# Atmosphere Loss by Aerial Bursts

Isabella L. Trierweiler1 Hilke E. Schlichting,2
1Department of Physics and Astronomy, University of California, Los Angeles,
CA 90095, USA
2Department of Earth, Planetary, and Space Sciences, University of California,
Los Angeles, CA 90095, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a simple analytic description of atmospheric mass loss by aerial
bursts and demonstrate that mass loss from aerial bursts becomes significant
when the maximum impactor size that leads to an aerial burst rather than a
ground explosion, $r_{o}$, is larger than the minimum impactor size needed to
achieve atmospheric loss, $r_{min}$. For vertical trajectories, which give the
most stringent limit, this condition is approximately satisfied when
$\rho_{o}/\rho_{i}\gtrsim 0.4v_{e}/v_{\infty}$, which implies atmospheric
densities need to be comparable to impactor densities for impactor velocities
that are a few times the escape velocity of the planet. The range of impactor
radii resulting in aerial burst-induced mass loss, $r_{o}-r_{min}$, increases
with the ratio of the atmosphere to the impactor density and with the
trajectory angle of the impactor. The range of impactor radii that result in
aerial burst-induced mass loss and the atmospheric mass lost is larger in
adiabatic atmospheres than isothermal atmospheres of equivalent total mass,
scale height, and atmospheric surface density. Our results imply that aerial
bursts are not expected to significantly contribute to the atmospheric mass-
loss history of Earth, but are expected to play an important role for planets
and exoplanets similar to Neptune with significant atmospheres. For Neptune-
like atmospheres, the atmospheric mass ejected per impactor mass by aerial
bursts is comparable to that lost by ground explosions, which implies that,
for impactors following a Dohnanyi size distribution, overall loss by aerial
busts is expected to exceed that by ground explosions by a factor of
$(r_{ground}/r_{aerial})^{0.5}$.

###### keywords:

planets and satellites: atmospheres

††pubyear: 2021††pagerange: Atmosphere Loss by Aerial Bursts–Atmosphere Loss
by Aerial Bursts

##  1 Introduction

The initial atmosphere of the Earth and other terrestrial planets was likely
established by an interplay of diverse loss and delivery processes. Relevant
processes that could deliver volatiles to a planet’s atmosphere include, for
example, degassing of volatiles during planetesimal impacts (e.g. Abe &
Matsui, 1985; Zahnle et al., 1988), volcanic outgassing (e.g. Craddock &
Greeley, 2009; Elkins-Tanton, 2012), tectonic activity Zahnle et al. 2007),
and accretion of hydrogen-helium gas from the primordial gas disk (e.g. Lammer
et al., 2014; Ginzburg et al., 2016).

As volatiles build up in the atmosphere, depending on composition and other
planet properties, atmospheric mass may be lost due to high energy radiation
from a planet’s host stars (e.g. Kulikov et al., 2007; Owen & Wu, 2013), core-
powered mass loss (Ginzburg et al., 2018; Gupta & Schlichting, 2019), or
impacts (e.g. Melosh & Vickery, 1989; Genda & Abe, 2003; Zahnle et al., 1992;
Schlichting et al., 2015; Wyatt et al., 2020).

Observed and inferred evidence for impacts of small planetesimals are
plentiful within our own solar system. Impacts can explain variations in
atmospheric mass between moons such as Titan, Ganymede and Callisto (Zahnle et
al., 1992), and may be responsible for the erosion of Mars’ atmosphere (Melosh
& Vickery, 1989). Evidence for Earth impacts can be seen in the presence of
our Moon (Cameron & Ward 1976, Canup & Asphaug 2001), the cratering on the
Moon and Earth, and the geochemical traces of the late veneer (Warren et al.
1999, Walker et al. 2004, Walker 2009). Aside from atmosphere loss, impacts
are important to Earth’s history as they are theorized to have delivered
volatiles to the surface (Owen & Bar-Nun 1995, Dauphas 2003). They are one
possible source of water for our oceans and for organic molecules (Chyba,
1990), though the types of impactors (asteroid, comet, planetary embryo, etc.)
that could have delivered this material are still under debate.

When considering atmospheric mass-loss by impacts, it is instructive to
distinguish between giant impacts and planetesimal impacts. A giant impact is
a planetary scale collision that can lead to global atmospheric mass-loss,
whereas a planetesimal impact can only eject mass locally above the tangent
plane. Although giant impacts can eject significantly more mass in a single
event than a planetesimal impact, planetesimal impacts are more efficient than
giant impacts, since they produce more atmospheric loss per impactor mass
(Schlichting et al., 2015).

Depending on impactor sizes, impact velocities and impact angles, volatiles
may be added to or removed from growing planetary embryos by planetesimal
impacts (e.g. Shuvalov, 2009; Schlichting et al., 2015; Wyatt et al., 2020;
Sinclair et al., 2020). Past works that have focused on evaluating the
collective impact on atmospheric erosion by a population of planetesimal
impactors of varying sizes have concentrated on ground explosions (Schlichting
et al., 2015; Wyatt et al., 2020), i.e. impactors large enough that they hit
the ground. In this paper we focus on planetesimal impactors that are
significantly decelerated in the atmosphere (Shuvalov et al., 2014, 2016) and
evaluate their contribution to atmospheric loss of terrestrial planets and
Neptune-like planets and exoplanets.

To study the role of small impactors in atmospheric loss and evolution, we
build on previous work by Schlichting et al. (2015), who derived a simple
analytic atmospheric mass-loss model for planetesimal impacts, which
successfully captured previous numerical results by Shuvalov (2009). We expand
on the work by Schlichting et al. (2015) first by accounting for impactors
which do not reach the ground and instead explode as aerial bursts, and second
by including both isothermal and adiabatic atmospheric profiles in our model.

By reasoning that aerial bursts deposit most of their energy at a location in
the atmosphere where they reach roughly half of their initial velocity, we
calculate the atmospheric properties required for a planet to lose atmospheric
mass by aerial bursts. We demonstrate that aerial burst-induced mass loss is
only significant for planets with atmospheric densities comparable to the
impactor density and quantify the resulting atmospheric mass loss by applying
our model to Neptune-like planets and exoplanets.

This paper is structured as follows: In section 2 we outline our approach for
calculating atmosphere loss by considering a simple impactor explosion model.
In section 3 we apply our model to aerial bursts and present results for
Neptune-like atmosphere in section 4. We outline general planet parameters for
which mass loss by aerial bursts is important in section 5. In section 6 we
discuss the contributions of aerial bursts to the total atmospheric mass loss.
Finally, we summarize our findings in section 7.

##  2 Methods

###  2.1 Atmospheric Profiles

Before presenting our mass loss model, we first describe the two atmospheric
profiles assumed throughout this work, and the mass loss limits associated
with each profile. We consider isothermal and adiabatic atmospheres with the
following density profiles:

$\begin{split}\rho(z)&=\rho_{o}e^{-z/h}\mbox{\quad(Isothermal)}\\\
\rho(z)&=\rho_{o}\left(1-\frac{\gamma-1}{\gamma}\frac{z}{h}\right)^{1/(\gamma-1)}\mbox{\quad(Adiabatic)}\end{split}$
(1)

Here $h=\frac{k_{B}T}{\mu m_{p}g}$ is the scale height, where $k_{B}$ is the
Boltzmann constant, $T$ is the temperature, $\mu$ is the mean molecular
weight, $m_{p}$ is the proton mass, and $g=\frac{GM_{p}}{R^{2}}$ is the
gravitational acceleration for a planet of mass $M_{p}$ and radius R. The
atmospheric density $\rho$ is a function of $z$, the height in the atmosphere
above the ground. $\rho_{o}$ is the density of the atmosphere at the planet’s
surface and $\gamma$ is the adiabatic index. Throughout this work we assume a
diatomic atmosphere with $\gamma=1.4$ so that the adiabatic profile is
$\rho(z)=\rho_{o}\left(1-\frac{2}{7}\frac{z}{h}\right)^{5/2}$. The atmosphere
extends from $z=0$ to $z=\infty$ for the isothermal profile and to $z=7h/2$
for the diatomic, adiabatic profile.

Figure 1: Diagram of an aerial burst. The planet has a radius $R$ and an
atmosphere with a scale height $h$. The planetesimal enters the atmosphere
along the grey dashed line with an angle $\phi$ relative to the normal above
the explosion site. It explodes as an aerial burst at height $H$ above the
ground. From the explosion site, we define $z$ to be the height above the
explosion ($z=0$ at the height $H$), and $s$ to be a column of atmosphere at
an angle of $\theta$ from $z$. For an aerial burst occurring at height $H$,
the maximum mass of atmosphere which can be ejected is roughly the cap of
atmosphere out to the tangent plane ($\theta=\pi/2$).

Using a coordinate system centered on the impact site (see figure 1), and
assuming $h\ll R$, we can write our coordinates as
$s=\sqrt{R^{2}\cos^{2}\theta+2zR}-R\cos\theta$ or
$z=(s^{2}+2Rs\cos\theta)/2R$. We note that the assumption of $h\ll R$ means
that the density of the atmosphere will decrease quickly with height, so that
the density profile will be close to zero for heights of $z\gg R$. All aerial
bursts will therefore occur at heights $H\ll R$. An additional benefit of the
assumption $h\ll R$ is that the height of the aerial burst $H$ only factors in
the density profile, and not in $z$ or $s$, simplifying many of the following
equations.

Assuming $h\ll R$, the total mass in the atmosphere for both the isothermal
and adiabatic profiles is well approximated by

$M_{tot}\simeq 4\pi R^{2}\int_{0}^{z(\rho\rightarrow
0)}\rho(z)dz=4\pi\rho_{o}hR^{2}.$ (2)

The maximum mass of atmosphere that can be ejected by an aerial burst at
height $H$ is roughly the cap of atmosphere above the tangent plane. We call
this mass $M_{cap}(H)$, and calculate it as

$M_{cap}(H)=2\pi\int_{0}^{\pi/2}\int_{0}^{z(\rho\rightarrow
0)}\rho(z+H)s^{2}\sin(\theta)dsd\theta.$ (3)

For an explosion on the ground ($H=0$), the cap mass for each profile is
$M_{cap,Iso}=2\pi\rho_{o}h^{2}R$ and
$M_{cap,Adi}=\frac{14}{9}\pi\rho_{o}h^{2}R$.

###  2.2 Mass Loss Model

We model each aerial burst as an explosion at the height in the atmosphere
where drag forces significantly decelerate the body. Specifically, we choose
the height where drag forces slow the impactor to half of its initial
velocity, such that the impactor’s momentum is halved and its kinetic energy
is $1/4$ of the initial kinetic energy. The assumption that the half velocity
height is the explosion height is a reasonable approximation because the
atmospheric density increases exponentially (or as a steep power-law in the
adiabatic case) as the body approaches the surface such that it experiences
most of its deceleration around this half-velocity height.

The direction of the impact trajectory is parameterized by the angle $\phi$,
which is measured from the normal above the explosion site (see figure 1),
such that $\phi=0$ represents a vertically falling impactor. The impactors’
initial velocity is $v_{\infty}$. We account for impactors approaching the
planet at any angle, so that the half-velocity height is a function of both
the radius of the impactor and the trajectory angle $\phi$. We outline this
half velocity calculation in section 2.3. Smaller impactors are decelerated to
half their initial velocity higher in the atmosphere than larger impactors.
Additionally, impactors travelling on nearly horizontal trajectories will pass
through more atmosphere than those following vertical trajectories, and so the
horizontal impactors will have relatively higher half-velocity heights. Bodies
which decelerate to half their initial velocity before hitting ground are
referred to as aerial bursts, while those that reach the ground with more than
half of their initial velocity are referred to as ground explosions.

At the calculated explosion height, we model the effect of each impactor as a
spherically symmetric point explosion. As the impactor travels through the
atmosphere, it creates a shock which heats the gas ahead of it (Zel’dovich &
Raizer 1967, Vickery & Melosh 1990). When the impactor decelerates, either by
hitting the planet’s surface or through drag forces, the heated gas expands as
a vapor plume. The plume expands in the atmosphere, transferring momentum to
the surrounding gas. Any gas in the plume that is accelerated to the escape
velocity of the planet is then ejected. This expanding plume can be well
modeled by an explosion placed at the point where the impactor deposits most
of its energy in the atmosphere (Vickery & Melosh 1990). The explosion is
modeled as an isotropic sphere of the same mass as the impactor, expanding at
the same velocity. The energy of this expanding mass is then transferred to
the surrounding atmosphere.

In each explosion, we distribute the energy of the impactor over the full
sphere around the half-velocity point. In this model, the atmospheric mass
that will be ejected is the cone of atmosphere which can be accelerated by the
impactor’s explosion to the planet’s escape velocity. We define the maximum
angle along which atmosphere can be ejected to be $\theta_{m}$. Assuming
energy conservation, this angle is where the atmospheric mass per solid angle
is equal to impactor mass ($M_{i}=\frac{4\pi}{3}r^{3}\rho_{i}$) distributed
over a sphere and scaled by the final and escape velocities:

$\int_{0}^{s(\rho\rightarrow
0)}\rho(z+H)s^{2}(\theta_{m})ds=\frac{M_{i}}{4\pi}\frac{v_{f}^{2}}{v_{e}^{2}}.$
(4)

Here $v_{f}$ is the velocity of the impactor when it reaches the explosion
site, $v_{e}$ is the escape velocity of the planet, and
$s(\theta_{m})=\sqrt{R^{2}\cos^{2}\theta_{m}+2zR}-R\cos\theta_{m}$.

We generally assume that the impactors are rocky bodies approaching the planet
at $2v_{e}$, the velocity we would expect for a body such as an asteroid. We
also consider comet-like bodies, which have lower densities and expected
initial velocities around $5v_{e}$.

The total mass lost is then found by integrating over the cone of material out
to $\theta_{m}$

$M_{eject}=2\pi\int_{0}^{\theta_{m}}\int_{0}^{s(\rho\rightarrow
0)}\rho(z+H)s^{2}\sin\theta dsd\theta.$ (5)

###  2.3 Half Velocity Heights

In the point explosion model for impactors, we assume the explosion occurs at
the height where the impactor is slowed to half its initial velocity by drag
forces. To calculate this half velocity height, we calculate the velocity of
the body as a function of the height of the impactor above the ground.

Bodies will always approach a planet with at least the escape velocity, and
asteroid and comet-like bodies are expected to be travelling at even higher
speeds. Because we are considering atmospheres for which $h\ll R$, any
significant deceleration the impactor experiences due to gas drag will occur
over a relativity short distance compared to R and within heights
significantly smaller than R above the ground. As such, in calculating the
impactor’s velocity we account for the drag force but neglect any additional
gravitational acceleration the impactor experiences as it is always either too
weak or acting on too short of a timescale to significantly change the body’s
velocity before the body reaches $1/2v_{\infty}\geq v_{e}$. We can verify that
this approximation is justified by taking the ratio of the drag force and
gravitational force, $(R/r)(\rho_{o}/\rho_{i})(v_{\infty}/v_{e})^{2}$, and
confirming that it is indeed greater than $\sim 1$, usually much greater, for
the atmospheric densities and aerial burst impactor sizes investigated here.

Drag forces will flatten the bodies as they travel through the atmosphere, so
we add a parameter $\alpha$ to the area of the body $(\rm Area=\alpha\pi
r^{2})$. While the body should flatten more over the course of its trajectory
through the atmosphere, we will set $\alpha$ to be a constant for now. Based
on numerical simulations (Shuvalov et al. 2014), assuming an increase in
radius of about 3 is reasonable, resulting in $\alpha=9$. We note that while
this flattening is due to fragmentation of the body, the impactor is still
expected to continue travelling as a single entity. Smaller fragmenting pieces
will fill in the gaps between larger fragments, allowing us to model the
impactor as a falling pancake (Hills & Goda, 1993). With the $\alpha$
parameter for flattening, the equation for the drag force acting on the
impactor becomes

$F_{d}=\frac{1}{2}C\rho(z)\alpha\pi r^{2}v^{2},$ (6)

where $\alpha\pi r^{2}$ is the impactor’s effective cross section, v is the
velocity, r is the impactor radius, and $C$ is the drag coefficient. For a
sphere, $C=0.5$, however the flattened shape of the impactor will change the
drag coefficient. Coefficients for non-spherical, rough bodies at high speeds
can reach 1 or higher, so we will assume $C=1$.

The equation of motion for the impactor is therefore

$\frac{vdv}{ds}=-\frac{1}{2}\alpha C\rho(z)\pi r^{2}v^{2}/M_{i}.$ (7)

We note that the impactor’s velocity will depend on the angle $\phi$ of the
impactor’s trajectory through the atmosphere, measured relative to the
vertical axis $z$ (see figure 1). For a given $\phi$, we can solve equation 7
for the velocity of the body at a height $H$ above the ground, finding

$\ln\left[\frac{v(H,\phi)}{v_{\infty}}\right]=-\frac{3}{8}\frac{\alpha
C}{r\rho_{i}}\int_{s(\rho\rightarrow 0)}^{0}\rho(z+H)ds,$ (8)

where $\rho_{i}$ is the density of the impactor,
$s=\sqrt{R^{2}\cos^{2}\phi+2zR}-R\cos\phi$, and $\phi$ is the angle from the
vertical along which the impactor moves through the atmosphere.

Finally, defining the height $H$ to be where the velocity of the body slows to
half of its initial velocity ($v(H,\phi)=\frac{1}{2}v_{\infty}$), we find

$\begin{split}\ln(1/2)&=-\frac{3}{8}\frac{\alpha
C}{r\rho_{i}}\int_{s(\rho\rightarrow 0)}^{0}\rho(z+H)ds\\\ &\mbox{or
equivalently, }\\\ \ln(1/2)&=-\frac{3}{8}\frac{\alpha
C}{r\rho_{i}}\int_{z(\rho\rightarrow
0)}^{0}\frac{\rho(z+H)R}{\sqrt{R^{2}\cos^{2}\phi+2zR}}dz,\end{split}$ (9)

where $H$ is the half velocity height.

For trajectory angles $\phi\ll\pi/2$ (i.e., vertical trajectories close to the
$z$ axis), we can approximate the half velocity height in equation 9 as a
function of the trajectory angle $\phi$, and the radius of the body $r$:

$\begin{split}\frac{H}{h}_{Iso}(r,\phi)&=\ln\left(\frac{3}{8}\frac{\alpha
C}{\ln 2}\frac{h}{r}\frac{\rho_{o}}{\rho_{i}}\frac{1}{\cos\phi}\right)\\\
\frac{H}{h}_{Adi}(r,\phi)&=\frac{7}{2}\left[1-\left(\frac{8}{3}\frac{\ln
2}{\alpha
C}\frac{r}{h}\frac{\rho_{i}}{\rho_{o}}\cos\phi\right)^{2/7}\right],\end{split}$
(10)

where $h$ is the scale height, the densities of the atmosphere and impactor
are $\rho_{o}$ and $\rho_{i}$, respectively, $C$ is the drag coefficient, and
the $\alpha$ is the flattening parameter.

Full numerical solutions for the half velocity heights in isothermal and
adiabatic profiles, illustrating two different $\phi$ angles, are found in
figure 2. These heights are calculated for an Earth-like atmosphere. We see
that larger impactors have lower half velocity heights. Comparing the
explosion heights for the vertical ($\phi=0$) and horizontal ($\phi=\pi/2$),
and isothermal and adiabatic cases, we see that the heights for horizontal
trajectories are higher by up to a factor of two for an Earth-like atmosphere.
This difference is because impactors on horizontal trajectories must pass
through more atmosphere than those on vertical trajectories. Impactors in an
adiabatic atmosphere also have an upper limit on the half velocity height as
the atmosphere extends only to $z=7h/2$.

Figure 2: Calculated half velocity heights as a function of impactor radius
(equation 10), for asteroids ($\rho_{i}=3300$ kg/m3, $v_{\infty}=2v_{e}$)
impacting an Earth-like atmosphere ($h=8$ km, $\rho_{o}=1.2$ kg/m3). The solid
curves show the isothermal profile, while the dashed curves show the adiabatic
profile. The black curve is calculated for a vertical impactor trajectory
($\phi=0$), and the red curve is for a horizontal trajectory ($\phi=\pi/2$).

###  2.4 Comparing analytical and numerical half velocity heights

We compare our derived half velocity heights with those obtained by Shuvalov
et al. (2014, 2016) using hydrodynamical simulations (figure 3). These works
investigated impactors entering two types of atmospheres - one analogous to
Earth’s current atmosphere ($h=8$ km, $\rho_{o}=1.2$ kg/m3) and one
representing a primordial atmosphere ($h=40$ km, $\rho_{o}=49$ kg/m3). The two
works included simulations for both asteroid and comet impactors, with a
variety of assumed initial velocities.

In figure 3, we compare the half velocity heights calculated in the numerical
simulations (Shuvalov et al., 2014, 2016) to our analytical half velocity
heights. The circular points indicate numerical heights calculated for the
$h=8$ km atmosphere while the triangular points represent the numerically
calculated heights for the denser, $h=40$ km atmosphere. Red points represent
simulations of comets ($\rho_{i}=1000$ kg/m3) and black points represent
asteroids ($\rho_{i}=3000$ kg/m3). Due to the sensitivity of the numerical
simulations to initial conditions, some impactor radii have multiple
corresponding half velocity heights.

The solid and dashed curves in figure 3 are the analytical half velocity
heights described by equation 10. The solid curves are calculated assuming an
isothermal atmosphere, while the dashed curves are for adiabatic atmospheres.
The thicker curves are heights calculated for the $h=40$ km, $\rho_{o}=49$
kg/m3 atmosphere while the thin curves at lower heights are for the $h=8$ km,
$\rho_{o}=1.2$ kg/m3 atmosphere. As with the numerical points, the red curves
correspond to half velocity heights for comets while the black curves
correspond to the half velocity heights calculated for impactors from the
asteroid belt. We find that our analytical expression for the half velocity
height is in reasonable agreement with numerical results for both type of
atmospheres as well as for cometary and asteroidal impactors.

In our mass loss model described below, we will assume that the half velocity
height is the location of the point explosion.

Figure 3: Comparison of analytical half velocity heights to those calculated
in hydrodynamics simulations. The analytical heights are calculated using
equation 10 and the simulated heights are from Shuvalov et al. 2014 and
Shuvalov et al. 2016. The circular points are from Shuvalov et al. 2016 and
correspond to numerical heights calculated for an Earth-like atmosphere ($h=8$
km, $\rho_{o}=1.2$ kg/m3). The triangular points are from Shuvalov et al. 2014
for a primordial Earth atmosphere ($h=40$ km, $\rho_{o}=49$ kg/m3). The solid
curves are calculated using an isothermal profile while the dashed lines are
calculated with an adiabatic profile. The thin curves are the analytical
heights for the Earth atmosphere while the thick curves are for the primordial
Earth atmosphere. The red curves and points represent impactors with comet
properties ($\rho_{i}=1000$ kg/m3) while the black curves and points are
asteroid-like impactors ($\rho_{i}=3300$ km/m3). We find that the numerical
half velocity heights for both the Earth and primordial Earth atmospheres are
in reasonable agreement with the analytical curves, for both comet and
asteroid impactors.

##  3 Mass loss by aerial bursts

The aerial burst atmospheric mass loss regime includes any impactors which can
eject atmosphere and which slow to half their initial velocity before reaching
the planet’s surface. As described in section 2, aerial bursts are modeled as
explosions at the height $H$ above the ground. At this height, the expanding
mass is distributed over the whole sphere around the explosion point.

To calculate the total mass loss due to aerial bursts, we first pick a height
$H$ and trajectory angle $\phi$ between $0$ and $\pi/2$, and then calculate
the impactor radius corresponding to this explosion height given the
trajecotry angle (equation 10). Following the explosion model outlined in
section 2, we then calculate the maximum angle along which an impactor of the
calculated radius can eject atmosphere ($\theta_{m}$ in equation 4). Noting
that the final velocity of the aerial burst is half of its initial velocity
($v_{f}=v_{\infty}/2$), we write equation 4 as:

$\int_{0}^{s(\rho\rightarrow
0)}\rho(z+H,z=s^{2}/2R+s\cos\theta_{m})s^{2}ds=\frac{r^{3}}{12}\rho_{i}\frac{v_{\infty}^{2}}{v_{e}^{2}}.$
(11)

Using this angle $\theta_{m}$ and equation 5, we calculate the mass ejected.
This yields mass loss as a function of impactor radius and trajectory angle at
a given height. We then sum over the explosion heights in order to find the
overall expected mass loss as a function of impactor radius and trajectory
angle.

###  3.1 Characteristic Radii

Two limiting impactor radii define the bounds of atmospheric mass loss by
aerial bursts. $r_{o}(\phi)$, which describes the impactor radius
corresponding to an explosion height of zero for a given trajectory angle,
sets the upper limit for the radius of aerial burst projectiles. At radii
larger than $r_{o}(\phi)$, impactors will result in ground explosions.
$r_{min}(\phi)$, the minimum impactor radius that can eject mass from a given
trajectory angle, sets the lower radius limit for planetesimals that will
result in any mass loss. As such, aerial burst mass loss can occur only when
$r_{min}<r<r_{o}$, where $r$ is the planetesimal radius.

Starting with the equations relating the half velocity height, trajectory
angle, and impactor radius (equation 10), we solve for the impactor radius
where the explosion height is zero. Rewriting equation 10 for the case where
$r_{o}(\phi)=r(\phi)|_{H=0}$ gives

$\frac{r_{o}}{h}(\phi)=\frac{3}{8}\frac{\alpha C}{\ln
2}\frac{1}{\rho_{i}}\int_{z(\rho\rightarrow
0)}^{0}\frac{\rho(z)R}{\sqrt{R^{2}\cos^{2}\phi+2zR}}dz.$ (12)

In the limit where $\phi\ll\pi/2$, we find

$\frac{r_{o}}{h}_{Iso,Adi}(\phi)=\frac{3}{8}\frac{\alpha C}{\ln
2}\frac{\rho_{o}}{\rho_{i}}\frac{1}{\cos\phi}.$ (13)

Note that the approximated form of $r_{o}(\phi)$ is equivalent for both the
isothermal and adiabatic profiles.

$r_{min}$ is calculated by considering the smallest planetesimal that will be
able to eject mass from it’s associated explosion height. At a given height,
the minimum mass that may be ejected in an explosion is the column of
atmosphere directly above the explosion site, so we first solve equation 4 for
the impactor radius required to eject mass along the vertical $z$ axis, or
equivalently along $\theta_{m}=0$. With the condition that
$s(\theta_{m}=0)=z$, we can rewrite equation 4 as

$\int_{0}^{z(\rho\rightarrow
0)}\rho(z+H(r_{min},\phi))z^{2}dz=\frac{\frac{4\pi}{3}\rho_{i}r_{min}^{3}}{4\pi}\frac{v_{f}^{2}}{v_{e}^{2}},$
(14)

where $r_{min}$ is the radius required to eject the column of mass directly
above an explosion site at height $H$, $\phi$ is the trajectory angle relative
to the $z$ axis, $\rho_{i}$ is the impactor density, $v_{f}$ is the final
velocity of the body at height $H$, and $v_{e}$ is the escape velocity of the
planet. As defined in section 2, the explosion height $H$ depends on both the
radius of the planetesimal and the trajectory angle $\phi$.

Substituting the expression of half velocity height for $H$, with a
planetesimal radius of $r_{min}$ and a trajectory angle of $\phi$
($H=H(r_{min},\phi))$ in equation 10, and noting that
$v_{f}=\frac{1}{2}v_{\infty}$ at the half velocity height, we calculate
$r_{min}(\phi)$ for $\phi\ll\pi/2$ as

$\begin{split}\frac{r_{min}}{h}_{Iso}(\phi)&=\sqrt{64\frac{\ln 2}{\alpha
C}\frac{v_{e}^{2}}{v_{\infty}^{2}}\cos\phi}\\\
\frac{r_{min}}{h}_{Adi}(\phi)&=12^{7/10}\left(\frac{v_{e}}{v_{\infty}}\right)^{7/5}\left(\frac{\rho_{i}}{\rho_{o}}\right)^{2/5}\left(\frac{8}{3}\frac{\ln
2}{\alpha C}\cos\phi\right)^{11/10}.\end{split}$ (15)

We note that for a given total atmospheric mass, $r_{min}(\phi)$ in an
adiabatic atmospheric profile will be somewhat lower than the $r_{min}(\phi)$
for the isothermal case. In other words, adiabatic aerial burst mass loss can
occur at smaller impactor radii than isothermal aerial burst mass loss.

##  4 Application to sub-Neptunes

To illustrate the effects of atmospheric mass-loss by aerial bursts, we apply
our model to a theoretical sub-Neptune atmosphere (figure 4). We assume
atmosphere properties similar to Neptune, with a scale height of 19 km, an
atmospheric density of $\rho_{o}\sim 1000$ kg/m3 at the base of the envelope,
and a planetary core radius of $R=6000$ km. We consider asteroid-like
impactors ($\rho_{i}=3300$ kg/m3) with an initial velocity of
$v_{\infty}=2v_{e}$. If the impactor density or initial velocity is greater
than these values, mass loss would increase in magnitude at all impactor
radii.

Figure 4 shows the ejected atmosphere per unit impactor mass as a function of
the projectile radius $r$ and the trajectory angle $\phi$ relative to the
vertical $z$ axis. For comparison, we also show the approximated forms of
$r_{o}(\phi)$ (solid curve) and $r_{min}(\phi)$ (dashed curve), as derived in
section 3.1. Aerial burst-induced mass loss occurs for the radii $r$ where
$r_{min}<r<r_{o}$. At radii larger than $r_{o}$, impactors result in mass loss
through ground explosions instead of aerial bursts (see section 6).

Figure 4: Ejected atmospheric mass per impactor mass as a function of the
impactor radius and trajectory angle for asteroid-like impactors entering a
sub-Neptune/Neptune like atmosphere. The trajectory angle $\phi$ is measured
relative to the vertical $z$ axis (see figure 1). The top figure shows mass
loss assuming an isothermal atmosphere and the bottom figure shows mass loss
for an adiabatic atmosphere. The solid curves show $r_{o}$ as a function of
trajectory angle and the dashed curves show $r_{min}$. Aerial burst mass loss
occurs between $r_{min}$ and $r_{o}$. Below $r_{min}$ (below the dashed line)
no mass loss can occur. We assume an atmospheric density at the surface of
$\rho_{o}\sim 1000$ kg/m3 and an impactor density of $\rho_{i}=3300$ kg/m3.
Mass loss by aerial bursts can occur at all trajectory angles, and occurs at
relatively smaller impactor radii for the adiabatic case than the isothermal
case.

A typical sub-Neptune/Neptune like planet has a large range of radii for which
aerial burst-induced mass loss occurs, with most aerial bursts resulting in
atmospheric mass loss equal to a few percent of the projectile mass. Because
projectiles pass through relatively more atmosphere, and therefore slow to
half velocity at higher heights, along horizontal ($\phi=\pi/2$) trajectories
than vertical trajectories, there is a much larger range of planetesimal radii
that can result in aerial burst-induced mass loss at large trajectory angles
than at small angles. Due to the differences in atmospheric mass distribution
in adiabatic and isothermal atmosphere profiles, aerial burst mass loss begins
at relatively lower radii in the adiabatic model than isothermal model.

##  5 Constraints on the aerial burst Regime

Recall that at a given trajectory angle $\phi$, aerial burst-induced mass loss
occurs for impactor radii greater than $r_{min}(\phi)$ and less than
$r_{o}(\phi)$. In other words, projectiles must be massive enough to provide
adequate energy for mass loss without being so large that they reach the
planet surface without decelerating. As shown in section 3.1, these radii
depend on the atmospheric profile. Therefore, our definitions of $r_{o}$ and
$r_{min}$ can be used to set limits on the properties of atmospheres that may
lose mass through aerial bursts. Because aerial burst-induced mass loss occurs
for planetesimal radii $r$ where $r_{min}<r<r_{o}$, for aerial burst mass loss
to be possible in an atmosphere with a given atmospheric surface density
$\rho_{o}$, $r_{o}(\phi)$ must be greater than $r_{min}(\phi)$ for at least
some values of $\phi$.

In figure 5, we illustrate the range of planetesimal radii that can cause
atmospheric-loss by aerial bursts for different atmospheric densities by
plotting $r_{o}(\phi)$ and $r_{min}(\phi)$. In this example, we begin with an
earth-like atmosphere, with $\rho_{o}=$ 1.22 kg/m3 and $h=8$ km, and show the
change in $r_{o}(\phi)$ as the atmospheric density is increased by consecutive
factors of ten. We assume impactor densities of $\rho_{i}=3300$ kg/m3 and
initial velocities $v_{\infty}=2v_{e}$.

As calculated in section 2.3, the approximated form of the $r_{o}(\phi)$ curve
is equivalent for both the adiabatic and isothermal profiles.
$r_{o}(\phi)\propto\rho_{o}$, so the upper radius limit of aerial bursts
$r_{o}$ increases with density as bodies must be larger in order to reach the
ground before decelerating. $r_{min}(\phi)$ is independent of atmospheric
density in the isothermal case, so the minimum radius limit of aerial bursts
stays constant with increasing atmospheric density. As the atmospheric density
increases, $r_{o}$ increases, and a larger range of aerial burst radii are
able to eject atmospheric mass.

Figure 5 shows that for atmospheres with densities close to Earth’s current
atmospheric density, $r_{o}<r_{min}$ at all $\phi$ angles. In other words,
only projectiles that reach the planet surface without decelerating will
typically provide enough energy to result in atmospheric mass loss. As a
result, for the current earth’s atmosphere we find no significant atmospheric
loss by aerial bursts.

However, when the atmospheric density is increased by a factor of ten,
$r_{o}>r_{min}$ for trajectories close to the horizontal ($\phi\sim\pi/2$). At
this point, some aerial bursts can eject mass if they approach the planet’s
surface on a nearly horizontal path. Increasing the surface density by a
factor of $10^{4}$ allows a range of aerial bursts approaching the planet on
any path to eject mass.

While the top panel of figure 5 shows the $r_{min}$ curve for an isothermal
atmosphere, we note that at high atmospheric densities we expect most
atmospheres to be closer to adiabatic than isothermal. The bottom panel of
figure 5 demonstrates the changes in $r_{min}$ with atmospheric density for an
adiabatic profile. As found in section 3.1, the $r_{min}$ curve in the
adiabatic case depends inversely on density
($r_{min,Adi}\propto\rho_{o}^{-2/5}$), so it would begin to intersect the
$r_{o}$ curve at lower atmospheric densities than the isothermal $r_{min}$.
This allows for mass loss by aerial bursts at relatively lower atmospheric
densities than we found in the isothermal case.

Figure 5: Maximum impactor radius that will result in an aerial burst,
$r_{o}$, as a function of trajectory angle $\phi$, for various atmospheric
densities ($\rho_{o}$). Here $\rho_{o}=1.22$ kg/m3, and we assume an impactor
density of $\rho_{i}=3300$ kg/m3 and an initial velocity of
$v_{\infty}/v_{e}=2$. The top figure demonstrates the change in characteristic
radii for an isothermal profile and the bottom shows the trends for an
adiabatic profile. The dashed curves show $r_{min}(\phi)$, the minimum radius
needed to eject any mass from a given trajectory angle. Note that $r_{min}$ is
independent of atmospheric density in the isothermal case, and varies
inversely with density in the adiabatic case. The solid curves show
$r_{o}(\phi)$, the maximum impactor radius that will result in an aerial burst
as opposed to a ground explosion. The $r_{min}$ and $r_{o}$ curves are colored
by their corresponding atmospheric density. The regime of mass loss by aerial
bursts for a given atmospheric density is found above $r_{min}$ and below
$r_{o}(\rho_{o})$. Grey shading shows regions where no mass loss is possible
(below $r_{min}$). Denser atmospheres increase $r_{o}$ at all trajectory
angles, increasing the extent of the aerial burst mass loss regime in both
impactor radius and trajectory angle. Because $r_{min}$ in the adiabatic case
decreases with increasing atmospheric density, mass loss through aerial bursts
can occur at relatively lower densities in the adiabatic atmosphere than in an
isothermal atmosphere.

With the knowledge that aerial burst-induced mass loss occurs when
$r_{o}(\phi)\leq r_{min}(\phi)$, we can calculate the minimum surface-level
atmospheric density ($\rho_{o}$) required to eject mass through aerial bursts
at a given trajectory angle ($\phi$). Equating the approximated forms of
$r_{o}$ and $r_{min}$ and solving for $\rho_{o}$ results in

$\begin{split}\frac{\rho_{o}(\phi)_{min,Iso}}{\rho_{i}}&=\frac{64}{3}\left(\frac{\ln
2}{\alpha C}\right)^{3/2}\frac{v_{e}}{v_{\infty}}\cos(\phi)^{3/2}\\\
\frac{\rho_{o}(\phi)_{min,Adi}}{\rho_{i}}&=\frac{32\sqrt{2}}{3}\left(\frac{\ln
2}{\alpha C}\right)^{3/2}\frac{v_{e}}{v_{\infty}}\cos(\phi)^{3/2},\end{split}$
(16)

where $\rho_{o}(\phi)_{min}$ is the minimum atmospheric density at the planet
surface required for mass loss by aerial bursts. Note that the limiting
densities have no dependence on the scale height or planet radius in the $h\ll
R,\phi\ll\pi/2$ approximation. A weak dependence of the limiting density on
scale height and planet radius can be recovered if we evaluate the integral
forms of $r_{o}$ and $r_{min}$ at $\phi=\pi/2$.

Taking the vertical limit as the most conservative indicator of whether aerial
burst-induced mass loss is possible, we conclude that consideration of aerial
bursts is necessary for atmospheres with densities of
$\rho_{o}/\rho_{i}\gtrsim 0.4v_{e}/v_{\infty}$.

Figure 6 illustrates the minimum densities in equation 16 for isothermal and
adiabatic atmospheres, assuming $\alpha=9$, $C=1$, and $v_{\infty}=2v_{e}$. We
also show the approximate $\rho_{o}/\rho_{i}$ values for Earth, Venus, and the
Neptune/sub-Neptune atmosphere, assuming impactor densities of $\rho_{i}=3300$
kg/m3. As seen in the example in figure 5, only denser atmospheres such as
that of Venus or Neptune will experience mass loss by aerial bursts for
projectiles on more vertical trajectories.

Of particular note are the minimum atmospheric densities at $\phi=\pi/4$, as
these trajectories are expected to represent the average collisions angle. We
find, to experience mass loss by aerial bursts for impact angles of
$\sim\pi/4$, atmospheres must have densities that are at least $\sim 10\%$ of
the impactor density.

Figure 6: Minimum atmospheric density required for the atmosphere to
experience mass loss through aerial bursts as a function of trajectory angle,
$\phi$ (equation 16). Note, the atmospheric density given on the y-axis is
normalized by the impactor density, $\rho_{i}$. The solid-black curve
represents the isothermal profile, while the dashed-back curve corresponds to
the adiabatic profile. Higher atmospheric densities are required for mass to
be ejected by aerial bursts in isothermal atmospheres and by projectiles
travelling vertically. As reference values, the atmospheric densities for
Earth (blue), Venus (yellow) and Neptune (green) are shown as dotted lines.

##  6 Contribution of aerial burst towards total mass loss

At a given trajectory angle, all planetesimals with radii larger than
$r_{o}(\phi)$ (equation 13) will not decelerate significantly before reaching
the planet surface, and will therefore result in ground explosions. We use the
same prescription outlined in section 2 to describe ground explosions by
simply setting the explosion height ($H$) in all equations to zero. Because
ground explosion impactors do not decelerate significantly in the atmosphere,
they will have impact velocities close to their initial velocities ($v_{f}\sim
v_{\infty}$).

The effects of ground explosions are described in detail in Schlichting et al.
(2015); in this section we explore how aerial burst-induced mass loss compares
to mass loss by ground explosions, and describe the overall differences
between the mass loss curves calculated for isothermal versus adiabatic
atmospheric density profiles.

###  6.1 Isothermal versus Adiabatic Atmospheres

In figure 7 we show the application of both the aerial burst and ground
explosion mass loss models to the same sub-Neptunian atmosphere discussed in
section 4 ($\rho_{o}=1000$ kg/m3, $h=19$ km), assuming projectiles of density
$3300$ kg/m3 with trajectory angles of $\phi=\pi/4$. The curve spans three
distinct categories of mass loss: aerial burst-induced mass loss at small
impactor radii ($r_{min}<r<r_{o}$) shaded in yellow, ground explosions at
intermediate radii ($r>r_{o}$) shaded in blue, and a tail at large radii where
impactors reach the surface and the impacts are massive enough to eject the
full cap of atmosphere above the tangent plane (where $\theta_{m}=\pi/2$)
shaded in red. Note that at radii exceeding $\sim 1000$ km we may be entering
the regime of giant impacts.

As demonstrated in section 3.1, figure 7 shows that aerial burst-induced mass
loss can occur at relatively smaller radii in an adiabatic atmosphere than in
an equal mass isothermal atmosphere. This is largely due to the difference in
half velocity heights between the two profiles (see figure 2), which result in
small particles decelerating relatively closer to the ground in adiabatic
atmospheres. Because of the dependence on explosion height, the difference
between isothermal and adiabatic mass loss is strongest in the aerial burst
regime, and we see that projectiles can eject significantly more mass per
impactor mass in the adiabatic case than bodies of the same radius in an
isothermal atmosphere. Once impactors are large enough to reach the planet
surface (the ground explosion regime), the gap between the mass loss curves is
significantly reduced, and the two curves converge as we reach the cap
ejection regime.

Figure 7: Atmospheric mass loss per impactor mass for a sub-Neptune/Neptune
like atmosphere ($h=19$ km, $\rho_{o}=1000$ kg/m3). We assume impactors of
density $\rho_{i}=3300$ kg/m3 approaching the planet with trajectory angles of
$\phi=\pi/4$. The solid-black curve shows mass loss assuming the atmosphere is
isothermal; the dashed-red curve assumes an adiabatic atmospheric profile.
Mass loss is separated into three distinct regimes based on the projectile
radius: aerial bursts (shaded yellow region), ground explosions (shaded blue
region), and ground explosions large enough to eject the full cap of
atmosphere above the tangent plane (shaded red region). For sub-Neptune and
Neptune-like atmospheres, aerial bursts can contribute significantly to
atmospheric mass loss, especially in the adiabatic case and when accounting
for the fact that smaller impactors are expected to be far more numerous than
larger ones.

###  6.2 Angle-Averaged Mass Loss

Figure 8 shows how the atmospheric mass-loss in figure 7 changes when the
trajectory angle is varied from $\phi=\pi/4$, to $\phi=0$ and $\phi=\pi/2$.
The general shape, progression from aerial burst to ground explosion to cap
ejection, and relative difference between the isothermal and adiabatic results
remain the same as in the $\phi=\pi/4$ case, with the curve simply extending
towards smaller sizes as the trajectory angle increases. As described in
section 2, this is because projectiles moving horizontally along the tangent
plane pass through relatively more atmosphere than those moving vertically,
and will therefore decelerate more quickly, broadening the range of impactor
radii that result in aerial bursts. This effect can also be seen in section
3.1, which shows that $r_{o}$, the maximum radius of projectile that results
in an aerial bursts, increases with trajectory angle $\phi$ while $r_{min}$
decreases with $\phi$.

In figure 9, we show the mass loss expected when we average over all
trajectory angles, from $\phi=0$ to $\phi=\pi/2$, for a sub-Neptune/Neptune-
like atmosphere. While the characteristic radii vary with trajectory angle,
the peaks in mass loss efficiency for the adiabatic and isothermal curves are
both located approximately at the aerial burst/ground explosion transition,
similar to figure 7. Note that $r_{min}$ only decreases significantly at
nearly-horizontal trajectory angles (figure 8), so that the range of radii
that allow for mass loss is relatively similar at all other angles. As such,
in the angle-averaged calculation the distribution of radii that result in the
highest overall mass loss efficiencies is relatively narrow, as it is largely
dominated by the radii that eject mass in the non-horizontal, $\phi\ll\pi/2$
cases. We therefore see a peak in mass loss efficiency at these radii,
accompanied by a tail of low efficiencies corresponding to aerial bursts
travelling the closer to the horizontal.

Overall, we find that aerial bursts are able to eject atmosphere with similar
maximum efficiencies to ground explosions. These maximum efficiencies are a
few to ten percent of atmospheric mass per impactor mass. Because of their
smaller size, in a given impact, an aerial bursts will typically eject less
mass than a single ground explosion. However, since smaller bodies are
expected to be more numerous, their collective contribution to the overall
atmospheric loss likely exceeds that by ground explosions. Assuming an
impactor size distribution given by $dN\propto r^{-3.5}dr$ (Dohnanyi, 1969),
we find that the more frequent aerial bursts would result in a total
atmospheric mass-loss that is larger by a factor of
$(r_{ground}/r_{aerial})^{0.5}$ compared to ground explosion of similar
efficiency.

Therefore, given the cumulative strength of aerial bursts, they should be
included in calculations of atmospheric mass-loss for thick atmospheres as
defined in section 5.

Figure 8: Dependence of ejected mass per impactor mass on trajectory angle,
$\phi$, for $\phi=0$ (dashed line), $\phi=\pi/4$ (dotted line), and
$\phi=\pi/2$ (solid line). We assume an atmosphere profile with $h=19$ km and
$\rho_{o}=1000$ kg/m3, and impactor densities of $\rho_{i}=3300$ kg/m3.
Impactors travelling horizontally pass through more atmosphere than those
vertically, so they decelerate higher in the atmosphere and result in a larger
range of possible impactor radii that can lead to aerial bursts.  Figure 9:
Ejected mass per impactor mass averaged over all trajectory angles ($\phi=0$
to $\pi/2$), for the isothermal and adiabatic profiles. We assume an
atmospheric profile with $h=19$ km and $\rho_{o}=1000$ kg/m3, and impactors of
density $\rho_{i}=3300$ kg/m3. The peak in each curve is located at
approximately the aerial burst/ground explosion transition, so that mass loss
to the left of the peak is largely due to aerial bursts while the right
corresponds mainly to ground explosions. The low efficiency tails of both
curves at small impactor radii are largely due to the increased range of radii
that can eject mass along horizontal trajectories. Meanwhile, the higher
efficiency portion of the curve is dominated by the impactor radii
corresponding to non-horizontal trajectories.

##  7 Discussion and Conclusions

In this work we presented a simple analytical model describing aerial bursts
and their associated atmospheric loss. Using this model, we have demonstrated
that aerial bursts can lead to significant atmospheric loss for planets with
sufficiently dense atmospheres, such as those expected for sub-Neptunes and
Neptune-like planets. For a typical sub-Neptunian atmosphere, we find mass
loss efficiencies of up to $\sim 10\%$ of the impactor mass. Specifically, the
main conclusions of this paper are as follows:

  1. 1.

Mass loss from aerial bursts becomes significant when the maximum impactor
size that will lead to an aerial burst rather than a ground explosion,
$r_{o}$, is larger than the minimum impactor size needed to achieve
atmospheric loss, $r_{min}$. For vertical trajectories, which give the most
stringent limit, this condition is approximately satisfied when
$\rho_{o}/\rho_{i}\gtrsim 0.4v_{e}/v_{\infty}$, which implies atmospheric
densities need to be comparable to impactor densities for impactor velocities
that are a few times the escape velocity of the planet. As a result, aerial
bursts are not expected to significantly contribute to the atmospheric mass-
loss history of Earth and Earth-like planets, but are expected to play an
important role for planets like Neptune with significant atmospheres.

  2. 2.

The range of impactor radii resulting in aerial burst-induced mass loss,
$r_{o}-r_{min}$, depends on the ratio of the density of the atmosphere to the
impactor density, and varies with the trajectory angle, such that the range
increases for larger impact angels and larger atmospheric densities.

  3. 3.

Because adiabatic atmospheres have larger densities for a given height above
the ground within in the first scale height compared to isothermal ones, the
range of impactor radii that result in aerial burst-induced mass loss is
larger in adiabatic atmospheres than isothermal atmospheres of equivalent
total mass, scale height, and atmospheric surface density. Similarly, aerial
burst impactors of equal size will eject relatively more mass in an adiabatic
atmosphere than in the isothermal atmosphere of equivalent total mass and
scale height.

  4. 4.

Increasing the density or initial velocity of impactors will increase the
expected mass loss per impactor mass in both the ground explosions and in
aerial bursts.

  5. 5.

The difference between isothermal and adiabatic atmospheric density profiles
matters most in the aerial burst-induced mass loss regime. The mass lost per
impactor mass is similar for both profiles in the ground explosion regime.

  6. 6.

For Neptune-like atmospheres, the atmospheric mass ejected per impactor mass
by aerial bursts is comparable to that lost by ground explosions. Therefore,
for impactors following a Dohnanyi size distribution, overall loss by aerial
busts is expected to exceed that by ground explosions by a factor of
$(r_{ground}/r_{aerial})^{0.5}$.

In addition to the contributions towards overall atmosphere mass loss
discussed thus far, the knowledge that a large number of projectiles will
result in aerial bursts as opposed to ground explosions may be important in
determining the final compositions of atmospheres.

Because aerial bursts are significantly decelerated high in the atmosphere,
they may be useful for depositing volatiles on a planet’s surface. In the sub-
Neptune atmosphere ($\rho_{o}\sim 1000$ kg/m3), we find that impactors with
radii up to tens of kilometers could be decelerated before they reach the
ground. These objects could therefore reach the surface of the planet without
creating a large crater and destroying the majority of the impactor.
Hydrodynamical simulations also show that for impactors with the density of
asteroids ($\sim 3300$ kg/m3), little to none of the impactor’s own mass will
be ejected over the course of its path through the atmosphere (Shuvalov et al.
2014). As such, these aerial bursts could trade small amounts of atmospheric
loss for delivery of their own volatiles to the planet.

## Acknowledgements

HES thanks Mark Wyatt for suggesting to work on this topic. This research was
supported by the National Aeronautics and Space Administration under grant No.
80NSSC18K0828.

## Data Availability

No new data were generated or analysed in support of this research.

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