Migrating Planets into Ultra-short-period Orbits during Episodic Accretion Events

The magnitude of the torque exerted on the planet due to net Lindblad and corotation torques (Goldreich & Tremaine 1979; Ward 1997) has the form

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{I}=-{C}_{I}{\left(\displaystyle \frac{{m}_{p}}{{M}_{* }}\right)}^{2}\pi \sigma {a}^{4}{{\rm{\Omega }}}_{\mathrm{kep}}^{2}{\left(\displaystyle \frac{a}{H}\right)}^{2},\end{eqnarray} \tag{ 5 }$$

where a is the semimajor axis of the planet, Ω_kep is the Keplerian rotation rate at orbital radius a, and H is the disk height at radius a (the ratio a/H varies with orbital radius and is set by Equation (3)). The dimensionless coefficient C_I depends on the disk properties and determines the direction of the torque. This torque leads to inward or outward motion of the planet (Goldreich & Tremaine 1979), depending on the disk properties as parameterized by the coefficient C_I (e.g., Paardekooper et al. 2010; Bitsch et al. 2019). If the torque is positive, which occurs in many scenarios (e.g., Dittkrist et al. 2014), including when radiative cooling is not efficient (Paardekooper & Mellema 2006; Yamada & Inaba 2012), the planet will migrate outward. If the torque is negative, then the planet migrates inward. Numerical experiments (Bitsch et al. 2014b) indicate that for planets with smaller masses (m_p < 10 M_⊕), Type I migration will generally be inward. For many cases, C_I can be computed using only the disk profile exponents p and q (Tanaka et al. 2002; D'Angelo & Lubow 2010; Paardekooper et al. 2010). For locations in the disk where p = 3/2, q = 3/4, Type I migration will tend to move small planets inward (Bitsch et al. 2014a, 2014b).

However, the typical scenario where Type I torques move planets inward does not hold once planets reach regions slightly exterior to a disk cavity (including the central cavity inside the truncation radius). Numerical simulations (Masset et al. 2006) indicate that near a cavity edge, where the disk surface density profile experiences a steep gradient, increasingly positive corotation torques will cancel out the negative differential Lindblad torques and provide the planet a positive net torque. The radius at which this occurs is often called a planet trap, since the torque balance will cause a planet to halt migration and become trapped at a fixed orbital radius. This behavior has been found in several other numerical studies (e.g., Morbidelli et al. 2008; Baillié et al. 2016; Romanova et al. 2019). Since the planets we consider in this work reside at (or near) the inner disk edge, slightly external to the interior cavity, Type I torques are expected to act outward, and under standard assumptions, C_I ≈ −0.6 (Ward 1997; Tanaka et al. 2002). However, as shown in Paardekooper (2009), sub-Keplerian gas flow can alter the relative positions of the Lindblad resonances (see their Figure 2), and C_I must be adjusted according to the disk scale height and the velocity of the gas flow. In general, this modification leads to a stronger Type I torque than predicted by standard models, with the exact scaling determined by the local disk surface density profile and disk aspect ratio. Nonetheless, in the inner disk regions, the Type I torque remains subdominant (see below). For the sake of definiteness, this work assumes C_I ≈ −1, but keep in mind that this value may vary for different disk parameters.

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As discussed in Section 2, magnetic fields thread through the protoplanetary disk and provide pressure support. The magnetic fields originate from two sources: a stellar magnetic field that scales with radius as B(r) ∝ r⁻³ and the remnant field dragged in during disk formation, which scales as B(r) ∝ r^−5/4 (Shu et al. 2007). These fields cause the gas to orbit at a sub-Keplerian rate. The torque that a sub-Keplerian gas disk exerts on a planet (Adams et al. 2009) can be written in the form

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{X}=-{C}_{D}\pi {r}_{p}^{2}a{\rho }_{\mathrm{gas}}{v}_{\mathrm{rel}}^{2},\end{eqnarray} \tag{ 6 }$$

where the planet itself is assumed to move at a Keplerian velocity. In this expression, C_D is a dimensionless drag coefficient analogous to C_I but has a value C_D ∼ 1 for a planet in a disk. ⁷ As before, r_p is the planetary radius, ρ_gas = Σ/2H is the gas density, and v_rel is the velocity differential between the gas and planet velocities, defined as v_rel = v_kep − v_gas. Following Shu et al. (2007), we define the parameter f to be the ratio between the true rotation rate of the gas and the expected Keplerian rotation rate, f ≡ Ω_gas/Ω_kep. For a planet in a Keplerian orbit in a disk rotating at a Keplerian speed, f = 1. This parameterization allows us to rewrite v_rel = (1 − f) v_kep. With these substitutions, the torque due to the headwind of a sub-Keplerian gas flow takes the form

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{X}=-{C}_{D}\pi {r}_{p}^{2}({\rm{\Sigma }}/2H){\left(1-f\right)}^{2}{{GM}}_{* }.\end{eqnarray} \tag{ 7 }$$

The relative importance of the two contributions to the total torque (Equations (5) and (7)) depends on the parameters of the planet. To compare the importance of these contributions for a particular object, we must assume a mass–radius relation for the planets. We construct a piecewise continuous mass–radius relation by computing the planetary bulk density ρ_p for three regimes:

$$\begin{eqnarray}&&{\rho }_{p}=\left\{\begin{array}{l}2.43+3.39\ {r}_{p}\ {\rm{g}}\ {\mathrm{cm}}^{-3}\ \ \ \mathrm{if}\ {r}_{p}\lt 1.5{R}_{\oplus }\\ 14.84\ {r}_{p}^{-1.7}\ {\rm{g}}\ {\mathrm{cm}}^{-3}\ \ \ \mathrm{if}\ 4\gt {r}_{p}\gt 1.5{R}_{\oplus }\\ 1.45\ {\rm{g}}\ {\mathrm{cm}}^{-3}\ \ \ \mathrm{if}\ {r}_{p}\gt 4{R}_{\oplus }.\end{array}\right.\end{eqnarray} \tag{ 8 }$$

The expression for planets with r_p < 1.5 R_⊕ is taken from the Weiss & Marcy (2014) mass–radius relation, and the expression for planets with 4 R_⊕ > R_p > 1.5 R_⊕ is derived from the Wolfgang et al. (2016) relation. For both cases, we ignore the possible range due to uncertainties and use only the best-fit line from each paper to simplify the calculation. For planets larger than 4 R_⊕, the density is taken to be a constant ρ = 1.45 g cm⁻³, which matches the upper end of the previous range. Using this conversion and values of C_I = −0.6 for a T Tauri star (Shu et al. 2007), C_D = 1 (Weidenschilling 1977), M_* = 1 M_⊙, and a disk with p = 3/2, q = 3/4, we can compute the ratio of torques for a variety of planet masses and orbital separations.

The resulting ratio between the magnitudes of the Type I and headwind-driven torques is shown in Figure 2 for a range of orbital locations and planet masses. The torques arising from the sub-Keplerian gas disk are important for close-in planets and those with lower planetary masses. For planets at moderate orbital separations, ∼1 au, the torque reduces to the Type I equation for nearly all planet masses.

Figure 2 shows that even though torques due to the headwind from sub-Keplerian gas flow are not important beyond 1 au, they can be important for planets at short orbital separations. For small planets at 0.05 au and within, the magnitude of the Type I torque can be several orders of magnitude smaller than that provided by the sub-Keplerian gas disk.

For planets to be affected by the headwind-driven torque, they must reside at fairly small orbital radii. To determine how often such torques are operative, we briefly describe the relative timescales. Planetesimals can form more rapidly than the grain migration timescale (Johansen et al. 2007) through the streaming instability (Youdin & Goodman 2005; Squire & Hopkins 2018), while a magnetized disk may accelerate this process (Seligman et al. 2019). Subsequently, planet formation from planetesimals can take place rapidly, but it is unlikely that planets will form exactly at the inner edge of the disk (Raymond & Morbidelli 2020). Planets residing near the disk edge are more likely to have formed further out in the disk and then migrated inward. As a result, the limiting timescale for placing planets at the disk edge will be their migration time. Here, as before, we consider smaller planets that migrate via Type I migration.

The timescale of Type I migration can be estimated using the result of Tanaka et al. (2002),

$$\begin{eqnarray}&&{t}_{I}=\displaystyle \frac{1}{4.35}\ \displaystyle \frac{{M}_{* }^{2}}{{\rm{\Omega }}\ {m}_{p}\ {\rm{\Sigma }}\ {a}^{2}}{\left(\displaystyle \frac{H}{a}\right)}^{2}.\end{eqnarray} \tag{ 9 }$$

The numerical coefficient is computed assuming a surface density profile with p = 3/2. For an Earth-mass planet that forms at a = 1–2 au, this timescale is of order (6–8) × 10⁴ yr for a surface density of 2000 g cm⁻³. Type I migration is rapid compared to disk dissipation timescales (Hernández et al. 2007). Provided that a planet forms early, Type I torques will generally be able to bring it close to the inner edge of the disk.

Although they move the disk edge inward, FU Ori outbursts are several orders of magnitude too short for a planet to migrate from 0.05 au to a USP orbit at 0.01 or 0.02 au via Type I migration. More importantly, because the inner disk edge has a repulsive effect on small planets, Type I torques alone cannot force a planet to remain at 0.01 au. From Figure 2, however, it is clear that at short orbital periods, the sub-Keplerian flow-driven torque will be much larger than the Type I torque. In this case, even though Type I migration rates are not fast enough to successfully move a planet down to ∼0.01 au in a few hundred years (over an FU Ori outburst), the combined effect of the two mechanisms might be sufficient. To assess this possibility, we compute the distance that a planet can migrate under the combined torque from both sources using the differential equation

$$\begin{eqnarray}&&\displaystyle \frac{{da}}{{dt}}=2\displaystyle \frac{{{\rm{\Gamma }}}_{t}}{{{am}}_{p}}\sqrt{{a}^{3}/{{GM}}_{* }},\end{eqnarray} \tag{ 10 }$$

where Γ_t = Γ_I + Γ_X is the total torque acting on the planet and can be computed using Equations (5) and (7). For a typical T Tauri accretion disk, the sub-Keplerian factor near the disk edge has been estimated to be f = 0.658, whereas for a star/disk system undergoing an FU Ori outburst, f = 0.386 (Shu et al. 2007). In this scenario, a planet formed at a moderate radius in the disk (1–2 au) migrates inward fairly rapidly. It will subsequently reside near the disk inner radius and, during FU Ori outbursts, be subject to headwind torques as the disk edge moves further inward according to Figure 1.

In Figure 3, we plot the distance that a planet starting at 0.05 au would migrate during one FU Ori outburst, computed by solving Equation (10) for t_max = 100 yr (corresponding to the lifetime of a single FU Ori outburst) and the standard star/disk parameters that we have used thus far in this work (f = 0.386 for an FU Ori outbursting star, $\dot{M}=5\times {10}^{-4}\ {M}_{\odot }$ yr⁻¹, M_* = 1 M_⊙, p = 3/2, q = 3/4, C_I = −0.6, and C_D = 1). In order for a planet to reach the original inner disk edge, it must form and migrate inward some distance, which requires a good fraction of the disk lifetime. As a result, due to the relative infrequency of FU Ori outburst events, by the time a planet reaches this location, only a few FU Ori outbursts may subsequently occur. We note that this process may not be broadly applicable due to both the timing of planet formation, migration, and FU Ori events and the particular parameters of individual systems. This mechanism may affect only a minority of systems.

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Figure 3 shows that planets with smaller masses migrate more readily to smaller orbital radii during an FU Ori event. Qualitatively, this makes sense; the effect of a headwind provided by a sub-Keplerian gas disk will be more important for smaller planets (unlike Type I torques, which increase in size as planet mass increases). For the smallest planets, those with masses below 1 M_⊕, a single FU Ori event could provide sufficient time for a planet to migrate into a USP orbit (at ∼0.02 au), while the disk truncation radius is temporarily reduced according to Equation (4) due to the increased mass accretion rate.

Considering that there is no canonical number of FU Ori events that a planet may experience, in Figure 4, we plot the time (in years) that a planet must reside at or within the disk truncation radius and in a disk subject to FU Ori–like mass accretion rates in order for the planet to migrate into a USP orbit. These results were again computed using Equation (10) and the same stellar and planetary parameters specified previously. We assume that after a single FU Ori event, the inner radius of the disk recedes quickly and decouples from the planet, which is left at its instantaneous orbital radius. When the next FU Ori event starts, the gas disk again rapidly decreases its inner radius (past the planet), and the headwind migration effect continues to migrate the planet further inward (it picks up where it left off). This piecemeal migration is possible in the picture of headwind-driven migration but not for Type I migration alone.

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Although the specifics will vary, Figure 4 indicates that in order for a planet to reach a USP orbit (0.01 au), more than one FU Ori outburst is often required. Intuitively, disks with inner edges closer to the star during the T Tauri phase will produce USP planets more readily. This process also has a weak dependence on mass, where larger planets require a longer phase of FU Ori events to reach USP orbits. Through this mass dependence, headwind migration predicts a higher abundance of smaller planets at the smallest orbital radii.

This migration mechanism requires that the gas in the inner disk orbits with a substantial departure from a Keplerian rotation curve. Shu et al. (2007) estimated the amount by which the gas velocity would be sub-Keplerian using typical values for the star and disk parameters in various phases of evolution (T Tauri, FU Ori, and protostars). However, the scaling factor f is sensitive to the number of magnetic fields dragged in, the disk surface density, and the mass-to-flux ratio in the disk, so that f will vary from system to system. A smaller value of f (a greater departure from Keplerian rotation) makes the mechanism outlined in this paper more effective. But even if the gas flow is substantially more Keplerian (larger values of f), the smallest planets will still be affected. Figure 5 shows the maximum planet mass that can be pushed inward by headwind-driven torques for a variety of values of the sub-Keplerian factor f.

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In this paper, we assume that two accretion states were possible for the young systems: a normal T Tauri accretion rate of 10⁻⁸ M_⊙ yr⁻¹ and an enhanced accretion rate during FU Ori outbursts, where the accretion rate increases to ∼10⁻⁴ M_⊙ yr⁻¹ for a short duration (∼100 yr). Each accretion rate corresponds to a different disk truncation radius (see Figure 1). Moreover, if planets can migrate quickly enough during the FU Ori outburst, they can reach the target orbital periods of less than a day to become USP planets. As shown in Figure 5, if the inner disk edge is closer to the host star during the quiescent phase of mass accretion, the system will form USP planets more readily (and possibly increase their chances of subsequent destruction). In contrast, larger values of the inner disk edge may significantly limit the masses of planets allowed to move inward through this mechanism.

However, the accretion rate must vary between the two aforementioned extremes as the disk evolves. A variety of processes have been proposed that will increase the accretion rate to some degree, but not to FU Ori outburst levels (e.g., MHD turbulence; Armitage et al. 2001). In addition, numerical modeling has shown that in systems with FU Ori outbursts, mass accretion rates vary both slowly and stochastically as the star evolves (Vorobyov & Basu 2006). These modes of disk operation move the truncation radius inward slightly and allow planets a closer starting position compared to the 0.05 au that we assumed in this work. This head start allows planets to migrate to the 0.01–0.02 au range through a smaller number of FU Ori outbursts. Additionally, as mentioned in Adams et al. (2009), another effect of episodic, rapid increases in the accretion rate is that the star cannot spin up fast enough, potentially contributing additional inward migration.

Additional work (as outlined in Hillenbrand & Findeisen 2015) to classify the specific parameters of FU Ori outburst stars, including the structure and temperature of the inner accretion disk and the frequency and duration of their outbursts (Rodriguez A. et al. 2021, in preparation), will improve estimates of how frequently the mechanism outlined in this work can affect the orbits of short-period planets. Identification of new FU Ori stars (Hillenbrand et al. 2018) will also improve our understanding of the demographics of sources that undergo FU Ori outburst events.

The mechanism outlined in this work makes distinct observational predictions regarding the mass distribution of USP planets. Headwind-driven migration will operate for small planets that are more affected by the velocity differential with the gas disk. Larger planets that migrate via Type II migration and/or clear a gap will not be affected by this mechanism. Figure 6 shows the observed distribution of orbital separation and planet mass for confirmed planets with orbital periods less than 1 day. Although many observational biases have been folded into this diagram (in particular, smaller planets are more difficult to find, so that completeness is lower for lower-mass planets), the planets with the smallest a/R_* ratio do appear to be those with smaller masses. As the plane shown in Figure 6 becomes populated with a greater number of exoplanets, and detected planets have improved observational characterization, more quantitative tests of the mechanism in this paper will be possible.

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The underlying physical parameters that define this migration scenario are subject to uncertainties. This work shows that magnetically enforced sub-Keplerian rotation curves will create some mass dependence for the occurrence rates of USP planets around stars that host FU Ori outbursts. However, the inner radius to which a planet can migrate depends sensitively on the properties of the system, including the magnetic field strength of the host star, the mass and radius of the planet, the surface density of the disk, and the duration and mass accretion rates of the FU Ori outbursts. As a result, a more complete assessment of the duty cycle of this migration mechanism requires more accurate determinations of those parameters.

As one example, consider a 1 M_⊕ planet starting at 0.05 au and typical values for the range of star, disk, and planet parameters. During a single FU Ori event, such a planet could migrate inward to anywhere between 0.02 and 0.04 au. The smaller final orbital radius results from disks with higher overall surface densities, higher H/r ratios, and stars with higher masses. Since these parameters vary widely over the population of T Tauri stars and their planets, the mechanism discussed in this work will operate to varying degrees in different systems.

Another complication arises for planets that reach the disk truncation radius too early. During sufficiently powerful FU Ori events, the disk truncation radius can reach the stellar surface. Any planet that experiences too many FU Ori events while residing on a short-period orbit could migrate inward too rapidly and collide with the star. It is likely that a sizable fraction of planets migrating via the mechanism described in this paper could meet this end. If planetary engulfment occurs too often, however, the planets would provide a source of heavy elements to the stellar photosphere, leading to metallicity enhancements that are potentially observable (Laughlin & Adams 1997). Since observations show that planetary architectures may correlate with stellar metallicity (Brewer et al. 2018; Anderson et al. 2021), pinning down the details of engulfment will serve as an important constraint on these processes.

Tidal dissipation provides another mechanism that can remove USP planets after they form and migrate to their final orbital positions. Tidal interactions with the central star act to move planets inward after FU Ori outbursts have finished and the mechanism of this paper is no longer operative. Energy could be tidally dissipated within the USP planet itself or its host star. The larger radius of the host star at early times increases the inward migration rate, as the tidal migration rate scales with Q_* ${R}_{* }^{5}$ (Goldreich & Soter 1966). This scaling may lead to the potential destruction of USP planets over time as their orbits decay.

Another observational diagnostic for USP formation mechanisms is the USP planet occurrence rate as a function of stellar age. In the scenario presented in this paper, USP planets attain their final orbital positions early, while the protoplanetary disk is still present. Although tidal effects (Goldreich & Soter 1966; Rasio et al. 1996) could subsequently remove some planets with appropriate physical properties (see above), the smaller planets more likely to migrate to USP orbits via our proposed mechanism should be found at approximately equal rates around young and old stars (consistent with the result of Hamer & Schlaufman 2020, who found no evidence for statistically different ages for USP planet hosts and field stars). In multiplanet systems, migration driven by obliquity tides can take the innermost planet in the system to a USP orbit (Millholland & Spalding 2020). Several other theories also involve dynamical interactions between the USP planet and its companions, including high-eccentricity migration (Petrovich et al. 2019) and low-eccentricity migration (Pu & Lai 2019). All of these mechanisms predict a stellar-age dependence of USP planetary occurrence, and barring eccentricity tides, the other mechanisms also require the USP planet to reside in a multiplanet system. In contrast, the mechanism described in this work allows for the presence of nearby planetary companions but does not require them and will not generate a stellar-age dependence on the USP planet occurrence rate.

Finally, we note a number of low-probability events that must occur for this mechanism to operate. The planet must form early and migrate to the inner disk edge in time to experience at least one FU Ori event. The magnetic field structure must result in sufficient sub-Keplerian rotation. And the resulting migration mechanism cannot move the planet too far inward, otherwise the planet will not survive. As a result, we do not expect a large population of planets to be placed on USP orbits through this mechanism. Thus, USP planets should be rare, a finding that is consistent with observational estimates of the USP planet occurrence rate, which is only ∼0.5% (Sanchis-Ojeda et al. 2014).