Rocky Planet Formation: Quick and Neat

Rocky Planet Formation: Quick and Neat

Scott J. Kenyon Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138 USA Joan R. Najita National Optical Astronomy Observatory, 950 Cherry Avenue, Tucson, AZ. 85719, USA Benjamin C. Bromley Department of Physics & Astronomy, University of Utah, 201 JFB, Salt Lake City, UT 84112 USA
Abstract

We reconsider the commonly held assumption that warm debris disks are tracers of terrestrial planet formation. The high occurrence rate inferred for Earth-mass planets around mature solar-type stars based on exoplanet surveys (%) stands in stark contrast to the low incidence rate ( 2%–3%) of warm dusty debris around solar-type stars during the expected epoch of terrestrial planet assembly ( Myr). If Earth-mass planets at AU distances are a common outcome of the planet formation process, this discrepancy suggests that rocky planet formation occurs more quickly and/or is much neater than traditionally believed, leaving behind little in the way of a dust signature. Alternatively, the incidence rate of terrestrial planets has been overestimated or some previously unrecognized physical mechanism removes warm dust efficiently from the terrestrial planet region. A promising removal mechanism is gas drag in a residual gaseous disk with a surface density of the minimum mass solar nebula.

planetary systems – planets and satellites: formation – protoplanetary disks – stars: formation – circumstellar matter



1 Introduction

The conventional picture of terrestrial planet formation begins with the growth of 1–10 mm pebbles from 1  dust grains (Chiang & Youdin, 2010; Birnstiel et al., 2010; Youdin, 2010; Windmark et al., 2012; Garaud et al., 2013; Birnstiel et al., 2016). Collisional and collective processes then convert pebbles into km-sized or larger planetesimals (Youdin & Goodman, 2005; Johansen et al., 2007; Youdin, 2011a; Johansen et al., 2015; Simon et al., 2016).

Various observations support the idea that planetesimals in protoplanetary disks grow rapidly (within 1 Myr), including (i) radiometric analyses of meteorites (Bizzarro et al., 2005; Kleine et al., 2009; Schulz et al., 2009; Dauphas & Chaussidon, 2011; Dauphas & Pourmand, 2011; Sugiura & Fujiya, 2014), (ii) comparisons of the mass distributions of solids in protostellar disks and known exoplanet populations (Najita & Kenyon, 2014), (iii) the trend in the abundance of HCN relative to HO as a function of disk mass (Najita et al., 2013), and (iv) coherent structure in the young HL Tau disk, as observed by ALMA (ALMA Partnership et al., 2015; Zhang et al., 2015).

Once planetesimals form, they quickly ( yr) merge into much larger protoplanets (Weidenschilling, 1974; Wetherill, 1980). Over 1–100 Myr, a series of giant impacts among the protoplanets then leads to several Earth-mass planets (Chambers & Wetherill, 1998; Chambers, 2001; Raymond et al., 2004; Kenyon & Bromley, 2006; Lunine et al., 2011; Raymond et al., 2014). Throughout this period, theory predicts a disk-shaped cloud of collisional debris that produces an observable infrared (IR) excess. These debris disks are expected to serve as signposts of ongoing rocky planet formation (e.g., Kenyon & Bromley, 2002; Zuckerman & Song, 2004; Kenyon & Bromley, 2004; Raymond et al., 2011, 2012; Leinhardt et al., 2015).

If warm debris is a dependable beacon, it provides a simple way to locate sites of ongoing terrestrial planet formation and to measure the frequency with which rocky planetary systems form. Compared to the challenges of identifying Earth-mass planets from direct imaging, microlensing, radial velocity, and transit observations (e.g., Gould et al., 2006; Cumming et al., 2008; Macintosh et al., 2014; Burke et al., 2015), measuring the magnitude of an IR excess is fairly straightforward and independent of viewing geometry (e.g., Carpenter et al., 2009b, a; Kennedy & Wyatt, 2013; Patel et al., 2014). IR observations are often sufficient to establish the temperature and location of the debris (e.g., Lisse et al., 2008; Currie et al., 2011). Theoretical models then allow us to use this information to construct a window into the planet formation process (e.g., Genda et al., 2015b; Kenyon & Bromley, 2016).

Here, we examine the reliability of warm debris disks as tracers of ongoing terrestrial planet formation. Current observations suggest the frequency of warm debris disks around young solar-type stars (§2) is much smaller than the frequency of Earth-mass planets around older solar-type stars (§3). Analytical calculations of dust emission during the final phases of planet assembly indicate that all popular scenarios of rocky planet formation predict detectable amounts of debris for stellar ages of 5–20 Myr (§4). Thus, there is a clear discrepancy between theoretical predictions and the observed frequencies of warm debris disks and Earth-mass planets.

In §5, we consider options for resolving this discrepancy. After constraining uncertainties in the frequency of Earth-mass planets (§5.1), dust emission (§5.2), and theoretical predictions (§5.3), we demonstrate that if stars retain a residual gaseous disk with a surface density of the minimum mass solar nebula (§5.4), gas drag and radiation pressure will rapidly remove small grains from the terrestrial planet region. This mechanism is plausible given current observational limits on gaseous disks among 5–20 Myr old stars (§5.5), We conclude with a set of suggestions to test the possibilities for reconciling theory and observations (§5.6) and a brief summary (§6).

2 Warm Excesses From Debris Disks Are Rare

Although opaque protoplanetary disks surround essentially all newly-formed stars (e.g., Kenyon et al., 2008; Williams & Cieza, 2011; Andrews, 2015, and references therein), the infrared excess emission produced by the disk vanishes roughly simultaneously at all wavelengths on a time scale of  Myr (Haisch et al., 2001; Kennedy & Kenyon, 2009; Mamajek, 2009; Williams & Cieza, 2011; Alexander et al., 2014). The ultraviolet excess produced by accretion onto the central star declines on a similar time scale (Hartmann et al., 1998; Kennedy & Kenyon, 2009; Sicilia-Aguilar et al., 2010; Ingleby et al., 2014).

Among older, solar-type main sequence stars, roughly 20% have optically thin IR or mm emission from debris disks (e.g., Hillenbrand et al., 2008; Trilling et al., 2008; Eiroa et al., 2013). Typically, this emission is comparable to or less than the stellar flux at 8–25  and a factor of 10 larger than the stellar flux at longer wavelengths. To relate excesses to a fractional luminosity, we assume the emission arises from a single-temperature, optically thin dust component with temperature located at a distance from a star with luminosity , mass , radius , and temperature :

(1)

The dust to star luminosity ratio is

(2)

where is the total cross-sectional area of all solid particles. Assuming the grains emit as blackbodies, we can relate to the flux ratio at any wavelength:

(3)

For 280 K,   5800 K, and at 12  (24 ), (). Debris disks are a factor 100–1000 less luminous than the protoplanetary disks surrounding T Tauri stars (see also Wyatt, 2008; Carpenter et al., 2009b).

Fig. 1 illustrates the variation of at 8–24  for = (violet curves) and (orange curves). Adopting detection limits from Spitzer (8, 16, 24; Carpenter et al., 2009b) and WISE (12; Luhman & Mamajek, 2012), 0.03–0.25, it is clear that detecting an excess at 8–12  from dust at  AU requires . At longer wavelengths, it is possible to detect warm dust with a substantially lower luminosity, to . Because “cold” dust ( 200 K) from beyond the terrestrial planet region (1–2 AU) can also contribute to the observed 24  emission, observations at shorter wavelengths (e.g., 8–16 ) or at longer wavelengths (e.g., 70 ) are required to constrain and .

To put an observed in perspective, we derive the required mass in solid particles. In standard models, the solids have a power-law size distribution, where is the radius of a particle and 3.5. Thus, , where is the mass density,  is the radius of the smallest particle, and  is the mass of the largest particle (Wyatt, 2008). For material with  = 1 ,  = 300 km, and at = 1 AU,  g. Thus, an observable excess at 24  requires roughly a third of a lunar mass of solid material at 1 AU.

Over the past decade, various surveys suggest a very low frequency of warm debris disks among solar-type stars (e.g., Stauffer et al., 2005; Silverstone et al., 2006; Currie et al., 2007; Carpenter et al., 2009b, a; Chen et al., 2011; Luhman & Mamajek, 2012; Kennedy & Wyatt, 2013; Cloutier et al., 2014; Matthews et al., 2014, and references therein). To illustrate current constraints on the incidence rate of warm dust, we describe in detail several studies using data from Spitzer and WISE.

The Spitzer Formation and Evolution of Planetary Systems (FEPS) program surveyed 314 solar-type stars at 3–70  (Meyer et al., 2006; Carpenter et al., 2009b). The sample included stars in clusters and the field, with ages ranging from 3 Myr to 3 Gyr. Within this group, only 5 have a measurable 16 excess. All five stars also have substantial excess emission at 8 , 24 , and 60–100  and prominent accretion signatures from hot gas close to the central star. Thus, all are primordial disks, the gas-rich disks commonly observed in T Tauri stars (Silverstone et al., 2006; Dahm & Carpenter, 2009; Carpenter et al., 2009b).

All of the non-primordial disk excess sources have the modest dust luminosities, , characteristic of debris disks. However, few if any sources have obvious emission from warm dust within 1 AU of the star (Carpenter et al., 2009b). Excluding the primordial disks, no source has an excess at () above 3% (16%) of the stellar photosphere. From , the ratio of the 24  to the 8  flux, 16% of all FEPS sources have robust detections of 24  dust emission 10% above the stellar photosphere. Among the 30 stars younger than 10 Myr, 3%, 7%, and 20% of the non-primordial disk sources have a detected excess at 50%, 30%, and 10% above the stellar photosphere.

Detailed fits of model spectral energy distributions for non-blackbody grains to the Spitzer IRS spectra at 8–35 m for excess sources of all ages yield average inferred dust temperatures  K to 200 K with a median at 112 K. Typical inner disk radii range from –3 AU to 40 AU with a median at 6 AU. Adopting a blackbody model for dust emission yields similar median values for and with somewhat larger ranges ( 50–280 K and 1–31 AU). Only a few (between 1 and 3) 24  excess sources have 200 K and 1–2 AU, demonstrating that there is little obvious evidence for warm dust in the terrestrial zones of FEPS targets. The excess limits at wavelengths shortward of constrain the fraction of the excess that could be produced by warm dust (§4).

Although the FEPS study has a high sensitivity to IR excess as a fraction of the stellar photosphere, it covers a wide range in stellar age and has a relatively small () set of sources with ages ( 10 Myr) relevant to terrestrial planet assembly. To complement these results, we examine the Spitzer and WISE statistics for the well-studied Upper Scorpius association (Upper Sco). Upper Sco, which is part of the nearby Sco-Cen association (Sco OB2; Preibisch & Mamajek, 2008), allows an accurate census of dust emission at ages of 7–12 Myr (Luhman & Mamajek, 2012; Pecaut et al., 2012; Rizzuto et al., 2016), when terrestrial planets accumulate most of their final mass (e.g., Dauphas & Chaussidon, 2011; Raymond et al., 2014; Quintana et al., 2016).

Analyses of Spitzer data alone indicate a very small frequency of warm debris disks among solar-type stars in Upper Sco (Carpenter et al., 2006; Dahm & Carpenter, 2009; Carpenter et al., 2009a; Chen et al., 2011). In a sample of 27 K-type pre-main sequence stars with masses of 0.8–1.3 , most of the excess sources (7/9) have IR colors consistent with primordial disks (Carpenter et al., 2006, 2009a). Although two additional stars have 24  excesses consistent with debris disks, both have cool dust ( 200 K). Within a much larger sample of 101 M-type stars with masses smaller than 0.8 , 9 (17) are debris (primordial) disks. The lack of short wavelength excesses among stars with debris disks suggest none of these have substantial amounts of warm dust.

To enlarge the sample of solar-type stars, Luhman & Mamajek (2012) added WISE data to previous Spitzer surveys. Among K0–M0 stars with masses of 0.7–1.3 , 28% (17/60) have a 22–24 excess and 13% (9/68) have a 12 excess at levels % of the stellar photosphere. There is little evidence for warm debris. All of the 12  excess sources are classified as “full, transitional, or evolved disks”; none are debris disks. When all detection statistics are considered (Luhman & Mamajek, 2012), fewer than 3% of Upper Sco sources have a 12  excess from a debris disk. Among the 22–24  excess sources without 8  or 12  excesses, eight are classified as “debris or evolved transitional” disks. For the two stars with large 22–24  excesses, the lack of an accompanying 8  or 16  excess indicates that most of the dust is cold. The maximum frequency of warm debris disks in this sample is then 10% (6/60).

Although there are no constraints on dust temperature for the other six stars, if the FEPS results are a guide, most contain cold dust. Among FEPS sources with 24  excesses that overlap in age with Upper Sco (5–20 Myr), roughly 80% have cold dust (Carpenter et al., 2009b). Adopting this scaling for Upper Sco, only 1–2 of the six 22–24  excess sources contain warm dust. Thus, a more realistic estimate for the fraction of 10 Myr old stars with warm dust is 2%.

Many other studies also conclude that warm excesses from debris disks around solar-type stars are extremely rare (e.g., Moór et al., 2009; Stauffer et al., 2010; Beichman et al., 2011; Smith et al., 2011; Zuckerman et al., 2011; Ribas et al., 2012; Urban et al., 2012; Zuckerman et al., 2012; Jackson & Wyatt, 2012; Kennedy & Wyatt, 2012; Ballering et al., 2013; Vican & Schneider, 2014). Although IRAS, Spitzer, and Herschel data suggest 10% to 30% of solar-type main sequence stars have IR excesses (e.g., Lagrange et al., 2000; Eiroa et al., 2013; Ballering et al., 2013, and references therein), nearly all sources have color temperatures characteristic of cold dust ( 300 K) at  AU. In addition to Upper Sco, Spitzer data for the young (15 Myr) clusters and Per suggest a small excess fraction of 1% to 2% at 8  (Currie et al., 2007; Cloutier et al., 2014). Analyses of WISE data for stars in the solar neighborhood yield similarly small ( 1%–2%) fractions of sources with a 12  excess (Kennedy & Wyatt, 2013; Patel et al., 2014).

3 Earth Mass Planets at 0.25–1 Au Are Fairly Common

In contrast to the small fraction of stars with warm debris disks, rocky planets within 1 AU appear to be fairly common companions to solar-type stars (Youdin, 2011b; Fang & Margot, 2012; Batalha et al., 2013; Foreman-Mackey et al., 2014; Silburt et al., 2015; Winn & Fabrycky, 2015). Although the false positive rate is uncertain, recent attempts to confirm Kepler candidates with ground-based and other space-based observations suggest false positive rates ranging from 10% to 75% for various ranges of planet masses (e.g., Morton & Johnson, 2011; Santerne et al., 2012; Fressin et al., 2013; Sliski & Kipping, 2014; Désert et al., 2015; Colón et al., 2015; Santerne et al., 2016; Coughlin et al., 2016; Mullally et al., 2016; Morton et al., 2016). Here, we focus on a comprehensive analysis of Kepler data which provides a detailed estimate for the occurrence rate of Earth-mass planets inside 1 AU. This approach is conservative: the formation of lower- and higher-mass rocky planets also produces observable amounts of debris. Assuming false positive rates are small, these analyses thus yield robust lower limits to the fraction of solar-type stars that produce detectable debris at ages of 5–20 Myr. We return to the false positive rate in §5.

Using Q1-Q16 Kepler data, Burke et al. (2015) estimate 0.77 planets with radii of 0.7–2.5  and orbital periods of 50–300 days ( 0.25–0.9 AU) per GK dwarf star (see also Petigura et al., 2013; Mullally et al., 2015; Coughlin et al., 2016, and references therein). Recent detailed analyses of transiting planets with radial velocity measurements suggest most planets with radii smaller than 1.5–2  have rocky compositions (e.g., Weiss & Marcy, 2014; Marcy et al., 2014; Buchhave et al., 2014). Applying these results to their complete Kepler samples, Burke et al. (2015) derive a probability of 0.1 for an Earth-mass planet (0.8–1.2 ) at 0.86–1.13 AU and a probability of 0.075 for an Earth-mass planet (0.8–1.2 ) at 0.61–0.8 AU. Taken at face value, these rates predict a somewhat larger frequency of planets (per unit area) at 0.7 AU than at 1 AU. Integrating over 0.6–1 AU, the probability is roughly 0.19 (assuming an intermediate rate for the missing region at 0.8–0.86 AU). Although Burke et al. (2015) do not quote a rate for Earth-mass planets within 0.25–0.6 AU, plausible extrapolations of their rates yield a probability of 0.22–0.25 (0.21–0.23) for an Earth mass planet (0.8–1.2 ) within 0.25–1 AU (0.4–1 AU) of a solar-type star.

Unless the false positive rate for the Kepler sample of Earth-mass planet candidates is much larger than 50%, the roughly 20% incidence rate for rocky planets with radii of 0.8–1.2  at 0.25–1 AU is much larger than the 2–3% rate of warm debris disks among solar-type stars with ages of 5–30 Myr. The discrepancy between the apparent formation rate of terrestrial planets and the detection rate for terrestrial debris disks is probably much larger than suggested by these estimates. We anticipate considerable debris from (i) the formation of rocky planets smaller than 0.8  and larger than 1.2  at 0.25–1 AU and (ii) the formation of any rocky planet at 0.25 AU and at 1–2 AU. If these formation channels yield a substantial population of rocky planets, the detection rate for debris disks is at least a factor of ten smaller than expected from the incidence rates of rocky planets.

4 Debris Generation From Planet Formation

To assess the significance of the different detection rates for warm debris disks and Earth-mass planets inside 2 AU, we must estimate the amount and lifetime of debris produced by terrestrial planet formation. To make this evaluation, we rely on theoretical estimates from popular scenarios that produce terrestrial planets on time scales consistent with the solar system and observations of protoplanetary disks. These scenarios are variants on models where continued agglomeration of small rocky solids yields a stable planetary system (e.g., Safronov, 1969; Wetherill, 1980). Solids not incorporated into planets provide material for excess emission from dust.

In the classical picture developed to explain the Solar System (e.g., Safronov, 1969; Lewis, 1972; Weidenschilling, 1974; Wetherill, 1980), the process starts with a disk of small solids having just enough mass (the Minimum Mass Solar Nebula, hereafter MMSN) to reproduce objects in the Solar System. Collisional processes merge small solids into km-sized or larger planetesimals, then Mars-mass protoplanets, and finally Earth mass planets. During the ‘giant impact’ phase when Mars-mass protoplanets merge into Earths, the surface density of the gaseous disk is probably 1% of the initial surface density; otherwise, gas drag circularizes the orbits of Mars-mass objects and prevents giant impacts (e.g., Kominami & Ida, 2002). Throughout the accumulation and “clean up” phases, high velocity collisions of leftover planetesimals, impacts of intermediate-sized protoplanets, and giant impacts of massive protoplanets convert 15% to 30% of the initial mass in solids into debris (Agnor et al., 1999; Kenyon & Bromley, 2004; Agnor & Asphaug, 2004; Goldreich et al., 2004; Raymond et al., 2011; Genda et al., 2015b).

In the pebble accretion scenario, dynamical processes within the gaseous disk concentrate cm-sized pebbles into large planetesimals with radii of 100–1000 km (e.g., Youdin & Goodman, 2005; Johansen et al., 2007; Youdin, 2010; Johansen et al., 2015; Simon et al., 2016). Continued accretion of pebbles and mergers of planetesimals eventually produce a set of stable planets (e.g., Johansen et al., 2015; Levison et al., 2015; Chambers, 2016). Current investigations of pebble accretion ignore the loss of material and debris production during mergers of large planetesimals. If fragmentation removes 5% to 10% of the initial mass (e.g., Johansen et al., 2015), neglecting this process has a limited impact on the formation of large planets. However, the mass lost through fragmentation is much larger than the sub-lunar mass of solids required to produce a detectable 24  excess (§2).

In the in situ planet formation scenario, Hansen & Murray (2012) suggest that high concentrations of solids inside 1 AU are required to explain various properties of the Kepler planet population111Volk & Gladman (2015) address aspects of this picture in the context of the solar system. (see also Kuchner, 2004; Chiang & Laughlin, 2013; Hansen & Murray, 2013; Hansen, 2015). Starting from an ensemble of protoplanets with a solid surface density 10 times larger than the MMSN, a series of giant impacts produces a stable system of several planets in a few Myr (Hansen & Murray, 2013). Because the calculations start with assumed ensembles of fully formed protoplanets, Hansen & Murray (2013) ignore the inefficiency of assembling solids into protoplanets and the debris produced by high velocity collisions of leftover planetesimals. As in recent explorations of pebble accretion, the numerical simulations also ignore the loss of material (and consequent debris production) in giant impacts.

For any of these scenarios, tidal torques can drive the radial migration of protoplanets through the gaseous disk (e.g., Ward, 1997; Ida et al., 2000; Masset & Papaloizou, 2003; O’Brien et al., 2006; Papaloizou et al., 2007; Ida & Lin, 2008; Bromley & Kenyon, 2011; Raymond & Cossou, 2014). Planets may then form at large and migrate to small . Although typical migration models do not consider collisional disruption of small solids inside 2 AU, large-scale destruction seems likely. When Earth-mass and super-Earth-mass planets migrate inside 2 AU, they excite pre-existing smaller solids onto high orbits (e.g., Armitage, 2003; Bromley & Kenyon, 2011; Walsh et al., 2011; Kenyon & Bromley, 2014). Destructive collisions among these objects then produce copious amounts of dust and a detectable 24  excess (e.g., Wyatt, 2008; Jackson & Wyatt, 2012).

4.1 Modes of Dust Production

To develop predictions for the magnitude of the IR excess produced during rocky planet formation, we identify likely epochs of dust production. In all of the scenarios discussed above, the growth of pebbles and larger planetesimals with 1 m to 1000 km takes place in an opaque gaseous environment where debris mixes with pre-existing small particles (see Dauphas & Chaussidon, 2011, and references therein). Because we cannot distinguish ‘primordial’ dust from debris, we ignore this phase of dust production.

As solids grow from planetesimals into protoplanets and then planets, there are three modes of dust formation. Mergers of planetesimals into protoplanets typically produce modest amounts of debris (e.g., Wetherill & Stewart, 1993; Kenyon & Luu, 1999; Weidenschilling et al., 1997). As protoplanets grow, they stir the orbits of leftover planetesimals which never become incorporated into a planet. Among the planetesimals, high velocity collisions then begin to produce numerous smaller particles with sizes ranging from 1  to tens of km (Greenberg et al., 1978; Wetherill & Stewart, 1993; Kenyon & Bromley, 2004; Weidenschilling, 2010; Raymond et al., 2011). This debris fuels a collisional cascade, where solids are ground down into sub-micron-sized particles which are ejected by radiation pressure. Eventually, giant impacts produce larger and larger protoplanets. Debris from giant impacts adds to the debris from collisions of leftover planetesimals and fuels the collisional cascade (Jackson & Wyatt, 2012; Genda et al., 2015b).

In all three scenarios, the growth of protoplanets, collisions of leftover planetesimals, and giant impacts generate debris. For simplicity, we ignore the modest amount of debris generated from the growth of protoplanets and derive separate estimates for dust produced from destructive collisions of leftover planetesimals (§4.2) and from giant impacts (§4.3).

4.2 Debris from Collisions of Leftover Planetesimals

In classical planet formation theory, semi-analytical and numerical calculations of collisions among ensembles of rocky planetesimals within 2–3 AU report debris production ranging from roughly 5% to almost 50% of the initial mass in solid material (Greenberg et al., 1978; Wetherill & Stewart, 1993; Kenyon & Bromley, 2004; Leinhardt & Richardson, 2005; Kenyon & Bromley, 2005; Chambers, 2008; Weidenschilling, 2010; Kenyon & Bromley, 2016). The typical mass in the debris is 10% to 20% of the initial mass. Significant numbers of destructive collisions begin early, at 0.01–0.1 Myr, and last until 10–100 Myr (see also Morbidelli et al., 2012; Raymond et al., 2014; Quintana et al., 2016, and references therein).

In these calculations, the timing of debris production overlaps epochs when we expect a significant decay in the surface density of the gaseous disk (Hartmann et al., 1998; Haisch et al., 2001; Mamajek, 2009; Williams & Cieza, 2011; Alexander et al., 2014). When is large, gas drag forces debris to spiral into the central star. As declines, the system retains a larger and larger fraction of the debris. For young stars with no gaseous disk at ages of 10–20 Myr, we conservatively estimate that 5% to 10% of the initial mass in solids is converted into debris. The formation of an Earth-mass planet thus generates 0.05-0.10  in debris, which is 10–20 times larger than the mass required to produce a detectable 24  excess.

4.3 Debris from Giant Impacts

To predict debris production from giant impacts, we first consider a simple analytical model. In this approach, we compare the collision energy to the binding energy of a pair of protoplanets (see also Davis et al., 1985; Housen & Holsapple, 1990; Davis & Ryan, 1990; Housen & Holsapple, 1999). We assume conservatively that all collisions are head-on (impact parameter = 0); more glancing collisions typically yield more debris (e.g., Leinhardt & Stewart, 2012). For two protoplanets with mass and radius , the center-of-mass collision energy is (e.g., Wetherill & Stewart, 1993; Kenyon et al., 2014; Kenyon & Bromley, 2016), where the impact velocity222In our approach, both protoplanets have the same velocity relative to a circular orbit. Thus, the impact velocity includes a contribution from each one. is

(4)

Here, is the velocity of each protoplanet relative to a circular orbit and is the mutual escape velocity of the pair at the moment of the collision, . The fraction of material ejected during a collision is

(5)

where  is the binding energy (e.g., Agnor et al., 1999; Benz & Asphaug, 1999; Canup & Asphaug, 2001; Leinhardt & Richardson, 2005; Leinhardt & Stewart, 2009; Genda et al., 2012, 2015a). When , half the mass of the merged pair is ejected to infinity.

Protoplanets typically have relative velocities of 25% to 75% of the escape velocity of the largest protoplanets. We set , where is the velocity of a circular orbit and is the eccentricity. When protoplanets have masses ranging from a lunar mass to an Earth-mass, 0.1–0.2 (e.g., Chambers, 2001; Raymond et al., 2004, 2005; Kenyon & Bromley, 2006; Kokubo et al., 2006; Kokubo & Genda, 2010; Chambers, 2013; Quintana et al., 2016).

For parameters appropriate for rocky objects at 1 AU (see §A.1), collisions between equal-mass protoplanets with 0.05-0.2 eject roughly 10% of the total mass (Fig. 2). Setting the relative velocity equal to 50% of the escape velocity of the protoplanet yields similar results. Although the accretion history of an Earth-mass object is complicated (e.g., Kenyon & Bromley, 2006; Kokubo et al., 2006; Chambers, 2013), final assembly requires several collisions of sub-Earth mass objects. If each collision loses roughly 10% of the initial mass, the total amount of lost mass exceeds 0.1 . This mass is roughly an order of magnitude larger than the mass required to produce a detectable debris disk (§2).

Detailed -body and SPH calculations support this simple estimate (e.g., Agnor & Asphaug, 2004; Asphaug et al., 2006; Raymond et al., 2011; Genda et al., 2012; Stewart & Leinhardt, 2012; Jackson & Wyatt, 2012; Chambers, 2013; Genda et al., 2015b). In SPH simulations, dust production depends on and . Head-on collisions with small and , yield little or no dust. When is large or the collision is oblique ( 0.3–0.4), dust production is substantial. Averaged over a complete -body simulation of an ensemble of growing protoplanets, protoplanet collisions disperse 10% to 20% of the initial mass into small fragments. Thus, giant impacts involved in the formation of a single Earth-mass planet yield at least 0.1–0.2  in debris.

4.4 Evolution of IR Excess from the Debris of Planet Formation

To derive predicted detection rates for warm debris disks, we adopt an initial disk mass and a model for the time evolution of the total mass  and cross-sectional area  of the debris produced through collisions of leftover planetesimals and giant impacts. In a real debris disk, stochastic collisions add and remove debris; thus,  and  decrease over the long term and increase and decrease on short timescales (e.g., Grogan et al., 2001; Kenyon & Bromley, 2004; Weidenschilling, 2010; Jackson & Wyatt, 2012; Genda et al., 2015b; Kenyon & Bromley, 2016). Instead of following this evolution in detail, we consider an analytical model where  and  decline monotonically with time (see §A.1). With this conservative assumption, we derive a lower limit for the expected IR excess from the debris at any time . If the analytical model predicts much larger IR excesses than observed, then more extensive numerical calculations of planet formation will also yield much larger excesses than observed.

As outlined in the Appendix (§A.1), the long-term evolution of the debris in the analytical model depends on the surface density of solids, the size of the largest object, and the orbital eccentricity of debris particles. The model assumes that collisions among objects with radii smaller than  are completely destructive, producing debris with particle sizes smaller than . The cascade of collisions maintains a power-law size distribution with from the smallest size  up to . Radiation pressure ejects smaller particles. Protoplanets with radii larger than  are ignored. With destructive collisions for all particles smaller than and ejection of material with , the mass in solids steadily declines with time.

To predict the time evolution of dust in the terrestrial zone, we consider a disk with initial solid surface density , inner radius   0.1 AU, and outer radius   1–2 AU. This inner radius lies between the value adopted in some numerical calculations (0.05 AU, e.g., Hansen & Murray, 2012, 2013) and the 0.25 AU inner boundary for the Burke et al. (2015) analysis of Kepler data. Our results are insensitive to the exact value of . Disks with scale factor  = 1 and = 10  at 1 AU have the surface density of the MMSN. For typical conditions in a disk with several protoplanets,   100–1000 km and 0.1 (Chambers, 2008; Raymond et al., 2011; Kenyon & Bromley, 2016). If  and remain fixed throughout the evolution, the surface density declines roughly linearly with time, , where is the collision time and is the orbital period. The relative luminosity of the debris disk is then an analytic function of the extent of the disk, the initial mass, and time (§A.1).

At the start of the evolution, depends only on  and the radial extent of the disk (Appendix, eqs. A14A15). The initial dust luminosity scales linearly with  and with radial distance as . When  = 0.1 AU and  = 2 AU, systems with have dust luminosity larger than the nominal detection limit, .

Once the evolution begins, the dust luminosity declines on the local collision time. For a disk with , . Although most of the mass is in the outer disk (), the material closest to the star has the largest brightness per unit surface area. Solids close to the star also have the shortest collision time. Thus, the dust luminosity begins to decline on the collision time of the inner disk, . The luminosity declines by a factor of roughly two on this time scale (see also Kennedy & Wyatt, 2010).

On time scales larger than , collisions remove material from larger and larger disk radii. At the outer edge of the disk, the collision time is . On time scales between  and , the dust luminosity declines rather slowly with time, with 0.3–0.9. Once the evolution time exceeds , the decline in the dust luminosity follows the decline of a narrow ring, with = 1.

At late times, the dust luminosity of disks with different  converges (e.g., Wyatt & Dent, 2002; Dominik & Decin, 2003). Although the initial disk luminosity scales with , disks with large  evolve more rapidly than disks with small . This convergent luminosity depends on , , and the extent of the disk.

The lower panel of Fig. 3 illustrates the evolution of at 0.1–50 Myr for 0.1–1 AU debris disks with  = 300 km, = 0.1, and a range of . Disks with  = 0.3 (30% of the MMSN) evolve from at  yr to at 0.1 Myr to at 3–4 Myr. These disks remain much brighter than the nominal detection limit until stellar ages of 30–40 Myr. At early times ( 0.1 Myr), the dust luminosity roughly scales with ; a disk with  = 0.003 has a dust luminosity 100 times smaller than a disk with  = 0.3. Because disks with smaller  have longer collision times, they evolve more slowly. It takes a disk with  = 0.003 roughly 15–20 Myr to decline from to .

The upper panel of Fig. 3 shows that larger disks are always brighter (eq. A15 of the Appendix). With a factor of 8 larger collision time at the outer edge, a disk extending to 2 AU declines more slowly than a disk extending to 1 AU. For disks with  = 0.3, the larger disk reaches the detection limit at 100 Myr instead of 30–40 Myr. As  decreases, however, the differences in evolution times become smaller. When  = 0.003, the 0.1–2 AU disk reaches the detection threshold at 25 Myr instead of 15–20 Myr for the 0.1–1 AU disk.

To derive predicted flux ratios for debris disks, we assume the solids radiate as blackbodies in equilibrium with radiation from the central star. For simplicity,  = 1  = 1 , and  = 5780 K at all times. The disk temperature is then = 280  K. With no analytic solution for the flux ratio, we divide the disk into a series of annuli, assign a temperature to each annulus, derive the time evolution of the cross-sectional area and emitted flux in each annulus, and add up the fluxes. In the Appendix, we show that numerical integrations of the dust luminosity agree well with analytic results.

Fig. 4 summarizes results for 0.1–2 AU disks with  = 1 ,  = 300 km and various . At current sensitivity limits, robust 8–12  detections of warm dust at 10 Myr require massive debris disks with   0.3; detections at 8  are strongly favored over those at 12 . Warm dust is much easier to detect at 16–24 . Spitzer 16  measurements of  Myr old stars can detect debris disks with   0.03. The sensitivity at 24  is fairly remarkable: debris disks with  as low as 1% of the MMSN are detectable.

We can use these results to interpret statistics for warm debris from §2. Among stars younger than 10 Myr in the FEPS sample, 3%, 7%, and 20% of the non-primordial disks have a detected excess above 50%, 30%, and 10% of the stellar photosphere. From Fig. 4, these results allow us to conclude that % of FEPS sources younger than 10 Myr have a warm excess consistent with . The 3% upper limit arises because cold dust beyond 2 AU could produce some of the observed excess. Similarly, fewer than 7% of sources younger than 10 Myr have a warm excess consistent with .

The FEPS excess statistics place a more stringent limit on . Because no FEPS source at any age (between 3 Myr and 3 Gyr) has a excess above 16% of the stellar photosphere, all sources younger than 10 Myr must have . Similarly, all sources younger than 5 Myr have .

For a given , reducing  reduces the predicted level of IR excess. Fig. 5 shows how the 24  excess varies with  for 0.1–2 AU debris disks with  = 0.1 and  = 1 . In systems with identical initial masses, swarms of particles with smaller  have larger initial  and smaller collision times. Although swarms with  = 3–10 km initially have larger 24  excesses than swarms with  = 1000 km, they also evolve much more rapidly. By 6 Myr (12 Myr), debris disks with  = 3 km (10 km) reach the nominal detection limit. Disks with  = 100–1000 km take 40–100 Myr to reach this limit.

4.5 Summary

Numerical calculations of planet formation suggest that the agglomeration of Earth-mass planets in the terrestrial zone creates copious amounts of debris. Current estimates suggest from 15% to 30% of the initial mass is potentially available to produce the debris signature of rocky planet formation. Debris production is probably independent of the mode – classical, in situ, or pebble – in which planets form. Unless planetesimal formation mechanisms assemble Mars-mass oligarchs directly, leaving behind 10% of the initial mass in debris is inevitable (e.g., Kenyon & Bromley, 2016). By generating smaller planetesimals, current formation paths appear to preclude this possibility (Johansen et al., 2015; Simon et al., 2016). All popular models for the formation of Earth-mass planets include a giant impact phase which also generates significant amounts of debris.

The debris is long-lived. In standard models of planet formation, the debris has a typical maximum size of 100–1000 km (e.g., Kenyon & Bromley, 2016). If the debris contains 15% to 30% of the initial mass required to assemble an Earth-mass planet, then the IR excess from the debris is detectable at 24  for 100 Myr (Fig. 4; see also Raymond et al., 2011; Genda et al., 2015b).

For stars with ages 10 Myr, it is challenging to detect debris from terrestrial planet formation with current sensitivity limits at 8–12 . For robust detections, recent surveys require an initial solid mass 30% of the MMSN at 8  and 100% of the MMSN at 12 . If the formation of several Earth-mass planets converts 15% to 30% of the initial mass into debris, detection at 12  is very unlikely: the debris simply is not luminous enough after 10 Myr. At 8 , the predicted level of debris from the formation of a single Earth-mass planet is somewhat smaller than the Spitzer limits. If Earth-mass planets are as common as suggested by Kepler (Burke et al., 2015), the low frequency, 3% (e.g., Luhman & Mamajek, 2012), of warm dust revealed by 8  observations of solar-type stars in the Upper Sco association is roughly consistent with theoretical predictions. However, if we include the excesses produced by the formation of more massive planets (), the expected 8  excess is larger than Spitzer sensitivity limits and potentially in conflict with observations.

If our nominal picture of debris production is correct (Fig. 4), current 16–24  sensitivity limits are low enough to detect every solar-type star  Myr engaged in forming Earth-mass planets. For 10 Myr old stars, current 16  data should detect debris disks with initial masses of 5% to 10% of the MMSN; however, the FEPS survey detected none of these systems. Similarly, % of FEPS sources younger than 10 Myr have a warm excess consistent with initial solid masses % of the MMSN.

Based on this analysis, there is a clear discrepancy between the observed frequency of warm debris from terrestrial planet formation ( 3%, §2) and the incidence rate of Earth-mass planets derived from Kepler ( 20%, §3). Based on Fig. 4, the observed frequency of 16–24  excess sources and the inferred incidence rate of Earth-mass planets imply that terrestrial planet formation must leave behind a very small fraction of the initial mass in debris-producing solids (% of MMSN).

5 Discussion

The low rate of warm excess among solar-type stars of terrestrial planet-forming age was previously noted in the early analysis of FEPS Spitzer/IRAC results. Because terrestrial planet formation should produce a detectable warm excess (Kenyon & Bromley, 2004), Silverstone et al. (2006) interpreted their non-detection of warm excesses as possible evidence that (i) terrestrial planets form infrequently or (ii) warm dust dissipates more quickly than expected (see also Carpenter et al., 2009b). At that time, the rarity of warm excesses was not particularly remarkable because the incidence rate of terrestrial planets was completely unknown. Our more recent understanding that Earth-mass planets may be fairly common now highlights the need to understand why warm excesses are rare.

In our analysis, the inferred discrepancy between the fraction of young solar-type stars with warm debris and the fraction of mature solar-type stars with Earth-mass planets () relies on our understanding of (i) recent Kepler planet detection statistics, (ii) updated statistics for 8–24  emission from disks around young stars, and (iii) new developments in planet formation theory. In this section, we consider each of these elements in turn to identify possible ways to resolve the discrepancy between and . We then consider physical mechanisms that might reconcile the two frequencies and suggest paths to test these ideas.

5.1 Frequency of Earth-Mass Planets at 0.25–2 AU Around Solar-Type Stars

When the frequency of false-positives among Kepler Earth-mass planet candidates is large, we overestimate . Current analyses infer 10% to 75% (e.g., Morton & Johnson, 2011; Santerne et al., 2012; Fressin et al., 2013; Sliski & Kipping, 2014; Désert et al., 2015; Colón et al., 2015; Santerne et al., 2016; Coughlin et al., 2016; Morton et al., 2016). Many studies focus on gas giant planet candidates; however, Désert et al. (2015) consider whether Spitzer observations confirm transits for Earth-mass candidates with orbital periods 100 d. For individual systems, they derive 1% to 42%; the complete sample suggests a typical 10%. Curiously, additional observations confirm planets in several candidate systems with large individual . Although larger samples are required to understand the true f for Earth-mass planet candidates, these data suggest is not overestimated by a factor of 10 (see also Coughlin et al., 2016; Morton et al., 2016).

If multi-planet systems occur commonly and with a large range in orbital inclination , the true is smaller than our estimates. Although transit surveys of such systems may detect only one of two or more possible planets, detecting a transiting planet in the system is more likely because a planet can be detected from multiple directions. As a result, the fraction of stars with planets in the habitable zone can be reduced while maintaining the same average number of Kepler planets per star overall.

Although observations do not yet provide a robust estimate of the frequency of multi-planet systems (e.g., Tremaine & Dong, 2012; Fabrycky et al., 2014; Van Eylen & Albrecht, 2015; Winn & Fabrycky, 2015, and references therein), comparing results from radial velocity and transit surveys provides useful constraints (S. Tremaine, private communication). Transit surveys can detect multi-planet systems only when the range of mutual inclinations is a few degrees; however, radial velocity surveys are sensitive to systems with a much broader range of mutual inclinations. The frequency of multi-planet systems from recent Kepler catalogs (0.19–0.20; Fabrycky et al., 2014; Mullally et al., 2015; Coughlin et al., 2016) is 2/3 of the frequency of 0.3 derived from radial velocity measurements (Limbach & Turner, 2015). The similarity between the two rates implies that Kepler does not significantly underestimate the fraction of multi-planet systems, i.e., mutual inclinations are small and planetary systems are fairly flat.

While this argument applies to mutual inclinations of planets of all masses, additional dynamical considerations can restrict the mutual inclinations of systems of Earth-mass planets. For known multi-planet systems, the typical range in is small, 1–5 (Tremaine & Dong, 2012; Fabrycky et al., 2014). With 2–4 (Fabrycky et al., 2014), 0.035–0.20. If this range in is typical of systems with Earth-mass planets, theory provides a guide to estimate a possible error in at = 0.25–1 AU (see also Tremaine, 2015). From numerical simulations, systems of several Earth-mass planets have 0.01–0.05 and (e.g., Chambers & Wetherill, 1998; Bromley & Kenyon, 2006; Raymond et al., 2007; Morishima et al., 2008; Chambers, 2013). These systems are dynamically unstable when the distance between the apocenter of the inner orbit and the pericenter of the outer orbit exceed 10 mutual Hill radii , where and () is the semimajor axis of the inner (outer) planet (Chambers et al., 1996; Yoshinaga et al., 1999; Fang & Margot, 2013; Pu & Wu, 2015; Petrovich, 2015). When all planets have = 0.03 ( = 0.10), the maximum number of stable Earth-mass planets within 0.4–1 AU is 7–8 (4–5). The maximum reduction in is then a factor of 5–7.

The number of transiting planets in a multi-planet system depends on the distribution of mutual inclinations (e.g., Tremaine & Dong, 2012; Fabrycky et al., 2014). To make estimates for ensembles of closely packed Earth-mass planets, we perform a simple Monte Carlo calculation. For adopted dispersions in and , and , our algorithm establishes a set of Earth-mass planets with orbital separations of 10 mutual Hill radii and semimajor axes between 0.25 AU and 1 AU. After randomly selecting one of these planets to have impact parameter 1, the code chooses random deviates for , infers impact parameters for the remaining planets, and counts the number of planets with 1. For Gaussian (Rayleigh) deviates with 0.035 and 1, closely packed systems have seven (six) planets and four (three) transits per system. When 0.175 and 5, tightly packed systems have 3–4 (2–3) planets and one (one) transit per system. Kepler should detect maximally packed Earth-mass planets on roughly circular orbits as multi-planet systems, but should fail to identify multiple planets in highly eccentric systems. Either way, these results suggest might be overestimated by factors of 2–4. Factor of ten overestimates are unlikely.

Overall, this analysis suggests that overestimates in are unlikely to reduce the fraction of stars with planets in the 0.25-1 AU region (%) to the fraction of stars with warm excesses (%). Moreover, our estimates are conservative. If there is a population of Earth-mass planets with 1–2 AU, then is larger than 20%. If the typical of multi-planet systems is roughly 4 instead of our adopted 2 (Fabrycky et al., 2014), then the maximum number of Earth-mass planets in Kepler systems with a single transit is roughly two. Thus, uncertainties in cannot reduce the discrepancy between and .

5.2 Emission from Small Dust Grains

Alternatively, we may have overestimated the expected IR excess produced by debris. To derive this excess (§4), we assume grains with sizes . In any debris disk model, the dust luminosity is sensitive to the size of the smallest grains, . Grains with sizes smaller (larger) than our nominal limit of 1  are more (less) luminous than predicted. Although  is clearly a free parameter, observations of comets in the solar system and warm debris around solar-type stars suggest 1  is a reasonable lower limit to the grain size (e.g., Lisse et al., 2006, 2007, 2008; Currie et al., 2011). Thus, increasing  seems an unlikely way to remove the discrepancy between and .

We also assume disks with power-law size distributions of small particles. In a real collisional cascade, the equilibrium size distribution has pronounced waves relative to our adopted power law (Campo Bagatin et al., 1994; O’Brien & Greenberg, 2003; Wyatt et al., 2011; Kenyon & Bromley, 2016). Adopting an analytic model for the waves which matches numerical simulations (Kenyon & Bromley, 2016) yields IR excesses a factor of 2–3 larger than the predictions of the basic analytical model outlined in the appendix. With this assumption, it is possible to detect debris disks with 10% to 30% (8–12 ), 1% (16 ), or 0.3% (24 ) of the MMSN.

We also assume small grains radiate as perfect blackbodies. In a real debris disk, small (1–10 ) grains are probably hotter and radiate less efficiently than perfect blackbodies (e.g., Backman & Paresce, 1993; Dent et al., 2000; Lisse et al., 2006). If we adopt an extreme model where all grains radiate inefficiently (e.g., with emissivity , with = 1  and b 0.8–1), the IR excess at 8–24  is roughly 50% smaller than the blackbody prediction. At 8–12 , this prediction has little impact on our ability to detect excesses around 10 Myr stars: blackbody grains are already at or below the nominal detection limits for Spitzer and WISE. At 16–24 , systems with initial masses of 10% (16 ) or 1% (24 ) of the MMSN still remain above the detection threshold even with an extreme emission model.

Considering the uncertainties in  and the radiative properties and size distribution of small particles, our estimates for the IR excesses of warm debris disks seem reasonable. Allowing for changes in  from pre-main sequence stellar evolution (e.g., Baraffe et al., 1998; Siess et al., 2000; Bressan et al., 2012; Chen et al., 2014; Baraffe et al., 2015) and corresponding variations in dust temperature also has little impact on predicted IR excess emission at 8–24 . Thus, the discrepancy between and remains.

5.3 Reconsidering Planet Formation Theory: Quick and Neat

If we accept the currently measured frequencies of Earth-mass planets and warm debris disks, we must then devise a theory that assembles Earth-mass planets with little detectable debris.

One way to achieve this goal is if planet formation is quick. If Earth-mass planets orbiting solar-mass stars reach their final masses during the T Tauri phase, debris ejected from giant impacts or collisions of leftover planetesimals would be rapidly mixed with the optically thick primordial dust left within the gaseous disk. Instead of producing a distinctive ‘debris disk signature,’ the debris simply contributes to the already large IR excess of the primordial disk. As viscous evolution and other processes remove the gas, the debris from Earth-mass planet formation leaves with the gas.

In the classical theory, planet formation is not quick. Although protoplanet formation might occur during the T Tauri phase (e.g., Wetherill & Stewart, 1993; Weidenschilling, 1997; Ohtsuki et al., 2002; Kenyon & Bromley, 2006; Kokubo et al., 2006; Kokubo & Genda, 2010; Kenyon & Bromley, 2016), growth then stalls (e.g., Weidenschilling et al., 1997; Chambers, 2001; Kenyon & Bromley, 2006; Raymond et al., 2007; Chambers, 2008, 2013; Raymond et al., 2014; Genda et al., 2015b; Quintana et al., 2016). Once the gaseous disk dissipates, giant impacts among Mars-mass protoplanets then build Earth-mass planets (Kominami & Ida, 2004), generating debris. The gaseous disk plays no further role in the evolution of the debris.

Pebble accretion is not intrinsically quick (e.g., Chambers, 2014; Johansen et al., 2015; Levison et al., 2015). Recent studies suggest the rapid formation of 100–1000 km planetesimals at 1 Myr. These planetesimals rapidly accrete leftover pebbles. Because these models begin with initial masses comparable to the MMSN, the giant impact phase begins well after the gaseous disk has dissipated. Giant impacts then occur on time scales when debris is easily detected.

The massive disks of solids proposed in the in situ theory ( 10 times the MMSN) enable rapid planet formation (Hansen & Murray, 2012, 2013; Hansen, 2015). Formation times scale inversely with the mass of solids; thus, protoplanets grow much faster than in the classical or pebble accretion theories. Even in a primordial gaseous disk, the large number of closely packed protoplanets generates a series of giant impacts, leading to multiple Earth-mass planets in a few Myr. The gas can then remove small particles produced in giant impacts. However, the gas probably cannot remove any leftover planetesimals or other 1–100 km particles generated during the giant impact phase. To produce a 24  excess that is % of the stellar photosphere in a system where the initial mass in solids is 10 times the MMSN, the giant impact phase must leave behind less than 0.1-0.2% of the initial mass in 100–1000 km objects (including any leftover planetesimals). Otherwise, the in situ theory makes too much debris at 10 Myr.

Thus, another way to satisfy the observational constraints is if planet formation is intrinsically neat: Earth-mass planets assemble with negligible dust emission. As described above, the classical, in situ, and pebble accretion scenarios are not neat enough to match observational constraints. To isolate the appropriate physical conditions for a ‘neat’ scenario, we rely on the analytical model outlined in the appendix (A.1) and published numerical simulations (e.g., Kenyon & Bromley, 2016). To match current sensitivity limits at 16  and 24 , we must reduce dust production by at least a factor of ten (i.e., reducing  from 15%–30% to %; Fig. 4). In all numerical simulations, dust production depends on , where is the impact velocity and  is the binding energy. Simple dynamics sets ; changing it significantly is unlikely.

The binding energy is based on analytical and numerical simulations using state-of-the-art equations of state. While increasing  seems unlikely, it is worth exploring the required physics. As a simple comparison, the gravitational binding energy per unit mass of a uniform sphere exceeds the  for basalt only when 10 . Thus, the internal degrees of freedom of protoplanets are important in setting . While it may be possible to modify these by factors of 2–3, order of magnitude changes seem unlikely (see also Housen & Holsapple, 1999; Holsapple et al., 2002, and references therein).

If dust production cannot be limited, the main alternative is to reduce the lifetime of the debris. In this approach, a shorter debris lifetime lessens the likelihood of detection. From eqs. A4 and A10, lowering  or increasing by a factor of ten shortens the lifetime of the debris by a similar factor. Having already ruled out factor-of-ten changes in , we consider the impact of changing .

In the analytical model, changing  modifies the detectability of the debris in two ways. For a fixed mass in debris, the cross-sectional area . Reducing  therefore increases the initial dust luminosity and shortens the collision time. As shown in Fig. 5,  must be quite small to reduce the predicted warm excess to the level of the 24 excess displayed by the brightest 20% of  Myr old stars. If  = 0.1 and  = 10 km, then at 10 Myr at 24. Approximately 20% of FEPS sources younger than 10 Myr have an excess brighter than at 24, a fraction comparable to the fraction of solar-type stars with Earth-like planets.

Although this solution seems attractive, it has several major drawbacks. Numerical simulations of giant impacts already yield 24  fluxes much larger than observed (Genda et al., 2015b); generating a smaller likelihood of detection at 10 Myr age in exchange for a much larger initial luminosity is probably a poor trade. For dust generated from planetesimals, numerical simulations suggest that collisional damping of small particles maintains large dust luminosities far longer than suggested by the analytical model (Kenyon & Bromley, 2016). Increasing the population of small particles by lowering  probably exacerbates this problem.

5.4 Neat Planet Formation in a Remnant Gas Disk

If we cannot substantially alter the observed frequency of terrestrial planets, models for the assembly of terrestrial planets, or the amount of debris generated by terrestrial planets, some process must remove small particles with 0.1–1 mm from the terrestrial zone. Possibilities include interactions with the stellar radiation field, the stellar wind, or a remnant gaseous disk. If one of these processes removes small particles faster than the collisional cascade produces them, the predicted IR emission from warm dust is reduced by the requisite factor of 10, eliminating the discrepancy between and .

Compared to aerodynamic drag from a remnant gaseous disk, other mechanisms probably have a limited role. Observations suggest radiation pressure may remove particles smaller than 1 , but larger particles are robustly detected (e.g., Lisse et al., 2008; Currie et al., 2011; Matthews et al., 2014, and references therein). For , Poynting-Robertson drag is ineffective (Wyatt, 2008). In the inner Solar System, stellar wind drag is roughly 30% as effective as Poynting-Robertson drag in removing small particles (Gustafson, 1994). Observations of the youngest solar-type stars suggest mass loss rates 10 times the mass loss rate of the current Sun (Wood et al., 2014, and references therein). Theoretical studies predict mass loss rates up to 100 (e.g., Cohen & Drake, 2014; Airapetian & Usmanov, 2016). Both of these estimates fall well below the 1000 required for stellar wind drag to remove small particles rapidly when . Finally, the sputtering rate for small particles is too small compared to the collision rate (Wurz, 2012).

In some circumstances, photophoresis333The photophoresis mechanism exploits the temperature gradient across an illuminated grain embedded in a low pressure gas. Momentum exchange with the gas results in the grain moving away from the radiation source. drives small particles radially outward through an optically thin gaseous disk (e.g., Krauss & Wurm, 2005; Herrmann & Krivov, 2007; Cuello et al., 2016). When photophoresis is effective, the time scale for radial drift is comparable to (and sometimes shorter than) the time scale for radial drift due to gas drag. However, the drift time is sensitive to the ratio of the heat conductivity to the particle asymmetry factor, which is uncertain (e.g., von Borstel & Blum, 2012).

Here, we focus on the impact of gas drag, where the physical properties of small particles are less important. Our goal is to identify a range of for the gaseous disk that allows giant impacts (0.1–1% or less of the MMSN) and removes 0.1–1 mm and smaller particles on time scales shorter than the collision time.

Within any gaseous disk, particles weakly bound to the gas rapidly drift inward when their ‘stopping time’ (, where is the drift velocity and is the drag force) is comparable to their orbital period (Adachi et al., 1976a; Weidenschilling, 1977a). At 1 AU, the shortest drift time is 50–100 yr (Fig. 6), shorter than the typical collision time of 1000 yr for 1  to 1 cm particles. In an optically thin disk, radiation pressure drives small, weakly coupled particles to large where the particles are colder and have much smaller dust luminosities (Takeuchi & Artymowicz, 2001). Effective radial drift thus eliminates particles that produce an IR excess444In a dense gaseous disk, interactions with the gas rapidly circularize the orbits of small particles (e.g., Adachi et al., 1976b; Weidenschilling, 1977a). During a collisional cascade in a low density disk, gravitational stirring by protoplanets raises faster than gas drag lowers (e.g, Kenyon & Bromley, 2004; Raymond et al., 2011; Genda et al., 2015b). Thus, we ignore this process..

To quantify radial drift at 1 AU in a low density gaseous disk, we combine the approaches of Weidenschilling (1977a) and Takeuchi & Artymowicz (2001). As outlined in the Appendix (§A.2), we solve for the radial and azimuthal velocity of particles relative to the gas in a protostellar disk with standard relations for the surface density, midplane temperature, vertical scale height, gas pressure, thermal velocity, and other physical variables (e.g., Kenyon & Hartmann, 1987; Chiang & Goldreich, 1997; Rafikov, 2004; Chiang & Youdin, 2010; Youdin & Kenyon, 2013; Armitage, 2013).

Fig. 6 illustrates the impact of a residual gas disk on the radial drift velocity of particles as a function of size at 1 AU under MMSN-like conditions. For this example, a disk with the surface density of the MMSN has = 2000 , = 278 K, and vertical scale height = 0.03 AU. Filled circles indicate inward drift; open symbols indicate outward drift. Other choices for these parameters – e.g., = 150–300 K and = 0.02–0.10 – change the maximum drift velocity and the particle size for this maximum drift by 25% to 50% but do not change the overall trends.

When the disk has comparable to the MMSN (Fig. 6, black symbols), particles with 50 cm have 1 and the largest drift velocity. Inflection points in occur when particles enter different drag regimes (e.g., Epstein, Stokes, quadratic). For particle sizes 30–40 , outward radiation pressure balances inward gas drag; these particles do not drift. Smaller particles drift outward555MMSN disks are probably optically thick. Small particles in the optically thin upper layers drift outward, while those in the optically thick midplane are unaffected. with maximum velocities slightly smaller than 0.1 

As the surface density of the gaseous disk declines, smaller particles drift more rapidly through the gas. Although the peak in  shifts to smaller sizes, the maximum drift velocity is always roughly 8000 . The balance between radiation pressure and gas drag remains fixed for 30–40  particles. However, in disks with lower , smaller particles have larger stopping times. With a weaker drag force, radiation pressure drives these particles outward at larger velocities. Once is roughly 0.001% of the MMSN, the outward drift velocities of 30  particles surpass the maximum inward drift velocities of 100  particles.

To judge the impact of radial drift on the collisional cascade, we compare the drift time to the collisional time. For disks with surface densities of 0.001% of the MMSN, 1–10  and 100  particles at 1 AU drift inward or outward on 50–75 yr time scales. In less (more) tenuous gas disks, particle lifetimes are longer (shorter; Fig. 6). For comparison, in a collisional cascade at 0.1–2 AU with , the collision time is roughly 1000 yr. In disks with 0.001% of the MMSN, radial drift from radiation pressure and gas drag removes particles smaller than 100  in yr, rapidly depleting the disk of the 1–100  particles that comprise 90% of the surface area. Gas drag also rapidly transports larger (0.1–1 mm) particles towards the central star.

These two processes have a dramatic effect on debris production. In a standard collisional cascade, mass flows at a roughly constant rate from the largest particles to the smallest particles (e.g., Wyatt, 2008; Kobayashi & Tanaka, 2010; Wyatt et al., 2011). When the gas drags 0.1–1 mm particles inward, the equilibrium population of these particles drops. Collisions among these objects are then less frequent, depressing the flow of mass to smaller particles. In turn, a smaller mass flow rate lowers the cross-sectional area of the swarm and weakens the infrared excess.

When gas drag and radiation pressure effectively remove small particles inside 2 AU, the IR excess drops dramatically. For disks with 0.001–0.01% of the MMSN, the dust luminosity drops by 2–4 orders of magnitude. If this remnant disk extends close to the stellar photosphere, particles with sizes 1 mm either evaporate or fall onto the photosphere. Thus, residual gas disks as dilute as 0.001% of the MMSN can remove the small solids () that make up 90% of the cross-sectional area of the debris, reducing the IR excess at 5–20 Myr to a level consistent with the observational constraints. Less dilute gas disks (0.01% of the MMSN) can reduce the population of small grains indirectly, by removing the 1–10 mm solids that generate them through collisions. More dilute gas disks are less efficient in removing solids and therefore less able to limit the IR excess.

We conclude that a residual gas disk in the range 0.001–1% of the MMSN that persists for  Myr is dilute enough to allow giant impacts to assemble terrestrial planets in classical planet formation, but dense enough to minimize the observable IR excess produced by the resulting debris.

5.5 Observational Constraints on Residual Gas Disks

Current observational constraints on the surface density of residual circumstellar gas disks cannot exclude our picture for reducing IR excesses around 10 Myr old solar-type stars. Constraints on the residual gas content of disks from in situ diagnostics are model-dependent (Gorti & Hollenbach, 2004) and not highly restrictive in this context. For example, analyses of Spitzer IRS emission line diagnostics for a sample of FEPS targets with little evidence for warm debris disks place a formal limit on the gas surface density % of MMSN beyond 1 AU (Pascucci et al., 2006). If this upper limit were to extend inside 1 AU, the surface density would be low enough to allow giant impacts in the classical planet formation picture (e.g., Kominami & Ida, 2002, 2004) and large enough to allow the removal of small particles by gas drag.

Measuring stellar accretion is an alternate way to search for evidence of a residual gas disk. While accretion indicates that residual gas is present, it does not directly measure . To obtain a rough scaling between stellar accretion rate and , we can consider the average accretion rates of classical T Tauri stars at 1–10 Myr age. Stellar accretion rates decrease from at the age of Taurus-Auriga (few Myr) to at 10 Myr (Sicilia-Aguilar et al., 2006), i.e., only a factor of 25. If the fractional decrement in indicates a similar reduction in , an accretion rate of corresponds to a surface density at 1 AU of if the initial state is an -disk with a surface density of at 1 AU (Hartmann et al., 1998) or if the initial state is the MMSN with a surface density of at 1 AU (Weidenschilling, 1977b). Non-accreting sources, with presumably lower , make up more than 90% of young stars at 5–10 Myr age (Fedele et al., 2010).

Based on the scaling relation, it appears that we need to probe very low accretion rates to reach of the MMSN and to exclude the possibility that 5–10 Myr old disks contain enough residual gas to erase a debris disk signature. For approximately solar mass stars with ages of 5–10 Myr, chromospheric emission limits our ability to measure stellar accretion rates below using -band excesses (Ingleby et al., 2011b) or hydrogen emission line fluxes (Manara et al., 2013). H emission line widths (Natta et al., 2004) and line profile analyses (Muzerolle et al., 1998b, a) can probe lower accretion rates (e.g., Muzerolle et al., 2000; Manara et al., 2013).

From the shape and velocity extent of the H line profile (White & Basri, 2003; Natta et al., 2004), Riviere-Marichalar et al. (2015) derive a broad range of accretion rates for 11 sources in the 8 Myr old Cha group. Six sources have from at least one spectrum; the other five sources have . While it may not be appropriate to apply the Natta et al. (2004) relation derived for low mass stars to a set of solar-type stars, these accretion rates are close to the rates required for gas drag to eliminate the IR excesses of debris disks.

There are multiple prospects for detecting a dilute reservoir of gas in solar-type stars. CO fundamental emission (e.g., Najita et al., 2003; Salyk et al., 2011, Doppmann, Najita, & Carr 2016) and UV transitions of molecular hydrogen (e.g., Ingleby et al., 2011a) probe molecular gas within an AU of young stars. Among A-type stars with debris disks, several have neutral or ionized gas close to the star (e.g., Hobbs et al., 1985; Ferlet et al., 1987; Welsh et al., 1998; Redfield, 2007; Montgomery & Welsh, 2012; Kiefer et al., 2014; Cataldi et al., 2014, and references therein). Evaporation of comets (Lagrange et al., 1987; Beust et al., 1990) and vaporization of colliding particles (e.g., Czechowski & Mann, 2007) might supply some of this material (see also Kral et al., 2016).

5.6 Testing the Possibilities

In our analysis, we have considered four main options to resolve the discrepancy between the frequencies of warm debris disks and Earth-mass planets among solar-type stars. Despite clear gaps in our understanding of the physics of terrestrial planet formation, there is no obvious way to modify the theories to limit the detectability of warm debris at 5–20 Myr. Uncertainties in our ability to relate IR excesses to debris from rocky planet formation are unlikely to change . Although overestimating the observed frequency of Earth-mass planets is a plausible cause for the difference between and , the ‘best’ explanation is rapid drift of small particles in a residual gaseous circumstellar disk.

Future observations and theoretical investigations can test all four explanations. We outline several possibilities.

  • Search for fainter warm excess signatures in large samples of young solar-type stars. Current sensitivity limits are not much lower than theoretical predictions. If warm excesses are simply a factor of 2–3 fainter than predicted, detecting these excesses with more sensitive surveys (e.g., using MIRI on JWST; Wright et al., 2003; Wells et al., 2006; Bouchet et al., 2015) may be able to distinguish neat scenarios from quick scenarios. For example, placing stricter limits on the level of 16  excess ( 10% of the stellar photosphere) would probe initial solid masses in debris of a few percent of MMSN (Fig. 4).

  • In Kepler systems with apparent Earth-mass planets, better limits on (i) the false-positive rate, (ii) the frequency of multiple planets, and (iii) the distributions of and in systems with multiple Earth-mass planets would reduce the uncertainties in the fraction of stars that host Earth-mass planets inside 1–2 AU. Theoretical studies into the stability of systems with multiple Earth-mass planets would place constraints on the ability of these systems to ‘hide’ from radial velocity and transit observations.

  • Hunt for planets around all young (5–20 Myr) solar-type stars. Several recent studies identify massive planets orbiting several pre-main sequence stars (e.g., Mann et al., 2016; Gaidos et al., 2016; David et al., 2016; Donati et al., 2016; Johns-Krull et al., 2016b, a). Identifying Earth-mass planets orbiting young stars would constrain the timescale of terrestrial planet formation relative to the production of debris. The K2 mission (Howell et al., 2014) might identify more short-period planets within several nearby young stellar associations666Among the candidates reported in recent analyses of the K2 data (e.g., Foreman-Mackey et al., 2015; Vanderburg et al., 2016), 205117205.01 matches the position of an M2 pre-main sequence star in the Upper Sco association, 2MASS J161014731919095 (Luhman & Mamajek, 2012). IR data suggest the star has a debris or evolved transitional disk. However, the transit depth is very uncertain. Clarifying the existence and depth of transits in this system would begin to place constraints on the frequency of planets in the youngest stars.. TESS (Ricker et al., 2015; Sullivan et al., 2015) can search for planets with a much larger range of orbital periods.

  • Search for evidence of tenuous residual gas disks, at the of MMSN level, around young solar-type stars. Direct detection of gas or robust measurement of very low mass accretion rates tests the idea that radiation pressure and aerodynamic drag remove the debris of terrestrial planet formation. In addition to surveys with the NIRSPEC or MIRI spectroscopic instruments on JWST (e.g., Wright et al., 2003; Wells et al., 2006), sensitive optical (e.g., Xu et al., 2016) or radio (e.g., ALMA, VLA) observations might reveal low mass gaseous disks in the terrestrial zones of 5–20 Myr old solar-type stars.

  • Examine quick and neat modes of planet formation. Although our current understanding appears to preclude these ideas, it is important to quantify the ability of planet formation scenarios to assemble planets quickly and neatly in the absence of a long-lived gaseous disk. Future theoretical calculations of terrestrial planet formation should include clear predictions of dust production for comparison with observations (e.g., Raymond et al., 2011; Genda et al., 2015b; Kenyon & Bromley, 2016).

  • Consider the late stages of protostellar disk evolution in more detail. Current mechanisms for disk dispersal (e.g., photoevaporation and viscous evolution) do not make firm predictions for the structure of gaseous material on time scales of 10–100 Myr (e.g., Gorti et al., 2015, and references therein). For example, including the gas from the evaporation of comets or vaporization of colliding planetesimals in photoevaporation models would help to constrain the long-term evolution of debris and gas in the terrestrial zones of young stars.

6 Summary

In the past decade, detailed analyses of Kepler, Spitzer, and WISE data have established estimates for the frequencies of Earth-mass planets and warm dust in the terrestrial zones of solar-type stars. Although rocky planets are fairly common, the expected dusty ‘signature’ of terrestrial planet formation among 5–20 Myr old stars is a rare phenomenon. Potential explanations for the discrepancy include the possibility that (i) terrestrial planets are much less common than believed, (ii) planet formation is quicker and/or neater than predicted, and (iii) some physical mechanism removes warm dust rapidly. Although we cannot preclude some combination of the first two options, gas drag and radiation pressure can efficiently eliminate warm dust particles from the terrestrial zone when a residual gaseous circumstellar disk has a surface density of of the MMSN. Current constraints on the gas surface density within 1 AU of 5–10 Myr old solar-type stars are consistent with this limit. Thus, tenuous reservoirs of gas may impact our ability to observe the debris produced by rocky planet formation. If they do, warm debris is not a reliable signpost of rocky planet formation.

Modifying planet formation theory to resolve the discrepancy is tenable if planet formation is much more efficient than currently predicted by theory and leaves behind little debris. This scenario echoes the results of several previous studies (e.g., Greaves & Rice, 2010; Najita & Kenyon, 2014) that attempt to reconcile the inventory of solids bound up in known populations of exoplanets with the solid masses of protoplanetary disks. Producing the known exoplanet systems from the limited solid reservoirs in protoplanetary (Class I) disks requires a planet formation efficiency of roughly 30%. If there is a substantial population of undiscovered planets or if planet formation is a messy process that discards solids by producing significant debris, the required efficiency of planet formation rises.

In the scenarios we investigate, typically % (20%) of the initial solid mass ends up in rocky planets (debris). To limit debris production, it is advantageous to start the planet formation process with an initial mass in solids reasonably close to the final mass in stable planets. Without some process that removes small grains from the disk, theoretical scenarios that invoke much larger initial mass reservoirs (e.g., Hansen & Murray, 2012, 2013; Hansen, 2015; Volk & Gladman, 2015; Levison et al., 2015) should produce very large IR excesses which violate existing constraints from observations of 5–20 Myr solar-type stars.

In this sense, collisional debris can place strong constraints on planet formation scenarios. As outlined in §45, stringent tests require (i) tighter observational constraints on warm dust and residual disk gas during the expected epoch of terrestrial planet formation and (ii) planet assembly simulations which include the effect of gas drag in a residual gas disk.

We acknowledge a generous allotment of computer time on the NASA ‘discover’ cluster. We thank G. Herczeg for valuable discussions of stellar accretion rates. Comments from and discussions with S. Andrews, J. Carpenter, M. Geller, A. Glassgold, G. Kennedy, N. Murray, I. Pascucci, D. Wilner, and an anonymous referee improved our presentation. Portions of this project were supported by the NASA Outer Planets Program through grant NNX11AM37G. The work of JN was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293. JN also acknowledges the stimulating research environment supported by NASA Agreement No. NXX15AD94G to the Earths in Other Solar Systems program.

Appendix A Long-term Evolution and Gas Drag in Debris Disks

a.1 Analytic Theory for an Extended Disk

Wyatt & Dent (2002) and Dominik & Decin (2003) first developed an analytic theory to track the long-term evolution of collisional cascades (see also Wyatt et al., 2007a, b; Heng & Tremaine, 2010; Kobayashi & Tanaka, 2010; Wyatt et al., 2011; Kenyon & Bromley, 2016). In this model, particles with sizes ranging from  to , orbital period , and orbital eccentricity occupy a single annulus with width at a distance from a central star with mass , radius , and luminosity . Particles collide at a rate , where is the number density, is the physical cross-section, and is the collision velocity. Setting the center-of-mass collision energy777In our approach, the center-of-mass collision energy is , where the reduced mass larger than the binding energy  of the largest particles ensures destructive collisions which produce debris. For an initial surface density of solids in the annulus, the collision time is

(A1)

where the correction factor is (Wyatt et al., 2007a; Heng & Tremaine, 2010; Kobayashi & Tanaka, 2010; Wyatt et al., 2011). The surface density of material in the annulus then evolves as

(A2)

Other collective properties of the particles – including the total mass, cross-sectional area, and relative luminosity – follow the decline of with time (Wyatt & Dent, 2002; Dominik & Decin, 2003; Wyatt et al., 2007a; Kobayashi & Tanaka, 2010; Wyatt et al., 2011, KB2016).

Here we expand the analytic theory to a disk888Kennedy & Wyatt (2010) developed a semi-analytic theory for extended debris disks around A-type stars. For the evolution of IR excesses at specific wavelengths, our approach is similar; our analytic derivation for the evolution of the luminosity is new. extending from an inner radius  to an outer radius . Particles with sizes ranging from  to  have an initial surface density

(A3)

where = 10  is the surface density of solids at = 1 AU in the MMSN and  is a scale factor (Weidenschilling, 1977b; Hayashi, 1981; Kenyon & Bromley, 2008). We assume that the correction factor . Setting , the collision time is then , where

(A4)

To derive the evolution of the disk, we assume the particles have a differential size distribution in each annulus (e.g., Dohnanyi, 1969; Williams & Wetherill, 1994; Tanaka et al., 1996; Kobayashi & Tanaka, 2010). Other choices lead to similar results (Wyatt et al., 2011). We can then relate the cross-sectional area of the swarm to the mass in each annulus:

(A5)

Setting , and , the general result of the analytic model in eq. (A2) and our relation for yields:

(A6)

To relate this evolution to an observable quantity, we set the relative luminosity = . We then have:

(A7)

Integrating eqs. (A6A7) over yields the time evolution of the cross-sectional area and relative luminosity for material between and .

Applying this theory to a real disk requires specifying parameters for the particles (, , and ) and the disk (, , , and ). Radiation pressure from the central star ejects particles with radii smaller than  (Burns et al., 1979); here, we set  = 1 . For equal mass objects with relative velocity , the collision energy is . The binding energy is

(A8)

where is the bulk component of the binding energy and is the gravity component of the binding energy (e.g., Benz & Asphaug, 1999; Leinhardt et al., 2008; Leinhardt & Stewart, 2009). Adopting parameters appropriate for rocky objects and setting yields (KB2016):

(A9)

Based on several simulations of planet formation in the terrestrial zone, 0.1 is a reasonable choice (e.g., Weidenschilling et al., 1997; Kenyon & Bromley, 2006; Raymond et al., 2007; Chambers, 2013). Setting  = 300 km, the collision time at 1 AU is:

(A10)

For a standard protostellar disk with , we adopt = 3/2 (e.g. Birnstiel et al., 2010; Windmark et al., 2012; Testi et al., 2014). In the standard analytic model, with = 5/6 (Wyatt et al., 2007a, b; Kobayashi & Tanaka, 2010). However, comprehensive numerical simulations suggest = 1 (KB2016). Here, we consider = 1 and = 0 ( = constant throughout the disk) or = 1 ( = constant throughout the disk). Adopting , the luminosity evolution is then

(A11)

for = 0 and

(A12)

for = 1, where

(A13)

At = 0, solutions for any yield the same initial luminosity:

(A14)

In terms of our adopted parameters:

(A15)

When , the initial luminosity of the debris disk exceeds our nominal detection limit, . At late times, we derive a single expression when = 0 or = 1:

(A16)

As in the standard model for a single annulus, the luminosity declines linearly with time when . For our adopted parameters and = 0:

(A17)

With  = 1 and = 10 Myr, . The relative luminosity reaches the nominal detection limit of at roughly 100 Myr.

Fig. 7 illustrates results for a disk with = 0.1 AU and = 2 AU. The two sets of curves compare the analytic model with numerical results derived from dividing the disk into 101 annuli and summing (eq. A7) from each annulus. There is superb agreement between the two approaches. For any time, the difference between the analytic and numerical results for = 0 or = 1 is less than 0.1%. At 10 Myr, all of the model curves lie above the approximate detection limit of .

To derive the predicted evolution of disk fluxes at specific wavelengths, we assume all particles radiate as blackbodies at the equilibrium temperature, , where  K. Although might be integrable, the result is very messy. Given the good agreement between the analytical and numerical results for , we integrate numerically.

a.2 Gas Drag in a Depleted Disk

As outlined in the main text, drag within a remnant gaseous disk can rapidly remove small particles from a debris disk around 10 Myr old stars. To quantify this drift, we adopt a model gaseous disk (e.g., Adachi et al., 1976b; Weidenschilling, 1977a; Kenyon & Hartmann, 1987; Chiang & Goldreich, 1997; Takeuchi & Artymowicz, 2001; Rafikov, 2004; Chiang & Youdin, 2010; Youdin & Kenyon, 2013; Armitage, 2013) with surface density, , midplane temperature , and vertical scale height . The midplane density is . Setting the sound speed – where is the ratio of specific heats, is Boltzmann’s constant, is the mean molecular weight, and is the mass of a hydrogen atom, the gas pressure in the midplane is . The gas has mean-free-path and viscosity