A Scaling for Atmospheric Heat Redistribution on Tidally-Locked Rocky Planets

TL;DR

This paper introduces an analytical scaling theory to predict atmospheric heat redistribution on tidally-locked rocky exoplanets, aiding interpretation of observations and improving 1D climate models.

Contribution

It presents a novel analytical scaling for heat redistribution based on planetary parameters, validated against GCM simulations, bridging theory and upcoming JWST observations.

Findings

Heat redistribution becomes significant above ~1 bar surface pressure.

Thermal phase curves deviate from bare rock predictions with increasing pressure.

The scaling applies to diverse atmospheres, including H2, CO, and lava worlds.

Abstract

Atmospheric heat redistribution shapes the remote appearance of rocky exoplanets but there is currently no easy way to predict a planet's heat redistribution from its physical properties. This paper proposes an analytical scaling theory for the heat redistribution on tidally-locked rocky exoplanets. The main parameters of the scaling are a planet's equilibrium temperature, surface pressure, and broadband longwave optical thickness. The scaling compares favorably against idealized general circulation model simulations of TRAPPIST-1b, GJ1132b, and LHS 3844b. For these planets heat redistribution generally becomes efficient, and a planet's observable thermal phase curve and secondary eclipse start to deviate significantly from that of a bare rock, once surface pressure exceeds O(1) bar. The scaling additionally points to planetary scenarios for which heat transport can be notably more or less efficient, such as H2 and CO atmospheres or hot lava ocean worlds. The results thus bridge the gap between theory and imminent observations with the James Webb Space Telescope. They can also be used to parameterize the effect of 3D atmospheric dynamics in 1D models, thereby improving the self-consistency of such models.

Tables

Table 1. Table 1: Planetary parameters for the GCM simulations.

	Radius	Period	Surf. gravity	Teq**Equilibrium temperature, for uniform heat redistribution and zero albedo.
	(R⊕)	(days)	(m/s2)	(K)
TRAPPIST-1b	1.12	1.51	7.95	391
GJ1132b	1.16	1.63	11.8	578
LHS3844b	1.32	0.46	12.9**** Assuming 2.3 $M_{\oplus }$, based on Chen & Kipping (2017).	805

Equations

$$T_{day}$$

$$\frac{c_p\omega }{g}\left(\frac{dT}{d\tau }-\beta \frac{T}{\tau }\right)$$

$$-\frac{c_p\omega }{g}\beta \frac{T_{skin}}{\tau }$$

$$F({\tau }_{LW})-F_n$$

$$F_n$$

$${\rho }_s{C}_d{U}^3=\eta Q.$$

$${\omega }_0$$

$$F_n\approx \chi \frac{c_p\beta }{gL}{\left(\frac{R{p}_s^2{\tau }_{LW}\sigma T_{eq}^8}{C_d}\right)}^{1/3},$$

$$F_n\approx \chi \frac{c_p\beta }{gL}{\left(\frac{R{p}_s^2{\tau }_{LW}}{C_d{\sigma }^2{T}_{eq}^4}\right)}^{1/3}\sigma T_{eq}^4,$$

$$F_n$$

$$F_{d,obs}$$

$$f$$

$$F$$

$${\tau }_{LW}$$

Full text

planets and satellites: atmospheres — planets and satellites: terrestrial planets — planets and satellites: individual (GJ 1132 b, LHS 3844 b, TRAPPIST-1 b, LHS 1140 b, 55 Cnc e, WASP-47 e, HD 219134 b, HD 15337 b, L 98-59 b, HD 213885 b, TOI-270 b, GL 357 b, K2-141 b)

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1 Introduction

Telescope observations have recently begun to probe the atmospheres of small, rocky, exoplanets around nearby stars (Demory et al., 2016; de Wit et al., 2018; Diamond-Lowe et al., 2018; Kreidberg et al., 2019). These observations will only get better in the near future with the launch of the James Webb Space Telescope (JWST) and the construction of extremely large ground-based telescopes. The data that we will be able to obtain with these instruments promise insight into fundamental questions such as how rocky exoplanets form, how many of them host atmospheres, and whether any of them might be habitable.

To interpret these observations correctly, however, we need to understand the chemical and physical processes that shape the observable features of rocky exoplanets. These processes include radiative transfer and molecular absorption (Seager & Sasselov, 2000), gas-phase chemistry (Moses et al., 2011; Hu & Seager, 2014), clouds and hazes (Hörst et al., 2018; Moran et al., 2018), as well as large-scale atmospheric dynamics (Joshi et al., 1997; Merlis & Schneider, 2010). Of these processes atmospheric dynamics has an outsized impact on thermal observations, because it determines a planet’s global heat redistribution which sets the depth of the planet’s secondary eclipse as well as the shape of the planet’s thermal phase curve.

Unfortunately heat redistribution remains poorly represented in most models that are being used to match and interpret exoplanet observations. The underlying reason is that these retrieval models have to be fast enough to be run $\sim 10^{5}-10^{6}$ times (Madhusudhan & Seager, 2009; Line et al., 2013), which is required to comprehensively map out which atmospheric scenarios can, or cannot, fit an observed dataset. As a consequence most retrieval models are idealized one-dimensional representations of a planet’s atmosphere (e.g., Tinetti et al., 2007; Madhusudhan & Seager, 2009; Benneke & Seager, 2012; Line et al., 2013; Kempton et al., 2017). By virtue of being 1D, these models cannot resolve 3D processes such as clouds and atmospheric heat redistribution, which has been shown to bias the results of atmospheric retrievals as well as model mean states (Line & Parmentier, 2016; Feng et al., 2016; Fauchez et al., 2018).

In principle 3D processes including atmospheric heat redistribution can be resolved by more complex models, such as general circulation models (GCMs). In practice 3D models are computationally far too costly to be used in place of 1D retrieval models. For example, the GCM used later in this paper requires about 24 hours of computation time on 8 processors. Even a dedicated supercomputer, running 32 GCM simulations in parallel for an entire month, would thus only be able to explore $\sim 10^{3}$ parameter combinations, falling far short of being useful for standard retrieval techniques.

An important question is therefore how exoplanet retrieval models should represent the basic physics that determines observable quantities such as a planet’s dayside emission or its thermal phase curve, while remaining computationally cheap. Most 1D models represent the effects of heat redistribution by adjusting the planet’s dayside energy budget, which can be written as (Burrows, 2014)

[TABLE]

Here $T_{day}$ is the observed dayside brightness temperature111$T_{day}$ is hotter than the average dayside temperature because the observer viewing geometry is skewed towards the hot substellar point (Cowan & Agol, 2008)., $T_{*}$ is the stellar temperature, $R_{*}$ is the stellar radius, $d$ is the planet’s semi-major axis, $\alpha_{B}$ is the planet’s bond albedo, and $f$ is the so-called heat redistribution factor. It is useful to re-express $T_{day}$ in terms of the equilibrium temperature $T_{eq}=T_{*}\sqrt{R_{*}/(2d)}(1-\alpha_{B})^{1/4}$, which is the temperature of an isothermal sphere with the same bond albedo and semi-major axis as the planet. The temperature $T_{eq}$ is closely related to the planet’s characteristic fluxes, in particular $\sigma T_{eq}^{4}$, the flux received from the host star averaged across the entire planet, $2\sigma T_{eq}^{4}$, the stellar flux averaged across the planet’s dayside only, and $4\sigma T_{eq}^{4}$, the incoming flux at the planet’s substellar point. The redistribution factor $f$ has to lie between $2/3$ for a planet without an atmosphere and $1/4$ for a planet that is extremely efficient at redistributing heat (Hansen, 2008), but currently there no easy way of expressing $f$ in terms of a planet’s physical properties.

The goal of this paper is to develop a simple scaling theory for $f$. The derivation is shown in Section 2. The results demonstrate that $f$ is sensitive to a planet’s equilibrium temperature, surface pressure, and its broadband optical thickness, although the scaling also describes the dependency on other physical parameters. In Section 3 the scaling is compared against GCM simulations of three nearby rocky planets: TRAPPIST-1b, GJ1132b, and LHS3844b (Gillon et al., 2016; Delrez et al., 2018; Berta-Thompson et al., 2015; Vanderspek et al., 2019), which are among the highest-priority targets for upcoming thermal observations with JWST. The scaling successfully captures the dominant processes that determine atmospheric heat redistribution on these planets. The scaling therefore lends insight into the atmospheric dynamics of tidally locked planets. Moreover, it can be used as a computationally efficient parameterization of large-scale dynamics in 1D models. Because the derivation and GCM simulations assume idealized semi-grey radiative transfer, Section 4 considers how this work can be applied to real gases with non-grey absorption. The results are discussed in Section 5, and the conclusions are in Section 6.

2 A scaling for atmospheric heat redistribution

Using the grey optical depth $\tau$ as the vertical coordinate, the balance can be written as (see Koll & Abbot, 2016)

[TABLE]

Here $T(\tau)$ is the atmosphere’s nightside vertical temperature profile, $F(\tau)$ is the net infrared flux, $\omega$ is the vertical velocity, $c_{p}$ is the atmosphere’s specific heat capacity, $g$ is the acceleration of gravity, $\beta\equiv R/(c_{p}n_{LW})$ is the dry adiabatic lapse rate in optical depth coordinates, $R$ is the atmosphere’s specific heat constant, and $n_{LW}=1$ if opacity is independent of pressure (e.g., if molecular line widths are set by thermal broadening) while $n_{LW}=2$ for pressure broadening. If one evaluates $F$ at the top-of-atmosphere one obtains the nightside’s flux of outgoing longwave radiation, $F_{n}\equiv F(\tau=0)$. equal the day-night heat transport by the atmosphere.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

To do so note that Equation 7 cannot be generally valid because it predicts that $F_{n}\rightarrow\infty$ as $p_{s}$ or $\tau_{LW}$ become large, whereas energy conservation requires that $F_{n}\rightarrow\sigma T_{eq}^{4}$ for a thick atmosphere. A general solution needs to reduce to the asymptotic solution as $p_{s}$ and $\tau_{LW}$ become small, but tend towards $\sigma T_{eq}^{4}$ as one or both parameters become large. following expression satisfies both requirements:

[TABLE]

There is no guarantee that Equation 8 will be exact when is of order unity. Nevertheless, because reduces to the correct limits for both a thin and a thick atmosphere, it is physically motivated.

To transform $F_{n}$ into a prediction for the dayside flux $F_{d}$ one can use the planet’s energy budget, At secondary eclipse the observer’s view is weighted towards the hot substellar point, which directly faces the observer, and is less sensitive to colder regions that lie close to the terminator, which tilt away from the observer (Cowan & Agol, 2008). The limiting expressions for the the observed dayside flux is thus

[TABLE]

Here $k=Lg/(\chi\beta c_{p})\times(C_{d}\sigma^{2}/R)^{1/3}(1\mathrm{bar})^{-2/3}(600\mathrm{K})^{4/3}$ captures all planetary parameters other than the optical thickness, surface pressure, and equilibrium temperature. $k$ is of order unity and planetary scenarios (e.g., radius and surface gravity tend to vary by less than a factor of two between different rocky planets).

To relate the main result back to the heat redistribution factor $f$ used in 1D models, one

[TABLE]

As expected, Equation 10 recovers the no-redistribution limit $f\rightarrow 2/3$ as $\tau_{LW},p_{s}\rightarrow 0$, and the uniform-redistribution limit $f\rightarrow 1/4$ as $\tau_{LW},p_{s}\rightarrow\infty$.

3 Testing theory with GCM Simulations

3.1 Numerical setup

To test the analytical the Flexible Model System (FMS) general circulation model (GCM) with dry thermodynamics. FMS is a widely-used model which has previously been applied to the atmospheres of Earth (Frierson et al., 2006), Jupiter (Liu & Schneider, 2011), hot Jupiters (Heng et al., 2011), tidally locked terrestrial planets (Merlis & Schneider, 2010; Mills & Abbot, 2013; Koll & Abbot, 2015, 2016; Hammond & Pierrehumbert, 2017), as well as rapidly rotating terrestrial planets (Kaspi & Showman, 2015). Consistent with dry thermodynamics, the radiative impact of clouds or photochemical hazes. of these modeling assumptions.

The FMS version simulates the atmosphere’s full large-scale dynamics coupled to semi-grey (shortwave versus longwave) radiative transfer. Convection is parameterized as instantaneous dry convective adjustment. Near-surface turbulence is parametrized using a standard Monin-Obukhov scheme which self-consistently computes the depth of the boundary layer as well as turbulent diffusion of heat and momentum. The surface is represented by a “slab layer”, that is a single layer with uniform temperature and fixed layer depth. Simulations are all tidally locked and orbits are assumed to be circular so that the stellar flux is fixed in space and constant in time.

, so all stellar energy is at the planet’s surface. Similar to Frierson et al. (2006) the longwave optical thickness depends quadratically on pressure in the lower atmosphere, which represents the effects of pressure broadening, while it depends linearly on pressure in the upper atmosphere, which represents the effect of thermal broadening and helps ensure that the stratosphere equilibrates within reasonable run times.

equivalent to about $64\times 128$ points in latitude and longitude, . As is standard in GCMs, FMS includes horizontal hyperdiffusion which acts as a kinetic energy filter at the smallest length scales resolved by the model. Although such a filter can be potentially problematic in modeling gas giant atmospheres, because the physical processes that lead to frictional dissipation in gas giants are often not explicitly modeled and thus hyperdiffusion acts as a stand-in for unresolved physics (Koll & Komacek, 2018), this issue is less important for rocky planets where friction from the solid surface is captured by the Monin-Obukhov boundary layer scheme.

Three rocky planets that orbit nearby M-dwarfs : TRAPPIST-1b, GJ1132b, and LHS3844b. The planetary parameters are shown in Table 1.

The three planets span a wide range of equilibrium temperatures, which is one of the dominant parameters in the (Section 2). In addition, the three planets span different rotational regimes. LHS3844b has an orbital period of about 11h, whereas TRAPPIST-1b and GJ1132b have orbital periods of 1.5 and 1.6 days, which translates into a nondimensional Rossby deformation radius of $a/L_{Ro}\approx 3$ for LHS3844b versus $a/L_{Ro}\approx 2$ for TRAPPIST-1b and GJ1132b. Given that the theoretical scaling does not account for the effect of planetary rotation, comparison between the three planets thus also provides a check on whether the is robust to changes in planetary rotation.

Consistent with the CFL criterion, are more likely to crash at higher stellar fluxes. therefore use a numerical diffusion coefficient of $2.31\times 10^{5}$ s*-1* (damping time of half a day) for TRAPPIST-1b and GJ1132b, and between $9.26\times 10^{4}$ to $2.31\times 10^{4}$ s*-1* for LHS3844b. In some cases . The default timestep is 120 s for TRAPPIST-1b, 60 s for GJ1132b, and 20 s for LHS3844b.

effect on the atmosphere’s heat redistribution of varying atmospheric surface pressure $p_{s}$ and optical thickness $\tau_{LW}$.

Motivated by the observation that solar system atmospheres have broadband optical thicknesses of about 1-10 at 1 bar (Robinson & Catling, 2014), a scenario in which with surface pressure $p_{s}$ using the relation $\tau_{LW}=p_{s}/(\text{1 bar})$.

3.2 GCM results

Figure 1 shows that thermal phase curves start to significantly deviate from a bare rock once surface pressure exceeds about 1 bar, in agreement with the analytical scaling. Shown are results for simulations with constant optical thickness, $\tau_{LW}=1$. across the JWST MIRI bandpass (5-12$\mu$m). To convert a planet’s simulated thermal emission to a planet-star flux ratio . At surface pressures of 0.1 bar or less phase curves are effectively indistinguishable from a bare rock. This is particularly the case once JWST’s instrumental precision , which cannot be any better than the photon noise limit but could optimistically be comparable to it (Fig. 1). Conversely, at surface pressures above 10 bar phase curves become increasingly uniform. This matches the theoretical prediction Moreover, for a fixed surface pressure, LHS3844b has the least efficient redistribution while TRAPPIST-1b has the most efficient redistribution. This matches the expectation that, at fixed surface pressure , redistribution

Figure 2 compares the theoretical prediction against the dayside brightness temperatures that an observer would see at secondary eclipse. To evaluate $k$ in Equation 10

Figure 2 shows that the scaling correctly captures the main variation in dayside thermal emission across several orders of magnitude variation in surface pressure, and more than one order of magnitude variation in optical thickness. The scaling is not perfect, however.

Nevertheless, the majority of simulations in Figure 2 fall inside the red envelope around the theoretical scaling. This indicates that the differences between and simulations can be largely explained by a factor-of-two inaccuracy in the derivation. One potential source of inaccuracy is the inefficiency factor $\chi$, which constant but which could vary if the atmosphere’s heat engine efficiency shifted in response to changes in surface pressure or optical thickness. Another potential source of inaccuracy is the effect of planetary rotation. For a given planet the rotation rate is constant, so one might naively expect a roughly constant mismatch between and simulations. A planet’s rotation interacts with the large-scale circulation, however, to induce spatially varying flow and temperature patterns which could shift with changes in pressure or optical thickness and thus create a varying mismatch between and simulations.

Figure 3 shows that the scaling also captures the main trends in nightside brightness temperatures. This is not surprising - given that the largely captures the dayside’s thermal emission, energy conservation implies that the should match the nightside thermal emission with similar accuracy. As on the dayside, the majority of nightside brightness temperatures agree with the scaling to within a factor of two change in $k$. This means the is able to a planet’s dayside and nightside thermal emission, which is important because one or both of these quantities can be directly measured via secondary eclipse and thermal phase curve observations.

4 Assigning an equivalent grey optical thickness

Up to now grey radiative transfer, but real gases have an optical thickness that varies strongly as a function of wavelength. How can one assign an equivalent optical thickness $\tau_{LW}$ to atmospheres with realistic compositions? tested Rosseland and Planck mean opacities but found that they provide a poor fit. Instead the degree to which an atmosphere diminishes the surface’s thermal emission, and thus reduces a planet’s observable surface emission below that of a bare rock.

For an atmosphere with spectrally varying absorption, the planet’s top-of-atmosphere thermal flux is equal to

[TABLE]

Here $\lambda$ is wavelength, $T_{s}$ is the surface temperature, the first term is the surface’s blackbody emission attenuated by the overlying atmosphere, and the second term is the atmosphere’s emission. For comparison, with a grey absorber the top-of-atmosphere thermal flux is equal to

[TABLE]

where $\tau_{LW}$ is now the atmosphere’s grey optical thickness.

By setting the two surface terms equal to each other, one can define the equivalent grey optical thickness for any atmospheric composition as

[TABLE]

This equivalent grey optical thickness has the desirable property that it exactly matches the extent to which an atmospheric column attenuates the surface’s thermal flux. It is therefore most appropriate for the thin-atmosphere limit, in which most of a planet’s thermal emission originates at the surface, and it correctly captures how the gradual increase of atmospheric mass then reduces the planet’s surface thermal emission below that of a bare rock.

Figure 4 shows $\tau_{LW}$ for four representative atmospheric compositions, namely pure CO2, H2O, O3, and CO. To calculate $\tau_{LW}$ a 1D radiative transfer model with line-by-line spectral resolution (Koll & Cronin, 2018)222Available at https://github.com/ddbkoll/PyRADS.. The surface temperature is set to the equilibrium temperature of LHS3844b and the overlying atmosphere follows a dry adiabat.

H2O is a very strong absorber, with 10*-2* bar of H2O enough to make the atmosphere optically thick. Next follow CO2 and O3, which become optically thick once surface pressure exceeds about 1 bar. Finally, CO is the poorest absorber out of these gases, and is optically thin even at a surface pressure of 1 bar. The rate at which $\tau_{LW}$ increases with pressure can be roughly understood as follows. At low surface pressure $\tau_{LW}$ is dominated by the behavior of individual molecular absorption lines, so absorption is proportional to pressure (see strong and weak line limits in Pierrehumbert, 2010). H2O and CO both follow this expectation below about $10^{-2}$ bar. At high surface pressures absorption becomes dominated by CIA, which scales with pressure squared. H2O follows this expectation above $1$ bar. Finally, for CO2 and O3, as well as CO above 0.1 bar, $\tau_{LW}$ is set by the closing of a molecule’s window regions and thus a molecule’s overall band shape. For a molecular band for which decays roughly exponentially away from the band center, this implies a logarithmic scaling with pressure (Pierrehumbert, 2010; Koll & Cronin, 2018).

Figure 4 shows that for an atmosphere with a moderately strong greenhouse effect, such as CO2, the equivalent optical thickness of a 1 bar atmosphere is of order unity. Combined with this implies that typical surface pressure above which the thermal phase curve of a rocky exoplanet with $T_{eq}\sim 600$ K starts to deviate from that of a bare rock is also about 1 bar.

5 Discussion

The scaling here provides first-order insight into the processes that determine the day-night thermal contrasts of tidally locked, rocky exoplanets. The scaling builds on the numerical results of Koll & Abbot (2015) and the theory of Koll & Abbot (2016), and translates them into observable quantities that can be measured in the near future with JWST via secondary eclipse and phase curve measurements.

Although the scaling relies on grey radiation for its derivation, it appears to agree with previous GCM studies that used comprehensive radiative transfer. Selsis et al. (2011) found that the transition between large day-night contrasts and uniform emission on short-period rocky planets occurs somewhere between 0.1 and 1 bars. The scaling here agrees with their results and predicts a transition at 0.4 bar, assuming their planetary parameters and $\tau_{LW}\sim 1$. Similarly, Yang et al. (2013) and Wolf et al. (2019) simulated planets with 1 bar . These studies found thermal phase curves that are fairly uniform relative to a bare rock, and thus also support the scaling here which predicts a transition at about 0.2 bar.

.

outstanding issue is that the thermal emission of rocky exoplanets can be affected by additional atmospheric and planetary physics not included here. should overpredict a planet’s day-night thermal contrast. First, the assumes dry thermodynamics, but an atmosphere with condensation can additionally transport heat via latent heat transport (Ding & Pierrehumbert, 2018). The derivation here does not capture this process, so it should tend to overpredict day-night thermal contrasts in atmospheres in which condensation starts to become important. some exoplanets might be able to sustain a surface ocean, in particular planets inside their host star’s habitable zone. In addition to , an ocean can additionally transport energy via its fluid motions which will again reduce the planet’s day-night thermal contrast (Yang et al., 2019).

6 Conclusion

an analytical scaling on tidally locked rocky exoplanets.

In addition to improving our understanding of , the scaling is useful in the context of interpreting thermal observations of rocky exoplanets. For favorable targets, JWST will be able to measure broadband thermal fluxes as well as emission spectra (e.g., Greene et al., 2016; Morley et al., 2017; Batalha et al., 2018; Kempton et al., 2018). To make sense of these measurements, models are needed that properly account for the physical processes which shape the observable features of rocky exoplanets. The scaling derived here offers a way of parameterizing the large-scale atmospheric heat transport in 1D models, and was used as such in a number of related studies (Kreidberg et al., 2019; Koll et al., 2019; Mansfield et al., 2019; Malik et al., 2019). Doing so is attractive because parameterization is many orders of magnitude faster than explicitly simulating an atmosphere’s large-scale fluid dynamics.

This work was supported by a James McDonnell Foundation postdoctoral fellowship.