# Forming Iron-rich Planets with Giant Impacts

Christian Reinhardt1, Thomas Meier1, Joachim G. Stadel1, Jon F. Otegi1,2,
Ravit Helled1
1Institute for Computational Science, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland
2Observatoire Astronomique de l’Université de Genève, 51 Ch. des Maillettes, –
Sauverny – 1290 Versoix, Switzerland
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We investigate mantle stripping giant impacts (GI) between super-Earths with
masses between $1\text{\,}\mathrm{M_{\oplus}}$ and
$20\text{\,}\mathrm{M_{\oplus}}$. We infer new scaling laws for the mass of
the largest fragment and its iron mass fraction, as well as updated fitting
coefficients for the critical specific impact energy for catastrophic
disruption, $Q_{\textnormal{RD}}^{*}$. With these scaling laws, we derive
equations that relate the impact conditions, i.e., target mass, impact
velocity and impactor-to-target mass ratio, to the mass and iron mass fraction
of the largest fragment. This allows one to predict collision outcomes without
performing a large suite of simulations. Using these equations we present the
maximum and minimum planetary iron mass fraction as a result of collisional
stripping of its mantle for a given range of impact conditions. We also infer
the radius for a given mass and composition using interior structure models
and compare our results to observations of metal-rich exoplanets. We find good
agreement between the data and the simulated planets suggesting that GI could
have played a key role in their formation. Furthermore, using our scaling laws
we can further constrain the impact conditions that favour their masses and
compositions. Finally, we present a flexible and easy-to-use tool that allows
one to predict mass and composition of a planet after a GI for an arbitrary
range of impact conditions which in turn allows to assess the role of GI in
observed planetary systems.

###### keywords:

planets and satellites: formation – planets and satellites: composition –
planets and satellites: terrestrial planets – hydrodynamics – equation of
state

††pubyear: 2022††pagerange: Forming Iron-rich Planets with Giant Impacts–A

##  1 Introduction

Since the discovery of the first exoplanet around a main-sequence star at the
end of the last century (Mayor & Queloz, 1995), over 5000111NASA Exoplanet
Archive: https://exoplanetarchive.ipac.caltech.edu/ planets orbiting other
stars than the Sun have been discovered. Their compositions are very diverse
and range from bodies that are dominantly made of refractory elements to ones
that are mostly composed of hydrogen and helium. Planets that have a
composition similar to Earth, i.e., consist mostly of iron and rock, seem to
be very common (e.g., Petigura et al. 2013; Bryson et al. 2020). Among those
rocky planets, several have very high bulk densities implying that they are
enriched in iron compared to the chondritic value and could have a composition
similar to Mercury. Some examples for such planets are K2-38b (Toledo-Padrón
et al., 2020), Kepler-107c (Bonomo et al., 2019), K2-229b (Santerne et al.,
2018), L 168-9b (Astudillo-Defru et al., 2020), K2-106b (Guenther et al.,
2017), Kepler-80d (Hadden & Lithwick, 2017), K2-291b (Kosiarek et al., 2019),
Kepler-406b (Marcy et al., 2014) and the recently discovered GJ 367b (Lam et
al., 2021) (see Table 1 for masses, radii and compositions). Their masses
range from $0.55\text{\,}\mathrm{M_{\oplus}}$ to
$9.4\text{\,}\mathrm{M_{\oplus}}$ so they are at least one order of magnitude
more massive than Mercury and therefore are often called super-Mercuries.

The origin of super-Mercuries is still unknown and several formation
mechanisms have been proposed. In a recent study Adibekyan et al. (2021) find
a link between the compositions of rocky planets and their host stars. In
their sample of exoplanets super-Mercuries are found to be more common around
stars that are enriched in iron. However, this is not the case for all iron-
rich planets. For example, the extreme composition of Kepler-107c cannot be
explained by the stellar abundance of rock-building elements (Schulze et al.,
2021). Furthermore, obtaining the high metal abundance of super-Mercuries
would require an unrealistically large iron enrichment of the host star. The
proto-planetary disc from which super-Mercuries are formed could be locally
enriched in iron, e.g., due to photophoretic forces (Wurm et al., 2013) or
different condensation temperatures of silicates and iron (Lewis, 1972;
Aguichine et al., 2020), but this usually requires very specific disc
conditions. It is unclear whether such conditions were met in the observed
planetary systems. Another possibility is that part of the mantle is lost due
to the photo-evaporation of silicates (Santerne et al., 2018). However, this
mechanism is limited to planets orbiting very close to the host star due to
the extreme temperatures required to evaporate rock. In addition, photo-
evaporation is unlikely to remove more than a few percent of the mantle for
planets with masses larger than the Earth (Ito & Ikoma, 2021). It is therefore
difficult to explain a Mercury-like composition by photo-evaporation alone.

Finally, giant impacts (GI) could strip a large fraction of the mantle
analogous to the scenario that was proposed for Mercury in the Solar system
(e.g., Benz et al. 2007; Asphaug & Reufer 2014; Chau et al. 2018; Clement et
al. 2021), leaving behind super-Mercuries. All of the super-Mercuries listed
above orbit very close (between $0.01\text{\,}\mathrm{ua}$ and
$0.06\text{\,}\mathrm{ua}$) to their host star and therefore have orbital
velocities between $130\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and
$260\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. As a consequence, high
velocity impacts during the formation and early evolution of these planetary
systems are possible. This in turn promotes disruptive collisions that can
strip a large fraction of the mantle, increasing the planet’s iron mass
fraction. Indeed, in the case of Kepler-107c Bonomo et al. (2019) demonstrated
that GI is the likely origin of its iron-rich composition using 3-D hydro
simulations to model the collision.

In the general context of super-Earths Marcus et al. (2009) (hereafter M09)
investigate mantle stripping GI using 3-D hydro simulations and derive scaling
laws for the mass of the largest fragment as well as its iron mass fraction,
$Z_{\textnormal{Fe}}$. The simulations focus on head-on impacts since these
impacts are the most erosive and grazing collisions occurring above the mutual
escape velocity tend to be hit-and-run collisions (Leinhardt & Stewart, 2012;
Timpe et al., 2020). Using these scaling laws Marcus et al. (2010) (hereafter
M10) then derive equations that relate the impact conditions, e.g., the impact
velocity, impactor-to-target mass ratio and total colliding mass, to the
fragment mass and iron mass fraction. This allows one to predict the fragment
mass and composition for given impact conditions without performing impact
simulations. Furthermore, they calculate each fragment’s radius from its mass
and composition using the interior models for super-Earths of Valencia et al.
(2007). From these results the minimum radius of a planet due to collisional
stripping of its mantle is inferred assuming an upper limit of
$80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ for the impact velocity.
This result is often used for interpreting the role of GI when explaining the
composition of observed exoplanets. Additionally, M10 also predict a maximum
mass of $5\text{\,}\mathrm{M_{\oplus}}$ for super-Mercuries based on the
scaling laws and the assumption that the initial pre-impact mass of a super-
Earth does not exceed $10\text{\,}\mathrm{M_{\oplus}}$. However, several
substantially more massive iron-rich planets like Kepler-107c or K2-106b have
been observed. Furthermore, all of the iron-rich exoplanets presented in Table
1 orbit very close to their host star so their orbital velocities exceed
$80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ allowing much larger impact
velocities than assumed in M10. Finally, M10 make several mistakes when
deriving their equations (see Appendix A for details). This in turn results in
errors in the minimum radius of exoplanets that are comparable to the accuracy
of current observations. We therefore reexamine such collisions considering
higher pre-impact masses and impact velocities.

In this work we use 3-D Smoothed Particle Hydrodynamics (SPH) simulations to
investigate mantle stripping in GI between super-Earths with masses of
$1\text{\,}\mathrm{M_{\oplus}}$ to $19\text{\,}\mathrm{M_{\oplus}}$. We then
derive scaling laws for the mass of the largest fragment and its iron mass
fraction that are applicable over a wide range of impact conditions,
particularly higher masses and impact energies, as well as updated fitting
coefficients for the critical specific impact energy for catastrophic
disruption, $Q_{\textnormal{RD}}^{*}$ . Furthermore, we combine these scaling
laws to obtain a general relationship between the impact conditions, i.e.,
target mass, impactor-to-target mass ratio and impact velocity, and the post-
impact fragment mass and composition following M10. For each fragment mass and
composition we also determine the radius, making the connection to
observational data. We then present a mass-radius (M-R) diagram where we
compare the results of our simulations to the iron-rich exoplanets listed in
Table 1 and show that their compositions are consistent with mantle stripping
GI. Finally, we propose a general criterion to constrain the minimum and
maximum iron mass fraction possible for exoplanets via the process of
collisional stripping based on the expected impact conditions and the total
colliding mass. This procedure is implemented in an easy-to-use tool which can
be applied to interpret present and future observations.

The paper is structured as follows: in Section 2 we discuss the initial
conditions for the collisions, the interior models used to obtain the fragment
radius and the suite of simulations. The results are presented and discussed
Section 3. Finally, we give a summary and conclusions in Section 4.

Planet | Mass [$M_{\oplus}$] | Radius [$R_{\oplus}$] | $Z_{\textnormal{Fe}}$
---|---|---|---
K2-38b | $7.3_{-1.0}^{+1.1}$ | $1.54\pm 0.14$ | $0.63_{-0.36}^{+0.29}$
Kepler-107c | $9.39\pm 1.77$ | $1.597\pm 0.026$ | $0.68_{-0.16}^{+0.14}$
K2-229b | $2.59\pm 0.43$ | $1.164_{-0.048}^{+0.066}$ | $0.67_{-0.32}^{+0.19}$
L 168-9b | $4.60\pm 0.56$ | $1.390\pm 0.090$ | $0.56_{-0.31}^{+0.25}$
K2-106b | $8.36_{-0.94}^{+0.96}$ | $1.520\pm 0.160$ | $0.74_{-0.36}^{+0.24}$
Kepler-80d | $3.70_{-0.60}^{+0.80}$ | $1.309_{-0.032}^{+0.036}$ | $0.63_{-0.36}^{+0.24}$
K2-291b | $6.49\pm 1.16$ | $1.589_{-0.072}^{+0.095}$ | $0.46_{-0.38}^{+0.23}$
Kepler-406b | $6.35\pm 1.40$ | $1.430\pm 0.030$ | $0.73_{-0.25}^{+0.15}$
GJ 367b | $0.546\pm 0.078$ | $0.718\pm 0.054$ | $0.82_{-0.32}^{+0.16}$
Table 1: Properties of the iron-rich exoplanets referenced in the paper. The
planet masses and radii are taken from the respective publication referenced
in Section 1. The iron mass fractions $Z_{\textnormal{Fe}}$ are obtained from
interior modelling as described in Section 2.2.

##  2 Methods

We perform a large suite of 168 simulations considering different target and
impactor masses and impact velocities. The mass of the target is
$10.5\text{\,}\mathrm{M_{\oplus}}$, $16.55\text{\,}\mathrm{M_{\oplus}}$ or
$19.04\text{\,}\mathrm{M_{\oplus}}$ and the impactor mass is
$1\text{\,}\mathrm{M_{\oplus}}$, $2\text{\,}\mathrm{M_{\oplus}}$,
$5\text{\,}\mathrm{M_{\oplus}}$ or $10\text{\,}\mathrm{M_{\oplus}}$. As the
most extreme case we also investigate impacts with an impactor-to-target mass
ratio of one, e.g., a $10.5\text{\,}\mathrm{M_{\oplus}}$ target colliding with
a $10.5\text{\,}\mathrm{M_{\oplus}}$ impactor. The impacts are head-on, i.e.,
occur at an impact parameter $b=0$. Such events are expected to be the most
erosive, they therefore provide a lower limit on the impact velocity required
to obtain a certain composition. The impact velocities range from two to six
times the mutual escape velocity which is given by:

$v_{\textnormal{esc}}=\sqrt{\frac{2G(M_{\textnormal{tar}}+M_{\textnormal{imp}})}{R_{\textnormal{tar}}+R_{\textnormal{imp}}}},$
(1)

where $M_{\textnormal{tar}}$, $M_{\textnormal{imp}}$, $R_{\textnormal{tar}}$
and $R_{\textnormal{imp}}$ are the mass and radius of the target and the
impactor, respectively, and $G$ is the gravitational constant.

###  2.1 Impact simulations

The impact simulations are performed using the Smoothed Particle Hydrodynamics
(SPH) code gasoline (Wadsley et al., 2004) with modifications for modelling
giant impacts (Reinhardt & Stadel, 2017; Chau et al., 2018; Reinhardt et al.,
2020; Meier et al., 2021). The particle representations of the colliding
planets are obtained using ballic (Reinhardt & Stadel, 2017; Chau et al.,
2018). We assume that the bodies are differentiated and have a chondritic
composition consisting of a $30\text{\,}\mathrm{\char 37\relax}$ by mass iron
core surrounded by a $70\text{\,}\mathrm{\char 37\relax}$ by mass rocky
mantle. The thermal profile is adiabatic and the surface temperature of all
models is set to $1000\text{\,}\mathrm{K}$. Due to noise inherent in SPH a
particle’s density and internal energy are not consistent with the imprinted
adiabatic thermal profile. We therefore first calculate all particles’
densities with gasoline and then correct their internal energies so that all
particles are on the same isentrope. The resulting models exhibit little noise
because the particles are close to their equilibrium state but we still evolve
each model in isolation in order to further relax the root mean square
velocities to below $1\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, which is
a hundred fold reduction in the velocities.

####  2.1.1 Equations of State

To model the materials of the colliding planets we use the iron (Stewart,
2020) and forsterite (Stewart et al., 2019) equation of state (EOS)
constructed from M-ANEOS (ANalytic Equation Of State) (Thompson & Lauson,
1974; Melosh, 2007; Thompson et al., 2019). These EOS are an improvement over
the standard M-ANEOS and ANEOS because the thermal part of the free energy is
more accurately modelled in the liquid phase (Stewart et al., 2020) providing
a significantly better fit to state-of-the-art high-pressure shock experiments
(Davies et al., 2020). This improvement is crucial when modelling impacts
involving bodies of more than $10\text{\,}\mathrm{M_{\oplus}}$ due to the huge
peak pressure (and temperature) reached during shock compression.

####  2.1.2 Resolution

Recently, Meier et al. (2021) demonstrated that the amount of mass stripped in
a collision shows little resolution dependence when ANEOS is used and has
already converged at a resolution of $10^{5}$ particles. We therefore choose a
resolution of $2\times 10^{5}$ particles for the
$16.55\text{\,}\mathrm{M_{\oplus}}$ body and adapt the resolution of all other
bodies so that the particle mass is constant ensuring numerical stability. In
all the simulations the target is therefore resolved with at least $10^{5}$
particles, which is sufficient to ensure that the bound mass has converged.

####  2.1.3 Analysis

The remaining bound mass, as well as the critical specific impact energy,
$Q_{\textnormal{RD}}^{*}$ , required to disrupt and gravitationally disperse
half of the total colliding mass are determined as in Meier et al. (2021). The
mass of the gravitationally bound group of particles after the impact is
calculated using the group finder skid222The source code is available at:
https://github.com/N-BodyShop/skid.. We find that the results are rather
insensitive to the choice of parameters of the group-finding algorithm and
that the fragment mass has converged in all simulations with nSmooth = 3200
and tau = 0.06. In a few cases skid finds multiple small fragments that did
not merge with the largest fragment until the end of the simulation
($53.1\text{\,}\mathrm{h}$ after impact). In these cases we first check
whether any of the smaller fragments are bound to the largest fragment. We
then calculate the bound mass and see if any other fragments are bound with
respect to the aggregate’s centre of mass. Those are again added to the bound
mass and the procedure is repeated until no additional bound fragments are
found. This algorithm usually converges after one or two steps since the
largest fragment contains most of the total bound mass and therefore dominates
the gravitational potential of the bound material. For a given target and
impactor mass, the specific impact energy $Q_{\textnormal{R}}$ is determined
for all impact velocities and is given by:

$Q_{\textnormal{R}}=\frac{1}{2}\frac{\mu
v_{\textnormal{imp}}^{2}}{M_{\textnormal{tot}}},$ (2)

where $M_{\textnormal{tot}}$ is the total colliding mass,
$v_{\textnormal{imp}}$ the impact velocity and $\mu$ the reduced mass. The
critical specific impact energy for catastrophic disruption
$Q_{\textnormal{RD}}^{*}$ is then calculated by linear interpolation between
the two data points that bracket the specific impact energy where exactly half
of the total colliding mass remains bound.

###  2.2 Interior models

After the collision we determine the bound mass as described above and assume
that the planet cools down to the pre-impact surface temperature of
$1000\text{\,}\mathrm{K}$. This temperature is comparable to the surface
temperatures of the close-in iron-rich exoplanets in our sample. The surface
temperature of the planet depends on the stellar type and age, the distance of
the planet from the host star, and the planetary thermal evolution. We find
that for values between $300\text{\,}\mathrm{K}$ and $1500\text{\,}\mathrm{K}$
the impact of the temperature on the inferred radius is very small
($<$1\text{\,}\mathrm{\char 37\relax}$$). For each fragment’s mass and
composition we determine the radius using the interior model of Otegi et al.
(2020), which assumes a pure iron core and a silicate mantle. The EOS used to
model the iron core is taken from Hakim et al. (2018). The constituents of the
mantle are assumed to be SiO2, MgO, CaO, Al2O3 and NaO2. The stable minerals
at a given pressure, temperature and composition are calculated by Gibbs
energy minimisation using the Perple_X code presented in Connolly (2009) and
the thermodynamical model of Stixrude & Lithgow-Bertelloni (2011). We assume
an adiabatic thermal profile for both the core and the mantle and that the
abundances of the mantle constituents are identical to the Earth’s mantle
which is consistent with the EOS we use for the impact simulations. For more
details about the model we refer the reader to Otegi et al. (2020) and
references therein.

In order to verify consistency between these models and the ones uses for the
impact simulations we also determine each post-impact fragment’s radius with
ballic and M-ANEOS using the same material parameters as for the initial
conditions of the impact simulations. We find that for most masses and
compositions, the radii of the resulting bodies agree within
$2\text{\,}\mathrm{\char 37\relax}$ for the two methods. Only for very small
masses and large iron mass fractions the agreement is worse but is still
within $5\text{\,}\mathrm{\char 37\relax}$. Generally, the models obtained
using ballic and M-ANEOS slightly overestimate the planet’s radius compared to
the interior model described above. Considering that M-ANEOS is designed for
modelling impact simulations and therefore aims at covering a large range of
densities and temperatures using analytic expressions for the free energy, we
consider this an excellent agreement. In any case, the dependence of the
impact simulation outcome on the precise interior model (considering mineral
level composition) is negligible.

##  3 Results and Discussion

###  3.1 Simulation result

Figure 1: The mass of the largest fragment for different specific impact
energies. The circles represent the simulation data and each colour
corresponds to the target’s pre-impact mass (violet:
$10\text{\,}\mathrm{M_{\oplus}}$, cyan: $16.55\text{\,}\mathrm{M_{\oplus}}$
and red: $19.04\text{\,}\mathrm{M_{\oplus}}$). The critical specific impact
energy $Q_{\textnormal{RD}}^{*}$ is determined from the simulation data for
each combination of target and impactor mass. Despite the large difference in
the total colliding mass ranging from $10\text{\,}\mathrm{M_{\oplus}}$ to
$40\text{\,}\mathrm{M_{\oplus}}$ the normalised fragment mass and specific
impact energies show little scatter and are in excellent agreement. We observe
that for $Q_{\textnormal{imp}}/Q_{\textnormal{RD}}^{*}\approx 1.26$ the bound
mass rapidly decreases and follows a power law. While the scaling law
presented in this work (solid black line) provides an excellent fit to the
simulation data the results deviate significantly both below and above
$Q_{\textnormal{RD}}^{*}$ from the behaviour predicted by the scaling laws of
SL09 and LS12. If ANEOS is used for identical initial conditions (grey
crosses), substantially less material is stripped due to the lack of an
accurate treatment of the liquid phase (for details see Section 3.1). Figure
2: The iron mass fraction of the largest fragment for different specific
impact energies. The circles represent the simulation data and the colours
correspond to the target’s pre-impact mass (violet:
$10\text{\,}\mathrm{M_{\oplus}}$, cyan: $16.55\text{\,}\mathrm{M_{\oplus}}$
and red: $19.04\text{\,}\mathrm{M_{\oplus}}$). As in Figure 1 the critical
specific impact energy $Q_{\textnormal{RD}}^{*}$ is determined from the
simulation data. The solid black line is our best fit and the dotted black
line the scaling law from M09. At lower impact energies the scaling law from
M09 overestimates the amount of mantle material lost due to mantle stripping.
The largest difference between the two scaling laws is at very high impact
energies due to the strong disruption above
$Q_{\textnormal{imp}}/Q_{\textnormal{RD}}^{*}\approx 1.26$ observed in our
simulation. Furthermore, our scaling laws correctly extrapolate to
$Z_{\textnormal{Fe}}=1$ in the limit of super-catastrophic disruption. These
differences have profound implications for the impact velocity required to
obtain a certain fragment mass and composition (see Section 3.3).

Figure 1 shows the mass of the largest fragment versus the specific impact
energy for all our simulations. Generally the impact conditions required to
strip a certain amount of the mantle depend on the target mass and the
impactor-to-target mass fraction
$\gamma=M_{\textnormal{imp}}/M_{\textnormal{tar}}$. However, once the bound
mass is normalised to the total colliding mass and the specific impact energy
to $Q_{\textnormal{RD}}^{*}$ (determined from the simulation data as described
in Section 2.1.3) the resulting dimensionless quantities are in excellent
agreement for all simulations. Only in the case of a super-catastrophic
disruption of the colliding bodies the normalised bound mass is more noisy as
the fragment is resolved with only a few thousand particles. At low normalised
specific impact energies very little mass is stripped in the collision. Around
the specific impact energy for critical disruption $Q_{\textnormal{RD}}^{*}$
we observe a linear dependence of the bound mass on the impact energy as
reported in SL09 and M09. However, in the critical disruption regime, around
$Q_{\textnormal{imp}}/Q_{\textnormal{RD}^{*}}\gtrsim 1.26$, we find that the
bound mass rapidly decreases and deviates from the linear behaviour reported
in Stewart & Leinhardt (2009) (hereafter SL09) and M09 following a power law.
In Leinhardt & Stewart (2012) (hereafter LS12) such a behaviour was observed
for super-catastrophic disruption, which was defined as
$M_{\textnormal{frg}}/M_{\textnormal{tot}}\lesssim 0.1$. Previous work was
limited to substantially smaller target–impactor masses as well as lower
impact velocities. Therefore, a reason for the difference in behaviour could
be the extreme impact energies probed in our simulations; energies for which
the peak shock pressures and temperatures are very high. Another difference to
previous work is that older versions of ANEOS and M-ANEOS were used. These EOS
model the thermal part of the free energy in the liquid phase as a solid and
therefore disagree with data from high-pressure shock experiments (Davies et
al., 2020). The modified version of M-ANEOS of Stewart et al. (2019) used in
this work substantially improves the treatment of the liquid phase and
provides a much better match to experiments in the region relevant for such
energetic impacts. In order to test this hypothesis we perform impact
simulations with ANEOS (the material parameters for iron and dunite can be
found in Meier et al. 2021) of the two most extreme cases: a
$10.5\text{\,}\mathrm{M_{\oplus}}$ target colliding with a
$1\text{\,}\mathrm{M_{\oplus}}$ impactor and a
$19.04\text{\,}\mathrm{M_{\oplus}}$ target colliding with a
$19.04\text{\,}\mathrm{M_{\oplus}}$ impactor for the entire range of impact
velocities investigated in this paper. We determine $Q_{\textnormal{RD}}^{*}$
for both cases and find that generally less mass is stripped compared to the
M-ANEOS simulations. This difference is most pronounced for large specific
impact energies where the normalised bound mass shows a linear dependency on
impact energy (see the grey crosses in Figure 1). Given that the impact
conditions are identical to the ones of the M-ANEOS simulations we conclude
that indeed the more accurate treatment of the liquid phase in M-ANEOS is
responsible for the different behaviour at very high impact energies. This
again demonstrates that the EOS plays a key role in faithfully modelling GI
(Stewart et al., 2019; Meier et al., 2021).

In order to test a possible connection between the transition to rapid mass
loss and the initial core mass fraction we perform an additional set of
simulations where we assume an iron mass fraction of
$Z_{\textnormal{Fe}}=0.5$. The initial conditions are identical to the other
simulations, and we again consider impacts between a
$10\text{\,}\mathrm{M_{\oplus}}$ target and a $1\text{\,}\mathrm{M_{\oplus}}$
impactor as well as collisions between two bodies of
$19.04\text{\,}\mathrm{M_{\oplus}}$. We find that the transition to rapid mass
loss indeed depends on the initial core mass fraction. The value of
$Q_{\textnormal{imp}}/Q_{\textnormal{RD}^{*}}$ at which the transition occurs
depends on the EOS of iron as well as the initial core mass fraction. However,
this dependency is limited to energies above the threshold for rapid mass loss
while the behaviour at lower specific impact energies does not depend on the
initial core mass fraction. We also find that the bound mass exhibits
substantially more noise at impact energies above the threshold for rapid mass
loss because the fragments are resolved with fewer particles. Therefore the
behaviour in this regime could also be resolution-dependent. Investigating
these two aspects is beyond the scope of this paper and we hope to address it
in future work.

We present the corresponding iron mass fractions in Figure 2. At low impact
energies, i.e., $Q_{\textnormal{imp}}/Q_{\textnormal{RD}^{*}}\leq 0.5$, little
mass is stripped and therefore the deviation of the iron mass fraction from
the pre-impact value is very small. For impact energies higher than
$Q_{\textnormal{imp}}/Q_{\textnormal{RD}^{*}}\approx 1.26$ on the other hand
the iron mass fraction massively increases as the bound mass decreases
following a power law. These findings have profound implications for the
stripping of a planet’s mantle due to a giant impact and therefore for the
likelihood of the formation and observation of iron-rich exoplanets. First,
since relatively large specific impact energies are required one would expect
that most GI are insufficient to produce super-Mercuries. Second, in order to
obtain a iron-rich massive planet, aside from large impact velocities, the
initial (pre-impact) mass must be extremely high since much of the mass is
lost in the critical disruption regime. This is consistent with the fact that
iron-rich exoplanets are rather rare and few with masses
$>$8\text{\,}\mathrm{M_{\oplus}}$$ have been observed.

Finally, except for the most disruptive collisions, we find that a fraction of
the impactor remains bound. The mass fraction of impactor material in the
post-impact planet $f_{\textnormal{imp}}$ increases with increasing impactor
mass and can be up to $50\text{\,}\mathrm{\char 37\relax}$ in the case of
equal mass collisions, as expected by the symmetry of the impact. For small
impactor-to-target mass ratios, i.e.,
$\gamma=M_{\textnormal{imp}}/M_{\textnormal{tar}}<0.2$ the mass fraction of
impactor material decreases with increasing impact velocity for a given
impactor mass. For the larger impactor masses we find that
$f_{\textnormal{imp}}$ is approximately equal to the impactor mass divided by
the total colliding mass and is independent of the impact velocity. A mantle
stripping impact can therefore leave an isotopic fingerprint if the impactor
accreted from a very different region of the proto-planetary disk and the
isotopic difference in composition is preserved after the collision.
Furthermore, the amount of impactor material in the post-impact planet in
principle allows one to predict the impactor mass and therefore could provide
constraints on the impact conditions.

###  3.2 Scaling laws

####  3.2.1 Fragment mass

As in previous studies we derive scaling laws from the simulation data. For
the mass of the largest fragment SL09 found that an expression of the form

$\frac{M_{\textnormal{frg}}}{M_{\textnormal{tot}}}=-\frac{1}{2}\left(\frac{Q_{\textnormal{R}}}{Q_{\textnormal{RD}}^{*}}-1\right)+\frac{1}{2}$
(3)

best fits their simulation data. While this fit is appropriate for the
relatively low mass and velocity collisions investigated in their paper it
does not accurately represent our data as shown in Figure 1. At specific
impact energies below $Q_{\textnormal{RD}}^{*}$ the bound mass is
underestimated because the massive targets considered in our study are more
resilient to erosion. Likewise, at specific impact energies above
$Q_{\textnormal{RD}}^{*}$ the bound mass is strongly overestimated because the
bound mass decreases rapidly following a power law beyond
$Q_{\textnormal{R}}/Q_{\textnormal{RD}}^{*}\approx 1.26$ as proposed in LS12
for super-catastrophic disruption.

We therefore propose a new scaling law for the mass of the largest fragment,

$\frac{M_{\textnormal{frg}}}{M_{\textnormal{tot}}}=\begin{dcases}\frac{1}{2}\cos{\left(\frac{\pi}{2}\frac{Q_{\textnormal{R}}}{Q_{\textnormal{RD}}^{*}}\right)}+\frac{1}{2}&\text{if
}\frac{Q_{\textnormal{R}}}{Q_{\textnormal{RD}}^{*}}\leq
q_{\textnormal{sc}},\\\
b_{\textnormal{M}}\left(\frac{Q_{\textnormal{R}}}{Q_{\textnormal{RD}}^{*}}\right)^{\eta}&\text{otherwise},\end{dcases}$
(4)

where,

$b_{\textnormal{M}}=\frac{\cos{\left(\frac{\pi}{2}q_{\textnormal{sc}}\right)}+1}{2q_{\textnormal{sc}}^{\eta}},$
(5)

is given by the requirement of continuity between the two functions. The
coefficients that best describe our results are $q_{\textnormal{sc}}=1.2645$
and $\eta=-8.1214$. The coefficient $q_{\textnormal{sc}}$ corresponds to the
specific impact energy at which the bound mass starts to decrease following a
power law and therefore marks the transition to extreme mass loss in
collisions which we observe earlier than in prior simulations that involved
lower mass bodies. Furthermore, the value of $\eta$ we find is smaller than
the $\eta\approx-1.5$ that was reported for super-catastrophic disruption in
LS12.

####  3.2.2 Iron mass fraction

For normalised specific impact energies
$Q_{\textnormal{R}}/Q_{\textnormal{RD}}^{*}$ below $q_{\textnormal{sc}}$ the
iron mass fraction of the remaining bound mass follows a power law as proposed
in M09. At larger normalised specific impact energies a power law provides a
poor fit to our simulation data and severely underestimates the iron mass
fraction due to the rapid mass loss beyond
$Q_{\textnormal{R}}/Q_{\textnormal{RD}}^{*}\approx 1.26$ (see Figure 2).
Additionally, the predicted iron mass fraction can exceed
$Z_{\textnormal{Fe}}=1$ for very disruptive collisions which is clearly
unphysical.

We find that a function of the form,

$Z_{\textnormal{Fe}}=\begin{dcases}Z_{\textnormal{Fe,i}}+a_{\textnormal{Fe}}\left(\frac{Q_{\textnormal{R}}}{Q_{\textnormal{RD}}^{*}}\right)^{b_{\textnormal{Fe}}}&\text{if
}\frac{Q_{\textnormal{R}}}{Q_{\textnormal{RD}}^{*}}\leq
q_{\textnormal{sc}},\\\
1-a_{\textnormal{Fe,sc}}\exp{\left[-{b_{\textnormal{Fe,sc}}\left(\frac{Q_{\textnormal{R}}}{Q_{\textnormal{RD}}^{*}}-q_{\textnormal{sc}}\right)}\right]}&\textnormal{otherwise,}\end{dcases}$
(6)

best fits our data. For our simulations the initial (pre-impact) iron mass
fraction is $Z_{\textnormal{Fe,i}}=0.3$. For lower impact energies the fitting
coefficients $a_{\textnormal{Fe}}=0.2099$ and $b_{\textnormal{Fe}}=2.1516$
differ from the ones found in M09 ($a_{\textnormal{Fe}}=0.25$ and
$b_{\textnormal{Fe}}=1.65$) corresponding to the lower iron mass fraction and
change in composition found in our simulations. For impact energies above
$q_{\textnormal{sc}}$ the coefficient

$a_{\textnormal{Fe,sc}}=1-\left(Z_{\textnormal{Fe,i}}+a_{\textnormal{Fe}}q_{sc}^{b_{\textnormal{Fe}}}\right)$
(7)

is determined by requiring continuity of $Z_{\textnormal{Fe}}$ at
$q_{\textnormal{sc}}$ and $b_{\textnormal{Fe,sc}}=6.4890$. The iron mass
fraction predicted by this scaling law not only is in excellent agreement with
our simulation data but also extrapolates to $Z_{\textnormal{Fe}}=1$ for very
large specific impact energies. The scaling laws derived from this equation in
Section 3.3 never provide unphysical values (i.e., $Z_{\textnormal{Fe}}>1$).

####  3.2.3 Critical specific impact energy for catastrophic disruption

Figure 3: The impact of different fitting coefficients for calculating the
critical specific impact energy from scaling laws on the normalised mass (top
panel) and iron mass fraction (bottom panel) of the largest fragment. The
circles represent the simulation data and the colours the different values of
$Q_{\textnormal{RD}}^{*}$ used to normalise the specific impact energy (black:
from the simulations, magenta: scaling law with fitting parameters derived in
this work and cyan: scaling law from SL09). If $Q_{\textnormal{RD}}^{*}$ is
calculated from the scaling law with the fitting parameters presented in SL09
the values deviate by up to a factor of $\sim 2.5$. This introduces large
scatter in the data when normalising the specific impact energy to the value
derived from the scaling laws. Furthermore, catastrophic disruption occurs
already at $\sim 0.5$ $Q_{\textnormal{RD}}^{*}$ and for specific impact
energies larger than $Q_{\textnormal{RD}}^{*}$ no mass remains bound. If the
updated fitting parameters derived in the present work are used,
$Q_{\textnormal{RD}}^{*}$ agrees within $\leq$33\text{\,}\mathrm{\char
37\relax}$$ with the value derived from the simulations and the scatter in the
data is substantially reduced. In turn catastrophic and super-catastrophic
disruption are in much better agreement with simulation data.

Finally, we compare the values of $Q_{\textnormal{RD}}^{*}$ calculated from
the velocity-dependent catastrophic disruption criterion proposed in SL09
(based on dimensional analysis following Housen & Holsapple (1990, 1999)) to
the values inferred from our simulations. SL09 find that in the gravity
dominated regime

$Q_{\textnormal{RD}}^{*}=q_{g}R_{C1}^{3\mu}v_{\textnormal{imp}}^{2-3\mu}$ (8)

where $R_{C1}$ is the radius of a sphere containing the total colliding mass
with a mean density of $1\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$ with
$q_{g}=10^{-4}$ (in cgs units) and $\mu=0.4$ is in good agreement with their
simulation data. However, these results cannot be extrapolated to collisions
involving orders of magnitude more massive bodies and the calculated values of
$Q_{\textnormal{RD}}^{*}$ differ by a factor of $\sim 2.5$ from the values
derived from our simulations. Given the huge range of extrapolation this
disagreement is not surprising. When normalising $Q_{\textnormal{R}}$ from our
simulations to $Q_{\textnormal{RD}}^{*}$ calculated with these fitting
coefficients one introduces large scatter and shifts the transition to super-
catastrophic disruption to below
$Q_{\textnormal{R}}/Q_{\textnormal{RD}}^{*}=1$ (see Figure 3). This is clearly
unrealistic and results in large errors when deriving additional equations
based on these scaling laws. We therefore determine the fitting coefficients
from our simulation data. For $q_{g}=$1.4679\text{\times}{10}^{-06}\text{\,}$$
(cgs) and $\mu=$0.6164\text{\,}$$ the values of $Q_{\textnormal{RD}}^{*}$
calculated from Equation 8 are in very good agreement with our results and the
largest relative error is smaller than $33\text{\,}\mathrm{\char 37\relax}$.
This substantially reduces the scatter in Figure 3 and allows a reliable
prediction of $Q_{\textnormal{RD}}^{*}$ without requiring a large suite of
simulations.

Figure 4: The impact velocity versus the mass of the largest fragment for
different iron mass fractions predicted by scaling laws. Shown are our results
(solid lines) and the ones from Figure 1b in M10 (dashed lines). The target-
to-impactor mass ratio is fixed to $\gamma=1$ here, and the colours represent
different iron mass fractions (see legend). For a given fragment mass the
impact velocity required to obtain a desired iron mass fraction is
substantially higher than predicted by M10. In order to obtain a
$10\text{\,}\mathrm{M_{\oplus}}$ pure iron body an impact velocity of about
$300\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ corresponding to three
times the Keplerian velocity of a planet orbiting a Sun like star at
$0.1\text{\,}\mathrm{ua}$ is required thus making the collisional formation of
such bodies very challenging. Figure 5: The iron mass fraction versus the mass
of the largest fragment for different impact velocities and target-to-impactor
mass ratios predicted by the scaling laws derived in this work. The colours
represent different impact velocities and the line styles different target-to-
impactor mass ratios (see legend). The earlier transition from normal to
super-critical disruption behaviour observed in our simulations is affecting
the slope of the curves and results in a kink where the function is continuous
but not differentiable. We also plotted the results from M10 for an impact
velocity of $80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and $\gamma=1$.
The curve shown in Figure 2b of M10 (solid black line) under and over predicts
the iron mass fraction by $\sim$10\text{\,}\mathrm{\char 37\relax}$$ at small
and larger fragment masses respectively. Even if the curve is corrected for
typos and other errors (dot-dashed black line, see Appendix A for details) the
predicted iron mass fraction is very different from our results. This affects
the minimum radius of a planet due to collisional stripping of the mantle
derived in M10 (see Figure 6).

Figure 6: The inferred radius (top) and target mass (bottom) versus the mass
of the largest fragment for different impact velocities predicted by our
scaling laws. The colours represent different impact velocities and the black
curves (solid: iron mass fraction in Figure 2b of M10, dashed: corrected as
described in Appendix A) correspond to the minimum radius predicted in M10
assuming an impact velocity of
$80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. The impactor-to-target
mass ratio is one for all curves. The largest difference in radius between M10
and this work is $6.5\text{\,}\mathrm{\char 37\relax}$ which is close to the
maximum uncertainty in radius in Otegi et al. (2020). Future missions are
expected to substantially reduce the uncertainty in the planetary radius
making such differences relevant when interpreting observations. For a given
fragment mass higher impact velocities require higher initial target masses,
e.g., for an impact velocity of
$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ the target has to have a
mass between $40\text{\,}\mathrm{M_{\oplus}}$ and
$93\text{\,}\mathrm{M_{\oplus}}$. Once the equations of M10 are corrected for
all errors, our scaling laws require larger initial target masses than
predicted in M10 due to the stronger disruption of the bodies found in our
simulations.

Figure 7: The post-impact iron mass fraction for different fragment masses.
The circles represent the simulation data and each colour corresponds to the
target’s pre-impact mass (violet: $10\text{\,}\mathrm{M_{\oplus}}$, cyan:
$16.55\text{\,}\mathrm{M_{\oplus}}$ and red:
$19.04\text{\,}\mathrm{M_{\oplus}}$). The most iron-rich and massive fragments
are obtained from collision onto a $19.04\text{\,}\mathrm{M_{\oplus}}$ target
so in order to obtain a metal-rich, massive planet large initial (pre-impact)
masses are required. The dashed and dotted grey curves correspond to the
original and corrected maximum iron content due to collisional stripping of a
planet’s mantle derived in M10 (see Appendix A for details). The shaded grey
area mark all possible outcomes for a restricted impact parameter space (dark
grey: $$0.1\text{\,}\mathrm{M_{\oplus}}$\leq
M_{\textnormal{tar}}\leq$10\text{\,}\mathrm{M_{\oplus}}$$ and
$$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$\leq
v_{\textnormal{imp}}\leq$100\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$,
light grey: $$0.1\text{\,}\mathrm{M_{\oplus}}$\leq
M_{\textnormal{tar}}\leq$20\text{\,}\mathrm{M_{\oplus}}$$ and
$$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$\leq
v_{\textnormal{imp}}\leq$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$
both with $\gamma=1$). The iron-rich exoplanets’s (see legend for details)
compositions are derived from interior modelling as described in Section 2.2
and are in good agreement with masses and compositions obtained from
simulations. Their large iron mass fractions therefore could be the result of
a GI that stripped a fraction of the mantle. The black marker shows the
composition of a hypothetical super-Mercury with a mass of
$1\text{\,}\mathrm{M_{\oplus}}$ and $Z_{\textnormal{Fe}}=0.7$ and
uncertainties of $10\text{\,}\mathrm{\char 37\relax}$ and
$3\text{\,}\mathrm{\char 37\relax}$ on the mass and radius, respectively, to
forecast the expected accuracy of future observations

.

Figure 8: The post-impact radius for different fragment masses corresponding
to the iron mass fractions shown in Figure 7. The circles represent the
simulation data and each colour corresponds to the target’s pre-impact mass
(violet: $10\text{\,}\mathrm{M_{\oplus}}$, cyan:
$16.55\text{\,}\mathrm{M_{\oplus}}$ and red:
$19.04\text{\,}\mathrm{M_{\oplus}}$). The dashed grey lines present a pure-
rock (top curve) and pure-iron (bottom curve) body. The pre-impact composition
is shown in black (dashed line) and the dashed red line corresponds to
$Z_{\textnormal{Fe}}=0.7$, i.e., a Mercury-like composition. The shaded grey
regions mark different impact parameter spaces (dark grey:
$$0.1\text{\,}\mathrm{M_{\oplus}}$\leq
M_{\textnormal{tar}}\leq$10\text{\,}\mathrm{M_{\oplus}}$$ and
$$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$\leq
v_{\textnormal{imp}}\leq$100\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$,
light grey: $$0.1\text{\,}\mathrm{M_{\oplus}}$\leq
M_{\textnormal{tar}}\leq$20\text{\,}\mathrm{M_{\oplus}}$$ and
$$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$\leq
v_{\textnormal{imp}}\leq$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$
both with $\gamma=1$). Shown are also the masses and radii of metal-rich
exoplanets which are all in good agreement with our simulation data. The black
marker shows a hypothetical super-Mercury with a mass of
$1\text{\,}\mathrm{M_{\oplus}}$ and $Z_{\textnormal{Fe}}=0.7$ and
uncertainties of $10\text{\,}\mathrm{\char 37\relax}$ and
$3\text{\,}\mathrm{\char 37\relax}$ on the mass and radius respectively to
forecast future observations.

###  3.3 Derived scaling laws

The above scaling laws (Equation (4), (6) and (8)) can be combined in a
similar way as in M10 to obtain additional scaling laws. Due to the more
complex nature of our fitting functions it is more convenient to use a
numerical root finder (SciPy: Virtanen et al. (2020)) in place of an analytic
solution as in M10. The resulting scaling laws allow one to predict the
conditions, i.e., impact velocity, impactor-to-target mass ratio and total
colliding mass, required to obtain a certain post-impact fragment mass and
composition without performing a large suite of simulations.

Figure 4 shows the impact velocity required to obtain a desired fragment mass
and composition for our scaling laws as well as the ones presented in M10. The
impactor-to-target mass ratio is fixed to $\gamma=1$ as such collisions strip
the most material for a given impact velocity and total colliding mass and
therefore provide a lower limit on the required relative velocities to obtain
a given fragment mass and composition. As expected the required impact
velocity increases with fragment mass and iron mass fraction. For example,
obtaining a fragment of $1\text{\,}\mathrm{M_{\oplus}}$ with a Mercury-like
composition, i.e., $Z_{\textnormal{Fe}}=0.7$, requires an impact velocity of
$44.14\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. This is comparable to
the orbital velocity at Mercury’s location. Generally, the impact velocities
predicted by our scaling laws are substantially larger than the ones in M10.
We also find that for large iron mass fractions the slopes of our curves are
steeper, which means that for higher fragment masses, even larger impact
velocities are required. The former is to some extent a result of the
different fitting functions as well as the errors made in M10 when deriving
the equations (see Appendix A for details). Furthermore, as shown in Section
3.2.3 the fitting coefficients from SL09 used to estimate
$Q_{\textnormal{RD}}^{*}$ in M10 are derived for low mass collisions and are
in poor agreement with values derived from high mass collision simulations.
The normalised specific impact energy required for super-catastrophic
disruption is therefore severely underestimated which in turn means that much
lower impact velocities are needed to obtain a desired fragment mass and
composition. The latter is a result of the simulations presented in M09 over-
estimating the bound mass compared to our results and therefore under-
estimating the iron mass fraction for a given specific impact energy. By pure
coincidence, the curves for $Z_{\textnormal{Fe}}=0.90$ from our simulations
and $Z_{\textnormal{Fe}}=0.99$ in M10 (their Figure 1b) are almost overlapping
(see Figure 4).

Figure 5 shows the iron mass fraction for a given fragment mass for different
impact velocities and impactor-to-target mass ratios. To compare with M10 we
also added the curve for $\gamma=1$ and an impact velocity of
$80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ from their paper (both as
shown in the paper and the correct version, see Appendix A for details) which
is used to determine the maximum iron mass fraction of an exoplanet due to
collisional stripping of the mantle. For a given impact velocity and fragment
mass larger impactor-to-target mass ratios result in more iron-rich bodies due
to the higher specific impact energy. Likewise increasing the impact velocity
for a given mass ratio $\gamma$ and fragment mass increases the iron mass
fraction. The normalised specific impact energy $q_{\textnormal{sc}}$ where
the fragment mass begins to abruptly decrease following a power law (see
Equation 4) results in a kink where the function is continuous but not
differentiable. Furthermore, the results of M10 severely over-predict the iron
mass fraction at large fragment masses. This is most pronounced for the curve
as shown in Figure 2b of their paper. At the same time the iron mass fraction
at small fragment masses is substantially smaller than in our simulations. If
one corrects for the mistakes made when deriving the equation in M10 the curve
is in better agreement with our results at large fragment masses but predicts
an unphysically large iron content, i.e., $Z_{\textnormal{Fe}}>1$, at small
masses. Due to the correct extrapolation to $Z_{\textnormal{Fe}}=1$ in
Equation 6 this is not the case for the scaling laws derived in this paper.

The fragment mass and composition obtained from these scaling laws can be used
to estimate the radius after a collision from interior modelling as described
in Section 2.2. The top panel of Figure 6 is an M-R diagram showing the radius
versus fragment mass for different impact velocities assuming the fragment had
time to cool to a surface temperature of $1000\text{\,}\mathrm{K}$ after the
collision. We consider fragment masses between
$0.3\text{\,}\mathrm{M_{\oplus}}$ and $40\text{\,}\mathrm{M_{\oplus}}$ and the
impact velocities vary between
$40\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and
$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. Again, the impactor-to-
target mass ratio is $1$; correspondingly the curves shown in the figure have
to be considered as a lower limit on the radius (see Section 3.2.2 for
details). For a given fragment mass, larger impact velocities result in
smaller radii since the remaining fragment contains more iron and is therefore
more compact. The difference in radius between an impact velocity of
$40\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and
$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ ranges between $8$ and
$21$ percent and is most pronounced at a mass of
$8\text{\,}\mathrm{M_{\oplus}}$.

The cyan curve shows the radius of a fragment obtained at an impact velocity
of $80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. This corresponds to the
upper limit M10 assumed on the possible relative velocity when deriving the
minimum radius of a fragment (and therefore an iron-rich exoplanet) due to
collisional stripping. We also calculated the fragment radius for the masses
and iron mass fractions obtained from the scaling laws presented in M10 (solid
black line: including all mistakes, dashed black line: corrected equation) at
identical impact conditions. This allows a direct comparison between the two
studies and how differences could affect implications made from observations.
As expected from Figure 5 the mass-radius curve inferred from the compositions
shown in Figure 2b of M10 overestimates the fragment radius at lower masses
and underestimates it at larger masses compared to our work. If the scaling
laws presented in M10 are corrected there is better overall agreement to our
work. However, for masses below $0.46\text{\,}\mathrm{M_{\oplus}}$ the iron
mass fraction is larger than $1$ making the validity of these results at lower
masses questionable. Generally, the radii inferred from our scaling laws agree
with those presented in M10 within $6.5\text{\,}\mathrm{\char 37\relax}$.
Given that future missions like PLATO (Rauer et al., 2014) are expected to
reduce uncertainty in the measured planetary radius below three to five
percent these differences are relevant for validating the effect of GI on
their composition. Another aspect when deriving a fragment mass and
composition for given impact conditions that has received less attention so
far is the total colliding mass involved. The bottom panel of Figure 6 shows
the (pre-impact) mass of the target for a given fragment mass and radius. As
expected for a given fragment mass the required target mass increases with
increasing impact velocity. In the most extreme case, for an impact velocity
of $160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, the target mass varies
between $40\text{\,}\mathrm{M_{\oplus}}$ and $93\text{\,}\mathrm{M_{\oplus}}$.

Clearly, defining a minimum radius (or maximum iron mass fraction) of an
exoplanet due to collisional stripping of its mantle based solely on the
impact velocity as proposed in M10 is incomplete as it does not account for
unrealistically large target and impactor masses that can be required to
obtain very massive super-Mercuries. It is therefore desirable to account for
the total colliding mass in addition to the impact velocity. Given the range
in architectures of exoplanetary systems and their host stars, realistic
impact conditions can vary greatly; a suitable criterion accounts for this
diversity.

###  3.4 Implications for iron-rich exoplanets

In Figure 7 we present the composition of the iron-rich exoplanets presented
in Table 1 and the post-impact fragments obtained in our simulations. The
exoplanets’ compositions are inferred from interior modelling as described in
Section 2.2. The grey lines represent the maximum iron mass fraction predicted
by M10 in Figure 2b and by the correct equations as described in the Appendix
A. The shaded grey areas show the possible compositions predicted by our
scaling laws for equal-mass collisions with
$$0.1\text{\,}\mathrm{M_{\oplus}}$\leq
M_{\textnormal{tar}}\leq$10\text{\,}\mathrm{M_{\oplus}}$$ and
$$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$\leq
v_{\textnormal{imp}}\leq$100\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$
(dark grey) and $$0.1\text{\,}\mathrm{M_{\oplus}}$\leq
M_{\textnormal{tar}}\leq$20\text{\,}\mathrm{M_{\oplus}}$$ and
$$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$\leq
v_{\textnormal{imp}}\leq$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$
(light grey). The curves that envelope this region are the minimum and maximum
iron mass fraction for these impact conditions. Because the minimum radius
curves derived in M10 do not depend on the total colliding mass they over-
predict the iron mass fractions for larger fragment masses compared to this
work and give unrealistically high values. For small fragment masses on the
other hand the iron mass fraction predicted by M10 is lower than in our work
because the impact velocities in M10 are limited to
$80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. We compare our results
with observations in the M-R diagram presented in Figure 8. The measured
masses and radii of the iron-rich exoplanets are taken from Otegi et al.
(2020) and are listed in Table 1. Also presented are the radii of the
fragments obtained in our simulations as well as the minimum and maximum
radius of the fragments inferred from our scaling laws. The radii are inferred
from the interiors models presented in section 2.2. The dashed curves show
different compositions ranging from pure-rock to pure-iron planets.

The inferred exoplanets’ compositions fall within the range of the predicted
compositions based on our impact simulations. Furthermore, our scaling laws
show that five out of nine exoplanets have a composition that is consistent
with mantle stripping involving a target mass of
$\leq$10\text{\,}\mathrm{M_{\oplus}}$$ and impact velocities below
$100\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. The other four agree with
these impact conditions within the compositional uncertainly but likely
require a target mass of $\leq$20\text{\,}\mathrm{M_{\oplus}}$$ and impact
velocities $\leq$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$ to
explain their larger masses and extreme compositions with GI. This suggests
that in order to further constrain the impact conditions more accurate
observational data are required.

Since all of the iron-rich exoplanets considered in this work are orbiting
very close to their host star and therefore have orbital velocities between
$130\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and
$260\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ the limiting impact
velocity of $80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ assumed in M10
is too restrictive. Even velocities up to
$160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ seem plausible. It is
possible that these metal-rich exoplanets formed farther out and migrated
inwards. If these planets formed in resonant chains (Izidoro et al., 2017)
such an instability can lead to high-velocity encounters and even larger
impact velocities (e.g., Scora et al. 2020; Clement et al. 2021). Therefore,
exoplanets with extreme iron mass fractions could be a few lonely survivors,
as has been suggested for Mercury (Clement et al., 2021). This hypothesis is
consistent with the fact that the iron-rich exoplanets in our sample are
single planets or correspond to the innermost planet in systems with a small
number of planets with significant distancing between them. Two exceptions are
the systems Kepler-80 and Kepler-107 that are rather compact and near
resonance.

However, a target mass of $\sim$20\text{\,}\mathrm{M_{\oplus}}$$ is more
difficult to explain. While the formation of such massive super-Earths may not
be excluded a priori, rather specific conditions are required. First, enough
mass in the form of refractory elements is required at the location where such
planets form. Second, once a body becomes so massive it is expected to accrete
a substantial layer of hydrogen (H) and helium (He) from the proto-planetary
disk and therefore have a composition similar to a giant planet rather than a
super-Earth. Due to the close proximity to the host star and the corresponding
high temperatures and stellar irradiation a fraction of this envelope could
have been lost over time (e.g., Owen 2019). Indeed, in some cases we find that
a massive iron-rich body has a sub-Neptune companion. If still present at the
time of the collision, a primordial H-He envelope can also be lost during the
mantle stripping GI (e.g., Kegerreis et al. 2020). However, the presence of a
massive envelope would increase the planetary mass and therefore more material
must be stripped in a given collision. As a consequence, higher impactor
masses and impact velocities are required to obtain a desired composition.

While the close proximity of the iron-rich exoplanets in our sample to their
host stars promote disruptive collisions, the corresponding short orbital
periods imply that part of the ejected debris could be re-accreted rapidly.
For Mercury, Benz et al. (2007) suggested that the ejecta could be in the form
of small droplets that can be swept away via Poynting-Robertson drag (Benz et
al., 2007) or Solar winds (Spalding & Adams, 2020). Furthermore, since the
observed exoplanets orbit close to their stars, their Hill radii are very
small and tidal forces could play an important role in preventing re-accretion
of ejecta.

Finally, the initial iron mass fraction (pre-impact) can vary, e.g., if the
host star is enriched in iron as proposed in Adibekyan et al. (2021). For
impact energies below the transition to rapid mass loss the bound mass is not
affected by the initial iron mass fraction (see Section 3.1 for details).
Likewise, the post-impact composition in this regime shows a similar behaviour
as the initial iron mass fraction enters Equation 6 as a constant. Generally,
for a higher initial iron mass fraction less material has to be removed in the
collision. Therefore, a lower total colliding mass and impact velocity are
required to obtain a desired iron mass fraction which facilitates the
formation of very massive super-Mercuries. Since the transition to rapid mass
loss shows some dependency of the initial iron mass fraction, it would be
desirable to investigate a range of different core mass fractions in future
studies.

##  4 Summary and conclusions

In this paper we present a large suite of SPH simulations of impacts between
super-Earths in order to investigate the efficiency of mantle stripping in GI.
Motivated by the observation of super-Mercuries with masses
$>$5\text{\,}\mathrm{M_{\oplus}}$$ we study head-on impacts between super-
Earths with masses between $1\text{\,}\mathrm{M_{\oplus}}$ and
$19.04\text{\,}\mathrm{M_{\oplus}}$ and impact velocities between two to six
times the mutual escape velocity. From these simulations we determine the
bound mass, which corresponds to the post-impact planetary mass, as well as
the iron mass fraction and radius. For each combination of target and impactor
mass we also calculate the critical specific impact energy for catastrophic
disruption $Q_{\textnormal{RD}}^{*}$. We then propose new scaling laws for the
post-impact mass and composition as well as $Q_{\textnormal{RD}}^{*}$ that
more accurately represent our data than prior work that investigated lower
mass bodies (SL09 and M09). Generally we find that at low specific impact
energies more mass remains bound and the iron mass fraction therefore remains
lower than predicted in M09. In the case of collisions occurring at specific
impact energies above $Q_{\textnormal{RD}}^{*}$ more mass is stripped and
correspondingly the iron mass fraction is higher than expected from M09.

We then combine these scaling laws to obtain equations that relate the impact
conditions, e.g., target mass, impactor-to-target mass ratio and impact
velocity, to the post-impact mass and composition of a planet. These equations
allow one to predict collision outcomes for different parameters without the
need of performing a large suite of impact simulations. We find that obtaining
a desired fragment mass and composition requires higher impact velocities than
predicted in M10. Furthermore, for a given impact velocity we predict higher
iron mass fractions for low fragment masses and lower iron mass fractions for
high fragment masses compared to M10. This is reflected in the minimum radius
of a super-Earth where we observe differences to M10 of up to
$6.5\text{\,}\mathrm{\char 37\relax}$. Additionally, we find that M10 predicts
an iron mass fraction larger than $1$ for smaller masses because the scaling
law for the iron mass fraction presented in M09 does not correctly extrapolate
to $1$ for very large impact energies. Considering that future missions are
expected to reduce the uncertainty in observed planetary radius to
$35\text{\,}\mathrm{\char 37\relax}$ these differences are relevant when
interpreting the origin of iron-rich planets. Finally, we find several errors
in the equations of M10 (see Appendix A for details) that can result in
differences of eight percent in the predicted planetary radius, once
corrected, and therefore are comparable to the uncertainty of current
observations. Future work should use our new results, or at the very least,
include the corrections to M10 presented in Appendix A.

In order to connect our findings to observations we compare the post-impact
bodies found in our simulations to a sample of iron-rich exoplanets and find
that they are in good agreement. A giant impact occurring in these exoplanets’
early history therefore could indeed explain their large iron mass fractions.
From the scaling laws derived in this work we also mark the outcome of
collisions for different ranges of pre-impact mass and impact velocity by
deriving a minimum and maximum radius and iron mass fraction respectively.
This allows to further constrain possible impact conditions required to obtain
these exoplanets’ compositions. We find that five out of nine exoplanets in
our sample are consistent with impacts involving a
$<$10\text{\,}\mathrm{M_{\oplus}}$$ target and impact velocities below
$100\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. The other four are
constrained by target masses $<$20\text{\,}\mathrm{M_{\oplus}}$$ and impact
velocities below $160\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$.

Finally, we propose the minimum and maximum radius for a rocky exoplanet of a
given mass due to collisional stripping of its mantle that accounts for both
the required impact velocities and total colliding mass. Between the two
curves are all possible compositions and radii of exoplanets due to mantle
stripping GI for given constraints on the impact conditions. Due to the
diverse architecture of exoplanetary systems and their host stars constraints
on the impact conditions can vary substantially. Therefore, we provide python
scripts (see data availability statement) that allow users to choose their
desired minimum and maximum impact velocity and total colliding mass
respectively, allowing one to constrain possible impact conditions for a broad
variety of observed exoplanetary systems.

Our main conclusions can be summarised as follows:

  * •

For collisions in the super-critical disruption regime
($Q_{\textnormal{R}}>1.26\leavevmode\nobreak\ Q_{\textnormal{RD}}^{*}$)
details of the EOS, in particular the treatment of the liquid phase, are
crucial for investigating mantle stripping GI.

  * •

Prior work has overestimated the mantle stripping of large rocky exoplanets at
lower specific impact energies, while underestimating this at higher energies.

  * •

Impacts occurring at specific impact energies below 0.5
$Q_{\textnormal{RD}}^{*}$ have little effect on a planet’s mass and iron mass
fraction.

  * •

The formation of very massive super-Mercuries requires both high impact
velocities as well as very massive pre-impact bodies. The existence of such
planets implies very specific formation conditions, and hence we expect such
planets to remain rare.

  * •

The observed masses and inferred compositions of super-Mercuries are
consistent with our simulation results implying that GI play an important role
during their formation.

  * •

The new M-R curves (for the software see data availability statement) allow
one to bracket the initial mass and velocity allowed for the GI formation of
rocky exoplanets.

  * •

Break-up of resonant chains may result in architectures favourable for the
formation of super-Mercury exoplanetary systems (close proximity to host star,
high relative velocities and large masses).

Our understanding of super-Mercuries is still incomplete and is limited by the
small number of observed metal-rich exoplanets and the data accuracy. Ongoing
and future space missions like TESS and PLATO will substantially reduce the
uncertainty of observations on mass and radius which gives tighter constraints
on the composition of observed exoplanets. This will allow to better
distinguish super-Mercuries from super-Earths and to assess the occurrence
rate of such planets in the galaxy. In combination with more accurate data
regarding the stellar properties and architecture of systems hosting super-
Mercuries this will tighten the constraints on possible formation scenario and
improve our understanding of the role of giant impacts.

## Acknowledgements

We thank John Chambers for valuable suggestions and comments that helped to
substantially improve the paper. We also thank Caroline Dorn for helpful
discussions. This work has been carried out within the framework of the
National Centre of Competence in Research PlanetS supported by the Swiss
National Science Foundation. The authors acknowledge the financial support of
the SNSF. We acknowledge access to Piz Daint or Eiger@Alps at the Swiss
National Supercomputing Centre, Switzerland under the University of Zurich’s
share with the project ID UZH4. This publication makes use of The Data &
Analysis Center for Exoplanets (DACE), which is a facility based at the
University of Geneva (CH) dedicated to extrasolar planets data visualisation,
exchange and analysis. DACE is a platform of the Swiss National Centre of
Competence in Research (NCCR) PlanetS, federating the Swiss expertise in
Exoplanet research. The DACE platform is available at https://dace.unige.ch.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author. The scaling laws based on this data are implemented in
python scripts that are available at Zenodo
https://doi.org/10.5281/zenodo.6592438.

## Software

In the present work we used the following software: Gasoline (Wadsley et al.,
2004), ballic (Reinhardt & Stadel, 2017), EOSlib (Meier & Reinhardt, 2021a,
b), M-ANEOS (Thompson et al., 2019; Stewart et al., 2019; Stewart, 2020),
skid333https://github.com/N-BodyShop/skid, numpy (Harris et al., 2020), scipy
(Virtanen et al., 2020), matplotlib (Hunter, 2007)

## References

  * Adibekyan et al. (2021) Adibekyan V., et al., 2021, Science, 374, 330
  * Aguichine et al. (2020) Aguichine A., Mousis O., Devouard B., Ronnet T., 2020, The Astrophysical Journal, 901, 97
  * Asphaug & Reufer (2014) Asphaug E., Reufer A., 2014, Nature Geoscience, 7, 564
  * Astudillo-Defru et al. (2020) Astudillo-Defru N., et al., 2020, Astronomy & Astrophysics, 636, A58
  * Batalha et al. (2011) Batalha N. M., et al., 2011, The Astrophysical Journal, 729, 27
  * Benz et al. (2007) Benz W., Anic A., Horner J., Whitby J. A., 2007, Space Science Reviews, 132, 189
  * Bonomo et al. (2019) Bonomo A. S., et al., 2019, Nature Astronomy, 3, 416
  * Bryson et al. (2020) Bryson S., et al., 2020, The Astronomical Journal, 161, 36
  * Chau et al. (2018) Chau A., Reinhardt C., Helled R., Stadel J., 2018, The Astrophysical Journal, 865, 35
  * Clement et al. (2021) Clement M. S., Chambers J. E., Jackson A. P., 2021, The Astronomical Journal, 161, 240
  * Connolly (2009) Connolly J. A. D., 2009, Geochemistry, Geophysics, Geosystems, 10, Q10014
  * Davies et al. (2020) Davies E. J., Carter P. J., Root S., Kraus R. G., Spaulding D. K., Stewart S. T., Jacobsen S. B., 2020, Journal of Geophysical Research: Planets, 125, e2019JE006227
  * Guenther et al. (2017) Guenther E. W., et al., 2017, Astronomy & Astrophysics, 608, A93
  * Hadden & Lithwick (2017) Hadden S., Lithwick Y., 2017, The Astronomical Journal, 154, 5
  * Hakim et al. (2018) Hakim K., Rivoldini A., Van Hoolst T., Cottenier S., Jaeken J., Chust T., Steinle-Neumann G., 2018, Icarus, 313, 61
  * Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357
  * Holman et al. (2010) Holman M. J., et al., 2010, Science, 330, 51
  * Housen & Holsapple (1990) Housen K. R., Holsapple K. A., 1990, Icarus, 84, 226
  * Housen & Holsapple (1999) Housen K. R., Holsapple K. A., 1999, Icarus, 142, 21
  * Hunter (2007) Hunter J. D., 2007, Computing in Science Engineering, 9, 90
  * Ito & Ikoma (2021) Ito Y., Ikoma M., 2021, Monthly Notices of the Royal Astronomical Society, 502, 750
  * Izidoro et al. (2017) Izidoro A., Ogihara M., Raymond S. N., Morbidelli A., Pierens A., Bitsch B., Cossou C., Hersant F., 2017, Monthly Notices of the Royal Astronomical Society, 470, 1750
  * Kegerreis et al. (2020) Kegerreis J. A., Eke V. R., Catling D. C., Massey R. J., Teodoro L. F. A., Zahnle K. J., 2020, The Astrophysical Journal, 901, L31
  * Kosiarek et al. (2019) Kosiarek M. R., et al., 2019, The Astronomical Journal, 157, 116
  * Lam et al. (2021) Lam K. W. F., et al., 2021, Science, 374, 1271
  * Leinhardt & Stewart (2012) Leinhardt Z. M., Stewart S. T., 2012, The Astrophysical Journal, 745, 79
  * Lewis (1972) Lewis J. S., 1972, Earth and Planetary Science Letters, 15, 286
  * Lopez & Fortney (2013) Lopez E. D., Fortney J. J., 2013, The Astrophysical Journal, 776, 2
  * Lopez & Fortney (2014) Lopez E. D., Fortney J. J., 2014, The Astrophysical Journal, 792, 1
  * Marcus et al. (2009) Marcus R. A., Stewart S. T., Sasselov D., Hernquist L., 2009, The Astrophysical Journal Letters, 700, L118
  * Marcus et al. (2010) Marcus R. A., Sasselov D., Hernquist L., Stewart S. T., 2010, The Astrophysical Journal, 712, L73
  * Marcy et al. (2014) Marcy G. W., et al., 2014, The Astrophysical Journal Supplement Series, 210, 20
  * Mayor & Queloz (1995) Mayor M., Queloz D., 1995, Nature, 378, 355
  * Meier & Reinhardt (2021a) Meier T., Reinhardt C., 2021a, ANEOSmaterial, Zenodo, doi:10.5281/zenodo.4705185
  * Meier & Reinhardt (2021b) Meier T., Reinhardt C., 2021b, EOSlib, Zenodo, doi:10.5281/zenodo.4704950
  * Meier et al. (2021) Meier T., Reinhardt C., Stadel J. G., 2021, Monthly Notices of the Royal Astronomical Society, 505, 1806
  * Melosh (2007) Melosh H. J., 2007, Meteoritics and Planetary Science, 42, 2079
  * Otegi et al. (2020) Otegi J. F., Bouchy F., Helled R., 2020, Astronomy & Astrophysics, 634, A43
  * Owen (2019) Owen J. E., 2019, Annual Review of Earth and Planetary Sciences, 47, 67
  * Petigura et al. (2013) Petigura E. A., Howard A. W., Marcy G. W., 2013, Proceedings of the National Academy of Sciences, 110, 19273
  * Rauer et al. (2014) Rauer H., et al., 2014, Experimental Astronomy, 38, 249
  * Reinhardt & Stadel (2017) Reinhardt C., Stadel J., 2017, Monthly Notices of the Royal Astronomical Society, 467, 4252
  * Reinhardt et al. (2020) Reinhardt C., Chau A., Stadel J., Helled R., 2020, Monthly Notices of the Royal Astronomical Society, 492, 5336
  * Rogers (2015) Rogers L. A., 2015, The Astrophysical Journal, 801, 41
  * Santerne et al. (2018) Santerne A., et al., 2018, Nature Astronomy, 2, 393
  * Schulze et al. (2021) Schulze J. G., Wang J., Johnson J. A., Gaudi B. S., Unterborn C. T., Panero W. R., 2021, The Planetary Science Journal, 2, 113
  * Scora et al. (2020) Scora J., Valencia D., Morbidelli A., Jacobson S., 2020, Monthly Notices of the Royal Astronomical Society, 493, 4910
  * Spalding & Adams (2020) Spalding C., Adams F. C., 2020, The Planetary Science Journal, 1, 7
  * Stewart (2020) Stewart S. T., 2020, Equation of State Model Iron ANEOS: Documentation and Comparisons (Version SLVTv0.2G1), Zenodo, doi:10.5281/zenodo.3866507
  * Stewart & Leinhardt (2009) Stewart S. T., Leinhardt Z. M., 2009, The Astrophysical Journal Letters, 691, L133
  * Stewart et al. (2019) Stewart S. T., et al., 2019, Equation of State Model Forsterite-ANEOS-SLVTv1.0G1: Documentation and Comparisons, Zenodo, doi:10.5281/zenodo.3478631
  * Stewart et al. (2020) Stewart S., et al., 2020, AIP Conference Proceedings, 2272, 080003
  * Stixrude & Lithgow-Bertelloni (2011) Stixrude L., Lithgow-Bertelloni C., 2011, Geophysical Journal International, 184, 1180
  * Thompson & Lauson (1974) Thompson S. L., Lauson H. S., 1974, Technical Report SC-RR-71-0714, Improvements in the CHART D Radiation-Hydrodynamic Code III: Revised Analytic Equations of State. Sandia Labs., Albuquerque, N.Mex. (USA)
  * Thompson et al. (2019) Thompson S. L., Lauson H. S., Melosh H. J., Collins G. S., Stewart S. T., 2019, M-ANEOS, Zenodo, doi:10.5281/zenodo.3525030
  * Timpe et al. (2020) Timpe M. L., Han Veiga M., Knabenhans M., Stadel J., Marelli S., 2020, Computational Astrophysics and Cosmology, 7, 2
  * Toledo-Padrón et al. (2020) Toledo-Padrón B., et al., 2020, Astronomy & Astrophysics, 641, A92
  * Valencia et al. (2007) Valencia D., Sasselov D. D., O’Connell R. J., 2007, The Astrophysical Journal, 665, 1413
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Wadsley et al. (2004) Wadsley J. W., Stadel J., Quinn T., 2004, New Astronomy, 9, 137
  * Wurm et al. (2013) Wurm G., Trieloff M., Rauer H., 2013, The Astrophysical Journal, 769, 78

##  Appendix A Discussion of the results from M10

Figure 9: The effect of the errors in the equations of M10 on their results.
The solid lines are the curves shown in Figure 1b and 2b of M10. The dashed
lines are the results obtained from the corrected equations derived from the
original scaling laws in M09. The top panel shows the impact velocity required
to obtain a desired fragment mass with an iron mass fraction of $0.7$ for
different impactor-to-target mass ratios $\gamma$ (purple: $\gamma=0.1$, cyan:
$\gamma=0.25$, yellow: $\gamma=0.5$ and red: $1.0$). The mistake in the
prefactor (see Appendix A for details) results in a severe underestimation of
the impact velocity required to produce a Mercury-like planet via collisional
stripping of the mantle. The bottom panel shows the iron mass fraction of the
largest fragment for an impact velocity of
$80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and different impactor-to-
target mass ratios (purple: $\gamma=0.1$, light green: $\gamma=0.25$ and red:
$\gamma=1.0$). Compared to the curve shown in Figure 2b of M10 the corrected
equations predict a lower iron mass fraction except for equal mass collisions
where it is higher for small fragment masses.

In M10 equations for the impact velocity required to produce a given fragment
mass and composition due to collisional stripping are derived by combining the
scaling laws presented in SL09 and M09. Trying to compare their results to our
work we find several typos and calculation errors that make it difficult to
reproduce the figures 1b and 2b in M10. First, we find that the prefactor of
$10.5$ in Equation 1 of M10 is wrong and should be $18.75$ because they did
not account for a factor $0.5$ that enters the equation from the kinetic
energy term. The impact velocities in Figure 1 of their paper are therefore
underestimated by approximately a factor of two. Second, the curves in Figure
2 of the paper can only be reproduced if the exponent in Equation 2 (in M10)
is $0.2606$ rather than $0.606$ (or $1/1.65$) as written in the paper. It
seems that M10 made a simple typo when calculating these curves for the plots.
Finally, many prefactors and exponents require higher precision than given in
the paper. Most importantly, the prefactor $-1.2$ in Equation 2 should be
$-1.1584$ in order to accurately reproduce the curves in M10.

These mistakes have a profound impact on the results of M10. In Figure 9 we
compare the curves obtained using the correct prefactors and exponents to the
ones shown in Figure 1b and 2b of M10. As mentioned above the impact velocity
required to obtain a desired fragment mass with an iron mass fraction of
Z${}_{\textnormal{Fe}}=0.7$ via collisional stripping of the planet’s mantle
is about twice as large as suggested in Figure 1b of M10. Likewise the iron
mass fraction of the largest fragment resulting from a collision at an impact
velocity of $80\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ is very
different once the correct equations are used. For example, in case of an
equal mass collision the iron mass fraction is $0.9$ rather than $0.84$ for a
fragment mass of $1\text{\,}\mathrm{M_{\oplus}}$ and $0.66$ rather than $0.58$
for a fragment mass of $10\text{\,}\mathrm{M_{\oplus}}$. This in turn affects
the fragment radius as well as the total colliding mass required to obtain a
given composition by collisional stripping (see Figure 6). The difference in
radius between the correct and the published results of M10 are
$\leq$8.5\text{\,}\mathrm{\char 37\relax}$$ which is comparable to the
uncertainty of current observations. Furthermore, the total colliding mass
predicted by the published results is massively overestimated compared to the
corrected equations as well as our results. Given the popularity and
importance of the minimum radius of super-Earth presented in M10 in
interpreting observational data (here some of the most cited papers: Holman et
al. 2010; Batalha et al. 2011; Lopez & Fortney 2013, 2014; Rogers 2015) we
suggest taking these corrections into consideration in future work.