The Astrophysical Journal, 804:50 (20pp), 2015 May 1 doi:10.1088/0004-637X/804/1/50 © 2015. The American Astronomical Society. All rights reserved. MODELING THE SURFACE TEMPERATURE OF EARTH-LIKE PLANETS Giovanni Vladilo1,2, Laura Silva1, Giuseppe Murante1,3, Luca Filippi3,4, and Antonello Provenzale3,5 1 INAF—Trieste Astronomical Observatory, Trieste, Italy; [email protected] 2 Department of Physics, University of Trieste, Trieste, Italy 3 Institute of Atmospheric Sciences and Climate—CNR, Torino, Italy 4 DIMEAS, Politecnico di Torino, Torino, Italy 5 Institute of Geosciences and Earth Resources—CNR, Pisa, Italy Received 2015 January 29; accepted 2015 April 3; published 2015 April 29 ABSTRACT We introduce a novel Earth-like planet surface temperature model (ESTM) for habitability studies based on the spatial–temporal distribution of planetary surface temperatures. The ESTM adopts a surface energy balance model (EBM) complemented by: radiative–convective atmospheric column calculations, a set of physically based parameterizations of meridional transport, and descriptions of surface and cloud properties more refined than in standard EBMs. The parameterization is valid for rotating terrestrial planets with shallow atmospheres and moderate values of axis obliquity ( ≲°45 ). Comparison with a 3D model of atmospheric dynamics from the literature shows that the equator-to-pole temperature differences predicted by the two models agree within ≈5 K when the rotation rate, insolation, surface pressure and planet radius are varied in the intervals 0.5 ≲≲WW⊕ 2, 0.75≲≲SS◦ 1.25, 0.3≲≲p (1 bar) 10, and 0.5 ≲≲RR⊕ 2, respectively. The ESTM has an extremely low computational cost and can be used when the planetary parameters are scarcely known (as for most exoplanets) and/or whenever many runs for different parameter configurations are needed. Model simulations of a test-case exoplanet (Kepler-62e) indicate that an uncertainty in surface pressure within the range expected for terrestrial planets may impact the mean temperature by ∼60 K. Within the limits of validity of the ESTM, the impact of surface pressure is larger than that predicted by uncertainties in rotation rate, axis obliquity, and ocean fractions. We discuss the possibility of performing a statistical ranking of planetary habitability taking advantage of the flexibility of the ESTM. Key words: astrobiology – methods: numerical – planetary systems 1. INTRODUCTION dependent indices of fractional habitability (e.g., Spiegel et al. 2008). The large amount of exoplanet data collected with the Modeling Tt(,)φ is a difficult task since many of the Doppler and transit methods (e.g., Batalha et al. 2013; Mayor physical and chemical quantities that govern the exoplanet et al. 2014, and references therein) indicate that Earth-size surface properties are currently not measurable. A way to cope planets are intrinsically more frequent than giant ones, in spite fi with this problem is to treat the unknown quantities as free of the fact that they are more dif cult to detect. Small planets parameters and use fast climate calculations to explore how are found in a relatively broad range of metallicities (Buchhave ) variations of such parameters affect the surface temperature. et al. 2012 and, at variance with giant planets, their detection General circulation models (GCMs) are not suited for this type rate drops slowly with decreasing metallicity (Wang & ) of exploratory work since they require large amounts of Fischer 2015 . These observational results indicate that computational resources for each single run as well as a ( Earth-like planets are quite common around other stars e.g., detailed knowledge of many planetary characteristics. Two ) Farr et al. 2014 and are expected to be detected in large types of fast climate tools are commonly employed in studies of numbers in the future. Their potential similarity to the Earth planetary habitability: single atmospheric column calculations makes them primary targets in the quest for habitable and energy balance models (EBMs). Atmospheric column environments ouside the solar system. Unfortunately, small calculations treat in detail the physics of vertical energy planets are quite difficult to characterize with experimental transport, taking into account the influence of atmospheric methods and a significant effort of modelization is required to composition on the radiative transfer (e.g., Kasting 1988). This cast light on their properties. The aim of the present work is to is the type of climate tool that is commonly employed in model the surface temperature of these planets as a contribution studies of the “habitable zone” (e.g., Kasting 1988; Kasting to the study of their surface habitability. The capability of an et al. 1993; Kopparapu et al. 2013, 2014; von Paris et al. 2013). environment to host life depends on many factors, such as the EBMs calculate the zonal and seasonal energy budget of a presence of liquid water, nutrients, energy sources, and planet using a heat diffusion formalism to describe the shielding from cosmic ionizing radiation (e.g., Seager 2013; horizontal transport and simple analytical functions of the Guedel et al. 2014). A knowledge of the surface temperature is surface temperature to describe the vertical transport (e.g., essential to apply the liquid water criterion of habitability and North et al. 1981). EBMs have been employed to address the can also be used to assess the potential presence of different life climate impact induced by variations of several planet forms according to other types of temperature-dependent parameters, such as axis obliquity, rotation period and stellar biological criteria (e.g., Clarke 2014). Here we are interested insolation (Spiegel et al. 2008, 2009, 2010; Dressing et al. in modeling the latitudinal and seasonal variations of surface 2010; Forgan 2012, 2014). By feeding classic EBMs with temperature, Tt(,)φ , as a tool to calculate temperature- multi-parameter functions extracted from atmospheric column 1 The Astrophysical Journal, 804:50 (20pp), 2015 May 1 Vladilo et al. Figure 1. Scheme of the Earth-like planet surface temperature model (ESTM). The zonal and seasonal EBM (central box) is fed by physical quantities (circled symbols) described in Section 2. At variance with classic EBMs, the physical quantities are multi-parameter functions modeled with the aid of atmospheric column calculations (I and A) and 3D climate experiments (D). calculations one can obtain an upgraded type of EBM that takes of the surface conditions. By assuming that the heating and into account the physics of vertical transport (Williams & cooling rates are balanced in each zone, one obtains an energy Kasting 1997). Following a similar approach, in a previous balance equation that is used to calculate Tt(,)φ . The most paper we investigated the impact of surface pressure on the common form of EBM equation (North et al. 1981; Williams & habitability of Earth-like planets by incorporating a physical Kasting 1997; Spiegel et al. 2008; Pierrehumbert 2010; treatment of the thermal radiation and a simple scaling law for Gilmore 2014) is the meridional transport (Vladilo et al. 2013, hereafter Paper I). T ⎡ T ⎤ Here we include the transport of the short-wavelength radiation ∂ ∂ 2 ∂ C − ⎣⎢Dx()1(1),(1)− ⎦⎥ +=IS − A and we present a physically based treatment of the meridional ∂tx∂ ∂x transport tested with 3D experiments. In this way we build up ( ) where x = sin φ and all terms are normalized per unit area. The an Earth-like planet surface temperature model ESTM in fi which a variety of unknown planetary properties can be treated rst term of this equation represents the zonal heat storage and as free parameters for a fast calculation of the surface describes the temporal evolution of each zone; C is the zonal ( ) habitability. The ESTM is presented in the next section. In heat capacity per unit area North et al. 1981 . The second term Section 3 we describe the model calibration and validation. represents the amount of heat per unit time and unit area Examples of model applications are presented in Section 4 and leaving each zone along the meridional direction (North the conclusions summarized in Section 5. et al. 1981, Equation (21)). It is called the “diffusion term” because the coefficient D is defined on the basis of the analogy with heat diffusion, i.e., 2. THE MODEL ∂T The ESTM consists of a set of climate tools and algorithms Φ≡−D ,(2) that interchange physical quantities expressed in parametric ∂φ form. The core of the model is a zonal and seasonal EBM fed where 2πR2Φ cos φ is the net rate of energy transport6 across a by multi-parameter physical quantities. The parameterization is circle of constant latitude and R is the planet radius (see obtained using physically based climate tools that deal with the ) I meridional and vertical energy transport. The relationship Pierrehumbert 2010 . The term represents the thermal between these ingredients is shown in the scheme of Figure 1. radiation emitted by the zone, also called outgoing longwave ( ) In the following we present the components of the model, radiation OLR . The right side of the equation represents the starting from the description of the EBM. fraction of stellar photons that heat the surface of the zone; S is In zonal EBMs the surface of the planet is divided into zones the incoming stellar radiation and A the planetary albedo at the delimited by latitude circles. The surface quantities of interest top of the atmosphere. All coefficients of the equation depend, are averaged in each zone over one rotation period. In this way, in general, on both time and latitude, either directly or the spatial dependence is determined by a single coordinate, the indirectly, through their dependence on T. latitude φ. Since the temporal dependence is “smoothed” over one rotation period, the time, t, represents the seasonal 6 With the adopted definitions of D and Φ it is easy to show that ⎡ ⎤ evolution during the orbital period.