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ABSTRACT and 00Wlu tet ut 0,Budr O832 USA 80302, CO Boulder, 400, Suite Street, Walnut 1050 , oeta h ehnssta rnaeteotrre- disk the actually outer photoevaporation) on the to (especially primarily truncate disk important act that the is mechanisms of it gions the picture, Although that self-consistent note isotopes. a solar radioactive paints early lived this our short pro- contaminate with at of to capable disk system able cluster the super-nova a truncate a with consistent to ducing is enough AU arguments dense 50 presents our around cluster of Adams star environment example, a birth For that the part understand as system. system to solar solar have effort the within an to edge of references outer and apparent (2010) this used Adams Indeed, disk. rbto fpaesadKO)rfr otedisk the to refers KBOs) and planets of tribution uikje l 05adseToms&Dna 2006 Duncan & Thommes 2005; see al. and 1996; et 2005 al. Alibert et al. 2003; Pollack solids et al. of Hubickyj et mass (e.g. Thommes total 1987; planets the equivalent Lissauer than, the is greater into assembled times 5 incorporated fully to left are up planetesimals or planets of to, current the mass total process; after the inefficient over that an forma- suggest is Jupiter’s of models formation time the Planet at disk tion. gaseous the of size the to edge an the disk. to in gas “edge” corresponds the an necessarily that disk claim planetesimal cannot Chiang planetesi- one & of Therefore Youdin (e.g. likelihood 2004). the formation the core enhancing and increasing formation in ratio, mal an useful gas in be even to born could have solid solids disk drifting authors of gaseous some reservoir extended large Indeed, a 1977a; that (Weidenschilling argued 2002). gas Shu the & and Youdin solids de- the of capable coupling mechanisms physical well-known are There nti ae epeeta ruetta constrains that argument an present we paper this In R D & ol aeectdKzioscillations Kozai excited have would ag yotcSre Telescope Survey Synoptic Large e 0A)i u oa ytma the at system solar our in AU) 80 nteslrsse.I standard In system. solar the in s ydisk. ry atrdasgicn ubrof number significant a cattered on tr aysgicnl in significantly vary stars young i indpn rtclyuo the upon critically depend tion hwta fteehdbe a been had there if that show & onc neitn planetary existing an connect oa ytmSre eare we Survey System Solar t ntantesz forown our of size the onstrain 50 hns.Uigteenon- these Using chanism. uvy aentdetected not have surveys France ◦ ,lwecnrct orbits low-eccentricity ), bet nteehigh- these in objects gas hl h aa(h dis- (the data the while , solids . 2 Kretke et al. for a review). Therefore during and shortly after the for- In this paper we investigate the behavior of planetes- mation of the giant planets, must have been a significant imals initially in the giant planet regions and model number of planetesimals left over in the planets’ feeding their gravitational interactions with the planets and the zones. These planetesimals are fated to undergo close en- gaseous disk. We initialize these simulations with four counters with the planets, and many will be scattered to giant planets in the compact configuration proposed by high eccentricity orbits. In a simple 3-body scattering, Tsiganis et al. (2005) as the initial positions of the plan- the perihelion of the scattered object remains roughly ets in the Nice Model, with Jupiter, Saturn, Uranus and unchanged, meaning that the planetesimal will return to Neptune at 5.45, 8.18, 11.5, and 14.2 AU, respectively. undergo additional close encounters. Eventually these However, as we discuss in §3, the results of this pa- objects will either collide with the planet or be ejected per are insensitive to this choice. For each simulation 3 by the system. we calculate the orbital evolution of NTP = 6 × 10 - However, in order to form the planets there must also 104 massless test particles initially distributed from 4- have been a circumstellar gas disk, a disk that poten- 15 AU with small eccentricities (e = 0.1) and incli- tially could have extended beyond the orbits of the plan- nations (0◦ orbit and an r −γ −t high-inclination, more circular orbit. During the high ec- Σ(r, t)=Σ0 Θ(r, RD)exp , (1) centricity phase the particle is susceptible to additional AU τD   close encounters with the planets, but during the low ec-   centricity phase, the particle’s orbit will no longer cross where Θ(r, RD) describes the functional form of the trun- the planets. After the disk disperses, statistically some cation of the outer disk at size RD, and τD is the disk depletion timescale. For the fiducial disk models we of the objects will be stranded at high inclination orbits. −2 This dynamical mechanism is capable of producing ob- use Σ0 = 2000g cm and γ = 3/2 consistent with the jects with large enough perihelia to place them in Kuiper benchmark minimum mass nebula profile (Hayashi 1981). Belt on orbits that are stable for the lifetime of the so- We investigate the effect of the sharpness of the disk trun- lar system with the planets in their present day orbital cation on distribution of planetesimals by comparing two configuration. But while these particles have semi-major truncation profiles, a sharp cutoff, axes that would place them in the Kuiper Belt, they 1, if 1 AU ≤ r ≤ R , ≡ D would have substantially higher inclinations than what Θ(r, RD)=Θcut(r, RD) 0, otherwise. has been observed so far. In this paper we explore this  (2) dynamical mechanism as a way to use the current dis- and an exponential cutoff tribution of Kuiper Belt Objects to place constraints on the early structure of the protoplanetary disk. exp(−r/R ), if r ≥ 1 AU, ≡ D In §2 we describe our numerical method and model pa- Θexp(r, RD) 0, otherwise. (3) rameters. In §3 we discuss the outcomes for various disk  parameters. In §4 we discuss how we can use these re- (c.f. Andrews et al. 2009, 2010). In all disk models we sults to constrain the size of the protosolar nebula using truncate the inner disk at 1 AU, but we vary the outer results from the Deep Ecliptic Survey (DES; Millis et al. disk cutoff (RD) from 30 AU to 100 AU. To preserve the 2002; Elliot et al. 2005; Gulbis et al. 2010) and the Palo- stability of the integration we use a “puffy” exponential mar Distant Solar System Survey (Schwamb et al. 2009, profile for the disk vertical profile, 2010). And finally, in §5 we summarize our results and z discuss the implications. ρ(r, z)=Σ(r)exp − , (4) H   2. where H is constant throughout the disk. This disk MODEL model is chosen for its simplicity and is not intended Constraining the Protosolar Nebula 3 to represent a detailed disk evolution model. the gas though its gravitational potential, not through There are five free parameters for the disks in these aerodynamic drag. As indicated in Brasser et al. (2007) simulations: Σ0, τD, RD, and γ, and the functional form the gas drag is only dynamically important for objects of Θ(r, RD). In total we present the results of the evo- with diameters smaller than about D =30 km. The im- lution in 35 different gas disks. Our fiducial parameters pact of the gas drag on these small bodies is to circularize −2 are Σ0 = 2000 g cm , τD = 2 Myr, Θ(r, RD)=Θcut, the orbits, and thus they do not participate in the effect RD = 100 AU, and γ = 3/2. With the fiducial disk pa- explored in this paper. Furthermore, as we discuss in §4, rameters, we perform 7 simulations varying Σ0 from 20 most of the mass is likely to be in 100 km, or larger, sized to 4000 g cm−2. We also perform 6 simulations varying objects and smaller objects would not be detectable in 4 6 τD from 2 × 10 to 4 × 10 years. Using the fiducial Σ0 current datasets. Therefore we only concern ourselves and τD we then perform simulations with RD =30, 40, with these larger bodies and neglect gas drag in these 50, 60, 80 and 100 AU. For the sharp cut off we allow simulations. γ =3/2 (the MMSN value) and γ = 1 (a value consistent 3. with some viscous evolution models). For the exponen- RESULTS tial cutoff we use γ = 1/2 and 1, appropriate values for In figure 1 we compare the orbital distribution of plan- observed disks (Andrews et al. 2010). In addition, we etesimals produced by disks of various sizes after the scat- compute one simulation in the fiducial disk with the four tering simulations, by which time the gas disks have dis- giant planets in the configuration of the modern day solar persed. Without the gravitational potential of a gaseous system. We run these scattering simulations for 5 Myr disk (left-hand panels), the planetesimals are scattered 6 (10 Myr for the disk with τD =4 × 10 ). by the planets into eccentric and moderately inclined or- In most of these simulations, some particles are cap- bits, but the perihelia (q) remain roughly unchanged. tured by the Kozai resonance and forced into orbits with Therefore, all of the particles remain on planet-crossing q> 30 AU; orbits that are potentially stable in the mod- orbits that are unstable on timescales short compared to ern day solar system. However, although a large number the age of the solar system. These unstable particles are of particles undergo oscillations with large perihelion val- indicated by gray points. In contrast, in middle and right ues at some point during their orbital evolution, at the panels we show the particles’ evolution in the presence of end of the scattering simulations only a handful of these a disk with a MMSN profile (γ =3/2) and a sharp cutoff particles happen to have been caught on the q phase of (Θ=Θcut), truncated at RD = 30 AU and 100 AU, re- their orbits. Still, the time at which a given particle is spectively. In both of these simulations, a subset of the scattered into one of these orbit is random, and could particles are placed into orbits with larger perihelia and as easily have happened right before the disk dispersed higher inclinations. In the small disk, some particles have as at any other time. Therefore instead of restricting their perihelia raised so that they have a high probability our stability calculation to only those particles still ac- of being stable in the early compact solar system, with tive at the end of the simulations, we randomly extract q > 20 AU (particles marked in blue), but it is only in the orbital elements (a,e, and i) of particles with large the larger disk that the perihelia are raised to the point perihelia (q > 20 AU) values throughout the simulations. where they would be stable in the modern solar system, We then randomly assign orbital angles and create a new q> 35AU (marked in red). suite of equally probable, potentially stable particles. We These particles get onto these high inclination, low ec- integrate these new particles in the presence of the four centricity orbits via Kozai oscillations. In figure 2 we giant planets, in their modern day configuration, for 4.5 show the temporal evolution of an example particle in Gyr. We use these particles to define the probability the 100 AU disk. The particle undergoes a series of close that a particle will survive as a function of its perihe- encounters with Jupiter and is scattered out to a high ec- lion distance at the beginning of the stability calculations centricity orbit. After about half a million years we see (PS (q)). We can then return to the initial scattering cal- a number of intervals in which the perihelion distance culations and use the probability that each particle (at increases and decreases in phase with oscillation in the its given perihelion distance) will be stable to calculate inclination while the semi-major axis remains constant, the expected fractional retention of particles, as is characteristic for Kozai oscillations. As further ev- idence, in the right hand panel we plot the eccentricity NTP and the argument of perihelion for this particle. The f ≡ PS (qi)/NTP. (5) large circles indicate a time interval (shaded in the left i=1 hand panel) of a single Kozai cycle. In this time the argu- X ment of perihelion oscillates, indicating that this is a true Here PS(q) is the function defined by the stability cal- Kozai cycle. Additionally, by comparing figure 2c with culation, and qi refer to the perihelion distance of each the Kozai dynamics exerted by the planets on objects particle at the end of the initial scattering simulations. with larger semi-major axes (c.f. Thomas & Morbidelli Because PS(q) goes to zero as qi approaches 20 AU in 1996) we can be confident that the perturbations seen all simulations, we are confident that the initial cut-off here are dominated by the external disk potential not of q = 20 AU does not impact the statistics in any part perturbations from the interior planets. During each of of this study. To determine our 95% confidence inter- the periods of high eccentricity (small q) the particle un- vals, we assume that our data is distributed according dergoes additional encounters with Jupiter and Saturn, to a binomial distribution and use a non-informative, causing sharp jumps in the particle’s semi-major axis. As Bayes-Laplace uniform prior (see Cameron 2011, for a the disk is depleted the timescale for the oscillation be- nice description of this method). come longer. In the end the period of oscillation becomes In these simulations, the particles only interact with infinite, stranding the particle on a high inclination orbit. 4 Kretke et al.

tions at large a, however the fact that the disk is not dominated by a single annulus of material at the outer edge means that the oscillations are shallower. Therefore the objects produced do not have very low eccentricity orbits. The shallower surface density profile (γ = 1/2) again more effectively generates these particles due to the fact that there is more mass at larger radii to excite Kozai oscillations. The effect of disk size on retention can be seen more clearly in figure 4 in which we show the fractional reten- tion (f, defined in eq. 5) as a function of the disk size and profile (using the parametric form shown in equation (1)). The downwards arrows indicate the upper limit on simulations in which no particle ever has its perihelia raised to a potentially stable orbit. There is a trend that larger disks tend to retain a larger fraction of particles. We note that in these various simulations we main- tained the same normalization of the surface density pro- Fig. 1.— The eccentricity (top), inclinations (middle), and per- −2 ihelion distance (bottom) for particles after 5 Myr in simulations file of the gas disk at 5 AU (Σ5 = 180g cm ). How- with no disk potential (left), and the potential of a MMSN disk ever, this arbitrary choice does not significantly impact truncated at 30 AU disk (middle) and 100 AU (right). In blue we the results. In figure 5 we show the fraction of parti- show particles stable in the compact early solar system and in dark cles retained as a function of disk mass (with the fiducial red we show particles stable in the present day solar system, while test particles on unstable orbits are in gray. depletion timescale of 2 Myr) and depletion timescale (with the fiducial disk mass of 0.05 M⊙). Other than at Regardless of the exact initial orbits of the planets, very small disk masses and very short disk lifetimes the at some point they must have moved into their cur- fraction of retained particles in remarkable insensitive to rently observed configuration. Therefore we are really these disk parameters. In order to understand this insen- only concerned with the orbital distribution of particles sitivity, we examine the timescale for a Kozai oscillation. that would be observable today due the fact that their For a<

Fig. 2.— In panel(a) we show the semi-major axis (black curve) and pericenter distance (Grey curve) for an example particle in the fiducial simulation. This particle is scattered by Jupiter and then undergoes a series of Kozai oscillations and close encounters. Eventually it is stranded in a high inclination orbit. In panel (b) we show the inclination of the particle. The shaded region indicates an example Kozai cycle. In panel (c) we show the eccentricity and argument of pericenter in a polar plot for the example particle. The small dots show these parameters at all times, while the large circles indicate quantities while the particle is in the shaded region of the left panels. tens of thousands of years, the disk mass decreased to the results are not sensitive to the other planets loca- less than a Jupiter Mass, we are insensitive to the actual tions. In these direct N-body simulation the potentially disk mass and lifetime. This means that we can safely stabilizing impact of the Additionally it is important to assume we are in a parameter space in which disk size note that the time t = 0 in these calculations corresponds and profile are important parameters, but disk mass and to the time of the formation of the giant planets, not the lifetime are not. initial time of the formation of the disk. One may also be concerned about the sensitivity of these results on our assumption about the position of 4. COMPARISON TO OBSERVATIONS the planets. There are two possible reasons the planetary In order to convert these retention fractions into a positions may be important, first the secular perturba- form suitable for comparisons with observations, we must tions from the interior planets may stabilize the parti- make a series of assumptions about the properties of cles against the Kozai resonance. Additionally, the loca- the initial population of planetesimals, specifically their tion of the planets may scatter the particles in different initial abundance, size distribution, and physical char- ways. Therefore, we perform the same calculation with acteristics. Ideally we would like these assumptions to the planets in the present-day solar system, as opposed be guided by the observed characteristics of populations to the Nice initial conditions. The fractional trapping is with the same source, unfortunately, we discuss later in virtually identical in this case (see the diamond in fig 5). this section, there is no observed population that we can We note that in the fiducial disk model, the total mass of be sure came from the same source region. Therefore the disk gas interior to 30 AU is over 13 MJup, therefore before we make a detailed comparison to observations the secular influence of the planets is minor compared to we will first present an order of magnitude estimate for that of the disk. This is because, in order to undergo a the number of high-inclination particles produced by this deep Kozai oscillation, a particle at a given semi-major mechanism. axis must either have a minimum eccentricity or inclina- Theoretical models suggest that in order to form the tion. As we are interested in using the Kozai resonance cores of the giant planets within the observed lifetime of to produce high perihelion (low eccentricity) objects, we the gaseous protoplanetary disks, the initial disk mass are interested in the initially eccentric objects. In or- must have been 2 to 5 or even 10 times larger than the der for the initially dynamically cold particles need to minimum mass solar nebula (e.g. Pollack et al. 1996; reach these minimum eccentricities, they must undergo Lissauer 1987; Thommes et al. 2003; Alibert et al. 2005; a close-encounter with a planet. After the scattering the Hubickyj et al. 2005 and see Thommes & Duncan 2006 particles orbit will have a perihelia close to the orbit of for a review). This means that fewer than half of the ini- the scattering planet (q = a(1 − e) ∼ ap). Therefore, if tial planetesimals were incorporated into the giant plan- we look at a given a, a particle scattered by a planet with ets, so they must have been ejected from the solar system a smaller aP will have a higher eccentricity than a par- or scattered into the sun. Therefore, we assume that the ticle scattered by a more distant planet. Starting with a total amount of mass in our solar system’s initial plan- low-inclination population, the planets beyond ∼ 10AU etesimal swarm must be at least equal to the mass of cannot efficiently produce particles with sufficiently high solids incorporated into the planets, ∼ 40M⊕, and is eccentricities to undergo Kozai oscillations at the semi- probably significantly larger. major axes of interest. Therefore as long as Jupiter (and It is less clear how this mass was distributed among Saturn, to a lesser extent) were not far from their present the planetesimals. Without direct evidence for the true day locations near the end of the gaseous disk’s lifetime, size distribution of these particles, we assume that the 6 Kretke et al.

(a) sharp cutoff and γ = 3/2 (b) sharp cutoff and γ = 1.

(c) exponential cutoff and γ = 1/2. (d) exponential cutoff and γ = 1.

Fig. 3.— The eccentricity and inclination of objects left in disk with disk size RD = 40, 50, 60, 80, and 100 AU (magenta stars, red diamonds, yellow squares, green hexagons, and blue circles, respectively). In the upper panels we shade the regions according to perihelia distance (q = 20, 30, 35 AU darker to lighter).

Fig. 4.— The fraction of objects left in stable high-inclination Fig. 5.— The fraction of objects left in stable high-inclination orbits after 5 Myr as a function disk size and profile for a disk orbits as a function disk mass (left), and disk lifetime (right) for with a sharp cutoff (left), and an exponential cutoff (right). The a disk with RD = 100AU and γ = 3/2. In the left-hand plot the circles, squares, and diamonds correspond to γ = 3/2, 1, and 1/2, light gray diamond indicates the results for a simulation in which respectively. The downwards arrows indicate simulations in which the scattering is done by planets on their present day orbits. no particle ever has its perihelia raised to a potentially stable orbit. this section. In general it appears that there is a rel- population has a size distribution similar to what is ob- atively steep size distribution for large particles, which served in other small body reservoirs in the outer so- then, at some size 50 . D . 150 km, rolls-over to a shal- lar system–an assumption we will revisit at the end of lower distribution (Bernstein et al. 2004; Fuentes et al. Constraining the Protosolar Nebula 7

(Schwamb et al. 2010), we can quantify this statement and make rigorous observational limits. The DES sur- vey observed over 800 deg2 with a mean limiting VR magnitude 23.6 (Elliot et al. 2005). Although this sur- vey targeted the ecliptic (±6◦), it was still sensitive to these high inclination objects during a fraction of their orbits. The Palomar survey is a shallower survey (mean limiting VR magnitude 21.3) but with a wider coverage of 12,000 deg2 up to ±40◦ ecliptic latitude. Using the a,e, and i from our population of stable particles, we 5 6 generate Ntest = 10 to 10 test particles and create a synthetic population of orbits by assigning randomized orbital angles. For each of these test particles we deter- mine the probability (pi,j ) that each particle (indexed by i) will be detected on each survey field (indexed by j) as a function of its intrinsic brightness (H). If a particle does not land on the field then pi,j (H) = 0. If it does Fig. 6.— The perihelion distance v. the inclination for all of the land on the field, we can define a critical absolute mag- simulations (blue, green, yellow, and red symbols are the same as nitude (Hcrit), a function of both the orbital properties in figure 3) compared to the Kuiper Belt objects (+ symbols) and and detection limit on the observation, centaur/scattered disk objects (x symbols) from the minor planet 2 2 center. di ri Hcrit,i,j ≡ mlimit,j − 2.5 log . (7) q(χi) 2009; Fraser & Kavelaars 2009). Particles with sizes near   In this expression d is the distance between the sun and this break dominate the mass of the whole population. i object, and r is the distance between the observer and Therefore if we assume a similar size distribution for i object (both measured in AU) and m is the back- our initial population, this corresponds to greater than limit,j ground limiting magnitude for the field. We parameterize 108 initial planetesimals of around D = 100 km in size. the phase function (q(χ )) of these small bodies with the With a capture efficiency around 10−3, we expect the i H G formalism (Bowell et al. 1989), ie. mass in our hypothetical reservoir of objects with incli- ◦ 5 nations larger than 50 to be > 0.05M⊕, or > 10 ob- q(χi)=(1 − G)φ1(χi)+ Gφ2(χi) (8) jects. This is of the same order as the upper limits on χ 0.63 the mass of the observed Kuiper Belt (Bernstein et al. φ1(χ)=exp −3.33 tan 2004; Fuentes & Holman 2008) and to date there are no 2     detected Kuiper belt objects with these inclinations. In- χ 1.22 φ (χ)=exp −1.87 tan tuitively it seems unlikely that we would have failed to 2 2 detect this massive of a population, implying that the     primordial gas disk must have had a small radial extent where χi is the phase angle and G is a parameter to avoid producing this population in the first place. describing the strength of the opposition spike (e.g. To emphasize the differences between our hypothetical Gehrels 1956; Hapke et al. 1993; Nelson et al. 1998; populations and the detected population we show the Rabinowitz et al. 2007) However due to the fact that perihelion distance versus the inclination for all stable the fields are always chosen to view objects near opposi- particles produced in all simulations as compared to the tion, even varying this parameter from G=0 to 1 changes detected KBOs in figure 6. The observed Kuiper Belt the critical value by less than 0.2 magnitudes so it does occupies a distinctly different region of parameter space not notably affect our conclusions. While the probabil- than any of the populations produced here. We note that ity of detection is a smoothed step function around this there are a few objects detected on the periphery of the critical magnitude (Hcrit), due to the uncertainties in area of parameter space that is populated by these mod- the population size distribution, a simple step function els; for instance 2005 NU125 (i = 56.5◦, a = 44.2 AU, will be adequate for these estimates. Therefore we use e = 0.026). However, fewer than 1% of particles pro- pi,j (H) = 1 if H

′ upwards triangles indicate the results if we assume a size −βH −β H −1 n(H)= n0(10 + c10 ) , (10) distribution consistent with the hot population (our pre- ferred model), while the open downwards triangles indi- where β is the slope on the bright end, β′ is the slope cate the limits assuming the size distribution of the cold on the faint end, and c is chosen so that the break be- population. The error bars here are the statistical error tween these two populations matches observations. Ob- bars (propagated through from figure 4). For disks that servations indicate at least two distinctly different pop- do not produce stable high-inclination particles we can ulations in the outer solar system. The dynamically hot not place any limits, and these disks are indicated with Kuiper belt objects (KBOs) have a fairly shallow slope upwards arrows. to their bright-end distribution, while the dynamically If we assume that the size distribution of scattered cold KBOs and the Jupiter Trojans have a steeper size planetesimals was similar to that of the hot population distribution. The different origins of these size distribu- then we can rule out 80 and 100 AU disks as the limit on tions are not well understood. There is evidence that the initial population mass is below the expected theo- the hot population may have accumulated closer to the retical amount. Indeed, for disks with shallower profiles sun (Levison & Stern 2001; Gomes 2003; Levison et al. we can marginally rule out 60 AU disks with this size dis- 2008), where the protoplanetary disk had shorter accre- tribution. However, if we relax this assumption on the tion times. For this reason we strongly prefer the as- size distribution then we cannot place as stringent limits sumption that the luminosity distribution of our hypo- on the disks with steeper surface density profiles. thetical population is similar to that of the hot popula- In order to significantly improve these constraints we tion, but we consider both for completeness. We revisit would need a survey with a limiting magnitude similar to this line of reasoning at the end of this section. that of the DES (r=23.5) but with an area comparable When looking at the bright end slope, Fraser et al. to the Palomar survey (10,000 deg2). A non-detection in (2010) found different values for this power-law index β this type of survey would allow us to rule out all disks in different populations, βcold = 0.8, and βhot = 0.35. larger than 60 AU, except the steep exponential profile. However, the data cannot constrain the shallow end The LSST, designed to cover 10,000 deg2 with a typi- slope. Bernstein et al. (2004) found a break in both the cal limiting magnitude of r=24.5 (Ivezic et al. 2008) will hot and cold populations. We include this break in the easily exceed this limit, allowing us to make meaningful ′ cold population and use βcold =0.38. However, we follow constraints down to the 50 AU level. Fraser et al. (2010) and do not change the slope of the ′ 4.1. Discussion of Size Distribution hot population after the “break” so βhot =0.35. We assume that particles extend from 30 km to 1000 In this paper we argue that the non-detection of this km in size with a break at D = 100 km. Additionally, hypothetical high-inclination population can be used to we assume that the particle size distribution extends to constrain the size of the protosolar nebula. Of course, 1 km for the estimates of the mass, but we note that another possibility is that the population does exist, but the calculation is insensitive to this choice as most of the that it has such a steep size-distribution that most of mass lies in larger particles. To convert the number of the mass is in bodies well below 100 km in size, and expected detection into an estimate of the mass in this thus undetectable. There are three main reasons why we population we assume a bond albedo p = 0.05 and a believe this is unlikely. density of ρ =2 gcm−3. First, there is another potential reservoir in which some Finally, we can create a synthetic population and use remnants from this same initial population of planetes- the previously described method to calculate the ex- imals are believe to have been stored: the Sedna popu- pected number of detections. For example, in our fiducial lation. The 1,000 km Sedna is so far unique in that it disk model, RD = 100 AU, γ = −1.5, and Θ=Θcut the has a perihelion distance of 76 AU, well detached from fractional trapping efficiency is greater than 2 × 10−3. If Neptune, but it has a semimajor axis around 500 AU, the scattered particles have the same size distribution of well interior to the main Oort cloud (Brown et al. 2004). the hot population, we expect the two surveys to have The observational difficulties in detecting an object on been able to detect ∼ 10−6 of the scattered objects. Now, this type of orbit implies there must be on the order of if we assume that the initial population of planetesimals 100 similar sized planetoids on similar orbits. The best was 40 M⊕, then, using the same size distribution, there dynamical mechanism to produce this type of orbit is would have been initially 108 planetesimals greater that to have an object scattered by a planet very early on D & 30 km. Therefore if our solar system had had a 100 in the solar system (at the same time considered in this AU disk gaseous disk we would expect the these two sur- paper) so that its pericenter can be lifted via an interac- Constraining the Protosolar Nebula 9

cycles excited by the presence of the gaseous protoplan- etary disk. If our own solar system’s disk had been ra- dially extended (RD & 50 AU) a significant number of scattered planetesimals would have been placed onto high inclination orbits which would be stable for the lifetime of the solar system. We find that the current observed lack of objects on these orbits in the DES and Palomar Distance Solar System Survey suggests that we can ex- clude very extended disks (& 80 AU) so long as we are correct in our assumption of a size distribution more sim- ilar to the hot KBOs than the cold KBOs. We argue that while we cannot technically rule out other options such Fig. 7.— Observational constraints on the upper limit on the ini- as that the planetesimals in the planet forming region tial mass of planetesimals in the planet forming region. As in figure were orders of magnitude less numerous than expected 4, the left panel is for sharp cutoff disks and the right panel is for by theory or that the size distribution was dominated by disks with exponential profiles. The color indicates the disk surface density profile (On the left, γ = 1, 3/2 in blue and black, respec- small planetesimals, we believe these options to be quite tively. On the right, γ = 1/2, 1 in blue and black, respectively.) unlikely. Upwards triangles indicate results for a shallow size distribution Future observations from deep all sky surveys (such consistent with the hot population, while the downwards triangles indicate the steeper profile consistent with the cold population (see as the LSST) will allow us to place more stringent con- text). Arrows indicate disk parameters for which we cannot limit straints on the size of the disk. With this type of sur- the initial mass. vey, if we are able to detect a population of Kuiper belt ◦ tion with the birth cluster potential or by the passage of objects with i ∼ 60 the orbital distribution of these nearby stars (Morbidelli & Levison 2004; Brasser et al. particles will reveal information on the shape of the pro- 2006). The large predicted number of 1,000 km-sized ob- toplanetary disks, especially revealing information about jects implies a shallow size distribution, and constraints the truncation. However, it is also possible (and perhaps from the TAOS survey indicate that this population must likely) that future observations will not reveal this pop- have β < 0.88(Wang et al. 2009), i.e. the size distribu- ulation. If this is the case we will be able to conclude tion cannot be much steeper than that of the cold popu- that the disk must have been smaller than 50 AU, but lation. we will not be able to place more stringent limits because Second, in the Kuiper belt where we can directly mea- if the disk had been smaller than 50 AU, the dynamical sure the size distributions, we see that the hot population mechanism presented in this paper cannot produce stable has a shallower size distribution than the cold popula- particles. tion. As mentioned previously, there is evidence that This method if the first technique to constrain the size the hot population may have accumulated closer to the of the gaseous protoplanetary disk of a known plane- sun (Levison & Stern 2001; Gomes 2003; Levison et al. tary system during the time of planet formation. The 2008), where the protoplanetary disk had shorter accre- disk sizes we find consistent with observations implies tion times. Therefore we would expect that the source that in some ways our solar system is typical. Although region for our hypothetical population, being even closer many of the first protoplanetary disks observed were to the sun than the hot population, may have a shallower quite radially extended, smaller disks with sizes below 60 population. At least it seems unlikely to be significantly AU are quite common according to recent observations steeper. (Andrews et al. 2010). Additionally, this small disk size Finally, we can be confident that the population that is consistent with other pictures of the early solar system. produced the comets could not have had a very steep size This work is consistent with the idea that a sharp cut- distribution. If it had, then mutual collisions would have off in the planetesimal disk at around 50 AU might have ground the comets down before they could have been corresponded to a sharp cut off in the gas disk. Future scattered into the Oort Cloud (Stern & Weissman 2001). deep all-sky surveys will allow us to probe closer to this Charnoz & Morbidelli (2003) demonstrated that this col- 50 AU radius and allow us to meaningfully constrain on lisional grinding problem can be avoided by assuming a the theoretical models of of solid-gas interactions during shallower size distribution up through 10s to 100s of km, the time of the formation of the solar system. such as the size distributions used in this paper. While the Oort cloud likely has a source region exterior to the population considered here (the Nice disk, Tsiganis et al. 2005) or even from other stars (Levison et al. 2010), the same argument holds for the Sedna region. For these reasons we believe that the size distribution of this pop- ulation are unlikely to have a steeper distribution than the cold population and very likely have been similar to, or even more shallow than, the hot population. 5. CONCLUSIONS AND DISCUSSION We thank M. Schwamb for guidance in our comparison In this paper we present a dynamical mechanism by with the Palomar Distant Solar System Survey. KAK which high inclination Kuiper Belt objects can be pro- and HFL are grateful for funding from NASA’s Origins duced during the planet formation process due to Kozai program. 10 Kretke et al.

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