# Takeout and Delivery: Erasing the Dusty Signature of Late-stage Terrestrial
Planet Formation

Joan R. Najita NSF’s NOIRLab, 950 N. Cherry Avenue, Tucson, AZ 85719, USA
Scott J. Kenyon Smithsonian Astrophysical Observatory, 60 Garden Street,
Cambridge, MA 02138, USA

###### Abstract

The formation of planets like Earth is expected to conclude with a series of
late-stage giant impacts that generate warm dusty debris, the most anticipated
visible signpost of terrestrial planet formation in progress. While there is
now evidence that Earth-sized terrestrial planets orbit a significant fraction
of solar-type stars, the anticipated dusty debris signature of their formation
is rarely detected. Here we discuss several ways in which our current ideas
about terrestrial planet formation imply transport mechanisms capable of
erasing the anticipated debris signature. A tenuous gas disk may be
regenerated via “takeout” (i.e., the liberation of planetary atmospheres in
giant impacts) or “delivery” (i.e., by asteroids and comets flung into the
terrestrial planet region) at a level sufficient to remove the warm debris.
The powerful stellar wind from a young star can also act, its delivered wind
momentum producing a drag that removes warm debris. If such processes are
efficient, terrestrial planets may assemble inconspicuously, with little
publicity and hoopla accompanying their birth. Alternatively, the rarity of
warm excesses may imply that terrestrial planets typically form very early,
emerging fully formed from the nebular phase without undergoing late-stage
giant impacts. In either case, the observable signposts of terrestrial planet
formation appear more challenging to detect than previously assumed. We
discuss observational tests of these ideas.

planet formation (1241) — circumstellar matter (241) — circumstellar gas (238)
— debris disks (363)

##  1 Introduction

The “classical” picture of terrestrial planet formation begins with the
formation of km-sized or larger planetesimals within a primordial gaseous disk
around a solar-type star (e.g., Lissauer, 1987; Wetherill & Stewart, 1993;
Weidenschilling et al., 1997; Chambers & Wetherill, 1998; Goldreich et al.,
2004; Kenyon & Bromley, 2006; Raymond et al., 2014). The planetesimals collide
and merge into lunar- to Mars-sized protoplanets, which interact
gravitationally once the gaseous disk dissipates, leading to collisions,
mergers, and ever-larger protoplanets (e.g., Iwasaki et al., 2001, 2002;
Kominami & Ida, 2002, 2004). Over the first 50–100 Myr,“giant impacts” among
the few remaining protoplanets—as in the moon-forming collision at $\sim$
30–50 Myr in the Solar System (e.g., Canup et al., 2021, and references
therein)—build the final set of terrestrial planets (e.g., Lammer et al.,
2021). Throughout this phase, giant impacts and collisions among remaining
planetesimals are expected to generate significant debris in the form of cm-
sized and smaller particles. The infrared emission from this warm debris,
which is expected to be bright and readily detectable, is the most anticipated
visible signpost of ongoing terrestrial planet formation (e.g., Kenyon &
Bromley, 2002, 2004; Raymond et al., 2011, 2012; Genda et al., 2015).

In Kenyon et al. (2016), we described how this picture leads to a current
conundrum: although planets like Earth and Venus are believed to occur
commonly around mature solar-type stars based on estimates from transit
surveys (e.g., $\sim 20$% occurrence rate; Petigura et al., 2013; Burke et
al., 2015; Bryson et al., 2021), the warm excesses expected to accompany their
formation (e.g., Kenyon & Bromley, 2004; Genda et al., 2015; Kenyon & Bromley,
2016; Kobayashi et al., 2019) are detected only rarely, in $\leq 2$%–3% of
young solar-type stars. The discrepancy can be resolved if terrestrial planets
are not as common as currently estimated. Alternatively, terrestrial planet
formation may work differently than we think, with planets completing their
formation on much shorter timescales than in our current picture, while they
are embedded in the primordial gaseous disk. A more prosaic possibility is
that there are unexamined processes that remove the dusty debris signature of
late-stage terrestrial planet formation.

Here we explore two possible dust removal mechanisms: gas drag and stellar
wind drag. As described in Kenyon et al. (2016), if the inner regions of
planetary systems evolve so that they end up possessing tenuous gas reservoirs
at the epoch of final terrestrial planet assembly, at a level of $\sim
10^{-5}-10^{-6}$ of the minimum mass Solar nebula (MMSN), the gas can remove
the dusty debris associated with terrestrial planet formation through a
combination of aerodynamic drag and radiation pressure (see also Takeuchi &
Artymowicz, 2001; Krumholz et al., 2020). A $10^{-5}$ MMSN column density of
0.02 $\rm g~{}cm^{-2}$ within 2 au corresponds to a gas mass of 0.01 $\rm
M_{\oplus}$ (equivalent to $\sim$ 43 Earth oceans).

Several possible pathways to such tenuous gas disks emerge naturally from our
current picture of terrestrial planet formation. Gaseous planetary
atmospheres, primordial or secondary, might be liberated through giant
impacts, the tragic loss of planetary atmospheres a boon to the removal of the
planet’s formation signature. Alternatively, gas may be brought in by
evaporating icy comets and asteroids that have been scattered into the
terrestrial planet region. Stellar wind drag is another possible dust removal
mechanism, in which dust grains in the terrestrial planet region are dragged
into the central star through interactions with a high velocity stellar wind
(e.g., Burns et al., 1979; Plavchan et al., 2005; Kla$v_{0}$cka, 2013;
Spalding & Adams, 2020). While this mechanism is unimportant for dust around
solar-type stars at Gyr ages and is weaker at large orbital distances, it can
be a much more potent dust removal mechanism at the small orbital radii
associated with terrestrial planet formation and at the enhanced wind mass
loss rates thought to characterize young stars (ages $\lesssim 100$ Myr).

In Section 2, we recap the predicted dusty signature that is predicted to
accompany late terrestrial planet formation and the current observational
limits. We then describe the gas reservoirs that are expected to be liberated
from planetary atmospheres by giant impacts (Section 3) and delivered to inner
solar systems by asteroids and comets (Section 4). Section 5 describes the
possible role of stellar wind drag in removing collisional debris from the
terrestrial planet region. These ideas and future directions are discussed in
Section 6 and summarized in Section 7.

##  2 Predicted IR Excesses and Observational Limits

To establish the conditions required for a detectable IR excess from solid
material at 1 au, we follow Kenyon et al. (2016). Spitzer observations at
16–24 $\rm\,\mu{\rm m}$ (8–12 $\rm\,\mu{\rm m}$) are sufficient to detect
solid material at $a$ = 1 au with a temperature $T_{d}\sim$ 300 K and a
fractional luminosity $L_{d}/L_{\star}\ =A_{d}/4\pi a^{2}\gtrsim 10^{-4}$
($\gtrsim 3\times 10^{-4}$), where $L_{d}$ is the thermal luminosity of the
solids, $L_{\star}$ is the luminosity of the central star, and $A_{d}$ is the
cross-sectional area of the solids. For a given sensitivity at 16–24
$\rm\,\mu{\rm m}$, hotter (colder) dust is harder (easier) to detect (see
Figure 1 of Kenyon et al., 2016).

Although 24 $\rm\,\mu{\rm m}$ excesses are fairly common among nearby solar-
type stars with ages of 10–100 Myr, 8–16 $\rm\,\mu{\rm m}$ excesses are rare
(e.g., Stauffer et al., 2005; Silverstone et al., 2006; Currie et al., 2007;
Rhee et al., 2008; Dahm & Carpenter, 2009; Carpenter et al., 2009b, a; Melis
et al., 2010; Stauffer et al., 2010; Chen et al., 2011; Beichman et al., 2011;
Smith et al., 2011; Zuckerman et al., 2011; Luhman & Mamajek, 2012; Ribas et
al., 2012; Urban et al., 2012; Zuckerman et al., 2012; Jackson & Wyatt, 2012;
Kennedy & Wyatt, 2012, 2013; Ballering et al., 2013; Cloutier et al., 2014;
Matthews et al., 2014; Vican & Schneider, 2014; Patel et al., 2014; Ishihara
et al., 2017; Melis et al., 2021; Moór et al., 2021). As discussed in Kenyon
et al. (2016), the great majority of 24 $\rm\,\mu{\rm m}$ excess sources lack
excess emission at shorter wavelengths, implying that they arise from cold
material with a dust temperature $T_{d}\lesssim$ 150 K. The high (low)
frequency of cold (warm) dust from short wavelength Spitzer and WISE studies
is consistent with the $\sim$ 25% frequency of cold excesses among solar-type
stars derived from other Spitzer and Herschel programs (e.g., Bryden et al.,
2009; Eiroa et al., 2013; Sibthorpe et al., 2018; Matrà et al., 2018; Najita
et al., 2022, and references therein). Thus, we conclude that the warm
excesses associated with solar-type stars with ages $\sim$ 10–100 Myr are
rare.

Converting an observational limit on fractional luminosity into an estimate of
the total mass of solids requires an adopted size distribution as a function
of particle radius $r$, $N(r)$. In a simple system, where all solids have the
same binding energy and the collision velocity is just large enough to
catastrophically fragment particles, $N(r)\propto r^{-3.5}$ (Kenyon & Bromley,
2020). The mass is then $M_{d}=(4/3)\rho A_{d}(r_{min}r_{max})^{1/2}$, where
$\rho$ is the mass density of a particle and $r_{max}$ is the radius of the
largest particle. Setting $r_{max}$ = 300 km, $\rho$ = 3 $\rm g~{}cm^{-3}$,
and $L_{d}/L_{\star}=10^{-4}$, the ensemble of solids has a total cross-
sectional area $A_{d}\approx 3\times 10^{23}\,{\rm cm}^{2}$ and a total mass
$M_{d}\approx 5.5\times 10^{25}$ g $\approx$ 0.7 lunar masses.

Systems with a size-dependent binding energy have wavy size distributions that
are somewhat steeper than $N(r)\propto r^{-3.5}$ (e.g., Campo Bagatin et al.,
1994; O’Brien & Greenberg, 2003; Kobayashi & Tanaka, 2010; Wyatt et al.,
2011). Adopting the bulk properties of basalt (Benz & Asphaug, 1999), a
collision velocity large enough to catastrophically fragment 300 km
planetesimals, and a collisional cascade in equilibrium from 300 km to 1
$\rm\,\mu{\rm m}$ (e.g., Wyatt et al., 2011; Kenyon & Bromley, 2020), the mass
is $\sim$ 65% smaller than estimated above: $M_{d}\approx 3.7\times 10^{25}$ g
(0.5 lunar masses). Although different choices for the bulk properties of
solids and the collision velocity result in somewhat different masses, 0.2–1
lunar masses in solids is sufficient to generate the cross-sectional area,
$A_{d}\approx 3\times 10^{23}\,{\rm cm}^{2}$, required for material with
$L_{d}/L_{\star}$$\approx 10^{-4}$ at $a\approx$ 1 au.

These estimates are consistent with solid mass and dust luminosity estimates
derived in numerical calculations of the giant impact phase (e.g., Kenyon &
Bromley, 2004; Jackson & Wyatt, 2012; Genda et al., 2015; Kenyon & Bromley,
2016; Kobayashi et al., 2019). In these simulations, collisions of lunar-mass
to Mars-mass protoplanets usually generate larger merged objects and ejected
fragments with radii of 100–1000 km and total masses of $\lesssim$ 1–2 lunar
masses. As protoplanets grow into larger and larger planets, the fragments
generate a collisional cascade with a detectable infrared excess at 16–24
$\rm\,\mu{\rm m}$ and a dust luminosity, $L_{d}/L_{\star}$ $\gtrsim 10^{-4}$,
throughout the 10–100 Myr phase of rocky planet building.

During the collisional cascade, high velocity collisions gradually grind down
larger solids into smaller solids, which are ejected from the system by
stellar radiation pressure (e.g., Wyatt, 2008; Kobayashi & Löhne, 2014;
Matthews et al., 2014; Kenyon & Bromley, 2016; Kobayashi et al., 2019; Marino,
2022). In the standard kinetic model, the evolution time for the cascade is
roughly equivalent to the time required for a collision between the two
largest particles in the swarm, which depends on the distance from the central
star, the radius $r_{max}$ of the largest solid involved in the cascade, and
the mass in solids with radii $r\leq r_{max}$ (e.g., Wyatt & Dent, 2002;
Dominik & Decin, 2003; Wyatt, 2008; Krivov et al., 2008; Kenyon & Bromley,
2017; Krivov & Wyatt, 2021). If the solids reside in an annulus with a radial
width $\delta a$ that is larger than the radial excursion of a typical solid,
$\delta a\gg ea$, where $e$ is the orbital eccentricity, the time scale to
remove the largest particles in the cascade is $t_{d}=r_{max}\ \rho
P\alpha/12\pi\Sigma_{d}$, where $P$ is the orbital period and $\Sigma_{d}$ is
the surface density of solids at a distance $a$ from the host star (e.g.,
Kenyon & Bromley, 2017, and references therein). In this expression, the
$\alpha$ term encodes the dependence of the time scale on the collision
velocity and the binding energy of the largest solids (for other approaches,
see Bottke et al. (2005), Kobayashi & Tanaka (2010), and Kennedy & Wyatt
(2010)). Defining $v$ as the center-of-mass collision velocity and
$Q_{D}^{\star}$ as the collision energy required to remove half the mass of a
pair of colliding planetesimals and disperse it to infinity, $\alpha$ is a
function of the ratio $v^{2}/Q_{D}^{\star}$ (Kenyon & Bromley, 2017). When
high velocity collisions just shatter the largest objects
($v^{2}/Q_{D}^{\star}$ $\approx$ 8), $\alpha\approx 5$. Factor of three larger
velocities yields $\alpha\approx 0.7$ (see Fig. 2 of Kenyon & Bromley, 2017).

Adopting $\Sigma_{d}\approx M_{d}/\pi a^{2}$, the lifetime of the cascade is

$t_{d}\approx{\rm 30~{}Myr}~{}\alpha\left(\frac{a}{\rm
1~{}au}\right)^{3/2}\left(\frac{r_{max}}{\rm
300~{}km}\right)^{1/2}\left(\frac{1~{}{\rm\,\mu{\rm
m}}}{r_{min}}\right)^{1/2}\left(\frac{10^{-4}}{L_{d}/L_{\star}}\right)~{}.$
(1)

As a result, the debris produced by giant impacts is expected to remain bright
and observable for an extended period of time, in the absence of removal
mechanisms other than collisions and stellar radiation pressure. When
$\alpha\approx$ 0.5–3, the collision time in eq. 1 is similar to the lifetime
of detectable warm dust in numerical calculations (Kenyon & Bromley, 2004;
Raymond et al., 2011; Genda et al., 2015; Kenyon & Bromley, 2016). In Fig. 6
of Genda et al. (2015), the 24 $\rm\,\mu{\rm m}$ flux from the debris of giant
impacts equals or exceeds the flux from the central star (and the detection
limit for Spitzer at 16-24 $\rm\,\mu{\rm m}$) for $\sim$ 100 Myr. Although the
bright pulse of emission generated in a single giant impact lasts for only a
few Myr, the broad pedestal of excess emission produced by dust from the
ensemble of giant impacts persists for $\gtrsim$ 100 Myr. Similarly,
calculations described in Kenyon & Bromley (2004, 2016) have dust luminosities
$L_{d}/L_{\star}$ $\gtrsim 10^{-4}$ for evolution times $\lesssim$ 100 Myr.
The rarity of infrared excess emission detections at short wavelengths (e.g.,
at 16 $\rm\,\mu{\rm m}$) demonstrates that such warm debris is rare.

##  3 Gas from Liberated Planetary Atmospheres

###  3.1 Primordial Atmospheres

Solar System formation models suggest that terrestrial planet formation was
well underway during the protosolar nebular phase. Planets like Earth and
Venus may have acquired a significant fraction of their mass (perhaps $>50$%)
prior to the dissipation of the nebula (Walsh & Morbidelli, 2011; O’Brien et
al., 2014; Raymond et al., 2014; Lammer et al., 2021), a perspective that is
supported by some interpretations of meteoritic data (Boyet & Carlson, 2005;
Yu & Jacobsen, 2011). Terrestrial planets that acquire a significant mass
while the nebular gas disk is present are expected to capture tenuous
primordial atmospheres whose mass depends on the nebular density (e.g.,
Rafikov, 2006). A protoplanet with mass $M_{p}=0.55\,{\rm M_{\oplus}}$ could
acquire an atmosphere that is $M_{\rm atm}/M_{p}=1$% of the planet’s mass
(Lammer et al., 2020a). A lower-mass protoplanet with $M_{p}\approx$ 0.1–0.3
$\rm M_{\oplus}$ may acquire an atmosphere that is 0.01% to 1% of the planet’s
mass (Ginzburg et al., 2016; Lammer et al., 2020a). These atmospheres are
similar to the 0.001–0.01 $\rm M_{\oplus}$ needed for aerodynamic drag to
remove dusty debris from the terrestrial planet region (Section 1).

As the nebula dissipates, the planetary atmosphere adjusts to the declining
external pressure from the nebula and the increasing exposure to direct
irradiation by the host star. Planets orbiting close to their stars are more
likely than planets at larger distances to lose some or all of their
primordial atmosphere before the giant impact phase commences.

Theory suggests that low mass planets ($\lesssim$ 5–10 $\rm M_{\oplus}$)
orbiting within $\sim 0.5$ au lose a significant fraction, perhaps all, of
their atmospheres through photoevaporation (Owen & Wu, 2013; Lopez & Fortney,
2014; Jin et al., 2014; Owen & Wu, 2016, 2017; Jin & Mordasini, 2018;
Mordasini, 2020; Venturini et al., 2020) or core-powered mass loss (Ginzburg
et al., 2016, 2018; Gupta & Schlichting, 2019). EUV/X-ray photoevaporation is
particularly efficient in reducing planetary atmospheres at orbital radii
$\lesssim 0.2$ au, with the transformation occurring on a timescale $\lesssim$
100 Myr (Owen & Wu, 2013). Core-powered mass loss operates over longer ($\sim$
Gyr) timescales (Ginzburg et al., 2018). Planetesimal impacts remove
atmospheres on even longer timescales. Among planets with orbital periods
$\lesssim$ 30 days, observations provide strong support for the ‘radius’ or
‘evaporation’ valley predicted from these studies111For an alternative
approach to the radius valley, see Lee & Connors (2021).  (e.g., Fulton et
al., 2017; Van Eylen et al., 2018; Fulton & Petigura, 2018; Hardegree-Ullman
et al., 2020; Petigura, 2020). Better statistics for the planetary mass–radius
relation at 0.2–0.6 au would test these theories in greater detail (e.g.,
Rogers et al., 2021).

Beyond 0.5–0.6 au, planets have a better chance of retaining their primordial
atmospheres. At these larger distances, photoevaporation and core-driven mass
loss are less effective (e.g., Mordasini, 2020). Planetesimal impacts—which
may remove or add material to the planet’s atmosphere—then become more
important (Wyatt et al., 2020; Kegerreis et al., 2020b, a). Although the
relative importance of these processes among exoplanets is uncertain,
abundance analyses of Venus and Earth provide some clues. Detailed models for
the atmospheric evolution of Earth (Venus) conclude that protoplanets with
masses exceeding $\sim$ 0.5–0.6 $\rm M_{\oplus}$ ($\sim$ 0.85–0.9 $\rm
M_{\oplus}$) retain their primordial atmospheres following the dissipation of
the disk, while those with lower masses do not (e.g., Lammer et al., 2020a).

###  3.2 Secondary Atmospheres

Whether they retain any portion of their primordial atmospheres, terrestrial
planets can also generate secondary atmospheres through outgassing. Recently
formed terrestrial planets are expected to possess significant magma oceans,
involving part or all of the silicate mantle (e.g., Urey, 1955; Elkins-Tanton,
2008; Elkins-Tanton et al., 2011; Elkins-Tanton, 2012; Lammer et al., 2018).
Melting is induced by heating from multiple processes: the decay of short-
lived radiogenic isotopes (e.g., 26Al; Neumann et al., 2014), the conversion
of gravitational potential energy into heat when the core and mantle decouple
(e.g., Rubie et al., 2015), and kinetic energy from protoplanet impacts (e.g.,
Tonks & Melosh, 1993; Elkins-Tanton et al., 2004; Monteux & Arkani-Hamed,
2014; Manske et al., 2021). As magma oceans solidify, they are expected to
expel copious amounts of gas. Some portion of the gaseous atmosphere is then
expected to be ejected in the giant impacts that build the final masses of
terrestrial planets.

Estimates from the literature aimed at explaining the known properties of
Solar System planets predict that a planet can outgas 70%–100% of its initial
carbon and water into the atmosphere (Elkins-Tanton, 2008; Hamano et al.,
2013; Erkaev et al., 2014; Lichtenegger et al., 2016; Odert et al., 2018;
Massol et al., 2016; Lammer et al., 2018, and references therein). We can
consider in a more general way the volatiles available to terrestrial planets
as they are assembled, independent of how the assembly works in detail. The
volatile budget of the solids from which planets form establishes an upper
envelope for the reservoir available for a regenerated gaseous atmosphere. The
water abundance of the bulk silicate Earth, 0.6%–3.6% by weight (e.g., Daly et
al., 2021), lies between that of carbonaceous chondrites (2%–13% by weight;
Alexander et al., 2012; Marty et al., 2016; McCubbin & Barnes, 2019; Shimizu
et al., 2021; Alexander, 2022) and that of enstatite and ordinary chondrites
(0.1%–0.5% by weight; Piani et al., 2018; Piani & Marrocchi, 2018; Piani et
al., 2020; Bates et al., 2021; King et al., 2021).

Taking a conservative approach, if Earth formed from material that was 1%
volatile by weight, and the volatiles are completely outgassed, as in some of
the detailed models above, the mass of the secondary atmosphere would be
similar to the gas reservoir needed to remove the debris signature (Section
1). If carbonaceous chondrite-like materials contribute significantly to the
planet’s mass budget, the volatile budget could be significantly larger. Given
the likely importance of inward pebble drift and pebble accretion in building
terrestrial planet cores (Lammer et al., 2020a, b), it seems plausible that
volatile-rich carbonaceous chondrites could contribute a significant fraction
of a terrestrial planet’s mass (Alexander, 2022).

Figure 1:  Schematic illustration of possible contributors to the regeneration
of a tenuous gas disk: gaseous atmospheres liberated by a sequence of giant
impacts (green dots) and their cumulative contribution (green line), as
described in Section 3, and the cumulative mass in volatiles delivered by
scattered comets (dashed blue line) and asteroids (dashed red line), as
described in Section 4. For illustration purposes, comets and asteroids are
assumed to deliver gas at constant mass input rates over the time intervals
shown.

###  3.3 Giant Impacts

Collisions between protoplanets and planets will liberate some or all of the
outgassed atmospheres (primordial and/or secondary; e.g., Ahrens, 1993; Genda
& Abe, 2003; Schlichting et al., 2015; Inamdar & Schlichting, 2015; Yalinewich
& Schlichting, 2019; Kegerreis et al., 2020b, a; Biersteker & Schlichting,
2019, 2021), regenerating a gaseous disk. As a concrete example, we imagine
starting with a 0.5 $\rm M_{\oplus}$ planet that experiences 5 successive
giant impacts with protoplanets, each of which has a mass 0.1 $\rm
M_{\oplus}$. For simplicity, we assume no primordial atmosphere and that the
volatiles in the planet and protoplanets have been completely outgassed into
their atmospheres. Following the discussion in Inamdar & Schlichting (2015),
we assume that each impact removes the atmosphere of the impactor completely.
The fraction of the planet’s atmosphere that is removed depends on the
quantity $v_{\rm imp}m_{\rm imp}/v_{\rm esc}M_{\rm core}$, where $v_{\rm imp}$
and $m_{\rm imp}$ are the velocity and mass of the impactor, $v_{\rm esc}$ is
the planet’s escape velocity, and $M_{\rm core}$ is the planet’s mass.
Assuming $v_{\rm imp}/v_{\rm esc}\simeq 1$ and $m_{\rm imp}/M_{\rm core}\simeq
0.2,$ the fraction of the planet’s atmosphere that is removed is $\sim 0.1$ as
long as the mass of the planet’s atmosphere is much less than the mass of its
core (Inamdar & Schlichting, 2015).

The successive collisions build up a tenuous gaseous disk. If all of the
colliding bodies have a volatile fraction of 1% by mass, each giant impact
contributes $10^{-3}\,{\rm M_{\oplus}}$ of gas from the impactor and $5\times
10^{-4}\,{\rm M_{\oplus}}$ to $3\times 10^{-4}\,{\rm M_{\oplus}}$ of gas lost
from the planet’s atmosphere; the amount declines with successive impacts as
the planet’s atmosphere is depleted. Each impact ejects $\sim 1.4\times
10^{-3}\,{\rm M_{\oplus}}$ of gas from the planet and impactor (Figure 1,
green points), and the total gas mass liberated after 5 impacts is $\sim
7\times 10^{-3}~{}{\rm M_{\oplus}}$ (Figure 1, green line). By regenerating a
gaseous inner disk through outgassing and giant impacts, the gas is available
exactly when needed to remove the dust signature generated by the same giant
impacts.

The liberated gas is comparable to the mass needed to remove the expected
debris signature (Kenyon et al., 2016). To estimate the removal time, we
assume the gas extends from $a_{in}\lesssim$ 0.5 au to $a_{out}\gtrsim$ 2–3
au, orbits the central star at the local circular velocity, and has a radial
surface density distribution $\Sigma\propto a^{-1}$. The central star has a
mass $M_{\star}$ = 1 $M_{\odot}$ and luminosity $L_{\star}$ = 1 $L_{\odot}$.
As in Kenyon et al. (2016), we combine the approaches of Weidenschilling
(1977) and Takeuchi & Artymowicz (2001) and solve for the radial and azimuthal
velocity of particles relative to the gas. Our approach assumes a vertical
scale height $H/a=0.03~{}(a/\rm{1~{}au})^{1/8}$, a gas temperature
$T_{g}=278~{}(a/{\rm 1~{}au})^{1/2}$, and other parameters appropriate for a
gaseous disk (see also Adachi et al., 1976; Kenyon & Hartmann, 1987; Chiang &
Goldreich, 1997; Takeuchi & Artymowicz, 2001; Rafikov, 2004; Chiang & Youdin,
2010; Armitage, 2013; Youdin & Kenyon, 2013). Additional details are available
in Appendix A.2 of Kenyon et al. (2016). Other choices for the exponents in
the relations for $\Sigma$, $T_{g}$, and $H$ would have a modest impact on the
results.

Figure 2 shows the drift time for solids of different sizes at 1 au in gaseous
reservoirs with a range of masses. Larger solids drift inward toward the star
under gas drag (solid symbols) while radiation pressure drives small, weakly
coupled particles outward, where they become colder (open symbols). For solids
that produce an IR excess (size $<300\,{\rm\,\mu{\rm m}}$), the drift times
are short ($<10^{2}-10^{4}$ yr) for gaseous reservoirs at the level
$10^{-6}-10^{-5}$ of MMSN (e.g., Weidenschilling, 1977; Rafikov, 2004; Kenyon
et al., 2016). Thus, for the amount of gas liberated in the first collision in
the example above ($1.5\times 10^{-3}\,{\rm M_{\oplus}}\sim 1.5\times 10^{-6}$
MMSN), the particles that produce an IR excess would have a drift time
$\lesssim 4000$ yr. If all of the gas liberated in each impact survives until
the end of the giant impact phase, the cumulative reservoir (of $7\times
10^{-3}\,{\rm M_{\oplus}}$ or $\sim 0.7\times 10^{-5}$ MMSN) would take $\sim
100$ years to remove the debris signature from the final impact. In either
case, the accompanying debris signature would be removed on a time scale that
is a small fraction of the lifetime of the giant impact phase ($\sim 100$
Myr).

Although a rigorous assessment of the removal time requires more detailed
calculations of the long-term evolution of gas liberated from planets and the
associated radial drift from gas drag and radiation pressure, these estimates
assume a fairly conservative volatile fraction of 1% by mass for the planet-
building material. Carbonaeceous chondrites have volatile fractions that are
as much as $\sim 10$ times larger (e.g., Alexander, 2022; Piani et al., 2020).
If the planets and protoplanets form from more volatile rich material, the
regenerated gas mass is correspondingly larger. In our model, more massive
regenerated gas disks have longer radial drift times (Fig. 2); however, the
drift times are still very short compared to the $\sim$ 100 Myr lifetime of
the giant impact phase.

Figure 2:  Radial drift time scales for solid particles within a gaseous disk
at 1 au. The legend lists the disk surface density relative to a MMSN with
$\Sigma_{0}$ = 2000 $\rm g~{}cm^{-3}$ at 1 au. Filled (open) circles indicate
inward (outward) drift. For this configuration, inward drift from gas drag
balances outward drift from radiation for particles with $r\approx$ 40
$\rm\,\mu{\rm m}$. As the gas surface density at 1 au falls, particles with
$r<$ 40 $\rm\,\mu{\rm m}$ drift outward more rapidly. The maximum inward drift
time for particles with $r\approx$ 100 $\rm\,\mu{\rm m}$ to 10–100 cm is
approximately independent of the gas surface density.

##  4 Gas Delivery by Comets and Asteroids

In contrast to the above self-reliant pathway, in which giant planet impacts
contribute the volatiles needed to remove their own debris signatures,
volatiles may also be delivered to the terrestrial zone by other mechanisms
(e.g., Greenberg & Nolan, 1989; Morbidelli et al., 2000; Vokrouhlický &
Farinella, 2000; Raymond et al., 2004; O’Brien et al., 2014). For example,
volatile-rich planetesimals scattered from beyond several au into the
terrestrial zone may outgas and/or evaporate, and occasionally bombard
terrestrial planets. While this mechanism is often invoked to transport water
and organic molecules to terrestrial planet surfaces, the difficulty of
hitting a planet implies that if this mechanism is successful, a much larger
reservoir of volatiles is delivered to replenish a gaseous inner disk.

Comets. Taking the Solar System as a guide, we consider a specific example
scenario in which comets deliver water to Earth. The mass in comets that
impact Earth is

$M_{\rm c,imp}=M_{\rm c}\,N_{\rm orb}\,P_{\rm c},$ (2)

where $M_{\rm c}$ is the total mass in comets sent to the inner Solar System,
$P_{\rm c}$ is the probability that the comet will hit Earth on a single
orbital passage, and $N_{\rm orb}$ is the number of times the comet orbits the
inner Solar System before it is destroyed or scattered away.

The collision probability $P_{\rm c}$ is very small. As described by Sinclair
et al. (2020), comets entering the inner Solar System will generally have
velocities $\sqrt{2}v_{\rm orb}$ when they reach 1 au, where $v_{\rm orb}$ is
the orbital velocity of the Earth (see also Schlichting et al., 2012; O’Brien
et al., 2018; Brunini & López, 2018; Martin & Livio, 2021). The velocity of
the comet relative to Earth is $v_{\rm rel}\simeq 0.4-1.7\,v_{\rm orb}$
depending on whether the comet encounters Earth at pericenter or far from
pericenter. These relative velocities correspond to an effective area for
capture of $A_{\rm eff}=A_{\oplus}[1+(v_{\rm esc}/v_{\rm rel})^{2}]$ where
$A_{\oplus}$ is the physical cross-sectional area of Earth ($\pi
R_{\oplus}^{2}$), and $v_{\rm esc}$ is its escape speed. Because Earth’s
escape speed is 11 $\rm km~{}s^{-1}$, there is no significant gravitational
focusing for comets, given their typical relative velocity of $\sqrt{2}v_{\rm
orb}.$ The Earth’s effective area is tiny compared to the size of the inner
Solar System, resulting in a collision probability of $P_{\rm c}=A_{\rm
eff}/\pi(1\,{\rm au})^{2}\simeq(R_{\oplus}/1\,{\rm au})^{2}=2\times 10^{-9}$.

As a result, if comets deliver some fraction of an ocean of water to Earth,
they also deliver a much larger mass of volatiles to the inner Solar System.
We can estimate the total mass in volatiles delivered to the inner Solar
System using equation 2, the above estimate for $P_{\rm c}$, and an estimate
for $M_{\rm c,imp}$ from the literature. Sinclair et al. (2020) estimate that
comets deliver $\sim 10^{-4}$ oceans to Earth, equivalent to a total mass
$\sim 2\times 10^{-8}f_{\rm vol}^{-1}\,{\rm M_{\oplus}}$ if comets have a
volatile fraction of $f_{\rm vol}$. This estimate implies that the combination

$M_{\rm c}\,N_{\rm orb}=M_{\rm c,imp}/P_{\rm c}=10\,{\rm M_{\oplus}}\,f_{\rm
vol}^{-1},$ (3)

or $M_{\rm c}\,N_{\rm orb}=20\,{\rm M_{\oplus}}$ if $f_{\rm vol}=0.5.$ That
is, comets with a total mass $M_{\rm c}$ orbit within the inner Solar System
an average of $N_{\rm orb}$ times each, which is equivalent to an effective
mass of $M_{\rm c}\,N_{\rm orb}=20\,{\rm M_{\oplus}}$ of comets each making
one pass through the inner Solar System.

We can estimate the volatiles lost by the orbiting bodies using Solar System
comets as a guide. If each comet has a mass $m_{\rm c}$ (in gas and dust) and
outgasses $\Delta m_{\rm g}$ of volatiles per passage, the effective mass of
$M_{\rm c}\,N_{\rm orb}$ corresponds to $M_{\rm c}\,N_{\rm orb}/m_{\rm c}$
comet passages that release a total of $M_{\rm c}\,N_{\rm orb}\,\Delta m_{\rm
g}/m_{\rm c}$ in volatiles. If each comet has the mass of Comet Halley ($\sim
2\times 10^{17}$g in gas and dust) and outgasses $2\times 10^{14}\,$g of
volatiles in one passage (as estimated for comet Halley’s appearance in 1910;
Hughes, 1985), the orbiting comets would release a total of 0.02 $\rm
M_{\oplus}$ in volatiles (i.e., $\sim 86$ Earth oceans $\simeq 2\times
10^{-5}$ MMSN) into the inner Solar System (see also Thomas & Keller, 1991;
Jewitt & Matthews, 1999; Lisse, 2002).

Similarly, we can estimate $M_{c}$, the total mass in comets sent to the inner
Solar System, by estimating $N_{\rm orb}$ from a consideration of the
lifetimes of the Jupiter Family Comets (JFCs). The orbital properties of JFCs
today suggest they have active lifetimes of 12,000 yr (Levison & Duncan,
1997). For a typical JFC orbital period of 6 yr, JFCs experience $\sim 2000$
returns to the inner Solar System (Weissman et al., 2020, and references
therein). Since comets lose 5–20 m of surface volatiles per orbit (e.g.,
Huebner et al., 1986; Marboeuf et al., 2016) and a Halley-like comet will lose
50% of its mass in the first 20% of these passes (Hughes, 1985), comets should
strike the Earth within the first few 100 orbits to deliver most of their mass
to Earth. If $N_{\rm orb}=400$, $M_{\rm c}=0.05\,{\rm M_{\oplus}}$, a small
fraction of the original reservoir of 10–20 $\rm M_{\oplus}$ from which JFC
comets originate.

These orbiting bodies can live as active comets for $\sim 2000$ orbits,
evaporating and shrinking away. If all of the $M_{c}=0.05\,{\rm M_{\oplus}}$
evaporates and $f_{\rm vol}=0.5$, the comets deliver $0.025\,{\rm M_{\oplus}}$
in volatiles ($\sim 2.5\times 10^{-5}$ MMSN) to the inner Solar System,
comparable to the mass needed to remove the dusty debris signature of
terrestrial planet formation (Figure 1, dotted blue line). If comets
contribute more than $10^{-4}$ of Earth’s oceans, they would deliver even more
gas into the inner Solar System. If some fraction of comets are ejected before
they can fully dispense their volatiles, comets would contribute less gas.

Asteroids. In lieu of comets as the source of Earth’s water, Morbidelli et al.
(2000) estimate that the asteroid belt beyond 2.5 au could have delivered more
than an ocean of water to Earth (see also Vokrouhlický & Farinella, 2000). In
their simulation, which assumes an initial reservoir of $M_{\rm ast}\sim 4$
$\rm M_{\oplus}$ in asteroids located beyond 2.5 au, the formation of Jupiter
disrupts the asteroids, sending some fraction inward to smaller radii. Each
asteroid has a probability of $1.3\times 10^{-3}$ of colliding with Earth. As
a result, $5\times 10^{-3}\,{\rm M_{\oplus}}$ of asteroids impact Earth,
delivering $5\times 10^{-4}\,{\rm M_{\oplus}}$ of water, if the asteroids have
the volatile fraction of carbonaceous chondrites (10%).

The excited asteroids would deliver much more water to the inner Solar System,
as can be inferred from the reported results. Figure 1 of Morbidelli et al.
(2000) indicates that a fraction $f_{\rm in}\sim$ 10–15% of all asteroids are
deflected onto orbits with perihelia $\lesssim 1$ au. If the asteroids
maintain these orbits long enough to completely outgas their volatiles, they
would deliver a total mass of $f_{\rm in}\,f_{\rm vol}\,M_{\rm ast}\sim
0.04-0.06\,{\rm M_{\oplus}}$ in volatiles to small radii, more than that
needed to remove dusty debris signatures (Figure 1, dotted red line). Their
contribution to the gas budget would be reduced if the asteroids are ejected
dynamically before they can evaporate or if they become inactive asteroids
before releasing all of their volatiles.

Timing. While both comets and asteroids can deliver a significant mass of gas
to the inner Solar System, the timing of the delivery depends on the dynamical
instabilities that trigger their journey. For comets, the gas is likely to
have been delivered at about the right time to sweep away the debris signature
of Earth’s formation. In the Nice model, the dynamical instability involving
the giant planets that led to the bombardment of the inner Solar System began
sometime in the first 100 Myr (Nesvorný et al., 2018), similar to the
timescale for the Moon-forming impact, i.e., the last major impact in forming
the Earth (Canup et al., 2021).

For asteroids, the situation depends on how the scattering occurs. If the
formation of a gas giant like Jupiter launches the asteroids (e.g., Morbidelli
et al., 2000; Vokrouhlický & Farinella, 2000), the gas is likely to be
delivered too early to be helpful in erasing the signature of terrestrial
planet formation. Gas giant formation likely occurs at $<10$ Myr, when a
primordial gas disk is present, much earlier than the Moon-forming impact is
thought to have occurred. The timing would work out better with asteroid belts
that are instead destabilized by the formation of lower mass ice giants (e.g.,
Neptunes) that form and migrate on a longer timescale. The time scale for
clearing an orbit is related to the local orbital period, i.e., $t_{\rm
clear}\sim 10^{9}~{}P~{}(M_{P}/{\rm M_{\oplus}})^{-2}$ (Tremaine, 1993;
Margot, 2015), or 30 Myr for a Neptune-mass planet at 4 au.

Applicability Beyond the Solar System. While the above examples are based on
scenarios for our own Solar System, similar events may have occurred in the
histories of other planetary systems, given the known properties of
protoplanetary disks and mature exoplanets. Firstly, the initial cometary
reservoirs of other stars are likely to be similar. At $\sim 1$ Myr, T Tauri
stars are surrounded by dust disks 10s of au in size with median dust masses
$\sim 6\times 10^{-5}$ of their stellar mass (e.g., Villenave et al., 2021;
Manara et al., 2022) or $\sim$ 20 $\rm M_{\oplus}$ for a $1\,M_{\odot}$ star.
Both theory and the demographics of debris disks suggest that protoplanetary
disks form planetesimals with modest efficiency $\sim 10\%-50$% (e.g., Simon
et al., 2016; Li et al., 2018; Abod et al., 2019; Carrera et al., 2020; Klahr
& Schreiber, 2020; Gole et al., 2020; Rucska & Wadsley, 2021; Najita et al.,
2022). As a result, it seems quite plausible that a large fraction of
protoplanetary disks have reservoirs of $\gtrsim 10\,{\rm M_{\oplus}}$ in
planetesimals at distances of tens of au. If giant planets are also present in
these disks and undergo migration, a small fraction of these solids may
eventually be delivered into the inner, terrestrial planet regions of the
system.

Indeed, a large fraction of stars like the Sun are known to harbor giant
planets at outer Solar System distances. Microlensing studies imply an average
of 0.8 planets per star for planets $>8\,{\rm M_{\oplus}}$ at orbital
distances 1–15 au (Suzuki et al., 2016; Poleski et al., 2021). Microlensing
observations also point to a large population of planets in the 5–60 $\rm
M_{\oplus}$ mass range that either orbit stars at Neptune-like or Uranus-like
distances (i.e., beyond $\sim 10$ au) or are free-floating, with about 3 times
as many 10 $\rm M_{\oplus}$ planets beyond 10 au as at 0.1–10 au (Poleski et
al., 2021; Gould et al., 2022). The above results are consistent with all
stars having 1–2 Neptune-mass planets at 5–30 au. Because the escape velocity
of a Neptune is 20–25 $\rm km~{}s^{-1}$ and the orbital velocity at 10 au is
13 $\rm km~{}s^{-1}$, a Neptune-mass planet is sufficient to scatter icy
planetesimals into the inner planetary system. Because all stars have one or
more Neptunes, this mechanism for delivering volatiles to the inner Solar
System could be generic.

Models of the growth and migration of giant planets find that as they rapidly
grow in mass (and migrate) during the rapid gas accretion phase, they
destabilize the orbits of nearby planetesimals, which are scattered into
eccentric orbits that deliver water to the inner few au of planetary systems
(Raymond & Izidoro, 2017; Ronnet et al., 2018). Dynamical interactions between
multiple (neighboring) giant planets would also deliver volatiles to inner
planetary systems. The broad distribution of orbital eccentricity among the
known exoplanets suggests that such instabilities are common in the histories
of other planetary systems (Chatterjee et al., 2008; Ford & Rasio, 2008; Jurić
& Tremaine, 2008). The much higher eccentricities of exoplanets compared to
the Solar System planets imply that they experienced more dramatic
instabilities (Raymond et al., 2010).

Thus, a tenuous gas disk can be acquired via “takeout” (i.e., the liberation
of planetary atmospheres in giant impacts) or “delivery” (i.e., by asteroids
and comets launched into the terrestrial planet region) at a sufficient level
to remove the dusty debris signature of terrestrial planet formation. While
larger solids will migrate under gas drag into the star, smaller solids will
migrate to the outer edge of the gas disk ($>3$ au; Figure 2), thereby
removing the warm excess signature of terrestrial planet formation. The strong
stellar winds that are expected to accompany young stars offer an additional
dust removal pathway via stellar wind drag.

##  5 Stellar Wind Drag

Throughout the giant impact phase (10–100 Myr), the local environment within
the terrestrial zone is severe. Compared to the Sun, young solar-type stars
have large magnetic fields ($B_{\star}\lesssim$ 3–5 kG; Johns-Krull, 2007;
Yang et al., 2008; Johnstone et al., 2014; Vidotto, 2021) and short rotational
periods ($P_{rot}\approx$ 2–8 days; Hartmann et al., 1986; Herbst et al.,
2001, 2007; Johnstone et al., 2015; Vidotto, 2021), which power substantial
mass loss rates ($\dot{M}\sim 1-5\times 10^{-11}$ $M_{\odot}~{}{\rm yr^{-1}}$;
Cohen & Drake, 2014; Ó Fionnagáin & Vidotto, 2018; Vidotto, 2021) and
considerable X-ray and EUV luminosities ($L_{X}\sim 10^{-4}{L_{\odot}}$;
$L_{EUV}\sim 10^{-3}{L_{\odot}}$; Tu et al., 2015; Johnstone et al., 2021;
Vidotto, 2021). As described by Vidotto 2021, there are few direct
observations of stars in the 10–100 Myr age, although multiple lines of
evidence point to the possibility of large wind mass loss rates, with a range
of $\sim 10^{-10}-10^{-12}\,{M_{\odot}~{}{\rm yr^{-1}}}$.

To estimate the importance of the stellar wind on small solids during the
giant impact phase, we adopt typical values for young solar-type stars, a mass
loss rate $\dot{M}\sim 1.5\times 10^{-11}{M_{\odot}~{}{\rm yr^{-1}}}$
($10^{15}$ ${\rm g~{}s^{-1}}$) and a wind velocity $v_{w}\approx$ 500 $\rm
km~{}s^{-1}$ (see also Chen et al., 2005a, b, 2006). Compared to the current
Sun, the wind velocity is the same, but the mass loss rate is 1000 times
larger. In the absence of gas within a circumstellar disk, the stellar wind
induces radial drift of small particles (e.g., Burns et al., 1979). In a frame
of reference moving with an orbiting particle, radiation and the wind from the
central star oppose the particle’s motion, dragging the particle inward. For a
central star with $L_{\star}\approx$ 1 $L_{\odot}$, the time scale for
radiation (Poynting-Robertson; PR) drag on particles with bulk density
$\rho_{s}$ = 3 $\rm g~{}cm^{-3}$ is

$t_{\rm PR}\approx 20\left(\frac{a}{\rm 1~{}au}\right)^{2}\left(\frac{r}{\rm
1~{}cm}\right)~{}{\rm Myr}~{}.$ (4)

For particles smaller than 1–2 $\rm\,\mu{\rm m}$, radiation pressure dominates
PR drag and ejects particles from the planetary system on the local dynamical
time.

For 1 $\rm\,\mu{\rm m}$ and larger particles orbiting the Sun, wind drag is a
factor of $\sim$ 5 times smaller than PR drag, $t_{w}\sim 5t_{\rm PR}$ (Burns
et al., 1979). However, for a young star with a factor of 100–1000 times
larger mass loss rate, wind drag is a factor of 100-1000 times more effective
than for the Sun (e.g., see also Plavchan et al., 2005; Chen et al., 2005a, b,
2006; Spalding, 2018; Spalding & Adams, 2020). For a 1 cm particle orbiting at
1 au around a 10 Myr-old solar-type star with a mass loss rate of $10^{15}$
${\rm g~{}s^{-1}}$, the time scale for wind drag is then $t_{w}\sim 10^{5}$
yr, and the wind drags micron-sized particles into the central star on a
decade time scale.

In a somewhat different approach from Burns et al. (1979), Spalding (2018) and
Spalding & Adams (2020) consider radial drift through a magnetic stellar wind
from a rapidly-rotating young star (see also Plavchan et al., 2005, 2009;
Kla$v_{0}$cka, 2013). The radial drift time scale is similar:

$\tau_{w}\approx 1.25\times 10^{8}\left(\frac{r}{\rm
1~{}cm}\right)\left(\frac{a}{\rm
1~{}au}\right)^{2}\left(\frac{\dot{M}}{10^{12}~{}{{\rm
g~{}s^{-1}}}}\right)^{-1}~{}{\rm yr}.$ (5)

For a typical mass loss rate throughout the giant impact phase,
$\dot{M}\approx 10^{15}$ ${\rm g~{}s^{-1}}$, the drift time for a 1 cm
particle at 1 au is then similar to the $\sim 10^{5}$ yr estimate from the
relations in Burns et al. (1979).

For the wind to be an effective removal mechanism, particles must drift
radially inward more rapidly than they collide with other particles. Within a
swarm of solids undergoing a collisional cascade at 1 au during the giant
impact phase, gravitational interactions and physical collisions deflect
solids and spread them radially inward and outward (e.g., Goldreich &
Tremaine, 1978, 1982; Hornung et al., 1985; Shu & Stewart, 1985; Stewart &
Ida, 2000). Among particles with radius $r$, physical collisions with other
particles of radius $r^{\prime}\gtrsim r$ completely destroy the particle;
collisions with much smaller particles yield a modest amount of debris (e.g.,
Wyatt et al., 2007; Wyatt, 2008; Kenyon & Bromley, 2016, 2017). The time scale
for physical collisions to grind solids to dust that can be removed by
radiation pressure is typically smaller than the time scale for radial
expansion. Thus, effective wind removal requires $t_{w}\lesssim t_{c}$, where
$t_{c}$ is the collision time.

To estimate an appropriate $t_{c}$, we consider the rate of physical
collisions between a particle with radius $r$ and all other
particles222Although smaller particles can catastrophically fragment particles
with radius $r$, including these collisions decreases the removal time by an
amount, $\sim$ 10% to 50%, that depends on the bulk properties of the solids
and the collision velocities. To avoid additional free parameters, we avoid
this complication for this first exploration of relative removal times. with
radii $\gtrsim$ $r$. For any particle in the swarm, the collision rate with
all other solids is $dn/dt=n\sigma v$, where n is the number density, $\sigma$
is the physical cross-section, and $v$ is the relative velocity of the solids.
For collisions between a particle with radius $r$ and other particles with
radius $r^{\prime}\geq r$, we write $n\sigma=N(\geq r)\sigma/V$, where $N(\geq
r)$ is the total number of particles with radius $\geq$ $r$ and $V$ is the
volume. We substitute $A_{d}(\geq r)$ for $N\sigma$ and $V=4\pi a^{2}H$ for an
annulus of solids with width $a$ and vertical scale height $H$ at a distance
$a$ from the central star. As additional approximations, $H=\imath a$,
$v=ev_{K}=ea\Omega$, and $e=2\imath$, where $e$ ($\imath$) is orbital
eccentricity (inclination) and $\Omega$ ($P=2\pi/\Omega$) is the angular
frequency (orbital period). Finally, we adopt a standard power-law size
distribution, $n(r)\propto r^{-3.5}$ for small particles. The required surface
area of the swarm is then $A_{d}(\geq r)=A_{d}(1{\rm\,\mu{\rm m}}/r)^{1/2}$,
where we assume that the smallest particle in the swarm has $r\approx$ 1
$\rm\,\mu{\rm m}$, consistent with radiation pressure removing smaller
particles. Setting $A_{d}=4\pi a^{2}L_{d}/L_{\star}$ yields a simple
expression for the collision rate, $dn/dt\approx(4\pi/P)(1{\rm\,\mu{\rm m}}\
/r)^{1/2}L_{d}/L_{\star}$. Defining the collision time as
$t_{c}=(dn/dt)^{-1}$,

$t_{c}=\left(\frac{P}{4\pi}\right)~{}\left(\frac{r}{1~{}{\rm\,\mu{\rm
m}}}\right)^{1/2}~{}\left(\frac{L_{\star}}{L_{d}}\right)~{}\approx~{}10^{5}\left(\frac{P}{\rm
1~{}yr}\right)~{}\left(\frac{r}{\rm
1~{}cm}\right)^{1/2}\left(\frac{10^{-4}}{L_{d}/L_{\star}}\right)~{}{\rm
yr}~{}.$ (6)

Figure 3:  Comparison of radial drift time scales for PR drag (dotted black
line) and for stellar wind drag (black, purple, dark green, and light green
solid lines) with the collision time for $L_{d}/L_{\star}$ = $10^{-4}-10^{-3}$
(orange band). The legend indicates the stellar mass loss rate in units of
$10^{12}$ ${\rm g~{}s^{-1}}$.

Fig. 3 illustrates the variation of $t_{w}$ and $t_{c}$ with particle radius
at 1 au for $L_{d}/L_{\star}$ = $10^{-4}-10^{-3}$ (orange band) and a range of
stellar mass loss rates, 1–1000 times the current mass loss rate of the Sun
(black, purple, dark green, and light green lines). As a reference, the dotted
line is the PR drag time scale. Collision times for small particles in a
system with $L_{d}/L_{\star}\ \gtrsim 10^{-4}$ are much shorter than $t_{\rm
PR}$. Poynting-Robertson drag is ineffective at removing small particles from
a collisional cascade around a solar-type star with $L_{d}/L_{\star}$$\gtrsim
10^{-4}$ (e.g., Wyatt et al., 2011, see below). For the current Sun, the time
scale for solar wind drag is roughly five times longer than $t_{\rm PR}$
(e.g., Fig. 11 of Burns et al., 1979). Thus, a wind with the current Sun’s
mass loss rate cannot remove small particles from a collisional cascade. When
the central star has a higher mass loss rate, $\sim 10^{13}$ ${\rm
g~{}s^{-1}}$, the wind is more effective at removing small particles than PR
drag. However, the collision time remains shorter than the drag time for all
particle sizes. As the stellar mass loss rate increases to $10^{14}$ ${\rm
g~{}s^{-1}}$, $t_{w}$ becomes comparable to $t_{c}$ for small particles,
$r\lesssim$ 1–10 $\rm\,\mu{\rm m}$. For the largest stellar mass loss rates,
$\sim 10^{15}$ ${\rm g~{}s^{-1}}$, the drift time scale for 1–100
$\rm\,\mu{\rm m}$ particles is shorter than the collision time. At these high
mass loss rates, the smallest particles drift through the wind before
colliding with any other similarly-sized particle in the swarm.

If collisional cascades in the terrestrial zone generate wavy size
distributions (e.g., Campo Bagatin et al., 1994; O’Brien & Greenberg, 2003;
Wyatt et al., 2011; Kenyon & Bromley, 2016, 2020), then the orange band in
Fig. 3 would have a wavy shape, with minima in the removal time where the size
distribution has maxima. For rocky solids with the bulk properties of basalt
(e.g., Benz & Asphaug, 1999), the wave amplitude is a factor of $\sim$ 10 at
1-100 $\rm\,\mu{\rm m}$ and damps quickly at larger radii (e.g., Figs. 4–5 of
Kenyon & Bromley, 2016). In Fig. 3, the removal time for collisions would then
be a factor of $\sim$ 10 smaller at 1 $\rm\,\mu{\rm m}$, a factor of $\sim$ 10
larger at 10 $\rm\,\mu{\rm m}$, and roughly the same at $\gtrsim$ 100
$\rm\,\mu{\rm m}$. Relative to the example in Fig. 3 with a very powerful
stellar wind (1000 times the solar wind), collisions would compete with wind-
induced drift for 1–3 $\rm\,\mu{\rm m}$ particles. However, the wind would
dominate removal at 3–100 $\rm\,\mu{\rm m}$. Deriving the size distribution of
a collisional cascade in the presence of a powerful stellar wind requires a
more detailed calculation that is beyond the scope of the present effort.

To estimate the impact of wind drag on the IR excess, we consider a simple
model for the removal of 1-100 $\rm\,\mu{\rm m}$ particles. For simplicity, we
ignore the effect of a wavy size distribution. Without solar wind removal and
in an equilibrium collisional cascade, the production and removal rates of
small particles from collisions is $\dot{M}_{d}=Nm/t_{c}$ = constant (e.g.,
Wyatt et al., 2011; Kenyon & Bromley, 2016), where $N$ ($m$) is the number
(mass) of particles with radius $r$. We adopt a wind removal rate,
$\dot{M}_{wd}=Nm/t_{w}$. Combining the two relations,
$\dot{M}_{wd}\approx\dot{M}_{d}~{}t_{c}/t_{w}$. In a system with
$t_{w}<t_{c}$, the wind removes solids faster than the collisional cascade.

In Fig. 3 with $t_{c}\approx t_{w}$ at $r\approx$ 100 $\rm\,\mu{\rm m}$ for
$L_{d}/L_{\star}$ $\sim 10^{-3}$ and a wind with $\dot{M}=10^{15}$${\rm
g~{}s^{-1}}$, $\dot{M}_{wd}\approx\dot{M}_{d}(r/100~{}{\rm\,\mu{\rm
m}})^{-1/2}$; at 1 $\rm\,\mu{\rm m}$ (1 cm), the wind mass loss rate is 10
(0.1) times the steady-state mass loss rate from the collisional cascade. With
these mass loss rates from wind removal, the number density of particles at 1
$\rm\,\mu{\rm m}$, 10 $\rm\,\mu{\rm m}$, and 100 $\rm\,\mu{\rm m}$ is 0.10,
0.33, and 0.5 times the number density in the equilibrium cascade. The total
cross-sectional area of small particles is then a factor of $\sim$ 5 smaller.
Using the same approach, the reduction in IR excess for a system with
$L_{d}/L_{\star}$ $\sim 10^{-4}$ is a factor of $\sim$ 50\. In a solar-type
star with a large mass loss rate, radial drift from the stellar wind
significantly reduces the IR excess from a collisional cascade.

While this analysis demonstrates the effectiveness of radial drift in removing
an IR excess at 1 au, it is important for collisions to dominate at $a\gtrsim$
10–20 au where $\sim$ 25% of solar-type stars are known to have bright IR
excesses (e.g., Carpenter et al., 2009b; Bryden et al., 2009; Sibthorpe et
al., 2018, and references therein). If solids at $a\gtrsim$ 10–20 au have the
same physical properties as those at 1 au, the drift time scale, $t_{w}\propto
a^{2}$, grows more rapidly than the collision time scale, $t_{c}\propto
a^{3/2}$. Extrapolating Fig. 3 to 30–40 au, the time scales for drift and
collisions are roughly equal for particles sizes 3–5 $\rm\,\mu{\rm m}$ when
$L_{d}/L_{\star}$ $\sim 10^{-3}$ and $\dot{M}_{w}\sim 10^{15}$ ${\rm
g~{}s^{-1}}$.

At large $a$, the lower bulk densities of icy particles yield shorter time
scales for both removal mechanisms: $t_{w}\propto\rho_{s}$ and
$t_{c}\propto\rho_{s}$. For the solar wind, observations suggest a roughly
constant wind speed and a particle density $n\propto a^{-2}$ (e.g., Bagenal et
al., 2016; Khabarova et al., 2018). Compared to 1 au, the drift time then
varies only with $a$ and $\rho_{s}$: $t_{w}\propto\rho_{s}a^{2}$.

In a collisional cascade, two factors modify $t_{c}$ at large $a$: (i) the
sizes of the largest objects involved in the cascade $r_{max}$ and (ii) the
ratio of the collision energy to the binding energy, which is encoded in
$\alpha=M_{d}/(\dot{M}_{d}~{}t_{c})$ (Kenyon & Bromley, 2016). At 30–40 au,
collision velocities (energies) are a factor of $\sim$ 6 (36) smaller than at
1 au for similar orbital eccentricities. Binding energies of small solids are
a factor of 20–300 smaller (e.g., Leinhardt & Stewart, 2012; Schlichting et
al., 2013), which results in $\alpha_{\rm 35~{}au}$ $\sim$ 0.25–1.1
$\alpha_{\rm 1~{}au}$ (see Fig. 2 of Kenyon & Bromley, 2017). With smaller
collision energies, large solids that were collisionally disrupted at 1 au are
‘safe’; only somewhat smaller particles take part in the collisional cascade,
and $r_{max}$ $\sim$ 30–100 km instead of $\gtrsim$ 300 km. With
$t_{c}\propto\alpha r_{max}$, time scales for a collisional cascade with icy
particles at 30–40 au are then a factor of $\sim$ 3–50 times shorter than
those with rocky particles. Combined with the variation of $t_{c}$ with $a$,
the collision time scales for 1 $\rm\,\mu{\rm m}$ and larger icy particles at
30–40 au are always shorter than the time scales for radial drift within a
stellar wind, even at the largest mass loss rates. Thus, if radial drift is
important at 1 au, it should have little impact on small solids at much larger
$a$, consistent with the more common appearance of IR excesses at larger
orbital distances.

##  6 Discussion

As described in the previous sections, the warm dusty debris signature that is
expected to accompany the formation of terrestrial planets may not be robust.
Processes that are central to our current picture of the assembly and
evolution of terrestrial planets—early planetary atmospheres and giant impacts
(Section 3), water delivery by asteroids and comets (Section 4)—can
potentially regenerate a tenuous gaseous disk that removes the dusty debris
via gas drag. In addition, drag from the strong stellar winds expected from
young stars (10-100 Myr old) can also efficiently remove dusty debris (Section
5).

These processes may help account for the puzzling rarity of warm debris
signatures at the expected epoch of terrestrial planet formation, despite the
$\sim 20$% inferred occurrence rate of terrestrial planets orbiting mature
solar-type stars. Because the strengths of gas drag and stellar wind drag are
expected to vary from system to system—stellar wind drag depends on the wind
mass loss rate, which can vary widely, and gas drag depends on how much of a
gas disk is regenerated—the small fraction of systems with a detectable excess
may simply reflect the fraction in which these mechanisms do not operate
efficiently, i.e., they have weak stellar winds or did not regenerate a
significant gas disk.

###  6.1 Photoevaporative Disk Wind

While our simple estimates point to the potential impact of gas drag and
stellar wind drag on debris signatures, understanding their true impact
requires detailed modeling that takes into account additional processes. In
particular, a photoevaporative disk wind driven by stellar X-ray and UV (XUV)
irradiation (e.g., Ercolano & Pascucci, 2017; Nakatani et al., 2018a, b) may
deplete a regenerated gas disk, reducing its effectiveness. Developed to
understand the evolution of much younger T Tauri stars surrounded by high-mass
hydrogen-rich disks, the theories generally predict winds driven from
distances beyond a few au. Irradiation heats the disk surface to a temperature
$T_{g}$, and the atmosphere becomes unbound beyond the gravitational radius

$R_{g}={GM_{*}\over c_{s}^{2}}\approx 10\,{\rm au}\left(T_{g}\over
10^{4}\,{\rm K}\right)^{-1}\,\mu\,f,$ (7)

where $c_{s}$ is the sound speed, and the mean particle weight $\bar{m}=\mu
m_{H},$ where $m_{H}$ is the mass of hydrogen, and the correction factor $f$
is $\sim 0.2$ (Liffman, 2003). Thus, gas in the terrestrial planet region is
more tightly bound, and more difficult to remove photoevaporatively, than gas
at larger distances. In addition, both high gas temperatures ($\sim 10^{4}$ K)
and low mean particle weights ($\mu\sim 1$) are needed to remove gas from the
terrestrial planet region. Such high temperatures typically require EUV
irradiation, in contrast to the $\sim 1000$ K temperatures more typical of FUV
irradiated disks (e.g., Gorti et al., 2009). Low mean particle weights would
be characteristic of hydrogen-rich gas but not gas rich in water vapor.

The impact of disk winds on a regenerated gas disk is difficult to estimate,
because empirical wind mass loss rates are poorly known for stars of any age,
and the predictions of disk wind theories are generally unverified
observationally. In addition, because they depend on XUV fluxes,
photoevaporative disk wind mass loss rates are also likely to vary greatly
even among stars of similar age. Approximately solar-mass stars in the age
range 10-100 Myr are estimated to have XUV fluxes that range from T Tauri
luminosities to values $\sim 100$ times lower ($L_{X}=3\times 10^{28}-3\times
10^{30}$ erg s-1), a consequence of both age and differing initial stellar
rotation rates (Tu et al., 2015).

Nevertheless, we can explore a few possibilities. For example, in the study by
Picogna et al. (2019), which explored photoevaporative disk wind mass loss
rate as a function of X-ray luminosity, the total mass loss rate, integrated
over all disk radii, is $\sim 10^{-9}\,{M_{\odot}~{}{\rm yr^{-1}}}$ for
$L_{X}\sim 10^{29}$ erg s-1. Although most of the mass loss will occur at
$R_{g}$ (eq. 7) and beyond, if $\sim 1$% of the total mass loss originates
within 1 au, it would take $\sim 300$ years to remove 0.001 $\rm M_{\oplus}$
of gas from the inner au, a short interval compared to the drift removal time
for the dust (Figure 2). If the mass loss rate is 10 times lower, the drift
rate is comparable to the gas removal time, and the gas released in giant
impacts could plausibly persist long enough to remove the debris signature
from a given giant impact.

Importantly, these models assume a hydrogen-rich gas disk, whereas regenerated
gas disks may be greatly enhanced in elements heavier than hydrogen, with a
corresponding increase in $\mu$ and $R_{g}$. For example, if the gas disk
originates as water vapor that is subsequently dissociated, $\mu=6,$ and
$R_{g}\approx 12$ au if $f=0.2$ and $T_{g}=10,000$ K. A metal-rich composition
will also affect the cooling of the gas, with lower temperatures increasing
$R_{g}$. Larger values of $R_{g}$ will better shield the terrestrial planet
region from photoevaporation and extend the time available to remove the warm
excess. Models tuned to the conditions in regenerated disks surrounding 10–100
Myr old stars would be very welcome.

Thus, although there are many uncertainties, it seems plausible that
photoevaporative disk winds could compromise gas drag as a dust removal
mechanism at high $L_{X}.$ Luckily, if stellar XUV is efficient in reducing
the effect of gas drag, the stellar wind associated with the same XUV activity
can potentially step in to remove dusty debris via stellar wind drag. Given
the multiple processes that are potentially involved, the range in $L_{X}$ as
a function of age and initial conditions, and the timing of terrestrial planet
formation, more detailed calculations are needed to understand which processes
dominate under what conditions, and the circumstances under which dust
removal, by gas drag and stellar wind drag, is effective.

###  6.2 Quick and/or Neat?

The above discussion illustrates how gas drag and stellar wind drag can
potentially act to remove the expected dusty signature from giant impacts and
account for the surprising lack of warm excesses that are expected to
characterize late-stage terrestrial planet formation. If processes like these
are effective, terrestrial planets may form in the traditional way, via giant
impacts, but leave little observational trace of these events (a tidy or
“neat” planet formation scenario).

An alternative interpretation of the rarity of warm excesses is that
terrestrial planets emerge fully formed from the nebular phase and only rarely
undergo late-stage giant impacts (a “quick” formation scenario). Possible
motivation for the latter scenario comes from the bimodal size distribution of
close in planets (the radius valley at $<0.3$ au), which is interpreted as
evidence that planets $>5\,{\rm M_{\oplus}}$ are commonly born with gaseous
envelopes that are eroded by stellar irradiation and core-powered mass loss,
to a greater or lesser extent, depending on their initial envelope mass as
well as planet mass and orbital distance (e.g., Owen & Wu, 2013; Lopez &
Fortney, 2014; Jin et al., 2014; Owen & Wu, 2016; Ginzburg et al., 2016; Owen
& Wu, 2017; Ginzburg et al., 2018; Jin & Mordasini, 2018; Gupta & Schlichting,
2019; Mordasini, 2020; Venturini et al., 2020; Rogers et al., 2021). In this
picture, multi-$\rm M_{\oplus}$ planets must clearly form quickly (perhaps via
pebble accretion), before the dissipation of the protoplanetary disk, i.e.,
within the first few Myr. The ability of these planets to reach masses above
$1\,{\rm M_{\oplus}}$ quickly at distances $<1$ au suggests that $\sim{\rm
M_{\oplus}}$ planets forming at $\sim 1$ au may also acquire all of their mass
in the nebular phase, emerging fully formed, with no need for late-stage giant
impacts (e.g., Mordasini, 2020; Venturini et al., 2020).

In the Solar System, evidence for the moon-forming impact (e.g., Asphaug et
al., 2021; Canup et al., 2021, and references therein) and analyses of
elemental abundances in Venus, Earth, and Mars (e.g., Lammer et al., 2021,
2022, and references therein) provide strong incentive for the role of giant
impacts in the evolution of at least one planetary system. However, this
evolutionary history may be rare if most planetary systems form their
terrestrial planets early, in the nebular phase.

###  6.3 Future Observations

Future work on exoplanet demographics and work that explores the plausibility
of dust removal mechanisms can help us understand whether planetary systems
form their terrestrial planets primarily “quickly” or “neatly.”

Gas reservoirs. If late-stage giant impacts are a common feature of
terrestrial planet formation but the debris they produce is swept away by gas
drag, we expect to see tenuous gas reservoirs around 10–100 Myr old stars.
Residual gas disks are generally assumed to be absent based on the lack of
detected stellar accretion signatures and in situ gas detections. But it is
difficult to rule out gas at a very low level. Stellar accretion rate
measurements are not commonly available below $\sim
10^{-11}-10^{-10}\,{M_{\odot}~{}{\rm yr^{-1}}},$ i.e., below 0.1%–1% of the
accretion rates typical of classical T Tauri stars (e.g., Riviere-Marichalar
et al., 2015). In terms of gas detections, a recent study using the UV
transitions of H2 to search for residual gas around $\sim 10$ Myr old stars
detected $10^{-10}-10^{-8}\,{\rm M_{\oplus}}$ of hot H2 ($\sim 1500$ K) within
1 au of TWA7 (Flagg et al., 2021). Just the tip of the iceberg, the hot H2 is
expected to be accompanied by a much larger mass of cool and/or atomic gas,
perhaps a total column density of $\sim 0.1$ g cm-2, similar to the column
density needed for efficient dust removal via gas drag. Further studies along
these lines would be of great interest. Given the low column densities of the
replenished gas disks, they are likely to be best probed with atomic
diagnostics. If they arise from secondary atmospheres or are delivered by
comets and asteroids, they will be enriched in volatiles (e.g., water) rather
than dominated by hydrogen, suggesting searches for atomic oxygen line
emission.

Stellar Winds of Young Stars. The major challenge in understanding the
efficiency of stellar wind drag is the current uncertainty in the mass loss
rates of young stars (e.g., Cranmer & Saar, 2011; Cohen & Drake, 2014;
Fichtinger et al., 2017; Ó Fionnagáin & Vidotto, 2018; Güdel, 2020; Vidotto,
2021, and references therein). With few direct measures of mass loss rates for
10–100 Myr old stars, published estimates are based on observations of older
stars (0.1–1 Gyr) and models that match the evolution of measured rotation
rates and proxies of coronal activity. Future direct measurements of mass loss
rates for stars with ages of 10–100 Myr would be extremely valuable.
Fortunately, indirect evidence for stellar wind drag may be found in the
combined X-ray/IR demographics of young stars (e.g., Plavchan et al., 2005;
Chen et al., 2005a, b, 2009; Plavchan et al., 2009; Sierchio et al., 2010).
Although sample sizes remain modest, some analyses suggest that young stars
with large (small) X-ray luminosities are less (more) likely to have IR
emission from a warm debris disk at distances of a $\lesssim$ 2–3 au from the
central star, a relation that is expected if stellar winds associated with
stellar activity act to remove dust signatures. Augmenting these studies with
more recent X-ray data from Chandra and the results of IR surveys with WISE
(e.g., Kuchner et al., 2016) and Herschel (e.g., Eiroa et al., 2013; Sibthorpe
et al., 2018) might enable a cleaner test of a correlation between X-ray
emission (a possible proxy for the stellar mass loss rate) and IR excess
emission.

Terrestrial Planet Atmospheres and Moons. If terrestrial planets with masses
$\sim$ 1–2 $\rm M_{\oplus}$ at $\sim 1$ au form quickly and emerge from the
nebular phase fully assembled, they will have primordial gaseous envelopes
with masses $\lesssim$ 1% to 10% of the mass of the planet (e.g., Mordasini et
al., 2012; Mordasini, 2020). Although various processes can remove primordial
atmospheres with masses $\lesssim$ 0.01 $\rm M_{\oplus}$ (e.g., Wyatt et al.,
2020; Kegerreis et al., 2020b, a; Lammer et al., 2021, and references
therein), the most massive envelopes may survive giant impacts and
evaporation. If this dichotomy is borne out by detailed calculations, we
expect ensembles of 1–2 $\rm M_{\oplus}$ planets at 1 au distances to consist
of a mixture of planets with negligible atmospheres and those with puffy
atmospheres, similar to the situation found for sub-Neptunes and super-Earths
at smaller orbital distances. Future searches for exoplanetary moons may
uncover valuable clues as to whether late-stage giant impacts play a
significant role in terrestrial planet formation. If the presence of large
moons (like that of Earth) is a signpost of late-stage giant impacts, the
incidence rate of moons associated with terrestrial exoplanets can reveal the
rarity or commonality of this formation pathway.

##  7 Summary and Conclusions

Giant impacts are a central theme in the story of terrestrial planet origins.
They play an important role in our ideas about how terrestrial planets gather
their mass and in the expectation that a dusty debris signature announces the
final stages of planetary assembly. As described here, giant impacts also
figure prominently in our ideas about how planets acquire (or lose) their
atmospheres and oceans, but it is perhaps less appreciated how the loss of
planetary atmospheres can also regenerate enough of a gas disk to compromise
our ability to detect the expected dusty debris signature of terrestrial
planet formation.

Similarly, impacts by comets and asteroids are often invoked to account for
the origin of Earth’s oceans. Because the Earth is a small impact target,
scenarios that deliver appreciable water to Earth would also launch many more
non-Earth-impacting bodies into the inner Solar System and regenerate a
tenuous gaseous disk. In either case, a regenerated gas disk, whether achieved
by “takeout” (by giant impacts liberating planetary atmospheres) or “delivery”
(by evaporating comets and asteroids), is plausibly hefty enough to whisk away
dusty debris via gas drag and erase the anticipated debris signature of
terrestrial planet formation.

The regenerated gas disk may be subject to photoevaporation by energetic XUV
radiation from the central star, potentially reducing its efficacy in removing
the dusty debris. However, the strong stellar wind that will accompany the XUV
radiation offers an alternative pathway to remove the dusty debris signature
of terrestrial planet formation: stellar wind drag. The major uncertainty
associated with stellar wind drag is the relatively poorly measured stellar
wind mass loss rates at ages of 10-100 Myr.

Thus, both gas drag and stellar wind drag provide pathways by which the
anticipated debris signature of terrestrial planet formation can be erased.
These processes may resolve the conundrum of why terrestrial planets like
Earth and Venus are found to occur relatively commonly around Sun-like stars,
but the warm dust debris signatures that are expected to accompany their
formation are rarely detected.

This conundrum might also be resolved in a different way. The size
distribution of close-in planets may suggest that terrestrial planet formation
departs significantly from the “classical” picture, with planets forming
completely in the nebular phase and late-stage giant impacts playing little or
no role. If this is the dominant pathway for terrestrial planet formation, the
primordial disk may obscure the assembly process, making it challenging to
detect and study. Thus, both the availability of multiple pathways to erase
the anticipated debris signature of terrestrial planet formation, as well as
the alternative possibility of very early terrestrial planet formation,
suggest that the observable signposts of terrestrial planets may be more
subtle and more challenging to detect than previously believed.

Exploring these ideas requires more detailed models that track the transport
of gas between planets, comets, asteroids, and a regenerated gas disk,
including important additional processes such as photoevaporation and other
gas removal mechanisms. Valuable insights can come from new observations,
e.g., measurements of stellar wind mass loss rates and confirmation of the
presence and longevity of tenuous regenerated gaseous reservoirs. Studies
along these lines may find evidence for dust removal mechanisms in action. In
parallel, future studies of the mass-size distribution of terrestrial planets
and the presence or absence of large moons can also reveal whether terrestrial
planets primarily form early, in the presence of a gas disk, and whether giant
impacts play a significant role in their formation history.

We thank Hilke Schlichting and the anonymous referee for valuable comments on
the manuscript and the Harvard EPS/CfA Planetary Journal Club community for
interesting discussions that stimulated some of the ideas considered here.
JN’s research was supported in part by a Fellowship from the John Simon
Guggenheim Memorial Foundation, the Harvard Radcliffe Fellowship Program of
the Radcliffe Institute for Advanced Study at Harvard University, and the
Institute for Theory and Computation at the Harvard-Smithsonian Center for
Astrophysics. JN’s research activities are also supported by the NSF’s
NOIRLab, which is managed by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. SK’s research is supported by the Smithsonian Institution.

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