra of known terrestrial exoplanets

We also present the reflected spectra for some of the well known habitable planets such as Kepler-442b, Kepler-62e, Kepler-22b, TOI-700d, Kepler-1649c, Teegarden’s Star b, Proxima Centauri b, TRAPPIST-1d and TRAPPIST-1e. These planets orbit stars of G, K and M spectral types. Their radii are in the range of 0.7 R⊕ and 2.4 R⊕. These planets lie in the habitable planets catalog in Hill et al. (2022). Although very little is known about their atmospheres at present, we expect them to have Earth-like atmospheric compositions with favourable temperature due to greenhouse effect. The input stellar flux at the surface of Kepler-442b, Kepler-62e and Kepler-22b were calculated by taking the stellar flux from Pickles (1998). We used PHOENIX model spectra for the cases of TOI-700d, Kepler-1649c and Teegarden’s Star b. For Proxima Centauri b, the stellar flux is taken from (Lin and Kaltenegger, 2020) and for TRAPPIST-1d and TRAPPIST-1e, we used the spectra from Burgasser et al. (2015).

Their equilibrium temperature Teq can be derived from the relationship given in equation 6 (Seager, 2010). The temperature at the bottom of the atmosphere (or surface temperature) with greenhouse effect is given by equation 7 (De Pater and Lissauer, 2015).

[TABLE]

[TABLE]

In equations 6 and 7, A is the Bond albedo, Rs is the radius of the host star, $a$ is the orbital distance, Teff is the effective temperature of the host star, Tsurf is the temperature at the surface of the planet with greenhouse effect and $\tau_{g}$ is the optical depth of the atmosphere at infrared wavelengths. We assumed it to be same as that for the Earth, i.e. $\sim$ 0.83. The surface temperature should not be less than 273 K for the planet to be habitable (Tsurf,min $\approx$ 273 K). And from equation 7, the minimum equilibrium temperature or the temperature at the top of the atmosphere (Teq,min) is about 242 K.

3.2.1 Kepler-442b

It is an Earth-like exoplanet orbiting its host star (K5V) within the habitable zone and about 366 pc away from the Earth. It is among all the detected rocky planets that is most similar to the Earth and has a very high habitability index value (Torres et al., 2015; Kane et al., 2016; Rodríguez-Mozos and Moya, 2017). This planet receives an incident stellar flux that is 0.9 times of the flux received by the Earth (Torres et al., 2015; Armstrong et al., 2016; Rodríguez-Mozos and Moya, 2017; Barbato et al., 2018). It is a promising candidate for search of biosignatures as K-type of stars maintain favourable circumstellar conditions for habitability (Cuntz and Guinan, 2016). Its density is very similar to the Earth and mean surface gravity is $\sim$ 12.5 m/s2, slightly higher than that of the Earth. According to Arney (2019), K-type stars present an advantage for the detectability of biosignatures. One of the reasons is that K dwarfs offer extended photochemical lifetime of methane as compared to G types stars. And the other reason is better signal-to-noise ratio (S/N) of K dwarfs than G dwarfs, due to which oxygen and methane can be strongly observed. We calculate the $T$-$P$ profile by the following method:

For tropospheric region ( up to 11 km), T = -mh + Tsurf where m is the adiabatic lapse rate.

Tsurf,min $\approx$ 273 K; For h=11 km, T $\approx$ 242 K; m = 2.83 K/km

Tsurf,max $\approx$ 290.1 K; For h=11 km, T $\approx$ 257 K; m = 3 K/km

[TABLE] 2. 2.

For Stratospheric region and above, T1 $\approx$ 242 K; T2 $\approx$ 257 K.

Similarly, we calculated the $T$-$P$ profile for all the other planets by calculating their adiabatic lapse rates. The possible range of $T$-$P$ profile for Kepler-442b is shown in Figure 12 and the temperature can lie anywhere in this range. The maximum value of Bond albedo (Amax), temperature at the top of the atmosphere ( Teq) and temperature at the bottom of the atmosphere including green-house effect ( Tsurf) were calculated in the same ways and are shown in Table 3. The atmospheric abundance was assumed to be the same as that of the Earth and shown in Table 1. We calculated the reflected spectra for the two $T$-$P$ profiles and found that the spectra does not alter with the variation in $T$-$P$ profile within the given range. The reflected spectra for the planet Kepler-442b for various surface compositions i.e. different surface albedos is shown in the Figure 12.

Figure 12 also shows the geometric albedo of Kepler-442b for different surface compositions of the planet. We note that the geometric albedo increases significantly with the increase in the surface Bond albedo or we can say that the surface albedo considerably affects the geometric albedo. This is because the surface also contributes in the total reflectivity of the planet. For example, for the zero surface albedo case, the geometric albedo is the least. And it is maximum for the present Earth-like surface components (0.14 surface albedo).

3.2.2 Kepler-62e

Kepler-62e also orbits within the classical habitable zone of the host star (K2V) and the orbital period is about 122 days (Borucki et al., 2013; Kaltenegger et al., 2013; Torres et al., 2015; Armstrong et al., 2016; Kane et al., 2016). The possible $T$-$P$ profile was calculated in the same way as in the case of Kepler-442b and is presented in Figure 12. The temperature and pressure can be anywhere between these limits.

The reflected spectra and the geometric albedo for Kepler-62e are also shown in Figure 12 for various surface compositions. It is highest for Earth-like surface composition (surface albedo 0.14) and lowest for no surface albedo at all. The green curve is for 50$\%$ ocean cover and the remaining covered with trees and grass. As the ocean cover is reduced from 70$\%$ to 50$\%$ by increasing the land cover, the geometric albedo decreases.

3.2.3 Kepler-22b

Kepler-22b is a super-Earth orbiting a G5V star , which is about 194.7 pc away from Earth. This planet is also orbiting within the habitable zone of the host star (Borucki et al., 2012; Neubauer et al., 2012; Torres et al., 2015; Kane et al., 2016). It is the first detected Earth-like exoplanet in the habitable zone of a solar-type star.

The atmospheric $T$-$P$ profile used to calculate the reflected spectra of the planet is shown in Figure 12. It is also calculated by assuming the atmosphere of the planet in hydrostatic equilibrium and considering greenhouse effect.

The effect of the surface Bond albedo (derived from the surface compositions) on the reflected spectra and the variation of the geometric albedo are also shown in Figure 12.

3.2.4 TOI-700d

It is TESS’s first Earth-size exoplanet , which lies in the habitable zone of its host star TOI-700 ( M dwarf). The planet is expected to be tidally locked as its eccentricity is close to zero (Gilbert et al., 2020; Rodriguez et al., 2020; Suissa et al., 2020; Kaltenegger et al., 2021). It receives about 86$\%$ of the insolation that the Earth receives (Gilbert et al., 2020). The possible range of the atmospheric $T$-$P$ profile for this planet is shown in Figure 12. The reflected spectra and the geometric albedo are also shown in this figure.

3.2.5 Kepler-1649c

This is an Earth-size planet lying in the habitable zone of its host star , which is of M5V spectral type. It is located at a distance of about 92 pc from the Earth (Vanderburg et al., 2020; Kane et al., 2020; Gvalani, 2022). The $T$-$P$ profile, the reflected spectra and the geometric albedo are presented in Figure 13.

3.2.6 Teegarden’s Star b

Teegarden’s Star was discovered by Teegarden et al. (2003) and it is at a distance of 3.831 pc and of spectral type M7V (Alonso-Floriano et al., 2015). It has two planets Teegarden’s Star b and c. Both of them are super Earths but Teegarden b is the most Earth-like planet or maximum ESI value (see Table 3), discovered till now (Wandel and Tal-Or, 2019; Zechmeister et al., 2019). This planet lies within the habitable zone and it is tidally locked. The T-P profile range, reflected spectra and the geometric albedo are presented in Figure 13.

3.2.7 Proxima Centauri b

Proxima Centauri b is a rocky planet that orbits within the habitable zone of our nearest neighbour Proxima Centauri (M5V), which receives about 65$\%$ of the total flux that our Earth receives from the Sun (Anglada-Escudé et al., 2016; Garraffo et al., 2016; Turbet et al., 2016; Ribas et al., 2017; Meadows et al., 2018; Lin and Kaltenegger, 2020; Galuzzo et al., 2021).

We modeled the reflected spectra of Proxima Centauri b by using the stellar flux presented by Lin and Kaltenegger (2020). The $T$-$P$ profile (derived in the same way) for the atmosphere of Proxima Centaui b is shown in Figure 13 where a range is given. The reflected spectra and the geometric albedo for Proxima Centauri b are shown in Figure 13 for various surface Bond albedo determined by different surface compositions. Here also, it is maximum for Earth-like surface composition and minimum for no surface albedo.

3.2.8 TRAPPIST-1d and e

TRAPPIST-1 is another M dwarf of spectral type M8V , which is about 12 pc away from us, hosts seven rocky planets out of which, three are in the habitable zone of the star (Gillon et al., 2016; Burgasser and Mamajek, 2017; Gillon et al., 2017; O’Malley-James and Kaltenegger, 2019; Lin and Kaltenegger, 2020). TRAPPIST-1e is most likely to have habitable surface conditions, as it receives about $\sim$ 66$\%$ of stellar radiation that the Earth receives from the Sun and needs very little greenhouse effect to have a surface temperature such that water can exist in liquid state (Kopparapu et al., 2013; Wolf, 2017, 2018; Fauchez et al., 2020). Also, TRAPPIST-1e is quite similar in size to the Earth. On the other hand, TRAPPIST-1d has a very high ESI value of 0.9 (see Table 3). So it becomes important to model both the planets.

The $T$-$P$ profile, the reflected spectra and the geometric albedo are presented in Figure 13 for both the cases. The reflected spectra and the geometric albedo were calculated for various surface materials. As the surface albedo increases, the reflected flux increases because the surface also contributes to the reflected flux. The geometric albedo is decreasing with the increase in wavelength because scattering becomes negligible at longer wavelengths. Also it decreases significantly with the decrease in the surface Bond albedo. For the zero surface albedo case, all the radiation is reflected only from the atmosphere.

3.2.9 Comparisons

The input stellar flux at the surface of the above planets is shown in Figure 14a. The reflected spectra for these planets for Earth-like surface albedo are shown in Figure 14b and the geometric albedo is shown in the Figure 14c. We can see that the reflected spectra follows the input stellar spectra in the visible wavelength region. The geometric albedo is highest for Kepler-22b and lowest for TRAPPIST-1e in the infrared. But in the optical, it is highest for the case of Teegarden’s Star b and lowest for Kepler-22b.

The geometric albedo in the optical region is not estimated for any of the planets around TRAPPIST-1 because it is a late M dwarf whose effective temperature is about 2400K. Its blackbody spectra peak lies at around 1 $\mu$m and thus the flux in the optical is very less in magnitude as compared to the flux in NIR. The nature of geometric albedo depends on the absorption and scattering co-efficients of the planetary atmosphere or the $T$-$P$ profile and the atmospheric composition of the corresponding planet.

3.3 The Transmission Spectra

When an exoplanet transits in front of its host star, it blocks some of the starlight along our line of sight resulting into a reduction in the observed stellar flux. During the transit, a fraction of the stellar radiation passes through the planetary atmosphere providing signatures of the gases present there. The stellar radiation that suffers absorption and scattering in the atmosphere of the planet is known as the transmission or transit spectra. The transmission spectra is usually presented by a wavelength dependent quantity called the transmission or transit depth.

3.3.1 Transit Depth

Transit depth is the ratio between the stellar flux obtained with and without transit. It can be expressed as

[TABLE]

where $F_{trans}$ is the stellar flux obtained during the planetary transit epoch and $F_{star}$ is the unblocked stellar flux or the stellar flux during the out of transit epoch. $F_{trans}$ can be written as (Kempton et al., 2017; Sengupta et al., 2020):

[TABLE]

where, $R_{pl,atm}$ is the radius of the planet including its atmosphere and $R_{star}$ is the radius of the star, $F_{atm}$ is the stellar flux , which gets transmitted through the planetary atmosphere along the line of sight. Transit depth corresponds to the ratio between the planetary radius $R_{pl,atm}$ and the stellar radius $R_{star}$.

In order to calculate $F_{atm}$, we used Beer-Bouguer-Lambert’s law given by:

[TABLE]

where $I(\lambda)$ is the intensity of the transmitted stellar radiation through the planetary atmosphere and $I_{0}(\lambda)$ is the intensity of the incident stellar radiation on the planet. In the above equation, $\tau_{\lambda}$ is the optical depth along the ray path and $\mu_{0}$ is the cosine of the angle between the direction of the incident radiation and the normal. The expression for the optical depth along the line of sight ($\tau(\lambda,z)$) is given by:

[TABLE]

where $\chi(\lambda,z)$ is the extinction coefficient , which is the sum of the absorption coefficient and the scattering coefficient, $\rho(z)$ is the density of the planetary atmosphere, $z$ is the height of the atmosphere from the planetary surface, $l$ is the distance covered by the radiation in the planetary atmosphere given by (Tinetti et al., 2013):

[TABLE]

where $R_{p}$ is the radius of the planet , below which the medium becomes opaque at all wavelength and $z_{max}$ is the maximum height (on the top of $R_{p}$) , above which photons do not suffer any absorption or scattering. We calculated the transmission spectra by using Exo-Transmit package (Kempton et al., 2017). Figure 15 shows the transmission spectra for the present and the prebiotic Earth-like exoplanets. Figure 15a presents the transmission spectra up to a wavelength of 4.5 $\mu$m while Figure 15b shows the same up to the wavelength 30 $\mu$m. The transmission depth due to absorption by O2, H2O, CO2 and O3 are marked in the spectra. For the early or prebiotic Earth-like exoplanets, the absorption lines of only CO2 molecules are seen.

In the transmission spectra of prebiotic Earth-like exoplanets presented in Figure 15(a) , which is the zoomed-in version of Figure 15(b), signatures of CO2 can be found at 1.4 $\mu$m, 1.6 $\mu$m, 2.0 $\mu$m, 2.7 $\mu$m and at 4.3 $\mu$m. On the other hand, the signatures of H2O at 0.72 $\mu$m, 0.82 $\mu$m, 0.94 $\mu$m, 1.10 $\mu$m, 1.87 $\mu$m and 2.70 $\mu$m are clear in the transmission spectra of modern Earth. In the transmission spectra of modern Earth, signatures of O2 can also be found at 0.63 $\mu$m, 0.69 $\mu$m, 0.76 $\mu$m and that of CO2 at 1.4 $\mu$m, 2.7 $\mu$m, 4.3 $\mu$m. The signature of O3 is visible at 3.3 $\mu$m. Figure 15(b) shows the transmission spectra for the whole wavelength region i.e. 0.3$\mu$m to 30.0$\mu$m.

Figure 16 shows the transmission spectra for the increased abundance of greenhouse gases in the atmosphere of the terrestrial exoplanet. Here we notice that the transmission depth increases with the increase in the abundance of greenhouse gases in the planetary atmosphere. However, this increase in the transmission depth is found to be confined only up to a certain wavelength region, which is again due to Rayleigh scattering.

3.3.2 Effects of cloud opacity

The observations of the various exoplanetary atmospheres indicate that the presence of clouds or hazes are common phenomenon in the planetary atmospheres (Kreidberg et al., 2014; Sing et al., 2016). This is one of the reasons for weak or no molecular feature observed in the transmission spectra of quite a few hot Jupiters (Sánchez-López et al., 2020). The same situation may arise for the terrestrial exoplanets if the upper atmosphere is covered by clouds or hazes. The presence of clouds or hazes however, increases the Rayleigh scattering.

For the gray cloud calculation, we selected a pressure layer in the atmosphere at which the cloud top is optically thick. We provided a threshold pressure within the pressure range of the $T$-$P$ profile and performed the radiative transfer calculations for pressures below that of the cloud deck. We used Exo-Transmit package (Kempton et al., 2017) for the calculation of cloud optical depth. Figure 17 shows the transmission spectra for clear sky and for the sky with 100$\%$ coverage of clouds at three different atmospheric heights i.e. 2.2 km, 9.5 km and 17.0 km from the surface of the planet.

4 Conclusions and Discussion

In the first part of this paper, we presented the numerical models of reflection spectra (in visible) for both the present and prebiotic Earth-like exoplanets orbiting within the habitable zone of main sequence stars of F, G, K and M spectral types. We also presented the model reflected spectra for the known exoplanets , which are orbiting around the stars of G, K and M spectral types.

We found that the nature of the reflected spectra is similar to that of the incident stellar spectrum i.e., the reflected flux peaks in the optical waveband but decrease significantly at longer wavelengths. However, Rayleigh scattering in the planetary atmosphere makes the reflected spectra comparatively steeper. The geometric albedo also decreases with the increase in wavelength because of the same reason i.e. Rayleigh scattering. The amount of reflected flux for the planets orbiting M dwarfs is significantly less compared to the stars of F, G and K spectral types. This is because the input stellar spectra peaks in the infrared wavelength region where Rayleigh scattering is negligible. The absorption lines of the biosignatures like O2, H2O, O3, etc. are dominant in the geometric albedo. Owing to the fact that prebiotic early Earth-like exoplanets have a greater percentage of greenhouse gas CO2, they scatter more radiation than the present Earth-like exoplanets do. A present Earth-like exoplanet with higher abundance of greenhouse gases also have greater albedo. We have also estimated the maximum possible values of Bond albedo for the known exoplanets and thus given a limit on Bond albedo for the planets to remain habitable.

We also investigated the effects of surface Bond albedo on the reflected spectra and geometric albedo for various solid and liquid surface composition. We considered several kinds of solid and liquid surfaces e.g., (1) present Earth-like surface composition, (2) early Earth-like surface composition, (3) 100$\%$ ocean cover, (4) 50$\%$ ocean and remaining with trees and grass and (5) 83$\%$ ocean and remaining with snow. The reflected flux and the geometric albedo increases with the increase in surface albedo. It is minimum for no surface albedo at all. The effect of the surface albedo becomes negligible for an atmosphere thick enough to obstruct the incident stellar radiation to reach the solid or liquid surface. Thus, surface composition plays a key role in determining the reflectivity of the planet. In the infrared region, the planetary surface with ocean, vegetation, desert etc. play important role in determining the reflected as well as the re-emitted thermal radiation. However we did not considered the re-emitted thermal radiation here.

In the second part of the work, we presented the transmission spectra for present and prebiotic Earth-like exoplanets. Since the transmission depth increases at the shorter wavelength due to scattering, an increase in the abundance of greenhouse gases yields into greater transmission depth. Also, the transmission depth reduces in magnitude with the increase in the height of the cloud level. Since the Exo-Transmit code does not incorporate diffused radiation by scattering, it just reduces the transmission depth. As the height of the cloud increases, the threshold pressure decreases. As a consequence, a comparatively smaller atmospheric region above the clouds yields a featureless transmission spectra. Since we assume a vertically homogeneous atmospheric abundance, the spectral feature remain the same, but the magnitude will change, with the change in the cloud height.

In the future, since many big-budget missions are coming like Habitable Worlds Observatory (HWO), GMT, Thirty Meter Telescope (TMT), Extremely Large Telescope (ELT), etc., our models will play an important role in the habitability study of the Earth-like exoplanets. By knowing their reflectivity, Bond albedo and the transmission spectra, we would be able to know about the factors like planet’s surface and atmospheric composition, atmospheric $T$-$P$ profile, presence of clouds, greenhouse gases, etc., which play a key role in determining the habitable planet.

Acknowledgements

We acknowledge the referee José A. Caballero for giving insightful comments and the improvement in the presentation of the paper. We thank Sukrit Ranjan for kindly providing model spectra for prebiotic Earth orbiting solar type of star and for many useful discussions. We also thank Adam Burgasser for providing the observed near infrared spectrum of TRAPPIST-1. MS would like to acknowledge Soumya Sengupta for fruitful discussions.