Albedos, Equilibrium Temperatures, and Surface Temperatures of Habitable Planets

Abstract

The potential habitability of known exoplanets is often categorized by a nominal equilibrium temperature assuming a Bond albedo of either ∼0.3, similar to Earth, or 0. As an indicator of habitability, this leaves much to be desired, because albedos of other planets can be very different, and because surface temperature exceeds equilibrium temperature due to the atmospheric greenhouse effect. We use an ensemble of general circulation model simulations to show that for a range of habitable planets, much of the variability of Bond albedo, equilibrium temperature and even surface temperature can be predicted with useful accuracy from incident stellar flux and stellar temperature, two known parameters for every confirmed exoplanet. Earth’s Bond albedo is near the minimum possible for habitable planets orbiting G stars, because of increasing contributions from clouds and sea ice/snow at higher and lower instellations, respectively. For habitable M star planets, Bond albedo is usually lower than Earth’s because of near-IR H₂O absorption, except at high instellation where clouds are important. We apply relationships derived from this behavior to several known exoplanets to derive zeroth-order estimates of their potential habitability. More expansive multivariate statistical models that include currently non-observable parameters show that greenhouse gas variations produce significant variance in albedo and surface temperature, while increasing length of day and land fraction decrease surface temperature; insights for other parameters are limited by our sampling. We discuss how emerging information from global climate models might resolve some degeneracies and help focus scarce observing resources on the most promising planets.

1. Introduction

The proliferation of newly discovered rocky planets has expanded the focus of exoplanet science from detection to characterization of their potential habitability. Initial assessments of habitability are based on a planet’s location relative to some definition of the habitable zone (see Kane et al. 2016). Often this is reported as a nominal planetary equilibrium (blackbody) temperature Tⁿ_eq, assuming a Bond albedo A ∼ 0.3 like Earth, or A = 0, an unrealizable value (e.g., Borucki et al. 2012; Anglada-Escudé et al. 2016; Dittmann et al. 2017; Gillon et al. 2017).

A habitable planet’s albedo, however, will be different when placed in a different orbit from Earth’s and/or in orbit around a different star, even if the planet is very Earth-like, precisely because such planets have water. An important feature of Earth’s climate is its existence near the triple point of H₂O. Surface ice, surface liquid, and liquid or ice suspended as cloud particles each make non-negligible contributions to Earth’s Bond albedo. Water vapor does not, but only because the Sun is a G star that emits much of its radiation at wavelengths at which H₂O does not absorb strongly (slightly more so as it ages). Thus, differences in the occurrence of the three phases of water and their interaction with starlight, if predictable, can shed light on actual equilibrium temperatures. Furthermore, equilibrium temperature differs from the actual parameter of interest for habitability, the surface temperature, depending on the atmosphere’s greenhouse effect, which varies with composition and pressure. Quantifying the relationship between equilibrium and surface temperature for habitable planets is necessary for assessing potential habitability.

Three-dimensional general circulation models (GCMs) have begun to address questions about conditions that are conducive to habitability for particular exoplanets (Turbet et al. 2016, 2018; Boutle et al. 2017; Wolf 2017, 2018; Del Genio et al. 2019b) and a range of hypothetical exoplanets (Leconte et al. 2013; Shields et al. 2013, 2014; Yang et al. 2013, 2014, 2019; Wolf & Toon 2014, 2015; Bolmont et al. 2016; Kopparapu et al. 2016, 2017; Fujii et al. 2017a; Wolf et al. 2017; Haqq-Misra et al. 2018; Lewis et al. 2018; Way et al. 2018; Jansen et al. 2019; Komacek & Abbot 2019). Nonetheless, such models are computationally expensive and not generally available to the larger community, and thus only a limited number of simulations exist to date. At the same time, the population of known rocky exoplanets is likely to grow dramatically as a result of upcoming spacecraft missions and new ground-based telescopes, yet the lengthy observing times needed to characterize small planets will limit such efforts to a small fraction of them. How might the meager available information about rocky exoplanets be used to choose the most promising subset for further study, given limited observing resources?

This paper describes the first steps of an approach to distill the information in GCM simulations into relationships that can be applied to known or probable rocky planets to refine initial assessments of their potential habitability. It cannot anticipate whether water, or any atmosphere, exists on any given planet whose estimated size and/or mass makes it likely to be rocky. It can only tell us what the plausible range of Bond albedos might be for a given rocky planet if it has water, and whether that range bodes well for its potential habitability.

Section 2 describes the GCM experiments used in our study, while Section 3 presents the analysis methods we apply to the ensemble. Section 4 explores factors that control Bond albedo, and thus the actual equilibrium temperature, for habitable planets and how predictable these may be from the sparse information currently available for exoplanets; it also estimates errors in surface temperature if predicted from the equilibrium temperature. Section 5 demonstrates the use of the predictor by applying it to a number of known exoplanets. Section 6 (plus Appendices A and B) explores additional parameters that may influence albedo and temperature and discusses how ambiguities that limit our predictive ability might be reconciled with future observations. Section 7 summarizes our findings and suggests future directions.

2. GCM Ensemble

We use simulations performed with the ROCKE-3D GCM (Way et al. 2017) in this analysis. All simulations, except one dry “land planet,” couple the atmosphere to a dynamic ocean. Ideally one would construct a large ensemble of simulations that systematically vary every relevant external parameter to represent all possible exoplanet climates. Such an ensemble does not yet exist. Instead, we employ a sparse “ensemble of opportunity” of 48 simulations already conducted with ROCKE-3D to illustrate the concept. Most simulations are for “Earth-like” atmospheres, with ∼1 bar of N₂ as the dominant atmospheric constituent, CO₂ and sometimes CH₄, and surface water. However, we do include several thinner and thicker atmospheres as well as CO₂-dominated atmospheres. The full diversity of habitable exoplanets is not represented by the ensemble, but it does include different instellations, stellar types (G and M), planet sizes and gravities, obliquities and spin–orbit states, land–ocean configurations, and ocean properties. In particular, several subensembles produce different albedos for reasons other than instellation, stellar type, or greenhouse gas concentrations. This illustrates degeneracies unrelated to radiative properties that are rarely considered in discussions of exoplanet climates.

Table 1 provides the relevant properties of all the simulated planets. Our ensemble consists of the following general classes of planets:

1. Proxima Centauri b (Del Genio et al. 2019b): 10 simulations using estimated planet size, gravity, stellar spectrum, distance from the star (Anglada-Escudé et al. 2016), and with different surfaces, atmospheric pressures and compositions, instellations, and spin–orbit configurations.

2. GJ 876 (Fujii et al. 2017a): 4 hypothetical aquaplanets in synchronous rotation using Earth size and gravity and incident stellar flux from 0.6 to 1.2× Earth’s solar constant S_o.

3. Hypothetical early Venus (Way et al. 2016): 3 simulations with Venus’ size, gravity, and modern rotation period (243 days), with different surfaces harboring liquid water, and with insolation and spectrum at different points in Venus’ past.

4. Early Earth: Four periods with different insolations, compositions and surfaces; 3 Archean Earths based on Charnay et al. (2013) (Del Genio et al. 2019a); 1 Huronian snowball Earth; 4 Sturtian equatorial waterbelt Earths (Sohl & Chandler 2007); 1 mid-Cretaceous equable Earth (Chandler et al. 2017).

5. Earth rotation-insolation experiments (Way et al. 2018): 5 simulations with different insolations, rotation periods, and zero obliquity and eccentricity, plus 6 other simulations with Earth’s actual obliquity and insolation and different rotation periods.

6. Obliquity experiments (Colose et al. 2019): 9 Earth-like aquaplanets with low or high obliquity, and different insolations and greenhouse gas concentrations, including examples with warm and cold start initial conditions that produce bistable behavior.

7. Hypothetical dry early Mars: Mars size, gravity, and orbital properties with 3.8 Ga instellation and 1 bar CO₂, but only trace amounts of water and no CO₂ surface ice.

8. Modified Kepler-1649 b analog (Kane et al. 2018): Aquaplanet in synchronous rotation with estimated planet size and gravity; modern Earth-like atmosphere but weaker instellation and longer orbital period than observed to place it inside the habitable zone.

Table 1.

Relevant Properties of Simulated Planets^a

Simulation	Sox	Tstar (K)	psurf (b)	Major Gases	Trace Gases (ppmv)	Prot (days)	Spin–Orbit	Surface	Φ(°)	A	Ts (K)
Proxima Cen b	Del Genio et al. (2019b)
(1) Control	0.65	3050	0.984	N2	CO2 (376)	11.2	1:1	Aqua	0	0.234	252
(2) Control-Hi	0.70	3050	0.984	N2	CO2 (376)	11.2	1:1	Aqua	0	0.232	260
(3) Archean-M	0.65	3050	0.984	N2	CO2 (900) CH4 (900)	11.2	1:1	Aqua	0	0.181	255
(4) Archean-H	0.65	3050	0.984	N2	CO2(104) CH4 (2000)	11.2	1:1	Aqua	0	0.161	263
(5)High Salinity	0.65	3050	0.984	N2	CO2 (376)	11.2	1:1	Aqua	0	0.178	263
(6) 3:2e30	0.65	3050	0.984	N2	CO2 (376)	7.5	3:2	Aqua	0	0.199	259
(7) Day-Land	0.65	3050	0.984	N2	CO2 (376)	11.2	1:1	Modern Earth	0	0.253	241
(8) Control-Thin	0.65	3050	0.1	N2	CO2 (376)	11.2	1:1	Aqua	0	0.310	236
(9) Control-Thick	0.65	3050	10	N2	CO2 (376)	11.2	1:1	Aqua	0	0.237	270
(10) Pure CO2	0.65	3050	1	CO2	None	11.2	1:1	Aqua	0	0.222	284
GJ 876	Fujii et al. (2017a)
(11) 0.6	0.60	3129	1	N2	CO2 (1)	32.3	1:1	Aqua	0	0.254	231
(12) 0.8	0.80	3129	1	N2	CO2 (1)	26.1	1:1	Aqua	0	0.255	257
(13) 1.0	1.00	3129	1	N2	CO2 (1)	22.0	1:1	Aqua	0	0.332	267
(14) 1.2	1.20	3129	1	N2	CO2 (1)	19.2	1:1	Aqua	0	0.311	303
Early Venus	Way et al. (2016)
(15) 1.5	1.48	5790	1.01	N2	CO2 (400) CH4 (1)	243	Asyn	Venus topog with ocean	2.6	0.521	284
(16) 1.9	1.90	5785	1.01	N2	CO2 (400) CH4 (1)	243	Asyn	Venus topog with ocean	2.6	0.609	289
(17) 2.4	2.40	5785	1.01	N2	CO2 (400) CH4 (1)	243	Asyn	Aqua	2.6	0.671	315
Early Earth	Del Genio et al. (2019a), Sohl & Chandler (2007), Chandler et al. (2017)
(18)Archean A	0.80	5710	1.01	N2	CO2 (900) CH4 (900)	1	Asyn	Modern Earth	23.5	0.492	235
(19)Archean B	0.80	5710	0.984	N2	CO2 (104) CH4 (2000)	1	Asyn	Modern Earth	23.5	0.378	266
(20)Archean C	0.80	5710	0.984	N2	CO2 (105) CH4 (2000)	1	Asyn	Sturtian Earth	23.5	0.286	304
(21) Huronian	0.84	5728	0.984	N2 (0.79) O2(0.21)	CO2 (40) CH4 (0.751) N2O (0.275)	1	Asyn	Sturtian Earth	23.5	0.504	230
(22) Sturtian 1	0.94	5760	0.984	N2 (0.79) O2 (0.21)	CO2 (40) CH4 (0.751) N2O (0.275)	1	Asyn	Sturtian Eartth	23.5	0.397	260
(23) Sturtian 2	0.94	5760	0.984	N2 (0.79) O2 (0.21)	CO2 (285) CH4 (0.791) N2O (0.275)	1	Asyn	Sturtian Earth	23.5	0.348	274
(24) Sturtian 3	0.94	5760	0.984	N2 (0.79) O2 (0.21)	CO2 (140) CH4 (0.751) N2O (0.275)	1	Asyn	Sturtian Earth	23.5	0.373	268
(25) Sturtian 4	0.90	5760	0.984	N2 (0.79) O2 (0.21)	CO2 (40) CH4 (0.751) N2O (0.275)	1	Asyn	Sturtian Earth	23.5	0.418	253
(26) Cretaceous	0.995	5773	0.984	N2 (0.79) O2 (0.21)	CO2 (40) CH4 (0.751) N2O (0.275)	1	Asyn	Cretaceous Earth	23.5	0.295	295
Earth rot-S0	Way et al. (2018)
(27) 1/1.1z	1.1	5787	0.984	N2	CO2 (400)	1	Asyn	Modern Earth	0	0.348	301
(28) 1/1.2z	1.2	5787	0.984	N2	CO2(400)	1	Asyn	Modern Earth	0	0.373	313
(29) 16/1.1z	1.1	5787	0.984	N2	CO2 (400)	16	Asyn	Modern Earth	0	0.278	298
(30) 16/1.2z	1.2	5787	0.984	N2	CO2 (400)	16	Asyn	Modern Earth	0	0.314	305
(31) 16/1.3z	1.3	5787	0.984	N2	CO2 (400)	16	Asyn	Modern Earth	0	0.358	312
(32) 1/1.0	1	5785	0.984	N2(0.79) O2 (0.21)	CO2 (400) CH4 (1)	1	Asyn	Modern Earth	23.5	0.306	289
(33) 8/1.0	1	5785	0.984	N2 (0.79) O2 (0.21)	CO2 (400) CH4 (1)	8	Asyn	Modern Earth	23.5	0.294	285
(34) 16/1.0	1	5785	0.984	N2 (0.79) O2 (0.21)	CO2 (400) CH4 (1)	16	Asyn	Modern Earth	23.5	0.273	287
(35) 64/1.0	1	5785	0.984	N2 (0.79) O2 (0.21)	CO2 (400) CH4 (1)	64	Asyn	Modern Earth	23.5	0.307	282
(36) 256/1.0	1	5785	0.984	N2 (0.79) O2 (0.21)	CO2 (400) CH4 (1)	256	Asyn	Modern Earth	23.5	0.328	277
(37) 365/1.0	1	5785	0.984	N2 (0.79) O2 (0.21)	CO2 (400) CH4 (1)	365	1:1	Modern Earth	23.5	0.372	265
Earth obliquity	Colose et al. (2019)
(38) 1	0.79	5785	1	N2	CO2 (1000)	1	Asyn	Aqua	75	0.251	277
(39) 1 cs	0.79	5785	1	N2	CO2 (1000)	1	Asyn	Aqua	75	0.450	245
(40) 2	1	5785	1	N2	CO2 (100)	1	Asyn	Aqua	20	0.302	288
(41) 2 cs	1	5785	1	N2	CO2 (100)	1	Asyn	Aqua	20	0.483	258
(42) 3	0.79	5785	1	N2	CO2(5×104)	1	Asyn	Aqua	20	0.452	256
(43) 4 cs	1	5785	1	N2	None	1	Asyn	Aqua	75	0.427	258
(44) 5	1	5785	1	N2	CO2 (5000)	1	Asyn	Aqua	75	0.312	315
(45) 6 cs	0.70	5785	1	N2	CO2 (105)	1	Asyn	Aqua	75	0.447	247
(46) 7	0.70	5785	1	N2	CO2 (1000)	1	Asyn	Aqua	20	0.500	225
Other planets
(47) Early Mars	0.32	5673	0.984	CO2 (0.87)	CH4 (104)	1.02	Asyn.	Modern Mars	25.2	0.248	194
(48) Kepler-1649b Kane et al. (2018)	1.4	3200	1.013	N2(0.12) N2	CO2 (376)	50	1:1	Aqua	0	0.408	332

Note.

^aS_ox = incident stellar flux relative to Earth; T_star = stellar effective temperature; p_surf = atmospheric surface pressure; P_rot = rotation period in Earth days, φ = obliquity; A = Bond albedo; all simulations have either zero or modern Earth eccentricity except Proxima Centauri b 3:2e30 (e = 0.30) and Early Mars (e = 0.093); obliquity simulations labeled “cs” (cold start) were initialized from a snowball Earth; all simulations have H₂O (only trace amounts for Early Mars).

Most of these planets are habitable based on the criterion of the presence of surface liquid water. The exceptions are Huronian Earth, a complete snowball; the early Mars analog, which has traces of water vapor and ice but is mostly a cold desert planet; and Kepler-1649 b, whose surface temperature stays below 335 K but with stratospheric H₂O mixing ratios an order of magnitude larger than the traditional “moist greenhouse” threshold of Kasting et al. (1993).

3. Methods

Our study applies three analysis approaches to the global mean climates of the GCM ensemble:

1. We use the method of Donohoe & Battisti (2011) to calculate the contributions of the surface and the atmosphere to the Bond albedo of each planet and thus partition the ensemble into planets whose albedos are dominated by clouds versus planets with significant contributions of surface scattering to albedo.

2. We develop a predictor for Bond albedo and equilibrium temperature that can be applied now to exoplanets for which the only available relevant information is the temperature of the star and the planet’s distance from it. This is accomplished via simple linear multiple regressions of Bond albedo versus incident starlight and stellar temperature for the subensembles described in (1) above. A separate regression of surface temperature versus equilibrium temperature is used to translate the Bond albedo prediction into a surface temperature estimate and to assess the accuracy with which surface temperature is predictable for habitable planets given current sparse information. The technique is then applied to some known exoplanets.

3. We then consider possibilities for constraining exoplanet albedo and surface temperature given information on additional planet parameters that might be obtained from future missions. For this higher-dimensional statistical analysis, we use the Alternating Conditional Expectations (ACEs) algorithm (Breiman & Friedman 1985). ACE iteratively solves for possibly nonlinear transforms of variables that maximize their correlation with the response variable in a generalized additive model (GAM; McCullagh & Nelder 1989), which is a linear sum of these nonlinear transforms. If the transform of a response variable is nonlinear, a curve fit to the ACE transform can both describe the behavior and enable back-transformation to the original units of the response. Multiple linear regression on these transforms then quantifies which variables are significant. It may be beneficial or desirable to fit simpler transforms to parametric curves (e.g., piecewise linear, log, polynomial). For confidence in coefficient estimates we use Student’s t; for goodness of fit we use the adjusted R² (which penalizes for number of parameters). For models with the same predictor variable (e.g., the raw data or the same transform), we use Akaike’s Information Criterion (AIC; Akaike 1981; Sakamoto et al. 1986) to distinguish the better model. Further details are given in Appendix A. Wang & Murphy (2004) show an example of the use of ACE to look for nonlinear relations.

4. Prediction of Albedo and Surface Temperature from S_ox and T_star

4.1. Controls on Bond Albedo and Equilibrium Temperature

Figure 1 illustrates the major effects on Bond albedo that cause planets to deviate from Earth’s A = 0.3. Figures 1(a), (b) and (c), (d) are for planets orbiting G and M stars, respectively. Sturtian Earth 4 (Figure 1(a)), irradiated by a dimmer Sun than modern Earth and with low CO₂ concentration, maintains an equatorial “waterbelt” region of open ocean but is largely covered by sea ice and snow poleward of ∼±30° latitude, giving it a fairly high Bond albedo (A = 0.418). In contrast, Archean Earth C (Figure 1(b)) has greatly elevated CO₂ and CH₄ and a very warm, almost ice-free climate despite the faint young Sun, making its albedo a bit darker than modern Earth’s (A = 0.286). Proxima Centauri b Control (Figure 1(c)) has a large dayside open ocean area, but this is significantly obscured by optically thick clouds that are typical of synchronously but slowly rotating exoplanets in GCM simulations (Yang et al. 2013). Nonetheless, it has a low Bond albedo (A = 0.234) because the incident flux from its very cool star is mostly in the near-IR and is strongly absorbed by atmospheric H₂O and CO₂, as well as by sea ice, which is darker in the near-IR than the visible (Joshi & Haberle 2012). Modified Kepler-1649 b (Figure 1(d)) is our most highly irradiated M star planet and a good example of the day–night circulation on synchronous slow rotators that produces thick dayside clouds. It thus has a high Bond albedo relative to our other M star cases (A = 0.408) but much lower than some of our G star planets because of the absorption of the mostly near-IR M star spectrum.

Figure 1.

Bond albedo maps for the (a) Sturtian Earth 4 (A = 0.418), (b) Archean Earth C (A = 0.286), (c) Proxima Centauri b Control (A = 0.234), and (d) modified Kepler-1649 b (A = 0.408) simulations. Gray areas represent the nightside for synchronously rotating planets.

All of these effects on Bond albedo are evident to varying degrees in the complete ensemble (Figure 2). A ∼ 0.3 is actually quite uncommon in the ensemble, despite it often being the default choice of astronomers (Figure 2(a)). For G star planets, the albedo dependence on instellation relative to that received by Earth (S_ox) exhibits something like a V-shape, with high albedos for planets much more strongly and weakly irradiated than modern Earth and a minimum near the value for modern earth (S_ox = 1). This is the result of Earth’s surface existing near the triple point of water—at much higher S_ox much more water vapor enters the atmosphere and thick clouds develop, obscuring the surface and increasing the Bond albedo, while at much lower S_ox sea ice and snow are more widespread, also increasing the Bond albedo (e.g., Figure 1(a)).

Figure 2.

(a) Bond albedo (A) versus incident stellar flux normalized by Earth’s solar constant (S_ox) for all members of the ensemble. (b) Fractional contribution of the ground albedo to A versus S_ox for the ensemble (A_surf/A). Red/blue symbols represent cool/warm star planets.

A few of our G star planets have albedos slightly lower than modern Earth; these are primarily planets with elevated greenhouse gas concentrations and thus less snow/ice (e.g., Figure 1(b)), or modestly slow rotators (8 or 16 days period) whose circulation is dominated by a broad Hadley cell rather than day–night contrasts, leading to a narrow equatorial band of reflective clouds rather than the extensive dayside cloud decks on very slowly rotating planets (Yang et al. 2014; Kopparapu et al. 2016; Way et al. 2018). Depending on these other properties, the minimum Bond albedo can occur at S_ox values somewhat higher or lower than that of modern Earth. Regardless of these other properties, though, it is difficult for a habitable G star planet to have a Bond albedo ≪0.3; far from being a representative value, it is close to a lower limit.

Figure 2(a) also shows that the situation is quite different for M star planets. Such planets can have albedos higher than Earth if they have high enough instellation and thick dayside clouds (e.g., Figure 1(d)), but most of our M star planets have lower albedos than modern Earth. This is due to the mostly near-IR spectrum of M stars and the resulting absorption of incident starlight by atmospheric H₂O and CO₂, as well as sea ice. As a result, Bond albedos for M star planets are more tightly clustered than for G star planets. Again, A ∼ 0.3 is not a representative value—it is instead not far from an upper limit for habitable planets orbiting such stars. We emphasize that the behavior described here is only for habitable planets. Planets without atmospheres or surface water can have low Bond albedos, for example, our dry early Mars, the darkest G star planet we simulate (A = 0.248). Likewise, very cold planets may have a hydrohalite crust with a high near-IR surface albedo (Shields & Carns 2018) and thus higher Bond albedo.

As implied by the previous discussion, Bond albedo is the result of contributions from both the atmosphere and surface. Donohoe & Battisti (2011) use upwelling and downwelling solar/stellar flux at the top of the atmosphere and the surface to separate the atmospheric (A_atm) and surface (A_surf) contributions to A based on the ground albedo A_g and the fractional reflection r and absorption a of shortwave flux for each pass of the radiation through the atmosphere:

$$A_{\text{atm}}=r$$ (1)

$$A_{\text{surf}}=A_g{\left(1-r-a\right)}^2/\left(1-A_{g}r\right).$$ (2)

Equation (2) shows that the surface contribution to Bond albedo depends not just on the ground albedo itself, but also on the extent to which starlight reflected from the surface is attenuated by the atmosphere above it. For modern Earth, Donohoe & Battisti (2011) find that A_atm = 0.88A and A_surf = 0.12A—that is, despite the fact that Earth’s surface is partly visible from space and partly covered by fairly bright desert or very bright sea ice and snow, most of its Bond albedo is due to scattering by clouds and to a lesser extent by Rayleigh scattering of the clear atmosphere. Figure 2(b) shows A_surf/A versus S_ox for the ensemble. All planets with S_ox > 1 have A_surf/A ≪ 1 (i.e., their Bond albedos are cloud-dominated). Most but not all planets with S_ox < 1 have A_surf/A > 0.1. For several G star partial or total snowball planets, the surface contribution controls the Bond albedo, but this is not true for any M star planet. The contribution of the surface to Bond albedo for weakly irradiated planets is a function mostly of the opacity of the overlying atmosphere and the albedo of the surface at wavelengths of maximum instellation.

4.2. Predicting A and T_eq from S_ox and T_star

Motivated by the qualitatively different roles of the surface and atmosphere for instellation stronger versus weaker than Earth’s in Figure 2(b) and the desire to use current observational constraints to assess potential habitability, we “predict” Bond albedo (A^p) for the ensemble with two linear regressions of A against S_ox and normalized stellar temperature $T_{\text{star}}^{\text{'}}=T_{\text{star}}/4500\text{K}$, one for strongly irradiated atmosphere-dominated planets and another for weakly irradiated planets with potentially significant surface contributions. The choice of breakpoint between the two regressions is limited by the sampling of S_ox in our GCM simulations. Our ensemble contains samples at S_ox = 0.94, 0.995, 1.0, and 1.1. By testing different breakpoints we find that including all planets with S_ox > 0.94 in the strong insolation subensemble explains the greatest amount of variance. Our parameter space samples only M and G stars with a wide gap in between, and preliminary exploratory regressions showed that the effect of T_star is significant across but not within the stellar types in our ensemble, so we segment the regression only on S_ox. The resulting regressions are

$$A^p=0.236S_{\text{ox}}+0.162T_{\text{star}}^{\text{'}}-0.101\left(2.40⩾S_{\text{ox}}⩾0.995\right)$$ (3)

$$A^p=0.347S_{\text{ox}}+0.162T_{\text{star}}^{\text{'}}-0.101\hspace{0.17em}\left(0.94⩾S_{\text{ox}}⩾0.32\right).$$ (4)

Table 2 gives the details for these regressions. Figure 3 shows the predicted albedo A^p versus actual albedo A for each simulation. The rms error in A^p is 0.07, versus 0.12 when A = 0.3 is assumed. It is evident that S_0x and T_star have some predictive skill for Bond albedo, but several degeneracies associated with the dependence of albedo on other external parameters limit this skill. These are visible as clusters of points with the same A^p but a range of actual A. For example, the Proxima Centauri b simulations, all but one having the same S_0x and T_star and thus the same predicted albedo (0.223), actually range in albedo from 0.161 to 0.310 because of differing greenhouse gas amounts, atmospheric thickness, ocean salinity, spin–orbit state, and the presence/absence of exposed land. Two pairs of otherwise identical Earth simulations that differ only in initial condition (modern Earth versus snowball state) equilibrate to different climates whose Bond albedos differ by ∼0.2. High obliquity planets (75°) tend to have lower albedos than low obliquity planets (20°). The same Earth-like planet with rotation periods from 1 to 365 days yields Bond albedos ranging from 0.273 to 0.372. Finally, planets with greater amounts of greenhouse gases tend to be darker.

Table 2.

Segmented Multiple Regression for A^p versus S_ox and $T_{\text{star}}^{\prime }$

	Estimated Coefficient	Std. Error	t Value	Pr(>|t|)	Confidence(1 − α) in Estimated Coefficienta	ANOVA(1) Fraction of SumSq	Pr(>F)	Significance α of ANOVA Modela
(Intercept)	−0.101	0.0547	−1.84	0.0718	.	…	…	…
Tstar.norm	0.162	0.0381	4.26	0.000107	***	0.325	9.02E−08	***
S0X:Lower	0.347	0.0608	5.71	8.93E−07	***	0.325	5.36E−07	***
S0X:Upper	0.236	0.0369	6.39	9.04E−08	***	…	…	…

Notes. Multiple R² = 0.650, adj. R² = 0.626; residual standard error 0.068, p = 4.14 × 10⁻¹⁰.

^aSignificance levels:

^“***”α ∼ 0,

^“**”α = 0.001,

^“*”α = 0.05,

^“.”α = 0.1, “ ”α = 1.

Figure 3.

(a) Predicted (A^p) from equations (3), (4) versus actual (A) Bond albedo for the ensemble.

The resulting predicted equilibrium temperature (T^p_eq) given A^p from Equations (3), (4) is

$${T^p}_{\text{eq}}={\left[S_o\left(1-A^p\right)S_{\text{ox}}/4\sigma \right]}^{1/4},$$ (5)

where S_o = 1361 W m⁻² is Earth’s solar constant and σ the Stefan–Boltzmann constant. The rms error in T ^p_eq using A^p is 9 K, versus 14 K when A = 0.3 is assumed. This improvement is potentially useful, as S_ox and T_star are known external parameters for every confirmed exoplanet, albeit with non-negligible uncertainties (Brown et al. 2018; Kane 2018).

The dependence of Bond albedo on S_ox for high instellation is primarily due to cloud feedbacks, which are considered the most uncertain aspects of predictions of future climate change. For Earth 21st Century CO₂-driven climate warming, most GCMs predict a positive low cloud feedback—that is, decreasing Bond albedo (e.g., Klein et al. 2017). For comparable increases in S_ox, ROCKE-3D also predicts a positive low cloud feedback at Earth’s rotation period (Figures 5 and 6 of Way et al. 2018), as do other GCMs (Leconte et al. 2013; Wolf & Toon 2015). For S_ox > 1.1 and for slower rotation, however, the feedback is negative. There are two reasons for this: an increase in higher level clouds, as seen in other GCMs (Yang et al. 2013, 2014; Wolf & Toon 2015; Komacek & Abbot 2019), and an increase in low clouds at higher latitudes where stratocumulus decks are less likely to break up into scattered shallow convective clouds (Way et al. 2018). Way et al. (2018) show that ROCKE-3D high clouds are fairly insensitive to removal of upward transport of cloud particles by convective updrafts, but given uncertainties in cloud and convection parameterization, and thus the physics of the feedback, this issue deserves further study.

Figure 5.

Frequency histogram of errors in predicted T_surf from (6) based on actual T_eq (blue), predicted T_eq from Equations (3)–(5) (red), and predicted T_eq assuming A = 0.3 (green).

Figure 6.

ACE transforms on nine parameters and a logical vector distinguishing G versus M star planets, with Bond albedo (“planetary.albedo”) as the response, pooling all n = 48 experiments. The sum of the transformed explanatory variables versus the transform of the response variable is also plotted. For Stellar.type, 1 indicates the intercept offset for G star planets, and 2 for M star planets. The color legend is given in Figure 7. Asterisks give the (1 − α) level of confidence of the transformed parameter fits in a linear fit, with “***,” 0.001, “**,” 0.01, “*,” 0.05, “.,” 0.1, “not,” ⩾0.1 to 1. While the overall ACE R² is 0.968, a linear fit of the transforms gives an R² of 0.971 and adjusted R² of 0.963.

4.3. Inferring Surface Temperature

Surface temperature cannot be fully constrained from S_0x and T_star alone without additional knowledge (e.g., about atmospheric composition and pressure), yet at the moment these are the only two parameters known for any rocky exoplanet that are of first-order importance to habitability. Our ensemble is dominated by Earth-like planets on which H₂O is the dominant absorber, but contains examples of un-Earth-like planets to allow us to estimate the errors that might be made in trying to draw inferences about habitability from incomplete information. We consider a surface temperature error of ∼±20–30 °C to be acceptable given the challenge inherent in characterizing exoplanet habitability. Applied to a remote observation of modern Earth as an exoplanet, such an error bar would allow observers to conclude that our planet, if retaining water, at least has open ocean in the tropics and at most is a hot but not runaway greenhouse planet.

Figure 4(a) shows the relationship between the actual T_eq and surface temperature T_surf for the GCM ensemble. The two quantities are strongly correlated for this subset of planets, but the standard deviation of T_surf over the ensemble is 28 °C, while it is only 18 °C for T_eq. Some of the biggest outliers are planets with non-condensing greenhouse gas abundances much greater or less than the typical (Earth-like) ensemble member. This illustrates the known inherent limitation of T_eq as an indicator of habitability in the absence of direct estimates of T_surf. A linear regression of the actual T_surf versus actual T_eq in Figure 4(a) (in K) gives

$$T_{\text{surf}}=1.33T_{\text{eq}}-51.93=T_{\text{eq}}+G_a,$$ (6)

where G_a is the atmospheric greenhouse effect, with a standard error of 0.12 in the slope and 29.19 K in the (0 K) intercept and multiple/adjusted R² of 0.729/0.723, respectively. Rearranging terms,

$$G_a=0.33T_{\text{eq}}-51.93=0.33\left(T_{\text{eq}}-{T^{\text{E}}}_{\text{eq}}\right)+{G^{\text{E}}}_a,$$ (7)

where $G_a^{\text{E}}=32.20\hspace{0.17em}\text{K}$ is the value of G_a for Earth implied by Equation (7) for $T_{\text{eq}}^{\text{E}}=254.95\hspace{0.17em}\text{K}$. This is close to the observed value $G_a^{\text{E}}=~33\hspace{0.17em}\text{K}$.

Figure 4.

(a) Actual surface temperature T_surf versus actual T_eq (in K) from the GCM ensemble; the dotted line is Equation (6). (b) Actual versus predicted T_surf; the dotted line is the 1:1 line, giving R² = 0.618 and adjusted R² = 0.592.

To interpret Equations (6) and (7), consider the simple Milne–Eddington two-stream approximation for radiative equilibrium in a gray atmosphere with only surface stellar heating (see, e.g., McKay et al. 1999 for a discussion). For that model T_surf = (1 + 3/4τ*)^1/4T_eq, where τ* is the total thermal IR optical thickness of the atmosphere. For Earth τ* ∼ 4, thus T_surf ∼ 1.41T_eq ∼ 361 K, much higher than Earth’s actual T_surf ∼ 288 K.

Equation (6) differs from the radiative equilibrium estimate in several ways. First, the near-surface radiative equilibrium temperature profile is usually superadiabatic and thus unstable to convection, which transports heat upward and cools the surface. This limits the lapse rate to the moist adiabatic value, resulting in a T_surf on Earth that is many tens of degrees cooler than radiative equilibrium predicts (Manabe & Strickler 1964). The intercept in Equation (6) is presumably the mean decrease in T_surf due to convective adjustment. Second, the slope of the T_surf − T_eq relationship in Equation (6) is slightly (though not significantly) smaller than that for radiative equilibrium. The Milne–Eddington model is only an approximation, but there is reason to expect a real difference in slope between it and the ensemble, as τ* in our ensemble generally increases with T_eq as water vapor concentration increases with S_ox. One reason for this possible lower sensitivity of T_surf to changes in T_eq is the negative lapse rate feedback on planets with water. On such planets convective adjustment of T_surf strengthens with warming because the moist adiabatic lapse rate decreases (becomes less steep) with increasing temperature. This partly offsets the positive water vapor feedback (e.g., Soden & Held 2006). Another reason for the lower sensitivity is that some of the incident starlight is absorbed within the atmosphere rather than at the surface, an “anti-greenhouse effect” (McKay et al. 1999) that limits the greenhouse warming due to thermal IR opacity. This is especially true for M star planets in the ensemble, even more so for those with higher H₂O amounts and/or CH₄. All of these effects reduce the error in implied T_surf relative to planets without water.

Equations (3), (5) and (6) predict A^p = 0.327 and T_surf = 9.3 °C for modern Earth, versus the observed 0.296 and 15 °C, respectively. For modern Mars, a planet outside the outer edge of the habitable zone with a thin mostly CO₂ atmosphere and no ocean, Equations (4)–(6) yield T_surf = −55.9 °C, close to its actual −59 °C. Applying Equation (6) to the ensemble using A^p and $T_{\text{eq}}^p$ predicted from Equations (3)–(5) gives the result shown in Figure 4(b).

Figure 5 shows the ensemble T_surf errors based on Equation (6) using 3 predictions of T_eq:

1. Actual T_eq from each simulation, which shows what might be possible when broadband thermal or reflected light phase curves become available for rocky planets

2. Predicted $T_{\text{eq}}^p$ from Equations (3)–(5), which can be used now with existing information on only S_ox and T_star

3. Predicted T_eq assuming A = 0.3 (i.e., current standard practice for assessing the potential habitability of newly discovered rocky exoplanets)

The rms error in T_surf is 14 °C using the actual T_surf, 17 °C for the S_ox − T_star predictor, and 21 °C for the A = 0.3 assumption (i.e., the regression removes about half the error caused by not knowing the Bond albedo of the planet). The actual errors may not be normally distributed, though. Figure 5 shows that if the actual A or T_eq are known, T_surf is predictable to within 10 °C for 27 of the 48 planets and within 20 °C for 40 of the 48 planets, if water was retained. At the present time, when only stellar temperature and instellation are known, the prediction based on the S_ox − T_star regression is within 20 °C of the actual T_surf for 34 of 48 planets. This is only slightly better than the default A = 0.3 assumption. The value of a predictor that anticipates tendencies toward high albedos for strongly illuminated planets and low albedos for weakly irradiated planets orbiting cool stars is instead its ability to limit the largest errors: the prediction based on the regression is off by >30 °C in only 2 cases, versus 8 cases when A = 0.3 is assumed.

5. Assessments of Known Exoplanets

To illustrate the potential use of our two-parameter predictor, we consider some confirmed exoplanets that have been advertised as potentially habitable. For several of these planets, Kane (2018) has recently provided updated estimates of instellation (S_ox) based on Gaia Data Release 2 (G-DR2). We exclude Proxima Centauri b, which is part of our GCM ensemble:

1. TRAPPIST-1 (Gillon et al. 2017): For TRAPPIST-1 e, Equations (4)–(6) predict T_surf = −12 or −9 °C using the original or G-DR2 S_ox values, respectively, implying a regionally habitable planet, somewhat warmer than our nominal Proxima Centauri b and Sturtian Earth GCM climates. By comparison, Wolf’s (2018) GCM predicts −32 to +63 °C for different compositions; Turbet et al. (2018) reach similar conclusions with a different GCM.

   For TRAPPIST-1 f, we predict T_surf = −44 or −41 °C for the original or G-DR2 instellation values, respectively, similar to our simulated uninhabitable (at the surface) snowball planets due to its weak S_ox = 0.38 or 0.40, unless several bars of CO₂ produce a “maximum greenhouse” habitable planet. This is consistent with GCM estimates (Turbet et al. 2018; Wolf 2018).

   Wolf (2017) finds TRAPPIST-1 d (S_ox = 1.14) to have a runaway greenhouse, whereas we predict T_surf = 29 or 32 °C, a very warm but habitable planet. This is likely a failure of our simple model. Our regression for S_ox > 1 is determined mostly by slowly rotating planets with thick dayside clouds, but TRAPPIST-1 d has a 4-day rotation period if it is synchronous, probably too fast for such clouds to occur, thus destabilizing its climate.

2. LHS 1140 b: Dittmann et al. (2017) classify this as a habitable zone planet, but with S_ox = 0.46, our predicted T_surf = −35 °C suggests a near-snowball state. G-DR2 greatly revised S_ox upward to 0.66, though, which increases this planet’s prospects for habitability (projected T_surf = −14 °C), making it potentially similar to Proxima Centauri b and TRAPPIST-1 e.

3. Kepler-186 f: This planet has received attention as a possible cold (S_ox = 0.32) but habitable Earth analog. Quintana et al. (2014) suggest that it could sustain liquid water even with an Earth-like atmosphere, while Bolmont et al. (2014) find that 0.5–5 bars of CO₂ are required for habitability, using a 1D model. We predict T_surf = −58 °C, much colder than our completely glaciated snowball planets. G-DR2, however, increases S_ox to 0.44 and our inferred T_surf to −39 °C, suggesting marginal habitability if the planet has a thick greenhouse gas atmosphere.

4. Kepler-452 b: This planet receives 11% more incident flux than Earth and was considered a habitable candidate (Jenkins et al. 2015), though its size of 1.6 R_E casts doubt on whether it is rocky. If it is, our predicted T_surf = 13 °C, similar to Earth, also suggests habitability.

5. Ross-128 b: Bonfils et al. (2017) characterize this M-star planet as “temperate.” They originally estimated S_ox = 1.38, but Souto et al. (2018) revise this upward to S_ox = 1.79. Our regression predicts T_surf = 36 and 45 °C for the old and new instellation estimates, respectively, suggesting a planet much warmer than Earth, in the class of our warmest early Venus or modified Kepler-1649 b analogs. However, Ross-128 b has a rotation period of only 9.9 days, making it more likely that this planet is hotter than those analogs, at least a moist greenhouse if not a runaway greenhouse planet, if it has an atmosphere and water.

6. K2–155 d: Hirano et al. (2018) use ROCKE-3D to show that this slowly rotating planet, if synchronously rotating with an Earth-like atmosphere but a very small CO₂ concentration, would have a moderate climate for S_ox as high as 1.5 but does not stabilize at the higher value actually observed (S_ox ∼ 1.67). For this value, our regression suggests T_surf = 39 °C, a very warm but not runaway scenario, unlike the ROCKE-3D result. Our closest analog is modified Kepler-1649 b, which is stable for S_ox = 1.4 but for which the regression greatly underestimates the actual T_surf (see Section 5b).

7. GJ 273 b (Astudillo-Defru et al. 2017): With S_ox = 1.06, our prediction of T_surf = 21 °C suggests a potentially Earth-like but warmer habitable planet if it is rocky.

8. GJ 3293 d (Astudillo-Defru et al. 2017): S_ox = 0.59, and our predicted T_surf = −22 °C, is consistent with a partly habitable planet if it is rocky, analogous to Proxima Centauri b.

9. K2–3 d (Crossfield et al. 2015): S_ox = 1.5 and our predicted T_surf = 36 °C argues for an inner edge, borderline habitable planet like our early Venus 2.4 analog or perhaps the warmest stable K2–155 d simulation of Hirano et al. (2018).

10. GJ 625 b: Suárez Mascareño et al. (2017) describe this as an inner edge planet and we predict T_surf = 45 °C, but given its red star and high instellation (S_ox = 2.1), a runaway greenhouse planet is more likely.

6. Effects of Other Parameters

The largest errors in Figure 5 are inherent in the use of the meager information presently available to characterize exoplanets, but the ensemble contains information about other factors that influence A and thus T_eq and T_surf. Here we use the ACE algorithm to estimate the effects of these factors and discuss how some of them might be constrained by future observations.

6.1. Effects of Unconstrained Parameters on Bond Albedo

Including additional planetary parameters in a GAM, we use ACE transforms to uncover nonlinear relations between Bond albedo A and 9 explanatory variables: incident stellar flux, stellar temperature, CO₂ partial pressure, CH₄ partial pressure, obliquity, land fraction of surface, rotation period, orbital period, and eccentricity (see Appendices A and B for details). We consider fits to all 48 GCM ensemble members, as well as separate fits for the 33 G star and 15 M star planets.

There are clearly at least two distinct regimes for the behavior of A^p. When combining all experiments into a single ACE fit (Appendix A, Figure 6), the response of A^p to the transformed variables is non-monotonic, with a damped response or transform at lower albedo ≲0.452. (The transition occurs between a Bond albedo of 0.452 and 0.483, and there are no other G star data points between these values.) The Venus planets seem to extend an increasing trend in the ACE transform at higher Bond albedo (when the Venus planets are removed, the ACE transform remains the same for the other planets). In separate ACE fits for the G and M star planets (Figure 7), the two stellar types seem to demonstrate parallel but offset behavior in the steep portion of the transform. The lower-albedo damping seen in the all-planets ACE fit is clearly driven by the G star planets, and the M star planets appear not to span a range covering this damped regime; the kink in the transform when experiments of both stellar types are in the combined ACE fit is a result of the offset behavior of the G and M star planets. For the G star planets, the transition between the damped and sensitive steep responses occurs at the same values for Bond albedo as for all planets combined.

Figure 7.

ACE transforms for the same variables as in Figure 6, but separately for G star and M star planets. Asterisks denote the (1 − α) level of confidence as in Figure 6. The R² values for the ACE fits for the G and M star planets are 0.966 and 0.861, respectively. Linear fits of the ACE transforms result in R² of 0.970 and adjusted R² of 0.958 for the G star planets, and 0.938 and 0.855 for the M star planets.

With all experiments pooled together, the overall ACE R² is 0.968, and a linear fit of the transforms gives an R² of 0.97 and adjusted R² of 0.96. This is high due to the strong influence of the early Venus experiments. Separate ACE fits for the G and M star planets give R² values of 0.966 and 0.861, respectively. Linear fits of the ACE transforms result in R² of 0.97 and adjusted R² of 0.96 for the G star planets, and 0.94 and 0.85 for the M star planets. For the M star planets, because there are only 15 experiments and 9 variables, this leaves only 6 degrees of freedom; given the few actually significant variables, the fairly high R² is due more to overfitting. The only robustly significant variable is log₁₀(CO₂), whereas the others appear slightly correlated according to various ANOVA comparisons (not shown).

Most variables are significant across all 48 experiments, but some are significant within only the G star or M star planet subsets. With all 48 experiments fitted together, all transformed variables are highly significant, except for eccentricity and land fraction. For the G star planets, all are significant except for rotation period. For M star planets, few variables are significant, with only log10(CO₂) significant at the α < 0.05 level (95% confidence); eccentricity at α < 0.1 (90% confidence), because only one planet has non-zero eccentricity; and rotation period at α < 0.1 (90% confidence). All obliquities are 0°, so no transform is fit. The logical vector distinguishing stellar classes is highly significant at α < 0.001. It is noteworthy that the intercept when all planets are combined is highly significant at α < 0.01 (99% confidence), implying again a systematic difference in Bond albedo between planets orbiting G versus M stars (see Figure 2).

As we cannot clearly eliminate any variables, for a final model, we perform a single regression to evaluate a GAM by combining the ACE transforms by stellar type (Figure 7), which yields an overall R² of ∼0.96 (Table 4). The highest t-values, in descending order, are for S_ox, log₁₀(CO₂), log₁₀(CH₄), land fraction, and T_star, while all transformed variables are highly significant at α ≪ 0.05. Back-transforming the ACE transforms to predict Bond albedo gives an R² of 0.82, adjusted R² of 0.75 (Figure 8).

Table 4.

Model “tAp.all”

Variable	Estimate	Standard Error	t Value	Pr(>|t|)	Confidence (1 − α) in Estimated Coefficient
(Intercept)	−4.89E−16	4.15E−02	0.000	1
acet(S0X)	1.01E+00	6.38E−02	15.826	<2e−16	***
acet(Tstar.K)	9.00E−01	1.39E−01	6.487	1.23E−07	***
acet(orb.period.Earthdays)	9.54E−01	1.26E−01	7.582	4.07E−09	***
acet(rotation.XEarth.day)	9.30E−01	1.21E−01	7.66	3.20E−09	***
acet(eccentricity)	9.66E−01	3.64E−01	2.651	0.011642	*
acet(Land.fraction)	8.23E−01	2.67E−01	3.079	0.003846	**
acet(log10CO2.mb)	1.03E+00	2.46E−01	4.179	0.000165	***
acet(log10CH4.mb)	8.90E−01	1.43E−01	6.237	2.69E−07	***
acet(obliquity.degrees)	8.89E−01	2.36E−01	3.772	0.000552	***

Note. Generalized additive model (GAM) fit to transforms in A^p, combining ACE transforms for the different stellar types, predicting the ACE transforms of Bond albedo. Residual standard error: 0.287 on 38 degrees of freedom. Multiple R-squared: 0.935. Adjusted R-squared: 0.919. AIC: 27.287.

Figure 8.

Model “tAp.all.” Left: predicted Bond albedo for the entire ensemble, from back-transform of GAM prediction combining the G and M star planet transforms in Figure 7. R² = 0.82, Adjusted R² = 0.75. Right: histogram of errors in predicted albedo.

Propagating this predicted Bond albedo through Equation (6) produces predictions for T_surf, with an R² of 0.745 and adjusted R² of 0.74 (if accounting for p = 13 parameters from 9 variables, plus 3 parameters for the G star planet back-transform, plus 1 parameter for the M star planet back-transform for Bond albedo).

Thus, the expanded GAM for Bond albedo, with adjusted R² of 0.75, accounts for ∼8% more of the variance than the prediction by Equations (3) and (4), whose adjusted R² is 0.67. Despite this small improvement, the distribution of residuals is improved such that the GAM Bond albedo propagation through calculation of T_eq and Equation (6) to predict T_surf accounts for 74% of the variance (adjusted R²) compared to only 59% for Equations (3)–(6).

6.2. Effects of Unconstrained Parameters on Surface Temperature

Directly fitting ACE transforms and GAM to predict surface temperature, rather than indirectly calculating it from Bond albedo and the Equation (6) regression on T_eq, shows that it may be easier to predict itself than Bond albedo. Full details of the model for T_surf are given in Appendix A. The ACE transforms to predict surface temperature, both with all experiments combined (Figure 9) and for stellar types separately (Figure 10), show that the response of surface temperature to the transforms of the other variables is monotonic and more nearly linear or simply piecewise linear, compared to the transform for Bond albedo. The transform of S_ox for G star planets shows persistence of the two-regime behavior with the discontinuity at S_ox = 1 as for Bond albedo. As with Bond albedo, we do a linear regression to evaluate a GAM combining the ACE transforms by stellar type (Figure 10). This yields an overall R² of ∼0.96 (Table 8). The highest t-values, in descending order, are for S_ox, log10(CO₂), log₁₀(CH₄), land fraction, and T_star, while all transformed variables are highly significant at a level α < 0.001.

Figure 9.

As in Figure 6, but for surface temperature as the response variable. ACE fit gives an R² of 0.925, while a linear fit of the transforms gives an R² of 0.930 and adjusted R² of 0.912.

Figure 10.

As in Figure 7, but for surface temperature as the response variable. The R² values for the ACE fits for the G and M star planets are 0.965 and 0.971, respectively. Linear fits of the ACE transforms result in R² of 0.968 and adjusted R² of 0.955 for the G star planets, and 0.989 and 0.974 for the M star planets.

Table 8.

Model “tTsurf.all”: Generalized Additive Model (GAM) Fit to Transforms in Figure 21, Combining ACE Transforms for the Different Stellar Types, Predicting the ACE Transforms of Average Surface Temperature

ACE-transformed Variable	Estimate	Std. Error	t Value	Pr(>|t|)	Confidence (1 − α) in Estimated Coefficient
(Intercept)	−4.15E−15	2.87E−02	0	1
acet(S0X)	1.07E+00	6.50E−02	16.425	<2e−16	***
acet(Tstar.K)	9.11E−01	1.20E−01	7.58	4.10E−09	***
acet(orb.period.Earthdays)	1.10E+00	1.92E−01	5.729	1.34E−06	***
acet(rotation.XEarth.day)	9.55E−01	1.37E−01	6.952	2.86E−08	***
acet(eccentricity)	1.05E+00	1.58E−01	6.622	8.02E−08	***
acet(Land.fraction)	1.02E+00	1.33E−01	7.636	3.45E−09	***
acet(log10CO2.mb)	1.04E+00	7.29E−02	14.304	<2e−16	***
acet(log10CH4.mb)	9.55E−01	9.47E−02	10.093	2.64E−12	***
acet(obliquity.degrees)	9.42E−01	1.57E−01	6.015	5.43E−07	***

Note. Residual standard error: 0.1991 on 38 degrees of freedom. Multiple R²: 0.969. Adjusted R²: 0.961. AIC: −7.95.

After back-transforming the ACE transforms of T_surf (details in Appendix B), the final prediction of T_surf is plotted in Figure 11, giving an R² of 0.947, adjusted R² of 0.929. The number of experiments with predicted T_surf error <1 °C is 10; <5 °C, 30; <10 °C, 45; and = 10 °C, 3. Compared to the adjusted R² of 0.59 for Equations (3)–(6), this is a major accounting of 34% more of the total variance in T_surf by the additional planetary parameters.

Figure 11.

(left) Predicted surface temperature (Tsurf, predicted) versus actual, R² = 0.947, adjusted R² = 0.929. (right) Distribution of T_surf errors, from model “tTsurf.all” of Table 8, with back-transforms to °C using the functions in Figures 22 and 23 in Appendix B.

6.3. Constraining Greenhouse Gas Abundances and Surface Pressure

Greenhouse gases influence Bond albedo, mostly for cool stars, but their biggest effect is the difference they cause between T_eq and T_surf (Equation (6)). In our ACE transforms, this is reflected in the tendency of CO₂ and CH₄ to decrease A^p and increase T_surf. As discussed earlier, some of our largest T_surf errors are for planets with high/low concentrations of greenhouse gases, as the difference between T_surf and T_eq is much larger/smaller for such planets than Equations (3)–(6), which are based on mostly Earth-like atmospheres, predict. For synchronously rotating planets, thermal phase curves may help diagnose these errors. The nightside temperature of such planets is highly correlated with their clear-sky greenhouse effect (Yang & Abbot 2014; Del Genio et al. 2019). The nightside greenhouse effect depends largely on the abundance of non-condensing greenhouse gases, which regulate the condensable gas H₂O (Lacis et al. 2010).

Table 3 shows the T_surf errors from Equations (3) to (6), the maximum/minimum thermal fluxes to space, and their fractional difference, for six planets that differ in atmospheric opacity. They fall into three groups: optically thin atmospheres (in the thermal IR), atmospheres with primarily H₂O opacity, and optically thick atmospheres with large CO₂ opacity. Proxima Thin and GJ 876 0.6 are optically thin for different reasons: the first because its N₂ atmosphere is only 100 mb thick and the second because there is only 1 ppmv of CO₂ in its weakly illuminated 1 bar atmosphere. Both planets exhibit a large (0.6) fractional flux contrast.

Table 3.

Day–Night Contrasts and Surface Temperature Errors^a

Simulation	Tsurf Error (°C)	Fmax (W m−2)	Fmin (W m−2)	ΔF/F
(8) Proxima Thin	22	260	96	0.63
(11) GJ 876 0.6	22	245	101	0.59
(14) GJ 876 1.2	−1	322	212	0.34
(48) Kepler-1649 b	−23	305	214	0.30
(9) Proxima Thick	−12	197	145	0.26
(10) Proxima Pure CO2	−26	183	158	0.14

Note.

^aF = thermal emitted broadband flux; F_max, F_min = maximum, minimum values of F on planet; ΔF/F = fractional difference between F_max and F_min.

GJ 876 1.2 also has only 1 ppmv CO₂ but is illuminated more strongly, leading to a larger dayside H₂O greenhouse effect but less on the cooler (but still warm in an absolute sense) nightside. The Kepler-1649 b case is even more strongly illuminated but with modern Earth concentration of CO₂. Both planets maintain a modest nightside clear-sky greenhouse effect and thus a moderate (0.3) contrast between maximum and minimum thermal flux. They are easily distinguished from the optically thin atmospheres by their large nightside thermal emission.

The final two planets have considerable CO₂ opacity. Proxima Thick is identical to Proxima Thin, except that its surface pressure is 10 bars, so its CO₂ amount is 100 times larger and the pressure broadening of its absorption lines is much stronger. Proxima Pure CO₂ is a 1 bar CO₂-only atmosphere. Both planets have a large nightside greenhouse effect, and thus a max–min thermal flux contrast ∼2–5 times weaker than our optically thin planets.

With one exception, the error in our predicted T_surf goes monotonically from strongly positive to increasingly negative as the fractional flux contrast decreases, suggesting that thermal phase curves may provide a useful additional constraint. The exception, Kepler-1649 b, is our hottest and most humid planet. Its T_surf prediction errs in the same sense as the cooler Proxima Pure CO₂ atmosphere while exhibiting more day–night flux contrast.

The largest possible contrast between maximum and minimum emission is expected for synchronously rotating planets that have lost their atmospheres due to various escape mechanisms (Airapetian et al. 2017; Dong et al. 2017; Zahnle & Catling 2017). It should be possible to differentiate such planets from those with atmospheres using the James Webb Space Telescope (Yang et al. 2013; Kreidberg & Loeb 2016).

6.4. Photochemical Hazes

One potential confounding feature is a thick photochemical haze that affects Bond albedo independent of the effects of water. Solar system examples are the H₂SO₄ solution haze on Venus (Hansen & Hovenier 1974) and the organic haze on Titan (Hörst 2017) and perhaps Archean Earth (Zerkle et al. 2012; Arney et al. 2017).

Venus’ H₂SO₄ haze is probably due to volcanic SO₂ emissions (Marcq et al. 2013), as is Earth’s episodic volcanic stratospheric haze (Sato et al. 1993). The fact that Earth’s volcanic hazes are optically thin and dissipate in a few years while Venus’ do not is explained by the presence of liquid water on Earth, which produces storms that wash out stratospheric aerosols soon after they sediment into the troposphere. Loftus et al. (2019) concludes that sulfur chemistry in an oxidized atmosphere with an ocean is probably incompatible with an optically thick haze. Thus we consider it unlikely for a rocky planet within or close to the inner edge of the habitable zone to have a Venus-like Bond albedo, unless it is a habitable planet with a thick water cloud.

Titan’s organic haze, the product of CH₄ photodissociation, is formed at altitudes far above the troposphere and produces a low Bond albedo (∼0.22) not too different from several habitable G star planets in our ensemble. Indeed Archean Earth, a habitable planet, may have had a somewhat similar haze. Such planets might be identifiable from the spectral dependence of fractal haze particle scattering (Wolf & Toon 2010) or its unusual phase angle dependence (Garcia Muñoz et al. 2017), but this would leave open the question of whether a habitable surface lay underneath. In principle the same problem might exist for modern Mars-like planets that develop global dust storms, but these should be time-variable and thus distinguishable from hazy or cloudy habitable planets. Furthermore, whether a planet with liquid water and precipitation washout of particulates could ever support a global dust storm is questionable.

6.5. Rotation

Planet rotation is a key factor in exoplanet habitability but cannot yet be directly observed, although techniques for doing so in the future from photometric variability have been explored (Fujii & Kawahara 2012; Snellen et al. 2014). In our ACE fits, rotation has weak effects on Bond albedo by stellar type, but otherwise has a strong tendency toward cooling surface temperature. Our strongly irradiated habitable planets have high albedos because they rotate slowly (either asynchronously, as for early Venus and Kepler-1649 b, or synchronously, as for S_ox = 1.2 GJ 876). Rapidly rotating strongly irradiated planets should enter a runaway greenhouse and be darker (e.g., Kopparapu et al. 2016). Habitable planets orbiting cool stars should be tidally locked, so their rotation periods should either match their orbital periods or be in a low order spin–orbit resonance. For weakly irradiated Proxima Centauri b, our 3:2 resonance planet is habitable despite its fairly rapid rotation. For a more highly irradiated planet, though, it is possible that resonances higher than 1:1 would compromise habitability. For planets outside the tidal locking radius, there is no way to constrain rotation period without direct observations.

6.6. Other Parameters

Bond albedo and thus climate can vary considerably in the face of large obliquity changes—for example, those in Mars’ past (e.g., Mischna et al. 2013; Wordsworth et al. 2015; Kite et al. 2017). Furthermore, for G star planets, climate at high obliquity can be bistable (Kilic et al. 2017; Colose et al. 2019), and thus without knowing the dynamical history of such planets (e.g., inward migration across the snow line versus in situ formation), it may be impossible to anticipate habitability. Obliquity is not well sampled in our ensemble and is currently unobservable for exoplanets, although it may eventually be inferred from the seasonal cycle of reflected starlight (e.g., see Kane & Torres 2017). Planet radius is correlated with orbital period in our ensemble sampling and is sampled over too small a range by our ensemble (other than the Early Mars simulation) to physically explain any albedo variance, so it was excluded from the ACE and GAM fits. Larger changes in radius can affect climate via the equator-pole temperature gradient (Kaspi & Showman 2015). Radius is a known quantity for exoplanets detected by the transit method. Eccentricity is not sampled well enough in our ensemble to capture its effect on climate and Bond albedo. It is estimated for some exoplanets but not others, and can be degenerate with albedo and obliquity (Barnes et al. 2015; Kane & Torres 2017). It can also directly affect albedo by changing the spin–orbit resonance state (e.g., our Proxima Centauri b 3:2e30 simulation). Land fraction directly influences Bond albedo and indirectly affects it by changing climate; retrievals of land–ocean distribution for exoplanets are challenging but may be possible in the future (Fujii et al. 2017b; Lustig-Yeager et al. 2018). In our GAM regressions, higher land fraction implies higher Bond albedo and lower T_surf, and is the fourth most significant influence on T_surf.

7. Conclusions

We have presented a simple approach for synthesizing a diverse set of GCM simulations to estimate the Bond albedo, equilibrium temperature, and surface temperature of habitable rocky exoplanets. It implicitly captures the effects of dayside cloud shielding on slowly rotating planets, sea ice-snow/albedo feedbacks on weakly irradiated planets, and near-IR absorption of incident starlight by water vapor on planets orbiting cool stars. Earth’s existence near the triple point of water makes its albedo close to the minimum possible for a habitable planet orbiting a G star, whereas the low albedo of water vapor and ice in the near-IR gives most habitable M star planets a Bond albedo lower than Earthʼs. We have also evaluated the role of as-yet unobserved parameters in explaining more of the variance of Bond albedo and surface temperature.

Relative to the default assumption of a fixed Bond albedo for all exoplanets, our observable two-parameter predictor removes about half the error in estimated equilibrium temperature and surface temperature associated with not knowing the Bond albedo, and allows surface temperatures to be anticipated to within ∼30 °C, and usually to much better accuracy, using only knowledge of the stellar flux incident on a planet and the stellar temperature (two known parameters for every confirmed exoplanet). Given the number of confirmed rocky exoplanets and the much larger population expected to be discovered in the next few years, our technique provides a quick way to identify a small number of the most promising candidates to be targeted for characterization, given the reality of long integration times required for useful rocky planet observations and scarce observational resources. At a minimum, we suggest that initial characterizations of the potential habitability of newly discovered rocky exoplanets should not assume a Bond albedo lower than ∼0.25 for planets orbiting G stars, or an albedo higher than ∼0.35 for planets orbiting M stars, unless they are irradiated more strongly than Earth. Our more extensive predictor identifies a number of parameters that could improve the prediction of surface temperature by up to 34%, motivating future attempts to constrain these parameters.

Based on our results, using the Gaia DR2 update, assuming these planets are rocky, and if they retained water, the known exoplanets TRAPPIST-1 e, Kepler-452 b, LHS 1140 b, GJ 273 b, GJ 3293 d, and Proxima Centauri b (which is in our ensemble) have the best chance to be habitable, as they can accommodate a fairly large range of greenhouse gas concentrations greater or less than Earth’s but still have a moderate climate. The planets TRAPPIST-1 f and Kepler-186 f have a chance to be habitable only if they have a thick enough atmosphere with a greenhouse gas such as CO₂ as the major constituent. K2–3 d is a possibly habitable planet at the very warm end of the spectrum, while TRAPPIST-1 d, Ross-128 b, K2 155 d, and GJ 625 b are too highly irradiated and/or too rapidly rotating to have a high likelihood of habitability.

Despite its success, it would make sense to refine our technique in several ways:

1. We did not add rotation period to the two-parameter predictor because it is not yet known for exoplanets. It can be inferred to some extent for tidally locked planets, although some of these may be in a higher order resonance rather than synchronous rotation. Expanding the ensemble to include rapidly rotating (<10 days), close-in planets that may lose water and develop low albedos would allow us to better anticipate runaway greenhouse conditions.

2. Any method to infer habitability from T_eq alone is limited by the absence of knowledge of the greenhouse effect that determines T_surf, given current observing capabilities. Our ensemble does not include planets with more than 1 bar of CO₂. Greenhouse warming continues to increase up to ∼5–8 bars of CO₂, beyond which Rayleigh scattering prevents further warming (e.g., Kasting et al. 1993; Wolf 2018). Whether habitable planets can sustain such thick CO₂ atmospheres is not known. It has been suggested that aquaplanets may not have the strong surface weathering sink that removes CO₂ (Abbot et al. 2012). Carbon cycle-climate modeling suggests, though, that seafloor weathering may be more effective than previously anticipated (Charnay et al. 2017; Krissansen-Totton et al. 2018), and for thicker ocean “water worlds” seafloor pressure may inhibit CO₂ buildup (Kite & Ford 2018). Early Mars, a planet at the outer edge of the habitable zone, appears to have only had 1–2 bars of CO₂ (Kite et al. 2014). Observations that reveal the efficacy of the carbonate-silicate cycle feedback that predicts increasing CO₂ retention as illumination decreases (Kasting et al. 1993) would be a useful constraint (e.g., Bean et al. 2017).

3. Our “ensemble of opportunity” is merely a first step in distilling general information about exoplanet habitability from GCMs. Multiple groups conduct rocky exoplanet GCM simulations (although most do not include dynamic oceans). Together these form a continually growing “grand ensemble” of hypothetical planets that our simple method or the more sophisticated ACE/GAM approach might utilize, both to limit the impacts of inaccuracies inherent to all models and uncertainties in cloud feedbacks, and to exploit the breadth of simulated planets and sampling of parameter space in this ever-expanding storehouse of information that cannot be matched by any single model. Future analyses may want to investigate Bayesian approaches to statistical inference on sparse data. Large “perturbed parameter” ensembles of GCM simulations that objectively sample all relevant external parameters, as has been done to estimate uncertainty in projections of terrestrial climate change (e.g., Stainforth et al. 2005), are a logical next step for exoplanet habitability estimates. A repository of such simulations available to the entire community, much like the Coupled Model Intercomparison Project protocol used by Earth GCM modelers for projections of 21st century climate change (Eyring et al. 2016), would accelerate the search for habitable planets.

Appendix A. Use of the ACE Algorithm to Develop GAMs for Bond Albedo and Surface Temperature

A.1. Statistical Models and Metrics

The ACE algorithm solves for possibly nonlinear transformations t_var of variables that maximize their correlations in a GAM of the form

$$t_y\left(y\right)=\sum _{i=1}^n{t}_{xi}\left(x_i\right),$$ (8)

where y is the predicted variable, x_i are the explanatory variables, and t_y and t_xi are the relevant transformations. The transforms are localized and provide no functional parameterizations, but can reveal relationships (positive, negative, nonlinearities). They can suggest functions, g(y) = E(t_y(y)) or f_i(x_i) = E(t_xi(x_i)), that can be fit as estimators of the transforms. Such functions can then be used in generalized linear model (GLM) regressions, in which the transformed explanatory variables are treated as linear terms, and the transformation of the response is treated as the link function of the GLM.

A GLM is a linear regression used to stratify regressions involving categorical variables to perform a logistical or logit regression that solves for the intercept b₀ and coefficients b_i of

$$\begin{gathered} g_y\left(y\right)=\sum _{i=1}^n{b}_0+b_i{f}_i\left(x_i\right) \\ +\sum _{\text{j}=1,\text{i}<>\text{j}}^n{b}_{i,j}{f}_i\left(x_i\right)f_j\left(x_j\right)+.... \end{gathered}$$ (9)

After examining various ACE experiments, we performed a series of GLM regressions (final selected models shown), each one removing non-significant parameters or introducing transformations that capture some behavior seen in the ACE transformations, or that normalize the residuals, to improve the fit.

We use some typical metrics for goodness of fit for linear models (Harrell 2015). The adjusted coefficient of determination (Adjusted R²) is used as a general measure of goodness of model fit, adjusted for the number of terms. It only increases if a term improves the prediction relative to what would be expected by chance. Starting from

$$R^2=1-\frac{{\sigma }^2}{\text{Var}\left(y\right)},$$ (10)

where σ is the residual variability and Var(y) the variance of the predictor, the adjusted R² is

$$\begin{gathered} {\widehat{R}}_{\text{adj}}^2=1-\frac{\frac{1}{n-p-1}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{\frac{1}{n-1}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2} \\ ={\widehat{R}}^2-\left(1-{\widehat{R}}^2\right)\frac{p}{n-p-1}. \end{gathered}$$ (11)

We also use Student’s T-test of standard error estimates as a measure of confidence of the fitted coefficients, and quantify the significance of each parameter in reducing GLM deviance (instead of variance, which assumes normally distributed variables) through the Chi-squared test.

To compare the goodness of fit for different GLMs, we use the Akaike Information Criterion (AIC; Akaike 1981; Sakamoto et al. 1986) to select parsimonious models,

$$\text{AIC}=-2*\text{log}-\text{likelihood}+k*n,$$ (12)

where log-likelihood is the value for the model, n is the number of parameters in the model, and k = 2 is used by AIC to penalize a model for overfitting, with the better model having a lower AIC. To quantify whether another model is significantly better, we calculate the likelihood L_m that model m minimizes information loss compared to a model with lower AIC (AIC_min):

$$L_m=\text{exp}\left(\frac{\text{AICmin}-\text{AIC}}{2}\right).$$ (13)

We also fit the same GLM as a simple linear regression to estimate R² and the F statistic, which quantifies the joint significance of the fitted coefficients (where a low probability that the fitted coefficients differ significantly from each other indicates a good model fit, and the statistic can be used to compare different models, similar to how the AIC score compares GLMs).

A.2. Planetary Variables

The experiments differ in these nine parameters as potential explanatory variables: S_0X, T_star (K), CO₂ partial pressure (mb), CH₄ partial pressure (mb), obliquity (degrees), land fraction of planet surface, rotation period (d), orbital period (d), and eccentricity. Prior to performing ACE fits, we applied a priori transforms based on prior knowledge about the effects of greenhouse gases. According to a GCM ensemble by Wolf & Toon (2013) for Archean Earth-like conditions, the warming effect of CO₂ scales approximately linearly with log₁₀ of its partial pressure. We assume that the response to CH₄ is similar. Because some experiments have no CO₂ and/or CH₄, the transform for CO₂ is done as log₁₀(CO₂ + 1e–06), and for CH₄ as log₁₀(CH₄ + 1e–07). The small offsets are selected to be an order of magnitude below the smallest significant figures in the sampled values, so as to not skew regression fits.

A.3. Bond Albedo Model

ACE transforms of all nine explanatory parameters and of Bond albedo A^p as the response are shown in Figure 6 for all 48 experiments, and in Figure 7 for G star (33 experiments) and M star (15 experiments) planets fitted separately. The (1 – α) level of confidence in secondary linear fits of the transformed parameters is denoted by asterisks above each panel.

With all experiments pooled together to fit A^p (planetary. albedo), the overall ACE R² is 0.968, and a linear fit of the transforms gives an R² of 0.97 and adjusted R² of 0.96. This is high due to the strong influence of the early Venus experiments. Separate ACE fits for the G and M star planets give R² values of 0.966 and 0.861, respectively. Linear fits of the ACE transforms result in R² of 0.97 and adjusted R² of 0.96 for the G star planets, and 0.94 and 0.85 for the M star planets. There are 15 M star planet experiments and nine variables, leaving only six degrees of freedom; given the few actually significant variables, the fairly high R² is due more to overfitting. The only robustly significant variable is log₁₀(CO₂), whereas the others appear slightly correlated according to various ANOVA comparisons (not shown).

To fit a GAM that approximates the ACE fits provided, because the G star versus M star planets exhibit distinct offset behaviors, we do the following for a final illustrative prediction:

1. Assume that the ACE transforms for the separate stellar types are the closest to reality or appropriately describe the data

2. Combine the separate stellar type transforms into single vectors, as total transforms for each variable for the entire ensemble, and use these transforms to fit a GAM

3. Fit functions to the ACE transforms just for the predicted variable, A, to enable back-transformation of the predicted transformed value to the original albedo units

The parametric fits to the transforms of A^p for the G star and M star planets are detailed in Appendix B. In our final GAM fit on the combined ACE transforms for both G and M star planets of all variables, all transformed terms are highly significant with high t-values (Table 4), giving an adjusted R² of 0.92 relative to the predicted transform. The back-transformed prediction of A^p is shown in Figure 8, which gives an R² of 0.82. To estimate the adjusted R², we counted the parameters in the inverse transform functions, giving a total of nine variables and an additional three and one more parameters from the G star and M star planet back-transform fits, or number of parameters being p = 13, which yields an adjusted R² of 0.75. This is a slight improvement over the adjusted R² of 0.67 from the segmented regression of Equations (3) and (4). The errors in the prediction show the largest error to be an overprediction by 0.17 and 3 planets with errors >0.1, while 45 planets have errors <0.1, and 40 have errors <0.05.

All transformed terms are highly significant in model tAp.all, with especially high t-values for S_0X, T_star.K, orbital period, rotation period, log₁₀(CO₂.mb), and log₁₀(CH₄.mb); therefore, dropping any of terms should worsen the AIC score. To understand how this model provides better prediction, see Appendix B for comparisons to a model without transforms (“tAp.no-x.all”) and a model with only two variables (“tAp. tS0X.tTstar.K.all”), all attempting to predict the same transforms as in model “tAp.all.”

A.4. Surface Temperature from Direct ACE Fit

The ACE transforms to predict T_surf (“Tsurf.actual.C”), both with all experiments combined (Figure 9) and for stellar types separately (Figure 10), show that the response of T_surf to the transforms of the other variables is monotonic and more nearly linear or simply piecewise linear, compared to the transform for A^p. As with A^p, while the average T_surf transform from pooling all experiments in one ACE fit can be predicted with a high R² of ∼0.93, fitting the data separately by stellar type shows some different, or offset, behaviors, by stellar type, that improve the prediction. The ACE transform of T_surf for the G star planets exhibits complex nonlinearity, nearly S-shaped, but is nearly linear for the M star planets. Mars, as an outlier, very likely is skewing the transform at the lower end of the G star experiments.

The transform of S_ox for G star planets also shows two-regime behavior with the discontinuity at S_ox = 1 as for Bond albedo, reflecting the transition from cold climates with sea ice/albedo feedbacks affecting T_surf to warm climates controlled mostly by clouds for slowly rotating planets. The M star planets appear not to be well sampled across both regimes. The transforms for T_star, orbital period, rotation period, and especially log₁₀(CH₄), which show nonlinear behavior for the full ensemble, produce simple linear relations for the separate stellar types.

Eccentricity is significant in explaining variation in T_surf, but we suspect this is an artifact of the specific experiments in the ensemble. Land fraction and log₁₀(CO₂) continue to exhibit complex nonlinearities, but smoother than for Bond albedo. The piecewise components in the land fraction transform for G star planets correspond to categories, in increasing order, aquaplanets, paleo-Earth continental scenarios, early Venus, and ancient Mars. Among the parameters, rotation period, eccentricity, and land fraction have transforms with negative trends (i.e., they are correlated with cooling of the planet surface).

As with Bond albedo, we do a linear regression for T_surf to evaluate a GAM combining the ACE transforms by stellar type (Figure 10). Back-transformed predictions of T_surf yield a very high adjusted R²of 0.961 (Table 5) and T_surf errors smaller than 10 °C for 45 of the 48 members of the ensemble (Figure 11).

Table 5.

Model “tTsurf.all”

ACE-transformed Variable	Estimate	Standard Error	t Value	Pr(>|t|)	Confidence (1 − α) in Estimated Coefficient
(Intercept)	−4.15E−15	2.87E−02	0	1
acet(S0X)	1.07E+00	6.50E−02	16.425	<2e−16	***
acet(Tstar.K)	9.11E−01	1.20E−01	7.58	4.10E−09	***
acet(orb.period.Earthdays)	1.10E+00	1.92E−01	5.729	1.34E−06	***
acet(rotation.XEarth.day)	9.55E−01	1.37E−01	6.952	2.86E−08	***
acet(eccentricity)	1.05E+00	1.58E−01	6.622	8.02E−08	***
acet(Land.fraction)	1.02E+00	1.33E−01	7.636	3.45E−09	***
acet(log10CO2.mb)	1.04E+00	7.29E−02	14.304	<2e−16	***
acet(log10CH4.mb)	9.55E−01	9.47E−02	10.093	2.64E−12	***
acet(obliquity.degrees)	9.42E−01	1.57E−01	6.015	5.43E−07	***

Note. As in Table 4, but for ACE transforms of surface temperature. Residual standard error: 0.199 on 38 degrees of freedom. Multiple R-squared: 0.969. Adjusted R- squared: 0.961.

Appendix B. GAM Details for Bond Albedo and Surface Temperature

This appendix details the full procedure for selecting multivariate GAMs to incorporate orbital and planetary parameters to predict planetary (Bond) albedo (A^p) and surface temperature (T_surf) from our ensemble of experiments conducted with the ROCKE-3D GCM. These data explorations use the ACE algorithm (Breiman & Friedman 1985) to diagnose nonlinear relations between the explanatory variables and the response variable. ACE solves for possibly nonlinear transformations of the variables that maximize their correlation in a GAM (McCullagh & Nelder 1989). We use the implementation of ACE in the acepack package of R software. In this appendix we provide descriptions of the general form of a GAM, as well as statistics metrics for goodness of fit and model comparison.

The experiments differ in these nine potential explanatory parameters: incident stellar flux (S_0X, relative to Earth), stellar temperature (T_star.K), CO₂ partial pressure (CO₂.mb), CH₄ partial pressure (CH₄.mb), obliquity (degrees), land fraction of planet surface (land.fraction), rotation period (in Earth days), orbital period (in Earth days), and eccentricity. Additional parameters included several that are not independent of orbital period, due to the sampling of our ensemble, and thus were excluded: planet radius, planet gravity (which is a function of planet radius for constant density), and surface pressure (which is correlated with planet radius in our ensemble).

Prior to performing ACE, we applied some a priori transforms based on prior knowledge about the effects of greenhouse gases. According to a GCM ensemble by Wolf & Toon (2013) for Archean Earth-like conditions, the warming effect of CO₂ scales approximately linearly with log₁₀ of its partial pressure. We assume that the response to CH₄ partial pressure is similar. Because some experiments have no CO₂ and/or CH₄, the transform for CO₂ is done as log₁₀(CO₂.mb + 1e–06), and for CH₄ as log₁₀(CH₄.mb + 1e–07). The small offsets are selected to be an order of magnitude below the smallest significant figures in the sampled values, so as to not otherwise skew regression fits. The log transforms do not necessarily improve the model fit over non-transformed values, but physics-based rationale takes precedence.

Not all parameters are broadly or evenly sampled, and the sampling is clumped with some variables correlated in certain subsamples of the data (Figures 12 and 13). Major distinct sets of data points are those for early Mars (only one point), early Venus, the G star (Sun) Earth-like planets, and the M star planets. While weighting of clumped data points is one way to avoid biased regression fits, we opted not to introduce weights, because not all clusters of different parameters are restricted to the same data points in this data set. This is therefore not an ideal data set for statistical fitting; nonetheless, this exercise may reveal some robust patterns.

Figure 12.

Data plots of the 48-member ROCKE-3D ensemble, showing response variables (planetary albedo, atmospheric contribution, surface contribution, T_surf) plotted versus planetary parameters, including the 9 parameters of interest (see text) as well as correlated parameters. Colors: black—G star Earth-size planets; red—M star planets; cyan—early Mars; pink—early Venus.

Figure 13.

Histograms of parameter distributions of the 48-member ROCKE-3D ensemble.

We performed initial ACE fits with the response variable being A^p and the explanatory variables being the nine parameters listed previously, adding a logical vector to distinguish G and M star planets. Because of evident differences between M star and G star planets in both behavior and sampling of the other parameters in these subgroups, we also performed separate ACE fits for the two different stellar types. Also, because the response of A^p showed a strong two-regime ACE transform due to switches in dominance of the surface versus atmospheric contributions to Bond albedo on cold versus warm planets, respectively, we also performed ACE fits for these two responses (where the surface and atmospheric contributions sum to Bond albedo). Finally, as the relation of A^p to T_surf is complex, we also performed ACE fits directly for T_surf.

Because ACE produces only numerical transforms and an overall R² of the resulting GAM but no statistics of significance, we then fit the ACE transforms themselves in a linear regression to calculate the Student’s t-value to obtain the (1 – α) level of confidence in transformed parameter fits and thus identify significant variables. The transforms of only the significant variables then were inspected for features indicating either sampling or physical explanations, and for behaviors that could be described with parametric fits, such as simple linear, piecewise linear, exponential, and so forth. Such parametric fits were then made, and the parametric transforms again fit to provide final predictions of the response variables, and their significance.

B.1. Results

B.1.1. Bond Albedo

ACE Transforms:

The ACE transforms of all 9 parameters as explanatory variables and of A^p as the response are shown in Figure 14 for all 48 experiments pooled, and are shown in Figure 15 for G star (33 experiments) and M star (15 experiments) planets fitted separately. Colors of dots distinguish G and M star planets and, within the G star planets, the one early Mars and three early Venus experiments. The (1 – α) level of confidence in secondary fits of the transformed parameters in a linear model are denoted by asterisks above each figure panel. With all experiments pooled, the overall ACE R² is 0.968, and a linear fit of the transforms gives an R² of 0.97 and adjusted R² of 0.96. This is high due to the strong influence of the early Venus experiments. Separate ACE fits for the G and M star planets give R² values of 0.966 and 0.861, respectively. Linear fits of the ACE transforms result in R² of 0.97 and adjusted R² of 0.96 for the G star planets, and 0.94 and 0.85 for the M star planets. For the M star planets, with only 15 experiments and 9 variables, this leaves only 6 degrees of freedom; given the few actually significant variables, the fairly high R² is due more to overfitting. The only robustly significant variable is log₁₀(CO₂.mb), whereas the others appear slightly correlated according to various ANOVA comparisons (not shown).

Figure 14.

ACE transforms on 9 parameters and a logical vector distinguishing G versus M star planets, with Bond albedo (“planetary.albedo”) as the response, pooling all n = 48 experiments. The sum of the transformed explanatory variables versus the transform of the response variable is also plotted. For Stellar.type, 1 indicates the intercept offset for G star planets, and 2 for M star planets. Color legend is given in Figure 15. Asterisks give the (1 − α) level of confidence of the transformed parameter fits in a linear fit, with “***,” 0.001, “**,” 0.01, “*,” 0.05, “.,” 0.1; “not,” ⩾0.1 to 1. While the overall ACE R² is 0.968, a linear fit of the transforms gives an R² of 0.971 and adjusted R² of 0.963.

Figure 15.

ACE transforms for the same variables as in Figure 14, but separately for G star and M star planets. Asterisks denote the (1 − α) level of confidence as in Figure 14. The R² values for the ACE fits for the G and M star planets are 0.966 and 0.861, respectively. Linear fits of the ACE transforms result in R² of 0.970 and adjusted R² of 0.958 for the G star planets, and 0.938 and 0.855 for the M star planets.

Most variables are significant across all 48 experiments, but some are significant within only the G star or M star planet subsets. With all 48 experiments fitted together, all transformed variables are highly significant except for eccentricity and land fraction. For the G star planets, all are significant except for rotation period. For M star planets, few variables are significant, with only log10(CO₂.mb) significant at the α < 0.05 level (95% confidence); eccentricity at α < 0.1 (90% confidence), because only one planet has non-zero eccentricity; rotation period at α < 0.1 (90% confidence); and there is no variation in obliquity from 0°, so no transform is fit. The logical vector distinguishing stellar classes is highly significant at α < 0.001. Noticeable is that the intercept when all planets are combined is highly significant at α < 0.01 (99% confidence), implying that there is a systematic difference in A^p between planets orbiting G versus M stars.

There are clearly at least two distinct regimes for the behavior of A^p. When combining all experiments into an ACE fit, this response of A^p to the transformed variables is non-monotonic, with a damped response or transform at A^p ≲ 0.452 (transition occurs between 0.452 and 0.483, and there are no other G star data points between these values). The Venus planets seem to extend an increasing trend in the ACE transform at higher A^p (when the Venus planets are removed, the ACE transform remains the same for the other planets). In the separate ACE fits for the G and M star planets, the two stellar types exhibit parallel but offset behavior in the steep portion of the transform. The lower-albedo damping seen in the all-planets ACE fit is clearly driven by the G star planets, and the M star planets appear not to span a range covering this damped regime; the kink in the transform when all experiments are combined in the ACE fit is a result of the offset behavior of the G and M star planets. For the G star planets, the transition between the damped and sensitive steep responses occurs at the same values of A^p as for all planets combined.

For S_0X, the ACE transforms show a similar dominance by G star planets in the 48-experiment ACE fit, and an offset parallel transform for M star planets relative to the G star planets. Opposing very linear trends occur with a minimum at S_0X = 0.94 for all planets, and at S_0X = 1.0 for G star planets only.

For Tstar.K, the ACE transforms for the G and M star planets are so distinct from each other, both combining all planets as well as fitting separately, that the effect of these two stellar types is clear. There are significant trends across stellar types as well as within the G stellar type planets, but any trend is not significant within the M star planets. The same can be said for the orbital period, which is dominated by Earth for G star planets.

The ACE transforms for orbital period, rotation period, eccentricity, and obliquity all appear to exhibit piecewise linearity or simple linearity. Thus these transforms can be emulated through segmented regressions or simple linear relations. Earth is often at the point of discontinuity.

The transform of Land.fraction exhibits an S-shape that results from the G star planets, which may be approximated as piecewise linear along segments covering aquaplanets, paleo-to modern Earth continents, then Venus and Mars. Only one M star planet is not an aquaplanet.

The transforms of log₁₀(CO₂.mb), for all planets combined as well as separated by stellar type, exhibit complex behavior. In the ACE fit with all 48 planets, the transform is piecewise linear negative, with a sharp dip in the transform at log₁₀(CO₂.mb) ∼ −0.4, which corresponds to Proxima Centauri b experiments at 376 ppmv CO₂ (log10(CO2.mb) ∼ −0.43), Kepler-1649b, early Venus, and a series of slow rotation with higher S_0x experiments for Earth-like planets. The transform of log₁₀(CO₂.mb) appears to be a catch-all for effects not captured by other parameter transforms, or the CO₂ greenhouse effect is due to interactions with other parameters—interactions we have not represented in these models. The ACE transforms for rotation period or S_0X apparently fail to capture the full effects of this parameter, whose influence on Bond albedo occurs in the dip in the transform for CO₂. The other planets in this dip have unique characteristics whose parameters we have not included, such as differences in ocean salinity and spin–orbit resonance state (Proxima Centauri b experiments), the only M star planet with high S_ox and slow rotation (Kepler-1649b), and the only planets with retrograde rotation and at an S_0x high enough to create a runaway greenhouse state, were it not for shielding by clouds (early Venus). The dip implies that these planets are all cooler than they might otherwise be given their CO₂ partial pressure.

The ACE transform for log₁₀(CO₂.mb) for the G star planets only offsets the dip upward and upper linear relation, with the discontinuity still at log₁₀(CO₂.mb) ∼ −0.40 corresponding to early Venus and the slow rotation and high S_0X experiments. With the M star planets, the discontinuity occurs at the Proxima b experiments at 376 ppmv CO₂ (log₁₀(CO₂.mb) ∼ −0.43) and K-1649 b (376 ppmv, log₁₀(CO₂.mb) = −0.42). It appears that CO₂ values closest to that of modern Earth again may be a transition point, or the statistical fits are heavily weighted by a higher sampling of planets possessing Earth-like parameter values.

The ACE transforms for log₁₀(CH₄.mb) appear to be close to linear, both with all experiments and separately for the G and M star planets. However, there is little variability in the sampled values, with possible discontinuities around the same G star planets as for log₁₀(CO₂.mb). Only two M star planets have non-zero CH₄—the Proxima b experiments with Archean-like atmospheres with very high CH₄—but these are not sufficient to show statistical significance.

In general, the negative relation between the transforms of the greenhouse gases and that of A^p is consistent with the near-IR absorption effects of these gases tending to reduce A^p.

Generalized Additive Model:

To fit a GAM that approximates the ACE fits above, and that takes into account both the significant relations across and within stellar types, there are a number of options. The ACE transform to Bond albedo for the fit to all 48 experiments is not straightforward due to its odd shape, while the transforms for the separate stellar types could be approximated by linear, linear-piecewise, or exponential fits; however, these vary in significance between and across the two stellar types. Ideally, one would fit parametric functions or segmented regressions for all nonlinear transforms, such that a GAM regression would then take into account the additional parameterizations to penalize the model score, and also the transforms could be calculated for new data. Given our sparse data sample, such additional functional fits to all the transforms would be excessive, as uncertainty in these transforms is unknown without more data.

Because it is clear that the G star versus M star planets exhibit distinct offset behaviors, we present the following method for a final illustrative prediction:

1. Assume that the ACE transforms for the separate stellar types are the closest to reality or appropriately describe the data

2. Combine the separate stellar type transforms into single vectors, as total transforms for each variable for the entire ensemble, and use these transforms to fit a GAM

3. Fit parametric functions to the transforms just for the predicted variable, Bond albedo, to enable back-transformation of the predicted transformed value to the original albedo units

4. Compare the AIC score of the GAM with versus without transformed explanatory variables, and compare residuals of the back-transformation to the original albedo units

For the parametric fits to the transforms of Bond albedo (step 3, above), the G star planets transform was done as a segmented regression to predict A^p for G star planets from the ACE transform, acet(Ap.G), with the segmentation between acet(Ap.G) < −0.1 and acet(Ap.G) = −0.1. We call this inverse function “invt.Ap.G” (Figure 16). For the M star planets, a simple linear fit was sufficient, which we call function “invtAp.M” (Figure 17).

Figure 16.

Function “invt.Ap.G”: back-transform of the ACE transform of Bond albedo for G star planets, done as a segmented linear regression. Residual standard error: 0.008696 on 29 degrees of freedom. Multiple R-squared: 0.9935.

Figure 17.

Function “invt.Ap.M”: back-transform of the ACE transform of Bond albedo for M star planets, as a simple linear regression. Residual standard error: 0.005632 on 13 degrees of freedom. Multiple R-squared: 0.9932.

A GAM fit on the combined ACE transforms for both G and M star planets, of all variables, is given in Table 6, which we call “model tAp.all.” All transformed terms are highly significant, giving an adjusted R² of 0.92 relative to the predicted transform. The back-transformed prediction of Bond albedo is shown in Figure 18, which gives an R² of 0.82. To estimate the adjusted R², we counted the parameters in the inverse transform functions, giving a total of 9 variables and an additional 3 and 1 more parameters from the G star and M star planet back-transform fits, or number of parameters p = 13, which yields an adjusted R² of 0.752. This is a slight improvement over the adjusted R² of 0.675 from the segmented regression of Equations (3) and (4) in the main text.

Table 6.

Model “tAp.all”: Generalized Additive Model (GAM) Fit, Combining ACE Transforms for the Different Stellar Types, Predicting the ACE Transforms of Bond Albedo

Variable	Estimate	Std. Error	t Value	Pr(>|t|)	Confidence(1 − α) in Estimated Coefficient
(Intercept)	−4.89E−16	4.15E−02	0.000	1
acet(S0X)	1.01E+00	6.38E−02	15.826	<2e−16	***
acet(Tstar.K)	9.00E−01	1.39E−01	6.487	1.23E−07	***
acet(orb.period.Earthdays)	9.54E−01	1.26E−01	7.582	4.07E−09	***
acet(rotation.XEarth.day)	9.30E−01	1.21E−01	7.66	3.20E−09	***
acet(eccentricity)	9.66E−01	3.64E−01	2.651	0.01164	*
acet(Land.fraction)	8.23E−01	2.67E−01	3.079	0.003846	**
acet(log10CO2.mb)	1.03E+00	2.46E−01	4.179	0.000165	***
acet(log10CH4.mb)	8.90E−01	1.43E−01	6.237	2.69E−07	***
acet(obliquity.degrees)	8.89E−01	2.36E−01	3.772	0.000552	***

Note. Residual standard error: 0.2873 on 38 degrees of freedom. Multiple R-squared: 0.9346. Adjusted R-squared: 0.9192. AIC: 27.287.

Figure 18.

Left: predicted Bond albedo for the entire ensemble, from back-transform of GAM prediction. R² = 0.82, Adjusted R² = 0.75. Right: histogram of errors in predicted albedo.

Comparison to Other Possible Statistical Models:

Do the additional variables and transforms make a better model statistically? All transformed terms are highly significant in model tAp.all, with especially high t-values for S_0X, Tstar.K, orbital period, rotation period, log₁₀(CO₂.mb), and log₁₀(CH₄.mb); therefore, dropping any of the terms should worsen the AIC score. To illustrate how this model provides a better prediction, we compare to a model without transforms (“tAp.no-x.all”) and a model with only the two variables of our original regression (“tAp.tS0X.tTstar.K.all”), all attempting to predict the same transforms in model “tAp.all.” The AIC score can only be compared among models with the same predicted variable, so we cannot compare directly to AIC scores for Equations (3) and (4), which directly predict A^p, whereas model tAp.all predicts the ACE transform of A^p.

The AIC score for tAp.all is 27. Without transformations on the explanatory variables (model “tAp.no-tx.all,” table not shown), the AIC score is 109, the R² is 0.64, the adjusted R² is only 0.555, and only S_0X and T_star.K are significant when no nonlinear transforms are made. Back-transforming to Bond albedo, the R² is actually negative, indicating that the fit without transforms to explanatory variables is worse than a horizontal line. With only the transform terms for S_0X and Tstar.K to predict the transformed Bond albedo (model “tAp.tS0X.tTstarK.all”), the AIC score is 82, while the R² is 0.727 and adjusted R² is 0.715. However, the back-transformed Bond albedo has a very poor R² of 0.435. In this case, it would be better to do direct prediction of the Bond albedo with Equations (3) and (4), or, better, re-fit the ACE transformations with only the two variables to fit an alternative transform for the Bond albedo; however, the model would not be directly comparable to tAp.all through the AIC, but only through R² and adjusted R² values. The various scores are summarized in Table 7. The log-likelihood for improvement of model tAp.all over the others is as expected highly significant, given the drastic differences in these models and the high significance of all terms in tAp.all.

Table 7.

Comparison of Models Predicting ACE Transform of Bond Albedo, Back-transformed Prediction, R², and Adjusted R², for Prediction of the ACE Transform of A, for Prediction of the Back-transform to A, AIC Score, and Log-likelihood That the Model Minimizes the Information Loss over the Model with the Lowest AIC Score

Model	Description	R2 for Prediction of acet(A)	Adjusted R2 Prediction of acet(A)	R2 after Back-transform to A Units	AIC	Log-likelihood
tAp.all	GAM of all ACE transforms	0.935	0.919	0.834	27	1.0
tAp.no-x.all	GAM without ACE transforms of explanatory variables in tAp.all	0.640	0.555	−0.211	109	1.6e−18
tAp.tS0X.tTstar. K.all	GAM with only two variables, transforms of S0X and Tstar.K	0.727	0.715	0.352	82	1.1e−12

In summary, the above exercise provides a model “tAp.all” with parametric functional transforms to predict Bond albedo that is a better predictor than the simple segmented linear model of Equations (3) and (4), and shows nonlinear relations between the Bond albedo and nine planetary parameters that are highly significant. At least four of the variables have very high t-values, which lends confidence to their strong role in influencing planetary Bond albedo. In the cases of S_0X, Tstar.K, log₁₀(CO₂.mb), log₁₀(CH₄.mb), and rotation period, the relations have clear physical explanations, whereas with orbital period the fit might be more due to the sampling pattern than a true physical relation. Nonetheless, this exercise demonstrates that these variables have explanatory significance for the differences in Bond albedo among our 48 experiments.

Surface Temperature from Equation (6):

Finally, our goal of course is to be able to predict the surface temperature. Using Equation (5) to calculate T_eq from the improved prediction of A^p, and then using this in Equation (6) to predict T_surf from T_eq, gives an R² of 0.745 or adjusted R² of 0.74 (Figure 19), which is little different from these metrics for the original Equation (6). The latter inherently underpredicts the higher T_surf values due to large scatter. Although Equation (6) demonstrates the strongly linear relation of T_surf to T_eq, the scatter implies that other parameters influence the variation in surface temperature.

Figure 19.

(Top left) T_eq calculated from Bond albedo in model “tAp.all.” (Bottom left) Prediction of surface temperature (T_surf) versus actual, using the predicted T_eq, in g Equation (6) to obtain T_surf. (Bottom right) Histogram of errors in the prediction.

While Bond albedo is a quantity that could be inferred from future observations of reflected light phase curves (or indirectly from thermal phase curves), its relation (via T_eq) to T_surf is itself worth further investigation. A direct prediction of T_surf (Figures 20 and 21) from the planetary parameters shows that it may be easier to predict than A^p. We investigate this next.

Figure 20.

ACE transforms on 9 parameters and a logical vector distinguishing G versus M star planets, with surface temperature (“Tsurf.actual.C”) as the response, pooling all n = 48 experiments. For Stellar.type, 1 indicates the intercept offset for G star planets, and 2 for M star planets. Colors and significance annotations as described in Figures 14 and 15. ACE fit gives an R² of 0.925, while a linear fit of the transforms gives an R² of 0.930 and adjusted R² of 0.912.

Figure 21.

ACE transforms for the same variables as in Figure 20 for surface temperature, but separately for G star and M star planets. Asterisks denote the (1 − α) level of confidence as described in Figure 14. The R² values for the ACE fits for the G and M star planets are 0.965 and 0.971, respectively. Linear fits of the ACE transforms result in R² of 0.968 and adjusted R² of 0.955 for the G star planets, and 0.989 and 0.974 for the M star planets.

Surface Temperature from Direct ACE Fit:

The ACE transforms to predict surface temperature (“Tsurf.actual.C”), both with all experiments combined (Figure 20) and for stellar types separately (Figure 21), show that the response of T_surf to the transforms of the other variables is monotonic and more nearly linear or simply piecewise linear, compared to the transform for Bond albedo. Again, as with Bond albedo, while the average planetary surface temperature can be predicted with a high R² of ∼0.93, fitting the data separately by stellar type shows some different behaviors, or offset behaviors, by stellar type, such that capturing these different responses improves the prediction. The ACE transform of T_surf for the G star planets exhibits complex nonlinearity, nearly S-shaped, but falling closely to the nearly linear transform for the M star planets. Ancient Mars, as an outlier, very likely is skewing the transform at the lower end of the G star experiments.

The transform of S_0X for G star planets shows persistence of the two-regime behavior with the discontinuity at S_0X = 1 as for Bond albedo, while the M star planets appear not to be sampled across both regimes. The transforms for stellar temperature (Tstar.K), orbital period (orb.period.Earthdays), rotation period (rotation.XEarth.day), and especially CH₄ (log₁₀(CH₄.mb)), which show nonlinear behavior in Figure 20, produce simple linear relations for the separate stellar types. While eccentricity is not a significant influence when combining all experiments, within the stellar type it is significant in explaining variation in surface temperature, and apparently linear in this ensemble, except for Mars as an outlier. Land fraction and log₁₀(CO₂.mb) continue to exhibit complex nonlinearities, but smoother than with Bond albedo. The piecewise components in the Land. fraction transform for G star planets correspond to categories, in increasing order, aquaplanets, paleo-Earth continental scenarios, early Venus, and Mars. Among the parameters, rotation period, eccentricity, and land fraction have transforms with negative trends, indicating that they are correlated with cooling of the planet surface. For rotation this feature is well-documented (Yang et al. 2014; Way et al. 2018). For land fraction it is plausibly an effect of the higher albedo of land than ocean surfaces. For eccentricity, though, it is likely to be an artifact of the small range sampled, the only exception being one moderately high eccentricity planet that is cool because it is in 3:2 spin–orbit resonance (Del Genio et al. 2019b).

As with Bond albedo, we do a linear regression for T_surf to evaluate a GAM combining the ACE transforms by stellar type (Figure 21). This yields an overall R² of ∼0.96 (Table 8). The highest t-values, in descending order, are for S_0X, log₁₀(CO₂.mb), log₁₀(CH₄.mb), land fraction, and stellar temperature, while all transformed variables are highly significant at a level α < 0.001.

To back-transform the ACE transforms of T_surf for the G star planets, due to the complex transform, we do simple linear interpolation (Figure 22). If the predicted transformed surface temperature from model tTsurf.all is outside the range of the original values, then the back-transform uses the nearest data extreme to extrapolate. (This happens only for one point at the high extreme.) For the M stars, we do a simple linear fit (Figure 23 and Table 9). The transform for the M stars is not perfectly linear, and it could be fit with a curve instead (e.g., a three-parameter curve fit can still give a higher adjusted R²); however, introducing more terms is excessive, given the already high R² and limited information from few data points, so we keep the simpler model.

Figure 22.

Back-transform from ACE transform of surface temperature (Tsurf) for G stars to surface temperature in °C. Because this is done through linear interpolation, there are no regression statistics.

Figure 23.

Back-transform of the ACE transform of surface temperature (Tsurf) for M star planets, as a simple linear regression.

Table 9.

Back-transform of the ACE Transform of Surface Temperature (Tsurf) for M Star Planets, as a Simple Linear Regression

ACE-transformed Variable	Estimate	Std. Error	t Value	Pr(>|t|)	Signif
(Intercept)	−8.1333	0.6137	−13.25	6.30E−09	***
tTsurf.M	24.7982	0.6137	40.41	4.71E−15	***

Notes. Residual standard error 2.377 on 13 degrees of freedom. Multiple R² and Adjusted R²: 0.99.

^aSignificance levels:

^“***”α ∼ 0,

^“**”α = 0.001,

^“*”α = 0.05,

^“.”α = 0.1, “α = 1.

The prediction of T_surf could be simplified by approximating the ACE transforms for T_surf as simply linear, thus avoiding the need to do back-transforms. A check of this result shows one can still obtain an R² of 0.91 or adjusted R² of 0.89 and significance of all terms at a level of α ≪ 0.001, except for obliquity’s significance decreasing to a respectable level of α < 0.01. However, residual errors exhibit a curvature that indicate the presence of the nonlinearity in relations. As a simpler model with a high R², it may be the more parsimonious or convenient model until the uncertainties of these predictions can be estimated.

Relation between Bond Albedo and Surface Temperature:

With an adjusted R² of 0.929 in our model tTsurf.all, we have shown that if additional planetary parameters are accounted for, an additional ∼25%–30% of the variance in surface temperature (Figure 24) could be explained over the adjusted R² of 0.63–0.65 from our simple two-variable models in Equations (3) and (4). However, Bond albedo alone provides an adjusted R² of 0.723.

Figure 24.

(left) Predicted surface temperature (Tsurf, predicted) versus actual, R² = 0.947, adjusted R² = 0.929. (right) Distribution of errors, from model “tTsurf. all” of Table 8, with back-transforms to °C using the functions in Figures 22 and 23.

Current exoplanet observing missions such as the Transiting Exoplanet Survey Satellite and the CHaracterising ExOPlanet Satellite will be used to characterize the planetary albedo of larger planets. In the near future a limited amount of direct imaging by the James Webb Space Telescope mission may do so for a small number of Earth-size planets, although most direct imaging of such planets is several decades away. From just transit observations, variables that are known or could be obtained are S_0X, T_star.K, planet radius, orbital period, and Bond albedo. If A is observed, then how well can we infer T_surf? Performing ACE fits on these variables to predict T_surf by stellar type shows ACE transforms that are all remarkably linear and significant for all (not shown). We find that a simple linear regression can provide an R² of 0.84 or adjusted R² of 0.82. Then T_surf can be predicted to within 10 °C accuracy for 39 of our 48 planets, within 5 °C accuracy for 37, and within 1 °C accuracy for 35 of our 48 planets (before propagation of uncertainty in the observations).

B.2. Discussion

There are a variety of other ways the data could have been fit with the techniques presented here; our results represent only one possibility. We do not include possible interaction effects among the variables, which could be considered by normalizing the variables and providing products of the normalized variables as interaction terms to ACE. As Bond albedo is a non-monotonic function of the other variables, one could examine predictions of the atmospheric and surface components of the Bond albedo to separate their effects and nonlinear responses to the planetary variables. More parsimonious models have not been ruled out. For example, while it is clear that nine of the planetary parameters are significant in predicting Bond albedo, other statistical fits could yield simpler models with fewer variables, which may or may not be better models, but their utility is also a matter of what observational constraints users of such a model will have available to them, and how important higher accuracy is.

The GCM data set is, of course, not ideal for any statistical analysis. However, it may be looked upon as an example of the biased, sparse observations that will be obtained from exoplanet observing missions. ACE may not be the best statistical approach to analyzing such data, but it is able to quantify patterns that we expect or can explain from qualitatively known physics. Future analyses may want to investigate Bayesian approaches to statistical inference on sparse data. In addition, large ensembles of well-sampled experiments will better inform the influence of different planetary parameters on A and T_surf. If A, or an indirect estimate of it from the emitted thermal flux, is the best observation to be obtained of a planet’s energy balance from missions in the near future, further direct analysis of the relationship between A and T_surf with respect to other variables could be a useful approach for inferring the potential habitability of exoplanets.