11institutetext: Leiden Observatory, Leiden University, Niels Bohrweg 2,
2333CA Leiden, the Netherlands 22institutetext: SRON Netherlands Institute for
Space Research , Niels Bohrweg 4, 2333 CA Leiden, the Netherlands
33institutetext: Eureka Scientific Inc, Oakland, CA 94602 44institutetext:
Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany
55institutetext: Department of Earth Sciences, Vrije Universiteit Amsterdam,
De Boelelaan 1085, 1081 HV Amsterdam, the Netherlands

55email<EMAIL_ADDRESS>
# Observability of evaporating lava worlds

M. Zilinskas 11 C.P.A. van Buchem 11 Y. Miguel 1122 A. Louca 11 R. Lupu 33 S.
Zieba 4411 W. van Westrenen 55

(Received June XX, 2021; accepted July XX, 2021)

Lava worlds belong to a class of short orbital period planets reaching dayside
temperatures high enough to melt their silicate crust. Theory predicts that
the resulting lava oceans outgas their volatile components, attaining
equilibrium with the overlying vapour. This creates a tenuous, silicate-rich
atmosphere that may be confined to the permanent dayside of the planet. The
James Webb Space Telescope (JWST) will provide the much needed sensitivity and
spectral coverage to characterise these worlds. In this paper, we assess the
observability of characterisable spectral features by self-consistently
modelling silicate atmospheres for all the currently confirmed targets having
sufficient sub-stellar temperatures (¿ 1500 K). To achieve this we used
outgassed equilibrium chemistry and radiative transfer methods to compute
temperature-pressure profiles, atmospheric chemical compositions, and emission
spectra. We explore varying melt compositions, free of highly volatile
elements, accounting for possible atmospheric evolution. Our models include a
large number of neutral and ionic species, as well as all up-to-date
opacities. The results indicate that SiO and SiO2 infrared features are the
best unique identifiers of silicate atmospheres, which are detectable using
the MIRI instrument of JWST. Detection of these two species in emission would
allow for strong constraints on the atmospheric thermal structure and possibly
the composition of the melt. We also propose that certain species, for example
TiO, may be directly tied to different classes of melts, possibly revealing
surface and interior dynamics. Currently, there are nearly a dozen confirmed
lava planets ideal for characterisation of silicate atmospheres using JWST,
with two of these already accepted for the initial General Observers
programme.

###### Key Words.:

Planets and satellites: atmospheres – Planets and satellites: terrestrial
planets – Techniques: spectroscopic

##  1 Introduction

The James Webb Space Telescope (JWST) Cycle 1 General Observers (GO) programme
has approved observations of 17 rocky exoplanets111Approved JWST Cycle 1 GO
rocky planets. Highly irradiated: K2-141 b, 55 Cnc e, GJ 367 b, TOI-178 b, HD
15337 b, L 168-9 b, TOI-836.02, and LHS 3844 b. Others: GJ 486 b, TRAPPIST-1
b, c, g, LHS 1140 b, TOI-776 b, GJ 357 b, LTT 1445 A b and L 98-59 c., with
eight most irradiated on the list having sub-stellar temperatures large enough
to sustain a molten surface crust. Close-in rocky planets, due to their short
orbital periods, are readily found by survey missions like the Transiting
Exoplanet Survey Satellite (TESS; Ricker et al. 2014). Because of their large
day-side temperatures, which allow for formation of silicate atmospheres, lava
planets are particularly interesting targets for follow up characterisation
with infrared spectroscopy. Figure 1 showcases all currently confirmed ¡ 2 R⊕
(along with TESS candidates) transiting exoplanets with equilibrium sub-
stellar temperatures above 1000 K (assuming zero albedo).

Figure 1: Confirmed transiting short period exoplanets (including current TESS
candidates) with radii ¡ 2 R⊕ and equilibrium sub-stellar temperatures ¿ 1000
K. Planets to the left of the ultra-short period (USP) line have orbital
periods ¡ 1 day. Named planets are approved for the JWST Cycle 1 GO programme.
The colour indicates the theoretical sub-stellar temperature of the planet.
The marker size is proportional to the radius of the planet. Over 800
confirmed planets and more than 150 TESS candidates are displayed. The
complete table of all shown planets with their parameters can be downloaded as
supplementary material.

Extensive research has been conducted on silicate atmosphere formation and the
possibility of their characterisation. Models by Schaefer & Fegley (2009)
showed that silicate atmospheres would be composed primarily of Na, O2, and
SiO. Based on their work, Miguel et al. (2011) studied the entire population
of Kepler super-Earths and found that these atmospheres are likely to be
diverse in their composition, which is strongly dependant on the given
composition of the melt and the irradiation of the planet. By additionally
considering radiative-transfer, Ito et al. (2015) investigated emission
properties of irradiated super-Earths and discovered that their atmospheric
temperature structure may be completely inverted. Their work also points to
the possibility of characterising lava planets through the emission features
caused solely by the presence of SiO.

It has also been proposed that because of their tenuous nature, silicate
atmospheres may be prone to temporal evolution (Schaefer & Fegley 2009; Kite
et al. 2016), as most volatile constituents could be driven to condense out in
the cooler regions of the planet or even escape the atmosphere entirely.
Because the atmosphere is essentially coupled to the underlying magma,
atmospheric evolution could irreversibly change the composition of the melt.
The effectiveness of this, however, is strongly dependant on the rapidity of
material recycling in the interior (Kite et al. 2016). Following this, Nguyen
et al. (2020) and Ito & Ikoma (2021) modelled evaporation-driven hydrodynamics
with radiative-transfer and found that atmospheres dominated by SiO are much
more likely to be stable against circulation and escape effects compared to
ones dominated purely by Na. The current consensus is that silicate
atmospheres may lack volatile Na and instead would be largely dominated by SiO
or SiO2, both of which could be characterised through emission features.

Following the approach from Miguel et al. (2011) and Ito et al. (2015) and
motivated by the possible evaporation evolution (Schaefer & Fegley 2009; Kite
et al. 2016; Nguyen et al. 2020), we extensively modelled outgassed
equilibrium chemistry and the atmospheric structure in order to investigate
JWST observability for the shown short period rocky planets (Fig. 1). We put
constraints on the current target list and indicate possible atmospheric
features that could help characterise lava planets through low resolution
spectroscopy. Our models are self-consistent, accounting for the changes in
thermal structure caused by the chemistry, and were done with the inclusion of
all up-to-date opacities that could potentially shape silicate atmospheres.

This paper is organised as follows. In Section 2 we describe our approach to
modelling silicate atmospheres, including the use of outgassed equilibrium
chemistry and radiative transfer methods. We present our findings for non-
evolved atmospheres in Section 3.1. For evolved atmospheres, the results are
outlined in 3.2. Section 3.3 highlights the results of models with different
initial melt compositions. Overall implications and observability of silicate
atmospheres are discussed in detail in Section 4. Finally, we conclude by
summarising our key findings in Section 5.

##  2 Methods

###  2.1 Outgassed chemistry

Sustained planetary surface magma will preferentially outgas its most volatile
components and eventually equilibrate itself with the overlying vapour,
forming a silicate atmosphere. We determine the total atmospheric elemental
budget by using the published results of the melt-vapour equilibrium code
MAGMA, which has been used to calculate outgassing on solar system bodies and
exoplanets (Fegley & Cameron 1987; Schaefer & Fegley 2004, 2009; Miguel et al.
2011; Kite et al. 2016). Its output is determined by the given initial
composition of the melt, surface temperature and evaporation percentage. The
involved chemistry considers the oxides of: Al, Ca, Fe, K, Mg, Na, Si and Ti.
We make the assumption that due intense irradiation from the host star, highly
volatile elements (H, C, N, S) are completely void from the atmosphere. In
such a case the atmospheric composition is controlled only by the continuous
evaporation of the underlying melt and not by past accretion of volatiles
(Schaefer & Fegley 2009; Kite et al. 2016).

Given the diverse population of stars, it is likely that melt compositions are
highly varied. Approximations can be made from the particular host star
(Brugman et al. 2021), this is however beyond the scope of this study. Bulk
Silicate Earth (BSE) has been widely used in research of silicate atmospheres
(Schaefer & Fegley 2009; Miguel et al. 2011; Ito et al. 2015; Kite et al.
2016). We adopt BSE as the main composition, but also briefly investigate a
selection of other possible, silicate based melts. All used compositions are
listed in Table 1.

In our models the surface temperature is solved for the sub-stellar point,
which is continuously adjusted using radiative-transfer methods that couple
the surface and the overlying atmospheric layer (described in Section 2.2).
This is process is done until convergence between the outgassed chemistry and
thermal structure is achieved.

Kite et al. (2016) has shown that for atmospheres with sub-stellar
temperatures ¿ 2400 K, atmospheric evolution is faster than surface-interior
exchange, implying that continuous evaporation would rapidly drive the surface
magma composition away from its initial values. While we do not directly
simulate loss of material from the atmosphere through condensation or escape,
we mimic the process using continuous removal of the most volatile components
from the system. As described in Schaefer & Fegley (2004), this is done in a
step-wise manner, until a set percentage of total melt mass is lost. We
explore continuous removal of up to 60 percent mass of the given melt
composition.

Table 1: Initial melt compositions |  SiO2 |  MgO |  FeO |  Fe2O3 |  Al2O3 |  CaO |  Na2O |  TiO2 |  K2O
---|---|---|---|---|---|---|---|---|---
BSEa | 45.97 | 36.66 | 8.24 | 0.0 | 4.77 | 3.78 | 0.35 | 0.18 | 0.04
Continentalb | 62.93 | 3.79 | 5.78 | 0.0 | 15.45 | 5.63 | 3.27 | 0.7 | 2.45
Oceanicc | 50.36 | 7.61 | 9.56 | 0.0 | 15.85 | 12.24 | 2.77 | 1.48 | 0.13
Komatiited | 47.1 | 29.6 | 0.0 | 12.8 | 4.04 | 5.44 | 0.46 | 0.24 | 0.09
Mercurye | 47.1 | 33.7 | 3.75 | 0.0 | 6.41 | 5.25 | 0.08 | 0.33 | 0.02
aBulk Silicate Earth (Schaefer & Fegley 2009); bContinental crust (Wedepohl
1995); cOceanic (Klein 2003); dKomattite (Schaefer & Fegley 2004); e Mercury
(Morgan & Anders 1980)

With elemental abundances established, atmospheric chemistry is solved using
the thermochemical equilibrium code
FASTCHEM222https://github.com/exoclime/FastChem (Stock et al. 2018) in
accordance with the thermal structure of the atmosphere. We consider the
following neutral components: Al, AlO, AlO2, Al2O, Al2O2, Ca, CaO, Fe, FeO,
FeO2, K, KO, K2, K2O, Mg, MgO, Na, NaO, Na2, Na2O, O, O^1, O2, O2^1, O3, Si,
SiO, SiO2, Si2, Ti, TiO, TiO2; as well as ions: Al+, Ca+, Fe+, K+, Mg+, Na+,
O+, O-, O2+, O2-, O3-, Si+, Ti+, TiO+. The thermal data is compiled from
Burcat NASA thermodynamics polynomial
database333http://garfield.chem.elte.hu/Burcat/burcat.html. We note that our
inclusion of various components is only limited by the availability of
thermodynamical data.

###  2.2 Temperature structure

We solve for the temperature structure using the radiative-transfer code
HELIOS444https://github.com/exoclime/HELIOS (Malik et al. 2017, 2019b).
Profiles are converged to radiative-convective equilibrium consistently with
the chemistry (convection has negligible effect for silicate atmospheres),
assuming no surface albedo. The surface temperature itself is affected by the
overlying atmospheric and is thus calculated according to the propagating
flux. The exact implementation of the surface layer for rocky planets in
HELIOS can be found in Malik et al. (2019a, Appendix A).

We consider 25 different opacity sources that could potentially affect the
temperature structure, which the list of can be seen in Fig. 2. Atomic
opacities (excluding Na) are calculated with the opacity calculator
HELIOS-K555https://github.com/exoclime/HELIOS-K (Grimm & Heng 2015; Grimm et
al. 2021), for which we use the Kurucz (Kurucz 1992) line list database.
Molecular opacities (excluding SiO) are sourced from the
DACE666https://dace.unige.ch/ database and/or computed with HELIOS-K. For
both, atoms and molecules, we use Voigt fitting profiles. Following the
methods from Grimm & Heng (2015); Grimm et al. (2021), we use a cutting length
of 100 cm-1 for molecular species and no cutting for atomic species (excluding
Na and SiO).

Since Na and SiO are likely to be major components of silicate atmospheres, we
compute these opacities separately. Na opacity is calculated using the Vienna
Atomic Line Database (VALD3) line list (Ryabchikova et al. 2015), approximated
using Voigt profiles for all but the 0.6 µm doublet lines. For the doublet,
standard Lorentzian fitting is inadequate. Instead, it is fitted using the
unified line-shape theory of Rossi & Pascale (1985); Allard et al. (2007a, b).
Our SiO opacity is constructed using the Barton et al. (2013) line list for
the ground state transitions in the infrared and with the Kurucz (1992) line
list for the A-X and E-X shortwave wavelength transitions.

We emphasise that these opacities are modelled with parameters applicable to
atmospheres of stars and gas giants, where the background is dominated by H2
and He (Marley et al. 2021). There are no currently existing equivalent
theories for silicate atmospheres. While we do find that inclusion such of
pressure broadening may have non-negligible effects on the temperature
structure, these are relatively minor for silicate atmospheres, typically
resulting in less than $\Delta T$ = 100 K close to the surface, but otherwise
nearly identical temperature-pressure profiles. To minimise the possibility of
overestimation we take opacity values at lowest available pressures, resulting
in Doppler-dominated solutions. Use of this assumption is one of the limiting
factors of the current work, emphasising the need for future development of
broadening theories for arbitrary atmospheres. The complete parameters and
sources of all the opacities can be found in the Appendix Table 2.

The last thing to account for is stellar irradiation. For cooler stars (e.g.,
M dwarfs), blackbody stellar spectra inaccurately approximates shortwave
energy flux. When dealing with high temperatures, shortwave absorption becomes
an important factor in shaping the thermal structure of an atmosphere. We thus
construct stellar spectra777All individual spectra are available for download
as supplementary material https://github.com/zmantas/LavaPlanets. for each of
the individual stars by combining PHOENIX models (Husser et al. 2013) with
scaled shortwave MUSCLES spectra (France et al. 2016; Youngblood et al. 2016;
Loyd et al. 2016). This method is only applied to low mass stars, with
temperatures covered in the MUSCLES database. For any hotter stars we scale
the shortwave VPL solar
spectrum888http://depts.washington.edu/naivpl/content/spectral-databases-and-
tools, in combination with PHOENIX models. All opacities and stellar spectra
are sampled at a resolution of $\lambda/\Delta\lambda=2000$, accounting for
wavelengths between 0.1 and 200 µm.

###  2.3 Synthetic emission spectra

Dayside confinement makes silicate atmospheres poor candidates for low
resolution transmission spectroscopy. However, the large temperatures make
secondary eclipse observations much more feasible. We generate emission
spectra using the radiative-transfer code
petitRADTRANS999http://gitlab.com/mauricemolli/petitRADTRANS (Mollière et al.
2019, 2020). Using the same opacities described in Section 2.2 we calculate
the spectra at a resolution $\lambda/\Delta\lambda=1000$ for wavelengths
between 0.3 - 28 µm. In many cases the spectra are convolved to a lower
resolution for better readability. The explored wavelength range encompasses
the coverage of all instruments of JWST.

For notable targets we assess JWST noise levels using
PANDEXO101010https://exoctk.stsci.edu/pandexo/ Batalha et al. (2017), which is
built on the Pandeia111111http://jwst.etc.stsci.edu engine. We only simulate
MIRI Low Resolution Spectroscopy (MIRI LRS with $\lambda/\Delta\lambda\approx
100$) in slitless mode, as it is likely to be the most suitable mode for
characterisation of silicate atmospheres. The wavelengths covered by the
instrument are 5 - 12 µm.

##  3 Results

###  3.1 Bulk silicate earth atmospheres

####  3.1.1 Opacities & Chemistry

The thermal structure in an atmosphere is shaped by wavelength dependant
absorption of the incoming stellar radiation. Silicate atmospheres possess
numerous different species that are good absorbers in the UV/Visible
(shortwave) and infrared (IR) wavelengths. Generally, if the opacity at short
wavelengths exceeds that of IR, the temperature of the absorbing region
increases, causing an inversion to occur (Gandhi & Madhusudhan 2019; Malik et
al. 2019b). In cases with no shortwave absorbers, efficient cooling through IR
will cause the temperature to decrease with increasing altitude.

Figure 2 shows example opacities of the 25 absorbing species investigated in
this study. All values are weighted to abundances of a non-evolved stage of an
atmosphere, formed from a BSE melt. Values are shown for a pressure of
$10^{-4}$ bar, a typical region where the photosphere begins to take shape.
For each given temperature and wavelength, the opacity contribution is
calculated relative to the total added opacity of all present species.
Different species will dominate distinct wavelength regions, allowing us to
easily track their effect on the thermal structure and emission spectrum.

Figure 2: Sampled opacities of a silicate atmosphere. Shown for a typical Bulk
Silicate Earth melt composition with a surface temperature of $\sim$2700 K and
atmospheric pressure of $10^{-4}$ bar. The individual species indicate
contributed absorption to the total opacity, with the minimum cutoff threshold
of 5%. The upper panel shows temperature-specific absorption contribution
relative to the maximum opacity for all wavelengths, with a minimum threshold
of $10^{-10}$. The curve is fitted to indicate the maximum, wavelength-
specific opacity for all temperatures. Note that the labelled temperatures are
atmospheric, which represent the temperature of the melt.

Most ions, even though increasingly abundant in hot atmospheres, can be
dismissed as potential absorbers. Their opacities are significantly lower in
value compared to their neutral counterparts. There are exceptions to this,
notably Fe+ and Ti+, that have meaningful inherent opacity; however, their
outgassed BSE abundances are relatively small, and thus do not contribute to
the models. On the neutral side, some of the strongest absorbers include Na
and SiO, both of which have been noted in previous studies as potentially the
most important constituents of silicate atmospheres (Schaefer & Fegley 2009;
Miguel et al. 2011; Ito et al. 2015).

Most of Na opacity comes from the strong 0.6 µm doublet (see Fig. 2). For
isolated atomic lines, the characteristics of line wings play a significant
role in its opacity value. Wings are difficult to model as their exact
broadening behaviour is dependant on collisions with the background species.
The required theory for arbitrary atmospheres (non-hydrogen) is currently not
well understood. In our case, the wings are modelled with an H2 atmosphere in
mind, however, as explained in section 2.2, pressure broadening effects are
ignored. To avoid opacity overestimation, the wings are also trimmed at an
arbitrary length. In terms of chemistry, Na is consistently the leading
atmospheric constituent until about 3000 K, at which point it starts to become
ionised. Yet, even at lower abundances, its strong opacity keeps it a major
shortwave absorber. That said, because Na is the most volatile constituent of
silicate atmospheres, its dominance may be short lived, as it is expected to
be readily lost through escape and condensation away from the sub-stellar
point. Evaporation results are outlined in detail in Section 3.2.

Silicon is the second most abundant element outgassed from the melt, mostly
residing in the form of SiO. Its abundance is closely followed by SiO2.
Contrary to Na, SiO is much less volatile and its abundance increases with
increasing temperature. Its opacity is strong in both shortwave and IR
regions. SiO shortwave band is strong enough to cause upper atmosphere thermal
inversions even if other major shortwave absorbers are excluded. As discussed
in Section 3.2, SiO long wavelength bands have distinct features that may
prove very useful in characterising silicate atmospheres using low resolution
spectroscopy. SiO2 is another major component of silicate atmospheres. It is
only recently that the vast SiO2 molecular list, consisting of 33 billion
transitions, has been computed (Owens et al. 2020). If sustained in gaseous
phase, SiO2 abundance is expected to be nearly $10^{-3}$ in volume mixing
ratio for atmospheres above 2500 K, about 2 orders of magnitude lower than the
abundance of SiO. A significant region of the IR opacity is dominated by SiO2.
As its bands do not overlap with SiO, the combination of the two molecules
creates a unique spectral pattern. If SiO2 is sustained in the atmosphere as a
gaseous species, it should be expected to be a major opacity source for all
tenable silicate atmospheres.

Many other species are important contributors. Notably, MgO and TiO are
crucial components that further shape the atmospheric structure. MgO is
expected to be highly abundant for the entire temperature range with its
opacity covering both shortwave and IR wavelength regions, acting as a
greenhouse continuum. TiO is present at the higher temperature range and is
one of the dominant shortwave absorbers in the upper atmosphere. Inclusion of
both molecules strongly affects the temperature structure and the expected
emission spectrum. Some other minor opacities that slightly influence the
atmosphere include Al, Ca and Mg, but all can be dismissed if necessary.

In Figure 2 the upper panel displays the total, wavelength and temperature
dependant opacity. The contribution for the given temperature is represented
by the colour intensity. The fitted curve shows which temperatures have
dominant opacity values per each defined wavelength. At $10^{-4}$ bar and
atmospheric temperatures above 2000 K, shortwave opacity will be greater than
its longer wavelength counterpart. This imbalance will shape the temperature-
pressure profile, causing an inversion to occur. This is only true for low
pressures. When the pressure increases, molecules become more abundant, thus
resulting in greater IR opacity. At high surface temperatures, the thermal
structure of silicate atmospheres should be expected to have distinct regions
with opposing temperature gradients.

####  3.1.2 Temperature-pressure structure

All temperature-pressure (T-P) profiles for lava planets with sub-stellar
temperatures ¿ 1500 K are shown in Figure 3. At 1500 K the atmosphere is
extremely tenuous, sustaining just above $10^{-7}$ bar surface pressure. At
these pressures the chemistry mainly consists of atomic, shortwave absorbers,
resulting in large surface temperature jumps, but otherwise a nearly
isothermal profile. Low pressure atmospheres are of little use for
observations using low resolution techniques. However, because outgassed
pressure scales exponentially with the temperature of the melt, at 2000 K it
reaches nearly $10^{-3}$ bar. This pressure is enough to allow for more
complex molecular composites, such as SiO, to exist in abundance. The effects
of this are twofold. First, the increased total opacity pushes the photosphere
away from the surface, eliminating the sharp temperature discontinuities.
Second, at high pressures IR opacity becomes larger than the shortwave
opacity, causing inversions to become confined only to the upper atmospheric
regions.

At 2500 K the IR opacity of silicate atmospheres increases substantially. The
surface pressure now reaches $10^{-2}$ bar. Lower regions follow a dry
adiabat, with a clear temperature minimum that is followed further up by an
inversion. The temperature minimum, otherwise defined as the tropopause, is a
good indication of the average photospheric pressure level. Without modelling
emission features, one should expect to see strong contributions from the
regions just above and below the tropopause.

Increasing the temperature further will render the atmosphere completely
opaque in shortwave and IR wavelengths. Regions below the photosphere are
trapped, no longer efficiently radiating. This results in a deep isothermal
layer that is inaccessible to observations. While models indicate that at 4000
K the surface pressure would equate to 10 bar, it is unclear whether such
strongly irradiated systems could be sustained. It is very likely that the
large deposition of irradiation would result in rapid loss of material caused
by erosion (Owen & Wu 2017; Owen 2019).

Figure 3: Temperature-pressure profiles of confirmed lava planets with sub-
stellar temperatures ¿ 1500 K. Calculated using non-evolved (0% fractional
vaporisation) Bulk Silicate Earth compositions. All profiles are converged to
radiative-convective chemical equilibrium using the opacities described in
Section 2.2. The colour is indicative of stellar irradiation temperature.

Observable emission features depend on the probed thermal structure of the
atmosphere (Zilinskas et al. 2021). The shape of the T-P profile is determined
by the shortwave/IR absorption of stellar irradiation. The spectral energy
distribution coming from hotter stars is much more weighted towards shorter
wavelengths. As opposing to cooler stars that peak in IR wavelengths, emitting
only a small fraction of energy in the UV/Visible. Figure 4 demonstrates how
the inversion strength scales with temperature of the host star and the sub-
stellar surface temperature of planet, which, for BSE, is generally within 50
K of the predicted theoretical value $T_{eq}$ 121212Depending on atmospheric
composition, the converged surface temperature of the planet may deviate from
$T_{eq}$, especially if the planet is strongly irradiated. The effect of this
is showcased in Sections 3.2 and 3.3. Inversion strength is defined by the
ratio of the maximum temperature achieved in the model to the temperature of
the surface. In most cases, the maximum temperature coincides with the
uppermost regions ($10^{-7}$ \- $10^{-5}$ bar), whereas the surface
temperature is close to the minimum temperature achieved in the entire
atmosphere. We find that inversion strength scales linearly with the
temperature of the host star. Thermal inversions will be strongest around hot
G and F stars. Planets around cool M or K dwarfs will only experience mild
inversions, as evident from purple T-P profiles in Fig. 3. Inversion strength
is anti-correlated with the surface temperature of the planet. This is
attributed not only to the increased chemical complexity and opacity that
occurs with greater pressures (larger melt temperatures), but also to thermal
dissociation and ionisation of important absorbers, e.g., Na or SiO. Inclusion
of photochemical destruction would likely result in even weaker thermal
inversions.

Figure 4: Thermal inversion strength relation with the temperature of the
stellar host and the sub-stellar temperature of the planet. Inversion strength
is the ratio of the maximum atmospheric temperature with the surface
temperature of the planet. In most cases the maximum temperature coincides
with the uppermost regions. Inversions become greater with increasing stellar
temperature and is anti-correlated with the temperature of the planet.
Silicate atmospheres become fully opaque at around 2600 K surface temperature,
causing a change in atmospheric structure (isothermal lower layer), indicated
via clustered planets separated via a distinct gap.

####  3.1.3 Observability

The high sensitivity and large spectral coverage (0.6-28 µm) of the JWST will
make characterisation of lava planets achievable. While the tenuous and
confined nature of silicate atmospheres render low resolution transmission
spectroscopy methods, which rely on atmospheric scale height, unreliable, the
scorching dayside temperatures and orbital periods of less than a day make
multiple orbit, thermal emission spectroscopy ideal. In addition to outgassed
chemical signatures, thermal emission directly probes the temperature
structure of the atmosphere, giving us insight into the atmospheric dynamics
and occurring radiative processes.

How good a planet is for silicate atmosphere characterisation is determined by
several important factors. First, the planet itself has to be of large enough
surface temperature to outgas and sustain a substantial atmosphere. Planets
with sub-stellar temperatures below 2500 K have pressures lower than $10^{-2}$
bar, which is barely enough to start forming a photosphere. Ideally, a sub-
stellar temperature of nearly 3000 K is required to produce an atmosphere that
would show large flux deviations from expected blackbody emission. The flux
ratio between the planet and the star has to be also large enough to be
detectable. This renders a large portion of currently known systems, where the
star is too hot or too large in comparison to the planet, unsuitable, even in
the IR, where most of the planetary emission comes from. Lastly, the observed
system has to be bright enough to have reasonable noise levels, but not so
bright that it would saturate the sensors. For MIRI LRS, the saturation limit
is expected to be around 5-6 J magnitude (Beichman et al. 2014). The lower
observable limit for the instrument is likely to be around 9-10 J magnitude.
MIRI LRS mode itself has spectral capabilities between 5 and 12 µm, making it
the ideal mode for characterising the presence of the 9 µm SiO feature that
has been previously shown to be possibly the best, if not the only
characterizable feature of silicate atmospheres (Ito et al. 2015).

In Figure 5 we showcase the observability distribution of all modelled lava
planets in terms of the above mentioned criteria. Majority of planets are too
dim and too cold to be considered for the JWST observations, however, there
are some excellent targets, as well as several TESS candidates that should be
considered. The standout, TOI-1807 b, is a recently discovered super-Earth
orbiting a K dwarf (Hedges et al. 2021). In the provided sample, it is by far
the best candidate for atmospheric characterisation. Its stellar host is
comparatively cool and small, resulting in expected planetary flux from the
dayside relative to the stellar flux (i.e. eclipse depth) at 9 µm of over 400
ppm. The sub-stellar temperature of the planet is expected to be just over
3000 K which would result in an outgassed atmosphere of $10^{-1}$ bar (BSE
melt with Na). In addition, with a J magnitude of 8.1, the system falls
perfectly within the range of sensitivity for MIRI LRS. Simulated noise levels
for 20 hours of observations (3 eclipses, binned to R=10) are just 21 ppm.
TOI-1807 b is an exceptional target with no current equal alternatives.

Figure 5: Emission observability of lava planets using the JWST MIRI LRS
slitless mode. Flux ratio is computed using radiative-convective models with
temperature profiles shown in Fig. 3. The flux is given for a wavelength of 9
µm, where the presence of SiO is expected to cause a broad emission feature.
Error bars are simulated for 20 hours of observations (3 eclipses) and binned
to R=10. The highlighted region indicates the optimal magnitude range for the
MIRI LRS mode. The colour of each planet denotes its sub-stellar temperature.
Ideal targets have high flux ratio, large surface temperature and fall between
9.5 and 5.5 J magnitude.

In terms of dayside flux, the closest two planets to TOI-1807 b are K2-141 b
and TOI-2431 b131313TOI-2431 b is at the time of writing still a TESS planet
candidate but the target is likely to be confirmed soon due to an accepted HST
programme (GO 16660, S. Quinn 2021).. As their sub-stellar temperatures are
comparable to TOI-1807 b, both of the planets are also expected to possess
substantial silicate atmospheres. With both systems being reasonably bright
for the JWST and emission flux reaching nearly 300 ppm, these planets are good
candidates for follow up characterisation. K2-141 b is a well studied target
(Barragán et al. 2018; Malavolta et al. 2018; Nguyen et al. 2020, ; Zieba et
al. in submitted) and is even approved for JWST GO Cycle 1 programme for over
20 hours with MIRI LRS (Dang et al. 2021) and 12 hours with NIRSpec (Espinoza
et al. 2021). While TOI-2431 b has not been selected for the JWST
observations, it may prove to be just as good of a target as K2-141 b for
future proposals.

Other notable targets within the brightness limits of MIRI LRS include: a 3300
K super-Earth orbiting a sun-like star TOI-561 b; a similar but cooler, 2300 K
super-Earth HD 3167 b and a very bright, 3000 K super-Earth HD 213885 b. HD
80653 b and Kepler-21 b are strongly irradiated planets that fall into the
MIRI range; however, their secondary eclipse flux values are below 100 ppm,
making atmospheric characterisation much more difficult. While GJ 367 b is a
nearby sub-Earth that is also approved for JWST, its equilibrium temperature
is lower than 2000 K, resulting in an outgassed atmospheric pressure of ¡
$10^{-4}$ bar. Such, relatively cool, targets are also unlikely to be good
candidates for characterisation of silicate atmospheres.

55 Cnc e currently stands as the brightest known super-Earth. The planet has
been under extensive investigation, with the current conclusion that it may
harbour a thick, volatile-rich atmosphere, possibly dominated by nitrogen,
carbon and/or oxygen bearing species (Demory et al. 2016; Angelo & Hu 2017;
Zilinskas et al. 2020, 2021). Its high surface temperature makes it also
likely to have a partially molten crust, which would result in a silicate-rich
atmosphere. Currently, there is not enough spectroscopic data to determine the
exact composition, however, the planet will be observed in thermal emission
using NIRCam and MIRI LRS modes during JWST GO Cycle 1 (Hu et al. 2021). If
successful, it would provide spectral coverage between 3.8 and 12 µm,
potentially revealing its atmospheric composition and thermal structure. The
large brightness of the system may result in oversaturation, partially
hindering the recovery of the full spectrum.

Among all of these targets there is a number of yet to be confirmed TESS
candidates. As can be seen in Fig. 5, many of these have expected sub-stellar
temperatures of 2500 K or more, have fluxes of over 100 ppm and lie perfectly
within the sensitivity range of the MIRI LRS mode.

Figure 6 shows synthetic spectra of four selected planets: HD 213885 b, 55 Cnc
e, K2-141 b and TOI-1807 b. While also being good targets, these planets cover
a range of different eclipse depth values. The spectra are generated for the
range of 0.3 - 28 µm. The wavelengths covered by the MIRI instrument are
between 5 - 28 µm. However, MIRI LRS mode already covers the sufficient range
(5 - 12 µm) for the major silicate atmosphere features. Thus, the error bars
are simulated only for the MIRI LRS slitless mode assuming 20 hours of
observations and binned to a resolution of R=10. All four planets have surface
temperatures of around 3000 K, with 55 Cnc e being slightly on the cooler
side, but they vary strongly in brightness and emission flux. K2-141 b and
TOI-1807 b orbit cooler K stars, resulting in weaker thermal inversions, but
otherwise similar thermal structure. Their respective temperature profiles are
given in the top left inset of the figure.

Figure 6: Synthetic emission spectra of four selected lava planets with
outgassed silicate atmospheres: HD 213885 b, 55 Cnc e, K2-141 b and TOI-1807
b. The spectra are computed for the entire JWST wavelength range and convolved
to a resolution of R=800. The blue curves denote blackbody emission of
numerically converged surface temperature. Corresponding temperature profiles
for each planet are shown in the inset. Error bars are simulated for the JWST
MIRI LRS mode, assuming 20 hours of observations (3 eclipses) and binned to
R=10. The indicated MIRI coverage region includes both, low resolution and
medium resolution modes. Shaded SiO (9 µm) and SiO2 (7 µm) are likely to be
strongest characterizable features of silicate atmospheres. Note that SiO also
has an additional band between 4 - 5 µm, however, its magnitude is, in most
cases, small. The noise at 4 - 5 µm is an artefact of generated synthetic
stellar spectra.

While these atmospheres are filled with many strong absorbers, the only two
relevant species that have broad, visible features are SiO and SiO2. SiO is
expected to be the major constituent of all silicate atmospheres. Equilibrium
chemistry shows that it is consistently abundant throughout the entire
atmosphere, making its opacity relevant even for upper regions, where thermal
inversions take place. SiO has a strong 9 µm band that is 3.5 µm wide. Because
its opacity comes from higher up in the atmosphere, the feature shows up as
increased flux. As inversions become weaker with increasing sub-stellar
temperature of the planet (see Fig. 4), the magnitude of SiO features will be
diminished for the hottest targets. The bands of SiO2, however, do not
manifest as emission, but rather as absorption. SiO2 is mostly confined to the
lower, cooler layers of the atmosphere. Its 7 µm SiO2 band does not overlap
with SiO, creating a large contrast between the two molecules. The combination
could prove to be the best characterisable feature of silicate atmospheres,
allowing to probe distinct thermal regions of the atmosphere.

In Figure 7 we show the opacity contribution for the atmosphere of TOI-1807 b.
Intensity shaded pressures represent the photosphere being probed by emission
spectroscopy. The opacity of the 9 µm SiO feature is not confined to single
atmospheric pressure, but rather becomes non-negligible at pressures as low as
$10^{-6}$ bar, extending for several orders of magnitude downwards. On the
contrary, SiO2 is confined to a region that is less than order of magnitude
wide. At $10^{-3}$ bar, it mostly coincides with the minimum of the
temperature profile, which results in an absorption feature. Going further
beyond of the 9 µm SiO band, the opacity comes mostly from MgO, which acts
more or less as a continuum for silicate atmospheres. As with SiO2, its
presence is also mostly confined to the lower temperature regions. Apart from
the small flux change at 13 µm, MgO does not have any distinct features. If
MgO was absent from the atmosphere, beyond 9 µm the spectrum would instead
show absorption of SiO2, which has a distinctive band between 11 - 16 µm.

Figure 7: Opacity contribution of an outgassed silicate atmosphere for
TOI-1807 b with a non-evolved (0% fractional vaporisation) Bulk Silicate Earth
melt composition. Intensity shaded pressures indicates optically thick
regions. All marked species are major contributors of the denoted wavelengths.
The corresponding temperature profile and the resulting spectrum are shown in
6.

While there are additional strong shortwave absorbers, such as Na and TiO,
their spectral features are extremely minor and likely undetectable with low
resolution spectroscopy. Even the 0.6 µm Na doublet results in only slightly
increased flux. In this study we omit pressure broadening effects, which can
result in broader Na features, but even then its characterisation might prove
difficult. Note that this is only true for this particular case. Other
outgassed compositions can have increased abundances of shortwave absorbers,
resulting in stronger emission features (see Section 3.3.

###  3.2 Evolved atmospheres

As shown by circulation models in Kite et al. (2016), planets with sub-stellar
temperatures ¿ 2400 K are likely to be ‘atmosphere-dominated’, meaning that
the surface melt composition is dragged away from its initial state by
atmospheric evolution itself. This is a direct consequence of material being
lost from the atmosphere either through escape to space or being carried to
the cooler regions and condensing out from the system. For strongly irradiated
planets this evolution is rapid and is expected to outpace mass recycling
between the interior and the surface melt.

In the outgassing code, MAGMA, atmospheric evolution is simulated with
fractional vaporisation. It is essentially sequential removal of the most
volatile constituents until a set percentage of the total mass has been
removed from the system. Na is the most volatile component and is in all cases
rapidly lost. At 3000 K sub-stellar temperature its abundance becomes
vanishingly small at less than 10% removed material. Because it is a major
vapour, its removal causes a large drop in atmospheric pressure, as well as
decrease in shortwave opacity. Continuing the evolution, abundances of K and
Fe elements slowly decrease, but neither are completely removed when the melt
temperature is 3000 K. Si is also lost from the system, albeit much more
slowly. While volatile species are reduced, the dominance of refractory
elements grows. Abundances of Mg, Ti, Ca and Al all steadily increase.
Appendix B contains figure for the evolution of major atmospheric abundances
with fractional vaporisation. Readers looking for a much more detailed
analysis of fractional vaporisation and possible removal of volatiles are
directed to Schaefer & Fegley (2004, 2009); Kite et al. (2016).

Figure 8 showcases T-P profiles at 60% fractional vaporisation, for surface
temperatures larger than 2000 K. The faint background curves indicate T-P
profiles with no vaporisation (from Fig. 3. Depletion of Na has the largest
effect on the thermal structure, causing the pressure to drop nearly two
orders of magnitude at sub-stellar temperatures near 2000 K. At higher
temperatures, Na is less dominant, hence its removal has a lesser effect on
the total vapour pressure. However, in all cases, its shortwave opacity is
completely removed, causing thermal inversions to weaken. With removal of
volatiles, MgO becomes more abundant, causing the photosphere and the
temperature minimum to shift upwards to lower pressures. The surface
temperature of the planet is also affected by the changes in opacity,
resulting in a greenhouse effect of up to 200 K. The severity of this is
greater for hotter planets that are subject to stronger shortwave irradiation.

Figure 8: Temperature profiles of evolved silicate atmospheres with sub-
stellar temperatures ¿ 2000 K, computed for a Bulk Silicate Earth composition
with 60% fractional vaporisation. All profiles are converged to radiative-
convective chemical equilibrium using the opacities described in Section 2.2.
Stellar irradiation temperature colour coded. Faint curves indicate
temperature profiles for non-evolved cases, (0% fractional vaporisation)
originally in Fig. 3.

The resulting differences in emission spectra are indicated in Figure 9. The
coloured spectra represent the original, non-evolved cases. The black spectra
show 60% fractional vaporisation models. While differences seem small, we find
that planets around cooler stars will have their spectral features severely
diminished. This is the combined result of the temperature profile shifting
closer to an isotherm (reduced inversions), as well as decreased abundances of
the absorbers, e.g., SiO and SiO2. The spectra now coincide closely with the
expected lower blackbody emission limit. The effect is slightly lesser for
planets around hotter stars, but generally similar. Needless to say, removal
of Na removes any possible emission from 0.6 µm doublet. Different
vaporisation percentages show similar results, with the general trend of
spectral features disappearing and the spectra moving closer to blackbody
emission. If the effects of vaporisation are prominent on lava worlds, it
would make characterisation of their atmospheres with low resolution
spectroscopy much more demanding.

Figure 9: Emission spectra of same four planets as in Fig. 6, but now showing
the differences caused by fractional vaporisation. Coloured curves indicate
atmospheres with no fractional vaporisation, while black curves show
atmospheres vaporised to 60%. All cases are for Bulk Silicate Earth melt
composition shown at R=800. Corresponding temperature profiles for each case
are shown in the inset. Blue shaded regions now denote blackbody emission
boundaries between the converged surface temperature of non-evolved and
evolved models.

###  3.3 Atmospheres of varying melt compositions

The bulk composition of lava planets can be estimated from their measured
density (Dorn et al. 2015), the properties of the specific host star (Santos
et al. 2017; Putirka & Rarick 2019), or even from polluted white dwarfs (Klein
et al. 2011; Putirka & Xu 2021). However, besides inherent degeneracy of
interior models complicating the matter, the formation history of the planet
can also strongly affect the composition (Bond et al. 2010; Carter-Bond et al.
2012). If the predictive assumption of silicate atmospheres is correct,
determining atmospheric abundances from observations could allow for better
constraints on melt compositions and possibly interiors. Until then, our best
examples come from our own solar system.

Hart & Zindler (1986) introduced the term Bulk Silicate Earth, representing
the combined composition of the early Earth’s crust and mantle, which excludes
the iron-rich core. While strongly irradiated rocky worlds may resemble early
Earth, currently there is no evidence for it. Thus, besides BSE, we consider
several other viable melts that vary in their oxide ratios.

Listed in Table 1 are: BSE, Continental, Oceanic, Komatiite, and Mercury.
Continental and Oceanic represent parts of the Earth’s upper lithosphere,
which formed from the molten silicate mantle. Compared to BSE, both
compositions are substantially less dominated by Mg oxides, but much richer in
Si, Al, Na, and Ti. Oceanic contains a large amount of Ti and Ca, while
Continental is more enriched in K. It has been argued the Komatiite lava,
erupted on the early Earth, might be more representative of molten, more
massive Earths (Miguel et al. 2011). Komatiite is resembled by BSE, but with
slightly higher Si and lower Mg content, while also being much more rich in
Fe. Mercury melt is uniquely poor in Fe and due to its close formation
proximity to the Sun, it is essentially void of two most volatile components,
Na and K. Its composition is otherwise similar to Komatiite.

Resulting atmospheric abundances for each of the compositions are indicated in
Figure 10. The selected cases are for 55 Cnc e, with sub-stellar temperature
ranging between 2600 - 2800 K. The showcased species are all major absorbers
of silicate atmospheres.

BSE and Komatiite melts result in nearly identical atmospheric abundances and
structure. This is not surprising, since the two melts have very similar
volatile reservoir. While Komatiite does have lower Mg content, diminishing
atmospheric pressure, its higher Fe abundance counteracts this effect. In
terms of characterisation, the two compositions are essentially identical.

With Mercury having nearly depleted NaO and K2O in the melt, the results are
somewhat similar to the previous compositions. Na is so volatile that even a
small fraction of it in the melt results in it being a major outgassed
species. Partial pressures are now more evenly divided between all different
elements (normally dominated by just Na or SiO). Since this melt is less
volatile, it results in slightly lower total pressure.

The two compositions that stand out from the rest are Oceanic and Continental
(crustal melts). Differences are caused mostly by their lack of MgO and high
abundance of TiO. Even while the crustal melts are much more abundant in Na,
diminished MgO leads to a substantial pressure drop and much lower surface
temperatures. Oceanic composition pressure is still largely dominated by Na,
whereas SiO is leading for Continental. Crustal compositions are uniquely rich
in TiO, enhancing shortwave absorption in the upper atmosphere. Both
compositions, in general, have much greater shortwave opacity, resulting is
stronger inversions. Altogether, we find that the variation in melt
composition results in moderate thermal structure and atmospheric abundance
changes.

Figure 10: Volume mixing ratio of silicate atmospheres outgassed from
Komatiite, Continental, Oceanic, Mercury and BSE melts for 55 Cnc e, with a
surface temperatures close to 2700 K. Exact melt compositions are given in
Table 1. The five listed species have the dominant opacities that shape the
thermal structure and emission spectrum of the atmosphere. Depending on the
exact surface temperature and volatile content, the total outgassed pressure
will vary between different melts.

The caused spectral differences are given in Figure 11. The three displayed
planets are TOI-1807 b, 55 Cnc e, and HD 80653 b; picked to cover a range of
different stellar and planet surface temperatures. The T-P profiles (shown in
the inset) vary only slightly between the bulk compositions (BSE, Komatiite
and Mercury), but are differentiated from crustal melts (Continental,
Oceanic), which cause a substantial change in the structure of the lower
atmosphere. This is reflected in the blackbody emission range, indicated via
shaded blue region, which is now much larger than in previous cases (see Fig.
6 and 9). The differences are smaller for planets orbiting cooler stars,
implying that the change in shortwave opacity is the most likely cause in this
discrepancy.

Figure 11: Emission spectra for different melt compositions at R=800. The
shown planets are: TOI-1807 b; 55 Cnc e; HD806563 b. For each planet different
colours indicate varying melt compositions, while the black curves represent
Bulk Silicate Earth melts. Corresponding temperature profiles for each of the
compositions are show in the inset. The blue shaded region represents
variation in surface temperature, which in this case is largest between Bulk
Silicate Earth and Oceanic comps.

From emission, bulk compositions are likely to be indistinguishable with low
resolution spectroscopy. However, outgassed crustal atmospheres have large
enough differences to be possibly identifiable. Crustal compositions have
stronger inversions and larger SiO abundance, which directly results in more
contrasting 9 µm SiO2/SiO features. For bulk atmospheres these features are
followed by continuous absorption of MgO. But in crust-like atmospheres MgO is
expected to be severely depleted. Since SiO2 has a distinct band between 11 -
16 µm, its opacity may replace MgO. Its features would show up as absorption,
coming from the lower regions of the atmosphere. Detected SiO2 features over
MgO would imply that the planetary atmosphere is being outgassed from an
Earth-like crust. Such melts are also very enriched in Ti, which results in a
larger atmospheric abundance of TiO. TiO is a strong absorber ¡ 0.9 µm. In our
models its presence causes a relatively large, continuous increase in flux,
which may indeed be observable for the very best targets with the JWST.

The situation is similar for evolved atmospheres. Figure 12 indicates melts
with 60% fractional vaporisation. Removal of volatile inventory causes
atmospheric composition to be more dominated by MgO and TiO, resulting in
increased temperatures. Vaporisation causes much larger atmospheric structure
difference between the two classes of melts, with the total vapour pressure
reduced significantly for crustal compositions. As abundances of Si composites
drop and inversions become weaker, its features are also diminished. However,
unlike for bulk melts, atmospheres produced from crustal melts still have a
strong 9 µm SiO feature. While SiO2 is still present in the atmosphere, its
detection in evolved atmospheres may be hindered for all plausible
compositions. TiO features are never removed with vaporisation, as its
abundance always grows.

Figure 12: Same as Fig. 11, but for 60% vaporised melts shown at R=800. For
each planet, different coloured curves represent different melt compositions.
The blue shaded regions represent blackbody emission boundaries between Bulk
Silicate Earth and Oceanic compositions. Temperature profiles are given in the
inset.

##  4 Discussion

###  4.1 Characterising lava worlds

Current theory predicts that lava planets are either leftover remnant cores of
sub-Neptunes, eroded away by intense irradiation, or are simply formed as
rocky worlds, without accreting much volatiles (Owen & Wu 2017; Lopez 2017).
Whether these planets form in situ or are scattered inwards through
gravitational interactions is unclear (Ogihara et al. 2015; Winn et al. 2018).
While there are several observed USP planets inconsistent with volatile-free
atmospheres (namely 55 Cnc e (Demory et al. 2016; Hammond & Pierrehumbert
2017)), most are found to have densities similar to that of Earth (Dai et al.
2019). It is expected that the permanently irradiated dayside will outgas and
sustain a silicate-rich atmosphere (Miguel et al. 2011). Characterisation of
short period planets is an intriguing next step in the exoplanet field. JWST
will provide much needed spectral coverage to make this possible. The
outstanding issue is what exactly do we look for.

Previous studies have put forward SiO as the primary detectable species in
silicate atmospheres (Schaefer & Fegley 2009; Miguel et al. 2011; Ito et al.
2015). As Ito et al. (2015) argue, due to strong thermal inversions, the 9 µm
SiO band would show up as a significant increase in flux coming from the
planet. In our models thermal inversions are not as prevalent and mostly
confined to just the upper regions. The discrepancy is mainly due to our
inclusion of up-to-date opacities while also using more accurate stellar
irradiation and self-consistent radiative-transfer models. However, because
SiO opacity is so dominant, it becomes significant even in very low pressures,
where inversions do occur. Thus we still see the 9 µm SiO band as emission,
though to a much lesser extent.

The additional inclusion of SiO2 in our models indicates that it would
manifest as absorption bands, possibly surrounding the 9 µm emission of SiO.
SiO2 abundance increases significantly towards the lower parts of the
atmosphere, which coincide with decreasing atmospheric temperatures. The
combination of the two molecules at different atmospheric regions may prove to
be unique, identifying features of silicate atmospheres. Detecting both as
absorption and emission would allow us to put strong constraints on the
temperature gradient. Presence of SiO2 may also indicate a lack of MgO, which
can be tied to the composition of the melt. Lack of SiO2 could also suggest
volatile depleted melts and/or possibly strong atmospheric dynamics.

Overall, we find that the variation of melt compositions and the stage of
evolution (volatile material removal through escape or condensation) will
impact characterisation. Depending on the melt composition, certain emission
features may be strongly enhanced or completely removed. Melts that deviate
substantially from the expected norm (crustal compositions) will result in
large spectral differences. Variation caused by fractional vaporisation is
also non-negligible. While with volatile removal the temperatures in the
atmosphere are generally increased, the total pressures are lowered. Shortwave
opacity reduction also makes inversions weaker. The combination of all these
effects can result in the largest emission features, e.g., 9 µm SiO band,
being reduced and certain bands entirely removed (SiO2 absorption). As
inversion strength decreases with decreasing stellar host temperature, planets
orbiting cooler M or K dwarfs may show weaker emission signs, if any at all.
Hotter stars will prove to be much better candidates for observing emission
features, however, their larger radii and increased temperatures weaken
planetary signal, rendering majority of currently known targets unfavourable.

JWST instruments will allow us to easily distinguish between volatile-rich and
silicate atmospheres. However, characterising spectral features in silicate
atmospheres may only be possible at longer wavelengths. MIRI LRS instrument
covering wavelengths between 5-12 µm is ideal for SiO and SiO2 bands. Its high
sensitivity will allow for detection of absorption and emission features in
just a few secondary eclipses. For the best possible outcome we recommend
focusing on targets with expected sub-stellar temperatures larger than 2500 K,
preferably orbiting K stars. Some of the best current targets for secondary
eclipse observations with MIRI LRS include: TOI-1807 b, 55 Cnc e, TOI-2431 b,
K2-141 b, HD 3167 b, TOI-561 b and HD 213885 b (see Figure 5).

###  4.2 Melts and interiors

The two melts that may be observationally distinguishable from the rest are
Earth’s oceanic and continental crust. The Earth’s oceanic crust is magnesium
and iron-rich, dense, and relatively thin ($\sim$8 km); efficient recycling
through the action of plate tectonics maximises the age of oceanic crust on
Earth today to about 200 million years (White & Klein 2014). In contrast, the
continental crust is richer in silicon and aluminium, less dense, and thicker
($\sim$40 km). Due to its low density, recycling of crust into the interior is
far less efficient, which leads to the fact that parts of the continental
crust on Earth today exceed 4 billion years in age (Ricker et al. 2014).
Although the crust of the Earth (oceanic+continental) constitutes only
$\sim$0.6% of the mass of the present-day silicate Earth, it contains between
20% and 70% of the highly incompatible141414(In)compatibility is a geochemical
term used to describe how easily a minor or trace element is able to replace
major elements in a mineral. trace element budget (Rudnick & Fountain 1995).
Hence, this explains the elevated Al2O3, CaO, Na2O, TiO2, and K2O values when
comparing the crustal to the other melts in Table 1. For such crustal melts to
co-exist, plate tectonics is likely required (Rudnick & Gao 2014). The
presence of Earth-like plate tectonics in turn necessitate the presence of a
weaker interior layer on which the plates can move, and liquid water is a
likely prerequisite to form such a weak layer (e.g. Mierdel et al. 2007).

Observed spectral signatures correlating with these compositions could
indicate that the planet in question has had a recent history of plate
tectonics and hence the presence of liquid water. The fact that lava oceans
are most likely well mixed with the mantle (Kite et al. 2016) means that the
melting of the crust must have occurred very recently due to, for example, a
recent giant impact or the host star entering the red giant phase. Therefore
it is unlikely that such signatures will be found on targets that are believed
to have a long-lived lava ocean. Alternatively, if such signatures are found
then it may be an indication that the upper layers of the melt are chemically
evolved in such a way that they resemble crustal composition or that the
planet has a mantle with a unique bulk composition.

###  4.3 Limitations of 1D equilibrium models

There are obvious predictive limitations of 1D equilibrium models. This is
especially true for lava planets where the atmosphere is likely to be confined
to the permanently irradiated dayside. Away from the sub-stellar point, the
sharp drop in temperature will induce an enormous pressure gradient, causing
strong horizontal winds towards the nightside (Castan & Menou 2011; Kite et
al. 2016; Nguyen et al. 2020). Material removed via winds is bound to condense
at cooler regions, settling down onto the surface, where it may or may not be
reincorporated into the outgassing cycle via surface and interior transport.
In this study we have mimicked this effect to some extent using fractional
vaporisation, but this is a very simplistic assumption. For a more accurate
assessment of observability, 2D Circulation models combined with simplified
outgassed chemistry and radiative transfer may prove to be the reasonable next
step.

Use of equilibrium chemistry models also ignores the potential effects of
photochemistry and vertical mixing. Photochemical destruction/ionisation of
strong absorbers, e.g., SiO or Na, may cause substantial changes in thermal
structure and observability of the specific species. Decrease in shortwave
absorption will result in weaker thermal inversions. This effect is apparent
with evolved cases, where Na is removed from the system while SiO abundance is
slightly reduced (Fig. 8). Approximations of SiO photochemistry done by Ito et
al. (2015) show that only pressures below $10^{-5}$ bar would be strongly
affected, however, the significance is unclear without employing full kinetic
photochemistry networks. The possibility of photochemical haze formation is
also not considered.

##  5 Conclusions

Characterisation of lava worlds is exciting new frontier to be explored in the
imminent future. Decade old studies have predicted formation of silicate-rich
atmospheres, outgassed from the molten surface of irradiated planets (Schaefer
& Fegley 2009; Miguel et al. 2011). With the launch of JWST, the topic has
received more interest than ever, including many predictive and observational
studies (Ito et al. 2015; Kite et al. 2016; Dai et al. 2019; Nguyen et al.
2020; Ito & Ikoma 2021, ; Zieba et al. submitted). Several lava planets have
been confirmed for the initial observers programme of JWST, which may lead to
first evidence of silicate atmospheres on exoplanets (Hu et al. 2021;
Brandeker et al. 2021; Dang et al. 2021; Espinoza et al. 2021).

As current theory predicts, these atmospheres should be depleted in highly
volatile material. Thus characterisation with low resolution spectroscopy is
likely to be only feasible via infrared emission coming from the dayside of
the planet. In this work we have modelled 1D outgassed equilibrium chemistry
consistently with radiative-transfer for all currently confirmed short period
rocky exoplanets. We have considered a large number of possible species,
including ions, as well as all up to date opacities. Finally, we have assessed
observability of the best targets with the MIRI LRS instrument. Our results
indicate the following:

  1. 1.

Thermal inversions may not be as dominant in silicate atmospheres as
previously thought. We find that inversions extend all the way to the surface
only for planets with sub-stellar temperatures below 2000 K. With larger
surface temperature, due to increasing IR dominance over shortwave opacity
inversions weaken and become confined to the upper atmospheric regions.
Inversion strength also decreases with decreasing stellar temperature,
implying that lava planets around M and K dwarfs may show only slightly
increased emission flux, if any at all. This severely impacts characterisation
of silicate atmospheres.

  2. 2.

The dominant opacity sources for non-evolved silicate atmospheres with sub-
stellar temperatures ¿ 2500 K are SiO, SiO2, MgO, Na and TiO. Excluding these
absorbers from models may result in inaccurate temperature-pressure profiles.

  3. 3.

The best observable features come from silicon oxides, specifically 7 µm SiO2
and 9 µm SiO bands. SiO2 is confined to the lower, cooler regions, manifesting
as absorption features. SiO is highly abundant in the low pressure, inversion
dominated regions. While our temperature result in relatively small
atmospheric features, we find that for the best targets, observations of only
a few eclipses with MIRI LRS are needed for a detection.

  4. 4.

The composition of the melt and possible volatile removal from the system will
impact observability. Fractionally evolved atmospheres will have reduced SiO
and SiO2 features and will prove much more difficult to characterise. Large
flux from TiO (¡ 1 µm) could suggest atmospheres outgassed from melts
resembling the Earth’s Continental or Oceanic crust. Detection 11-16 µm SiO2
features imply depletion of MgO, which is also characteristic of crust-like
compositions. Detection of certain atmospheric abundances may allow us to
constrain melt compositions, surface activity and even interior dynamics.

  5. 5.

The recommended targets for silicate atmosphere characterisation with low
resolution infrared spectroscopy are lava planets with sub-stellar
temperatures of at least 2500 K that ideally orbit cooler K dwarfs. Some of
the currently confirmed planets to keep an eye on include: TOI-1807 b, 55 Cnc
e, TOI-2431 b, K2-141 b, HD 3167 b, TOI-561 b, HD 213885 b. 55 Cnc e and
K2-141 b are confirmed for the JWST Cycle 1 GO programme.

We stress the fact that 1D equilibrium models are limited in their
predictability. The large temperature contrast between the day and the night
side, circulation and interior effects, photochemistry and kinetics are all
likely to influence the thermal structure and atmospheric abundances. There is
also a need for more accurate modelling of opacities, accounting for pressure
broadening effects and the high temperatures of silicate atmospheres.

###### Acknowledgements.

We thank Renyu Hu for his insightful comments that helped improve the paper.
This work has made use of the VALD database, operated at Uppsala University,
the Institute of Astronomy RAS in Moscow, and the University of Vienna.

We are also very grateful to all the authors of the following packages making
this work possible: HELIOS-K (Grimm & Heng 2015; Grimm et al. 2021);
HELIOS(Malik et al. 2017, 2019b); FASTCHEM (Stock et al. 2018); MAGMA (Fegley
& Cameron 1987; Schaefer & Fegley 2004); petitRADTRANS (Mollière et al. 2019,
2020); PANDEXO Batalha et al. (2017); numpy (Harris et al. 2020); matplotlib
(Hunter 2007); seaborn (Waskom 2021).

Supplementary material, including star spectra, temperature-pressure profiles,
emission spectra or planetary parameters can be downloaded at
https://github.com/zmantas/LavaPlanets.

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##  Appendix A Opacity Data

Table 2: Sources and parameters of opacities used in this work | Source | Line list | Cutting Length (cm-1)
---|---|---|---
Al | This worka | Kurucz1 | -
AlO | DACEb & This work | ATP2 | 100
Ca | This work | Kurucz | -
CaO | DACE & This work | VBATHY3 | 100
Fe | This Work | Kurucz | -
K | This Work | Kurucz | -
Mg | This Work | Kurucz | -
MgO | DACE & This work | LiTY4 | 100
Na | This Workc | VALD35 | Varied (300 max)
O | This Work | Kurucz | -
O2 | This Work | HITRAN6 | 100
Si | This Work | Kurucz | -
SiO | This Work | Kurucz (shortwave) & EBJT7 | Varied (300 max)
SiO2 | DACE | OYT38 | 100
Ti | This Work | Kurucz | -
TiO | DACE & This Work | Toto9 | 100
Al+ | This Work | Kurucz | -
Ca+ | This Work | Kurucz | -
Fe+ | This Work | Kurucz | -
Mg+ | This Work | Kurucz | -
Na+ | This Work | Kurucz | -
O+ | This Work | Kurucz | -
Si+ | This Work | Kurucz | -
Ti+ | This Work | Kurucz | -
a Opacities are computed with HELIOS-K https://github.com/exoclime/HELIOS-K
(Grimm & Heng 2015; Grimm et al. 2021) ; b DACE database
https://dace.unige.ch/; c Na doublet wings computed using the unified line-
shape theory of Rossi & Pascale (1985); Allard et al. (2007a, b); 1 Kurucz
(Kurucz 1992); 2 ATP (Patrascu et al. 2015); 3 VBATHY (Yurchenko et al. 2016);
4 LiTY (Li et al. 2019); 5 VALD3 (Ryabchikova et al. 2015); 6 HITRAN (Gordon
et al. 2017; Chubb et al. 2021); 7 EBJT (Gordon et al. 2017; Chubb et al.
2021); 8 OYT3 (Owens et al. 2020); 8 Toto (McKemmish et al. 2019). We note
that we only use opacities of main isotopes. In this work we make use of the
DACE (Grimm & Heng 2015; Grimm et al. 2021), ExoMol (Chubb et al. 2021),
HITRAN (Gordon et al. 2017), Kurucz (Kurucz 1992) and VALD3 (Ryabchikova et
al. 2015) opacity databases.

##  Appendix B Evolution of Atmospheric Abundances

Figure 13 displays the changes in key atmospheric abundances caused by
fractional vaporisation (Schaefer & Fegley 2004, 2009; Miguel et al. 2011;
Kite et al. 2016). The process sequentially removes the most volatile species
from the system with Na, in all cases, being removed first. Since the
outgassed abundances depend on the surface temperature, the vaporisation
percentage to remove a certain species is also temperature dependant.
Generally, at higher temperatures it takes a larger percentage to remove
certain species. With loss of volatiles, refractories become more dominant.
This can have substantial effect on the thermal structure of the atmosphere as
well as the observable spectrum.

Figure 13: Atmospheric abundance changes with fractional vaporisation (removal
of volatile material). Shown for a Bulk Silicate Earth Composition for a range
of different surface temperatures. Abundances are taken as averages for all
pressures larger than $10^{-5}$ bar, assuming an isothermal temperature
profile. Only the most relevant, typically high in abundance, species are
shown.