Modelling the Atmosphere of Lava Planet K2-141B: Implications for Low and High Resolution Spectroscopy

MNRAS 000,1–8 (2020) Preprint 28 October 2020 Compiled using MNRAS LATEX style ﬁle v3.0

Modelling the atmosphere of lava planet K2-141b: implications for low and high resolution spectroscopy

T. Giang Nguyen,1★ Nicolas B. Cowan,2 Agnibha Banerjee3 and John E. Moores1 1Centre for Research in Earth and Space Sciences, York University, 4700 Keele st., Toronto, Ontario, M3J 1P3, Canada 2Department of Earth and Planetary Sciences, and Department of Physics, McGill University, 3550 Rue University, Montréal, Québec, H3A 2A7, Canada 3Department of Physical Sciences, Indian Institute of Science Education and Research, Kolkata, Mohanpur, 741246, India

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT Transit searches have uncovered Earth-size planets orbiting so close to their host star that their surface should be molten, so-called lava planets. We present idealized simulations of the atmosphere of lava planet K2-141b and calculate the return flow of material via circulation in the magma ocean. We then compare how pure Na, SiO, or SiO2 atmospheres would impact future observations. The more volatile Na atmosphere is thickest followed by SiO and SiO2, as expected. Despite its low vapour pressure, we find that a SiO2 atmosphere is easier to observe via transit spectroscopy due to its greater scale height near the day-night terminator and the planetary radial velocity and acceleration are very high, facilitating high dispersion spectroscopy. The special geometry that arises from very small orbits allows for a wide range of limb observations for K2-141b. After determining the magma ocean depth, we infer that the ocean circulation required for SiO steady-state flow is only 10−4 m/s while the equivalent return flow for Na is several orders of magnitude greater. This suggests that a steady-state Na atmosphere cannot be sustained and that the surface will evolve over time. Key words: planets and satellites: atmospheres – instrumentation: detectors – methods: numerical

1 INTRODUCTION We also use the results to anticipate future observations with space telescopes as well as ground-based high-dispersion spectroscopy. Lava planets are a class of rocky exoplanets that orbit so close to their star that parts of their surface are molten (for a review of ultra-short- period planets, see Winn et al. 2018). Indeed, dayside temperatures can be hot enough to maintain a rock vapour atmosphere detectable 2 THEORY through transit and eclipse spectroscopy, as well as phase curves 2.1 The surface and below (Schaefer & Fegley 2009). Theoretical studies of lava planets and their relatively volatile sodium atmospheres have been published We adopt planetary parameters from Malavolta et al.(2018) and for CoRot-7b (Léger et al. 2011), Kepler-10b, 55 Cnc-e (Castan & Barragán et al.(2018). The subsolar equilibrium temperature (the Menou 2011), KIC 12556548, and HD 219134b (Kite et al. 2016). planet’s irradiation temperature) is 3038 ± 64 K; we adopt the round These studies utilized a 1D model that solves for the ﬂow of mass, number of 3000 K for the remainder of this study. Previous estimates momentum, and energy from the hot dayside to the cold nightside. of the surface temperature (푇푠) by Castan & Menou(2011) and Kite We study K2-141b, the highest signal-to-noise lava planet dis- et al.(2016) accounted for the zero-albedo local radiative equilibrium covered to date, as its phase variations have been measured by K2 temperature and geothermal heating:

arXiv:2010.14101v1 [astro-ph.EP] 27 Oct 2020 (Malavolta et al. 2018; Barragán et al. 2018), Spitzer (Kreidberg ( 1/4 et al. 2019), and possibly in the future with the James Webb Space (푇푠푠 − 푇푎푠) cos (휃) + 푇푎푠 for 휃 ≤ 90°, Telescope (JWST) (e.g Stevenson et al. 2016). We use essentially 푇푠 = (1) 푇푎푠 for 휃 > 90°, the same model as the above studies, but we employ more accurate surface temperature calculations. We consider three compositional where 푇푠푠 and 푇푎푠 are the temperature at the subsolar and antisolar end-members for the atmosphere: 1) pure sodium (Na), due to its point, respectively, and 휃 is the angular distance from the subsolar 1/4 high volatility, 2) silicon monoxide (SiO), or 3) pure silica (SiO2) point. The cos 휃 expression, however, is only valid in the limit of due to their abundance in the crust of rocky planets. These three cases a distant star, 푅∗/푎 << 1. K2-141b has such a tight orbit that the bracket the range of plausible atmospheric compositions (Schaefer star has an angular diameter of 50 degrees as seen from the planet & Fegley 2009). We focus on the mass exchange between surface and the surface can still be illuminated well past 휃 = 90°. Here, we and atmosphere to provide insights on the magma ocean return flow. employ a more accurate analytic expression for the surface temperature originally developed for binary star systems (Kopal 1954). The surface temperature of K2-141b calculated using these two schemes is shown in Fig.1. Beyond 휃 ' 115°, the illumination flux is so low ★ E-mail: [email protected] that the temperature is mainly dependent on a geothermal flux, which

tions are therefore dependent on the surface temperature and solely 106 a function of the angular distance from the sub-solar point (휃푠푠). ] 1/4 2 cos We assume a hydrostatically bound “thin" atmosphere with negli- 4 Kopal (1954) gible absorption of radiation. Although radiative transfer simulations 3 of mineral-vapour atmospheres suggest that this is not the case (Ito 2 et al. 2015), it is a useful first approximation and makes the problem tractable. We also neglect Coriolis forces to keep the problem one- 1 dimensional although our estimated Rossby number of 10−1 suggests Insolation flux [W/m 0 that Coriolis forces may be stronger than on other lava planets (Cas- 0 20 40 60 80 100 120 140 160 180 tan & Menou 2011). Lastly, we assume that the atmosphere is tur- bulent and well-mixed, leading the vertically-uniform wind velocity, 3000 푉. Other state variables, pressure 푃 and temperature 푇, are values calculated at the top of the boundary layer (Ingersoll et al. 1985). We therefore use the shallow-water equations, commonly used for 2000 hydrodynamical calculations of non-linear evaporation/sublimation processes (e.g Ingersoll et al. 1985; Ding & Pierrehumbert 2018). 1000 We compute the surface temperature ahead of time because, as Temperature [K] stated above, we assume it is unaffected by the atmosphere. The 0 atmosphere is idealized to be pure and condensible because of the 0 20 40 60 80 100 120 140 160 180 Angular distance ( ) from subsolar point [°] model’s limits. We first consider a pure Na atmosphere, neglecting dissociation processes that unbind Na from other molecules as is done by previous studies of lava planets (e.g., Castan & Menou 2011; Kite Figure 1. Top: the illumination received by K2-141b as a function of angular et al. 2016). Na is relatively scarce in a rocky planet’s crust (Schaefer distance from the sub-stellar point. Bottom: equilibrium temperature. In both & Fegley 2009) whereas SiO2 is the dominant constituent. Therefore, panels, the black line shows the cos1/4 휃 flux dependence adopted by previ- we also run the model using SiO in the same spirit as Na. Since ous researchers while the purple line shows more accurate formulation that gas-phase SiO will likely condense as SiO2 and may recombine accounts for the large angular size of the star as seen from the planet. with oxygen even before condensing, we finally use a pure SiO2 atmosphere. The model was derived from the shallow-water equations by In- we neglect. Our model will show that the atmosphere fully condenses gersoll et al.(1985) and adapted by Castan & Menou(2011) for out well before this point. exoplanets. It describes the conservation of mass (Eq.3), momentum With the surface temperature now calculated, we can infer physical (Eq.5) and energy (Eq.6). Mass conservation is as follows: properties of the magma ocean by making a few assumptions and simplifications. As silicate materials dominate the composition of 휕푡 ℎ + ∇ · (휌ℎ푣) = 0, (2) rocky planets (Schaefer & Fegley 2009) and K2-141b’s bulk density is comparable to that of the Earth (Malavolta et al. 2018), we assume where 휌 is the fluid density, ℎ is the fluid’s column thickness, and that K2-141b’s crust is entirely silica (SiO2). We further assume that 푣 is the horizontal flow. Applying this equation to our case requires the temperature is vertically constant below the surface for the top time invariance, 휕푡 ℎ = 0, hydrostatic equilibrium, ℎ = 푃/(휌푔), and 150 km. We neglect geothermal heating, which is reasonable if the a source/sink to account for evaporation/condensation such that the ocean is relatively shallow and vertically well mixed. right hand side of Equation (2) is no longer 0 but some mass flux − − To determine the magma ocean depth, we combine the planet’s rate, 퐸 [molecules s 1 m 2]. The final step is to change to spherical 2 known surface gravity (21.8 m/s ) with the density of SiO2 (2650 coordinates which gives our mass conservation equation: kg/m3) to obtain the pressure as a function of depth. These calculated pressures and temperatures then allow us to predict the phase (liquid 1 푑 푉 푃 sin(휃) = 푚 퐸, (3) vs solid) of the SiO2 as a function of depth using the phase diagram 푟 sin(휃) 푑휃 푔 shown in Fig. A1(Swamy et al. 1994). Using these data, we calculate the depth of magma ocean as a function of the surface temperature at where 푟 is the planet’s radius (9.62 × 106 m), 푚 is the mass per the top of the column and hence as a function of 휃, also shown in Fig. molecule, and 푔 is the surface gravity (21.8 m/s2). We can do the A1. In Section 3.2 we will close the loop between the atmosphere and same for momentum, which is affected by pressure gradient forces the surface: mass must be redistributed via ocean currents to account and surface drag 휏 [ kg m−1 s−2]. This relation has the general form: for the constant evaporation and condensation. 2 휕푡 (휌푣) + ∇ · (휌푣 ) = −∇푃 + 휏. (4) A steady state flow means that 휕 (휌푣) is 0. Substituting the mass 2.2 The atmosphere 푡 for a product of 푉, 푃 and 푔, integrating ∇푃 vertically, collecting like As a short-period planet on a circular orbit, K2-141b is expected to be terms, and changing to spherical coordinates yields the momentum tidally locked into synchronous rotation, leading to a permanent day- conservation equation: side and nightside. Not only does this introduce symmetry in how the planet is illuminated, it also leads to equilibrium dynamics (Schaefer 2 1 푑 (푉 + 훽 퐶푝 푇)푃 sin(휃) 훽 퐶푝 푇 푃 & Fegley 2009). The atmosphere of a lava planet originates from the = + 휏, (5) 푟 sin(휃) 푑휃 푔 푔 푟 tan(휃) sublimation/evaporation of the surface and the flow is maintained by the pressure gradient between the nightside and dayside. The solu- where 퐶푃 is the heat capacity, 훽 is the ratio 푅/(푅 + 퐶푝), 푅 is the

MNRAS 000,1–8 (2020) T. G. Nguyen et al. (2020) 3 gas constant. Parameters specific to each atmospheric constituent are included in AppendixB. Energy is conserved if the vertically inte- 5 Atmospheric pressure grated kinetic energy flux, internal heat energy, gravitational energy, 10 and work done via adiabatic expansion is balanced out by energy exchanged with the surface, 푄 [W m−2]. The sum of the internal 100 heat, gravity, and work terms becomes the enthalpy per unit mass at Na SiO the top of the boundary layer: 퐶 푇. Considering 1D steady state flow Pressure [Pa] SiO 푝 2 in spherical coordinates yields the equation for energy conservation: 10-5 0 20 40 60 80 100

2 Wind velocity and Mach number 1 푑 (푉 /2 + 훽 퐶푝 푇)푉 푃 sin(휃) = 푄. (6) 푟 sin(휃) 푑휃 푔 2000 Na 30 SiO 1500 SiO Calculations of ﬂuxes 퐸, 휏, and 푄 are included in AppendixC. 2 20 Detailed derivation and steps shown between the generalized equa- 1000 10 Speed [m/s] tions and our conservation equations can be found in Ingersoll et al. 500 Mach number

(1985). As the system only depends on 휃 through the state variables 0 0 푃 [Pa], 푉 [m/s], and 푇 [K], the equilibrium solution is obtained by 0 20 40 60 80 100 iteratively integrating the ﬂuxes. This computationally solves the or- Atmospheric and surface temperature dinary diﬀerential equations as an initial boundary condition problem 3000 (details in AppendixD). 2000

1000

3 RESULTS Temperature [K] Surface 0 0 20 40 60 80 100 3.1 The atmosphere Angular distance ( ) from sub-solar point We used the model described above to determine the steady-state flow of a pure Na, SiO, and SiO2 atmosphere for lava planet K2-141b. Atmospheric pressure, wind velocity, and temperature for the three Figure 2. Top panel: atmospheric boundary-layer pressure for a pure sodium cases are shown in Fig.2. Na is the most volatile molecule, making the (Na), silicon monoxide (SiO), and silica (SiO2) atmosphere. Middle panel: Na atmosphere the thickest of the three with the maximum pressure wind velocity (left/black) and mach number (right/purple). Bottom panel: at the subsolar point of 13.8 kPa. The Na atmosphere fully condenses atmospheric and surface temperatures (solid purple line). out at 휃 = 102° where wind speed can reach up to 2.3 km/s (Mach 30). The atmospheric temperature of Na can drop to near zero before fully collapsing. The ocean return flow, 푣표, that balances the atmospheric mass SiO is not as volatile as Na making the SiO atmosphere thinner with transport is therefore calculated via: a maximum pressure of 5.25 kPa. The wind, however, accelerates much faster than the Na case with velocities reaching up to 2.2 km/s 푣 (휃) = 푀 (휃)/(휌 퐴(휃)), (8) and the atmosphere condenses out at 휃 = 92°. The temperature also 표 푇 3 drops much faster. Running the model for SiO2 produces the thinnest where 휌 is the density of the magma ocean: 2650 kg/m for SiO2 3 atmosphere with a maximum pressure of 240 Pa and collapses at or 2270 kg/m for Na2O (Na likely exists as Na2O; Miguel et al. 휃 = 89°. The wind is slower with a maximum speed of 1.75 km/s 2011). 퐴(휃) is the cross-sectional area of the magma ocean given the but the SiO2 atmosphere is much warmer as the temperature remains magma ocean depth (estimated in section 2.1 and shown in Fig. A1). above 1500 K. Eqs.7 and8 predict the ocean velocity without accounting for the Evaporation and condensation rates vary significantly between dissociation of oxygen. This approximation may be fine for the SiO2 the Na, SiO, and SiO2 cases. Although Na and SiO have similar atmosphere, but not for the SiO and Na scenarios. To account for the evaporation rates near the subsolar point, the SiO atmosphere starts oxygen dissociated from SiO2 or Na2O, we simply scale the 푚 value to condense sooner than its Na counterpart. The evaporation rates for in Eq.7 to conserve the mass of oxygen during the evaporation and the three scenarios are shown in Fig.3. condensation processes: scaling factor = (1+푚푂/푚푆푖푂) for SiO and (1 + 0.5 × 푚푂/푚푁 푎) for Na. As the magma is predominantly SiO2, the ocean circulation is 3.2 The magma ocean dictated by our SiO atmosphere (recall that SiO has a much greater vapour pressure than SiO2). These calculations yield a maximum To maintain steady-state atmospheric flow, material must be trans- magma ocean return flow of 2-3×10−4 m/s, very slow compared to ported back to the sub-stellar region along the surface, most likely ocean circulation on Earth. The evaporation rate and inferred ocean through currents in the magma ocean. Horizontal mass transport is velocities are shown as Fig.3. determined by calculating the cumulative mass that has evaporated If we assume a SiO to Na ratio similar to that of a Bulk Silicate and subsequently condensed. Therefore, the atmospheric mass trans- 2 Earth (where Na2O accounts for 0.7% of the BSE), then Na accounts port (푀푇 ) at some 휃 is given by: for 1.52% by mass of our idealized crust (Miguel et al. 2011). Ap- plying the same ratio to the magma ocean, we calculate the ocean ∫ 휃 circulation required to maintain mass balance for Na which resulted 푀푇 (휃) = 푚퐸 (휃)푑휃. (7) 0 in a maximum velocity of 2-3 cm/s, two orders of magnitude faster

MNRAS 000,1–8 (2020) 4 T. G. Nguyen et al.

10-5 589 nm Na 3000

] 0.15 0 Surface 2 1 Na atmosphere 0.1 -0.01 2000 *

0.05 -0.02 F/F 0.5 1000 0 Evaporation rate -0.03 Velocity [m/s] Ocean flow without O Bright. Temp. [K]

Mass flux [kg/s/m with O -0.05 Ocean flow -0.04 0 0 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1

10-4 3.4 m SiO 10-4 2 3000

] 0.15 0

2 Surface 1.5 SiO atmosphere 0.1 2 2000

-1 * 1

0.05 F/F 1000 -2 0.5 0 Velocity [m/s] Bright. Temp. [K]

Mass flux [kg/s/m 0 0 -0.05 -3 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 -4 10.1 m SiO 10 -3 2 -6 3000 10 10 2

] 2 0 Surface 2 SiO atmosphere 2000 1 -1 * 1 F/F 1000 0 -2 Bright. Temp. [K] Velocity [m/s] 0 0 Mass flux [kg/s/m -1 -3 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 t/t Angular distance ( ) from sub-solar point period

Figure 4. Top: predicted blackbody radiation of K2-141b at 589 Na spectral Figure 3. Left/black: evaporation rate per square metre (negative numbers feature (left/black) and phase-dependent disk-integrated planetary brightness denote condensation). Right/purple: velocity of magma ocean return flow. temperature (right/purple). Middle: the same for the 3.4 micron SiO2 spectral Negative values denote mass flow towards the subsolar point (in the negative feature. Bottom: the same for the 10.1 micron SiO spectral feature. Note 휃 direction). Solid purple lines denote calculations neglecting the transport that eclipses and transit have been omitted but would occur at 0 and 0.5 of oxygen while dashed lines includes oxygen to maintain mass balance. respectively. The phase curve is shown as the flux ratio between the planet and star. Solid lines refer to the surface (a wavelength just outside the spectral feature) and dashed lines refer to the atmosphere (a wavelength in the spectral than the expected circulation in the magma ocean. While not much feature). mass condenses beyond the magma ocean for our SiO and SiO2 cases, a significant amount of Na is deposited onto solid ground. Here, material can no longer be transported by ocean circulation, but 10.1 휇m SiO feature (Rawlings 1968). Phase variations are deter- rather by solid flow akin to glaciers flowing into the ocean on Earth. mined by calculating the expected wavelength-dependent blackbody radiative flux of the planet throughout an orbit. This process, detailed in AppendixE, involves transforming our 1D results to a 2D map, which we then integrate over the planet’s visible hemisphere to get 4 DISCUSSION the expected flux measurements. As seen in Fig.4, the nightside flux and brightness temperature 4.1 Implications for observation of both the surface and the Na and SiO atmospheres do not go to We presented three scenarios where the atmosphere of K2-141b is 0 at inferior conjunction (orbital phase of 0.5). This is because the either purely Na, SiO, or SiO2 and modelled the pressure, flow, and visible hemisphere is still somewhat illuminated, and the Na and SiO temperature for each. While qualitatively similar, the results differ atmospheres have not fully condensed. Despite being the thinnest, significantly depending on the atmospheric constituent, suggesting the SiO2 atmosphere produces the highest radiative flux as it is sig- that observations could distinguish between the three scenarios. For nificantly warmer than the other two. However, the SiO2 atmosphere example, the James Webb Space Telescope (Gardner et al. 2006) will have fully condensed out before 휃 = 90°. This suggests that tran- could observe the near infrared (NIR) phase curve of K2-141b while sit spectroscopy is impossible for a SiO2 atmosphere, unlike Na and the Hubble Space Telescope could observe it in the visible spectrum. SiO, as the atmosphere is entirely hidden from view on the planet’s Spectroscopy spanning the NIR SiO2, IR SiO , or visible Na features dayside. We show below that this is not necessarily the case. could simultaneously probe the planetary surface and atmospheric In addition to its effect on illumination patterns and surface tem- temperature. perature, the especially close orbit of K2-141b has implications for In Fig.4, we show phase variations at wavelengths in and out of the transit geometry. It is generally assumed that transit spectroscopy 589 nm Na feature; we likewise show phase variations at wavelengths only probes the atmosphere 90° from the subsolar point. As illus- in and out of the 3.4 휇m SiO2 feature (Society & Craver 1977) and trated in Fig.5, regions probed during transit can be quite far from

MNRAS 000,1–8 (2020) T. G. Nguyen et al. (2020) 5

−4 the ' 10 m/s provided by the SiO2 magma ocean. This implies that a more realistic Na atmosphere is much thinner than what we predicted above because the steady-state ﬂow cannot be sustained, leading to the “evolved" state anticipated by Kite et al.(2016). Also, any precipitation that falls beyond the magma ocean shore (휃 > 79°) must be brought back to maintain mass conservation. This may occur via solid state ﬂow analogous to glaciers on Earth, or via isostatic adjustment. If mass is transported back too slowly, the resulting mass imbalance could lead to reorientation of the planetary spin (Leconte 2018).

4.2 Model Limitations Our model makes promising predictions for observation of the lava planet K2-141b. However, our idealized approach to characterize Figure 5. Illustration of the observed longitude during a transit (the observer K2-141b’s atmosphere and magma ocean has limits which must be is at the bottom of the page). The star and planetary orbit are drawn to scale. addressed. This subsection will discuss these limitations and list the caveats that come with our expected observation measurements as seen in Figures4 and6. Our proposed pure or nearly pure SiO2 magma ocean is likely 20 100 affected by interior dynamics which we neglect. The calculated ocean 18 80 flow only accounts for the bulk mass transport along 휃 to balance the 16 60 evaporation and condensation rates. We also neglect Coriolis forces despite the fast rotation of K2-141b. Qualitatively, Coriolis force 14 40 would deflect the wind en route to the cold nightside and the mass 12 20 transport of both the atmosphere and the ocean would be constrained 10 0 to a smaller area, presumably near the equator. Despite this, there H [km] 8 -20 should still be net mass transport in the atmosphere as the evaporated Velocity [km/s] 6 -40 volatiles would still move from an area of high vapour pressure to Na -60 4 SiO areas of low vapour pressure. This also applies to the magma ocean as SiO -80 mass would need to recirculate to maintain net mass balance between 2 2 -100 the surface and atmosphere. 0 -20 -15 -10 -5 0 5 10 15 20 If atmospheric dynamics were dictated by both vapour pressure solar longitude (°) gradients and Coriolis forces, then it would lead to the formation of Rossby waves and possibly weather systems. The ocean flow is slower Figure 6. Anatomy of a transit of K2-141b, in degrees from transit. Left/black: than the atmosphere, so it would be even more affected by Coriolis forces. Ultimately, a Global Circulation Model (GCM) is required to observed scale height for the Na, SiO, and SiO2 cases for regions along the equator. Even though the SiO2 atmosphere collapses before 휃 = 90° and accurately infer atmospheric dynamics that arise from the planet’s hence is invisible in the centre of eclipse, the scale height of the SiO2 atmo- rotation. However, this is difficult for a lava planet because the GCM sphere is bigger and observable for most of the planet’s transit. Right/purple: would need to handle supersonic winds in an atmosphere that fully the planet’s velocity along the observer’s line of sight (negative velocities collapses far away from the sub-solar point. denote motion towards the observer). Another approximation is the fixed surface temperature which al- lowed for the energy of the surface and atmosphere to be decoupled. This adds significant stability to the model especially as supersonic 휃 = 90° for ultra-short period planets. The start and end of transit flow is guaranteed. Coupling the energy budgets of the surface and therefore probe regions with a significant atmosphere even in the atmosphere in post-processing can help to estimate the actual surface SiO2 case. For the specific case of K2-141b and assuming an impact temperature. We calculated how much energy it would take to heat parameter of zero (consistent with observational constraints), ingress up the advected ocean mass to the equilibrium surface temperature −1 and egress probe equatorial regions at an angle tan (푅∗/푎)' 26° from the model’s results. This is done by taking the bulk transported away from the usually assumed 휃 = 90° (for sight lines offset from mass per 휃 and multiplying it with the heat capacity and the temper- the planetary equator, the observed 휃 is closer to 90°). ature gradient. This energy peaks at 102 W/m2 which is minuscule This opens up an avenue for high-dispersion spectroscopy where compared to other energy contributions. Doppler shifts from the changing radial velocity of the planet can help Latent heat, may be a more significant source of energy for the disentangle planetary and stellar absorption features and potentially surface. But the maximum latent heat produced by the evaporation detect atmospheric winds (Snellen 2014). Note that the extreme orbit of silica (∼102 W/m2), silicon monoxide, and sodium (both ∼105 of K2-141b leads to significant changes in radial velocity over the W/m2) are at least an order of magnitude smaller than the instellation course of the transit. The radial velocity and the observed scale height (∼106 W/m2). This suggests that latent heat would barely affect the along the planetary equator are shown in Fig.6. surface temperature, in agreement with Castan & Menou(2011) If we assume the composition of K2-141b to be similar to the Bulk and Kite et al.(2016). It remains to be seen whether the energy of Silicate Earth, then there is 100x more silicate material than Na, by dissociation (e.g., SiO2 -> SiO + O) is as important for lava planets mass (Miguel et al. 2011). However, our results show that the mass as it has proven to be for ultra hot Jupiters (Bell & Cowan 2018; Tan transport required for a steady-state flow of Na is much greater than & Komacek 2019; Mansfield et al. 2020).

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Another important limitation if our model is the lack of consider- DATA AVAILABILITY ation for radiative processes in the atmosphere. Although the work The data underlying this article will be shared on reasonable request of Ito et al.(2015) suggests that there should be strong radiation to the corresponding author. absorption in the upper atmosphere, their results are sensitive to specific atmospheric compositions, stellar spectra in the ultraviolet, and atomic/molecular line lists, all of which are difficult to know for exoplanets. REFERENCES The composition of the atmosphere is also very idealized as we Barragán O., et al., 2018, Astronomy & Astrophysics, 612, A95 ignore chemical processes where SiO2 or Na2O dissociate to form O and O , both of which are more volatile than SiO but less volatile Bell T. J., Cowan N. B., 2018, The Astrophysical Journal Letters, 857, L20 2 2 Castan T., Menou K., 2011, The Astrophysical Journal Letters, 743, L36 than Na. This has huge implications on transit spectroscopy as each Chase Jr M. W., 1998, J. Phys. Chem. Ref. Data, Monograph, 9 molecule has its own distinct spectral features. Even in our idealized Cowan N. B., Agol E., 2008, The Astrophysical Journal Letters, 678, L129 case where we focused on evaporation and condensation at great Cowan N. B., Agol E., 2011, The Astrophysical Journal, 729, 54 length, we neglected macro-scale meteorological processes, chiefly Ding F., Pierrehumbert R. T., 2018, The Astrophysical Journal, 867, 54 cloud formation. This also greatly affects observations as clouds can Gardner J. P., et al., 2006, Space Science Reviews, 123, 485 trap heat via longwave radiation absorption but reflect shortwave Ingersoll A. P., Summers M. E., Schlipf S. G., 1985, Icarus, 64, 375 radiation. Ito Y., Ikoma M., Kawahara H., Nagahara H., Kawashima Y., Nakamoto T., 2015, The Astrophysical Journal, 801, 144 Kite E. S., Fegley Jr B., Schaefer L., Gaidos E., 2016, The Astrophysical Journal, 828, 80 Kopal Z., 1954, Monthly Notices of the Royal Astronomical Society, 114, 5 CONCLUSIONS 101 Kreidberg L., Cowan N., Malavolta L., Louden T., Lupu R., Stevenson K., K2-141b is the best target to study an exotic situation where a 2019, AAS, 233, 123 planet’s interior, ocean, and atmosphere are compositionally similar. Leconte J., 2018, Nature geoscience, 11, 168 It will soon be observable with the James Webb Space Telescope and Léger A., et al., 2011, Icarus, 213, 1 with ground-based telescopes equipped with high-resolution spectro- Lim H., Kim T., Eom S., Sung Y.-i., Moon M. H., Lee D., 2002, Journal of graphs. Although previous studies focused on more volatile species Analytical Atomic Spectrometry, 17, 109 such as Na or K (Castan & Menou 2011; Kite et al. 2016), we have Malavolta L., et al., 2018, The Astronomical Journal, 155, 107 Mansfield M., et al., 2020, The Astrophysical Journal Letters, 888, L15 shown that the significantly less volatile SiO2 atmosphere may be easier to observe due to the extreme geometry of this system. Transit Miguel Y., Kaltenegger L., Fegley B., Schaefer L., 2011, The Astrophysical Journal Letters, 742, L19 spectroscopy of K2-141b is possible due to its proximity to its star Rawlings I., 1968, Journal of Physics D: Applied Physics, 1, 733 which make dayside regions accessible at the start and end of transit, Schaefer L., Fegley B., 2009, The Astrophysical Journal Letters, 703, L113 and leads to large acceleration throughout transit. Doppler shifts in Snellen I., 2014, Philosophical Transactions of the Royal Society A: Mathe- transmission spectra could confirm the km/s wind speed of the atmo- matical, Physical and Engineering Sciences, 372, 20130075 sphere. Spectral phase measurements should be able to distinguish Society C., Craver C., 1977, The Coblentz Society Desk Book of In- between the planetary surface and the somewhat cooler atmosphere. frared Spectra. The Society, https://books.google.ca/books?id= Since radiative transfer simulations predict a dayside temperature _tpvrgEACAAJ inversion (Ito et al. 2015), phase curves may show a transition from Stevenson K. B., et al., 2016, Publications of the Astronomical Society of the inverted to non-inverted spectra. Pacific, 128, 094401 Our calculated return flow of SiO through the magma ocean is Swamy V., Saxena S. K., Sundman B., Zhang J., 1994, Journal of Geophysical 2 Research: Solid Earth, 99, 11787 slow (10−4 m/s), so mass balance can be easily achieved. This sup- Tan X., Komacek T. D., 2019, The Astrophysical Journal, 886, 26 ports a steady-state silicate flow whereas the volatile Na atmosphere Winn J. N., Sanchis-Ojeda R., Rappaport S., 2018, New Astronomy Reviews, would be limited by the return flow through the magma ocean. Al- 83, 37 though K2-141b is an especially good target for atmospheric observation, our results regarding atmospheric dynamics, replenishment via the magma ocean, and transit geometry may apply to other lava planets. APPENDIX A: SIO2 PHASE DIAGRAM See Fig. A1.

ACKNOWLEDGEMENTS APPENDIX B: GAS SPECIFIC PARAMETERS We would like to thank Raymond Pierrehumbert for his comments on the pre-submission draft as well as mentorship in hydrodynamical This subsection includes molecule speciﬁc parameters of Na, modelling. This work was made possible by the Natural Science and SiO, and SiO2. The mass per molecule for the three constituents −26 −26 Engineering Research Council (NSERC) of Canada’s Collaborative are: 푚푁 푎 = 3.82 × 10 kg/molecule, 푚푆푖푂 = 7.32 × 10 −26 Research and Training Experience Program (CREATE) for Tech- kg/molecule, and 푚푆푖푂2 = 9.98 × 10 kg/molecule. With this, nology for Exo-planetary Sciences (Teps). This work is also made you can compute the respective gas constant 푅 with 푅 = 푘/푚 where possible by Mathematics of Information Technology and Complex 푘 is the Boltzmann constant. Systems (MITACS) Globalink Intership program. NBC acknowl- The heat capacity 퐶푝 of Na, SiO, and SiO2 is 904 J/K/kg (Castan edges support from the McGill Space Institute (MSI) and l’Institut & Menou 2011), 851 J/K/kg (evaluated from the Shomate equation at de recherche sur les exoplanètes (iREx) 2400 K provided by Chase Jr 1998), and 1428 J/K/kg (Lim et al. 2002)

MNRAS 000,1–8 (2020) T. G. Nguyen et al. (2020) 7

where 휔푎 [푚/푠], the transfer coefficient as defined by Ingersoll et al. 0 0 (1985), is dependent on two velocity values: 푉푒 and 푉푑. There is a negative sign to denote that this force always acts against the flow. The 20 2 vapour density, 휌 is determined via the ideal gas law. 푉푒 represents 40 the mean flow and is calculated by: 푉푒 = 푚 퐸/휌푠. 푉푑 represents the 60 2 4 effects of eddies which is calculated by: 푉푑 = 푉∗ /푉 where 푉∗ is the 80 friction velocity; this must be calculated iteratively from:

Pressure [GPa] 100 6

120 Magma ocean depth (km) 푉 푉∗,푛+1 = (C3) 8 140 2.5 푙표푔(9 푉∗,푛 퐻 휌푠/(2휂)) 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 Temperature [K] where 퐻 is the scaled height, 휂 is the dynamic viscosity calculated −1 −1 −5 3/2 by: 휂(푇푠)[푘푔 푚 푠 ] = 1.8 × 10 (푇푠/291) (411/(푇푠 + 120)) 3000 0 (Castan & Menou 2011). -20 2500 -40 Energy -60 2000 The energy ﬂux 푄, a combination of enthalpy at surface and top -80 of boundary layer as well as kinetic energy, is calculated by the -100 Surface temp. [K] 1500 following:

-120 Magma ocean depth [km]

1000 -140 2 0 20 40 60 80 100 푄 = 휌푠 (푤푠 퐶푝 푇푠 − 푤푎 (푉 /2 + 퐶푝 푇)) (C4) Angular distance ( ) from sub-solar point where 푤푠 is the other transfer coeﬃcient deﬁned by Ingersoll et al. (1985). Both 푤푎 and 푤푠 are evaluated from Equation (C5) to (C8).

Figure A1. Top: phase diagram (black) and depth (purple) of a SiO2 magma 2 − + 2 ocean. Left axis shows the pressure while the right shows the depth at which 푉푒 2푉푑 푉푒 2푉푑 푤푎 = , 푉푒 < 0 (C5) the phase will change between solid and liquid given the temperature and pres- −푉푒 + 2푉푑 sure. The inflection points reflect transitions between different SiO2 crystal structures which are, in increasing pressure: cristobalite, 훽-quartz, coesite, 2 and stishovite (Swamy et al. 1994). Bottom: surface temperature (left/black) 2푉푑 푤푎 = , 푉푒 ≥ 0 (C6) and magma ocean depth (right/purple) as a function of angular distance from 푉푒 + 2푉푑 subsolar point. 2 2푉푑 푤푠 = , 푉푒 ≤ 0 (C7) respectively. With this, 훽 can be calculated with the gas constant 푅 −푉푒 + 2푉푑 where 훽 = 푅/(푅 + 퐶푝). The final component is the saturated vapour pressure approx- 2 − + 2 푉푒 2푉푑 푉푒 2푉푑 imated from the model of Schaefer & Fegley(2009) where: 푤푠 = , 푉푒 > 0 (C8) 푁 푎 9.6 −38000/푇푠 푆푖푂 13 −69400/푇푠 푉푒 + 2푉 푃푣 [푃푎] = 10 푒 , 푃푣 [푃푎] = 6.16 × 10 푒 , 푑 푆푖푂2 12 −66600/푇푠 and 푃푣 [푃푎] = 1.07 × 10 푒 . APPENDIX D: NUMERICALLY SOLVING THE SYSTEM OF EQUATIONS APPENDIX C: CALCULATING THE FLUXES The state variables (푃, 푉, 푇) can be calculated given initial values at Mass the boundary (subsolar and antisolar point) by integration of fluxes 퐸 is the mass flux from Eq.3, which is essentially calculated by in the right-hand side of Eq. (3)-(6). If we treat these equation as a the evaporation/sublimation of the surface through the difference single equation, we get: between the saturated vapour pressure and the atmospheric pressure: 푑 ( 푓 (휃)) = 푓 (휃) (D1) (푃 − 푃)푣 푑휃 푠푡푎푡푒 푓 푙푢푥푒푠 퐸 = √ 푣 푐 . (C1) 2휋 푚 푅 푇푠 The state variable can be calculated at the next spatial step (the change in angular distance Δ휃) through finite differences: the kinetic speed of sublimation/evaporation is denoted by 푣푐, √︁ which for any species sublimating into a near vacuum, is 푘푇푠/푚, where 푘 is the Boltzmann’s constant. 푓푠푡푎푡푒 (휃 + Δ휃) = 푓 푓 푙푢푥푒푠 (휃) × Δ휃 + 푓푠푡푎푡푒 (휃) (D2) We solve for a solution through finite-differences using a Runge- Momentum Kutta scheme to the second order accuracy. After acquiring 푓푠푡푎푡푒, the state variables can be solved arithmetically but due to the The momentum flux 휏 is calculated by the following: quadratic nature of 푉 in the equations, two branches can be ex- tracted corresponding to the flow being either subsonic or super- 휏 = 휌푠 (−푉 푤푎), (C2) sonic. Although, a higher order scheme could remedy the sensitivity

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of modelling fluid transitioning from subsonic to supersonic flow, Brightness temperature 푇푏 is calculated from (Cowan & Agol the calculations to extract the state variables from Eq. (3)-(6) would 2011): take longer. Therefore, we elect to employ other schemes to control the instability inherent in the model, as described below. ℎ푐/휆푘푇 ∗ −1 ℎ푐 푒 푏 − 1 We use the shooting method where we guess the initial 푃(휃 = 푇 = log 1 + , (E4) 푏 휆푘 휓 0) and find a solution that fits the boundary conditions: 푉 is 0 at 휃 = 휃 = 푇 (휃 = ) = 푇 ∗ 0° and 180°, 0 푠푠. The instability of the non- where 푇푏 is the star’s effective temperature (4599 K) and 휓 is the linear system of equations can lead to solutions either exponentially radiative flux ratio between the planet and the star. expanding or decaying. The goal, therefore, is to solve for a solution that exists in between the bounds of solutions that blew up (upper) or This paper has been typeset from a TEX/LATEX file prepared by the author. decayed to 0 (lower) exponentially. Conditions that define whenever a solution expand or decay exponentially is explicitly described by Ingersoll et al.(1985). Solving for velocity will yield two solutions, one for subsonic flow and the other for supersonic flow. Getting to the critical point where the flow transition between different subsonic and supersonic regimes can be difficult. Therefore, we dynamically increase the spatial resolution whenever the upper bound and lower bound solution start to diverge past a set parameter. A solution close to the "true" solution (which lies in between the upper and lower bound) will approach Mach 1 more reliably, after which we extrapolate the flow into the supersonic regime. This extrapolation process is done by linearizing the state variables with respect to 휃 until Mach 1 is achieved. The flow is then calculated using the supersonic branch of the solutions.

APPENDIX E: OBSERVATIONAL INFERENCE CALCULATIONS The predicted blackbody radiation of K2-141b required to produce Fig.4 is calculated using the methods from Cowan & Agol(2008) and Cowan & Agol(2011). The ﬁrst step is to convert temperature, 푇 to blackbody radiative intensity, 퐼 [W m−2 m−1 sr−1] at wavelength 휆:

2ℎ푐2 1 퐼 = , (E1) 휆5 푒ℎ푐/(휆푘푇 ) − 1 where ℎ is the Planck constant, 푐 is the speed of light, and 푘 is the Boltzmann constant. The next step is to create an intensity map with latitude (훼) and longitude (휙) from angular distance (휃) from the subsolar point using the Spherical Pythagorean Theorem:

퐼(훼, 휙) = 퐼(휃 = cos−1 [cos(훼) cos(휙)]). (E2)

Now that each latitude (훼) and longitude (휙) coordinate has a radiative intensity, we then integrate the intensity across the hemisphere visible to the observer. Assuming an edge-on viewing geometry where the observer is on the planet’s orbital plane, the observed radiative ﬂux, 퐹, is calculated by:

∫ 휋 ∫ 휉+휋/2 퐹(휉) = 퐼(훼, 휙) sin2 훼 cos(휙 − 휉) 푑훼 푑휙, (E3) 0 휉−휋/2 where 휉 is the planet’s position around the orbit. The stellar ﬂux at the corresponding wavelength is calculated in the exact same way but with a uniform temperature of 4599 K (Malavolta et al. 2018) and 2 scaled to the ratio of (푟푠푡푎푟 /푟 푝푙푎푛푒푡 ) . This therefore yields the ratio of planetary to stellar ﬂux at that point as a function of the planet’s orbital motion about its star, the so-called phase curve.

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