Comparative Climates of TRAPPIST-1 planetary system: results from a simple climate-vegetation model

TL;DR

This study uses a simple climate-vegetation energy-balance model to analyze the potential habitability of TRAPPIST-1 planets, identifying conditions under which liquid water could exist and highlighting planet TRAPPIST-1d as the most stable candidate.

Contribution

It introduces a simple climate-vegetation model to evaluate the habitability of TRAPPIST-1 planets, focusing on surface water potential under various conditions.

Findings

TRAPPIST-1d is most likely to sustain liquid water.

Outer planets f, g, h are unlikely to have surface liquid water.

The model captures key climate features despite its simplicity.

Abstract

The recent discovery of the planetary system hosted by the ultracool dwarf star TRAPPIST-1 could open new perspectives into the investigation of planetary climates of Earth-sized exoplanets, their atmospheres and their possible habitability. In this paper, we use a simple climate-vegetation energy-balance model to study the climate of the seven TRAPPIST-1 planets and the climate dependence on the global albedo, on the fraction of vegetation that could cover their surfaces and on the different greenhouse conditions. The model allows us to investigate whether liquid water could be maintained on the planetary surfaces (i.e., by defining a "surface water zone") in different planetary conditions, with or without the presence of greenhouse effect. It is shown that planet TRAPPIST-1d seems to be the most stable from an Earth-like perspective, since it resides in the surface water zone for a wide range of reasonable values of the model parameters. Moreover, according to the model outer planets (f, g and h) cannot host liquid water on their surfaces, even for Earth-like conditions, entering a snowball state. Although very simple, the model allows to extract the main features of the TRAPPIST-1 planetary climates.

Tables

Table 1. Table 1: Values of the model parameters.

Symbol							Value		Units
$C_T$							500		W yr K-1 m-2
${\alpha }_v$							0.1
${\alpha }_g$							0.4
${\alpha }_{max}$							0.85
${\alpha }_{min}$							0.25
$T_{low}$							263		K
$T_{up}$							300		K
$T_{opt}$							283		K
$k$							0.004		yr-1 K-2
$\gamma$							0.1		yr-1

Equations

$$C_T\frac{dT}{dt}$$

$$\frac{dA}{dt}$$

$${\alpha }_o(T)={\alpha }_{max}+\left({\alpha }_{min}-{\alpha }_{max}\right)\left[\frac{T-T_{low}}{T_{up}-T_{low}}\right]$$

$$R(T)=\left[1-m\tanh {\left(\frac{T}{T_0}\right)}^6\right]\sigma T^4,$$

$$\text{SWZ}(\{\theta\})=\left\{\begin{array}[]{cc}1&\mbox{if }273\leq T\mbox{K }\leq 373\mbox{K}\;,\\ 0&\mbox{otherwise}\;.\\ \end{array}\right.$$

$$S(a,L_{\star })=\frac{0.0005}{a^2}{S}_{\odot },$$

Full text

It is shown that planet TRAPPIST-1d seems to be the most stable from an Earth-like perspective, since it resides in the surface water zone for a wide range of reasonable values of the model parameters. Moreover, according to the model outer planets (f, g and h) cannot host liquid water on their surfaces, even for Earth-like conditions, entering a snowball state. Although very simple, the model allows to extract the main features of the TRAPPIST-1 planetary climates.

planets and satellites: atmospheres, planets and satellites: terrestrial planets

1 Introduction

The sharp acceleration of exoplanets discovery in the recent years (NASA Exoplanet Archive, 2017; Mayor & Queloz, 1995; Marcy & Butler, 1996; Petigura et al., 2013; Gillon et al., 2016, 2017) and the presumed habitability of some of them (Kasting et al., 1993; Scharf, 2009; Spiegel et al., 2008; Kopparapu et al., 2013; Gillon et al., 2017), are changing our point of view on planetary science.

According to the usual definition (Kasting et al., 1993; Kopparapu et al., 2013), a planet resides in the so-called circumstellar habitable zone (HZ) if, being a terrestrial-mass planet with a CO${}_{\text{2}}$-H${}_{\text{2}}$O-N${}_{\text{2}}$ atmosphere, it can sustain liquid water on its surface (Kasting et al., 1993; Kopparapu et al., 2013). The above requirements, coupled with the assumption of an Earth-like geology for the resulting greenhouse effect and carbon-silicate weathering cycle, imply that the surface temperature must be in the range $0-100$ ∘C. Typically, apart from orbital features (e.g., eccentricity, period, transit time, inclination) and rough estimates of mass and radius, little information is directly known about exoplanets. For instance, the planetary surface temperature can be roughly estimated by using equilibrium conditions from energy-balance climate models depending on the distance of the planet from the hosting star and planetary outgoing energy. However, in such cases these estimates could be incorrect, since no information about the planetary atmosphere is included in these models (as for Venus which has an estimated temperature of $\sim$ 300 K while the true surface temperature is about 737 K). Nevertheless, more complex energy-balance climate models, including the greenhouse effect and/or heat diffusion, provide insight into the climate on a planet (Alberti et al., 2015, and references therein).

Despite some recent comments on the metrics used to define the habitable zone in relation to public interest in scientific results (Tasker et al., 2017; Moore et al., 2017), the recently discovered TRAPPIST-1 system (Gillon et al., 2016, 2017), formed by seven temperate (with equilibrium temperatures $\lesssim$ 400 K) Earth-sized planets orbiting around a nearby ultracool dwarf, increased the attention to studying climate conditions of terrestrial exoplanets (Bolmont et al., 2017; O’Malley-James & Kaltenegger, 2017; Bourrier et al., 2017; Wolf, 2017). Since, as pointed out above, the estimates of equilibrium temperatures which do not take into account the greenhouse effect and the albedo feedback (so-called null Bond albedo hypothesis) cannot be sufficiently reliable, improved and advanced climate models, considering the planetary albedo and the atmospheric composition, are required (Gillon et al., 2017; Wolf, 2017). By using both a 1-D radiative-convective climate model and a more sophysticated 3-D model, Gillon et al. (2017) found that inner planets, T-b, T-c, and T-d (in the following we indicate as T-x the x-th TRAPPIST-1 planet), show a runaway greenhouse scenario, while outer planets, T-e, T-f, and T-g, could host water oceans on their surfaces, assuming Earth-like atmosphere. Concerning the seventh planet T-h, due to the low stellar irradiance received, it cannot sustain surface liquid water oceans. However, since only little is known about the planetary system, several approaches and hypothesis can be helpful in investigating both planetary climates and atmospheric composition (De Wit et al., 2016; Bolmont et al., 2017; O’Malley-James & Kaltenegger, 2017; Wolf, 2017).

One of the drawbacks of the more detailed climate models is the necessary large pool of assumptions of atmospheric and surface conditions. In this paper we investigate the possible climates of the TRAPPIST-1 planetary system by using a simple zero-dimensional energy-balance model (Rombouts & Ghil, 2015; Alberti et al., 2015), which allows the extraction of global information on the the climate evolution by using the actual knowledges about the planetary system. This model has the advantage of transparency through minimal assumptions, allowing a comparative sets of models to be studied. We study several situations, from completely rocky planets to Earth-like conditions, both neglecting and considering the greenhouse effect, to explore different possible climates and make a comparative study of TRAPPIST-1 planetary system climates.

2 The climate model

The main features of the climate dynamics of Earth-like planets can be recovered through a zero-dimensional model based on two equations describing the time evolution of the global average temperature $T$ and of the fraction of land $A$ covered by vegetation:

[TABLE]

Here $C_{T}$ is the planet heat capacity, $\alpha(T,A)$ is the planetary albedo, $S(a,L_{\star})=L_{\star}/(4\pi a^{2})$ is the mean incoming radiation which depends on the star-planet distance $a$ (in au) and on the star luminosity $L_{\star}$, $R(T)$ is the outgoing energy from the planet, $\beta(T)$ and $\gamma$ are the vegetation growth and death rates, respectively (Watson & Lovelock, 1983; Rombouts & Ghil, 2015; Alberti et al., 2015). The albedo of the planet depends on the fraction of land $p$, namely $\alpha(T,A)=(1-p)\alpha_{o}(T)+p[\alpha_{v}A+\alpha_{g}(1-A)]$, where $\alpha_{o}$, $\alpha_{v}$ and $\alpha_{g}$ represent the albedos of ocean, vegetation and bare-ground, respectively. The albedo of the ocean is assumed to be linearly dependent on temperature as

[TABLE]

in a range of temperatures $T\in[T_{low},T_{up}]$, resulting in $\alpha_{o}(T)=\alpha_{max}$ for an ocean completely covered by ice ($T\leq T_{low}$) and $\alpha_{o}(T)=\alpha_{min}$ for an ice-free ocean ($T\geq T_{up}$).

The outgoing energy is described by a black-body radiation process, modulated by a grayness function, in order to take into account the greenhouse effect

[TABLE]

where $\sigma=5.67\times 10^{-8}$ W m*-2* K*-4* is the Stefan-Boltzmann constant, $m\in[0,1]$ is a grayness parameter ($m=0.5-0.6$ for an Earth-like planet (Sellers, 1969; Alberti et al., 2015)), and $T_{0}$ represents the mean global planetary temperature. The growth-rate $\beta(T)$ of vegetation is a quadratic function of temperature, $\beta(T)=max\left[0;1-k(T-T_{opt})^{2}\right]$ (being $k$ a parameter for the growth curve width and $T_{opt}$ an optimal temperature), while the death-rate $\gamma$ is assumed to be constant (Watson & Lovelock, 1983; Alberti et al., 2015).

The free parameters $k$, $T_{opt}$, $\gamma$ and $\alpha_{v}$ are related to vegetation (as in Rombouts & Ghil, 2015; Alberti et al., 2015). In the following, since we assume an Earth-like vegetation, these parameters are set to Earth’s conditions. A complete list of the used parameters and their corresponding values is shown in Table 1.

As shown in previous studies (Sellers, 1969; Watson & Lovelock, 1983; Rombouts & Ghil, 2015; Alberti et al., 2015, and references therein), this set of parameters produces results in agreement with the observed Earth’s surface temperature. In particular, the model also shows oscillatory solutions which can reproduce the observed sawtooth-like behavior of paleoclimate changes (see Rombouts & Ghil, 2015, for more details).

By using the above set of parameters, we perform a parametric study of the solutions as functions of the initial fraction $A_{0}$ of land covered by vegetation and of the bare-ground albedo $\alpha_{g}$. Moreover, we use different values of $p$ and $m$ in order to investigate the effect of land/ocean distribution and the role of the greenhouse effect on planetary climates. We define a “surface water zone (SWZ)” as the circumstellar region where the planetary surface temperature ranges between $273$ K and $373$ K. It depends on the set $\{\theta\}$ of the variable parameters of the model and can be expressed as a a step-wise function

[TABLE]

Note that $\text{SWZ}(\{\theta\})$, in the parameter space $\{\theta\}$, generally defines a range where equilibrium temperatures calculated from the model are compatible with the presence of liquid water on planetary surface, independently from their atmospheric composition.

3 TRAPPIST-1 planetary climates

The possible climates of the TRAPPIST-1 planetary system are investigated by numerically solving Eq.s (1)-(2) through a second order Runge-Kutta scheme for time integration and looking at the stationary equilibrium solutions (Alberti et al., 2015). The luminosity of the star is set to

[TABLE]

where, based on stellar properties of TRAPPIST-1 (Gillon et al., 2016, 2017), we assumed that $L_{\star}=0.0005L_{\odot}$, $d_{p}=a*d_{\odot}$ (being $d_{\odot}=1$ au the Sun-Earth distance), and $S_{\odot}={L_{\odot}}/{4\pi d_{\odot}^{2}}=342.5$ W m*-2* is the mean solar radiation observed at the top of the Earth’s atmosphere. The initial temperatures are set equal to the equilibrium temperatures obtained by assuming a null Bond albedo (see Table 1 in Gillon et al., 2017) and the scale parameter $a$ is chosen as the mean distance of each T-x planet to the TRAPPIST-1 star (Gillon et al., 2017).

First of all we consider the case of rocky planets ($p=1$) with no vegetation ($A=0$) and no greenhouse effect ($m=0$). In Figure 1 we show the stationary solutions for the temperature of the planets, obtained from Eq.s (1)-(2), as functions of the star-planet distance, for different values of the bare-ground albedo $\alpha_{g}$.

When $\alpha_{g}$ is set to zero (i.e., for black dots in Figure 1), solutions reported in Table 1 of Gillon et al. (2017) are obtained, since they are equilibrium solutions of an energy-balance climate model with null Bond albedo hypothesis. Moreover, in this case only T-c and T-d reside in the surface water zone (SWZ). However, as the bare-ground albedo $\alpha_{g}$ is changed, different conditions can be observed, that is, as $\alpha_{g}$ increases some T-x planets can enter in or exit from the surface water zone. For example T-b could host surface liquid water on its surface only for a range of $\alpha_{g}$ close to $\alpha_{g}\simeq 0.5$. This suggests that in the simple case when planets are mainly rocky and without atmospheres, their residence in the surface water zone is dependent on their surface albedo. This consequently implies that the vegetation coverage is a main feedback acting as a thermal regulator for planetary temperature.

For the above reasons, in the following we investigate the climate properties of the planetary system when planetary conditions, related to different surface vegetation coverage and bare-ground albedo, are considered. We will show, in detail, T-x temperatures for three different situations: i) rocky planets without oceans or ice ($p=1$) and without greenhouse effect ($m=0$); ii) Earth-like land distribution ($p=0.3$) without greenhouse effect ($m=0$); iii) Earth-like land distribution ($p=0.3$) with a greenhouse effect similar to that observed on Earth ($m=0.6$). This gradual approach is useful to investigate planetary climates starting from different conditions, in order to make a comparative study on the possible climates of TRAPPIST-1 planets by considering several possible situations.

The stationary solutions for temperatures in the plane $(\alpha_{g},A_{0})$ are shown in Figure 2 for the case of rocky planets ($p=1$).

As said before, only the first three planets can be in the surface water zone when no atmosphere is considered ($m=0$). Moving from low to high values of $A_{0}$, planetary surface temperature changes accordingly, increasing as $A_{0}$ increases. More specifically, for $A_{0}\geq 0.5$ T-c displays only a weak temperature variation with $\alpha_{g}$. This indicates that vegetation acts as a feedback to maintain conditions for which $\text{SWZ}(\{\theta\})=1$. A similar behavior is recovered for T-d but for higher values of $A_{0}$ ($A_{0}\gtrsim 0.8$), suggesting that this exoplanet should be almost completely covered by vegetation to reside in the surface water zone. Conversely, due to its lower distance from the star, T-b shows an opposite behavior, entering the surface water zone for lower values of $A_{0}$, namely $A_{0}\lesssim 0.5$. For planets T-e, T-f, T-g and T-h, although the equilibrium temperature increases with $A_{0}$, the stellar irradiance is not enough to have global temperatures compatible with a surface water zone. We remark that the presence of liquid water in a planet without atmosphere does not make intuitive sense. The atmospheric envelope is indeed a fundamental component of a climate system to develop and maintain conditions for life on a planet. Therefore the case just discussed is presented mainly for a comparison of surface temperature conditions with the more Earth-like cases which are reported below.

We turn now to a situation where planets have an Earth-like land distribution. This is done by setting $p=0.3$, corresponding to a planet covered by land, ocean and ice. We also assume that the greenhouse effect is negligible ($m=0$). Figure 3 shows the stationary solutions for temperatures when $p=0.3$ and $m=0$.

The main difference with the previous case concerns planet T-d. Indeed, when an Earth-like land distribution is considered, the global surface temperature is always lower than $273$ K, that is, T-d is not in the surface water zone. For both T-b and T-c there are wide ranges of the parameters for which $\text{SWZ}(\{\theta\})=1$, while the other planets cannot host surface liquid water on their surfaces. The observed changes in equilibrium temperatures suggest that oceans play a primary role in setting the thermal equilibrium conditions for planetary surface temperature, at least when the greenhouse effect is not considered.

Let us now consider the same land distribution ($p=0.3$), but with an Earth-like greenhouse ($m=0.6$). The stationary solutions for temperatures are shown in Figure 4.

The planetary climates change with respect to the previous case, since temperatures increase. In particular, T-d again resides in the surface water zone for a wide range of $A_{0}$ and $\alpha_{g}$, although for $A_{0}<0.5$ the planet leaves that zone when $\alpha_{g}>0.5$. As for the previous cases, even using the Earth’s value of $m$, outer exoplanets cannot reach global surface temperatures in the range [273 K,373 K]. This implies that these planets need a different greenhouse effect with respect to the Earth, and consequently a different atmospheric composition, to enter the surface water zone. These results are quite different from those by Gillon et al. (2017) who showed that outer planets, with Earth-like atmospheres, could host water oceans on their surfaces. This could be related to the main difference between our model and that used in Gillon et al. (2017). While in our model the greenhouse effect is included by using a parametric approach (Sellers, 1969; Alberti et al., 2015), Gillon et al. (2017) utilized a 1-D radiative-convective cloud-free model in which greenhouse effect is taken into account by considering the contribution of several types of greenhouse gases (e.g., CO${}_{\text{2}}$, H${}_{\text{2}}$O, N${}_{\text{2}}$) with different partial pressures (Wordsworth et al., 2010). However, our results are quite in agreement with that reported by Wolf (2017), who showed, by using a 3-D climate model, that outer planets (i.e., T-f, T-g, and T-h) are not warmed enough, falling beyond the habitable zone and entering a snowball state. On the other hand, inner planets T-b and T-c show higher surface temperatures, so that to reside in the surface water zone their greenhouse effect should be similar or less efficient than on the Earth. In particular, T-b cannot be in the surface water zone for Earth-like greenhouse effect, while, when low values of $A_{0}$ and high value of $\alpha_{g}$ are considered, T-c is in the surface water zone for a narrow range of the model parameters. These results on inner planets are also in agreement with Wolf (2017) who reported that the inner three planets (T-b, T-c, and T-d) could reside in the traditional liquid water habitable zone but only with runaway greenhouse conditions.

The surface water zone changes significantly with the planetary atmospheric composition, such that, for instance, the first three planets can reside in the surface water zone for a wide range of the parameters, or, in some cases, even none of them can be there. In particular, it is interesting to see whether T-e, which is at the center of the system, could enter the surface water zone, since previous studies by Wolf (2017) have suggested this planet has the best chance to have water oceans on its surface. For this reason, we investigate changes in the surface water zone keeping fixed $\alpha_{g}=0.4$ and varying both $p$ and $m$.

In Figure 5 we show the stationary solutions for temperatures in the plane $(p,m)$. As expected, the planetary surface temperature strongly depends on the greenhouse effect conditions. In particular, T-b has a very narrow range of surface liquid water for very low values of $m$, while planets T-c and T-d have a wide range of parameters for which $\text{SWZ}(\{\theta\})=1$. Interestingly, for high values of $p$ and $m$, T-e enters the surface water zone. In an Earth-like situation this planet cannot have surface liquid water for lower values of $m$, suggesting that its atmosphere must have higher levels of greenhouse gases, such as CO${}_{\text{2}}$ or N${}_{\text{2}}$, with respect to those observed on Earth. This result is quite different from that obtained by Wolf (2017) according to which T-e has the best chance to be a habitable ocean-covered planet. This discrepancy could be related to the fact that our model considers the greenhouse effect in a parametric way, while the 3-D model used in Wolf (2017) directly uses the contribution of several types of greenhouse gases, as N${}_{\text{2}}$ and CO${}_{\text{2}}$ (similarly to Wordsworth et al., 2010). Indeed, a zero-dimensional climate model can determine the effective planetary emissivity of long wave radiation emitted to space, while a radiative-convective model considers different processes of energy transport, from radiative transfer through atmospheric layers to heat transport by convection. This allows to directly investigate the effects of varying greenhouse gas concentrations on thermal energy balance. However, although the results are different, the number of unknowns is such that it is not possible to know which climate model is more likely. Finally, outer planets (T-f, T-g and T-h) seem to be not in the surface water zone, entering a snowball state (Wolf, 2017), even if the greenhouse effect increases to higher levels than those observed on Earth.

4 Conclusions

In this paper we investigated the climate of the TRAPPIST-1 planetary system by using a zero-th order energy-balance model which allows us to outline the main features of the different planets. We found that the surface water zone, defined as the circumstellar region where a planet can host liquid water on its surface, is strongly dependent on the different parameters of the model and, in particular, on the initial fraction of vegetation coverage, the bare-ground albedo and the presence of oceans. More specifically, the “inner” three planets T-b, T-c, and T-d seem to be located in the surface water zone for several values of the parameters, as described before, while planet T-e, at variance to what have been reported in Gillon et al. (2017), can present water oceans only for greenhouse effect conditions different from the Earth. The climate of planet T-d seems to be the most stable from an Earth-like perspective, because this planet resides in the range of $\text{SWZ}(\{\theta\})=1$ for a wide interval of reasonable values of the different parameters. This result is not in agreement with that reported by Wolf (2017) for which the best candidate for a habitable ocean-covered surface is the planet T-e. This difference could be related to the different models employed, as in our energy-balance model a parametric description of the greenhouse effect is used, while in the 3D climate model by Wolf (2017) the contribution of several types of greenhouse gases is taken into account. However, since the number of unknowns makes difficult to choose one model with respect to another, different approaches, based either on simple or more complex climate models, can be useful. In this framework, our model has the advantage of transparency through minimal assumptions, allowing a comparative sets of cases to be studied.

Here we showed that the TRAPPIST-1 system can have different climates and that equilibrium temperatures depend on the global albedo, that is, on the mean physical conditions of the planetary surface. However, this parameter is strongly variable, since the vegetation could cover only a fraction of the surface, as for example the case of Earth. Moreover, also the greenhouse effect needs to be properly considered since it is one of the main feedback in regulating thermal energy balance. Investigating these features require more sophisticated models, extended to space variables, at least with a description of the atmospheric heat diffusion. The model is actually under investigation and results will be reported in a forthcoming paper.

We acknowledge S. Savaglio for useful discussions and for her interest in our work. We thank the anonymous reviewer for fruitful and helpful suggestions.