# Hidden water in magma ocean exoplanets

Caroline Dorn University of Zurich, Institute for Computational Science,
Winterthurerstrasse 190, CH-8057, Zurich, Switzerland  Tim Lichtenberg
University of Oxford, Atmospheric, Oceanic and Planetary Physics, Department
of Physics, Parks Road, Oxford OX1 3PU, United Kingdom

###### Abstract

We demonstrate that [the deep volatile storage capacity of] magma oceans [has
significant] implications for the bulk composition, interior and climate state
inferred from exoplanet mass and radius data. [Experimental petrology
provides] the fundamental properties on the ability of water and melt to mix.
So far, these data have been largely neglected for exoplanet [mass-radius
modeling]. Here, we present an advanced interior model for water-rich rocky
exoplanets. The new model allows us to test the effects of rock melting and
the redistribution of water between magma ocean and atmosphere on calculated
planet radii. Models with and without rock melting and water partitioning lead
to deviations in planet radius of up to 16% for a fixed bulk composition and
planet mass. This is within current accuracy limits for individual systems and
statistically testable on a population level. Unrecognized mantle melting and
volatile redistribution in retrievals may thus underestimate the inferred
planetary bulk water content by up to one order of magnitude.

Exoplanet – magma ocean – sub-Neptune – super-Earth – Ocean planet –
Trappist-1 – $\nu^{2}$ Lupi b – TOI-1266c – Kepler-10b – Kepler-36b

##  1 Introduction

Many exoplanets discovered to date likely host globally molten mantles – magma
oceans – because the distance to their stars or their thick atmospheres
prohibit efficient cooling and solidification of their rocky mantles (Massol
et al., 2016; Grenfell et al., 2020). Magma oceans are substantial reservoirs
for volatiles [(Schaefer et al., 2016)], especially water, as water solubility
in magma is high compared to other volatile species (Fegley et al., 2020).
Above pressures of a few GPa, water even becomes [fully] miscible in melt,
such that magma oceans can take up significant amounts of water (Ni et al.,
2016). Here, we investigate the effect of water in molten rocky mantles on the
total radius of the planet and the total volatile abundance that can be
inferred from exoplanet observations. [We use the term global magma ocean to
highlight that the magma ocean is not locally concentrated or focused on one
hemisphere, but its extent may be limited in depth such that deeper parts of
the mantle are solid due to increasing pressure in the interior.]

The volatile abundance of rocky planets alters their structural and dynamical
properties (Dorn et al., 2018), such as differentiation between core and
mantle (Bonati et al., 2021), geodynamic regime (Meier et al., 2021),
atmospheric composition (Gaillard et al., 2021), and thus long-term climate
(Spaargaren et al., 2020). The amount of water in particular sensitively
controls the rheology of the mantle (Lenardic et al., 2016) and surface state
of the planet, for instance the potential for liquid oceans (Miyazaki &
Korenaga, 2021), and the recycling of life-essential elements between the
atmosphere and mantle over geologic time (Foley & Driscoll, 2016). The right
water abundance to establish habitable conditions on a planet, however, is not
a given: too [little] of it and the planet remains a desert world; too much of
it and the planet turns into a drowned ocean planet. On clement orbits, where
water can condense at the surface, more than about $\sim$1 wt% creates high-
pressure ice phases at the mantle-ocean interface (Journaux et al., 2020), but
even a few 0.1 wt% exceed the storage capacity of solid planetary mantles
(Elkins-Tanton, 2008) and prevent dry land masses on the surface (Kite & Ford,
2018). Inside the runaway greenhouse threshold surface water evaporates in a
feedback loop that drives the planet into a global hothouse climate and melts
the surface (Ingersoll, 1969), either directly as a consequence of planetary
formation (Hamano et al., 2013), or due to stellar brightening at a later
stage (Kopparapu et al., 2013). The water content of rocky exoplanets is thus
an important environmental marker to assess potential habitability and
interpret putative biosignatures within the context of a given planetary
system (Meadows et al., 2018; Krissansen-Totton et al., 2021).

The orbital transition between clement and runaway greenhouse climates is
expected to be abrupt, not continuous, because the outgoing longwave radiation
of planets governed by steam atmospheres is controlled by the tropospheric
radiation limit [at surface temperatures between $\sim$1000–2000 K for
approximately Earth-like water ocean inventories in the atmosphere (Boukrouche
et al., 2021)]. Observationally, this predicts that in a system with [an
initially similar bulk abundance of water with changing] heliocentric
distance, planets inside the runaway greenhouse threshold should, on average,
either (i) be larger than their counterparts outside the runaway greenhouse
threshold; or (ii) be of similar size but significantly water-depleted. In
case (i), planets with similar atmospheric quantities of water in a runaway
greenhouse phase are expected to be hot and their atmospheres to be thermally
stratified [and thus larger], while planets outside the runaway greenhouse
limit can condense their water into surface oceans (Turbet et al., 2019, 2021)
and [potentially] recycle it into the mantle via tectonic processes. In case
(ii), runaway greenhouse climates expose their atmospheric water vapor to
high-energy radiation from the host star, which desiccates the planet via H2O
photolysis and hydrogen loss (Hamano et al., 2013; Wordsworth & Pierrehumbert,
2013; Luger & Barnes, 2015; Schaefer et al., 2016; Arnscheidt et al., 2019).
The left-over atmospheres, potentially made-up primarily of CO2 [ (Ortenzi et
al., 2020)] or O2 (Schaefer et al., 2016), have varying tropospheric radiation
limits (Lichtenberg et al., 2021a), which permits [the atmospheres] to cool
down and shrink significantly relative to the steam atmosphere case. If mass
and radius of an exoplanet are the only two reasonably[-well] constrained
observational properties, the two end-member cases (i) and (ii) [can] lead to
order-of-magnitude different interpretation of the maximum water content of
planets [inside and outside] the runaway greenhouse threshold.

However, the equilibrium state of runaway greenhouse climates, where the
temperature can increase above the tropospheric radiation limit, lies above
the melting point of basaltic rock already for a fraction of the Earth’s
surface water inventory (Boukrouche et al., 2021). Terrestrial and super-Earth
exoplanets that start out with significant quantities of nebular H/He can be
globally molten for hundred millions to billions of years (Kite & Barnett,
2020; Lichtenberg et al., 2021a). Therefore, rocky planets inside the runaway
greenhouse state can be expected to be at least partially molten at their
surface and in their interiors, if not globally from the primordial magma
ocean phase after accretion (Bonati et al., 2019). Atmospheric volatile
compounds can dissolve into liquid or solid rock to various degrees. Water in
particular is highly soluble in molten rock, and if a substantial fraction of
the planet’s interior is molten, most of it will reside in the interior,
rather than in the atmosphere (Elkins-Tanton, 2012; Vazan et al., 2020;
Gaillard et al., 2021).

Retrieving the volatile inventory of rocky planets is a major goal of
exoplanetary science to provide quantitative measures of surface habitability
(Kaltenegger, 2017; Catling et al., 2018) and inform theories on the origin of
life on the viability of prebiotic chemical reaction networks on early Earth
(Sasselov et al., 2020). Therefore, we here test the influence of water
partitioning and rock melting on the observed radius of planets within the
runaway greenhouse threshold from transit surveys such as TESS (Ricker et al.,
2015), CHEOPS (Benz et al., 2021), PLATO (Rauer et al., 2014), and ARIEL
(Tinetti et al., 2021; Helled et al., 2021), [as an important prerequisite on
the path to inferring the surface state and climate of terrestrial exoplanets
with] future direct imaging surveys (Gaudi et al., 2020; The LUVOIR Team,
2019; Quanz et al., 2019, 2021).

##  2 Method

Our planetary structure model is based on Dorn et al. (2015) and Dorn et al.
(2017b) with significant improvements that are explained in the following.
Major improvements are the inclusion of liquid phases both in mantle and core
materials and the ability of water to dissolve in magma. The first addition of
liquid phases mainly follows the approach taken in Haldemann et al. (in
prep.).

###  2.1 Interior model

Our 1-D interior model describes a planet in hydrostatic equilibrium that is
composed of three main components: an iron-dominated core, a silicate-rich
mantle, and a water layer of condensed water or steam, depending on the
pressure ($P$) and temperature ($T$) conditions. For the purpose of this
study, we do not include an atmosphere of other high mean molecular weight
species or H/He, but only account for water. We solve the equations of mass
conservation, the equation of hydrostatic equilibrium, thermal transport, and
the equation of states for different materials and their phases.

####  2.1.1 Metal core

We assume a core made of Fe and FeS. Many previous studies assume pure iron
(Zeng & Seager, 2008; Dorn et al., 2015; Rogers & Seager, 2010). However, the
addition of light alloys in the Earth’s core (Dziewonski & Anderson, 1981) can
reduce the core density between 5 to 10%. As data precision increases with
ongoing missions like CHEOPS, the addition of light alloys thus becomes
important. Possible light alloys include S, Si, O, C, and Ni (Hirose et al.,
2013). Here, we consider the core to be made of a uniform mixture of Fe and
FeS (Valencia et al., 2007). To stay consistent with experimental results we
use a molar fraction of 10%, which is well within the maximum molar fraction
of 23.4% that is constrained from laboratory measurements (Ichikawa &
Tsuchiya, 2020).

For pure Fe, we use the equations of state for hexagonal close packed (hcp)
solid iron at $P\geq 310$ GPa (Hakim et al., 2018) and at $P<310$ GPa (Miozzi
et al., 2020); face-centered cubic (fcc) solid iron (Dorogokupets et al.,
2017); and liquid iron at $P\geq 116$ GPa (Ichikawa & Tsuchiya, 2020) and at
$P<116$ GPa (Kuwayama et al., 2020). The phase transitions and the melting
curve are calculated using Anzellini et al. (2013). The presence of sulfur
reduces the melting temperature according to (Stixrude, 2014)

$T_{\rm melt}^{\rm FeS}(P)=T_{\rm melt}^{\rm Fe}(P)\cdot(1-{\rm ln}{x_{\rm
Fe}})^{-1},$ (1)

where $x_{\rm Fe}$ is the weight fraction of pure Fe in the mixture of Fe-FeS,
i.e., $x_{\rm Fe}+x_{\rm FeS}=1$. For pure FeS, we distinguish between the
equations of state for solid FeS (Hakim et al., 2018) and liquid FeS (Ichikawa
& Tsuchiya, 2020). To compute mixtures of Fe and FeS, we use the additive
volume law.

The core thermal profile is assumed to be adiabatic throughout the core. At
the core-mantle boundary (CMB), there can be a temperature jump as the core
can be hotter than the mantle due to the residual heat released during core
formation. Following Stixrude (2014), this temperature jump depends on the
melting temperature of the silicate mantle. If the initially calculated
temperature at the CMB is less than the melting temperature of the mantle
material, the CMB temperature is increased up to the melting temperature.

####  2.1.2 Mantle

The mantle is made of MgO, SiO2, and FeO that form different minerals.
Previous models (Dorn et al., 2015) included minor elements like Ca, Al, and
Na. Here, the limited availability of high pressure data on their liquid
phases lead us to neglect the minor elements that only account for about 7% of
the Earth’s mantle mass (Workman & Hart, 2005). Hence, they introduce little
uncertainty to the calculated interiors but add a lot of model complexity.

For the solid mantle, we use the thermodynamical model of Connolly (2009),
Perple_X, to compute stable mineralogy for a given composition, pressure and
temperature. This model employs equations of state from Stixrude & Lithgow-
Bertelloni (2011). For the liquid mantle, we use the equations of state for
Mg2SiO4 from Stewart et al. (2020), for SiO2 at $P\geq 20$ GPa from Faik et
al. (2018), for SiO2 at $P<20$ GPa from Melosh (2007), and for FeO from
Ichikawa & Tsuchiya (2020). Mg2SiO4 was chosen instead of MgO, since Stewart
et al. (2020) recently published an updated version of M-ANEOS with parameters
for forsterite, which self-consistently covers a large range in pressure and
temperature, which is not available for MgO to our knowledge. To compute
mixtures of the above components, we use the additive volume law.

The melting curve is calculated following Belonoshko et al. (2005) ($P<189.75$
GPa) and Stixrude (2014) ($P\geq 189.75$ GPa):

$T_{\rm melt}^{\rm
MgSiO_{3}}(P)=\begin{cases}a_{1}\cdot(1+P/a_{2})^{a_{3}}&{\rm
if\quad}P<189.75{\rm GPa}\\\ b_{1}\cdot(P/b_{2})^{b_{3}}&{\rm if\quad}P\geq
189.75{\rm GPa},\end{cases}$ (2)

where $a_{1}=1831$ K, $a_{2}=4.6$ GPa, $a_{3}=0.33$, and $b_{1}=5400$ K,
$b_{2}=140$ GPa, $b_{3}=0.48$. The melting temperature is influenced by the
composition, specifically the addition of Fe lowers the melting temperature,
for which we follow Dorn et al. (2018c):

$T_{\rm melt}^{\rm FMS}=\begin{cases}T_{\rm melt}^{\rm
MgSiO_{3}}+c_{1}\cdot(c_{2}-x_{\rm FeO})\\\ T_{\rm melt}^{\rm
MgSiO_{3}}+(c_{3}+c_{4}*P-c_{5}*P^{2})\cdot(c_{2}-x_{\rm FeO}),\end{cases}$
(3)

where $P$ is in GPa, $c_{1}=360$K and $c_{2}=0.0818$ (which is the Earth’s
iron mantle content (Workman & Hart, 2005)), and other fitting constants are
$c_{3}=102$, $c_{4}=64.1$, and $c_{5}=3.62$. [Partial melting is neglected.]

####  2.1.3 Water layer

The water layer is made of pure H2O for which we use the EOS of Haldemann et
al. (2020). Depending on pressure and temperature, water can create a steam
atmosphere or reside in one of the many condensed ice phases including liquid,
ice, and high pressure ice phases.

The transit radius of a planet is assumed to be at a pressure of $P_{\rm
Transit}=1$ mbar. This is a simplification as the transit radius depends on
temperature, however, the effect on the planets of interest is small (Turbet
et al., 2020). The thermal profile is assumed to be fully adiabatic, except
for pressures [less than the pressure at the] tropopause $P_{\rm iso}$, where
we keep an isothermal profile. We test two different cases for the tropopause
location, i.e., $P_{\rm iso}$ equals 0.1 bar or 1 bar. When assuming $P_{\rm
iso}$ equals 0.1 bar, our results are very similar to those of Turbet et al.
(2020) within 0.5%. If not mentioned otherwise, the temperature at the transit
radius is fixed at 400 K to assure water vapour in the atmosphere.

####  2.1.4 Water-magma mixtures

For one of our model scenarios (C, see below), we allow the water to be mixed
into the magma. Water naturally dissolves into magma as it thermodynamically
equilibrates with silicate melt, leading to hydrous magma oceans. The
partitioning between the melt and the steam layer is determined by a modified
Henry’s law that accommodates a power-law relationship between the pressure at
the mantle-steam layer boundary (MSB) and the water mass fraction in the melt,

$x_{\rm H2O}=\alpha\cdot(P_{\rm MSB})^{1/\beta},$ (4)

where $\alpha$ and $\beta$ are fitting parameters which are given by Bower et
al. (2021) (for pressures of 0-300 bar and different rock types) and
Lichtenberg et al. (2021a) (for pressures up to 1GPa). For pressures above 1
GPa, [solubilities reach the so-called second critical endpoint (Kessel et
al., 2005). If the second critical endpoint is reached, there is no limit on
the amount of water that the magma can take up. Thus, if the second critical
endpoint is reached for a given interior model, any addition of water will
only be made to the magma reservoir and not to the surface water reservoir. ]
We have combined the available fits into an empirical solubility function
depending on $P$ (Figure A1).

Water plays a crucial role in melt composition, with important consequences
for the properties of hydrous silicate melt and the melt fraction. Water
increases melt fraction by lowering the melting temperature of rock, for which
we use the approach of Katz et al. (2003):

$T_{\rm melt}^{\rm FMS,wet}=T_{\rm melt}^{\rm FMS}-d_{1}*x_{H2O}^{d_{2}}$ (5)

where $d_{1}=43$ K and $d_{2}=0.75$. [This relation is limited to the
saturation concentration of water in the melt, a regime that we do not enter
in our tested cases.]

Water also changes the density of magma and thus modifies the total radius of
a planet. However, this effect is small for low water mass fractions. The melt
density decrease per wt% water depends on the melt composition and is 0.036
g/cm3 for basalts, 0.035 g/cm3 for enstatite and 0.03 g/cm3 for silica. Here,
we use a constant value of 0.036 g/cm3 per wt% water (Bajgain et al., 2015).
According to Bajgain et al. (2015), the density contrast between pure melts
and hydrous melts is nearly independent of pressure and temperature. At large
water mass fractions, a constant density reduction becomes invalid. Thus, we
calculate also the density of the rock-water mixture with the additive volume
law and use the maximum density of both values.

In principle, water can also be taken up by solid mantle rocks. However, the
solubility of water in solid rocks is orders of magnitudes lower than for
magma (below few wt%, Elkins-Tanton, 2008; Hamano et al., 2013). Thus, we
neglect the effect of hydrated solid rock on the total radius as it accounts
only for a few percent at maximum and only in extreme cases (Shah et al.,
2021).

###  2.2 Interior scenarios

Figure 1: Three model scenarios employed in our study. Model (A) is very
similar to models presented in Dorn et al. (2015) and Dorn et al. (2017b) and
other commonly used exoplanet interior models follow model (A), where liquid
rock phases are neglected. Characteristically, the total radius is largest for
model (B), where solid and melt phases are present in the core and the mantle.
Generally, the radius is smallest for model (C), where additionally the effect
of water partitioning into the magma ocean is taken into account. Model C most
accurately reflects our current knowledge of mineral physics and exoplanet
interiors.

In order to study the effect of interior models on the amount of inferred
water mass, we employ three different interior scenarios:

  1. (A)

rocky interior with no melts, and a separate water layer;

  2. (B)

rocky interior with possible (dry) melt in both mantle and core, and a
separate water layer;

  3. (C)

rocky interior with possible melt in both mantle and core, and water being
distributed between mantle melt and a separate surface (steam) water layer
depending on the solubility relationship.

Figure 1 illustrates all model scenarios, demonstrating that model (B)
generally features the largest radii for a given planetary mass, while model
(C) generally features the smallest radii. All interior scenarios follow the
model described in section 2.1 with few exceptions for (A) and (B). In
interior scenario (A), we artificially set the adiabatic temperature profile
in the rocky interior to low temperatures, i.e., the temperature at the water-
mantle boundary is fixed to a maximum of 1500 K. This prevents the occurrence
of melts in the planets of interest and thus water can only be present in a
separate layer on top of the mantle. In interior scenario (B), water
solubilities are fixed at zero.

##  3 Results

We explore the effect of water in the model scenarios A, B, and C on exoplanet
transit radii. The calculated mass-radius curves in Figure 2 illustrate that
transit radii differ significantly for a given planetary mass and water mass
fraction. These deviations are on the order of commonly achieved observational
uncertainties for individual planets. In general, calculated radii are largest
if a planet’s interior accounts for liquid rock phases (internal magma oceans)
and all water is surface water (scenario B). Imposing the assumption that all
mantle rocks are in solid phase reduces the calculated radii by up to several
percent (scenario A). A further critical reduction in radius is obtained if
water is distributed between the surface and the magma ocean as determined by
its equilibrium stage (scenario C).

Figure 2: Mass-radius diagram for exoplanets from the PlanetS database (Otegi
et al., 2020) (as of Oct 2021), TOI-1266c (see discussion) and curves for
different composition and interior models: the rocky interior is composed of
Mg-Si oxides and silicates (66% in mass) and an iron core of 33% in mass. In
addition, varying mass fractions of [bulk] water are added: 5.4 wt% (dotted
lines), 0.1 wt% (dashed lines), or = 0 wt% (solid black line). The black (A),
red (B), and purple (C) coloring of the lines refers to the different model
scenarios in Figure 1. Largest radii are achieved when accounting for magma
oceans while neglecting the solubility of water in magma (scenario B). Taking
water solubility into account reduces the calculated radii by $\sim$5–15%. The
transit radius on this figure is fixed at 400 K and 1 mbar. Individual planets
are colored according to their equilibrium temperature.

The radius reduction of hydrous magma ocean planets (C) compared to magma
oceans with only surface water (B) of identical mass fraction is because water
dissolved in the magma ocean contributes less to the total radius compared to
the same mass in the surface/atmospheric reservoir. On the atomic scale, water
is a relatively small molecule that easily fits inbetween the molecules of
silicate melts and thus the increase in volume of hydrous magma compared to
dry magma is limited.

The calculated radius deviations between the three model scenarios (A, B, C)
depend on bulk planetary mass, water mass fraction, and the melt fraction of
the silicate mantle. Figure 3 quantifies relative radii differences between
the three scenarios. We find increasing differences of up to 16% for
decreasing planetary masses and between 1–10 wt% water mass fractions, when
comparing scenario B and C (left panel). Increasing the planetary mass reduces
the differences in transit radius. However for a 6.7 M⊕ planet, differences in
radius are still significant with a deviation of 7% for $P_{\mathrm{iso}}=0.1$
bar. When comparing model scenarios A and C, radius differences are a factor
$\sim$ 2 smaller. This is because different simplifications imposed in
scenario A partly cancel each other out. More specifically, scenario A
neglects the effect of liquid rock, which decreases the radii, and it neglects
the effect of water dissolution in the magma underneath, which increases the
radii. Both simplifications compensate each other which is why the radii
differences between model A and C are smaller than between B and C.

Figure 3: Calculated planet radii depending on the interior model scenarios
(A, B, C) and as a function of [bulk] water mass fraction. Left: Comparing
scenario C to B indicates that errors of up to 16% are produced when correctly
accounting for liquid rock phases but neglecting water solubility. Right:
Comparing scenario C with the most commonly used model in the exoplanet
literature, scenario A. When neglecting both liquid rock phases and water
solubilities, the error on the radius is 1–8% for the cases shown. Largest
errors are identified for small planet masses. The transit radius in all
scenarios is fixed at 400 K and 1 mbar.

In Figure 3 we also test the effect of the thermal profile in the steam
atmosphere. We test a potentially varying location of the tropopause by
changing the pressure $P_{\rm iso}$, below which we assume an isothermal
profile. Steam atmospheres with $P_{\rm iso}$ of 0.1 bar are thus overall
warmer than those with $P_{\rm iso}$ of 1 bar. There are two consequences on
the calculated interior profiles: Firstly, warmer atmospheres lead to larger
atmospheric scale heights and thus larger total radii (see Section 4.1 for
examples). Secondly, warmer atmospheres allow a larger portion of the mantle
underneath to be molten and thus more water can be dissolved in the magma. For
the relative radii differences in Figure 3, the latter effect can be clearly
seen as radii differences of cooler atmospheres (with $P_{\rm iso}$ of 1 bar,
dotted lines) are a factor of $\sim$ 2 smaller than the differences among
warmer atmospheres (with $P_{\rm iso}$ of 0.1 bar, solid lines).

Figure 3 shows a characteristic behaviour: for small water mass fractions
below 1 wt%, the relative differences between the radii decrease [from the
peak between $\sim$3–10 wt% water]. [Similarly,] for large water mass
fractions of tens of percent, the relative differences decrease from the peak
differences, although they never reach zero. This is because for small water
mass fractions, an increase in $x_{\rm H2O}$ leads to higher surface
temperature at the top of the magma ocean and thus the melt fraction of the
mantle increases and thus more water can be dissolved in the mantle. In
consequence the radius difference increases as more water from the surface
water reservoir can be stored in the magma ocean mantle.

At large water mass fractions (tens of percent), the melt fraction of the
mantle becomes 100%, the mantle is completely molten and therefore a further
increase in relative radius deviation between model scenarios is not possible.
In this stage gravitational compression becomes the dominant effect. In
essence, the radius does not increase linearly with surface water fraction.
For example, the radius difference between planets of 0 and 5 wt% steam
envelope is $\sim$3 times larger than between planets of 5 and 10 wt% steam
envelope. Thus for large $x_{\rm H2O}$, different surface water budgets only
yield small radius differences. The increasing values of relative radii
differences for large $x_{\rm H2O}$ in Figure 3 have reached this regime.

In other words, the differences in radii are caused by (1) different mantle
melt fractions and their ability to store large amounts of water as well as
(2) the surface water budgets. Wherever relativ radii differences decrease
with increasing water mass fractions in Figure 3, the first effect dominates
(small $x_{\rm H2O}$), while the second effect dominates where relative radii
differences increase (large $x_{\rm H2O}$).

As these results indicate, the calculated radii sensitively depend on the
interior model scenario used. In consequence, for a given planetary mass and
radius from exoplanet observations of individual planets, the inferred water
mass fraction will depend on the interior model. Figure 4 illustrates how the
inferred water mass fraction for a fixed (observed) exoplanet radius differs
for the model scenarios (A, B, C). The differences between water mass
fractions between scenarios B (lowest $x_{\rm H2O}$) and C (highest $x_{\rm
H2O}$) are on the order of one magnitude. The differences are a factor of two
smaller when comparing scenarios A and C. Figure 4 demonstrates that the phase
state of the planetary interior affects the estimated water mass fraction of
an exoplanet significantly. Hence, estimated water mass fractions obtained
using different interior models are not easily comparable. In Section 4, we
discuss selected planets and their estimated water mass budgets.

Figure 4: Inferred water mass fraction depending on interior model scenarios
(A, B,C ) and as a function of planet transit radius. The inferred [bulk]
water mass fraction can vary by up to one order of magnitude depending on the
interior scenario used. This is clear from the vertical difference between
different colors for planets of the same mass (different line styles). Largest
differences between the scenarios are seen when comparing scenario B (red) to
C (purple), while differences between scenarios A (black) and C are a factor 5
at most. If degeneracies of rocky interior composition were added, the effect
on inferred water mass fraction would be similar. [The curves for 0.3 and 0.8
M⊕ cross for high water mass fractions, as steam atmosphere become
gravitationally unbounded for low planet masses.] Again, the transit radius in
all scenarios is fixed at 400 K and 1 mbar.

##  4 Discussion

A large portion of detected super-Earths may be magma ocean worlds. Out of the
54 exoplanets shown in Figure 2, 12 (22%) of them have equilibrium
temperatures above 1600 K which is already hot enough to melt surface rocks,
independent of any atmosphere. Furthermore, 47 planets (87%) have equilibrium
temperatures above 400 K. If these planets host even small amounts of
greenhouse gases (a few tens of bars in case of H2O or H2, Kite & Barnett,
2020; Lichtenberg et al., 2021a; Boukrouche et al., 2021), their surface rocks
can be molten, allowing a deep magma ocean underneath. At a fraction of one
Earth ocean (EO) (Earth hosts about $\sim$3–11 oceans in its mantle+atmosphere
and up to 100 oceans locked in H in the core, Peslier et al., 2018)
irradiation above the runaway greenhouse limit ($\sim$280 W m-2) will drive
the planet into the globally molten phase. For instance, at 50 bar surface
pressure of water, the surface equilibrium temperature would be 1500 K, at 260
bar ($\sim$1 EO) the surface temperature would be $\sim$1800 K (Boukrouche et
al., 2021). From an astronomical perspective, atmophile compounds are abundant
in star-forming and thus planet-forming regions (Öberg & Bergin, 2021).
Migration of planets, icy pebbles, and snowlines suggest that inner planetary
systems are easily enriched in major volatile compounds (Lichtenberg et al.,
2019; Bitsch et al., 2019). The water snowline is the preferred nucleation
region for the growth of planets (Drążkowska & Alibert, 2017), including the
terrestrial planets of the Solar System (Lichtenberg et al., 2021b). Even in
the absence of exogenous volatiles, super-Earth mantles may generate H2O by
internal geochemical reactions (Kite & Schaefer, 2021). Therefore, super-
Earths with outgassed secondary atmospheres may preferentially harbour magma
oceans unless they are bare rocky worlds. Although the absence of any
atmosphere is very difficult to achieve, relatively thick atmospheres can be
ruled out in some cases. For the case of LHS 3844b, atmospheres with surface
pressures above 10 bars could be ruled out (Kreidberg et al., 2019); and
indeed a 10 bar atmosphere of steam would not be sufficient to raise the
surface temperature high enough for rocks to be molten on LHS 3844b.

If many super-Earths can in principle be magma ocean worlds, they may host
consequentially large volatile reservoirs in their interiors. So far, these
deep mantle reservoirs are not taken into account when inferring volatile (or
water) budgets from planetary data. Previous studies have published inferred
surface water contents (Rogers & Seager, 2010; Dorn et al., 2017a; Unterborn
et al., 2018; Turbet et al., 2020; Brugger et al., 2017; Madhusudhan, 2019;
Mousis et al., 2020; Aguichine et al., 2021) or are limited by only accounting
for surface water reservoirs when modelling interiors (Venturini et al., 2020;
Owen & Wu, 2017; Fortney et al., 2013; Rogers & Owen, 2021). Our study
highlights that there is a clear difference in bulk water contents versus
surface water content. Taking magma ocean reservoirs into account can increase
water budget estimates by one order of magnitude for a given radius or it can
reduce calculated planetary radii by up to 16%. This is equivalent to a change
in planetary density of 75%. Clearly, there is a need to reinvestigate
possible water budgets for the populations of observed super-Earths and sub-
Neptunes.

Here, we focus on water in magma, as solubilities of other volatiles (e.g.,
CO2, H2, CO, CH4) are 1–6 orders of magnitudes lower than for water and
partial pressures below 1 GPa. Thus, unless large quantities are available of
other species, water will be the dominant volatile in the magma.

H2 can in fact be available in large quantities when a planet has accreted
primordial H/He envelopes. Hydrogen-dominated envelopes are likely shaping the
radii of sub-Neptunes (Owen & Wu, 2017; Fortney et al., 2013), although
additional large water budgets are possible (Mousis et al., 2021; Venturini et
al., 2020; Kite et al., 2020). But what about deep water reservoirs in molten
mantles of sub-Neptunes?

Primordial hydrogen may reduce mantle oxides and produce H2O (potentially
dominant when Fe reaches the core, Kite & Schaefer, 2021) or FeH, but it is
unclear if reduced Fe would merge with the core (Lichtenberg, 2021). In global
chemical equilibrium, mantle oxygen may be able to produce water in mole
number comparable to or more than hydrogen (Schlichting & Young, 2021). Thus,
primordial H/He atmospheres may be enriched with water vapour (Kimura & Ikoma,
2020). In consequence, the water will in large parts dissolve into the mantle.
Thereby, [atmospheric water abundances of sub-Neptune envelopes would
consequently become sub-solar]. For now, there is no comprehensive interior
model that accurately accounts for the chemical reactive atmosphere-magma
ocean boundary as well as the partitioning of water and other volatiles in the
deep interior of sub-Neptunes. Only few studies have investigated individual
aspects of it (Kite et al., 2020; Lichtenberg, 2021; Schlichting & Young,
2021; Vazan et al., 2020; Kimura & Ikoma, 2020; Olson & Sharp, 2018; Chachan &
Stevenson, 2018). As volatile partitioning in the deep interior has been
neglected for inference studies of exoplanets, previous interior predictions
have thus far generally underestimated the amount of water and hydrogen for
sub-Neptunes. [Sub-Neptune envelopes may possess compositional gradients
(Ormel et al., 2021; Helled & Stevenson, 2017), e.g., water might only be
mixed within a hydrogen layer up to heights where water condenses. This effect
itself influences the calculated radii and should ideally be considered in
parallel with the partitioning of volatiles in the deeper planetary parts.]
This is particularly important for very water-rich atmospheres as a potential
explanation for the radius valley (Mousis et al., 2020; Aguichine et al.,
2021): most of the atmospheric water will be dissolved in the deep interior
and does not contribute to the atmospheric layer thickness.

Our presented model is a static model and does not focus on evolutionary
aspects. Atmospheric escape can efficiently reduce the amount of surface water
by high energy stellar flux [(Johnstone, 2020)]. A reduction of surface water
mass will shift the equilibrium state between magma ocean and surface
reservoir according to Eq. 4, such that fractions of dissolved water outgas
and are added to the steam atmosphere. A planet with water in its magma ocean
(scenario C) will experience less water-loss compared to a planet with only
surface water (scenario A and B). This is because the partitioning of water in
the interior [further decreases the upper-atmosphere water mixing ratio due to
condensation (Graham et al., 2021)]. Hence, high solubility of H2O in magma
oceans may enable its safe storage over long time spans. Time-dependent
coupled models of magma ocean evolution, outgassing and atmospheric escape are
necessary to tackle these questions (Kite & Barnett, 2020; Lichtenberg et al.,
2021a).

Here, we have not addressed the inference of interior parameters from specific
exoplanet data but have outlined the physical reasoning and implications of
magma ocean interiors on transit radii and physical state of the planet. This
interior model introduces a higher degree of non-linearity compared to other
interior models (Unterborn et al., 2016; Dorn et al., 2017b; Mousis et al.,
2020), as there are stronger inter-dependencies between interior parameters.
It does not, however, introduce a higher degree of degeneracy. Thus using our
model in an inference scheme does not lead to larger uncertainties on
predicted interior properties. The new model comes with a higher computational
cost: it is two times slower than the previous models from Dorn et al.
(2017b). However, inference schemes can be done more efficiently111Some model
parameters can be computed extremely fast (e.g., planet mass), while others
require a detailed structure computation (e.g., planet radius). For a more
efficient inference, a detailed structure computation is only done, if a newly
proposed set of model parameters is not rejected, assuming a perfect fit for
model parameter that would require an expensive computation. such that the
overall computational cost for an inference analysis that includes our new
model is two times faster than previously (Dorn et al., 2017b).

In the following we consider the potential influence of magma oceans on
specific exoplanets.

###  4.1 Trappist-1 planets

The Trappist-1 system (Gillon et al., 2017) hosts a fascinating chain of small
and cool planets. Our current knowledge on their masses and radii (Agol et
al., 2021) allows to gain an improved understanding of their interiors and
atmospheres. Possible scenarios that fit the slightly lower densities compared
to an Earth-like interior include: (1) rocky interiors that are iron-depleted
compared to the Earth, (2) a core-free rocky interior with oxidised iron, (3)
surface water reservoirs on top of a rocky interior, or possibly (4) H-rich
cores (Schlichting & Young, 2021).

Interestingly, single [composition mass-radius] curves exist that can fit all
seven planets (Agol et al., 2021). This is possible for the purely rocky,
iron-depleted scenarios (1 & 2). A single interior model that can explain them
all, seems to be intriguing as the least complex solution.

For the water-rich scenario (3), the amount of water that is compatible with
each individual planet strongly depends on their individual equilibrium
temperatures. While the inner planets b, c, and d seem to feature small water
mass fractions ($<10^{-5}$) in the form of steam atmospheres, the outer
planets can host up to several wt% of condensed water (Agol et al., 2021).
Thus, if the lower densities of the Trappist-1 planets are due to a higher
amount of water compared to the Earth (Schoonenberg et al., 2019; Dorn et al.,
2018b; Lichtenberg et al., 2019), there is no single interior model that can
explain all planets.

The timing of magma ocean crystallization sensitively depends on the initial
amount of volatiles inherited from planetary formation. Hence, this begs the
question whether the inner planets can have a magma ocean water reservoir at
present day. Is it possible that all planets actually started off with similar
water contents while the inner planets have most of their water lost to space
or stored in molten mantles? Figure 5 shows that within 2-$\sigma$
uncertainty, planets b and c are consistent with 0.01 wt% of water at maximum
(dotted purple line). For water mass fractions below 0.01%, no hydrous magma
ocean is present. This means that for lower water mass fractions, the steam
atmosphere does not allow to raise the temperature at the surface high enough
to keep mantle rocks molten, or $P_{\rm MSB}$ is too low and dissolved water
budgets are marginal. In these cases, the interior model (C) yields identical
results compared to model (B) (masses below $\sim$0.4 M⊕ in Figure 5). Thus,
if no other interior heating mechanism is considered, it is unlikely that the
inner planets posses magma oceans at present day in which large quantities of
water could be stored. This is consistent with the planets being depleted in
atmospheric water by H2O photolysis and atmospheric hydrogen escape.

Indeed, heating mechanisms are actually proposed to maintain magma oceans on
the inner planets (Bolmont et al., 2020; Kislyakova et al., 2017; Barth et
al., 2021). However, even if we force the interior model (C) to allow for
magma oceans by fixing the temperature of the mantle surface to 1800 K (Figure
5, dotted lines), the amount of dissolved water is marginal. In this case, the
curves of scenarios (C) and (B) appear indistinguishable. Hence, if the inner
planets have magma oceans, the amount of dissolved water must be limited to
minor amounts.

Figure 5: Mass-radius diagram showing the seven confirmed Trappist-1 planets
(Agol et al., 2021) and curves for different interior models: the rocky
interior is composed of Mg-Si oxides and silicates (66% in mass) and an iron
core of 33% in mass. In addition 0.01 % mass fraction of [bulk] water are
added. [The letters refer to rocky interiors being (A) solid & dry, (B)
possibly molten & dry, (C) possibly molten & wet, see] Figure 1. For the outer
planets (e-h), steam atmospheres do not apply, instead water is condensed on
the surface (blue curve). A magma ocean with dissolved water only establishes
for $P_{\rm iso}$ = 0.1 bar and high masses ($>$0.5M⊕). For $P_{\rm iso}$ =
0.1 bar and masses below 0.5 M⊕, no magma ocean is present and thus model
scenario (C) provides identical results compared to scenario (B). Dotted lines
show interiors for which we artifically imposed a mantle surface temperature
of 1800 K (B & C).

###  4.2 $v^{2}$ Lupi b

Previous models for $v^{2}$ Lupi b estimate a total water amount of
$12.6^{+14.7}_{-11}$ wt% and a low core mass fraction of 14% (Delrez et al.,
2021). Such large amounts of water can only be achieved if water is assumed to
be fully condensed. Given the equilibrium temperature of 905 K, this is
unlikely to be the case. The planetary density is significantly below an
Earth-like interior (see Figure 2). The low density can be explained by a
molten mantle and the presence of water.

Water will form a steam atmosphere and be in large portions hidden in the
underlying magma ocean. More specifically, the planet can be described with a
steam atmosphere of about 0.07 wt%, while the mantle contains 2 wt% of water
in the melt (melt fraction of 40%). For this case, we assume an Earth-like
core mass fraction. A smaller core mass fraction similar to Delrez et al.
(2021) would further decrease the inferred total water budget. Our estimates
thus differ significantly from those of Delrez et al. (2021), which can be in
part attributed to the different interior models but also to the treatment (or
the inclusion) of a steam atmosphere. A complete inference analysis within the
context of our model is beyond the scope of this paper.

###  4.3 55 Cnc e

The intensely irradiated super-Earth 55 Cnc e features an unusually low
density ($\rho=1.164\pm 0.062\rho_{\oplus}$ at a mass of $M_{p}=8.59\pm
0.43$M⊕, Crida et al., 2018b, a). Its interior has previously been interpreted
to be due to a hydrogen-rich or nitrogen-rich volatile layer (Modirrousta-
Galian et al., 2020; Hammond & Pierrehumbert, 2017), an absent or a thin
metal-rich atmosphere (Ridden-Harper et al., 2016; Morris et al., 2021) on top
of a magma ocean (Bower et al., 2019), or an interior enriched in those
minerals (e.g., Al, Ca) condensing at high temperatures and thus depleted in
iron (Dorn et al., 2018a). Observations in Ly-$\alpha$ indicate the absence of
a primordial atmosphere, while the presence of water could not be excluded
although its presence was evaluated to be unlikely (Bourrier et al., 2018).

Any atmosphere for this hot planet ($T_{\rm eq}\approx 2000$ K) is in chemical
equilibrium with the rocky and necessarily molten interior. Volatiles like
water could be contained in the magma ocean. Previous estimates on water loss
were limited to _surface_ water reservoirs (Bourrier et al., 2018) and did not
account for deep reservoirs where water is shielded from XUV irradiation. We
thus hypothesize that water could still be present in the interior of 55 Cnc
e, mainly in the deep mantle reservoir. If this were the case, we expect non-
zero partial pressures for water in a metal-rich atmosphere, or more
specifically ionized water [given its stellar irradiation environment]. Hence,
a plausible atmosphere could contain H, O, Mg, Si, S, as well as other minor
metal elements. A more tailored escape model would be needed to estimate the
possible loss of hydrogen in light of deep magma ocean reservoirs for water.

Given the data of 55 Cnc e, it is possible that the planet has several
percents of water in the melt. For a melt fraction of 50% in the mantle, this
would account for 1 wt% of dissolved water relative to the total mass of the
planet. In that case, the surface reservoir would only contain less than 0.01
wt% of water. Such an interior can explain the low density of 55 Cnc e while
only little water is exposed on the surface to evaporative loss. It remains to
be seen whether 55 Cnc e must have lost all its water during its lifetime or
if deep water reservoirs in the molten interior can prolong the retention of
it.

###  4.4 Kepler-10b and Kepler-36b

Both Kepler-10b and Kepler-36b have high equilibrium temperatures (2169 K and
979 K, respectively) and both planets are less dense than Earth (Figure 2).
Hence water is a possible component to explain their bulk densities.

For Kepler-10b, a purely rocky interior of 3.3 M⊕ in mass that fits the
stellar abundance proxy (Dorn et al., 2015) with [Fe/H] = -0.14, [Mg/H] =
-0.01, and [Si/H] = -0.1, yields a radius of 1.429 R⊕, which is 3% lower than
the median observed radius of $1.47\pm 0.03$ R⊕ (Adibekyan et al., 2021). The
addition of a thin steam atmosphere of $1.7\times 10^{-6}$ wt% allows to match
the observed radius. In that case, $\sim$0.01 wt% of the planet’s mass is
contained as dissolved water in the mantle. Our estimates for the surface
water reservoir are similar to previous estimates (Dorn et al., 2017a), which
employed a model that is close to model scenario (A) with a dry solid-only
rocky interior. Of course, other volatiles could dominate the atmosphere
(e.g., CO2), which would imply their presence in the magma by necessity.

For Kepler-36b, a pure rocky interior of the planet’s mass of 3.9 M⊕ (Otegi et
al., 2020) yields radii of 1.44 to 1.48 R⊕ (for core mass fractions between
0.35 and 0.25, respectively) which is slightly below the observed radius of
$1.5\pm 0.1$ R⊕. The addition of $1\times 10^{-4}$ wt% surface water would
allow to fit the radius (for a core mass fraction of 0.33), and it would imply
an additional water reservoir in the molten mantle of $\sim$0.01 wt% of the
total mass. The presence of water for Kepler-36b is further supported by its
proposed formation path (Owen & Morton, 2016), where it would have formed with
a thick H/He envelope. Any primordial envelope is chemically reducing and
would lead to the formation of water (Kimura & Ikoma, 2020).

###  4.5 TOI-1266c

The planet candidate TOI-1266c was recently hypothesized to host a steam
atmosphere (Harman et al., 2021). Although TOI-1266c has a large uncertainty
on mass and thus can be fit with very different interiors, it is likely a
volatile-rich planet given its peculiar low density (Figure 2). Indeed, the
nominal data of TOI-1266c (1.9 M⊕, 1.673 R⊕, Stefansson et al., 2020) allows
for fully molten mantle with a total water budget of 31 wt% (for a core mass
fraction of 33%). In this case, the total water budget of 31 wt% is divided
between the surface reservoir (5 wt%) and the magma ocean reservoir (26 wt%).
Our proposed interior composition is significantly different from those models
with 50–77% water proposed by Harman et al. (2021), who refer to different
mass-radius-relationships (Fortney et al., 2007; Noack et al., 2016; Zeng et
al., 2019). The large differences stems from the fact that these pre-
calculated mass-radius relationships do not account for liquid phases, nor the
dissolution of water in melt, nor the specific equilibrium temperature of
TOI-1266c. Hence, these mass-radius-relationships are unable to provide self-
consistent interiors for the conditions of TOI-1266c.

##  5 Summary and Conclusions

The majority of observed super-Earths and sub-Neptunes are subject to
irradiation that promotes molten rocky interiors, i.e. magma oceans, either
due to direct surface melting or greenhouse forcing from water or other
volatiles. Global magma oceans represent potentially vast volatile reservoirs
that can lock them in the deep interior of planets and consequently shrink the
atmospheric reservoir mass. Here, we focused on water and its ability to be
stored in large quantities in magma oceans due to its high [solubility] in
molten rock relative to other volatile compounds. In contrast to previous work
[(Schaefer et al., 2016; Bower et al., 2019; Lichtenberg et al., 2021a)], we
account for the entire range of low water solubilities up to the miscibility
regime and calculate the effect on mass-radius relationships by providing a
comprehensive interior model.

We demonstrate that magma ocean planets with volatiles locked up in the
interior differ fundamentally from their solid or molten mantle counterparts
with all volatiles stored in the atmospheric reservoir. The differences
concern not only interior structure, but also likely their evolution. [Our
work complements previous studies that suggest water-dominated secondary
atmospheres to emerge for rocky planets evolving from sub-Neptunes with
primordial H2-dominated atmospheres (Kite & Schaefer, 2021).]

Introducing a quantitative treatment of volatile storage in molten rock phases
is essential for correctly inferring bulk [volatile mass budgets], and thus
being able to retrieve the formation and volatile loss chronology of specific
exoplanets and their environment. This will be essential to probe the
atmospheric and climate diversity of rocky exoplanets, and to be able to use
short-period exoplanet abundances as environmental proxies for their more
distant, potentially habitable siblings in multi-planet systems.

Observational efforts are increasing data precision on planetary mass and
radius. These efforts must be accompanied by theoretical effort to improve
precision in structural and compositional models. Our updated interior models
include a more profound understanding of [petrology]. Compared with Dorn et
al. (2017b), the updated model includes (1) liquid rock phases in mantle and
core, (2) light core elements, (3) improved equations of state, and most
importantly (4) the dissolution of water in the magma ocean depending on
solubility equilibria. Future model improvements may include the addition of
other volatiles in both interior and atmospheric structure (Graham et al.,
2021) and the dependency of solubility on mantle composition.

For short-period exoplanets inside the runaway greenhouse transition, our
results suggest the following. [The bulk water budget will be divided between
a surface and the magma ocean reservoir. While common interior models include
surface reservoirs, the deep magma ocean reservoir is usually neglected with
severe consequences for inferred water mass fractions.] By necessity, any
inferred surface water reservoir generally implies an even larger water
reservoir in the mantle. Hence, inferred water mass fractions can vary by up
to an order of magnitude depending on whether deep magma ocean water
reservoirs are included or not. The differences depend on planet mass, the
parameterisation of the thermal structure of a steam atmosphere, as well as
the actual water mass fraction. We have tested three different interior models
of various degrees of complexity. For a given interior composition, calculated
radii can vary by up to 16%, [which can be well above the precision obtained
from transit surveys (e.g., TESS, CHEOPS)]. The potential for a much larger,
undetected storage capacity of volatiles inside nominally rocky super-Earths
indicates that they may host a larger volatile budget than previously
estimated. We will examine this question in more detail in future work.

Exoplanet population studies may [be sensitive to the effect of deep water
reservoirs. Current data already allow to] infer the fraction of short-period
exoplanets in runaway greenhouse states and their total bulk volatile budgets.
In summary, our results suggest that volatile inferences from retrievals need
to account for the additional mass that can be locked up in the interior in
highly molten states. The presence or absence of magma oceans in planetary
mantles affects the bulk volatile abundances and climatic conditions of rocky
exoplanets to first order. Ultimately, deep volatile reservoirs are crucial to
consider for a better understanding of exoplanet interiors, with important
implications for their formation and evolution.

We thank an anonymous reviewer for help in improving the manuscript. C.D. was
supported by the Swiss National Science Foundation (SNSF) under grant
PZ00P2_174028. T.L. was supported by the Simons Foundation (SCOL Award No.
611576). This work was in part carried out within the framework of the
National Centre for Competence in Research PlanetS supported by the Swiss
National Science Foundation and benefited from information exchange within the
program ‘Alien Earths’ (NASA grant No. 80NSSC21K0593) for NASA’s Nexus for
Exoplanet System Science (NExSS) research coordination network.

##  Appendix A Solubility law

A solubility law relates the abundance of water in the magma to its abundance
in the atmosphere, according to the partial pressure of water at the surface
of the magma ocean.

Figure A1: Solubility of water in silicate melts as a function of pressure
from different petrological experiments. The fit in solid black is used in our
study and connects the region of moderate solubilities ($<1$GPa) to the region
above the critical second endpoint where water becomes fully miscible in melt.
The solubility curve neglects any effects of melt composition.

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# The invariance of inelastic overlap function

S.M. Troshin, N.E. Tyurin
NRC “Kurchatov Institute”–IHEP
Protvino, 142281, Russia

We consider the symmetry property of the inelastic overlap function and its
relation to the reflective scattering mode appearance.
Keywords: Unitarity; Symmetry property of inelastic overlap function;
Reflective scattering mode.

## Introduction: unitarity of partial amplitude

Unitarity written for the partial wave amplitude $f_{l}(s)$ of elastic
scattering (spin degrees of freedom are neglected) has familiar form at high
energies, i.e.

$\mbox{Im}f_{l}(s)=|f_{l}(s)|^{2}+\eta_{l}(s),$ (1)

where the function $\eta_{l}(s)$ represents contribution of the intermidiate
inelastic states into the product $SS^{+}$ ($SS^{+}=1$). Respective elastic
S–matrix element is $S_{l}(s)=1+2if_{l}(s)$.

Unitarity in the impact parameter representation ($b=2l/\sqrt{s}$) connects
the elastic and inelastic overlap functions introduced by Van Hove [1] with
$h_{tot}(s,b)\equiv\mbox{Im}f(s,b)$ by the following relation

$h_{tot}(s,b)=h_{el}(s,b)+h_{inel}(s,b)$ (2)

The respective cross–sections are determined by the integrals of the functions
$h_{i}$ over $b$:

$\sigma_{i}(s)=8\pi\int_{0}^{\infty}bdbh_{i}(s,b)$ (3)

where $i=tot,el,inel$.

This note is devoted to the inelastic overlap function, $h_{inel}(s,b)$,
namely, its energy behavior at $b=0$. Based on unitarity we consider this
quantity as a function of elastic scattering amplitude.

Unitarity allows variation of the scattering amplitude in the interval
$0\leq|f|\leq 1$ which covers both the absorptive and reflective scattering
modes when $|f|\leq 1/2$ and $|f|>1/2$, respectively. Consideration of the
inelastic overlap function provides valuable insight on the nature of the
scattering. Transition to the reflective scattering mode results in the
peripheral behavior of the inelastic overlap function as well as it changes
the structure of the hadron interaction region [2].

##  1 Symmetry of inelastic overlap function

We consider a pure imaginary case of elastic scattering amplitude with
replacement $f\to if$. Inelastic overlap function can be expressed then
through the scattering amplitude $f(s,b)$ in the following form due to
unitarity

$h_{inel}(s,b)=f(s,b)[1-f(s,b)].$ (4)

To clarify the symmetry property of inelastic overlap function consider energy
variation of the scattering amplitude $f$ under fixed value of the impact
parameter. It is enough to consider the case of $b=0$. The region of variation
of the scattering amplitude covers the range $0\leq f\leq 1$ and
$h_{inel}\equiv h_{inel}(s,0)$ invariant under replacement

$f\leftrightarrow 1-f.$ (5)

Thus, the Eq. (4) is invariant for the two amplitude variation intervals (0,
1/2] and [1/2, 1), i.e.

$(0,1/2]\leftrightarrow[1/2,1)$ (6)

and the both ranges of amplitude variation correspond to a single range of
inelastic overlap function variation $(0,1/4]$. These two intervals are
equivalent in the sense that $h_{inel}$ repeats its values. Note, that the use
of $U$–matrix unitarization [3] provides a continuous transition from
absorptive to reflective scattering mode covering the whole range of the
amplitude variation allowed by unitarity.

The energy evolution of the elastic $S$–matrix scattering element $S$
($S\equiv S(s,0)$), the elastic scattering amplitude $f$ ($f\equiv f(s,0)$)
and the inelastic overlap function from some initial energy $s_{i}$ to a final
value $s_{f}$ across the energy $s_{m}$ where a $h_{inel}$ has its maximal
value, ( $h_{inel}^{m}=1/4$):

$s_{i}\to s_{m}\to s_{f}$ (7)

has been discussed in [3]. It is illustrated by the following relations:

$S^{i}>0\to S^{f}<0,\,\mbox{i.e.}\,f^{i}<1/2\to f^{f}>1/2,$ (8)

and

$h_{inel}^{i}<1/4\to h_{inel}^{f}<1/4.$ (9)

Thus, the inelastic overlap function $h_{inel}$ can perform a loop variation
with increasing energy in accordance with Eq.(7). It varies as

$h_{inel}^{i}\to 1/4\to h_{inel}^{f}$ (10)

and $h_{inel}^{f}=h_{inel}^{i}$ provided that $f^{i}+f^{f}=1.$

Eq. (9) means the appearance of the reflective scattering mode and it takes
place regardless of the scattering amplitude form. Indeed, such behavior of
$h_{inel}$ implies an onset of decrease (i.e. $\partial h_{inel}(s,b)/\partial
s$ at $b=0$ becomes negative) of the inelastic overlap function and is
associated with appearance of reflection at the LHC energy range111The
$U$-matrix form of unitarization and its relation to the symmery property of
$h_{inel}$ was used for continuous extrapolation to higher energies in [3]. It
incorporates both scattering modes providing a ground for their simultaneous
presence.. Impact parameter profile of $h_{inel}(s,b)$ becomes peripheral when
$s>s_{m}$, i.e. $\partial h_{inel}(s,b)/\partial b$ at $b=0$ is strictly
positive. Thus, this invariance of the inelastic overlap function $h_{inel}$
is in favor of coexistence of the absorptive and reflective scattering modes
in elastic scattering (see for discussion of the latter in [2, 3]) and
corresponds to the transition of the elastic scattering matrix element:

$S\leftrightarrow-S.$ (11)

It is not clear whether this property has a meaning of a separate physical
concept. However, it is coherent with saturation of unitarity limit for the
amplitude $f$ implied by the principle of maximal strength of strong
interactions proposed long ago by Chew and Frautchi [5, 6]. They noted that a
“ characteristic of strong interactions is a capacity to “ saturate unitarity
condition at high energies”. Factor $-1$ in Eq. (11) is interpreted as a
result of a reflection by analogy with optics. It can also be considered as a
result of an analogue of Berry phase appearance [2] or of color–conducting
matter formation in the interaction region of reaction [4]. The
color–conducting matter formation can be used in its turn for explanation of
the correspondence of above symmetry property to a quark–hadron duality
(confinement) [7]. For discussion of correlation of the $S$–matrix unitarity
and confinement see [8]. It is also suggested to associate this mode with
effective resonance formation resulting from “ an exceptional intermidiate
state that unites correlated partons” [9, 10, 11].

##  2 Real part of the scattering amplitude

It should be recollected that the above symmetry property for the inelastic
overlap function takes place for the pure imaginary scattering amplitude.

This section discusses the changes in the symmetry properties due to the real
part of the scattering amplitude. Relaxing requirement of a pure imaginary
elastic scattering amplitude and taking $b=0$, unitarity condition in the
impact parameter representation can be rewritten in the form:

$\mbox{Im}f[1-\mbox{Im}f]=h_{inel}+[\mbox{Re}f]^{2}.$ (12)

It is evident from Eq. (12) that

$[\mbox{Re}f]^{2}\leq 1/4-h_{inel}.$ (13)

Thus, $|\mbox{Re}f|\to 0$ in both cases: when $h_{inel}\to 1/4$ (full
absorption, $S=0$) and/or $\mbox{Im}f\to 1$ (full reflection, $S=-1$). Without
neglect of the small real part of the scattering amplitude, one should
consider invariance of the function $h_{inel}+[\mbox{Re}f]^{2}$ under
replacement

$\mbox{Im}f\leftrightarrow 1-\mbox{Im}f.$ (14)

Account of a small real part of the scattering amplitude makes picture less
transparent but does not change it qualitatively222Numerical calculations
based on the model–independent analysis of available experimental data [12]
give $h_{inel}=1/4-\alpha$ with small positive function $\alpha$, $\alpha\ll
1/4$, at the LHC energies..

## Conclusion

The symmetry property of the inelastic overlap function and simultaneous
presence of the two scattering modes is a consequence of unitarity. It is in
favor of coexistence of the absorptive and reflective scattering modes at
small impact parameter values333When addressing the asymptotics with the use
of the respective relations, for example, Gribov–Froissart projection formula
[13, 14], one should not expect the symmetry realization since the scattering
is reduced to purely absorptive mode at large impact parameter values.. In
contrast, the considered symmetry property disfavors an ad hoc exclusion of
one of the scattering modes (i.e the reflective mode) under approaching the
asymptotic limit $s\to\infty$. Predominance of the particular mode is to be
correlated with the energy and impact parameters ranges under consideration.

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