Making Terrestrial Planets: High Temperatures, FU Orionis Outbursts, Earth, and Planetary System Architectures

TL;DR

This paper explores how high-temperature events like FU Orionis outbursts influence the formation of dry silicate planetesimals and the resulting architectures of planetary systems, addressing current gaps in planet formation theory.

Contribution

It proposes molten grain collisions as a solution to dry silicate planetesimal formation and links accretion events to planetary system configurations.

Findings

Molten grain collisions may produce dry silicate planetesimals.

FU Orionis outbursts influence planetesimal location and system architecture.

Systems with FU Orionis events resemble our Solar System.

Abstract

Current protoplanetary dust coagulation theory does not predict dry silicate planetesimals, in tension with the Earth. While remedies to this predicament have been proposed, they have generally failed numerical studies, or are in tension with the Earth's (low, volatility dependent) volatile and moderately volatile elemental abundances. Expanding on the work of Boley et al. (2014), we examine the implications of molten grain collisions and find that they may provide a solution to the dry silicate planetesimal problem. Further, the source of the heating, be it the hot inner disk or an FU Orionis scale accretion event, would dictate the location of the resulting planetesimals, potentially controlling subsequent planetary system architectures. We hypothesize that systems which did undergo FU Orionis scale accretion events host planetary systems similar to our own, while ones that did not instead host very close in, tightly packed planets such as seen by Kepler.

Equations

$$\partial_{t}\mbox{\boldmath$v$}{}=-\frac{\mbox{\boldmath$v$}{}-\mbox{\boldmath$u$}{}}{\tau},$$

$$\text{St}\equiv \tau {\Omega }_K.$$

$${\text{St}}_0=\frac{\pi a{\rho }_{\bullet }}{2{\Sigma }_g},$$

$${\text{St}}_0=\frac{2}{9}\frac{{\rho }_{\bullet }{a}^2}{m_g}\frac{{\sigma }_g}{H},$$

$$v_{th}=⟨v⟩=\sqrt{\frac{8k_{B}T}{\pi m_g}}$$

$$H=\sqrt{\frac{\pi }{8}}\frac{v_{th}}{{\Omega }_K}$$

$$\delta v=\frac{\pi }{16}s\frac{v_{th}^2}{R}{\Omega }_K^{-1}\simeq 50{\text{ m s}}^{-1}.$$

$$v_r\simeq 2\text{St}\delta v\simeq \text{St}\times 10^4{\text{ cm s}}^{-1},$$

$${\tau }_R=\frac{R}{v_r}\simeq \frac{R}{2St\delta v}=\frac{1}{2\pi s\text{St}}\frac{R^2}{H^2}\text{Orb},$$

$$v_b$$

$$\simeq f\times 3\times 10^{-4}{\tilde{R}}^{5/4}{\text{St}}_2^{-5/6}{\text{ cm s}}^{-1},$$

$$v_0\simeq \sqrt{\alpha \text{St}}{c}_s=38\sqrt{{\alpha }_5{\text{St}}_2}{\tilde{R}}^{-1/4}{\text{ cm s}}^{-1}.$$

$$S\simeq \sqrt{2\pi }n{a}^2{v}_0{\left(\frac{v_b}{v_0}\right)}^4,$$

$$n=\frac{ϵ{\rho }_g}{m}.$$

$$\sqrt{2\pi }n{a}^2{v}_0{\left(\frac{v_b}{v_0}\right)}^4=\frac{v_r}{R},$$

$$f\simeq 6.6\times 10^4\times {\text{St}}_2^{41/24}{\alpha }_5^{3/8}{\widetilde{ϵ}}^{-1/4}{\tilde{R}}^{-11/8}.$$

$${\partial }_{t}a=\frac{ϵ{\rho }_g}{4{\rho }_{\bullet }}\delta v\simeq \frac{3}{8}{\tilde{R}}^{-11/4}{\text{ cm yr}}^{-1}.$$

$$\eta \lesssim 6\times 10^7\left(\frac{a}{\text{cm}}\right){\left(\frac{v}{{\text{cm s}}^{-1}}\right)}^{-1/5}\text{ P},$$

$$\eta \gtrsim 4\times 10^{-12}{\left(\frac{a}{\text{cm}}\right)}^3{\left(\frac{v}{{\text{cm s}}^{-1}}\right)}^5\text{ P},$$

$$\left(\frac{a}{\text{cm}}\right){\left(\frac{v}{{\text{cm s}}^{-1}}\right)}^{2.6}<4\times 10^9$$

$$v\simeq 2\sqrt{\text{St}}\times 10^4{\text{ cm s}}^{-1},$$

$$a_E\lesssim 5.7{\tilde{R}}^{-0.85}\text{ cm},$$

$$a_S\lesssim 6.3{\tilde{R}}^{0.54}\text{ cm},$$

$${\text{St}}_E\lesssim 0.016{\tilde{R}}^{0.65},$$

$${\text{St}}_S\lesssim 0.015{\tilde{R}}^{-0.42},$$

$${\eta }_E\simeq 7.7\times 10^7{\tilde{R}}^{-0.9}\text{ P},$$

$${\eta }_S\simeq 8.5\times 10^7{\tilde{R}}^{0.6}\text{ P},$$

$$\frac{\partial a}{\partial t}=\frac{ϵ{\rho }_{g}v}{4{\rho }_{\bullet }}.$$

$$St=\sqrt{\frac{\pi }{8}}\frac{a{\rho }_{\bullet }{\Omega }_K}{{\rho }_g{v}_{th}}.$$

$$\frac{\partial \text{St}}{\partial t}=\sqrt{\frac{{\pi }^3}{32}}ϵ\left(\frac{v}{v_{th}}\right){\text{Orb}}^{-1}\simeq ϵ\left(\frac{v}{v_{th}}\right){\text{Orb}}^{-1}.$$

$$2\sqrt{\text{St}(t)}=\sqrt{\alpha }ϵ\frac{t}{\text{Orb}}+2\sqrt{\text{St}(0)}.$$

$$\text{St}(t)\simeq \frac{\alpha {ϵ}^2}{4}{\left(\frac{t}{\text{Orb}}\right)}^2=2.25\times 10^{-6}{\alpha }_2{\widetilde{ϵ}}^2{\left(\frac{t}{\text{Orb}}\right)}^2,$$

Full text

1. Introduction

One of the more significant difficulties facing modern dust coagulation theory is that it seems to prohibit the straightforward formation of dry (water poor) planetesimals or planets. That is, distinctly, in contradiction with clear observational evidence both from within our Solar System and from extrasolar planetary systems (Marty & Yokochi, 2006; Klein et al., 2011). The fundamental problem is that rocky material is not particularly sticky. As a consequence, the dust coagulation phase for non-icy grains in protoplanetary disks is predicted to end long before the grains grow large enough to permit the formation of planetesimals. While ice-rimmed grains are much stickier, potentially allowing for the formation of wet planetesimals (Birnstiel et al., 2016), it is not clear how to make a dry planet from wet planetesimals.

Several potential ways around this difficulty have been proposed. We could be dramatically underestimating the effective stickiness of silicates (Kimura et al., 2015; Arakawa & Nakamoto, 2016). Similarly, dust grains could acquire sticky organic coatings (Hill & Nuth, 2000; Flynn et al., 2013), increasing their stickiness above that of bare silicates. Even though bouncing stalls growth, the Solar Nebula contained vast numbers of grains, some of which would have been astoundingly lucky in their collisions (Windmark et al., 2012). Such lucky grains could potentially grow large enough to begin sweeping up small scale material, growing straight to planetesimal sizes. Even if silicate growth is indeed prohibited, wet planetesimals formed outside the frost line could be scattered inwards, subsequently accreting local dry dust. If that dry accretion proved sufficient to overwhelm their wet cores, those planetesimals could go on to form dry planets.

In this paper we examine the possibility that high temperatures, which can dramatically increase the stickiness of the dust, leads to dry planetesimal formation, as suggested by Boley et al. (2014). As we will show, this route to dry planetesimal formation has interesting consequences for planetary system architectures, and offers an “in situ” model for Mars’ small size wherein planetesimal formation naturally ceases to operate efficiently significantly outside of Earth’s orbit (Chambers, 2014). Taking this line further, it also suggests that there might be two dominant initial planetary system architectures, one of which looks like our own, and the other more like systems with tightly spaced inner planets (or STIPs, Lissauer et al., 2011; Boley et al., 2014).

2. Statement of the difficulty

The challenge bouncing poses to the formation of rocky planetesimals has been known since its introduction to dust coagulation theory (Zsom et al., 2010). In a recent epic endeavor Estrada et al. (2016) verified that the problem persists even taking into account lucky grains (Windmark et al., 2012; Garaud et al., 2013). In this section we quickly sketch how large modifications to our understanding would be required to allow dry silicates to form planetesimals.

2.1. Stokes numbers

The dynamics of dust grains in a protoplanetary disk are controlled by their aerodynamical drag:

[TABLE]

where $v$ is the dust grain’s velocity, $u$ the gas velocity at the dust grain’s location, and $\tau$ the drag based stopping time. It is conventional to non-dimensionalize $\tau$ through the local Keplerian orbital frequency $\Omega_{K}$ to define a Stokes number:

[TABLE]

In regions associated with the formation of the Earth, the dust grains we will consider are small enough to be in the Epstein drag regime, with midplane Stokes numbers

[TABLE]

where $a$ and $\rho_{{\bullet}}$ are the dust grain radius and solid density, and $\Sigma_{g}$ the disk’s gas surface density.

For the inner disk we will also consider solids larger than the gas mean-free-path, in the Stokes drag regime, which depends on the gas mean molecular mass rather than the gas density. Such grains have Stokes numbers

[TABLE]

where $m_{g}\simeq 2.3$ amu and $\sigma_{g}\simeq 2\times 10^{-15}$ (Chapman & Cowling, 1970) are the gas mean molecular mass and molecular collisional cross-section, respectively. Further,

[TABLE]

is the local gas thermal speed and

[TABLE]

defines the local vertically isothermal pressure scale height.

2.2. Radial drift

Due to their radial pressure gradient, disks orbit slower than Keplerian, with a velocity deficit $\delta v$. If the pressure is a power-law in radius with power $s$, $\delta v$ is given by:

[TABLE]

where $s\simeq 13/4$ for a Hayashi MMSN (Hayashi, 1981). Note that $\delta v$ is independent of $R$ in a disk with $T\propto R^{-1/2}$. A dust grain with $\text{St}\ll 1$ drifts inwards at a speed

[TABLE]

allowing us to define a drift time scale

[TABLE]

where Orb is the local orbital period. Local structures such as pressure bumps alter $\delta v$, potentially reversing its sign, allowing the trapping of particles (Dittrich et al., 2013).

2.3. Streaming Instability

The Streaming Instability (or SI) is a mechanism that produces planetesimals from the minimal constituent size, significant because dust coagulation theory does not predict large dust grains. The trigger conditions for the SI are nonetheless relatively large dust grains of $\text{\text{St}}_{2}\equiv\text{St}/0.01\simeq 1$ combined with elevated local dust-to-gas mass ratios of $\tilde{\epsilon}\equiv\epsilon/3\times 10^{-2}\simeq 1$ (Carrera et al., 2015). While that value is above the expected overall value of $\epsilon=5\times 10^{-3}$ for the Solar Nebula inside of the frost line (Lodders, 2003), there are several ways to effectively enhance $\epsilon$, including strong settling, or radial pressure or temperature traps (Dittrich et al., 2013; Hubbard, 2016b).

2.4. Bouncing Barrier

Bouncing sets the upper size limit for locally coagulating reasonably compact (non-fractal) dust grains. We assume that the critical bouncing velocity $v_{b}$ is significantly below the fragmentation velocity, allowing us to neglect the latter. Fragmentation velocities only modestly above the bouncing velocity would further reduce the upper size limit. The bouncing barrier and its consequences has been explored numerically (e.g.  Zsom et al., 2010; Windmark et al., 2012; Garaud et al., 2013; Estrada et al., 2016), and found to prohibit grain growth to SI triggering dust sizes.

From Güttler et al. (2010), the critical bouncing velocity for porous dry silicates is

[TABLE]

where $m$ is the dust grain mass, $\tilde{R}$ the orbital position in a.u., and $f$ an arbitrary scaling factor which parameterizes the dust’s surface parameter uncertainties. While strict sticking/bouncing velocity cut-offs such as Equation (10) are simplifications, we can estimate the rate of sticking events as the rate of collisions at velocities $v<v_{b}$. In Equation (10), following Güttler et al. (2010), the grains have volume filling factors of $\phi=0.12$ and monomer densities of $\rho=2$ g cm*-3*, slightly at odds with the canonical molten solid density we will adopt later of $\rho_{{\bullet}}=3$ g cm*-3* (Friedrich et al., 2015).

Assuming turbulent stirring, we can estimate the collision velocity scale as (Voelk et al., 1980)

[TABLE]

In this section, to give dust coagulation the best chance, we assume that the disk is locally barely turbulent, approximating an $\alpha$ disk with $\alpha_{5}\equiv\alpha/10^{-5}\simeq 1$, appropriate for quiescent dead zone midplanes (Oishi & Mac Low, 2009). In the limit of $v_{b}\ll v_{0}$ we can estimate the rate at which a given grain collides with like grains at velocities $v<v_{b}$ to be (Hubbard, 2016a)

[TABLE]

where the number density of the grains is

[TABLE]

Equating the drift rate $v_{r}/R$ from Equation (9) with the estimated sticking rate $S$ from Equation (12) gives an extremely optimistic estimate for the maximum size dust grains can reach before radially drifting out of our zone of interest. That becomes

[TABLE]

which reduces to

[TABLE]

Equation (15) implies that our estimate of the critical bouncing velocity (Equation 10) would need to be low by $4$ to $5$ orders of magnitude to allow the SI (i.e. $\text{\text{St}}_{2}\simeq 1$). Thus, dust coagulation theory predicts that dry planetesimals will not form.

2.5. Possible remedies

That prediction is at odds with evidence from the Solar System. One obvious remedy would be poor estimates of the sticking parameters of dry silicates (Kimura et al., 2015), or of the appropriate monomer size (Arakawa & Nakamoto, 2016), but Equation (15) makes it clear however how extreme those mis-estimates would need to be. Magnetic interactions could also increase the sticking rates of dust grains, but Hubbard (2016a) showed that the critical magnetic velocity is expected to be well below the collisional velocity $v_{0}$ from Equation (11) for $\text{St}\gtrsim 10^{-2}$, so magnetic interactions should die off well before the SI can be triggered.

Fischer-Tropsch-like processes could have lead to the synthesis of complex organic molecules within the Solar Nebula (Hill & Nuth, 2000; Flynn et al., 2013), depositing layers of sticky organic solids on the dust (Kuga et al., 2015). Those layers would have dramatically increased the dust’s stickiness. However, the Earth is not merely dry, but also strongly depleted in carbon. While the degree of that depletion is uncertain, it seems likely that adding more than just a few mass percent of chondritic material would oversupply the Earth’s entire carbon budget (Marty et al., 2013), to say nothing of even more strongly carbon rich organic gunk. Thus, any organic gunk model would need to demonstrate that it can operate without oversuppling carbon. Indeed, if recent laboratory work on catalyzing Fischer-Tropsch like processes is confirmed (Nuth et al., 2016), models for the formation of the Earth will likely need to explain the destruction of such layers!

Lucky grains, which undergo a series of low probability low velocity sticking collisions, can grow significantly larger than their more abundant staid counterparts, but numerical studies have found that lucky grains are too rare to trigger planetesimal formation (Estrada et al., 2016). That also rules out importing large dust grains from outside the frost line. Importing sufficient numbers of fully formed planetesimals from outside the frost line would allow planet formation to proceed, but those planetesimals would need to accrete sufficient local dry material to drop their bulk volatile abundances.

Thanks to the radial pressure gradient, planetesimals move with a speed of about $\delta v=50$ m s*-1* with respect to the local gas and small scale dust (Equation 7). Assuming perfect, purely geometric sweep-up of ambient dust, those planetesimals would grow at a rate of about

[TABLE]

Thus, purely geometrically accreting planetesimals will only gain a few km of dry surface over a Myr, which seems unlikely to sufficiently overwhelm their volatile rich core. Pebble accretion could speed the process as long as the available grains are comparable in size to chondrules (Johansen et al., 2015), but pebble accretion’s size sensitivity poses its own difficulties.

3. High temperature dust collisions

The purpose of this paper is to discuss how high temperatures could act as a mechanism for allowing dry dust to trigger the SI. Dust grains collide and interact as liquids as long as the temperature of the solids is above about $1100$ K (Ciesla et al., 2004). In that regime, the grains are much stickier and less prone to fragmentation as long as the grain sizes are small enough and the temperature not too high. Such temperatures are associated with sufficient thermal ionization of potassium to allow for full MRI activity (Balbus & Hawley, 1991; Gammie, 1996); so we normalize through $\alpha_{2}\equiv\alpha/0.01$ for high temperatures. Boley et al. (2014) suggested that the inner disk region hot enough to allow liquid grain collisions could have seen direct dust coagulation to planetesimal sizes, producing STIPs-like systems. However, as we will show, molten collisions are not immune to bouncing and fragmentation.

3.1. Liquid bouncing and splashing

Highly viscous molten grains collide as solids, rather than liquids. As long as those effectively solid grains are not extremely sticky ($f\sim 6.6\times 10^{4}$, Equation 15), the bouncing barrier remains in place at high viscosity. If the grains are instead barely viscous, and too large for surface tension to contain the collisional energy, they will splash upon collision. The viscosity condition for colliding as liquids is (Ciesla et al., 2004; Hubbard, 2015):

[TABLE]

where P stands for Poise, the cgs unit of dynamical viscosity.

While the condition for avoiding splashing is not certain, from Jacquet & Thompson (2014) we have:

[TABLE]

where we have assumed a surface tension of $400$ dyn cm*-1* ($0.4$ N m*-1*) for molten chondritic material (Susa & Nakamoto, 2002). Combining Equations (17) and (18) we find the condition

[TABLE]

for there existing a viscosity for which liquid, non-splashing collisions are possible.

At temperatures near $T\simeq 10^{3}$ K, the thermal speed is approximately $c_{s}\simeq 2\times 10^{5}$ cm s*-1* and the turbulent collision speed is about (Eq 11):

[TABLE]

where we have assumed $\alpha\sim 10^{-2}$. In a Hayashi MMSN (Hayashi, 1981), with $\Sigma_{g}=1700\,\tilde{R}^{-1.5}$, assuming $T\simeq 10^{3}$ K, we can combine Equations (3), (4), (19), and (20) to find the largest possible grain sizes:

[TABLE]

for the Epstein (dominant outwards of $0.93$ au) and Stokes (dominant inwards of $0.93$ au) drag regimes, respectively.

Equations (21) and (22) imply largest possible Stokes numbers of

[TABLE]

respectively. Those sizes require viscosities of

[TABLE]

to reach. Thus, the liquid bouncing/fragmentation barriers prevent growth to $\text{St}\gg 10^{-2}$, with critical viscosities to reach $\text{St}\simeq 10^{-2}$ of $\eta\sim 5-10\times 10^{7}$ P. The precise viscosities of Solar Nebula solids are unknown, but those values would have required temperatures near $T=10^{3}$ K (Hubbard, 2015).

Being molten is not a panacea for the formation of terrestrial planets, and the collision of heated solids does not allow direct coagulation to $\text{St}\gg 10^{-2}$. However, it does seem likely that temperatures in a relatively narrow range near $T\sim 10^{3}$ K permit nearly perfect sticking up to $\text{St}\sim 10^{-2}$, conspicuously the critical value for triggering the SI. Unfortunately, we lack detailed laboratory experiments measuring the effective viscosity of chondritic melts of centimeter grains at the relevant temperatures, with thermal histories, and we lack detailed zero-g, low pressure splashing experiments. That limits how thoroughly molten grain growth can be modeled beyond the simple estimates above.

3.2. Growth rates and comparison to radial drift

Neglecting bouncing and fragmentation, the growth rate of a given dust grain is the same as sweep-up up to the details of the velocity (Equation 16):

[TABLE]

From Equations (3) and (4) we can see that the Stokes number’s quadratic dependence on $a$ means that the Stokes number increases more rapidly in the Stokes drag regime, and we need only consider the growth time scales in the Epstein regime where

[TABLE]

gives the Stokes number at all altitudes.

Combining Equations (27) and (28) we find

[TABLE]

We can insert Equation (11) into Equation (29) to find

[TABLE]

Neglecting $\text{St}(0)$ we arrive at

[TABLE]

or alternatively, the time to reach a given $St$ is

[TABLE]

where we recall that $\alpha$ and St are both normalized to $10^{-2}$: $\text{\text{St}}_{2}=\text{St}/10^{-2}$ and $\alpha_{2}=\alpha/10^{-2}$.

Comparing Equations (9) to (32) we find that dust growth outpaces radial drift as long as

[TABLE]

Dust growth can outpace radial drift up to SI triggering dust grain sizes for reasonable dust-to-gas mass ratios even for modest turbulence and relatively thick disks.

4. Sources of heating and planetary system architecture

4.1. Inner disk

Boley et al. (2014) proposed the inner disk, with $T\sim 1500$ K, as a region where partially molten grains could coagulate well beyond the conventional bouncing and fragmentation limits, envisioning coagulation up to planetesimal sizes. As we showed in Section 3.1 though, new versions of the bouncing and fragmentation barriers show up as dust grains grow beyond about $\text{St}\simeq 0.01$. However, that approximate Stokes number of $\text{St}=0.01$ is sufficiently large to allow the SI (Johansen et al., 2007) to trigger, albeit in a narrow radial annulus where the temperature is modestly above $T=10^{3}$ K and the corresponding viscosity of the molted grains several times $10^{7}$ P. That would naturally occur at around $R=0.1$ au, and the temperature range is particularly interesting as it is close to the temperature required for sufficient thermal ionization to allow MRI activity (Gammie, 1996). Thus, the region should occur near the inner edge of the dead zone, a location theorized to concentrate dust particles (Lyra et al., 2009).

We have then the potential for rapid planetesimal formation through the SI if that concentrated dust can be brought into regions of the correct temperature. That would occur either if those temperatures are the normal background temperature at the edge of the dead zone, or through secular evolution of the disk’s radial profile. Thus, the picture of Boley et al. (2014), with terrestrial planetesimals forming in the hot inner disk, survives in a modified fashion. Significantly, it can potentially explain the recent observation of systems with tightly spaced inner planets (or STIPs) representing a significant fraction (more than $10$%) of stellar systems.

4.2. FU Orionis type events

However, that inner disk picture is a poor match for the formation of the Earth, and due to the large amount of migration that would be required, Solar System like planetary architectures in general. The Earth shows a smooth volatility dependent depletion pattern across a broad condensation temperature range of $T\sim 700-1400$ K (Palme, 2000; McDonough, 2003), which is ill-fit by planetesimal formation in a narrow temperature range. Hubbard & Ebel (2014) put forth a model to explain the Earth’s abundance pattern through time-dependent heating and cooling from an FU Orionis type event in the early Solar System (Hartmann & Kenyon, 1996).

FU Orionis type events are dramatic, long lived accretion events, associated with luminosity increases of $4-6$ magnitudes and time scales of $50-100$ yr (Hartmann & Kenyon, 1996; Green et al., 2016). FU Orionis events (or FUors) can raise the temperature at Earth’s orbital position to about $1350$ K (Hubbard & Ebel, 2014), and drive the frost line out to about $40$ au (Cieza et al., 2016). An FUor that strong would raise the temperature above $1000$ K out to Mars’ orbit, while a weaker, $4$ magnitude FUor would still raise the temperature above $1000$ K out to about Venus’ orbit. A long lived, $100$ yr life time FUor outburst can satisfy the time scale constraint from Equation (32) out to a bit past Earth’s orbit, while a $50$ yr FUor would still satisfy Equation (32) out to Venus’. While the frequency of FU Orionis events is not yet certain, statistics are consistent with a large fraction of protoplanetary disks undergoing one or several FU Orionis outbursts (Hartmann & Kenyon, 1996).

We therefore propose that molten collisions during an FU Orionis event were the solution to the bouncing and fragmentation barriers for dry silicates at the locations of the terrestrial planets in our Solar System. Thermal processing and molten collisions could explain both the existence of dry planetesimals at Earth’s position, and the Earth’s volatile abundance trend (Hubbard & Ebel, 2014). The scenario suggests an “in situ” explanation for Mars’ small size (Chambers, 2014): the temperature and growth time scales required mean that the mechanism ceases to operate somewhere between Earth and Mars’ orbital positions (see Equation 32), naturally reducing the number density of dry planetesimals beyond Earth’s orbit. By cycling the temperature high enough to burn off organic surface layers, the model would also help explain why the Earth is carbon-poor even if carbon deposition is expected (Marty et al., 2013; Nuth et al., 2016). In our picture, molten collisions and evaporation/recondensation mechanisms (Hubbard, 2017), rather than conventional dust coagulation, put the Solar System’s architecture in place very early in the Solar Nebula’s existence. In addition to terrestrial planets, the duration and heating of an FUor leads to evaporation/recondensation cycles that might have driven icy planet formation in the outer disk (Hubbard, 2017).

The briefer (up to a few year duration) accretion events associated with Ex Lupi and Exors in general (Herbig, 2007) are too brief for any given eruption to allow molten collisions to drive significant growth far beyond the inner edge of the dead zone, and are generally weaker than FUors. If Exors survive as episodic accreters long enough to accumulate decades of integrated outburst time, then they might drive intermittent molten dust growth to SI triggering sizes. Note however that dust drifts radially during quiescent periods as well as during active ones, so the radial drift condition (Equations 33) becomes more stringent.

5. Discussion and Conclusions

Thermal processing of dust and molten grain collisions at $T\gtrsim 10^{3}$ K provides dry silicate dust a way past the bouncing and fragmentation barriers (Zsom et al., 2010) to the Streaming Instability (or SI), although it likely does not permit direct coagulation to Stokes numbers $\text{St}\gg 10^{-2}$. Significantly, most other mechanisms that have been proposed to bypass those barriers that have not been ruled out by numerical study (Estrada et al., 2016) are inconsistent with the volatile abundances of the terrestrial planets in the Solar System (Flynn et al., 2013; Marty et al., 2013). This thermal processing seems likely to take two forms depending on the host disk dynamics, and those forms would set the planetary system architecture in an in situ planet formation scenario.

The first form, suggested by Boley et al. (2014), appeals to the inner disk where temperatures are naturally elevated. This region is also attractive due to being close to the inner edge of the dead zone, a location prone to concentrating solid material (Lyra et al., 2009). This form is inconsistent with the composition of the Earth (Palme, 2000; McDonough, 2003), but is attractive for explaining the systems with tightly spaced inner planets (or STIPs) seen by Kepler in a large fraction of stellar systems (Lissauer et al., 2011).

The second form appeals to the century scale (Hartmann & Kenyon, 1996; Green et al., 2016) heating from an FU Orionis scale accretion event, which would provide the required heating for the required time out to an orbital location between Venus’ orbit and Mars’. The heating time scale and resulting radial planetesimal distribution would be consistent with Earth’s composition (Hubbard & Ebel, 2014) and with Mars’ small size (Chambers, 2014); and appeals to a process that we have proposed to promote the formation of Jupiter and Saturn-like gas giants (Hubbard, 2017). This suggests that stellar systems which hosted FU Orionis outbursts would tend to host planetary systems like our own, with full-sized dry terrestrial planets in an orbital band extending from the inner edge of the dead zone into the habitable zone. FUors naturally lead to planetesimal formation through the inner habitable zone (Venus to Earth). Mar’s size suggests that modestly sized planets can still form too far out for immediate planetesimal formation via molten grain collisions, fully populating the habitable zone with planets. Thus, we next predict a planet gap extending from the outskirts of the habitable zone to the frost line. Outside the frost line, the accretion events would have trigged gas giant formation.

The trigger mechanism(s) for FU Orionis scale accretion events are not yet known, and given their long duty cycles, those mechanism(s) will remain uncertain for the foreseeable future. However, they appear to be order-unity events in protostellar evolution (Hartmann & Kenyon, 1996), while STIPs are order-unity outcomes of planet formation. If STIPs and FU Orionis hosting systems are mutually exclusive, as we have suggested, then FU Orionis mechanisms would have to divide protostellar systems into two roughly even categories. This is not unreasonable: as an example, Hall MRI (Balbus & Terquem, 2001) depends on whether the background magnetic field is aligned with or anti-aligned with the protoplanetary disk’s rotation. It is unclear which orientation would be more likely to lead to massive accretion events, but numerical simulations suggest that the difference between aligned and anti-aligned systems could easily determine the fate of the disk (Bai, 2014). The idea that planetary system architectures may ultimately depend on something as innocuous as the chance relative orientation of the ambient magnetic field and their protoplanetary disk’s rotation is intriguing. Future observations of the relative rates of planetary system architectures will provide us with a new window on protoplanetary disk dynamics, including constraints on the trigger mechanism for massive accretion events.

Acknowledgements

The research leading to these results was funded by NASA OSS grant NNX14AJ56G. Mordecai-Mark Mac Low, Denton S. Ebel, and the anonymous referee provided advice, assistance, and valuable suggestions for improvements to the manuscript.