# Atmosphere Loss in Oblique Super-Earth Collisions

Thomas R. Denman1, Zoë M. Leinhardt1 and Philip J. Carter1,2
1School of Physics, H.H. Wills Physics Laboratory, University of Bristol,
Bristol BS8 1TL, UK
2Department of Earth and Planetary Sciences, University of California, Davis,
One Shields Avenue, Davis, CA, 95616, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form 2021 ???)

###### Abstract

Using smoothed particle hydrodynamics we model giant impacts of Super-Earth
mass rocky planets between an atmosphere-less projectile and an atmosphere-
rich target. In this work we present results from head-on to grazing
collisions. The results of the simulations fall into two broad categories: 1)
one main post-collision remnant containing material from target and
projectile; 2) two main post-collision remnants resulting from ‘erosive hit-
and-run’ collisions. All collisions removed at least some of the target
atmosphere, in contrast to the idealised hit-and-run definition in which the
target mass is unchanged. We find that the boundary between ‘hit-and-run’
collisions and collisions that result in the projectile and target
accreting/merging to be strongly correlated with the mutual escape velocity at
the predicted point of closest approach. Our work shows that it is very
unlikely for a single giant impact to remove all of the atmosphere. For all
the atmosphere to be removed, head-on impacts require roughly the energy of
catastrophic disruption (i.e. permanent ejection of half the total system
mass) and result in significant erosion of the mantle. We show that higher
impact angle collisions, which are more common, are less efficient at
atmosphere removal than head-on collisions. Therefore, single collisions that
remove all the atmosphere without substantially disrupting the planet are not
expected during planet formation.

###### keywords:

Planetary systems – planets and satellites: atmospheres – planets and
satellites: dynamical evolution and stability – planets and satellites:
formation – methods: numerical

††pubyear: 2021††pagerange: Atmosphere Loss in Oblique Super-Earth
Collisions–F

##  1 Introduction

Both observations and numerical simulations indicate that planet formation
often results in multiple planet systems. Gravitational interactions with the
proto-planetary disc cause planets embedded within to migrate. This migration
is typically inward and is dependent on planet mass (Ward, 1986). In systems
with multiple Super-Earth mass planets this process forms resonant chains at
the inner edge of the disc. The large rocky cores of these Super-Earths can
accrete significant primordial atmospheres from the proto-planetary disc
(Rogers et al., 2011).

When the proto-planetary disc dissipates it ceases to provide the drag force
that circularises the orbits of the planets. Small orbital perturbations can
destabilise any resonant chain of close orbiting Super-Earths that may have
formed within the disc (Barnes & Raymond, 2004) as they are often already on
the borders of stability (Fang & Margot, 2013). Eventually, the instability
will lead to collisions between the planets or gravitational ejection of a
planet from the system. As such, the systems we observe with Super-Earth mass
planets typically only have a few planets. Observed Super-Earths in multiple
planet systems are often detected orbiting close to, but not in, mean motion
resonance (Fabrycky et al., 2014). Very rarely we observe a system of multiple
planets in a meta-stable resonant chain (for example Trappist-1, Gillion et
al. 2017, and Kepler-223, Mills et al. 2016). Volk & Gladman (2015) suggest
such resonant chains of close orbiting planets are common in planetary
formation and potentially even occurred in the early stages of our own solar
system’s formation but that over long time periods they become unstable
leading to either destruction or collisional consolidation.

Giant impacts are not only thought to be an important formation mechanism for
Super-Earths. The dissipation of the proto-planetary disc can result in orbit
destabilisation for objects from planetesimals all the way up to giant
planets, leading to orbital crossings and collisions. A similar resonant chain
breaking scenario to the Super-Earth formation process is thought to be a
possible formation scenario for Hot Jupiter systems (Raymond & Morbidelli,
2020). Giant impacts are thought to be a common process in planet formation,
and may allow us to observe ongoing planet formation in young planetary
systems (e.g. Watt et al., 2021). Our own solar system has multiple examples
of planets for which giant collisions are the most probable formation
mechanism, for example there is strong evidence for a giant impact having
formed the Earth-Moon system (e.g. Hartmann & Davis, 1975; Asphaug, 2014; Lock
et al., 2018) and also for a giant collision causing Uranus’ unusual axis of
rotation and magnetic field (Kegerreis et al., 2018).

Another impetus for the study of giant collisions between Super-Earth sized
objects is the large amount of diversity in their measured densities, both
overall and between Super-Earths observed in the same system. This density
diversity as a whole cannot solely be accounted for by other methods, such as
XUV radiation from the central star. XUV erodes the less dense outer layers of
close orbiting planets. XUV radiation should lead to a simple orbital distance
density relation due to the reduction in XUV flux with orbital distance
(Inamdar & Schlichting, 2015). XUV radiation can explain differences in
density between planets orbiting different stars but not a large amount of
density difference within a multiple planet system, especially when the
density does not correlate with orbital distance (e.g. Bonomo et al., 2019).

Inamdar & Schlichting (2016) shows theoretically that giant collisions are a
potential explanation to the observed density diversity because collisions can
cause a significant percentage of a planet’s atmosphere to be ejected. In
addition, Bonomo et al. (2019), present observational evidence for a giant
impact in the Kepler 107 system. In this system planet Kepler-107b is less
dense ($\rho=5.3\,\mathrm{g\,cm^{-3}}$) than its neighbour Kepler-107c
orbiting externally to it ($\rho=12.6\,\mathrm{g\,cm^{-3}}$). The K-107 system
exhibits exactly the opposite situation as we would expect to occur due to
XUV, which has a stronger effect on planets closer to the central star.
However, as shown in the paper a large collision could have stripped the
lighter material from Kepler-107c, increasing its density.

Our research in this paper involves modelling such giant impacts between
Super-Earth mass exoplanets. In particular we focus on collisions involving a
rocky target which has accreted a thick hydrogen atmosphere from the proto-
planetary disc and an atmosphere-less rocky projectile.

###  1.1 Previous Work

Due to the difference in density between mantle and atmosphere, and because it
is preferable in SPH for (neighbouring) particles to have similar masses most
previous numerical work on giant impacts have modelled the targets without
atmospheres. As an approximation this works well for smaller terrestrial
planets with masses $\lesssim 1\mathrm{M_{\oplus}}$ which typically have low
atmosphere mass fractions (of the order of $1\%$ or less). However, Kegerreis
et al. (2020) modelled these thin atmospheres, which required simulations with
resolutions at the upper end of what is currently typically feasible at order
$10^{7}$ particles.

The assumption of a thin atmosphere is not accurate for Super-Earth mass
objects, however, which can often accrete large amounts of gas from the proto-
planetary disk. Fortunately these thicker atmospheres do not require as high a
total simulation resolution to model, as the higher atmosphere fraction means
a greater percentage of the total particles in the simulation are in the
atmosphere.

Whilst not going so far as to include atmosphere, Gabriel et al. (2020)
investigate the effects of density stratification in planetary bodies on
collisions involving Earth mass objects, i.e the effect of a planet with an
internal density that decreases with radius from its centre, either due to
pressure or composition. They look at the effects of this stratification in
objects that are comprised of a combination of iron, silicate, and water. They
show that density stratification can lead to hit-and-run style collisions at
lower impact angles than constant bulk density models predict. The densities
of some Super-Earths analysed in our paper are even more highly stratified
than those covered in Gabriel et al., in part due to the higher masses
involved and also due to the density contrast between the core and mantle
material and the atmosphere.

Whilst we make comparisons to Gabriel et al. (2020), this paper does not
follow the same method; they use raw relative kinetic energy, $K$, and its
relationship to the binding energy of material in the system, as suggested by
Movshovitz et al. (2016). In this paper, we instead use the specific relative
kinetic energy measure of Leinhardt & Stewart (2012), which is given by:

$Q_{\mathrm{R}}=\frac{1}{2}\mu\frac{v^{2}}{M_{\mathrm{tot}}},$ (1)

where $M_{\mathrm{tot}}$ is the total mass in the system, $\mu$ the reduced
mass, and $v$ the impact velocity. Specific energy is commonly used to
construct collision scaling laws (e.g. Benz & Asphaug, 1999; Stewart &
Leinhardt, 2009; Leinhardt & Stewart, 2012). This choice of $Q_{\mathrm{R}}$
is consistent with our previous work (Denman et al., 2020), which examined
solely head-on collisions involving similar mass bodies to those covered in
this paper. This paper expands on our previous work by examining collisions at
many different impact parameters.

###  1.2 Collision Outcomes

The head-on impacts from our previous work (Denman et al., 2020) resulted in
either the majority of the projectile merging with the target, or, at higher
energies, one or both being disrupted. Real collisions are unlikely to be
head-on, in general we would expect an impact angle closer to $45^{\circ}$
(Shoemaker & Hackman, 1962).

At these higher impact angles additional impact outcomes can occur. For
example, if a collision is sufficiently glancing a hit-and-run can occur in
which the projectile bounces off the target (and in the idealised definition
does not erode it Leinhardt & Stewart 2012). Another possible impact outcome
is graze-and-merge (Genda et al., 2012), these begin similarly to hit-and-run
collisions but the collision removes sufficient kinetic energy from the
projectile that it can not escape the gravitational influence of the target
and the projectile and target eventually merge.

##  2 Methods

Here we provide a brief summary of the numerical methods used, for more
details see Denman et al. (2020).

###  2.1 Numerical code

The simulations in this paper were carried out using the same build of the SPH
code GADGET-2 (Springel, 2005) as used in Denman et al. (2020) (code available
from Carter 2022). This version of GADGET-2 has been modified so it can use
tabulated equations of state to model planets (Marcus et al., 2009; Ćuk &
Stewart, 2012). GADGET-2 was run in ’Newtonian mode’ with timestep
synchronisation and the standard relative cell-opening criterion. We use the
standard timestep criterion described in Springel (2005), with a Courant
factor of $0.1$. We also use the standard artificial viscosity formulation
with a strength parameter of $0.8$.

###  2.2 Initial Conditions

Planets were modelled as either two or three component objects, with an iron
core, a forsterite mantle, and a hydrogen atmosphere (target only). For all
planets we built an iron core surrounded by a forsterite mantle of double the
mass of the core. This choice of Earth-like composition is consistent with
many previous Super-Earth studies including Marcus et al. (2009); Liu et al.
(2015) and Hwang et al. (2017). We ran preliminary simulations to equilibrate
each planet in isolation to ensure they were stable, as in Marcus et al.
(2009). For the target object we then added a hydrogen atmosphere as an outer
layer and re-equilibrated as per Denman et al. (2020). The mass of this
atmosphere, $M_{\mathrm{atmos}}$, was determined via the following relation:

$\frac{M_{\mathrm{atmos}}}{M_{\oplus}}=0.01\times\left(\frac{M_{\mathrm{c\&m}}}{M_{\oplus}}\right)^{3},$
(2)

where $M_{\mathrm{c\&m}}$ is the combined mass of the core and mantle. This
rule was determined empirically from the results of Bern model population
synthesis simulations (Alibert et al., 2005; Mordasini, 2018) for planets
$\sim 10\,\mathrm{Myr}$ old (a sufficiently early time that planetary systems
should still be dynamically active and thus collisions should still be
occurring), and is the same as was used in Denman et al. (2020).

All collisions used the same target, which had a total mass of
$6.25\,\mathrm{M_{\oplus}}$. This was the same mass as the intermediate mass
target from Denman et al. (2020), and had a significant atmosphere. For
computational run-time reasons, using only one target meant we could test
collisions at a larger range of velocities and impact angles. This target
planet consisted of a $1.67\,\mathrm{M_{\oplus}}$ iron core,
$3.33\,\mathrm{M_{\oplus}}$ forsterite mantle and $1.25\,\mathrm{M_{\oplus}}$
hydrogen atmosphere. We used 5 different projectiles evenly spaced in mass
from $1\,\mathrm{M_{\oplus}}$ to $5\,\mathrm{M_{\oplus}}$. We did not
implement atmospheres in the smaller projectiles since, due to the
preferability in SPH of equal mass particles, we were not simulating at a high
enough resolution for the smaller atmospheres to be well resolved. We kept the
larger mass projectiles without atmosphere for ease of comparison. Target and
projectiles had a $1:2$ ratio by mass of iron core to forsterite mantle with
equal particle masses.

We used tabulated ANEOS/MANEOS equations of state (from Melosh & Vickery 1989)
to describe the iron and forsterite (full tables are available from Carter et
al. 2019). The hydrogen atmospheres on the other hand were modelled using the
ideal gas equation of state (built in to GADGET-2) for simplicity. Initial
radial density and temperature profiles for the core and mantle were taken
from Valencia et al. (2006), and we used the same initial profiles for the
atmosphere as in Denman et al. (2020). Planets were constructed buy splitting
each planet into radial shells and placing a number of particles at random
positions in each shell proportional to the density at the shell’s radius. The
temperature profiles were used for initial estimates before equilibration
simulations were run. During equilibration particles were forced to follow
isentropes, and particle velocities were damped by a restitution factor of
$50\%$ each timestep. A planet was considered settled if both the mean and
maximum distance of the particles of each material from the planet centre of
mass changed by less than $10^{-3}\,\mathrm{R_{\oplus}}$ between output
snapshots (5000 s).

We used the same equilibration times as given in Denman et al., $4\times
10^{5}\mathrm{s}$ for preliminary core equilibration simulations and twice
this for equilibrating the atmosphere on top. Core and mantle specific
entropies were set to $1.3\,\mathrm{kJ\,K^{-1}\,kg^{-1}}$ and
$3.2\,\mathrm{kJ\,K^{-1}\,kg^{-1}}$ respectively. Unlike in our previous work,
here the initial value of the atmosphere’s pseudo-entropy was set to the
higher value of $1.3\times 10^{11}\mathrm{Ba\,g^{-\gamma}\,cm^{3\gamma}}$
(where $\gamma=\frac{5}{3}$ was the value of the adiabatic index). This higher
pseudo-entropy meant a more extended atmosphere than Denman et al. (2020)
which better matched the initial radial density profile after equilibration
was completed.

Most simulations had a resolution of $10^{5}$ particles in the target core and
mantle as this allowed us to run a large number of them. We also ran a smaller
group of simulations at both higher and lower resolutions, these typically
showed good agreement with the equivalent intermediate resolution results that
our analysis is based on. The standard deviation of resolution test results
for total mass of the largest remnant was $2\,\%$ the result for our standard
resolution, whereas atmosphere mass in the largest remnant had a standard
deviation of $7\,\%$ (a more in depth discussion of this is given in appendix
A). Collisions were run using the University of Bristol’s phase 3 and phase 4
Bluecrystal supercomputers. Phase 3 nodes were 16 core 2.6 GHz SandyBridge
processors (Gardiner, 2015), whereas phase 4 nodes have two 14 core 2.4 GHz
Intel E5-2680 v4 (Broadwell) CPUs (Gardiner, 2017). For a typical collision
simulation we used all processor cores in a single compute node; with this
configuration our standard resolution runs took $\sim 7\,\mathrm{hrs}$.

###  2.3 Run Parameters

Figure 1:  Collision geometry showing the parameters used to describe both
highly grazing impacts in which the projectile only passes through the
atmosphere (top) and mantle collisions where the mantles collide (bottom). A
normalised impact parameter is used, $b=B/(R_{\mathrm{p}}+R_{\mathrm{t}})$.
The relative velocity of an impact is $v=v_{\mathrm{t}}-v_{\mathrm{p}}$. The
brown solid lines indicate the surface of the mantle of the projectile and
target and the blue dotted line represents one scale height within the
target’s atmosphere.

The simulations were run for about four times the gravitational dynamical
time, $t_{\mathrm{dyn}}$, to allow enough time for gravitational resettling of
material after the collision. This interaction time (one dynamical time) is
given by:

$t_{\mathrm{dyn}}=\sqrt{\frac{R^{3}}{GM_{\mathrm{tot}}}},$ (3)

where $R$ is initial separation between target and projectile centres of mass,
$M_{\mathrm{tot}}$ the total mass of the system, and $G$ is the gravitational
constant.

The other time constraint we considered was the time required to observe the
secondary impact for graze-and-merge collisions. At higher impact parameters
as energy increases from perfect merging to hit-and-run, graze-and-merge
collisions can occur where the remnants of the projectile have sufficient
energy to escape from the initial collision but not enough energy to escape
from the gravitational influence of the largest remnant (Genda et al., 2012;
Emsenhuber & Asphaug, 2019). In such a situation, these projectile remnants
will eventually fall back to the target for a secondary collision. Our
collisions were run for sufficiently long that we could simulate re-collision
orbits of the size of the Hill sphere of the target at $0.4\,\mathrm{au}$ from
the central star (see appendix C for details). We do not include the effects
of the central star in our simulations, however, so graze-and-merge collisions
are removed from most analyses in the rest of this work.

For all collisions we used the same $6.25\,\mathrm{M_{\oplus}}$ target (the
same mass as the middle mass target of our previous work Denman et al. 2020).
The projectiles were $1,\,2,\,3,\,4\,\&\,5\,\mathrm{M_{\oplus}}$. We chose
eight different velocities:
$20,25,30,35,40,50,60\,\&\,70\,\mathrm{km\,s^{-1}}$, the slowest of these is
approximately escape velocity (mutual escape velocity was in the range
$19.2$–$21.6\,\mathrm{km\,s^{-1}}$ for all collisions), whereas the fastest is
approximately the expected collision velocity of a prograde and retrograde
planet at $0.1\,\mathrm{au}$; lower collision velocities are more probable
(Gabriel et al., 2020) so we carried out a wider range of collisions at low
velocities.

In most previous work on planetary impacts the point at which collision occurs
is straightforwardly defined by the time of first contact between the
planetary surfaces. Atmospheres, however, do not have a well defined outer
“edge". So that we do not need to define such an edge to the atmosphere, for
most collisions we define a collision to occur when the mantle surfaces of
target and projectile touch. For highly grazing collisions, in which the
projectile only passes through the target atmosphere, however, we define the
point of collision to be the point of closest approach. We also use the radius
of the surface of the mantle to define the radius of the target,
$R_{\mathrm{t}}$, and projectile $R_{\mathrm{p}}$. Likewise the impact
parameter, $b$, is normalised by the sum of the target and projectile mantle
surface radii following

$b=\frac{B}{R_{\mathrm{t}}+R_{\mathrm{p}}},$ (4)

where $B$ is the vertical distance between the centres of mass of target and
projectile as shown in Figure 1. This normalisation means that highly grazing
collisions where the mantles do not touch will have impact parameter $b>1$,
this is in contrast to previous atmosphere-less works on planet collision
where $b>1$ meant a collision would not occur. (Impact angle is defined as
$\theta_{\mathrm{imp}}=sin^{-1}(b)$, because this is undefined for $b>1$ we do
not use it in our analyses.)

We chose a range of seven different impact parameters roughly evenly spaced,
from head-on ($b=0$) to a series of collisions that only passed through the
atmosphere at $b=1.3$ (these were
$b\,=\,0,\,0.1,\,0.34,\,0.5,\,0.71,\,0.94,\,1.1\,\&\,1.3$). Full data of all
of these collisions and their precise collision parameters is given in Tables
2–6 in appendix E.

To determine the required positions of planets at the beginning of our
simulation runs we used the above collision parameters, assumed target and
projectile were perfectly spherical and thus well represented by a point mass,
and used a Verlet integrator to trace the path of the projectile backwards in
time until the centres of mass of both target and projectile were separated by
the distance of five times the sum of their mantle surface radii. This choice
of initial separation allowed both objects time to tidally distort before
collision.

For determining the amount of material bound in each remnant after a collision
we use the iterative procedure outlined in Benz & Asphaug (1999). In this
method, first the particle lowest in the potential is found, then all the
particles gravitationally bound to it are identified. The total mass and
centre of mass of this group are calculated and the particles bound to it
detected. This process continues until the bound group no longer increases in
mass each iteration.

##  3 Results

Figure 2: Snapshots showing cross-sectional slices of collisions in progress
for four different types of collision between a $6.25\,\mathrm{M_{\oplus}}$
target and a $2\,\mathrm{M_{\oplus}}$ projectile. Colours indicate material
type, with different shades to distinguish between projectile and target: iron
cores are red, forsterite mantle particles are orange/yellow, and atmosphere
particles are blue. Colours in the first panel indicate the final location of
each particle, black for unbound material, shades of grey are for the second
largest remnant, and largest remnant particles retain the colours described
previously. The four different collisions are as follows: A) a head-on like
collision – the majority of target and projectile merge to form a new planet
as described in Denman et al. (2020), the remaining mass is ejected. B) A
collision in the transition region between merge and hit-and-run – this
particular example is a graze-and-merge in which the remnants of the
projectile undergo a secondary collision with the target; tidal forces tear
apart the projectile as it comes back for the second collision resulting in
chunks of projectile remaining in orbit. C) An erosive hit-and-run collision –
the mantles of both objects collide causing both to lose mass during the
collision; a stream of mantle debris is left between the two objects after the
collision. D) A highly grazing collision – the projectile only passes through
the atmosphere of the target, shockwaves eject some of the atmosphere but the
projectile and target lose a negligible mass of core and mantle.

In this paper we simulate collisions between Super-Earths at a wide array of
impact angles, from head-on, to the projectile barely grazing the target’s
atmosphere, as well as a wide variety of impact speeds, from the escape
velocity, to twice the orbital velocity of a target at $0.1\,\mathrm{au}$
(i.e. the expected velocity of a collision between prograde and retrograde
planet). An array of these results is shown in Figure 2 detailing some of the
more common outcomes.

Overall we find that the results are a strong function of specific impact
energy, $Q_{\mathrm{R}}$. We see a separation of the results into two regimes.
At higher impact parameters and energies a large percentage of the projectile
has enough energy to escape the larger target remnant resulting in two large
remnant objects with masses of a similar order of magnitude (for example
collisions C and D in Figure 2). At lower energies and impact angles the
projectile remnant cannot escape and we observe behaviour similar to the head-
on collisions described in Denman et al. (2020), where we obtain a single
large remnant containing core material from both target and projectile (e.g.
collision A in Figure 2). We develop scaling laws for these two main regimes,
these are summarised in appendix F. In between these two regimes we observe a
narrow band of graze-and-merge collisions in which the projectile escapes the
target on its first collision but does not have enough energy to escape the
gravitational influence of the target and thus merges in a later secondary
collision (e.g. collision B in Figure 2).

###  3.1 Largest Remnant Mass

Figure 3:  Left: The mass of the largest remnant as a fraction of total mass
compared with impact energy. The collisions with impact velocities below
$v_{\mathrm{split}}$, which are well described by Denman et al. (2020), are
coloured in blue, other collisions are orange and partially transparent as
their masses are not well described using this normalisation. Centre: The mass
of the largest remnant as a fraction of target mass compared to impact energy.
Collisions with impact velocities greater than $v_{\mathrm{split}}$ are
coloured orange. The orange points being constrained to a small wedge
indicates this normalisation is worth investigating as in it the data for
remnant mass caused by collisions of different mass projectiles overlaps (see
Figure 4) . Other collisions are coloured blue and made partially transparent.
Right: Second largest remnant mass in terms of the projectile mass compared
with the ratio of the impact velocity to the critical velocity,
$v_{\mathrm{split}}$. The shaded region shows the values we chose to ignore
for our fits to either regime; this selection gives the circled points in all
three graphs.

We begin by examining the overall mass of the largest post-collision remnants.
The two graphs on the left and centre in Figure 3 show the dependence of the
mass of the largest remnant on the relative specific kinetic energy of the
collision, $Q_{\mathrm{R}}$ – a parameter that has been shown in multiple
previous works (e.g. Stewart & Leinhardt 2009; Marcus et al. 2009; Leinhardt &
Stewart 2012) to be useful for constructing scaling relations. In the
rightmost graph of Figure 3 we observe two separate collision regimes which
result in different masses for the second largest collision remnant. These two
collision regimes correspond to situations where there is either one or two
resultant large post-collision remnants, i.e. situations in which the cores of
target and projectile either merge with one another or survive and continue on
separate trajectories after the impact. The simulations in the first of these
regimes (blue points) result in one single large remnant (leftmost graph,
Figure 3) formed from the combined cores of both target and projectile and a
low or negligible mass secondary remnant. The second of these regimes (orange
points) is similar to hit-and-run, with a significant mass secondary remnant
present due to the projectile bouncing off the target. The difference between
these collisions and the idealised Leinhardt & Stewart (2012) definition of
hit-and-run (i.e. no erosion of the target) is that, due to the presence of
the lightly bound atmosphere, some erosion from the target is always observed
(centre graph Figure 3). The energy cutoff between these two collision regimes
is dependent on impact parameter with hit-and-run like collisions at higher
$b$ occurring at lower impact energies.

We find that the transition between these two regimes can be well approximated
by a simple escape velocity measure,

$v_{\mathrm{split}}=\sqrt{\frac{2GM_{\mathrm{tot}}}{b(R_{\mathrm{p}}+R_{\mathrm{t}})}}.$
(5)

This equation assumes, firstly, that each planets trajectory is well
approximated by describing each planet as a point mass, and also that the
impact parameter, $b$, is a good approximation for the distance of closest
approach.

These two different collision outcomes are the reason why the remnant mass is
best described using two different mass normalisations (left and centre panels
in Figure 3). At lower velocities and impact parameters we have a regime with
one single large remnant formed by a combination of both projectile and target
so we normalise by the combined mass of projectile and target,
$M_{\mathrm{tot}}$, as shown by the blue points in the leftmost graph of
Figure 3. The largest remnant mass, $M_{\mathrm{LR}}$, in these collisions
behaves like the head-on collisions described in Denman et al. (2020) as shown
in Figure 6. At higher impact energies, when most of the atmosphere is
removed, the largest remnant mass in this regime behaves similarly to that
described in Leinhardt & Stewart (2012), that is decreasing linearly with
increasing $Q_{\mathrm{R}}$; at low energies atmosphere removal is less
efficient and we, therefore, have a shallower gradient. The second regime
occurs at higher velocities for each impact angle. All largest remnants in the
second regime are less massive than the target – the target always suffers
some erosion. We normalise largest remnant mass by target mass in this regime
(see orange points in the middle graph of 3) because the largest remnant mass
can be considered to be less than the target mass by some impact energy
dependent erosive factor as will be shown later in Figure 4.

While the rightmost graph in Figure 3 shows that $v_{\mathrm{split}}$ is a
good approximation for the regime boundary, there are a few misdiagnosed
results. There are three potential causes for these outliers: firstly the
assumption of spherical symmetry, in reality there is some tidal distortion at
this point; secondly, graze-and-merge collisions should occur at slightly
lower velocity than this boundary, those on long orbits may not have had time
to re-collide yet, meaning they may appear as hit-and-run collisions in our
data even if their final result would be a merger; or thirdly, material
dependent impact effects such as deflection or drag.

Another confusing factor for the boundary between collision regimes is the
possibility of secondary collisions which occur when the secondary remnant
does not have the energy to escape the target after collision. These secondary
collisions could occur close to the end of the simulation, meaning we would
not be able to measure the equilibrium bound mass. For planets orbiting close
to the central star the simulation timescale we have used is sufficient for
them to escape the Hill sphere of the largest remnant, but because we do not
model a central star, they undergo a secondary collision. To keep our
collision analysis agnostic to orbital distance and to account for the
imperfect nature of this regime boundary measure, we have elected to ignore
all simulations with velocities in the range $0.94<v_{\mathrm{split}}<1.07$
when fitting.

####  3.1.1 Erosive Hit-and-Run

Figure 4:  Top: Mass of the largest remnant relative to the target mass
compared with specific impact energy. Erosive hit-and-runs are shown with
filled symbols, open symbols indicate head-on or transition region collisions.
Projectile mass is indicated by symbol shape; different impact angles are
given in different colours. This graph illustrates the projectile mass
independence of erosive hit-and-run collisions. The data has been fitted
linearly with an intercept of the target mass and a gradient that is a power
law function of impact parameter (Equation 6). The black horizontal line
indicates the original target mass. Bottom: Mass of atmosphere in the largest
remnant normalised by the initial atmosphere mass of the target compared to
$Q_{\mathrm{R}}$. The fit shown by the solid lines is a power law fit with a
shared minimum energy for atmosphere loss $Q_{\mathrm{0}}$, and a power law
coefficient that is itself a power law function of impact angle (Equation 13).

For collisions with a velocity above $v_{\mathrm{split}}$ the projectile
rebounds off the target causing erosion in one or both; following Gabriel et
al. (2020) we call such collisions erosive hit-and-runs.

In these erosive hit-and-run collisions we observe an approximately linear
decrease in largest remnant mass (normalised by the target mass) with
increasing impact energy (top graph, Figure 4). This linear relationship is
independent of projectile mass, but with a gradient that decreases with
increasing impact parameter, meaning that more grazing collisions remove less
material.

The gradient of this linear relationship $\Gamma_{\mathrm{EHR}}$ can be shown
to have a power law dependence on impact parameter (see appendix B). If we fit
to all of the impact parameters simultaneously we obtain:

$\frac{M_{\mathrm{LR}}}{M_{\mathrm{t}}}=-10^{\left(-1.28\pm
0.02\right)b-8.47\pm 0.01}Q_{\mathrm{R}}+1.$ (6)

We have set this model to pass through $M_{\mathrm{LR}}=M_{\mathrm{t}}$ at
$Q_{\mathrm{R}}=0$ so that we have no mass loss for zero input energy. A fit
with a loose intercept (see appendix B) gives an intercept slightly below one
indicating that low energy mass loss is potentially non-linear, despite this
we have still used a linear fit so as to provide the simplest description of
the results.

Decrease in mass ejected with increasing impact parameter is observed in many
previous collision works on atmosphere-less collisions (e.g. Leinhardt &
Stewart, 2012; Movshovitz et al., 2016). To determine the effects of the
atmosphere on the impact angle dependence of the largest remnant we compare
our results to those of Leinhardt & Stewart (2012). They found that their
observed largest remnant mass could be well described by a simple relation
based on the geometry of the impact that described the dependence of the
amount of impact energy the projectile deposits in the target on the impact
parameter. They show the effective specific kinetic energy of a collision at a
particular impact angle $Q^{{}^{\prime}}_{\mathrm{R}}$, is related to the
specific kinetic energy of a head-on collision at the same mass and velocity,
$Q_{\mathrm{R}}$, by

$Q^{{}^{\prime}}_{\mathrm{R}}=\frac{\mu_{\mathrm{\alpha}}}{\mu}Q_{\mathrm{R}},$
(7)

where $\mu$ is the reduced mass and $\mu_{\mathrm{\alpha}}$ a modified version
of the reduced mass determined from the mass of the projectile that interacts
with the target. $\mu_{\mathrm{\alpha}}$ is given by

$\mu_{\mathrm{\alpha}}=\frac{\alpha M_{\mathrm{p}}M_{\mathrm{t}}}{\alpha
M_{\mathrm{p}}+M_{\mathrm{t}}},$ (8)

where $\alpha$ is found by calculating the fraction of the projectile that is
below the uppermost point of the target at the point of impact (see figure 2
in Leinhardt & Stewart 2012):

$\alpha=\frac{3R_{\mathrm{p}}l^{2}-l^{3}}{4R_{\mathrm{p}}},$ (9)

$l$ here is the vertical distance between the base of the projectile and the
top of the target. Note that $l=(R_{\mathrm{p}}+R_{\mathrm{t}})(1-b)$. In this
way the energy involved in the oblique impact is always smaller than (or equal
to) the energy of the head-on impact.

This correction to the impact energy can be related to our gradient,
$\Gamma_{\mathrm{EHR}}=-10^{-1.28b-8.47}$, above by considering the
simultaneous equations of equation 6 and the equivalent equation we would get
with the corrected impact energy, $Q^{*}_{\mathrm{R}}$:

$\frac{M_{\mathrm{LR}}}{M_{\mathrm{t}}}=-\Gamma_{\mathrm{0}}Q^{*}_{\mathrm{R}}+1.$
(10)

This leaves us with

$Q^{*}_{\mathrm{R}}=\frac{\Gamma_{\mathrm{EHR}}(b)}{\Gamma_{\mathrm{0}}}Q_{\mathrm{R}},$
(11)

or in other words the reduced mass ratio we expect to observe in our
collisions should be directly proportional to $\Gamma_{\mathrm{EHR}}$ and some
constant scaling factor $\Gamma_{\mathrm{0}}$,

$\frac{\Gamma_{\mathrm{EHR}}(b)}{\Gamma_{\mathrm{0}}}=\frac{\mu_{\mathrm{\alpha}}}{\mu}.$
(12)

The exact correction from Leinhardt & Stewart (2012) cannot be used for our
collisions with an atmosphere because atmospheres do not have a defined outer
edge, instead they decrease roughly exponentially in density with distance
from their base. Instead we assume the radius of the target is the outer
radius of its mantle. The relation between reduced mass ratio and impact
parameter predicted using this modified model is shown by the coloured lines
in Figure 5.

When $\Gamma_{\mathrm{EHR}}(b)/\Gamma_{\mathrm{0}}$ is scaled up such that
$\Gamma_{\mathrm{0}}$ is the largest measured gradient
($\Gamma_{\mathrm{EHR}}(b=0.34)$, red points in Figure 5), the low impact
parameter collisions ($b\leq 0.8$) agree well with the modified version of the
Leinhardt & Stewart (2012) interacting mass measure using the mantle surface
radius as the target radius (see Figure 5). This agreement indicates that the
corrected Leinhardt & Stewart prediction of the relation between impact
parameter and impact energy of the interacting mass still works well for
collisions in which the majority of interaction between projectile and target
is between core and mantle, even though the prediction does not include the
effects of the atmosphere. The shock caused by the projectile passing through
the atmosphere however means that the amount of material removed at higher
impact angles is greater. The atmosphere-less bodies in Leinhardt & Stewart
(2012) would no longer impact one another at impact parameters greater than
one so there would be no energy deposition and no mass loss, whereas, due to
our definition of the point of impact, the projectile dives into the target’s
atmosphere at these impact parameters (1.0 – 1.3) and we observe atmospheric
erosion. This atmosphere material is removed preferentially to the mantle for
two reasons: firstly, it is significantly less tightly bound than the core and
mantle material, secondly, the impedance mismatch between atmosphere and
mantle mean that any shockwaves in the mantle caused by atmospheric shocks are
substantially weaker than the initial shock that caused them. This low level
of mantle interaction is especially true for highly grazing ($b>1$)
collisions.

Figure 5:  Two different scalings of the total mass loss gradient as a
function of impact parameter. The scaling in black is with respect to the
predicted gradient at $b=0$ from the fit in equation 6. The data points are
from the measured gradients for linear fits to individual projectile masses
and impact parameters (see appendix B). The red line and data points are
scaled with respect to the greatest measured gradient. These are compared to
our modified version of Leinhardt & Stewart (2012)’s reduced mass ratio
modification to $Q_{\mathrm{R}}$. Our results at the lower impact angles
follow a pattern consistent with Leinhardt & Stewart (2012), but they diverge
at higher impact angles due to the presence of an atmosphere.

####  3.1.2 Atmosphere Loss in the Erosive Hit-and-Run Regime

To understand how much atmosphere gets removed in these erosive hit-and-run
collisions between atmosphere rich and atmosphere-less Super-Earths we compare
the specific impact energy, $Q_{\mathrm{R}}$, with the fraction of atmosphere
remaining on the largest remnant after a collision,
$M_{\mathrm{LR}}^{\mathrm{H}}/M_{\mathrm{t}}^{\mathrm{H}}$, where
$M^{\mathrm{H}}$ is the mass of the hydrogen atmosphere and the subscripts
$\mathrm{t}$ and $\mathrm{LR}$ indicate the target and the largest remnant
respectively. Figure 4 shows that the amount of atmosphere in the largest
remnant decreases with increasing impact energy and also with decreasing
impact parameter. The amount of extra material removed for a small increase in
energy also decreases with increasing energy. The data for each separate
impact parameter thus follow convex curves downwards.

In the erosive hit-and-run regime each of the separate projectile masses
(shown by different shapes in Figure 4) all follow the same loss curve for
each impact parameter. This means that any projectile mass dependency in the
atmosphere mass loss from these impacts is characterised by $Q_{\mathrm{R}}$.

In all collisions the amount of material ejected is proportional to the
difference between the input kinetic energy of the impact, and the
gravitational potential energy of the material being ejected, minus any losses
due to heat generating processes. Physically the lack of any additional
dependence on projectile mass beyond specific impact energy is likely due to
the projectile rebounding off the target sufficiently quickly that the
material it causes to be ejected from the target only needs to escape the
target’s gravitational potential. It should be noted that for projectile-
target mass ratios smaller than the range tested here ($0.16-0.8$) one might
expect this symmetry to be broken, as smaller projectiles have lower momenta
and are thus likely to deposit more of their kinetic energy in the atmosphere.
This phenomena is likely the reason why the $1\,\mathrm{M_{\oplus}}$ results
(the triangles in Figure 4) are not in agreement with the higher masses for
our higher impact parameter simulations.

Initial power law fits for each impact parameter (appendix B) crossed the
initial atmosphere mass at a non-zero energy, this suggests that there is a
minimum input energy required for atmosphere to be ejected from the planet.
The precise value for this minimum energy, $Q_{\mathrm{0}}$, for each of the
initial fits is given in graph C Figure 12 and is due to the energy required
to accelerate atmosphere particles to a velocity where they can escape the
largest remnant’s Hill sphere. As such the precise value is likely to be
dependent both on how efficiently collision energy is spread between
atmosphere particles and also potentially for simulations, their resolution.

The power law coefficient was also found to itself have a power law dependence
on impact parameter (see appendix B). Fitting for all projectile masses and
impact angles simultaneously with the function

$\frac{M_{\mathrm{LR}}^{\mathrm{H}}}{M_{\mathrm{t}}^{\mathrm{H}}}=-10^{\delta
b+\epsilon}\log_{10}\left(\frac{Q_{\mathrm{R}}}{Q_{\mathrm{0}}}\right)+1,$
(13)

(solid lines in the bottom graph of Figure 4) we obtained values of
$\delta=-0.70\pm 0.01$, $\epsilon=0.13\pm 0.01$ and $Q_{\mathrm{0}}=(1.8\pm
0.1)\times 10^{6}\,\mathrm{J\,kg^{-1}}$.

The physical meaning of the impact parameter dependent power law coefficient,
$\gamma_{\mathrm{atmos}}=-10^{\delta b+\epsilon}$, is related to the
efficiency of the collision in removing atmosphere from the target. Collisions
closer to head-on remove more atmosphere than more highly grazing ones as a
greater amount of the projectile’s momentum can be transferred to the target’s
atmosphere. In addition, for collisions with low $b$ the projectile causes
shockwaves that travel through the core and mantle, which can cause atmosphere
ejection from all over the planet not just the parts close to the trajectory
of the projectile. Atmospheric shockwaves from high $b$ impacts, on the other
hand, are unlikely to cause strong shocks in the mantle due to the impedance
mismatch between atmosphere and mantle.

####  3.1.3 The Head-On-Like Regime

Figure 6:  Top: Largest remnant mass as a function of normalised specific
impact energy for all collisions in the head-on-like regime. Results are
compared to the model from Denman et al. (2020) (the black line). We observe
both the transition energy $Q_{\mathrm{piv}}$ between atmosphere dominated
loss at low energies and core dominated loss at high energies, and also the
loss gradients to be in close agreement with this prediction. Bottom: The
fraction of atmosphere lost by the largest remnant for all collisions with
velocity below $v_{\mathrm{split}}$ (head-on-like) compared with impact energy
normalised by $Q_{\mathrm{piv}}$ from Denman et al. (2020) (see appendix F for
calculation). The black line shows their predicted atmosphere loss fraction.
The two results agree well with one another.

For collisions below the critical velocity, $v_{\mathrm{split}}$, we compare
the mass in the largest remnant to the specific impact energy normalised by
the catastrophic disruption threshold $Q_{\mathrm{RD}}^{*}$ outlined in Denman
et al. (2020), as shown in Figure 6. We observe an initial shallow decrease in
mass with increasing impact energy, which sharply increases at about
$0.5Q_{\mathrm{RD}}^{*}$; this is strongly consistent with the model derived
in Denman et al. (black lines in Figure 6), as such we label all these
collisions as ‘head-on-like’. Denman et al. (2020) identify this transition
with the minimum energy at which mantle will be excavated by the impact in
addition to atmosphere.

We have also compared the fraction of atmosphere lost from the largest remnant
for this set of head-on-like collisions to our predictions in Denman et al.
(2020) (see bottom panel, Figure 6). Again the results are strongly consistent
with the predicted values. There is, however, an impact angle dependent
divergence from predicted atmosphere loss which increases at low impact
energies. More grazing collisions ($b\gtrsim 0.3$) remove less atmosphere than
predicted by the model (blue and green points). Glancing collisions between
planets without atmospheres (such as those detailed in Leinhardt & Stewart
(2012) section 3.1.2) have been shown to have less efficient energy deposition
at higher impact parameters. A less energetic shockwave in the core means less
atmosphere being pushed away by the shock as it reaches the target surface
again on the other side of the planet.

This consistency between the results of this work and the model from Denman et
al. (2020) occurs despite this paper using a significantly increased
atmosphere pseudo-entropy, $1.3\times
10^{12}\mathrm{Ba\,g^{-\gamma}\,cm^{3\gamma}}$ as opposed to $5\times
10^{11}\mathrm{Ba\,g^{-\gamma}\,cm^{3\gamma}}$, which resulted in a warmer,
more extended atmosphere. Looking at these collisions in terms of energy, this
consistency is likely because although the higher pseudo-entropy means the
outermost atmosphere particles are higher up the potential well, the
difference in energy required for them to be ejected is small compared to the
differences in how efficiently the impact energy is spread between particles.

###  3.2 Remnant Composition

Figure 7:  Top: Mass fraction of iron in the largest remnant as a function of
the remnant mass. Iron fraction increases at low remnant masses because
collisions preferentially remove material that is less well bound. Middle:
Silicate mass fraction of the largest remnant. The mantle is less tightly
bound than core material yet more strongly bound than atmosphere, we thus
observe a turnover at the point where the majority of atmosphere has been
eroded and mantle erosion begins in earnest. Bottom: The fraction of the
largest remnant that is hydrogen atmosphere. Atmosphere is the least strongly
bound part of the planet and thus is removed preferentially. We also show the
predictions for head-on collisions from Denman et al. (2020) as dotted lines
(open symbols at the end represent the projectile mass of the prediction).
Colours and symbols have the same meanings as in previous Figures. For each of
these graphs we have fit the $v_{\mathrm{imp}}>v_{\mathrm{split}}$ data with a
power law (solid black lines).

Preferential erosion or accretion of planet components during collisions will
result in compositional change. Figure 7 shows the fraction of largest remnant
that is comprised of each constituent material after a collision. The original
mass fractions were, $26\%$ iron, $53\%$ forsterite and $20\%$ hydrogen for
the target, and, $33\%$ iron and $67\%$ forsterite for the projectile. We
observe two different patterns: head-on-like collisions with
$v<v_{\mathrm{split}}$ tend to clump around separate curves dependent on
projectile mass; whereas the erosive hit-and-run collisions, for which
$v>v_{\mathrm{split}}$, tend to all follow one single curve, this is true for
the fractions of all material types (core, mantle, and atmosphere). The single
line for erosive hit-and-runs is due to the collisions only eroding material
from the target and barely depositing any.

Comparing the three graphs in Figure 7, looking at the erosive hit-and-run
collisions, we see iron core content increasing with decreasing largest
remnant mass (top), Hydrogen atmosphere content by comparison decreases with
decreasing final mass (bottom), while forsterite mantle content increases with
mass removed initially it reaches a turning point at $5.2\pm
0.2\,\mathrm{M_{\oplus}}$ and starts to decrease again as more material is
removed (middle). Whether a material fraction increases or decreases with
decreasing final remnant mass depends upon if material is preferentially
removed in an impact or not. In our simulations, first the lightly
gravitationally bound outer atmosphere is removed, then the more tightly bound
mantle layer underneath, and finally the strongly bound iron core material.
This explains the turning point in mantle fraction, at lower final masses the
majority of atmosphere has already been removed and mantle is then removed
preferentially to iron core, but at higher masses atmosphere is still being
removed preferentially to mantle.

An important result to take from all erosive hit-and-runs following the same
pattern of final material fraction, is that the atmosphere fraction after an
erosive hit-and-run collision can be calculated from the final remnant mass
and vice versa.

The material fraction results were fitted with power laws; for the iron core
we obtained:

$f_{\mathrm{Fe}}=10^{(-0.091\pm 0.002)M_{\mathrm{LR}}-0.01\pm 0.01},$ (14)

while for the hydrogen atmosphere we obtained,

$f_{\mathrm{H}}=10^{(0.39\pm 0.01)M_{\mathrm{LR}}-3.14\pm 0.05},$ (15)

finally we combine equations 14 and 15 for the fraction of the largest remnant
that is silicate mantle:

$f_{\mathrm{Si}}=1-f_{\mathrm{FE}}-f_{\mathrm{H}}.$ (16)

These relations are shown in Figure 7 with solid black lines.

This result implies that, in the erosive hit-and-run regime, no matter the
projectile-target mass ratio (within the range tested) a collision that
results in the same largest remnant mass (i.e. the same normalised impact
energy) will result in the same material fractions.

For the atmosphere fraction we have also added lines detailing the
prescription from Denman et al. (2020) – the dotted lines with open symbols
representing their projectile mass in the bottom panel of Figure 7. Because
our head-on results show strong consistency with the model for total and
atmospheric mass in the largest remnant, we again observe consistency here,
with some deviation occurring for the $b=0.34$ collisions which are close to
the transition region, $v\approx v_{\mathrm{split}}$.

###  3.3 The Second Largest Remnant

Figure 8:  Top: Mass of the second largest remnant as a function of impact
velocity in terms of the transition velocity, $v_{\mathrm{split}}$, showing
the dependence on both projectile mass (different symbols) and impact
parameter (different colours). More grazing impacts (e.g. blue and purple)
result in less material lost from the projectile, as do lower velocity
impacts, although sufficiently low velocity impacts will result in a merger
($v<v_{\mathrm{split}}$). Points with a black outline indicate those
collisions in which the second largest remnant has increased in mass,
typically by accreting atmosphere from the target. Bottom: The mass of target
atmosphere accreted on to the second largest remnant as a function of impact
velocity. The amount of accreted atmosphere peaks at a low energy close to the
edge of the grazing erosion regime at $v=v_{\mathrm{split}}$.

The top panel of Figure 8 shows the dependence of the mass of the second
largest remnant on impact velocity. At low relative velocities and impact
angles we observe head-on like behaviour with no secondary remnant of more
than a few particles in most cases. The most massive second largest remnants
we observe are found at velocities just above the transition velocity,
$v_{\mathrm{split}}$. As the energy increases the projectile gets more and
more eroded by the impact. For lower impact parameters, $b\sim 0.3-0.5$,
eventually at sufficiently high energy the projectile will get disrupted. The
degree to which the projectile is eroded is dependent on impact parameter,
with more grazing impacts requiring more energy to remove the same amount of
material. For the highly grazing impacts ($b>1$) the amount of mass lost is
negligible, and at lower impact energies the projectile can even gain mass.
This mass gain comes from accretion of some of the comparatively lightly bound
atmosphere of the target as the projectile passes through it (these collisions
are highlighted by black outlines in Figure 8).

The bottom panel of Figure 8 shows the amount of atmosphere that is accreted
onto the secondary remnant from the target. Like for the total secondary
remnant mass, accreted atmosphere mass peaks at low impact velocities just
above the critical impact velocity ($v_{\mathrm{split}}$). The amount of
atmosphere accreted decreases a lot more sharply after this point than the
total mass of the secondary remnant though, with little to no atmosphere being
accreted at velocities twice that of peak accretion.

All second largest remnants which have greater mass than the projectile have
accreted atmosphere from the target during the collision. The outermost target
atmosphere particles are loosely bound so the gravitational pull of the
projectile can be sufficient to pull them away if they are not completely
ejected by atmospheric shocks. This effect is likely exaggerated by the
projectiles in our study being atmosphere-less, this means target atmosphere
particles can approach closer to the projectile and thus experience a stronger
gravitational pull towards it.

The collisions where mass increases also occur at the highest impact
parameters where the projectile only barely grazes the target mantle, if at
all, as collisions where the mantles collide are sufficiently more erosive to
the projectile that it will erode more than it will accrete. The collisions in
which the secondary remnants accrete the most atmosphere, however, tend to be
those of lower impact parameters (around $0.3<b<0.75$) where more of the core
and mantle of the projectile is eroded, but the projectile passes through more
of the atmosphere and thus can accrete more.

##  4 Discussion

###  4.1 The Erosive Hit-and-Run Regime

An important difference between collisions involving planets with and without
an atmosphere is that, due to the lightly bound nature of the upper
atmosphere, there is a very low probability of getting a ‘true’ hit-and-run
collision in which target and projectile mass are both affected negligibly.
Instead we get ‘erosive hit-and-runs’ (Gabriel et al., 2020) where collisions
resulting in two large remnants also erode the target.

In this erosive hit-and-run regime we normalise the mass of the largest post-
collision remnant with respect to target mass as opposed to total mass. This
is because in erosive hit-and-run collisions the largest remnant is formed
from the partially eroded target with the remnants of the projectile
rebounding, as opposed to projectile and target cores combining similar to
what happens in a head-on collision.

Our results for the mass of the largest remnant in the erosive hit and run
regime show negligible dependence on the projectile mass for our particular
target mass. Because of this lack of projectile mass dependence we have
elected not to describe our results in terms of accretion efficiency (Asphaug,
2010). In this regime the dependence of the amount of material excavated by
the projectile on projectile mass is completely characterised by the specific
impact energy, $Q_{\mathrm{R}}$. The efficiency of this excavation is a strong
function of impact parameter, with grazing impacts removing a lower proportion
of the target’s mass. Resolution test simulations showed minimal differences
to the standard resolution this analysis is based on, being on average within
$2\%$ (see Appendix A). This relation is also seen in the data provided by
Gabriel et al. (2020) for atmosphere-less silicate-iron planets.

###  4.2 The Transition to Hit-and-Run

Figure 9:  Top: Different methods of calculating the velocity of transition
between merging and ’hit-and-run’ and the dependence on $b$. Our model
provides a close match to the Gabriel et al. (2020) prediction for impact
angles greater than their predicted minimum hit-and-run impact angle (dashed
lines). Middle: Comparison between the Gabriel et al. method for determining
the erosive hit-and-run transition and our model applied to our simulations.
Our results are in agreement apart from some collisions we classify as erosive
hit-and-run (green points). Bottom: A comparison between our method for
determining the erosive hit-and-run transition and the Gabriel et al. method,
applied to their two component iron-silicate planets. Again our predictions
identify more erosive hit-and-run collisions.

We find the transition between the merging and hit-and-run collision regimes
to be strongly related to the mutual escape velocity of the two bodies at the
point of closest approach, $v_{\mathrm{split}}$ defined in equation 5.

Earlier works on impacts such as Leinhardt & Stewart (2012) use the Asphaug
(2010) grazing angle measure of
$b_{\mathrm{crit}}=R_{\mathrm{t}}/\left(R_{\mathrm{p}}+R_{\mathrm{t}}\right)$
to approximate the transition to hit-and-run. $b_{\mathrm{crit}}$ is the
boundary impact parameter where the velocity vector of the projectile’s centre
of mass no longer intersects the target. Gabriel et al. (2020) show that for
larger planets $\approx 1\,\mathrm{M_{\oplus}}$, where there can be
significant stratification of density due to both gravity and material types,
this can cause the boundary between merging and hit-and-run to occur at
substantially lower impact parameters. The smallest planets in this paper are
$1\,\mathrm{M_{\oplus}}$ and the targets all have large hydrogen atmospheres
which are significantly less dense than the cores, we thus observe a
transition to hit-and-run collisions at less than $b_{\mathrm{crit}}$.

We use a different parametrisation of this boundary, one that has a simple
functional form and physical motivation, i.e. the velocity required for the
secondary remnant to escape the system from its point of closest approach
($v_{\mathrm{split}}$). The top panel in Figure 9 shows comparisons between
our model and the models from Gabriel et al. (2020), Genda et al. (2012) and
Leinhardt & Stewart (2012) where we have replaced the target full radius with
that of its surface due to the presence of an atmosphere. In the simplest
model, like that used by Leinhardt & Stewart (2012), there is a sharp (mass
ratio dependent) impact parameter cutoff above which all collisions are hit-
and-run (dotted vertical lines). Both our model and that of Genda et al.
(2012) (solid lines and dash-dotted lines respectively) describe the
transition to the hit-and-run regime as impact parameter dependent, occurring
at increasingly higher velocities at lower impact parameters, following a
convex function. The model of Gabriel et al. (2020) (dashed lines) uses
aspects of both descriptions, below a particular impact parameter threshold
hit-and runs will not occur, but above that they will occur if they are above
some transition velocity which decreases as a convex function of increasing
impact parameter.

The bottom two panels compare which collisions both our method and that of
Gabriel et al. (2020) classify as merge or hit-and-run for both our work and
the two component bodies from their work. The central condensation parameter
$\Lambda$ for each projectile-target pair used our simulations was calculated
to be between $0.35$ for the smallest projectile and $0.58$ for the largest.
Red points are collisions that both methods classify as erosive hit-and-run,
blue points are collisions in which both methods predict accretion or
disruption, and green points are collisions that only this work classifies as
erosive hit-and-run. Only one point (given in orange) was classified as
erosive hit-and-run by Gabriel et al. but not by our model. The two panels
show the two methods give broadly consistent results, with
$v_{\mathrm{split}}$ from this work classifying more collisions as hit-and-
run.

The minimum possible velocity of a collision between two planets with $b<1$ is
their mutual escape velocity, if the two planets are in isolation this is
$v_{\mathrm{esc}}\,=\,\sqrt{2GM_{\mathrm{tot}}/(R_{\mathrm{p}}+R_{\mathrm{t}})}$.
For highly grazing collisions ($b>1$) the escape velocity from the point of
closest approach is $v_{\mathrm{split}}$ which implies that they can only be
erosive hit-and-run collisions. In our simulations, however, we observe some
highly grazing collisions to result in mergers. These highly grazing mergers
occur because the atmosphere also has a drag force on the projectile as it
passes through it, if the projectile is moving slowly enough, this drag force
can be sufficient to cause it to become a graze-and-merge collision. In the
real world collisions slightly slower than mutual escape velocity can also
occur due to the presence of the other bodies in the system, if we also
consider the presence of the central star for example we only need to consider
the projectile starting at zero velocity at the edge of the Hill sphere
(rather than at infinity).

Material properties may affect the accuracy of $v_{\mathrm{split}}$ as a
predictor of the transition between head-on-like and erosive-hit-and-run
collisions, for example by changing how much drag or deflection the projectile
undergoes as it passes through the target. $v_{\mathrm{split}}$ seems to work
well for both simulations despite differences in equation of state and the
presence of an atmosphere. This agreement is likely due to all simulated
collisions being deep in the gravity-dominated regime (Housen & Holsapple,
1990).

###  4.3 Atmosphere Removal

Denman et al. (2020) show that the likelihood of a collision removing all the
atmosphere from a planet in one go is low, and if it does, it will be
sufficiently energetic to cause catastrophic disruption. The head-on
collisions in this work are in strong agreement with their predictions for
atmosphere loss, as shown in Figure 6.

Comparing our results for the mass remaining in the largest remnant for head-
on collisions to those at higher impact parameters (given in Figure 4) we see
that, for all cases where $b>0.3$, higher impact parameters mean a lower
efficiency of atmosphere erosion. Considering that higher impact parameters
are more likely than head-on (Shoemaker & Hackman 1962 predict a most likely
value of $45^{\circ}$), this means that a single impact removing the entirety
of a Super-Earth’s atmosphere is very unlikely. This implies that when we
observe a Super-Earth mass planet with no substantial atmosphere it has likely
either undergone multiple collisions or formed sufficiently close to its
central star for photo-evaporation to have stripped the entire atmosphere.

In our resolution tests we observe an average difference of $7\%$ from our
standard resolution for atmosphere mass (see appendix A). Our higher
resolution simulations typically show slightly less atmosphere erosion than
our lower resolution ones, so we would expect full atmosphere stripping in a
real system to be less likely than in our simulations.

###  4.4 Final Planet Radii

Figure 10:  Top: The radii of the largest remnants generated by our collisions
as a function of their masses. The colour of points indicates the impact
parameter whereas the shape indicates the projectile mass. Lines indicate the
expected results for total atmosphere erosion (solid line), and for partial
atmosphere erosion in both the erosive hit-and-run (dash-dot) and head-on-like
regimes (dotted lines, from Denman et al., 2020). The open symbol at the end
of each dotted line shows which projectile mass the line represents. Bottom:
Corresponding densities as a function of largest post-collision remnant mass.
Note that for real systems the upper limit for density is likely also
dependent on material types in the core. Highly erosive collisions will strip
mantle more readily than core, leaving a denser remnant than would be expected
if the same ratio of core to mantle was maintained.

As in Denman et al. (2020), we have used Lopez & Fortney (2014)’s results to
predict the final radii and densities of planets from their percentage
composition post collision once all the bound material has settled back down,
see Figure 10.

Lopez & Fortney (2014) model a planet’s radius by splitting it into 3
contributions, these are: the core and mantle (modelled as a power law
function of mass), the convective envelope (which is dependent on temperature,
incident flux and planet age), and the radiative atmosphere (also dependent on
temperature). For our results we used a stellar age of $5\,\mathrm{Gyr}$ (the
most common age for nearby stars) and a flux of $100\,\mathrm{F_{\oplus}}$
(the expected flux of a planet orbiting at $0.1\,\mathrm{au}$ about a sun like
star). The contribution of the radiative atmosphere is small for the regime we
are dealing with ($\sim 0.1\,\mathrm{R_{\oplus}}$) so we ignore its
contribution.

The majority of higher impact parameter results in Figure 10 (greens, blues
and purples) cluster in a single group, these are the erosive hit-and-run
collisions, the dash-dotted line here is Lopez & Fortney (2014)’s model
applied to the material composition fit of equation 15. Lower impact parameter
collisions on the other hand separate into multiple groups depending on their
projectile mass (symbol shape); these groups agree with radius and density
predictions using the Denman et al. (2020) prescription to predict material
composition (dotted lines, with the shape of the open symbol at the end
denoting projectile mass). The black line denotes the expected radius if the
entirety of the atmosphere was eroded.

All simulated collisions result in ejection of atmosphere causing a reduction
in planet radius typically leading to an increase in final density. Figure 7
shows that the the atmosphere fraction for the majority of collisions clusters
around a single function of total mass, as such so do the final radii of
planets. The majority of collisions that are not close to this erosive hit-
and-run prediction line instead follow the predictions for atmosphere mass
fraction from Denman et al. (2020) for merging collisions.

Although from Figure 10 it appears that these collisions could generate Earth
analogues, i.e. objects with the Earth’s mass and size, Figure 7 reveals that
to erode enough material to reach this mass these collisions eject a
significant fraction of the mantle and as such they result in a much higher
iron fraction than the Earth. The model we are using to predict radius doesn’t
include core fraction for sake of simplicity, so we would expect these Earth
mass planets to actually be smaller and denser. Considering the relationship
between final iron content and final mass detailed in Figure 7, we expect the
predictions here to underestimate density and overestimate radius by a factor
that increases with decreasing final remnant mass.

Overall, Figure 10 shows that a collision between a single target-projectile
pair (these are each denoted by a different symbol) can result in a wide
variety of radii and densities in the remnant planets. These simulations
therefore provide direct support to the conclusions of Inamdar & Schlichting
(2016) that giant impacts can enrich the density diversity of Super-Earth
planets.

##  5 Conclusions

Collisions between a Super-Earth sized target with an atmosphere, and a second
Super-Earth sized projectile (with no atmosphere) tend to fall into two
different collision regimes. At high impact parameter and velocity they have
erosive hit-and-run collisions in which the target and projectile cores mostly
remain intact and separate, but erosion and sometimes accretion occurs during
the collision. For low impact angles and velocities we instead observe the
projectile and target combining into a single large remnant, following Denman
et al. (2020). Between these two regimes we observe graze-and-merge collisions
where the target and projectile eventually merge into one large remnant, but
not on the first collision.

We observe the transition between these two regimes to be well described by a
simple escape velocity measure, the minimum velocity required for an erosive
hit-and run being

$v_{\mathrm{split}}=\sqrt{\frac{2G(M_{\mathrm{p}}+M_{\mathrm{t}})}{b(R_{\mathrm{p}}+R_{\mathrm{t}})}}.$

For erosive hit-and-run collisions in which large secondary remnants arise,
the dependence of the mass of the largest remnant on projectile mass is solely
contained within the specific impact energy. In other words, all collisions in
the erosive limit appear to follow the same pattern with regards to specific
relative kinetic energy of impact regardless of projectile mass.

We have used our results to derive scaling laws for both the mass of
atmosphere and also the total mass lost for both regimes for collisions within
the parameter space tested. These scaling laws are summarised in appendix F
which details how they expand the prescription of Denman et al. (2020) to
oblique impacts.

The results of the simulations outlined in this paper reinforce the conclusion
from Denman et al. (2020) that it is impossible to remove the entirety of the
atmosphere of the target planet without removing a significant percentage of
the mantle as well. Considering the higher probabilities of higher impact
angles and lower velocity collisions, a giant impact being sufficiently
energetic to remove the entirety of a planet’s atmosphere is likely a rare
occurrence.

Despite the low likelihood of removing the entire atmosphere of a planet with
a single collision, the planets resulting from giant impacts cover a wide
array of different radii and densities, reinforcing the hypothesis that
planet-planet collisions are a cause of the large density diversity we observe
in Super-Earth populations.

## Acknowledgements

This work was carried out using the computational facilities of the Advanced
Computing Research Centre, University of Bristol -
http://www.bristol.ac.uk/acrc/. TD acknowledges support from an STFC
studentship (grant number: ST/R504634/1). ZML and PJC acknowledge support from
UKRI. PJC acknowledges financial support from the Simons Foundation. We thank
Christoph Mordasini for his work generating the atmosphere profiles.

This research has made use of NASA’s Astrophysics Data System. We acknowledge
the use of the python libraries matplotlib (Hunter, 2007), numpy (Harris et
al., 2020), and scipy (Virtanen et al., 2020)

## Data Availability

A summary of simulation results for this paper are provided in the appendices.
Full simulation output is available from the authors on reasonable request.
The modified version of Gadget-2 is available from Carter (2022). The equation
of state tables for use with Gadget-2 are available from Carter et al. (2019).

## References

  * Alibert et al. (2005) Alibert Y., Mordasini C., Benz W., Winisdoerffer C., 2005, Astronomy and Astrophysics, 434, 343
  * Asphaug (2010) Asphaug E., 2010, Chemie der Erde, 70, 199
  * Asphaug (2014) Asphaug E., 2014, Annual Review of Earth and Planetary Sciences, 42, 551
  * Barnes & Raymond (2004) Barnes R., Raymond S. N., 2004, The Astrophysical Journal, 617, 569
  * Benz & Asphaug (1999) Benz W., Asphaug E., 1999, Icarus, 142, 5
  * Bonomo et al. (2019) Bonomo A. S., et al., 2019, Nature Astronomy, 3, 416
  * Carter (2022) Carter P., 2022, PhilJCarter/gadget2-planetary: v1.0.0: Gadget2-Planetary initial versioning release, doi:10.5281/ZENODO.5879324, %****␣Paper_2.bbl␣Line␣50␣****https://doi.org/10.5281/zenodo.5879324#.YfFfQVFmPQ9.mendeley
  * Carter et al. (2019) Carter P. J., Lock S. J., Stewart S. T., 2019, Replication Data for: “The energy budgets of giant impacts”, doi:10.7910/DVN/YYNJSX, https://doi.org/10.7910/DVN/YYNJSX
  * Ćuk & Stewart (2012) Ćuk M., Stewart S. T., 2012, Science, pp 1047–1053
  * Denman et al. (2020) Denman T. R., Leinhardt Z. M., Carter P. J., Mordasini C., 2020, Monthly Notices of the Royal Astronomical Society, 496, 1166
  * Emsenhuber & Asphaug (2019) Emsenhuber A., Asphaug E., 2019, The Astrophysical Journal, 881, 102
  * Fabrycky et al. (2014) Fabrycky D. C., et al., 2014, Astrophysical Journal, 790
  * Fang & Margot (2013) Fang J., Margot J. L., 2013, Astrophysical Journal, 767, 115
  * Gabriel et al. (2020) Gabriel T. S. J., Jackson A. P., Asphaug E., Reufer A., Jutzi M., Benz W., 2020, The Astrophysical Journal, 892, 40
  * Gardiner (2015) Gardiner C., 2015, Blue Crystal Phase 3, https://www.acrc.bris.ac.uk/acrc/phase3.htm
  * Gardiner (2017) Gardiner C., 2017, Blue Crystal Phase 4, https://www.acrc.bris.ac.uk/acrc/phase4.htm
  * Genda et al. (2012) Genda H., Kokubo E., Ida S., 2012, Astrophysical Journal, 744, 4
  * Genda et al. (2015) Genda H., Fujita T., Kobayashi H., Tanaka H., Abe Y., 2015, Icarus, 262, 58
  * Gillion et al. (2017) Gillion M., et al., 2017, Nature, 542, 456
  * Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357
  * Hartmann & Davis (1975) Hartmann W. K., Davis D. R., 1975, Icarus, 24, 504
  * Housen & Holsapple (1990) Housen K. R., Holsapple K. A., 1990, Icarus, 84, 226
  * Hunter (2007) Hunter J. D., 2007, Computing in Science & Engineering, 9, 90
  * Hwang et al. (2017) Hwang J., Chatterjee S., Lombardi J., Steffen J., Rasio F., 2017, The Astrophysical Journal, 852, 41
  * Inamdar & Schlichting (2015) Inamdar N. K., Schlichting H. E., 2015, Monthly Notices of the Royal Astronomical Society, 448, 1751
  * Inamdar & Schlichting (2016) Inamdar N. K., Schlichting H. E., 2016, The Astrophysical Journal, 817, L13
  * Kegerreis et al. (2018) Kegerreis J. A., et al., 2018, The Astrophysical Journal, 861, 52
  * 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
  * Leinhardt & Stewart (2012) Leinhardt Z. M., Stewart S. T., 2012, Astrophysical Journal, 745, 79
  * Liu et al. (2015) Liu S. F., Hori Y., Lin D. N., Asphaug E., 2015, Astrophysical Journal, 812, 164
  * Lock et al. (2018) Lock S. J., Stewart S. T., Petaev M. I., Leinhardt Z., Mace M. T., Jacobsen S. B., Cuk M., 2018, Journal of Geophysical Research: Planets, 123, 910
  * Lopez & Fortney (2014) Lopez E. D., Fortney J. J., 2014, Astrophysical Journal, 792, 1
  * Marcus et al. (2009) Marcus R. A., Stewart S. T., Sasselov D., Hernquist L., 2009, Astrophysical Journal, 700, 118
  * Marcus et al. (2010) Marcus R. A., Sasselov D., Stewart S. T., Hernquist L., 2010, Astrophysical Journal Letters, 719, 45
  * Melosh & Vickery (1989) Melosh H. J., Vickery A. M., 1989, Nature, 338, 487
  * Mills et al. (2016) Mills S. M., Fabrycky D. C., Migaszewski C., Ford E. B., Petigura E., Isaacson H., 2016, Nature, 533, 509
  * Mordasini (2018) Mordasini C., 2018, in , Handbook of Exoplanets. Springer International Publishing, pp 2425–2474, doi:10.1007/978-3-319-55333-7_143
  * Movshovitz et al. (2016) Movshovitz N., Nimmo F., Korycansky D. G., Asphaug E., Owen J. M., 2016, Icarus, 275, 85
  * Raymond & Morbidelli (2020) Raymond S. N., Morbidelli A., 2020, Planet formation: key mechanisms and global models, http://arxiv.org/abs/2002.05756
  * Rogers et al. (2011) Rogers L. A., Bodenheimer P., Lissauer J. J., Seager S., 2011, Astrophysical Journal, 738
  * Shoemaker & Hackman (1962) Shoemaker E. M., Hackman R. J., 1962, Symposium - International Astronomical Union
  * Springel (2005) Springel V., 2005, Monthly Notices of the Royal Astronomical Society, 364, 1105
  * Stewart & Leinhardt (2009) Stewart S. T., Leinhardt Z. M., 2009, Astrophysical Journal, 691, L133
  * Valencia et al. (2006) Valencia D., O’Connell R. J., Sasselov D., 2006, Icarus, 181, 545
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Volk & Gladman (2015) Volk K., Gladman B., 2015, Astrophysical Journal Letters, 806, L26
  * Ward (1986) Ward W. R., 1986, Icarus, 67, 164
  * Watt et al. (2021) Watt L., Leinhardt Z., Su K. Y. L., 2021, Monthly Notices of the Royal Astronomical Society, 502, 2984

##  Appendix A Resolution Testing

Figure 11: Top: The ratio of largest remnant mass for various simulation
resolutions to the remnant mass for our standard resolution ($N=10^{5}$)
compared to impact velocity. The majority of runs here give results within
$2\%$ of the standard resolution chosen. Bottom: Atmosphere mass remaining as
a fraction of the atmosphere mass obtained at our standard resolution at
different impact velocities. Due to the lower resolution of the atmosphere
compared to the total mass this shows less close agreement, the majority of
results still fall within $6\%$ of the values from the chosen resolution.

Many previous papers, e.g. Genda et al. (2015), have focused on the effect of
resolution on the results of SPH simulations, despite this the precise
dependence of loss overestimation is unknown. We provide here the results of
an array of different resolution simulations to measure these effects for our
collisions. The resolutions used for these tests were $10^{4},2\times
10^{4},5\times 10^{4}$ and $10^{6}$ particles in the $5\mathrm{M_{\oplus}}$
target core and mantle compared to the standard resolution of $10^{5}$
particles in the target core and mantle.

A comparison of the results for these different resolutions is given in Figure
11. For mass of the largest remnant our results show agreement between the
resolutions to within $2\%$ for all but two tests. For atmosphere mass
remaining in the largest remnant the results detailed in Figure 11 show a
greater spread between resolutions; the majority of results agree to within
$7\%$, however there are some significant outliers which differ by up to
$25\%$.

We also observe in our results that the amount of atmosphere remaining seems
slightly dependent on resolution, with the higher resolution simulations
typically retaining more atmosphere after an impact. What is likely happening
is that the greater spatial resolution for the higher resolution simulations
means that atmosphere particles interact more frequently and there is thus
more efficient energy partitioning between particles.

Overall these resolution tests show a good agreement with our standard
resolution results both mass of the largest remnant and for atmosphere mass.

##  Appendix B Erosive Hit-and-run fitting process

Figure 12:  A: Atmosphere mass in the largest remnant as a function of
$Q_{\mathrm{R}}$; this has been fit for each impact parameter and projectile
mass by a series of power law fits (dotted lines). Erosive hit-and-run
collisions are given by filled symbols, all other collisions are open symbols.
B: Largest remnant mass normalised by target mass compared to
$Q_{\mathrm{R}}$. The erosive hit-and-run collisions for impact parameter and
projectile mass have been fit with linear fits. C: Minimum energy required for
atmosphere to be ejected, $Q_{\mathrm{0}}$, for each power law fit in A as a
function of impact parameter. There is some scatter with no obvious physical
cause so the final fit (red line) uses a common value for all impact angles
and projectiles. The mean $Q_{\mathrm{0}}$ is shown in black. D: The intercept
of the linear fits to largest remnant mass versus impact energy as a function
of impact parameter. Many of these intercepts fall below the target mass (the
mean value is in black), implying that mass could be lost without input
energy, this is un-physical so we set the intercept to the target mass (unity)
for all fits. E: The coefficient for power law fits to atmosphere mass in the
largest remnant versus impact energy, this itself follows a power law function
of the impact parameter. An initial power law fit to these coefficients is
shown in black. The final fit from figure 4, which fits for impact parameter
and impact energy dependence at the same time, is shown in red. F: The
gradient of linear fits to projectile mass in the largest remnant versus
impact energy fixed to go through zero mass loss for zero impact energy. These
gradients follow a power law (black line).

This section summarises the reasoning behind the functional forms of the final
fits used for the total mass and atmosphere masses of the largest remnant in
sections 3.1.1 and 3.1.2.

###  B.1 Atmosphere fitting

The left side of Figure 12 shows atmosphere mass in the largest remnant as a
function of specific impact energy and the dependence of fit parameters on the
impact parameter, $b$. The top left panel shows atmosphere mass normalised by
initial target atmosphere mass compared to impact energy, $Q_{\mathrm{R}}$.
Data for each individual projectile mass and impact parameter are fit with the
following power law:

$\frac{M^{\mathrm{H}}_{\mathrm{LR}}}{M^{\mathrm{H}}_{\mathrm{t}}}=\gamma_{\mathrm{atmos}}\log_{10}\left(Q_{\mathrm{R}}\right)+c_{\mathrm{atmos}},$
(17)

where $\gamma_{\mathrm{atmos}}$ and $c_{\mathrm{atmos}}$ are fit parameters.

Each of these fits (dotted lines in the top left graph of Figure 12) cross the
target atmosphere mass at an impact energy of approximately $2\times
10^{7}\,\mathrm{J\,kg^{-1}}$. The exact value of this collision energy is
given by

$Q_{\mathrm{0}}=10^{\frac{1-c_{\mathrm{atmos}}}{\gamma{\mathrm{atmos}}}},$
(18)

for each fit. This value has a reasonably low amount of scatter, that does not
appear to show any impact parameter or projectile mass dependence, so we
elected to use an impact parameter dependant fit parameter for the final
combined fit (section 3.1.2. The $Q_{\mathrm{0}}$ measured in the final fit is
given by the red line in the left middle graph in Figure 12, the mean value of
$Q_{\mathrm{0}}$ from our separate fits for each proectile mass and impact
parameter is given by the black line.

The power law coefficient $\gamma_{\mathrm{atmos}}$ for each of these
individual fits is given in the bottom left graph of Figure 12. Apart from the
head-on like collisions ($b<0.2$), these have a power law dependence on impact
parameter; the black line shows a power law fit to these coefficients, whereas
the red line is the impact parameter dependence of the coefficient obtained
for our final fit (equation 13) where we fit all the projectiles and impact
angles simultaneously.

###  B.2 Largest Remnant mass fitting

The top right graph of Figure 12 compares largest remnant mass and impact
energy, the erosive hit-and-run collisions are shown with solid symbols. These
erosive hit-and-run collisions show a linear dependence of remnant mass on
impact energy, each separate impact parameter and projectile mass was
therefore fit with:

$\frac{M_{\mathrm{LR}}}{M_{\mathrm{t}}}=\Gamma_{\mathrm{EHR}}Q_{\mathrm{R}}+C,$
(19)

where $\Gamma_{\mathrm{EHR}}$ and $C$ are the gradient and the intercept
respectively.

The intercept for each of these fits is shown in the middle right graph of
Figure 12. Many of these values are below the target mass (less than 1), as is
the mean intercept (the black line), this is unphysical as it implies material
can be lost from the target without any impact energy. What we actually
observe at low impact energies is a curve upwards away from the linear fit, so
it is likely that mass loss for low impact energies, close to the transition
region, is non-linear. However, we chose to use the simplest possible model to
describe the data and so used a linear fit fixed to the target mass for zero
impact energy (red line), i.e. $C=1$, despite the low energy differences this
shows good agreement with the majority of the data.

The bottom graph in Figure 12 details the dependence of the gradient of the
largest remnant mass to impact energy relation on impact parameter. The
gradient becomes less negative by a decreasing amount with increasing impact
parameter, so we fit it with a power law (black line) which shows good
agreement. The dependence of gradient on impact parameter obtained from our
final fit (equation 6), which simultaneously fit to all projectile masses and
impact parameters, is shown in red.

##  Appendix C Graze-and-Merge re-collision time

To approximate the longest possible orbit time that could occur before re-
collision, without the secondary remnant escaping the largest remnant’s
gravitational influence, we first need to calculate the Hill radius of the
target planet $r_{\mathrm{H}}$:

$r_{\mathrm{H}}=a_{\mathrm{i}}\sqrt[3]{\frac{M_{\mathrm{t}}}{3M_{\mathrm{0}}}},$
(20)

where $a_{\mathrm{i}}$ is the semi-major axis of the target about the star it
orbits and $M_{\mathrm{0}}$ that star’s mass. The semi-major axis of an orbit
about the target after a grazing collision is half the sum of the apoapsis and
periapsis distance and so the longest possible orbit of the secondary remnant
without it escaping is
$a_{\mathrm{ii}}=(r_{\mathrm{H}}+R_{\mathrm{t}}+R_{\mathrm{p}})/2$. From this
semi-major axis we can calculate the orbital period of this limiting orbit of
the projectile about the target to be

$T=2\pi\sqrt{\frac{a^{3}_{\mathrm{ii}}}{G(M_{\mathrm{t}}+M_{\mathrm{p}})}}.$
(21)

Applying this to our scenario, for a close orbiting super Earth
$0.1\,\mathrm{au}$ from a sun-like star (like we use in section 4.4 for
density estimates) this gives us re-collision times of $5-7\,\mathrm{hrs}$ for
Super-Earth mass projectiles. This time period is well within the time limit
of $27.7\,\mathrm{hrs}$ of our simulations. So we would be able to model such
re-collisions in our simulations. We do not model the central star’s gravity
in our simulations, however, so our simulations include graze-and-merge
collisions on longer re-collision timescales, as such we removed the graze-
and-merge collisions from our collision fits so that they could remain
agnostic to orbital distance.

##  Appendix D Remeasuring the point of collision

Figure 13:  Variation of the specific impact energy as a function of other
collision parameters. The colour of the points indicates the impact parameter,
the shape indicates the mass of the projectile. Top: Ratio of measured to
expected $Q_{\mathrm{R}}$ vs measured impact velocity. There is a significant
decrease in energy compared to the prediction for low velocities where a
change in momentum due to drag can cause a much larger proportional change in
velocity and thus kinetic energy. Middle: Projectile mass compared with
measured to expected $Q_{\mathrm{R}}$ ratio. Although there is a slight
decrease in the mean impact energy for smaller projectiles due to lower
momenta, projectile mass does not have a significant effect. Bottom: The
measured to expected $Q_{\mathrm{R}}$ ratio compared with measured impact
parameter. Impact parameter difference does not appear to cause a significant
change to impact energy; note, however, that impacts with the largest change
in kinetic energy tend to show the largest change from expected impact
parameter (shown with dotted lines).

For ease of comparison to previous works on cratering and giant impacts (for
example Leinhardt & Stewart 2012; Marcus et al. 2009, 2010) and also to give a
simple, reproducible approach that is independent of atmosphere scale height,
we have chosen to measure our impact parameters at the point of collision
between the mantle surfaces of projectile and target, or at the point of
closest approach for highly grazing collisions. This method is not without its
drawbacks however, the main one which needs to be considered is the effect of
atmospheric drag.

To understand the difference caused by this drag we re-simulate the point of
collision at a high snapshot output frequency and remeasure the relative
velocity, separation, and impact parameter of the centre of masses at the
point where the mantles collide (or reach their point of closest approach if
$b>1$).

Because of this atmospheric drag, the velocity at the point of collision for
an impact is reduced slightly. This means that the measured specific relative
kinetic energy of impact, $Q_{\mathrm{R}}^{\mathrm{meas}}$, is typically less
than that initially predicted $Q_{\mathrm{R}}^{\mathrm{pred}}$. Figure 13
compares the ratio of measured to expected $Q_{\mathrm{R}}$ with various
collision parameters it is likely to be dependent on. The top two graphs
compare the energy ratio to relative velocity and projectile mass; as might be
expected, lower velocities and projectile masses typically mean a greater
difference in impact energy, because the change in momentum due to the drag
force is a larger percentage of the overall momentum of the projectile. The
decrease we see in measured $Q_{\mathrm{R}}$ for lower impact velocities is
stronger than that for lower projectile masses, this is due to $Q_{R}$ having
a squared dependence on velocity (equation 1). The dependence of $Q_{R}$ on
projectile mass is somewhat more complicated, being somewhere between
approximately zero proportionality for $M_{\mathrm{p}}\approx M_{\mathrm{t}}$
and approximately linear dependence for $M_{\mathrm{p}}<<M_{\mathrm{t}}$.

Predicted impact parameter does not appear to have a significant effect on the
amount of kinetic energy lost to atmospheric drag (see bottom panel Figure
13). We do, however, observe that, especially at higher impact parameters, the
greater the measured deflection from the intended impact parameter the larger
the percentage decrease in impact energy. Atmospheric pressure is deflecting
the projectiles away from the target causing them to spend more time passing
through the atmosphere before collision, thus producing a greater decrease in
projectile momentum and therefore impact energy.

For the majority of collisions, apart from those at the lowest impact
velocity, the drag effects on kinetic energy are relatively small, typically
causing a decrease of $<6\%$. Since this effect is small we therefore used the
expected parameters for our analyses in this paper.

##  Appendix E Data tables

We present here data tables, Table 1 provides details for each of the bodies
we simulated collisions with, the other Tables each list the results of our
simulations for a particular projectile mass. All collisions used a
$6.25\,\mathrm{M_{\oplus}}$ target, the projectile mass for each of the
collision results tables are as follows: Table 2 is for
$1\,\mathrm{M_{\oplus}}$ projectiles, Table 3 is for $2\,\mathrm{M_{\oplus}}$
projectiles, Table 4 is for $3\,\mathrm{M_{\oplus}}$ projectiles, Table 5 is
for $4\,\mathrm{M_{\oplus}}$ projectiles, and Table 6 is for
$5\,\mathrm{M_{\oplus}}$ projectiles.

Table 1: Initial masses, $M$ of each planet used in our simulations, as well as the masses of each component (Fe – iron, Si – forsterite, H – hydrogen) in each body, and the radius of the outermost simulation particle in each material layer. T indicates the planet was used as a target, whereas P were projectiles. The final column details the measured gravitational binding energy of each object.  Type | $M$ | $M_{\mathrm{Fe}}$ | $R_{\mathrm{Fe}}$ | $M_{\mathrm{Si}}$ | $R_{\mathrm{Si}}$ | $M_{\mathrm{H}}$ | $R_{\mathrm{H}}$ | $U_{\mathrm{g}}^{\mathrm{n}}$
---|---|---|---|---|---|---|---|---
| $\mathrm{M_{\oplus}}$ | $\mathrm{M_{\oplus}}$ | $\mathrm{R_{\oplus}}$ | $\mathrm{M_{\oplus}}$ | $\mathrm{R_{\oplus}}$ | $\mathrm{M_{\oplus}}$ | $\mathrm{R_{\oplus}}$ | $\mathrm{J}$
T | 6.26 | 1.66 | 0.76 | 3.34 | 1.51 | 1.25 | 5.34 | 5.06$\times$1033
P | 1.00 | 0.33 | 0.50 | 0.67 | 0.96 | 0.00 | - | 2.53$\times$1032
P | 2.00 | 0.67 | 0.60 | 1.33 | 1.17 | 0.00 | - | 8.40$\times$1032
P | 3.01 | 1.01 | 0.68 | 1.99 | 1.32 | 0.00 | - | 1.70$\times$1033
P | 4.01 | 1.32 | 0.72 | 2.69 | 1.43 | 0.00 | - | 2.79$\times$1033
P | 5.01 | 1.66 | 0.77 | 3.34 | 1.52 | 0.00 | - | 4.13$\times$1033
Table 2: A summary of the results of the simulations of collisions of the $1\,\mathrm{M_{\oplus}}$ projectile with the $6.25\,\mathrm{M_{\oplus}}$ target. Full planet data is given in Table 1, the rest of the collision results are in Tables 3, 4, 5, and 6.  $b$ | $v$ | $M_{\mathrm{LR}}$ | $f_{\mathrm{Fe}}$ | $f_{\mathrm{Si}}$ | $f_{\mathrm{H}}$ | $M_{\mathrm{SLR}}$
---|---|---|---|---|---|---
| $\mathrm{km\,s^{-1}}$ | $\mathrm{M_{\oplus}}$ |  |  |  | $\mathrm{M_{\oplus}}$
0.00 | 20.0 | 7.09 | 0.28 | 0.57 | 0.15 | -
0.00 | 25.0 | 6.95 | 0.29 | 0.58 | 0.14 | -
0.00 | 30.0 | 6.73 | 0.30 | 0.59 | 0.11 | -
0.00 | 35.0 | 6.52 | 0.31 | 0.61 | 0.08 | -
0.00 | 40.0 | 6.32 | 0.32 | 0.62 | 0.06 | -
0.00 | 45.0 | 5.81 | 0.34 | 0.62 | 0.04 | -
0.00 | 50.0 | 5.08 | 0.39 | 0.58 | 0.02 | -
0.10 | 20.0 | 7.11 | 0.28 | 0.56 | 0.15 | -
0.10 | 25.0 | 6.96 | 0.29 | 0.58 | 0.14 | -
0.10 | 27.5 | 6.87 | 0.29 | 0.58 | 0.13 | -
0.10 | 30.0 | 6.79 | 0.29 | 0.59 | 0.12 | -
0.10 | 35.0 | 6.58 | 0.30 | 0.60 | 0.09 | -
0.10 | 40.0 | 6.28 | 0.32 | 0.61 | 0.07 | -
0.10 | 50.0 | 5.02 | 0.38 | 0.59 | 0.03 | -
0.34 | 20.0 | 7.11 | 0.28 | 0.56 | 0.15 | -
0.34 | 25.0 | 6.98 | 0.29 | 0.57 | 0.14 | -
0.34 | 27.5 | 6.86 | 0.29 | 0.57 | 0.13 | -
0.34 | 30.0 | 6.69 | 0.30 | 0.57 | 0.13 | -
0.34 | 35.0 | 6.38 | 0.31 | 0.57 | 0.11 | -
0.34 | 40.0 | 6.07 | 0.33 | 0.57 | 0.10 | -
0.34 | 45.0 | 5.40 | 0.32 | 0.59 | 0.09 | 0.28
0.34 | 50.0 | 5.09 | 0.33 | 0.59 | 0.08 | 0.20
0.50 | 20.0 | 7.12 | 0.28 | 0.56 | 0.16 | -
0.50 | 25.0 | 6.90 | 0.29 | 0.56 | 0.15 | -
0.50 | 27.5 | 6.72 | 0.30 | 0.56 | 0.14 | -
0.50 | 30.0 | 6.44 | 0.30 | 0.56 | 0.14 | 0.20
0.50 | 35.0 | 5.99 | 0.30 | 0.57 | 0.14 | 0.18
0.50 | 40.0 | 5.80 | 0.30 | 0.58 | 0.13 | 0.18
0.50 | 45.0 | 5.62 | 0.31 | 0.58 | 0.12 | 0.14
0.50 | 50.0 | 5.43 | 0.31 | 0.58 | 0.10 | 0.10
0.71 | 20.0 | 7.12 | 0.28 | 0.56 | 0.16 | -
0.71 | 25.0 | 6.28 | 0.27 | 0.56 | 0.17 | 0.79
0.71 | 27.5 | 6.20 | 0.27 | 0.56 | 0.17 | 0.80
0.71 | 30.0 | 6.13 | 0.28 | 0.56 | 0.16 | 0.80
0.71 | 35.0 | 5.99 | 0.28 | 0.57 | 0.15 | 0.81
0.71 | 40.0 | 5.91 | 0.28 | 0.57 | 0.15 | 0.78
0.71 | 45.0 | 5.84 | 0.29 | 0.58 | 0.14 | 0.77
0.71 | 50.0 | 5.77 | 0.29 | 0.58 | 0.13 | 0.73
0.94 | 20.0 | 7.00 | 0.29 | 0.56 | 0.15 | -
0.94 | 25.0 | 6.18 | 0.27 | 0.55 | 0.18 | 0.94
0.94 | 27.5 | 6.13 | 0.27 | 0.55 | 0.18 | 0.95
0.94 | 30.0 | 6.10 | 0.27 | 0.55 | 0.17 | 0.96
0.94 | 35.0 | 6.04 | 0.28 | 0.56 | 0.17 | 0.96
0.94 | 40.0 | 6.00 | 0.28 | 0.56 | 0.16 | 0.97
0.94 | 45.0 | 5.96 | 0.28 | 0.56 | 0.16 | 0.96
0.94 | 50.0 | 5.92 | 0.28 | 0.57 | 0.15 | 0.96
1.10 | 20.0 | 6.65 | 0.27 | 0.55 | 0.18 | 0.54
1.10 | 25.0 | 6.16 | 0.27 | 0.55 | 0.18 | 0.98
1.10 | 27.5 | 6.13 | 0.27 | 0.55 | 0.18 | 1.00
1.10 | 30.0 | 6.11 | 0.27 | 0.55 | 0.18 | 0.99
1.10 | 35.0 | 6.09 | 0.27 | 0.55 | 0.18 | 1.00
1.10 | 40.0 | 6.06 | 0.27 | 0.55 | 0.17 | 0.99
1.10 | 45.0 | 6.03 | 0.28 | 0.55 | 0.17 | 0.99
1.10 | 50.0 | 6.00 | 0.28 | 0.56 | 0.17 | 0.99
1.30 | 20.0 | 6.23 | 0.27 | 0.54 | 0.19 | 0.98
1.30 | 25.0 | 6.19 | 0.27 | 0.54 | 0.19 | 1.00
1.30 | 27.5 | 6.18 | 0.27 | 0.54 | 0.19 | 1.00
1.30 | 30.0 | 6.16 | 0.27 | 0.54 | 0.19 | 1.00
1.30 | 35.0 | 6.15 | 0.27 | 0.54 | 0.18 | 1.00
1.30 | 40.0 | 6.13 | 0.27 | 0.55 | 0.18 | 1.00
1.30 | 45.0 | 6.11 | 0.27 | 0.55 | 0.18 | 1.00
1.30 | 50.0 | 6.08 | 0.27 | 0.55 | 0.18 | 1.00
Table 3: A summary of the results of the simulations of collisions of the $2\,\mathrm{M_{\oplus}}$ projectile with the $6.25\,\mathrm{M_{\oplus}}$ target. Full planet data is given in Table 1, the rest of the collision results are in Tables 2, 4, 5, and 6. $b$ | $v$ | $M_{\mathrm{LR}}$ | $f_{\mathrm{Fe}}$ | $f_{\mathrm{Si}}$ | $f_{\mathrm{H}}$ | $M_{\mathrm{SLR}}$
---|---|---|---|---|---|---
| $\mathrm{km\,s^{-1}}$ | $\mathrm{M_{\oplus}}$ |  |  |  | $\mathrm{M_{\oplus}}$
0.00 | 20.0 | 7.94 | 0.29 | 0.59 | 0.12 | -
0.00 | 25.0 | 7.72 | 0.30 | 0.60 | 0.09 | -
0.00 | 30.0 | 7.51 | 0.31 | 0.62 | 0.07 | -
0.00 | 35.0 | 6.93 | 0.34 | 0.62 | 0.05 | -
0.00 | 40.0 | 6.07 | 0.38 | 0.58 | 0.04 | -
0.00 | 45.0 | 5.09 | 0.44 | 0.55 | 0.02 | -
0.00 | 50.0 | 3.82 | 0.48 | 0.52 | - | -
0.10 | 20.0 | 7.98 | 0.29 | 0.59 | 0.12 | -
0.10 | 25.0 | 7.77 | 0.30 | 0.60 | 0.10 | -
0.10 | 27.5 | 7.69 | 0.30 | 0.61 | 0.09 | -
0.10 | 30.0 | 7.52 | 0.31 | 0.61 | 0.07 | -
0.10 | 35.0 | 7.05 | 0.33 | 0.61 | 0.06 | -
0.10 | 40.0 | 6.20 | 0.38 | 0.59 | 0.04 | -
0.10 | 50.0 | 3.85 | 0.48 | 0.51 | 0.01 | -
0.34 | 20.0 | 8.04 | 0.29 | 0.58 | 0.13 | -
0.34 | 25.0 | 7.85 | 0.30 | 0.59 | 0.11 | -
0.34 | 27.5 | 7.54 | 0.31 | 0.59 | 0.10 | 0.14
0.34 | 30.0 | 7.25 | 0.32 | 0.58 | 0.09 | -
0.34 | 35.0 | 5.75 | 0.31 | 0.60 | 0.09 | 1.12
0.34 | 40.0 | 5.36 | 0.32 | 0.60 | 0.08 | 0.96
0.34 | 45.0 | 4.98 | 0.34 | 0.60 | 0.06 | 0.81
0.34 | 50.0 | 4.61 | 0.36 | 0.59 | 0.05 | 0.56
0.50 | 20.0 | 8.07 | 0.29 | 0.58 | 0.13 | -
0.50 | 25.0 | 7.85 | 0.30 | 0.59 | 0.11 | -
0.50 | 27.5 | 7.81 | 0.30 | 0.59 | 0.11 | -
0.50 | 30.0 | 6.08 | 0.30 | 0.57 | 0.13 | 1.64
0.50 | 35.0 | 5.77 | 0.30 | 0.58 | 0.12 | 1.45
0.50 | 40.0 | 5.57 | 0.30 | 0.59 | 0.11 | 1.35
0.50 | 50.0 | 5.15 | 0.32 | 0.60 | 0.08 | 1.04
0.71 | 20.0 | 8.11 | 0.29 | 0.58 | 0.14 | -
0.71 | 25.0 | 6.19 | 0.27 | 0.57 | 0.16 | 1.85
0.71 | 27.5 | 6.13 | 0.27 | 0.57 | 0.16 | 1.82
0.71 | 30.0 | 6.06 | 0.28 | 0.57 | 0.15 | 1.82
0.71 | 35.0 | 5.91 | 0.28 | 0.58 | 0.14 | 1.79
0.71 | 40.0 | 5.78 | 0.29 | 0.58 | 0.13 | 1.77
0.71 | 50.0 | 5.63 | 0.30 | 0.59 | 0.12 | 1.72
0.94 | 20.0 | 7.94 | 0.29 | 0.57 | 0.13 | -
0.94 | 25.0 | 6.14 | 0.27 | 0.55 | 0.18 | 1.96
0.94 | 27.5 | 6.10 | 0.27 | 0.55 | 0.17 | 1.97
0.94 | 30.0 | 6.06 | 0.27 | 0.55 | 0.17 | 1.97
0.94 | 35.0 | 6.01 | 0.28 | 0.56 | 0.17 | 1.98
0.94 | 40.0 | 5.95 | 0.28 | 0.56 | 0.16 | 1.97
0.94 | 50.0 | 5.86 | 0.28 | 0.57 | 0.15 | 1.98
1.10 | 20.0 | 8.06 | 0.29 | 0.56 | 0.15 | 0.13
1.10 | 25.0 | 6.14 | 0.27 | 0.55 | 0.18 | 2.00
1.10 | 27.5 | 6.12 | 0.27 | 0.55 | 0.18 | 2.01
1.10 | 30.0 | 6.10 | 0.27 | 0.55 | 0.18 | 2.01
1.10 | 35.0 | 6.06 | 0.27 | 0.55 | 0.17 | 2.00
1.10 | 40.0 | 6.04 | 0.28 | 0.55 | 0.17 | 2.00
1.10 | 50.0 | 5.98 | 0.28 | 0.56 | 0.16 | 2.00
1.30 | 20.0 | 6.20 | 0.27 | 0.54 | 0.19 | 2.01
1.30 | 25.0 | 6.18 | 0.27 | 0.54 | 0.19 | 2.01
1.30 | 27.5 | 6.17 | 0.27 | 0.54 | 0.19 | 2.00
1.30 | 30.0 | 6.16 | 0.27 | 0.54 | 0.19 | 2.01
1.30 | 35.0 | 6.15 | 0.27 | 0.54 | 0.19 | 2.00
1.30 | 40.0 | 6.12 | 0.27 | 0.55 | 0.18 | 2.00
1.30 | 50.0 | 6.08 | 0.27 | 0.55 | 0.18 | 2.00
Table 4: A summary of the results of the simulations of collisions of the $3\,\mathrm{M_{\oplus}}$ projectile with the $6.25\,\mathrm{M_{\oplus}}$ target. Full planet data is given in Table 1, the rest of the collision results are in Tables 2, 3, 5, and 6. $b$ | $v$ | $M_{\mathrm{LR}}$ | $f_{\mathrm{Fe}}$ | $f_{\mathrm{Si}}$ | $f_{\mathrm{H}}$ | $M_{\mathrm{SLR}}$
---|---|---|---|---|---|---
| $\mathrm{km\,s^{-1}}$ | $\mathrm{M_{\oplus}}$ |  |  |  | $\mathrm{M_{\oplus}}$
0.00 | 20.0 | 8.75 | 0.31 | 0.60 | 0.09 | -
0.00 | 25.0 | 8.72 | 0.31 | 0.61 | 0.08 | -
0.00 | 30.0 | 8.41 | 0.32 | 0.62 | 0.06 | -
0.00 | 35.0 | 7.45 | 0.36 | 0.60 | 0.04 | -
0.00 | 40.0 | 6.39 | 0.41 | 0.56 | 0.03 | -
0.00 | 45.0 | 5.01 | 0.44 | 0.54 | 0.01 | -
0.00 | 50.0 | 3.51 | 0.52 | 0.48 | - | -
0.10 | 20.0 | 8.86 | 0.30 | 0.60 | 0.10 | -
0.10 | 25.0 | 8.69 | 0.31 | 0.61 | 0.08 | -
0.10 | 27.5 | 8.60 | 0.31 | 0.62 | 0.07 | -
0.10 | 30.0 | 8.41 | 0.32 | 0.62 | 0.06 | -
0.10 | 35.0 | 7.64 | 0.35 | 0.60 | 0.05 | -
0.10 | 40.0 | 6.53 | 0.41 | 0.56 | 0.03 | -
0.10 | 50.0 | 3.21 | 0.52 | 0.48 | - | -
0.34 | 20.0 | 9.01 | 0.30 | 0.59 | 0.11 | -
0.34 | 25.0 | 8.76 | 0.31 | 0.61 | 0.09 | -
0.34 | 27.5 | 8.60 | 0.31 | 0.61 | 0.08 | -
0.34 | 30.0 | 8.24 | 0.32 | 0.62 | 0.05 | 0.07
0.34 | 35.0 | 5.62 | 0.31 | 0.61 | 0.08 | 2.08
0.34 | 40.0 | 5.19 | 0.33 | 0.61 | 0.06 | 1.78
0.34 | 45.0 | 4.76 | 0.35 | 0.60 | 0.05 | 1.57
0.34 | 50.0 | 4.24 | 0.39 | 0.58 | 0.03 | 1.31
0.50 | 20.0 | 9.04 | 0.30 | 0.59 | 0.11 | -
0.50 | 25.0 | 8.79 | 0.30 | 0.61 | 0.09 | -
0.50 | 27.5 | 8.82 | 0.30 | 0.60 | 0.09 | -
0.50 | 30.0 | 5.91 | 0.29 | 0.59 | 0.12 | 2.82
0.50 | 35.0 | 5.64 | 0.30 | 0.59 | 0.11 | 2.67
0.50 | 40.0 | 5.35 | 0.31 | 0.59 | 0.09 | 2.47
0.50 | 45.0 | 5.20 | 0.32 | 0.60 | 0.08 | 2.29
0.50 | 50.0 | 4.98 | 0.33 | 0.60 | 0.07 | 2.09
0.71 | 20.0 | 9.11 | 0.29 | 0.59 | 0.12 | -
0.71 | 25.0 | 6.11 | 0.27 | 0.57 | 0.15 | 2.91
0.71 | 27.5 | 6.05 | 0.28 | 0.58 | 0.15 | 2.88
0.71 | 30.0 | 5.98 | 0.28 | 0.58 | 0.14 | 2.85
0.71 | 35.0 | 5.83 | 0.29 | 0.58 | 0.14 | 2.81
0.71 | 40.0 | 5.69 | 0.29 | 0.58 | 0.13 | 2.80
0.71 | 45.0 | 5.62 | 0.30 | 0.58 | 0.12 | 2.75
0.71 | 50.0 | 5.55 | 0.30 | 0.59 | 0.11 | 2.74
0.94 | 20.0 | 9.11 | 0.29 | 0.58 | 0.12 | -
0.94 | 25.0 | 6.12 | 0.27 | 0.55 | 0.17 | 2.99
0.94 | 27.5 | 6.06 | 0.27 | 0.56 | 0.17 | 3.00
0.94 | 30.0 | 6.03 | 0.28 | 0.56 | 0.17 | 2.99
0.94 | 35.0 | 5.98 | 0.28 | 0.56 | 0.16 | 2.99
0.94 | 40.0 | 5.93 | 0.28 | 0.56 | 0.15 | 2.99
0.94 | 45.0 | 5.88 | 0.28 | 0.57 | 0.15 | 2.99
0.94 | 50.0 | 5.83 | 0.29 | 0.57 | 0.14 | 2.98
1.10 | 20.0 | 9.15 | 0.29 | 0.58 | 0.13 | -
1.10 | 25.0 | 6.12 | 0.27 | 0.55 | 0.18 | 3.02
1.10 | 27.5 | 6.11 | 0.27 | 0.55 | 0.18 | 3.02
1.10 | 30.0 | 6.10 | 0.27 | 0.55 | 0.18 | 3.01
1.10 | 35.0 | 6.06 | 0.27 | 0.55 | 0.17 | 3.01
1.10 | 40.0 | 6.03 | 0.28 | 0.56 | 0.17 | 3.01
1.10 | 45.0 | 5.99 | 0.28 | 0.56 | 0.16 | 3.00
1.10 | 50.0 | 5.97 | 0.28 | 0.56 | 0.16 | 3.00
1.30 | 20.0 | 6.19 | 0.27 | 0.54 | 0.19 | 3.03
1.30 | 25.0 | 6.18 | 0.27 | 0.54 | 0.19 | 3.01
1.30 | 27.5 | 6.18 | 0.27 | 0.54 | 0.19 | 3.01
1.30 | 30.0 | 6.16 | 0.27 | 0.54 | 0.19 | 3.01
1.30 | 35.0 | 6.14 | 0.27 | 0.55 | 0.18 | 3.01
1.30 | 40.0 | 6.12 | 0.27 | 0.55 | 0.18 | 3.01
1.30 | 45.0 | 6.10 | 0.27 | 0.55 | 0.18 | 3.01
1.30 | 50.0 | 6.08 | 0.27 | 0.55 | 0.18 | 3.01
Table 5: A summary of the results of the simulations of collisions of the $4\,\mathrm{M_{\oplus}}$ projectile with the $6.25\,\mathrm{M_{\oplus}}$ target. Full planet data is given in Table 1, the rest of the collision results are in Tables 2, 3, 4, and 6. $b$ | $v$ | $M_{\mathrm{LR}}$ | $f_{\mathrm{Fe}}$ | $f_{\mathrm{Si}}$ | $f_{\mathrm{H}}$ | $M_{\mathrm{SLR}}$
---|---|---|---|---|---|---
| $\mathrm{km\,s^{-1}}$ | $\mathrm{M_{\oplus}}$ |  |  |  | $\mathrm{M_{\oplus}}$
0.00 | 20.0 | 9.73 | 0.31 | 0.62 | 0.08 | -
0.00 | 25.0 | 9.69 | 0.31 | 0.62 | 0.07 | -
0.00 | 30.0 | 9.40 | 0.32 | 0.63 | 0.05 | -
0.00 | 35.0 | 8.32 | 0.36 | 0.60 | 0.04 | -
0.00 | 40.0 | 7.11 | 0.42 | 0.56 | 0.03 | -
0.00 | 45.0 | 5.45 | 0.44 | 0.55 | 0.01 | -
0.00 | 50.0 | 3.70 | 0.53 | 0.47 | - | -
0.10 | 20.0 | 9.83 | 0.30 | 0.61 | 0.09 | -
0.10 | 25.0 | 9.62 | 0.31 | 0.62 | 0.07 | -
0.10 | 27.5 | 9.51 | 0.31 | 0.63 | 0.06 | -
0.10 | 30.0 | 9.41 | 0.32 | 0.63 | 0.05 | -
0.10 | 35.0 | 8.42 | 0.35 | 0.61 | 0.04 | -
0.10 | 40.0 | 7.20 | 0.41 | 0.57 | 0.02 | -
0.10 | 50.0 | 3.08 | 0.53 | 0.47 | - | 0.13
0.34 | 20.0 | 9.96 | 0.30 | 0.60 | 0.10 | -
0.34 | 25.0 | 9.68 | 0.31 | 0.62 | 0.07 | -
0.34 | 27.5 | 9.52 | 0.31 | 0.63 | 0.06 | -
0.34 | 30.0 | 9.27 | 0.32 | 0.64 | 0.04 | -
0.34 | 35.0 | 5.50 | 0.31 | 0.62 | 0.07 | 3.62
0.34 | 40.0 | 5.05 | 0.33 | 0.61 | 0.05 | 2.91
0.34 | 45.0 | 4.49 | 0.37 | 0.60 | 0.04 | 2.44
0.34 | 50.0 | 3.86 | 0.42 | 0.56 | 0.02 | 2.13
0.50 | 20.0 | 10.01 | 0.30 | 0.60 | 0.10 | -
0.50 | 25.0 | 9.74 | 0.31 | 0.62 | 0.08 | -
0.50 | 27.5 | 9.45 | 0.32 | 0.63 | 0.06 | -
0.50 | 30.0 | 5.75 | 0.30 | 0.60 | 0.11 | 3.94
0.50 | 35.0 | 5.56 | 0.30 | 0.60 | 0.10 | 3.73
0.50 | 40.0 | 5.28 | 0.32 | 0.60 | 0.08 | 3.51
0.50 | 45.0 | 5.01 | 0.33 | 0.60 | 0.07 | 3.31
0.50 | 50.0 | 4.83 | 0.34 | 0.60 | 0.06 | 3.11
0.71 | 20.0 | 10.09 | 0.30 | 0.60 | 0.11 | -
0.71 | 25.0 | 9.97 | 0.30 | 0.60 | 0.10 | -
0.71 | 27.5 | 5.95 | 0.28 | 0.58 | 0.14 | 3.96
0.71 | 30.0 | 5.89 | 0.28 | 0.58 | 0.14 | 3.93
0.71 | 35.0 | 5.75 | 0.29 | 0.58 | 0.13 | 3.86
0.71 | 40.0 | 5.62 | 0.30 | 0.58 | 0.12 | 3.82
0.71 | 45.0 | 5.52 | 0.30 | 0.59 | 0.11 | 3.78
0.71 | 50.0 | 5.48 | 0.30 | 0.59 | 0.10 | 3.74
0.94 | 20.0 | 10.16 | 0.29 | 0.59 | 0.11 | -
0.94 | 25.0 | 6.08 | 0.27 | 0.56 | 0.17 | 4.02
0.94 | 27.5 | 6.04 | 0.28 | 0.56 | 0.17 | 4.02
0.94 | 30.0 | 6.01 | 0.28 | 0.56 | 0.16 | 4.01
0.94 | 35.0 | 5.95 | 0.28 | 0.56 | 0.16 | 4.00
0.94 | 40.0 | 5.90 | 0.28 | 0.57 | 0.15 | 4.00
0.94 | 45.0 | 5.85 | 0.28 | 0.57 | 0.14 | 4.00
0.94 | 50.0 | 5.81 | 0.29 | 0.57 | 0.14 | 4.00
1.10 | 20.0 | 10.07 | 0.30 | 0.60 | 0.11 | -
1.10 | 25.0 | 6.12 | 0.27 | 0.55 | 0.18 | 4.03
1.10 | 27.5 | 6.10 | 0.27 | 0.55 | 0.18 | 4.03
1.10 | 30.0 | 6.09 | 0.27 | 0.55 | 0.18 | 4.02
1.10 | 35.0 | 6.06 | 0.27 | 0.55 | 0.17 | 4.01
1.10 | 40.0 | 6.02 | 0.28 | 0.56 | 0.17 | 4.01
1.10 | 45.0 | 5.98 | 0.28 | 0.56 | 0.16 | 4.01
1.10 | 50.0 | 5.95 | 0.28 | 0.56 | 0.16 | 4.01
1.30 | 20.0 | 6.18 | 0.27 | 0.54 | 0.19 | 4.04
1.30 | 25.0 | 6.18 | 0.27 | 0.54 | 0.19 | 4.02
1.30 | 27.5 | 6.17 | 0.27 | 0.54 | 0.19 | 4.01
1.30 | 30.0 | 6.17 | 0.27 | 0.54 | 0.19 | 4.01
1.30 | 40.0 | 6.11 | 0.27 | 0.55 | 0.18 | 4.01
1.30 | 45.0 | 6.09 | 0.27 | 0.55 | 0.18 | 4.01
Table 6: A summary of the results of the simulations of collisions of the $5\,\mathrm{M_{\oplus}}$ projectile with the $6.25\,\mathrm{M_{\oplus}}$ target. Full planet data is given in Table 1, the rest of the collision results are in Tables 2, 3, 4, and 5. $b$ | $v$ | $M_{\mathrm{LR}}$ | $f_{\mathrm{Fe}}$ | $f_{\mathrm{Si}}$ | $f_{\mathrm{H}}$ | $M_{\mathrm{SLR}}$
---|---|---|---|---|---|---
| $\mathrm{km\,s^{-1}}$ | $\mathrm{M_{\oplus}}$ |  |  |  | $\mathrm{M_{\oplus}}$
0.00 | 20.0 | 10.63 | 0.31 | 0.62 | 0.06 | -
0.00 | 25.0 | 10.66 | 0.31 | 0.62 | 0.06 | -
0.00 | 30.0 | 10.40 | 0.32 | 0.63 | 0.05 | -
0.00 | 35.0 | 9.35 | 0.36 | 0.61 | 0.03 | -
0.00 | 40.0 | 7.99 | 0.42 | 0.56 | 0.02 | -
0.00 | 45.0 | 6.25 | 0.44 | 0.55 | 0.01 | -
0.00 | 50.0 | 4.45 | 0.52 | 0.48 | - | -
0.10 | 20.0 | 10.81 | 0.31 | 0.62 | 0.08 | -
0.10 | 25.0 | 10.56 | 0.32 | 0.63 | 0.06 | -
0.10 | 27.5 | 10.47 | 0.32 | 0.63 | 0.05 | -
0.10 | 30.0 | 10.40 | 0.32 | 0.63 | 0.05 | -
0.10 | 35.0 | 9.48 | 0.35 | 0.61 | 0.03 | -
0.10 | 40.0 | 8.08 | 0.41 | 0.57 | 0.02 | -
0.10 | 50.0 | 4.09 | 0.51 | 0.49 | - | 0.01
0.34 | 20.0 | 10.93 | 0.30 | 0.61 | 0.09 | -
0.34 | 25.0 | 10.67 | 0.31 | 0.62 | 0.07 | -
0.34 | 27.5 | 10.61 | 0.31 | 0.63 | 0.06 | -
0.34 | 30.0 | 10.29 | 0.32 | 0.64 | 0.04 | -
0.34 | 35.0 | 5.36 | 0.31 | 0.63 | 0.06 | 4.86
0.34 | 40.0 | 4.88 | 0.34 | 0.62 | 0.04 | 4.36
0.34 | 45.0 | 4.24 | 0.38 | 0.59 | 0.03 | 3.71
0.34 | 50.0 | 3.52 | 0.45 | 0.53 | 0.02 | 3.15
0.50 | 20.0 | 11.03 | 0.30 | 0.61 | 0.09 | -
0.50 | 25.0 | 10.72 | 0.31 | 0.62 | 0.07 | -
0.50 | 27.5 | 10.43 | 0.32 | 0.63 | 0.05 | -
0.50 | 30.0 | 5.67 | 0.29 | 0.61 | 0.10 | 4.98
0.50 | 35.0 | 5.47 | 0.30 | 0.61 | 0.09 | 4.83
0.50 | 40.0 | 5.16 | 0.32 | 0.60 | 0.07 | 4.63
0.50 | 45.0 | 4.90 | 0.34 | 0.60 | 0.06 | 4.39
0.50 | 50.0 | 4.64 | 0.35 | 0.60 | 0.05 | 4.20
0.71 | 20.0 | 11.08 | 0.30 | 0.60 | 0.10 | -
0.71 | 25.0 | 10.95 | 0.30 | 0.61 | 0.09 | -
0.71 | 27.5 | 5.90 | 0.28 | 0.58 | 0.14 | 5.00
0.71 | 30.0 | 5.81 | 0.29 | 0.58 | 0.13 | 4.99
0.71 | 35.0 | 5.66 | 0.29 | 0.58 | 0.12 | 4.93
0.71 | 40.0 | 5.53 | 0.30 | 0.58 | 0.12 | 4.87
0.71 | 45.0 | 5.49 | 0.30 | 0.59 | 0.11 | 4.83
0.71 | 50.0 | 5.42 | 0.31 | 0.59 | 0.10 | 4.75
0.94 | 20.0 | 11.15 | 0.30 | 0.60 | 0.10 | -
0.94 | 25.0 | 6.04 | 0.28 | 0.56 | 0.17 | 5.05
0.94 | 27.5 | 6.01 | 0.28 | 0.56 | 0.16 | 5.04
0.94 | 30.0 | 5.97 | 0.28 | 0.56 | 0.16 | 5.03
0.94 | 35.0 | 5.92 | 0.28 | 0.56 | 0.15 | 5.01
0.94 | 40.0 | 5.88 | 0.28 | 0.57 | 0.15 | 5.01
0.94 | 45.0 | 5.83 | 0.29 | 0.57 | 0.14 | 5.00
0.94 | 50.0 | 5.79 | 0.29 | 0.58 | 0.14 | 5.00
1.10 | 20.0 | 11.09 | 0.30 | 0.60 | 0.10 | -
1.10 | 25.0 | 6.10 | 0.27 | 0.55 | 0.18 | 5.05
1.10 | 27.5 | 6.09 | 0.27 | 0.55 | 0.18 | 5.03
1.10 | 30.0 | 6.08 | 0.27 | 0.55 | 0.18 | 5.03
1.10 | 35.0 | 6.05 | 0.28 | 0.55 | 0.17 | 5.02
1.10 | 40.0 | 6.02 | 0.28 | 0.56 | 0.17 | 5.01
1.10 | 45.0 | 5.98 | 0.28 | 0.56 | 0.16 | 5.01
1.10 | 50.0 | 5.94 | 0.28 | 0.56 | 0.16 | 5.01
1.30 | 20.0 | 6.17 | 0.27 | 0.54 | 0.19 | 5.05
1.30 | 25.0 | 6.17 | 0.27 | 0.54 | 0.19 | 5.02
1.30 | 27.5 | 6.17 | 0.27 | 0.54 | 0.19 | 5.02
1.30 | 30.0 | 6.17 | 0.27 | 0.54 | 0.19 | 5.02
1.30 | 35.0 | 6.15 | 0.27 | 0.54 | 0.19 | 5.01
1.30 | 40.0 | 6.12 | 0.27 | 0.55 | 0.18 | 5.01
1.30 | 45.0 | 6.09 | 0.27 | 0.55 | 0.18 | 5.01
1.30 | 50.0 | 6.06 | 0.27 | 0.55 | 0.17 | 5.01

##  Appendix F A Modified Prescription For The Mass Of the Largest Remnant

Using our new results we can now add to the collision prescription from Denman
et al. (2020) to account for different impact angles. It should be noted that
we have not yet tested the effects of target mass and target atmosphere mass
on oblique collisions. The new prescription is as follows:

  1. 1.

For a given collision scenario with a $6.25\,\mathrm{M_{\oplus}}$ target with
$20\%$ atmosphere by mass ($M_{\mathrm{p}}$, $v$ and $b$), first calculate the
specific kinetic energy of the impact:

$Q_{\mathrm{R}}=\frac{1}{2}\mu\frac{v^{2}}{M_{\mathrm{tot}}}.$ (22)

  2. 2.

Then calculate the transition velocity between erosive hit-and-run and
accretion/disruption events, $v_{\mathrm{split}}$, which is given by

$v_{\mathrm{split}}=\sqrt{\frac{2G(M_{\mathrm{t}}+M_{\mathrm{p}})}{b(R_{\mathrm{t}}+R_{\mathrm{p}})}}.$
(23)

  3. 3.

If $v<v_{\mathrm{split}}$ then follow the prescription from Denman et al.
(2020), outlined here, otherwise skip to 4.

    1. (a)

Calculate the specific kinetic energy of the transition between the atmosphere
loss and core and mantle loss regimes,

$Q_{\mathrm{piv}}=\left(\frac{M_{\mathrm{tot}}}{M_{\oplus}}\right)\left(-2.45\frac{M_{\mathrm{p}}}{M_{\mathrm{t}}}+14.56\right)\;\;[10^{6}\,\mathrm{J\,kg^{-1}}].$
(24)

    2. (b)

Calculate the catastrophic disruption threshold,

$Q_{\mathrm{RD}}^{*}=c^{*}\frac{4}{5}\pi\rho_{\mathrm{1}}GR_{\mathrm{C1}}^{2},$
(25)

where $\rho_{\mathrm{1}}=1000\,\mathrm{kg\,m^{-3}}$, and
$R_{\mathrm{C1}}=\left(\frac{3M_{\mathrm{tot}}}{4\pi\rho_{\mathrm{1}}}\right)^{\frac{1}{3}}$
is the radius of a spherical body with a density of $\rho_{\mathrm{1}}$ and
mass $M_{\mathrm{tot}}$. We use the value $c^{*}=2.52$ predicted by Denman et
al. (2020) to determine the catastrophic disruption threshold
($Q_{\mathrm{RD}}^{*(\mathrm{New})}$), as well as the value $c^{*}=1.9$ from
Leinhardt & Stewart (2012) for atmosphere-less planets which is used to
predict energy dependence in the core and mantle dominated loss regime
($Q_{\mathrm{RD}}^{*(\mathrm{LS12})}$).

    3. (c)

Then, calculate the specific energy gradient for largest remnant mass for the
core and mantle loss regime:

$m_{\mathrm{c\&m}}=\frac{-0.5(1+f_{\mathrm{H}}^{\mathrm{t}})}{Q_{\mathrm{RD}}^{*(\mathrm{LS12})}}.$
(26)

where $f_{\mathrm{H}}^{\mathrm{t}}$ is the mass fraction of the target which
is atmosphere.

    4. (d)

Next, calculate the gradient for the atmosphere loss dominated regime from the
core and mantle loss gradient assuming zero impact energy means zero mass
loss,

$m_{\mathrm{atmos}}=\frac{m_{\mathrm{c\&m}}(Q_{\mathrm{piv}}-Q_{\mathrm{RD}}^{*(\mathrm{New})})-0.5}{Q_{\mathrm{piv}}}.$
(27)

    5. (e)

The next step is to calculate the super-catastrophic disruption threshold
($M_{\mathrm{LR}}<0.1M_{\mathrm{tot}}$), we use the Leinhardt & Stewart (2012)
method for this,

$Q_{\mathrm{supercat}}=Q_{\mathrm{RD}}^{*(\mathrm{New})}-\frac{0.4}{m_{\mathrm{c\&m}}}.$
(28)

    6. (f)

From the above three steps the total mass in the largest remnant must
therefore be:

$\frac{M_{\mathrm{LR}}}{M_{\mathrm{tot}}}=\begin{cases}m_{\mathrm{atmos}}Q_{\mathrm{R}}+1&0<Q_{\mathrm{R}}<Q_{\mathrm{piv}}\\\
\\\
m_{\mathrm{c\&m}}(Q_{\mathrm{R}}-Q_{\mathrm{RD}}^{*(\mathrm{New})})+0.5&Q_{\mathrm{piv}}<Q_{\mathrm{R}}<Q_{\mathrm{supercat}}.\\\
\end{cases}.$ (29)

As neither this study nor Denman et al. (2020) probed the super-catastrophic
disruption regime we do not predict results here, we recommend the Leinhardt &
Stewart (2012) prescription for collisions in this regime.

    7. (g)

Finally, the mass of atmosphere in the largest remnant is given by:

$\frac{M^{\mathrm{H}}_{\mathrm{LR}}}{M^{\mathrm{H}}_{\mathrm{t}}}=\begin{cases}1+\frac{0.94^{2}}{4}\left(\frac{Q_{\mathrm{R}}}{Q_{\mathrm{piv}}}\right)^{2}-0.94\frac{Q_{\mathrm{R}}}{Q_{\mathrm{piv}}}&\frac{Q_{\mathrm{R}}}{Q_{\mathrm{piv}}}<2.12\\\
0&\frac{Q_{\mathrm{R}}}{Q_{\mathrm{piv}}}>2.12.\end{cases}$ (30)

  4. 4.

If, on the other hand, $v>v_{\mathrm{split}}$ then we have an erosive hit-and-
run collision and the following process should be used to predict largest
remnant mass (note we have not tested the dependencies of these power laws on
target mass or target atmosphere fraction):

    1. (a)

The total atmosphere mass remaining in the largest remnant is given by

$\frac{M^{\mathrm{H}}_{\mathrm{LR}}}{M^{\mathrm{H}}_{\mathrm{t}}}=-10^{-0.76b+0.18}\log_{10}(Q_{\mathrm{R}})+10^{-0.54b+1.02},$
(31)

    2. (b)

and the mass remaining in the largest remnant is

$M_{\mathrm{LR}}=-10^{-1.11b-8.56}Q_{\mathrm{R}}+M_{\mathrm{t}}.$ (32)