The Impact of Planetary Rotation Rate on the Reflectance and Thermal Emission Spectrum of Terrestrial Exoplanets around Sunlike Stars

ROCKE-3D is the generalization and adaptation of the NASA Goddard Institute for Space Studies (GISS) ModelE2 GCM to planetary and exoplanetary worlds (Way et al. 2017). ModelE2 is a terrestrial climate model currently being used for the Coupled Model Intercomparison Project 5 and 6 (CMIP5, CMIP6; Schmidt et al. 2014). Much of the ROCKE-3D model's code, including the dynamical core and many of the physical parameterizations, are identical between the parent terrestrial GCM and the modified planetary/exoplanetary version, termed "Planet 1.0," that we employed for these simulations. The model is freely available to the public and can be downloaded from the NASA GISS website (https://simplex.giss.nasa.gov/gcm/ROCKE-3D/). Among other ongoing work, ROCKE-3D has been used previously to simulate hypothetical ancient Venus configurations to study the inner edge of the habitable zone (Way et al. 2016), the habitability of Proxima Centauri b (Del Genio et al. 2019), and the moist greenhouse limit (Fujii et al. 2017).

While ROCKE-3D derives the bulk of its code heritage from the GISS ModelE2 GCM, some changes were required to make the code adaptable to, e.g., changing orbital parameters, different stellar types and insolation rates, varying atmospheric compositions and pressures, etc. The largest of these changes was through the incorporation of the Suite of Community Radiative Transfer codes (SOCRATES) radiation scheme (e.g., Amundsen et al. 2017). SOCRATES is flexible to insolation and gaseous absorption/scattering variations from the terrestrial baseline to known planetary (e.g., Mars) and hypothetical exoplanet environments. SOCRATES employs a correlated-k method for gaseous absorption in the longwave and shortwave streams (Edwards 1996; Edwards & Slingo 1996). The number of bands chosen in each stream can be changed based on the required precision of the radiative transfer, stellar spectrum, and abundances of various radiatively active gas species in the simulated atmosphere.

Our simulations are investigating the variability in the reflectance and thermal emission spectra due to rotation rate alone, so we limited the (potentially very large) parameter space that we explored to that factor in exclusion of others (e.g., such as surface land/water distributions, atmospheric composition, etc.). All simulations were run with 4° latitudinal and 5° longitudinal resolution, 40 atmospheric vertical layers spanning the surface (∼1000 mb) to 0.1 mb, and 13 layers in a fully coupled dynamical ocean with depths reaching ∼5000 m in places. To prepare each simulation, we altered the sidereal rotation period (i.e., sidereal day length) in the model's "rundeck." The "rundeck" is simply the input file that prepares a given simulation's parameters. Our rundecks are archived at Zenodo (doi:10.5281/zenodo.3727183). Table 1 specifically defines the sidereal rotation period of each simulation in seconds and the names of each simulation that we will use for the remainder of this paper. For comparison, Earth's sidereal rotation period is 86164.094 s and sidereal year is 31558150.23 s. Our focus in this work is on slower rotation periods and G-type stars as rotation rate (specifically tidally locking or resonant rotation) has been discussed thoroughly in the literature especially relating to M-type stars and their planets (e.g., Kopparapu et al. 2017; Wolf et al. 2017) and the inherent limitations of the GCM at faster rotation rates. While the ROCKE-3D GCM can simulate modestly faster rotation and maintain stability (e.g., a sidereal day length half that of Earth's), there is more rotation rate parameter space that can be simulated by the GCM at slower rotation rates. Additionally, using the examples provided by nature in our own solar system, there is a bias toward slower rotation among the terrestrial planets with even the best estimates of early Earth's day length only slightly shorter than the modern value (i.e., ∼21–22 hr; Williams 2004). In addition, work by Barnes (2017) has shown that if Earth had an initial rotation period of ∼3 days and never experienced a moon impactor, it may have reached synchronous rotation after ∼4.5 Gyr, hence, spending much of its spin evolution in the slowly rotating regime (period >16 sidereal days). Although, atmospheric tidal forcing may continue to drive asynchronous slow rotation (e.g., Leconte et al. 2015).

Table 1.  List of Simulations

Name	Sidereal Rotation Period (s)	Obliquity
1×	86164.094	235
4×	344656.38	235
8×	689312.75	235
16×	1378625.5	235
32×	2757251.0	235
64×	5514502.0	235
128×	11029004.0	235
243×	21038480.0	235
256×	22058008.0	235
365×	31558150.23	235
365 × 0°	31558150.23	0°

Download table as:  ASCIITypeset image

Other customizable factors in the simulation were left as earthlike as possible. The background atmosphere was identical to Earth's with CO₂ present at 400 ppm and ${{\rm{O}}}_{3}$ and CH₄ present in their default terrestrial configurations that exhibit a prescribed slight seasonal variation in abundances with latitude and pressure. The input stellar spectrum was the modern solar spectrum, and the remainder of Earth's orbital parameters were retained except for the "365 × 0°" simulation where obliquity was set to 0° to simulate a truly tidally locked world. The topographic map was slightly altered from the terrestrial baseline by eliminating most islands (e.g., Japan, New Zealand, New Guinea, the Caribbean islands, etc.) and closing narrow sea passages (e.g., the Mediterranean, Red Sea, Persian Gulf, and Hudson Bay). All vegetation was eliminated from the land, and the surface soil was set to be a 50/50 mix of sand and clay with a set albedo of 0.3. Ocean bathymetry was simplified to a "bathtub" ideal with a set 1360 m ocean depth and limited ocean shelves. This prevents the ocean freezing to the bottom in shallow seas, which crashes the simulation. See Way et al. (2018) for more details on these modifications, which are the same as those used herein. The model self-consistently calculates the water cycle to include dynamically variable lakes and snow and ice coverage. We excluded all aerosols (e.g., dust, black carbon, and volcanic aerosols) from the simulations.

Each simulation was run until it reached radiative equilibrium, defined as being less than ±0.2 W m⁻², and then simulated for an additional 100 yr (Figure 1). For the 243 × 256×, and 365× simulations, the resonant, near-resonant, or synchronous rotation rates, respectively, of the planets introduce semi-regular multi-decadal variations in the global net radiative balance (Figure 1, bottom row). These runs were evaluated as being in radiative equilibrium over the long term by simulating for an additional length of time and having consistent and stable global mean surface temperatures. Except where specified, the data shown below is an average of the last 10 yr of each simulation.

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To investigate the spectroscopic observables simulated in this work that may not be present in the lower-resolution radiative transfer that is native to ROCKE-3D, we developed an interface between the ROCKE-3D GCM and the SMART model (Meadows & Crisp 1996). This model builds upon and generalizes the Virtual Planetary Laboratory (VPL) 3D Spectral Earth Model (Robinson et al. 2010, 2011, 2014) to allow for the spectroscopic visualization and disk-integration of planets described by GCM output states. We refer to this new code as the VPL SPM, which has three coupled modules that are used in series to: (1) interface the essential ROCKE-3D output variables with SMART input variables; (2) simulate the atmospheric radiative transfer with SMART for each resolved planet pixel; and (3) integrate the visible pixels to create disk-integrated spectra. We describe these three modules below.

2.2.1. ROCKE-3D to Smart Translation

We interface ROCKE-3D with SMART by translating various parameters from the GCM's output files into files readable by SMART. These parameters include surface temperature, atmospheric composition as a function of pressure, atmospheric temperature structure, and liquid water and ice cloud optical depths as a function of pressure, cloud covering fractions, and ground albedos that are independent of wavelength (i.e., "gray"). These quantities are binned into Hierarchical Equal Area isoLatitude Pixelization (HEALpix; Górski et al. 2005) pixels, weighting the contribution by the GCM's latitude–longitude pixel area, which varies with latitude. We employ the HEALPix scheme, which covers the planet globe with equal area pixels, to avoid the geometrical instabilities, such as diminishing pixel size and the increasingly triangular pixel shape that arise near the poles of a latitude–longitude grid and to simplify our spectral disk-integration calculations. Our nominal simulations are run with NSIDE, a parameter that defines the resolution of a HEALPix grid, equal to two for a total of 48 pixels across the planet globe. An example of a HEALPix grid with this resolution is shown in Figure 2. The final binned quantities are written to ASCII files that are used as inputs to SMART. Note that the module to interface ROCKE-3D with SMART can be swapped out at a later time with interfaces for other GCMs, to expand the functionality of this model over time.

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2.2.2. Running SMART and Database Creation

2.2.3. Production of Disk-integrated Spectra

Given a sub-observer and substellar points on the ROCKE-3D latitude–longitude grid, the SMART radiance database from the previous module is processed to produce a disk-integrated planet flux as it would be seen by a distant exoplanet observer with a direct-imaging telescope. We set the substellar latitude and longitude based upon the illumination pattern of the tidally locked planet that includes obliquity (365×) because the subsolar point in this case does not move with respect to longitude. The sub-observer longitude and latitude are user-specified quantities that are selected depending on the specific experiment under consideration, and they control the desired planet phase angle. With all angles specified, the gridded radiance field for each pixel is linearly interpolated to the observer position on the sky plane. We note that at lower resolutions (NSIDE ≤ 4), HEALPix pixels are large enough that many of the pixels will be partially illuminated and/or visible to the observer. This can cause pixels near the planet's terminator/limb to have a solar/observer zenith angle greater than 90° while still being partially illuminated/visible, resulting in an underestimation of the number of pixels that contribute to the disk-integrated spectrum. To address this issue, we employ a HEALPix "sub-grid", which splits each (macro)pixel into 1024 sub-pixels. An example of sub-gridding is seen in Figure 2. We compute the same geometric values for each of these sub-pixels as we do for the larger HEALPix pixels, allowing us to determine what fraction of each pixel is visible and/or illuminated. Finally, the disk-integrated spectrum is produced by summing the HEALpix radiances weighted by the illumination and visibility in the following equation:

$$\begin{eqnarray*}&&{f}_{\mathrm{disk}\mathrm{int}.}(\lambda )=\sum _{i=0}^{{N}_{{HPIX}}}{A}_{i}(\hat{p}\cdot {\hat{a}}_{i}){S}_{i}(\lambda )+{B}_{i}(\hat{p}\cdot {\hat{b}}_{i}){T}_{i}(\lambda ).\end{eqnarray*}$$

Here, A_i is the fraction of sub-pixels in pixel i that are both illuminated and visible to the observer (orange sub-pixels in Figure 2), and B_i is the fraction of sub-pixels in pixel i that are visible to the observer (orange and red sub-pixels in Figure 2). $\hat{p}$ is the observer's position vector in relation to the HEALPix grid. ${\hat{a}}_{i}$ is the vector describing the illuminated and visible area of pixel i, and ${\hat{b}}_{i}$ is the vector describing the visible area of pixel i. S_i(λ) and T_i(λ) are the spectra produced by interpolating the reflectance and emission radiance fields, respectively, for pixel i to the observer's position. The result is that we weight both illumination and viewing geometries in the calculation of a pixel's contribution to the disk-integrated spectrum.

The influence of planetary rotation rate on climate has been well known for decades (Williams & Hollway 1982; Del Genio & Suozzo 1987; Navarra & Boccaletti 2002; Showman et al. 2015) as it naturally flows from the foundational equations of fluid motion on a rotating planet, termed the "primitive equations" by Andrews et al. (1987), among others. That rotation rate, often represented as Ω, is directly included in the Coriolis parameter and through the centrifugal forces imparted to the fluid by the rotating planet. In our simulations, we have slowed the rotation rate of the planet and, hence, reduced Ω by the integer factors shown in Table 1. This equivalently reduces the Coriolis parameter and centrifugal force for our simulated planets. The impact of this reduced Coriolis force is well known as it broadens the overturning Hadley circulation (Figure 4), changes and eventually eliminates the characteristic westerly jet streams (Figure 6) of Earth (and Mars) and associated transient weather systems (baroclinic waves), and, hence, feeds into the pattern of clouds and precipitation around the planet (Figure 3). It is this latter factor that is particularly relevant to the observation of exoplanets by large space telescopes such as Origins, LUVOIR, or HabEx and the habitability of terrestrial exoplanets (e.g., Yang et al. 2013, 2014a; Way et al. 2016) as it modifies the reflected and thermal light spectra of the planet (see Sections 3.2, 3.3, and 4).

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Our focus in this section is to present the climate of our simulated planets to both provide a point of reference for our subsequent discussion on how the reflectance and thermal spectra are changed and to demonstrate that ROCKE-3D produces the expected changes to planetary climate due to a slowed rotation rate. Way et al. (2018) provides a more comprehensive description of the influence of rotation rate (and other factors) on the climate and, hence, habitability of terrestrial planets as simulated in ROCKE-3D.

Following Kopparapu et al. (2017), Figure 3 presents the surface air temperature, planetary albedo, column-integrated cloud-condensed water, and total cloud cover for each simulation. Each plot in Figure 3 represents the average of the last 10 yr of the model simulation, except for the 256× simulation, where the average of the last 50 yr is shown. The 243× simulation is not shown in Figure 3 as it is substantially similar to 256×.

As the rotation rate slows and the days lengthen, the planet initially cools (from a global mean surface temperature of $16.2^\circ {\rm{C}}$ for 1× to $11.6^\circ {\rm{C}}$ for 32×) while also becoming less cloudy. This is due to the broadening of the Hadley circulation (Figure 4) and a larger area of descending drier air in the mid-latitudes, resulting in reduced cloud cover and increased outgoing longwave radiation at middle and high latitudes. Indeed, between 8× and 16× day lengths, the Hadley cell becomes fully equator-to-pole (Figure 4) and the counter-rotating Ferrel cells are not present. Navarra & Boccaletti (2002) note that their simulations lost the presence of a Ferrel cell between a 72 and 144 hr day length. Kaspi & Showman (2015) lose indications of a Ferrel cell between their ${{\rm{\Omega }}}_{{\rm{e}}}$/4 and ${{\rm{\Omega }}}_{{\rm{e}}}$/8 simulations with otherwise earthlike conditions (corresponding to our 4× and 8× simulations). Our ROCKE-3D simulations clearly show a Ferrel cell in the 4× (i.e., 96 hr day length) simulation and even a hint of one in the 8× simulation before being entirely absent by 16×. This broad Hadley cell reduces the equator-to-pole temperature gradient, causing the tropics to cool while the poles warm. This is also reflected in the planetary albedo (second column from the left in Figure 3) via darker (reduced) albedo in high latitudes due to diminished snow and ice cover (Figure 5).

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Near a day length of 32×, cloud cover begins to expand around the planet and becomes increasingly predominant through the 256× simulation where most of the planet (save for specific isolated locations over continents) experiences cloud cover more than 70% of the time (Figure 3). The 32× day length also marks a transition point between continental and ocean mean surface temperatures. The increasingly weak near-surface wind field (not shown) with increased day length reduces the mixing of air between land surfaces and open ocean, and the far larger heat capacity of the ocean keeps the ocean surface as much as $30^\circ {\rm{C}}$ warmer than the neighboring land surface, which cools much more than the ocean surface during the long night. The expansion of continental snow cover and elimination of permanent sea ice is also of key importance in increasing the land-ocean temperature differences (Figure 5). Even with a 4× day length, the Arctic ocean becomes permanently ice free (Figure 5) and most sea ice near Antarctica is eliminated as the mean surface temperature increases above freezing (Figure 3). Land snow cover begins to noticeably expand again at a 32× day length, with the exception of Antarctica where snow cover declines at all increased rotation periods. By 365×, as expected, the nightside of the planet (and part of the dayside near the terminator) is nearly entirely covered in ice and snow. Note that we did not run our simulations to a point where the hydrological cycle reached equilibrium. It is conceivable that over geological times, nearly all of the water on the dayside (i.e., in the Pacific Ocean and surrounding land surfaces) would be transported to the nightside as snow and ice and the dayside water reservoir would only be replenished by glacial flows returning to the dayside. Such a scenario is discussed by Yang et al. (2014a; see also Menou 2013; Leconte et al. 2013; Turbet et al. 2016), but ROCKE-3D both lacks a dynamic land ice model that accounts for the glacial flow that could possibly replenish the dayside with water (to some degree), and the duration of the simulation time necessary to largely evacuate the Pacific Ocean is well beyond reasonable simulation run durations.

The 243× and 256× simulations occur at or near, respectively, the 3:2 spin–orbit resonance for Earth. This results in unique climate impacts on yearly and decadal timescales where a small area of the planet (well less than one full hemisphere as in the 365× simulations) is in permanent night. As expected, snow and ice build up near this naturally colder location, but this region of snow and ice cover moves eastward around the planet, particularly in the northern hemisphere, at slow timescales. This produces a regular "beat" to the climate (e.g., Figure 1) driven by the land and ocean distribution on the planet and the corresponding impact of relatively warmer or colder surface temperatures and snow and ice coverage as the small permanently shadowed portion of the planet moves across that land and ocean distribution.

The tidally locked/Sun-synchronous 365× simulations produce familiar patterns (Yang et al. 2013, 2014a, 2019a, 2019b; Kopparapu et al. 2017; Fauchez et al. 2020) of a very thick substellar cloud, primarily composed of ice particles at high altitudes (Figure 3), which is not perfectly round but instead modulated by the presence of continents and a dynamical ocean transporting heat poleward. There is an intense dayside-to-nightside temperature gradient (with a global mean surface temperature below freezing) that drives radial wind flow away from the subsolar point (not shown) and also transports moisture that falls in the form of snow (Figure 5). Note that because of this, the nightside of the 365× simulations are far from cloud-free.

The atmospheric dynamics of our simulated planets manifests in many of the ways previously discussed in the literature. Namely, as the rotation rate slows and day length increases, the planet begins to exhibit an equatorial super-rotating jet stream, and the westerly jet streams in the mid-latitudes are diminished or absent entirely. Indeed, the flow in the stratosphere (particularly the middle and upper stratosphere) switches between easterly in the 1× simulation to westerly and super-rotating even in the 4× simulation. Laraia & Schneider (2015) noted that equatorial upper tropospheric flow became westerly when Earth's rotation rate was slowed by a factor of two (i.e., a "2×" day length simulation). The vertical and latitudinal extent of the super-rotating jet stream expands through the 32× simulation (Figure 6), the same point at which the tropospheric westerly jet streams at mid-latitudes are entirely absent. By the 64× simulation, the super-rotating jet stream is absent, with only a very weak westerly flow in the stratosphere and weak easterly flow in the troposphere (Figure 6). This pattern persists until the 365× day length where radial flow away from the subsolar point dominates the troposphere, and the stratosphere is essentially stagnant. Interestingly, our 128×, 243×, and 256× simulations show no super-rotation, while Del Genio & Zhou (1996) showed strong super-rotation in these ranges, and Venus exhibits very strong super-rotation at a (retrograde) 243× day length (e.g., Read & Lebonnois 2018). It has been suggested that the high opacity of Venus' atmosphere at high altitudes may help drive the super-rotation there due to enhanced wave and tide forcing (e.g., Lebonnois et al. 2010).

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We touched on the changes to the tropospheric overturning circulation above, but we note that the stratospheric circulation also goes through a substantial change. Carone et al. (2018) describe an "Anti-Brewer–Dobson circulation" in the stratosphere of tidally locked exoplanets, and the same general pattern develops in our 8× and longer-day-length simulations. As seen in Figures 4 and 3, distinct overturning circulation cells form in the vertical column in each hemisphere, with generally an equator-to-pole Hadley circulation in the troposphere, a counter-rotating cell (i.e., pole-to-equator) near the tropopause, and then a second equator-to-pole cell in the stratosphere and mesosphere. As Carone et al. (2018) discussed, this likely has implications for photochemically active species and their distributions in the atmospheres of slowly rotating terrestrial planets, but it also may play a role in the escape of water from slowly rotating planets. Specifically, the upper tropospheric/lower stratospheric tropical cold trap present on Earth is weakened for the slower rotating simulations, leading to a stratosphere that contains 30%–70% more water vapor (e.g., ∼0.003 g kg⁻¹ at the 100 mb pressure level in the 32× simulation) relative to the Earth baseline (i.e., the 1× simulation with ∼0.002 g kg⁻¹ at the 100 mb pressure level) for the non-Sun-synchronous simulations and 100%–200% more water vapor in the 365× simulations (not shown, ∼0.0043 g kg⁻¹ at the 100 mb pressure level).

We computed the reflected-light (visible and NIR) spectrum of the ROCKE-3D simulated planets using SMART to determine the observable changes driven by planetary rotation rate. Our discussion focuses on the 1× (earthlike case), 64×, and 365 × 0° simulations as they are representative of fast rotators, slow rotators, and tidally locked planets, respectively. Our analysis assumes a "face-on" orbital plane from the observer's point of view to simplify the dimensionality of our presented results. However, we note that variations in planet phase/inclination can also affect the observable reflectivity spectrum (e.g., Cahoy 2010; Nayak et al. 2017), which may be degenerate with our spectral results.

Figure 7 shows reflected-light spectra for 1 day, 64 day, and 365 day rotation cases in the months of June and December. The reflection spectra are shown as the "apparent albedo", which is the geometric albedo the planet would have if it were a Lambertian sphere and is calculated by dividing the phase-dependent reflectance spectrum of the planet by the albedo of a Lambertian-scattering sphere at the same phase as the planet. This attempts to remove the effect of phase by scaling the spectrum so it approximates the planet at full phase and, therefore, approximates the geometric albedo. Note that the phase effects of cloud scattering still remain in the apparent albedo spectra, which is slightly elevated at quadrature compared to the geometric albedo due to cloud forward scattering, which increases the reflectivity over that of a diffusely scattering Lambertian sphere (Robinson et al. 2010).

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In both Figures 7 and 8, we see differences in the planet albedo spectra as a function of season for a given rotation rate, and as a function of rotation rate for a given season. In Figure 7, there is a relatively large difference (about 0.1) in the apparent albedo continuum levels of the 1× case between June and December compared to the 64× and 365 × 0° cases (both about 0.02 change in apparent albedo), indicating that the spectral variance is larger for faster rotating planets in our sampled cases. The continuum albedo of the 365 × 0° planet consistently exceeds the faster rotating cases, and the water absorption bands (e.g., 0.82, 0.94, 1.12, 1.4, 1.85, 2.65 μm) are not as deep as the other faster rotation states. Additionally, the intermediate 64× case has deeper water absorption features than the 1× and 365 × 0° cases in both months. These observable effects are nearly all a result of cloud coverage, cloud optical thickness, and cloud vertical extent, which we discuss in greater detail in Section 4.

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Figure 8 further visualizes the seasonal spectral changes of the 1×, 64× and 365 × 0° cases by depicting the mean spectrum and standard deviation over all 12 months for each case. Note that at 1 μm, the 1σ deviations of our cases are 0.035 (or 13.5% coefficient of variation, or CV), 0.013 (6.0% CV), and 0.007 (2.4% CV) for the 1×, 64×, and 365 × 0° cases, respectively. The diminishing seasonal variation in the continuum with slowing rotation rate is due to differences in cloud properties and cover as well as surface albedo, which we will discuss in Section 4.

The thermal infrared (IR) emission spectra is modeled using SMART in an edge-on observer angle configuration that is amenable to the analysis of the primary and secondary transits by an exoplanet. We again focus on the 1×, 64×, and 365 × 0° simulations to provide representative cases for terrestrial planet rotation rates.

Figure 9 shows the thermal emission spectra of the 1×, 64×, and 365 × 0° rotation cases, viewed at full phase with an edge-on inclination. The most notable change is that the thermal continuum between 8 and 13 μm varies with rotation rate. The nominal Earth (1×) has a peak thermal flux of ∼18 W m⁻² μm⁻¹, the 64× simulation peaks at ∼20 W m⁻² μm⁻¹, and the 365 × 0° simulation peaks at ∼16 W m⁻² μm⁻¹, all at ∼11 μm. The small absorption features present throughout this range are residual lines in the longwave tail of the 6.3 μm water band and the shortwave tail of the 50 μm water band. The prominent 15 μm CO2 band and the 9.6 μm O3 band both exhibit negligible changes in the bottom of their bands with rotation rate for the cases examined. Like the Earth's true thermal emission spectrum, our models show an emission peak at 15 μm because temperature increases with altitude in the stratosphere, due to ${{\rm{O}}}_{3}$ absorption. The central absorption peak of the 15 μm CO₂ band is so strongly absorbing that emission comes from near the top of the stratosphere, compared to the lower flux troughs surrounding the peak (at ∼14 and 16 μm), which emit from near the top of the troposphere, which is colder than the top of the stratosphere. Removing ${{\rm{O}}}_{3}$ or self-consistently modeling the photochemistry could change this behavior. However, the size of these absorption bands does vary between our rotation cases simply due to the aforementioned thermal continuum offsets.

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Figure 10 shows the thermal emission from the tidally locked and synchronously rotating planet as the planet orbits around the Sun as seen from an edge-on inclination observer. As the synchronously rotating Earth orbits the Sun, the edge-on observer sees different sub-observer longitudes at different planet phases, which may be sensitive to different depths into the atmosphere, due to the longitudinal distribution of clouds. The planet's thermal continuum peaks at full phase (light red or dark red) when the dayside of the planet is in view, and the thermal continuum is at its minimum, at new phase (bright red), when the nightside of the planet is in view. The maximum flux of the thermal continuum (solid red line in the right panel of Figure 10) is seen to be much lower than the maximum that occurs when we exclude clouds from the radiative transfer calculations (solid blue line in the right panel of Figure 10), but the minima of these two cases are much closer. This indicates that clouds are effective at muting thermal emission from the dayside, but they negligibly affect emission from the nightside for our synchronously rotating Earth case. Both the ${{\rm{O}}}_{3}$ band at 9.6 μm and the ${{\rm{H}}}_{2}{\rm{O}}$ lines near 11.75 μm show similar sinusoidal trends with orbital longitude. However, the water lines transition from absorption to emission as the planet moves from full phase to new phase. This can be seen in the zoomed inset axis in the left panel of Figure 10. The CO₂ band at 15 μm, which is not one of the sampled wavelengths, exhibits negligible change throughout phase. The phase-resolved thermal flux curves in Figure 10 are shown for individual wavelength resolution elements from our line-by-line radiative transfer calculations, which correspond to a spectral resolving power of R ∼ 900 near 11 μm. The water lines in this spectral region that exhibit transitions with phase from absorption to emission would require a spectrograph with at least R = 200 at 11 μm to be individually resolved by a single spectral resolution element.

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As discussed in the introduction, rotation rate can affect the habitability of a planet through changes in the global mean surface temperatures (Way et al. 2018) and carbonate-silicate weathering process (Jansen et al. 2019). Our work also highlights this aspect, as shown in Figures 3–5. Our ROCKE-3D simulations recreate the expected climate behavior for slowing planetary rotation rate seen in previous theoretical and modeling work (Williams & Hollway 1982; Del Genio & Suozzo 1987; Navarra & Boccaletti 2002; Showman et al. 2015; Way et al. 2018). Specifically, the equator-to-pole temperature gradient weakens with increased day length/slower rotation rates due to an expanded Hadley circulation and the eventual elimination of the Ferrel cells between 8× and 16× day lengths. The climate cools between 1× and 32× day lengths and steadily becomes cloudier at longer day lengths/slower rotation speeds as the planet becomes an "all-tropics" planet from a dynamical regime. Sea ice is reduced from the Earth baseline in all cases until a tidally locked/Sun-synchronous rotation rate is reached and a permanent nightside allows the ocean to freeze. Land snow and ice coverage decreases with increasing day length until temperatures become sufficiently cold (and nights sufficiently long) to allow snow to cover all continental areas seasonally/diurnally. These changes to cloud cover and snow/ice cover are of particular importance when evaluating how rotation rate drives changes to a planet's reflectance or thermal emission spectrum.

Figure 7 illustrates how some of the climatic phenomena discussed above are revealed in the reflected-light spectra of these planets. In particular, the slower rotating planets in our sample tend to exhibit less month-to-month variability in their reflected-light spectra compared with our sample's faster rotators. The 1× case's continuum albedo spectrum is higher in June than in December, in part due to the visibility of the northern ice cap. In June, the ice cap is at its smallest, but Earth's obliquity tilts the ice cap toward the Sun where it reflects strongly in the visible. In December, even though the northern ice cap is at its largest, it is tilted away from the Sun, making most of the reflecting surface continent and ocean, both of which are much less reflective than ice in the visible. This trend is also seen for the 64× case, though the decrease is less pronounced. However, because the 365 × 0° lacks obliquity and is tidally locked, the surface of the planet and cloud coverage, and, therefore, its spectral features, are seen to experience negligible seasonal change.

The effect of these seasonal changes in surface albedo, or lack thereof, can also be seen in Figure 8. Sampling the continuum of the apparent albedo spectra at 1 μm over all 12 months in each of our simulations, we observe that apparent albedo spectra values are 0.257 ± 0.034 (13.4% coefficient of variation, or CV), 0.224 ± 0.013 (5.6% CV), and 0.314 ± 0.007 (2.3% CV) for the 1×, 64×, and 365 × 0° cases, respectively. This decrease in continuum variability with decreasing rotation rate is the result of seasonal changes in planetary surface features, like the northern ice cap, as discussed above.

There are also changes in the depth of absorption features as a function of both planetary rotation and season. As can be seen in Figure 7, the water bands for the 1× Earth rotation case in the NIR at 1.4 and 1.8 μm are deeper in June than in December, and the top panel of Figure 8 shows this feature to have an apparent albedo value of 0.111 ± 0.025 (22.4% CV). This is due to increased cloud coverage during northern winter. Increased cloud coverage obscures the lower atmosphere via scattering optical depth, which competes with the water absorption optical depth and decreases the size of the water bands. In contrast, the 64× case has less cloud coverage and optically thinner clouds in December than in June, allowing reflected light to probe deeper into the atmosphere where there is more water, producing the deeper water features we observe in December. The large variance of the 64× case's 1.4 μm water feature in the middle panel of Figure 8, with values of 0.094 ± 0.042(45.1% CV), is due to these changes in both cloud coverage and cloud optical depth. While the synchronously rotating case with zero obliquity (365 × 0°) shows little seasonal change in the continuum, the water bands seen in Figure 7 are deeper in June than in December. This change in water absorption can also be seen in the bottom panel of Figure 8, where the water feature near 1.4 μm has an apparent albedo value of 0.148 ± 0.038 (25.6% CV) around 1.4 μm in Figure 8. Cloud coverage in this case varies very little (5% CV) over the course of a full year, and the variance in water absorption of the 365 × 0° is due to differences in water and ice cloud optical depths across the planet's surface.

The ${{\rm{O}}}_{3}$ Chappuis band between 0.4–0.65 μm and ${{\rm{O}}}_{2}$ A-band (0.76 μm) appear more prominent in the reflected-light spectrum in the presence of significant cloud coverage, which we see consistently in our simulations on the dayside of the 365× planet throughout the entire year. This occurs because water clouds increase the continuum reflectivity in the optical, which increases the equivalent width of gaseous absorption features that occur at or above the altitude of the cloud deck (e.g., Rugheimer et al. 2013).

The relationship between rotation rate and observed thermal flux in Figure 9 is the result of many factors including atmospheric temperature structure, molecular composition, and surface temperature. However, we find that clouds exert more influence on the emerging spectra than any of the phenomena listed above. Because clouds reflect strongly in the visible and absorb and re-emit heavily in the NIR/IR, many spectral features that would arise from phenomena taking place below the cloud deck are not seen by the observer. Comparing the 1× (modern Earth) and 64× cases, although the 1× case has less uniform cloud coverage than the 64× case (17.6% CV versus 10.2% CV across all pixels), the 1× water clouds extend higher into the troposphere (reaching an optical depth of 10⁻³ at 0.2 bar versus 0.4 bar) and, at 0.8 bar where the clouds reach their maximum optical depth, are twice as optically thick. The result of this is that the observer sees deeper into the atmosphere, where it is hotter, in the 64× case than in the 1× case, and the thermal continuum is correspondingly higher. The 365 × 0° case (synchronously rotating Earth) has high-altitude (reaching an optical depth of 10⁻³ at 0.2 bar), very optically thick (10 at 0.8 bar) clouds covering most of the planet's dayside, which truncate the atmosphere at a higher, colder altitude in the troposphere, and so, the observer sees a lower thermal continuum. Clouds extend higher in the 1× case relative to the slower rotation cases (until 365×) due to the deeper Hadley Circulation (Figure 3) and a generally more stable troposphere in the slow-rotating planets.

The interplay between cloud altitude and the atmospheric vertical thermal/compositional structure also affects molecular absorption features in the thermal emission spectrum. Figure 9 shows numerous water absorption features throughout the mid-IR for all of the rotation cases, but they are slightly stronger for the 1× (modern Earth) and 64× cases than for the 365 × 0° case (tidally locked Earth). This is mainly due to cloud coverage, where the clouds in the 365 × 0° case are much more optically thick and extend higher in the atmosphere than for the 1× and 64× cases. These thick clouds control the altitude from which the thermal continuum emerges, which is above the bulk of the water column since the water abundance decreases with altitude. Even though the 365 × 0° case has a more humid stratosphere than the other rotation cases, the radiative effect of clouds dominates and, as a result, the water features are slightly weaker in the synchronous rotation case.

In Figure 9, we also see the effect of clouds on the 9.6 μm ${{\rm{O}}}_{3}$ band. The bottom of this band is mostly independent of rotation rate, without the strengthened secondary emission core in the 64× case. This is mainly due to the fact that, as a function of rotation rate, ${{\rm{O}}}_{3}$ abundance in the stratosphere changes very little (<0.3% CV across a year for any given HEALPix pixel; note that our simulations do not include active photochemistry that could alter the ${{\rm{O}}}_{3}$ distribution based on planetary rotation rate and potentially lead to larger variations in stratospheric gas abundances between the day- and nightsides of the planet; see Chen et al. 2018, 2019), and for all rotation cases, the ${{\rm{O}}}_{3}$ abundance peaks at altitudes above the clouds. Combining this with the minor fluctuations in the temperature of the stratosphere where ${{\rm{O}}}_{3}$ is absorbing most heavily, the result is that the bottom of the ${{\rm{O}}}_{3}$ band remains constant throughout our rotation cases. However, the depth of this band is still dependent on rotation rate due to the effect of clouds on the continuum as discussed above. The result is that planets with low, optically thin clouds like the 64× case planet have a higher thermal continuum and, therefore, deeper ${{\rm{O}}}_{3}$ bands than planets like that of the 365 × 0° case, which have high-altitude, optically thick clouds with a lower thermal continuum.

Interestingly, the effect of clouds leads to opposing trends in the detectability of ${{\rm{O}}}_{3}$ as a function of rotation rate for reflected light versus thermal emission. As previously stated, in reflected light, the large dayside cloud on the synchronously rotating Earth strongly reflects light and effectively backlights the ozone's Chappuis band, making it appear more prominent than the fiducial Earth. However, in thermal emission, the large dayside cloud decreases the temperature contrast between the thermal continuum and bottom of the 9.6 μm ${{\rm{O}}}_{3}$ band, making it appear less prominent than the fiducial Earth. Cloud particle properties control the scattering and absorption of light by the clouds, and they could modify these effects if the properties are substantially different than what the model produces. These properties are (and will likely remain) unconstrained on terrestrial exoplanets.

The thermal flux emitted by the synchronously rotating Earth (365 × 0°) shown in Figure 10 exhibits interesting trends in flux versus orbital longitude (phase). In Figure 3, we see that the 365 × 0° case has a large, high-altitude dayside cloud. As we noted for Figure 9, these clouds block outbound thermal radiation from the atmosphere below them, and therefore, much of the flux we see on the dayside of the planet is emitted from this cloud layer at higher, colder altitudes in the troposphere. This is seen near the peak of the emission spectra near 0/360° (dark purple and yellow spectra) orbital longitude where the water features are weak and the continuum is at its maximum. As a note, the blue line in the right panel of Figure 10 samples the same wavelength and orbital space as the solid line, but clouds were removed from the radiative transfer calculations to assess their spectral effects (clouds were not removed from the GCM simulations). We see that the continuum is much higher for the clear sky than for the cloudy case on the dayside of the planet (0/360° orbital longitude), further supporting that these clouds block emission from the lower troposphere. As the planet orbits from full phase to new phase (0° → 180° orbital longitude, or dark purple → magenta), the dayside and its large, subsolar cloud rotate out of view, and the nightside comes into view for the edge-on observer. Recall from Figure 5 that the nightside is covered almost entirely in ice due to snowfall caused by winds that carry hot, moist air from the dayside to the cold nightside. This cold surface temperature, along with a colder atmosphere due to a complete lack of stellar insolation, corresponds to a large dip in the thermal continuum. Additionally, the observer begins receiving flux from lower in the atmosphere from the nightside of the planet, which is significantly less cloudy than the dayside. The fact that emission from the cold nightside of the planet emanates from deeper, warmer layers in the atmosphere, and emission from the hot dayside emanates from higher, cooler layers acts to decrease the amplitude of the planet's thermal phase curve, which is in agreement with previous findings for synchronously rotating planets (e.g., Yang et al. 2013).

In addition to clouds for the synchronously rotating planet reducing the day/night temperature contrast (or inverting it due to cloud behavior in the case of Yang et al. 2013) that would be seen in a phase curve, we find an interesting analog of this effect to the inversion of weak water lines in the mid-IR, which switch from absorption on the dayside to emission on the nightside (Figure 10). As explained above, the water features at full phase are particularly weak, but as the nightside becomes visible, we are able to see deeper into the atmosphere. The temperature structure of the nightside of the tidally locked Earth is morphologically similar to that of modern Earth, except that there is an additional low-altitude temperature inversion, where the surface is much colder (∼40 K) than the atmosphere directly above it until about 700 mbar, at which point the atmosphere begins to cool again until the tropopause. This phenomenon is due to the circulation and transport of warm air from the dayside to the nightside, meaning water molecules at these mid-tropospheric altitudes are at a higher temperature than the air and surface below them, and so, water is seen in emission. A similar phenomenon for water has been observed for the day/night difference on Earth (Prakash et al. 2019, their Figure 4), where the surface is hotter than the near-surface atmosphere in the day, and the opposite is true at night. This effect is only seen near the peak of the thermal emission where the atmospheric opacity is low, and thus, the thermal continuum comes from the cold inversion layer near the surface. Longward of ∼17.5 μm, the water opacity is higher such that the continuum emission probes above the near-surface inversion, and as a result, water is only seen in absorption. At these longer wavelengths, the thermal phase curve does invert slightly (stronger emission from the nightside due to day–night cloud behavior), as seen in Yang et al. (2013).

The reflected-light spectra of slowly rotating Earths have important implications for the search of habitable and inhabited exoplanets with next-generation direct-imaging missions, such as LUVOIR and HabEx (and possibly ground-based observations from large telescopes like the Extremely Large Telescope (Turbet et al. 2016; Boutle et al. 2017) and Very Large Telescope (Lovis et al. 2017)). Our finding that the synchronously rotating Earth case has a consistently higher continuum albedo than both the true Earth and intermediate slow rotation cases may help to better detect such planets and characterize them because (1) higher albedo planets will yield higher signal-to-noise ratio observations in a fixed exposure time than lower albedo planets, and (2) the higher continuum albedo increases the contrast on many key spectral features such as for the biosignature gases ${{\rm{O}}}_{2}$ and ${{\rm{O}}}_{3}$. This same cloud-induced effect decreased the contrast on ${{\rm{H}}}_{2}{\rm{O}}$ bands by raising the bottom of the band, which could be a telltale sign that a directly imaged planet has thick dayside clouds and a cold trap, perhaps as a result of synchronous rotation. However, a planet with a high albedo surface and low water abundance may be degenerate with this interpretation. Feng et al. (2018) reported a retrieval degeneracy between surface albedo and cloud optical thickness, which may limit the unambiguous interpretation of cloudy atmospheres in reflected light. However, the extent to which wavelength-dependent information, such as aerosol scattering, surface albedo, and gaseous absorption, may further exacerbate or help to break these degeneracies remains an open question for future works.

Additionally, muted seasonal changes to the albedo spectrum continuum seen in observations at different orbital epochs could indicate a slowly or synchronously rotating planet. Although, like spectral absorption features, such an interpretation of the spectral continuum may be degenerate with other factors that could also reduce observable seasonal variations. These factors may include planets with zero obliquity that may not have strong seasonal variations (e.g., Olson et al. 2018), planets inclined such that they are viewed equator-on where north–south hemispheres are averaged out (e.g., Schwieterman et al. 2018), and atmospheres with global aerosol coverage that entirely obscures the lower atmosphere.

We identified numerous interesting thermal emission effects for slowly rotating planets, but these are likely infeasible to observe for true Earth–Sun analogs due to the poor signal contrast of eclipse spectroscopy in this planet-to-star-radius ratio regime. For example, the inverting water features discussed in the last paragraph of Section 4.2 exhibit a contrast flux signal of 10⁻² ppm and require a spectral resolving power of R > 200 at 11 μm. However, if future observations of M-dwarf Earths show similar thermal emission features in secondary eclipse or phase curve measurements, they could potentially be explained by the observables of slowly rotating planets presented here. While dayside clouds on synchronously rotating planets in many ways increase the prospect of atmospheric characterization for planets imaged in reflected light, they can be a detriment in the thermal emission spectra. This is because (1) clouds lower the dayside thermal continuum flux and decrease the thermal phase curve amplitude, both of which would require longer exposure times to be accounted for, and (2) this has a secondary effect of decreasing the contrast on mid-IR absorption features, such as those from ${{\rm{O}}}_{3}$. Intriguingly, our result that water lines near 11 μm switch from absorption on the dayside to emission on the nightside could, in principle, be used in the future to identify a synchronously rotating planet orbiting an M dwarf in medium to high spectral resolution data (R > 200) using the Doppler cross-correlation technique (R > 25,000) in the mid-IR (e.g., Snellen et al. 2013, 2017; Brogi et al. 2014), phase-resolved emission spectroscopy (e.g., Stevenson et al. 2014), or phase-resolved IR interferometry (e.g., Bracewell 1978; Angel et al. 1986; Cockell et al. 2009; Defrére et al. 2018).