MAKING PLANET NINE: A SCATTERED GIANT IN THE OUTER SOLAR SYSTEM

To set the stage for relocating a scattered planet from 5–15 au into the outer solar system, we model the Sun's gas disk with the following prescription for surface density Σ, scale height H, and midplane mass density ${\rho }_{{}_{{\rm{gas}}}}$:

$$\begin{eqnarray}{\rm{\Sigma }}(a,t)=\left\{\begin{array}{ll}{{\rm{\Sigma }}}_{{}_{0}}{\left(\tfrac{a}{{a}_{{}_{0}}}\right)}^{-1}{e}^{-t/\tau }, & \mathrm{if}\ {a}_{{}_{{\rm{in}}}}\leqslant a\leqslant {a}_{{}_{{\rm{out}}}},\\ 0 & \mathrm{otherwise},\end{array}\right.\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&H(a)={h}_{{}_{0}}a{\left(\displaystyle \frac{a}{{a}_{{}_{0}}}\right)}^{2/7},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{\rho }_{{}_{{\rm{gas}}}}(a,t)=\displaystyle \frac{{\rm{\Sigma }}(a,t)}{H(a)}\,.\end{eqnarray} \tag{ 3 }$$

Here, ${{\rm{\Sigma }}}_{{}_{0}}$ sets the surface density at distance ${a}_{{}_{0}}\equiv 1\,{\rm{au}}$, ${a}_{{}_{{\rm{in}}}}$ and ${a}_{{}_{{\rm{out}}}}$ are the inner and outer edges of the disk, and ${h}_{{}_{0}}=0.05$ establishes the scale height of the flared disk (Kenyon & Hartmann 1987; Chiang & Goldreich 1997; Andrews & Williams 2007; Andrews et al. 2009). The global surface density decay parameter $\tau =1$–10 Myr enables a homologous reduction in the surface density (Haisch et al. 2001). To allow the inner edge of the disk to expand as in a transition disk, we adopt an expansion rate ${\kappa }_{{}_{{\rm{in}}}}$:

$$\begin{eqnarray}&&{a}_{{}_{{\rm{in}}}}(t)={a}_{{}_{{\rm{in}}}}(0)+{\kappa }_{{}_{{\rm{in}}}}t,\end{eqnarray} \tag{ 4 }$$

where the initial size of the inner cavity is ${a}_{{}_{{\rm{in}}}}(0)\equiv 20\,{\rm{au}}$. Observations of transition disks (Calvet et al. 2005; Currie et al. 2008; Andrews et al. 2011; Najita et al. 2015) suggest opening rates of O(10) au Myr⁻¹.

Toward estimating dynamical friction and gas drag, we assume that the sound speed in the gas is ${c}_{{\rm{s}}}\approx {{Hv}}_{{}_{{\rm{Kep}}}}/a$, where ${v}_{{}_{{\rm{Kep}}}}$ is the circular Keplerian speed at orbital distance a from the Sun. We also assume that both H and ${c}_{{\rm{s}}}$ are independent of time. Armed with these variables, we can determine the Mach number of a planet moving relative to the gas, and hence derive drag forces on the planet. These estimates include the effect of pressure support within the gas disk, which makes the bulk flow in the disk sub-Keplerian, with an orbital speed that is reduced from ${v}_{{}_{{\rm{Kep}}}}$ by a factor of $(1-{H}^{2}/{a}^{2})$ (e.g., Adachi et al. 1976; Weidenschilling 1977a; Youdin & Kenyon 2013).

Table 1 lists disk model parameters. We distinguish two types of disks: static and evolving. Static disks have fixed surface density profiles and small total mass, $0.002{M}_{\odot }\lt {M}_{{}_{{\rm{disk}}}}\lt 0.06{M}_{\odot }$ (2–60 ${M}_{{\rm{J}}}$, where ${M}_{{\rm{J}}}$ is the mass of Jupiter), which extend to 1600 au. These models enable us to consider the possibility of long-term (≳100 Myr) planet–disk interactions. Evolving disks extend to 800 au with larger initial surface density and mass. In the most extreme case (${{\rm{\Sigma }}}_{{}_{0}}=1000{\rm{g}}\,{\mathrm{cm}}^{-2}$), the disk mass is half that of the Sun. Although improbable (see, e.g., Andrews et al. 2013), this extreme disk mass allows us to explore the possibility of rapid, strong orbital damping. Because dynamical friction depends on the gas density, ${\rho }_{{}_{{\rm{gas}}}}\sim {\rm{\Sigma }}/H$, we can scale results to less massive disks with smaller scale heights.

Table 1.  Disk Parameters

Name	Symbol	Value or Range	Units
Radial length scale	a0	1	au
Scale height factor	${h}_{{}_{0}}$	0.05	⋯
static disk
Surface density	${{\rm{\Sigma }}}_{{}_{0}}$	2, 10, 20, 50	g cm−2
Initial inner edge	${a}_{{}_{{\rm{in}}}}$	50, 100, 200	au
Outer edge	${a}_{{}_{{\rm{out}}}}$	1600	au
evolving disk
Surface density	${{\rm{\Sigma }}}_{{}_{0}}$	50, 100, 200, 500, 1000	g cm−2
Initial inner edge	${a}_{{}_{{\rm{in}}}}$	20, 60, 100, 140, 180	au
Outer edge	${a}_{{}_{{\rm{out}}}}$	800	au
Opening rate (${\dot{a}}_{{}_{{\rm{in}}}}$)	${\kappa }_{{}_{{\rm{in}}}}$	20, 40, 60, 80	au Myr−1
decay time	τ	2, 4, $\infty$	Myr

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Our evolving model disks have large radial extent (800 au) to enhance the interaction between the gas and a scattered planet. This choice is consistent with observations of disks in young stellar associations where measured radii are commonly in the range of 400–1000 au (e.g., Andrews et al. 2009; Guilloteau et al. 2013). However, the solar system's protoplanetary disk may have been truncated from a stellar encounter if—as radiometric evidence suggests (Adams & Laughlin 2001)—the Sun formed in a more populous star cluster (Clarke & Pringle 1993; Ostriker 1994; Kenyon & Bromley 2004; Malmberg et al. 2011; Muñoz et al. 2015; Portegies Zwart 2016). Yet cluster membership does not guarantee disk truncation. Vincke et al. (2015) demonstrate that 15%–50% of extended (${a}_{{}_{{\rm{out}}}}\sim 1000$ au) protoplanetary disks can survive cluster dispersal, depending on cluster properties and the host star's location. Even if the Sun's disk was truncated, our models might still apply to a disk that had little or no flaring. Then, the central gas density dropped off less steeply with radius, enhancing the disk–planet interactions in the outer regions and compensating for a reduction in its outer radius.

Compared with the evolving models, our static disks span an even larger radial distance (1600 au), but with low surface densities and total masses that are as small as a fraction of Jupiter's mass. No such long-lived, low-mass disks are observed; however, they are challenging to detect (Andrews et al. 2009; Dent et al. 2013; Andrews 2015). Long-lived disks must survive the hard photons and winds from massive stars in the Sun's birth cluster (e.g., Adams et al. 2004; Desch 2007; Gaidos et al. 2009). Disk accretion—hot gas flowing onto the young Sun—also generates radiation that photoevaporates the disk over a wide range of orbital distances (e.g., Gorti et al. 2009, 2015; Alexander et al. 2014). Nonetheless, after the Sun leaves its birth cluster and its inner disk erodes, thereby quenching hard photons from disk accretion (e.g., Haworth et al. 2016), a long-lived reservoir of cool gas might remain. The static disk models reflect this possibility, in addition to offering a simple testbed to understand disk–planet interactions.

A planetary core scattered from the gas giant region typically undergoes a complex set of encounters with competing cores and growing giant planets (e.g., Thommes et al. 1999; Chatterjee et al. 2008; Izidoro et al. 2015). The onset of scattering occurs as planets migrate or their Hill spheres expand from accretion, causing orbits to cross. A smaller body flung on an eccentric orbit can return to experience additional strong interactions with a growing gas giant, typically leading to ejection or a merger. However, these fates may be avoided if an outwardly scattered planet's perihelion is raised by other influences such as interactions with gas (Marzari et al. 2010; Moeckel & Armitage 2012), planetesimals (Levison et al. 2008), or other cores (Ford & Chiang 2007). Similarly, the orbit of the larger scatterer can also change, preventing subsequent close encounters (e.g., Cresswell & Nelson 2008).

While previous work on orbital evolution in planet–planet scattering has mostly focussed on dynamics within the gas giant region, we expect that the long orbital period of scattered cores of interest here will lessen the likelihood of a repeated scattering. Our own simulations of gas giant formation (Bromley & Kenyon 2011), which track chaotic interactions between multiple growing cores, confirm that super-Earth-mass planets can scatter to the outer solar system on eccentric orbits and remain there for millions of years or longer without being ejected.

Here, we model only the final outcome of the scattering process, simulating planets scattered to large ($\gt 1000$ au) distances for each disk configuration in Table 1. As summarized in Table 2, each planet is assigned a mass ${m}_{\bullet }$ and mean density ${\rho }_{\bullet }$, from which we infer a physical radius, ${r}_{\bullet }$. In our orbital dynamics code, the planet is launched from a perihelion distance of ${r}_{{}_{\mathrm{peri}}}=10\,{\rm{au}}$ with a speed that would take it to a specified aphelion distance ${r}_{{}_{\mathrm{apo}}}$ if it were on a Keplerian orbit about the Sun. We track the subsequent dynamical evolution with orbital elements calculated geometrically, since the disk potential can complicate the interpretation of osculating orbital elements. While the planet's orbit typically comes close to the nominal starting aphelion, it never makes it to ${r}_{{}_{\mathrm{apo}}}$ exactly due to dynamical friction with the gas and the disk's overall gravitational potential.

Table 2.  Planet Parameters

Name	Symbol	Value or Range	Units
Mass	${m}_{\bullet }$	1, 5, 10, 15, 20, 30, 50	${M}_{\oplus }$
Mean density	${\rho }_{\bullet }$	1.33	g cm−3
Initial perihelion	${r}_{{}_{\mathrm{peri}}}$	10	au
Initial aphelion	${r}_{{}_{\mathrm{apo}}}$	1600, 2000, 2400, ..., 3600	au
Inclination	i	0	rad

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To evolve planetary orbits in a gas disk, we follow Bromley & Kenyon (2014). We use the orbit integrator in our hybrid n-body-coagulation code Orchestra to calculate the trajectory of individual planets around the Sun in the midplane of the disk. We calculate disk gravity using 2000 radial bins spanning the planet's orbit, assigning a mass to each bin according to Equation (1). We initially solve the Poisson equation by numerical integration, storing the results. The saved potential is updated as the disk evolves.

To estimate acceleration from dynamical friction, we adopt a parameterization similar to Lee & Stahler (2014) in the absence of gas accretion (see also Dokuchaev 1964; Ruderman & Spiegel 1971; Ostriker 1999):

$$\begin{eqnarray}&&\displaystyle \frac{d{\boldsymbol{v}}}{{dt}}=-\displaystyle \frac{{G}^{2}{\rho }_{{}_{{\rm{gas}}}}{m}_{\bullet }\mu }{{c}_{{\rm{s}}}^{2}}\displaystyle \frac{{(1+4{\pi }^{2}{C}_{{}_{{\rm{df}}}}^{2}{\mu }^{2})}^{1/2}}{{(1+{\mu }^{2})}^{2}}\displaystyle \frac{{{\boldsymbol{v}}}_{{}_{{\rm{rel}}}}}{| {{\boldsymbol{v}}}_{{}_{{\rm{rel}}}}| },\end{eqnarray} \tag{ 5 }$$

where ${{\boldsymbol{v}}}_{{}_{{\rm{rel}}}}$ is the planet's velocity relative to the gas, $\mu \equiv | {{\boldsymbol{v}}}_{{}_{{\rm{rel}}}}| /{c}_{{\rm{s}}}$ is the mach number, and ${C}_{{}_{{\rm{df}}}}$ is a constant that depends on the geometry of the disk in the plane perpendicular to the planet's motion.

In evaluating the coefficient ${C}_{{}_{{\rm{df}}}}$, we previously only considered contributions from gas more distant than $H/2$ from a planet (Bromley & Kenyon 2014). Here we are less restrictive and include contributions from material closer to the planet. The drag acceleration thus has a piece from distant disk material in slab geometry (Bromley & Kenyon 2014), along with a Coulomb logarithm (e.g., Binney & Tremaine 2008):

$$\begin{eqnarray}&&{C}_{{}_{{\rm{df}}}}\approx 0.31+\mathrm{ln}\left[\max \left(1,\displaystyle \frac{H}{2R}\right)\right],\end{eqnarray} \tag{ 6 }$$

where the radius R is

$$\begin{eqnarray}&&R\equiv \max ({r}_{\bullet },{R}_{{\rm{s}}})\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&{R}_{{\rm{s}}}=\displaystyle \frac{1}{{\mu }_{x}^{2}-1}\displaystyle \frac{2{{Gm}}_{\bullet }}{{{cs}}^{2}}\ \ \ \ \ [{\mu }_{x}^{2}=\max ({\mu }^{2},1.0001)]\end{eqnarray} \tag{ 8 }$$

is an effective sonic radius (e.g., Thun et al. 2016).

With this formulation, our goal is to map how scattered planets with large eccentricities ($e\lesssim 1$) damp to modest values ($e\sim 0.5$) when the planet moves at supersonic speeds. Once the planet achieves low eccentricity, its subsequent evolution is complicated by differential torque exchange with the disk (Goldreich & Tremaine 1980; Ward 1997) and accretion of disk material (Hoyle & Lyttleton 1939; Lee & Stahler 2011). We do not attempt to track this behavior. Our prescription underestimates dynamical friction in the subsonic and transonic regimes (see, e.g., Ostriker 1999). Thus, we follow a planet as its orbit damps, but stop the integration if it manages to fully circularize before the gas disappears.

While the main focus here is on how a scattered planet can settle beyond 300 au, we also check to see how this process impacts objects beyond the gas giant region. In several scattered planet simulations, we include 40,000 tracer particles set up on circular, coplanar orbits with semimajor axes between 20 and 80 au. For computational speed, we ignore the effects of gas on the tracers; we simplify the calculation of the planet's orbital evolution by requiring that the semimajor axis and eccentricity change linearly in time from specific starting conditions to final values that are consistent with the full disk–planet calculations. Experiments with more realistic parameterizations for a and e show that properties of the final tracer orbits are most sensitive to the average changes in a and e, and not the details of the planet's orbital evolution.

In this set of simulations, we set up static disks with low surface density (${{\rm{\Sigma }}}_{{}_{0}}=2$–50 g cm⁻²), large radial extent (${a}_{{}_{{\rm{out}}}}=1600\,\mathrm{au}$), and big inner cavities (${a}_{{}_{{\rm{in}}}}=50$–200 au). We evolve scattered planets over a time t = 100 Myr. Figure 1 shows several outcomes with planets of different masses. All planets follow the same track in a–e space as they circularize; more massive planets evolve further along the track.

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Figure 2 illustrates how evolution depends on the disk configuration. The plot shows three separate evolutionary tracks in a–e space, each corresponding to a different inner edge of the disk (${a}_{{}_{{\rm{in}}}}$). With a smaller inner edge, the disk causes a planet to settle more quickly (because there is more disk material to interact with) and closer to the Sun. The markers in the plot designate how far each planet evolves. Planets in low surface density disks make less progress along their track than those in high surface density disks.

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These calculations establish an approximate degeneracy between mass and the surface density parameter ${{\rm{\Sigma }}}_{{}_{0}}$ in the formula for dynamical friction acceleration (Equation (5)). As a result, the progress that a planet makes along its a–e track in a fixed amount of time depends only on the product ${\text{}}{m}_{\bullet }\times {\text{}}{{\rm{\Sigma }}}_{{}_{0}}$. Thus, data points showing the orbital evolution as a function of planet mass in Figure 1 can also represent the progress of a planet of fixed mass in disks with different surface densities. Similarly, points in Figure 2 can represent outcomes with different planet masses at fixed surface density.

If long-lived disks are responsible for settling a planet on the type of orbit inferred by Batygin & Brown (2016), then our suite of simulations suggests the following condition leads to successful Planet Nine-like outcomes:

$$\begin{eqnarray}&&\left(\displaystyle \frac{{{\rm{\Sigma }}}_{{}_{0}}}{10\,{\rm{g}}\,{\mathrm{cm}}^{-2}}\right)\left(\displaystyle \frac{{m}_{\bullet }}{20\,{M}_{\oplus }}\right)\approx {\left(\displaystyle \frac{{a}_{{}_{{\rm{in}}}}}{100\mathrm{au}}\right)}^{1/2}{\left(\displaystyle \frac{t}{100\mathrm{Myr}}\right)}^{-1}\,.\end{eqnarray} \tag{ 9 }$$

This expression applies as long as ${a}_{{}_{{\rm{out}}}}\gg {a}_{{}_{{\rm{in}}}}$. While it is only approximate, this relation suggests that a persistent, low-mass disk (${{\rm{\Sigma }}}_{{}_{0}}\approx 0.3$ g cm⁻²; about a quarter of a Jupiter mass in gas) can modestly damp a scattered Neptune-size body within the age of the solar system.

Observations indicate that the youngest stars are surrounded with opaque disks of gas and dust (see Kenyon & Hartmann 1995; Kenyon et al. 2008; Williams & Cieza 2011; Andrews 2015). Surface densities vary; the "Minimum Mass Solar Nebula" value of ${{\rm{\Sigma }}}_{{}_{0}}\,\approx 2000$ g cm⁻² (Weidenschilling 1977b; Hayashi 1981) is at the upper end of the range observed in the youngest stars (e.g., Andrews et al. 2013; Najita & Kenyon 2014). These disks globally dissipate on a timescale, τ, of millions of years (Haisch et al. 2001), and may also erode from the inside out at a rate of ${\kappa }_{{}_{{\rm{in}}}}\gtrsim O(10)$ au Myr⁻¹, as in transition disks (e.g., Andrews et al. 2011; Najita et al. 2015). The resulting behavior of a scattered planet as it settles depends sensitively on how mass is distributed in these disks as they evolve.

Figure 3 illustrates the dependence of planetary settling on disk parameters τ, ${\kappa }_{{}_{{\rm{in}}}}$, the initial disk surface density, and the inner disk edge. Adopting a baseline model where (${{\rm{\Sigma }}}_{{}_{0}}$, ${a}_{{}_{{\rm{in}}}}$, ${\kappa }_{{}_{{\rm{in}}}}$,τ) = (1000 g cm⁻², 60 au, 40 au Myr⁻¹, 4 Myr), we vary individual disk parameters for planets with masses of 15–30 ${M}_{\oplus }$, scattered to starting distances of ${r}_{{}_{\mathrm{apo}}}=2000$–2800 au. The general trends are clear. Dynamical settling to small orbital distance and low eccentricity is more effective in massive, slowly evolving disks with small inner cavities.

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Figure 4 summarizes the outcomes of all of our evolving disk models (see Tables 1 and 2). The trends that emerged in Figure 3 are apparent: long-lived, massive disks lead to significant dynamical evolution, while short-lived, low-mass disks do not. Figure 4 also shows an extended "sweet spot" in a–e space, labeled with "Planet Nine," roughly corresponding to orbital elements of the massive perturber hypothesized by Batygin & Brown (2016) and Brown & Batygin (2016). Several hundred models yield planets that lie in the sweet spot, suggesting that the scattering mechanism can explain the inferred orbit of Planet Nine in the outer solar system.

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Despite the trends revealed in Figures 1–3, it is difficult to tell which set of model parameters leads to successful Planet-Nine-like orbits. To distinguish models in a way that highlights successful ones, we define two variables:

$$\begin{eqnarray}&&P\equiv \left(\displaystyle \frac{{m}_{\bullet }}{10\,{M}_{\oplus }}\right)\left(\displaystyle \frac{{{\rm{\Sigma }}}_{{}_{0}}}{1000\,{\rm{g}}\,{\mathrm{cm}}^{-2}}\right){\left(\displaystyle \frac{{r}_{{}_{\mathrm{apo}}}}{1000\mathrm{au}}\right)}^{-1}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&Q\equiv \left(\displaystyle \frac{\tau }{1\,\mathrm{Myr}}\right){\left[\left(\displaystyle \frac{{\kappa }_{{}_{{\rm{in}}}}}{60\mathrm{au}{\mathrm{Myr}}^{-1}}\right)\left(\displaystyle \frac{{a}_{{}_{{\rm{in}}}}}{20\mathrm{au}}\right)\left(\displaystyle \frac{{r}_{{}_{\mathrm{apo}}}}{1000\mathrm{au}}\right)\right]}^{-1}.\end{eqnarray} \tag{ 11 }$$

Roughly, the first variable is a mass-dependent damping rate, determined by planet mass and the disk mass, along with a factor of $1/{r}_{{}_{\mathrm{apo}}}$ that reduces this rate if the planet is launched further away from the Sun. The second one measures the disk lifetime, based on the global disk decay time and the time for the disk to clear from the inside out, along with geometric factors involving the disk's radial extent and the planet's initial orbit. For models where τ is formally infinite, we set $\tau =10\,{\rm{Myr}}$, the simulated duration of the evolving disk models.

The variables P and Q help to isolate the parameters necessary for settling a scattered giant planet on a Planet-Nine-like orbit. Our choice for defining these quantities, along with the mass, length, and timescales in Equations (10) and (11), are based more on simplicity than anything else; other combinations of model parameters may serve the same purpose. Nonetheless, our choice yields a nicely compact region in P–Q space for models that succeed in matching Adams & Laughlin (2001) criteria for Planet Nine.

Figure 5 shows a swath of points in the P–Q plane that correspond to successful models. These points have just the right balance between the masses of the disk and the planet (P), on the one hand, and disk lifetime (Q) on the other. Models without this balance tend to produce planets that circularize at small semimajor axes (high P and high Q; large masses and long disk lifetimes) or remain highly eccentric at large semimajor axes (low P and low Q; small masses and short disk lifetimes).

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A rough quantitative relationship between P and Q for successful models is $Q\sim {P}^{-3/2};$ in terms of model parameters, this condition translates to

$$\begin{eqnarray}\left(\displaystyle \frac{{m}_{\bullet }}{1\,{M}_{\oplus }}\right)\left(\displaystyle \frac{{{\rm{\Sigma }}}_{{}_{0}}}{1000\,{\rm{g}}\,{\mathrm{cm}}^{2}}\right) & \approx & {\left(\displaystyle \frac{{\kappa }_{{}_{{\rm{in}}}}}{40\mathrm{au}{\mathrm{Myr}}^{-1}}\right)}^{2/3}{\left(\displaystyle \frac{{a}_{{}_{{\rm{in}}}}}{40\mathrm{au}}\right)}^{2/3}\\ & & \times {\left(\displaystyle \frac{\tau }{1\mathrm{Myr}}\right)}^{-2/3}{\left(\displaystyle \frac{{r}_{{}_{\mathrm{apo}}}}{1000\mathrm{au}}\right)}^{-1/3},\end{eqnarray} \tag{ 12 }$$

from which the inverse relationship between disk lifetime and mass factors is apparent, as is the sensitivity to disk and orbit geometries.

These simulations suggest a broad range of outcomes in a–e space for 1–50 ${M}_{\oplus }$ planets scattered from 10 au into the outer part of a gaseous disk. For many combinations of input parameters, planets remain on $e\gtrsim 0.80$ orbits at $a\gtrsim 400\,\mathrm{au}$. Another set of parameters yields massive planets on nearly circular orbits at 100–200 au from the host star. Specific combinations of disk and planet properties result in "successful models," with planets on orbits consistent with the massive perturber of Batygin & Brown (2016).

• 1.  
  A planet scattered at low inclination into a low-mass, long-lived disk damps at a rate proportional to the planet mass and the disk's surface density. The final semimajor axis depends on ${a}_{{}_{{\rm{in}}}}$, the inner radius of the disk; successful models have ${a}_{{}_{{\rm{in}}}}\gtrsim 50\,{\rm{au}}$. A power-law disk (${\rm{\Sigma }}\sim 1/a$) with low surface density (${\rm{\Sigma }}\sim 3\times {10}^{-3}$g cm⁻² at 100 au, extending to ∼1000 au) can produce a Neptune-size Planet Nine on a moderately eccentric orbit within the lifetime of the solar system. Higher planet masses or larger surface densities lead to success in less time.
• 2.  
  Scattering in a massive, short-lived disk leads to Planet-Nine-like orbits when there is a balance between the damping rate and the disk evolution timescale. In successful models, planet masses are typically 10 ${M}_{\oplus }$ or more, although 5 ${M}_{\oplus }$ planets can acquire a Planet-Nine-like orbit in the most massive, long-lived disks. Most successful models experience either slow global decay ($\tau =4\,\mathrm{Myr}$) or none at all. The disk then evolves primarily through inside out erosion, as in a transition disk. This feature helps successful planets settle at large semimajor axes.
• 3.  
  In all of our simulations, successful models tend to have semimajor axes that lie within a = 600 au. Batygin & Brown's preferred model has a higher orbital distance, with $a\approx 700\,{\rm{au}}$ (see also Brown & Batygin 2016; Malhotra et al. 2016). While their analysis accommodates a wide range of possibilities, our current models do not. If a massive perturber were to have a semimajor axis firmly established beyond 700 au, our mechanism would require a disk with more mass beyond a few hundred au.
• 4.  
  In successful models, the scattered planet's perihelion sweeps through the Kuiper Belt. Representative simulations with tracers in a cold disk configuration explore the impact of the planet on Kuiper Belt objects or their precursors. Planets with mass as high as 30 ${{\rm{M}}}_{\oplus }$, scattered to 1600 au and settling at a = 300 au and e = 0.4 in 10 Myr leave at least 90% of the tracers on low eccentricity orbits, with $e\lt 0.05$. Slower evolution stirs up more particles; a 30 ${{\rm{M}}}_{\oplus }$ planet that evolves in a and e at half the rate leaves only about one-third of the tracers with $e\lt 0.05$. However, in this case, realistic predictions require calculations that include gas drag and collisional damping between small solids, both of which will cool the disk. We conclude that a scattered Planet Nine can settle while preserving a population of objects like the cold classical Kuiper Belt.