Ultrashortperiod planets from secular chaos
Abstract
Over a hundred rocky planets orbiting Sunlike stars in very short orbital periods ( day) have been discovered by the Kepler mission. These planets, known as ultrashortperiod (USP) planets, are unlikely to have formed locally, or have attained their current orbits when their birth protoplanetary disks were still present. Instead, they must have migrated in later in life. Here we propose that these planets reach their current orbits by higheccentricity migration. In a scaleddown version of the dynamics that may have been experienced by their high mass analog, the hot Jupiters, these planets reach high eccentricities via chaotic secular interactions with their companion planets and then undergo orbital circularization due to dissipation of tides raised on the planet. This proposal is motivated by the following observations: planetary systems observed by Kepler often contain several superEarths with nonnegligible eccentricities and inclinations, and possibly extending beyond AU distances; while only a small fraction of USP planets have known transiting companions, and none closely spaced, we argue that most of them should have companions at periods of days. The outer sibling planets, through secular chaos, can remove angular momentum from the inner most planet, originally at periods of days. When the latter reaches an eccentricity higher than , it is tidally captured by the central star and becomes an USP planet. This scenario naturally explains the observation that most USP planets have significantly more distant transiting companions compared to their counterparts at slightly longer periods ( days), a feature unaccounted for in other proposed scenarios. Our model also predicts that USP planets should have: (i) spinorbit angles, and inclinations relative to outer planets, in the range of ; (ii) several outer planetary companions extending to beyond AU distances, both of which may be tested by TESS and its followup observations.
Subject headings:
planets and satellites: dynamical evolution and stability1. Introduction
Ultrashortperiod planets (or USP planets), the rare and enigmatic class of transiting exoplanets with orbital periods shorter than one day, have an unknown origin and are the topic of this study. For reference, a day orbital period for a solartype star corresponds to AU and a blackbody temperature of .
About a hundred of these planets have been discovered by the Kepler transit mission, and they are inferred to exist around of stars (SanchisOjeda et al., 2014), making them slightly less abundant than hot Jupiters (Jovian planets orbiting closer than days, frequency , e.g., Mayor et al., 2011; Howard et al., 2012; Wright et al., 2012). These planets appear statistically different from the more populous Kepler systems, their closest analog, in that they are either the only transiting planet in the system, or in cases when they have transiting outer companions (e.g., in Kepler10 and 55 Cancri systems, Batalha et al., 2011; Butler et al., 1997; Marcy et al., 2002; McArthur et al., 2004; Fischer et al., 2008), the latter orbit at periods times longer (Steffen & Coughlin, 2016), i.e., much further away than in typical Kepler multiplanet systems.
It is almost certain that USP planets did not form at their current locations. These planets fall within the dust sublimation radius for even the most refractive minerals (iron sublimates at ). Moreover, the radii of host stars during the premain sequence phase were several times larger than their current values (e.g., Palla & Stahler, 1991; D’Antona & Mazzitelli, 1994) and would have swallowed a number of the closerin USP planets.
Several formation models have been proposed to explain the origin of these planets, with varying degrees of success. One theory is that they are the exposed cores of giant planets after their gaseous atmospheres have been stripped off by photoevoporation or tidal forces (Jackson et al. 2013, Valsecchi et al. 2014, Jackson et al. 2016). While there are theoretical objections to this scenario (e.g., MurrayClay et al., 2009), empirically, Winn et al. (2017) compared the metallicities of stars harboring USP planets and hot Jupiters and found that they are significantly different, with the hot Jupiters preferentially orbiting around metalrich stars (Gonzalez, 1997; Santos et al., 2004; Fischer & Valenti, 2005). Instead, the USP planets’ hosts’ metallicities are indistinguishable from those of Kepler planets’ hosts, for which there is no notable association with high metallicity (Udry et al. 2006, Schlaufman & Laughlin 2011, Buchhave et al. 2012). They went on to posit that USP planets may be the exposed cores of Neptunes (Valencia et al., 2010; Owen & Wu, 2013; Lundksvist et al., 2016; Lee & Chiang, 2017). However, even if the latter proposal is correct, one still needs to understand how the planets get so close to the stars in the first place. Proposals like that in Mandell et al. (2007), where USP planets are formed from the accretion of material “shepherded” inwards by outer giant planets, suffer similar setbacks.
In this work, we propose that most of the ultrashortperiod planets were initially the innermost planets in typical multiplanet Kepler systems. They reach their current orbits from a combination of secular chaotic excitation of their eccentricities and efficient tidal dissipation in the planets at high eccentricities. This is akin to one of the proposals to form hot Jupiters, secular chaos (Wu & Lithwick, 2011). Consider a planetary system with a large number of planets (). If these planets are spaced far enough from each other such that their interactions are mostly secular in nature (as opposed to meanmotion resonances), and if the orbits of these planets have some moderate amounts of eccentricity and inclination, secular interactions can become nonperiodic and chaotic, leading to diffusive angular momentum transfer among planets that tends to raise the eccentricity and inclination of the innermost planet (Laskar, 1996; Lithwick & Wu, 2011, 2014). As this planet’s pericenter approaches the central star with an everdecreasing range, tidal interactions enter at some point. This dissipates the orbital energy of the planet, bringing it to close circular orbits around the star, and snatching it away from the forcing by other planets. Eventually, we are left with a planet that is orbiting at a close range from the star and is dynamically detached from the outer system.
This proposal is motivated by multiple lines of arguments:
1. analogy with hot Jupiters
there are many observational parallels between USP planets and hot Jupiters. Both are rare classes compared to their more populous cousins: hot Jupiters occur in of FGK stars, while cold Jupiters occur around ; USP planets occur in of stars, while Kepler systems (planets with radii less than that of Neptune and inward of days) occur around of stars (Zhu et al., 2018). They also tend to have lower masses compared to these cousins: USP planets have radii , on the low end of the size spectrum even among those closein Kepler planets that have presumably suffered photoevaporation and are bare rocky cores; while masses of hot Jupiters are a factor of or more below the average cold Jupiters (e.g., Dawson & Johnson, 2018). Lastly, they both lack close neighbors: hot Jupiters have been known to be mostly ‘lonely’ (e.g., Steffen et al., 2012; Huang et al., 2016), though lately they have been shown to possess an abundance of distant ‘friends’ (Knutson et al., 2014); analogously, most USP planets are either apparently single or have distant (orbits of tens of days) outer companions (Steffen & Coughlin, 2016). These similarities propel us to invoke a common mechanism for their formation.
2. highmultiplicity systems are common
about of stars host lowmass, multiplanet systems inward of 400 days, with an average multiplicity of (Zhu et al., 2018). Further than days, transit searches are highly incomplete, but there is evidence suggesting that the planet ladder goes on (ForemanMackey et al., 2016). Microlensing observations also show that Neptunes are common in longperiod orbits, at least around Mdwarfs (e.g., Suzuki et al., 2016).
3. multiplanet systems often have significant eccentricities and inclinations
the Kepler sample shows that systems hosting three or fewer planets in subyear orbits (also those most likely to interact secularly) have significant inclination dispersions ( (); Zhu et al. 2018). These systems are generally observed as singletransiting systems. Interestingly, these very systems also appear to exhibit large eccentricity dispersions (; Xie et al. 2016).
This paper is organized as follows. In §2 we present the sample of USP planets discovered by the Kepler survey and study the possible orbital properties of their outer planetary companions. In §3 we set analytical constraints on our proposed mechanism and in §4 we present the results of some numerical experiments. Finally, §5 provides a discussion of our results and a critique of previous works, and we summarize our main findings in §6.
2. Ultrashortperiod planets likely have distant companions
Among the currently known USP planets, only a small fraction are in multipletransiting systems. We use this to infer how likely USP planets are to have distant companions, and the orbital configurations (in terms of orbital periods and mutual inclinations) of these companions. We conclude that most should have companions, but the outer companions are either distant ( days), or are highly mutually inclined ().
inner period  = mult./sing.  () 

days  
days  
days 
2.1. The sample of veryshortperiod planets ( days)
In Table 1 we show the number of stars with single and multitransiting planets discovered by Kepler with and days from the NASA exoplanet archive as of January 2018^{1}^{1}1Confirmed planets from NASA exoplanet archive based on Quarters 117, https://exoplanetarchive.ipac.caltech.edu. We notice that the number of stars with singletransiting planets remains roughly constant for the different bins, while the number of stars with multitransiting planets increases abruptly with period.
By looking in more detail at the sample of 7 systems with USP planets in multitransiting systems, we notice that there seem to be two distinct classes depending on the stellar types:

for there are 4 systems (out of 11) in preferentially compact configurations, including Kepler42 (3 planets with ), Kepler32 (5 planets with ), and Kepler80 (6 planets with ). The exception to this trend is Kepler732 (2 planets with ).

for there are 3 systems (out of 44) with dynamically detached USP planets, including Kepler10 (2 planets with ), Kepler653 (2 planets with ), and Kepler990 (2 planets with ).
The first class is reminiscent of Mdwarfs which are known to have more miniature systems (Dressing & Charbonneau, 2015). Although the USP planets are still more detached compared to other planets, we ignore this class here and focus instead on systems around FGK stars. Thus, we apply an arbitrary cut in host stellar mass of , which is roughly equivalent to making a cut in effective temperature of K. This sample contains 3 USP planets in multitransiting systems and 44 in singletransiting systems.
Defining a multiplicity ratio
(1) 
we have for our USP planets. In contrast, systems with an inner planet at days have .
We note that the recent study by Adams et al. (2016) including various transit surveys (Kepler, K2, WASP) finds a similar ratio of . Similarly, the previous work by SanchisOjeda et al. (2014) analyzing the Kepler data with their own detection pipeline finds a slightly larger number of companions with , consistent with our sample without the host star mass cut () and marginally consistent with the systems with .
The multiplicity ratio for USP planets ( for our preferred sample) is so low one may worry that many USP planets are truly singles. However, this is unlikely (also see Steffen & Hwang, 2015; Steffen & Coughlin, 2016). Even if every USP planet has a companion, but at large periods as is observed for the 3 systems, the chance of observing the companion transit is low. This is made worse if the mutual inclinations are large. This is quantified in Figure 2 and discussed in detail below. Moreover, since the USP planets are rare with an occurrence rate nearly two orders of magnitude lower than that of average Kepler systems (SanchisOjeda et al., 2014), which are in turn very common and harbor multiple planets (Zhu et al., 2018), it seems reasonable to investigate whether some unusual architecture in the latter systems can lead to USP planet formation, as opposed to assuming that USP planets are a class of their own and are truly single.
2.2. Constraints from period ratios
In Figure 1 we show the period ratios of adjacent planets for systems with an inner planet inside a day orbit. The set of three USP planets have periods ratios , while the planets in the period range of days have a median period ratio of . This result indicates that the USP planets are more dynamically detached than their widerorbit counterparts.
This result that veryshortperiod planets have larger periods ratios was previously pointed out by Steffen & Farr (2013) and the authors were able to place a boundary at . Their results are based on the Quarters 112 KOI catalog, while ours are based on the Quarters 117 catalog with confirmed planets and a cut in host star mass (see previous section).
2.3. Constraints from multiplicity ratios:
We compute the multiplicity ratio expected by Kepler assuming that all systems with USP planets have at least one companion inside days.
We shall assume that for host stars with detected USP planets, the Kepler pipeline has a high detection efficiency at detecting planets with in the period range of days and the ratio mainly depends on the transit probabilities. This is a reasonable assumption as the average detection probability of planets with radii () with periods days is near unity () (e.g., Burke et al. 2015; Petigura et al. 2017).
We estimate the transit probabilities following the method and notation in Tremaine & Dong (2011) and provide the necessary details to reproduce our results in the Appendix. From Equation (A9) we compute the expected multiplicity ratio assuming that the USP planet has one outer planetary companion as
(2) 
where , , and is related to the meansquare value of through Equation (A3). The function is given by Equation (A7).
In Figure 2 we show the contours of the multiplicity ratio from Equation (2) by integrating over the observed range of for USP planets (left panel) and planets with in the period range of days (right panel). Thus, depends only on the orbital separation of the outer planets and the inclination dispersion (or through Eq. [A3]). As expected, decreases for higher inclination dispersions and longerperiod outer planets. We include the error bars for the observed ratio from Table 1 with and observe that the USP planets are constrained to large and/or long . In particular, if the , then days. If days, then .
Our estimate of assumes that there are only two planets in the system. If the intrinsic multiplicity inside days is higher than 2, then the expected ratio would increase, demanding for longer periods and higher inclinations to match the observations. Therefore, considering only two planets is a conservative assumption to put constrains on the minimum values of .
If we further assume that the outer companions do not know about the presence of an inner USP planet and they follow an orbital distribution from the Kepler sample (e.g., Tremaine & Dong 2011)
(3) 
we can integrate in Equation (2) over . We show the results for these multiplicity ratios in Figure 3 as a function of and for different period ranges. We observe that if we limit the period range to days (green line), then nearly isotropic inclination distributions are required to explain the data. For periodranges of days (solid blue line), a dispersion of or larger is required to explain the data while by limiting ourselves to days, then is preferred by the data.
In conclusion, the USP planets have outer companions with typical periods of days and/or days but which are very highly inclined (). If the period of the outer planet is drawn from the observed distribution in Kepler up to 50 days, then the preferred inclination dispersion is .
2.4. Comparison with veryshortperiod planets ( days)
In the right panel of Figure 2 we show the contours of for planets days. We observe that, unlike the USP planets (left panel), these systems are consistent with having at least one companion with (even down to days) in nearly coplanar orbits ().
We note, however, that the planet multiplicity inside 50 days might be higher than 2 planets (see Figure 1), in which case planets can be placed at larger orbital distances and still be consistent with the observed . This is unlikely as nearly half of the observed companions are inside 10 days (Figure 1).
3. USP planets produced by secular chaos: analytical preliminaries
Having argued that USP planets likely reside in multiplanet systems, we proceed to discuss other physical constraints in order for secular chaos to produce USP planets. We find that secular chaos can naturally lead to the formation of an USP planet in generic Kepler systems because:

the requisite eccentricity and inclination values are likely common;

precession from general relativity and tidal bulges can be overcome for planets initially orbiting beyond day obits;

secular chaos excites the eccentricities slowly enough that tidal captures can occur, giving rise to planets with final periods of day.
In what follows, we justify each of these statements separately.
3.1. Required angular momentum deficit to produce USP planets
The angular momentum deficit (AMD), defined as follows (e.g., Laskar, 1997)
(4)  
where is the stellar mass, and the mass, semimajor axis, eccentricity and inclination for planet , describes the deficit in orbital angular momentum relative to that of a coplanar and circular system. AMD is an important index for the strength of secular interactions. Only when it exceeds a certain threshold can secular chaos occur. Moreover, there must be a minimum amount of AMD for the inner planet to be excited to a highly eccentric orbit and be tidally captured into a tight orbit. We consider this latter constraint below.
Since secular interactions do not modify the orbital energies, AMD is conserved. Thus, for the innermost planet to migrate from to a final circular orbit (assuming angular momentum conservation during the circulation process, ), the minimum AMD required is
(5) 
where we keep the term , ignored in the case of hot Jupiter migration (Wu & Lithwick, 2011) ( is not generally much larger than in our scenario). By setting and assuming that all planetary orbits have some typical r.m.s. eccentricity and inclination , we get the following condition for migration:
(6) 
Typically in our model, the planets start migration from AU to AU, so . By assuming that all the outer planets () have the same mass and follow a simple spacing law with constant period ratio , the above condition can be expressed as
(7)  
So to produce a given USP planet, the required eccentricities and inclinations are lower if we assume more outer planets which are widely spaced and have higher masses.
In Figure 4 we show the minimum from Equation (7) for as a function of the mass ratio and the period ratio . We observe that for , the required eccentricities and inclinations are relatively large with for . In turn, by increasing the mass ratio to we get , or in case of equipartition ().
For reference, there are a few systems where the USP planet has a mass measurement and there is an outer planet with a mass constraint. Kepler10 has (Dumusque et al., 2014; Weiss et al., 2016), while Kepler407b (likely ) has a likely nontransiting outer giant planet (Marcy et al., 2014). Other nonKepler systems include 55 Cancri with Nelson et al. (2014) and CoRoT7 with (Queloz et al., 2009). We caution that these systems might not be representative of the whole sample as there is a bias towards detecting the most massive planets from radial velocity measurements.
In conclusion, based on the conservation of angular momentum deficit, the formation of an USP planet by secular chaos roughly requires eccentricity and inclination dispersions at the level of for systems with superEarths mass planets (USP planets have Earth masses). For Jupitermass planets the required dispersions can be much lower. We shall confirm this result with numerical experiments in §4.
3.2. Secular excitation vs apsidal precession forces
We consider whether the diffusive growth in eccentricity for the inner planet, once secular chaos is initiated, can be stalled by other precessional forces. This can limit the maximum eccentricity the planet can reach and prevent USP planet formation. The shortest timescale at which its pericenter distance is forced to vary is given by the quadrupole forcing from the closest outer companion (planet 2) and is
(8)  
where is the starting distance for the USP planet and we have evaluated the fiducial values of and as those for Kepler10c (Weiss et al., 2016). If the planet reaches , but its inclination is still moderate () then the quadrupole forcing vanishes and has a longer (octupole) timescale by a factor of .
3.2.1 Relativistic precession
The relativistic precession can change the argument of periapsis in a characteristic timescale given by
(9)  
As the eccentricity increases and decreases, the above apsidal precession rate rises, leading to a quenching of the secular perturbations from the outer planet when , which occurs at
For instance, for Kepler10b to reach its current semimajor axis AU, it should have started migration from AU ( days), which we determine by setting .
3.2.2 Precession by tidal bulges
The tidal deformation of both the planet and the star by their mutual gravitational perturbations leads to apsidal precession. Assuming that , the characteristic timescale for the tidal bulge on the planet is given by (Sterne, 1939)
(11)  
where is the tidal Love number and we scale it by that of the Earth. Similarly, the tidal bulge on the star gives rise to
(12)  
where is the tidal Love number of the star and we scale it by the solar value. Thus, the precession rate is generally dominated by the tidal bulge on the planet instead of the star. As argued by Liu et al. (2015), the maximum eccentricity allowed for the tidal bulges in threebody interactions is reached when , which occurs at
(13)  
This expression sets the minimum allowed by tidal bulges. We note that for Kepler10b with and , the current location implies that migration should have started from AU ( days), comparable to that obtained from relativistic precession.
3.3. Tidal captures
We assess whether the USP planets can attain their current detached orbits by tidal decay and whether these can prevent their tidal disruptions. We describe the roles of equilibrium tides and dynamical tides separately.
3.3.1 Equilibrium tides
Since the planet’s spin synchronizes with the orbit in short timescales compared with migration timescales, the planet needs to be in an eccentric orbit for friction to extract orbital energy.
We describe the tidal effects on the orbital evolution of the planet using the weak friction theory of equilibrium tides (e.g., Hut 1981), according to which the rate of decay of the semimajor axis for a pseudosynchronized planet can be written as
where is the final circularization radius, is the tidal Love number, is the tidal lag time (assumed constant in the weak friction theory), and
(15)  
From Equation (LABEL:eq:tau_a_hut) the timescale to form an USP planet with final period days (0.02 AU) starting from a period around a Sunlike star becomes
(16)  
For Earthlike planets we assume s (Lambeck, 1977; Neron de Surgy & Laskar, 1997).
Using the tidal quality factor with we obtain
(17)  
Evidently this timescale is short enough so that if planets can attain eccentricities large enough to reach day ( AU), then circularization is possible. However, since relativistic precession and tidal bulges do not efficiently limit the eccentricity growth, the pericenter can continue shrinking until the planet gets tidally disrupted.
The disruption can be prevented by a tidal capture, meaning that the planet can shrink its orbit significantly by tidal dissipation before the pericenter continues to approach the disruption distance. This possibility seems promising because, as shown by Muñoz et al. (2016), rocky planets can survive secular migration for a wide range of parameters compared to gaseous planets.
For Kepler10b with we get the following condition for a capture at its current location (i.e., AU)
(19) 
Thus, it is possible that Kepler10b has achieved its current location by a tidal capture if it started migration from AU.
We stress that the final semimajor achieved by a tidal capture is the minimum value allowed by tidal dissipation in the planet. The actual value for a planet undergoing higheccentricity migration might be longer for the following separate reasons:

tidal dissipation shrinks the orbit after several secular cycles, not one as it is assumed in a tidal capture;

the secular forcing can have a longer timescale than the one used above ( in Eq. [8]) because it can be driven by the octupole moment, not the quadrupole, from the outer planetary orbit. If so, increases by a factor of , increasing by a factor of .
3.3.2 Dynamical tides: diffusive mode excitation preventing disruptions
As discussed above, the tidal dissipation rate in the planet from equilibrium tides might be efficient enough to tidally capture the protoUSP planet and prevent its disruption. However, the dissipative properties of the shortperiod planets are quite uncertain, and the values of can potentially be large enough that they invalidate our previous statement (e.g., GJ 876d, Puranam & Batygin, 2018).
Fortunately, even in the limit of an inviscid planet (), there is salvation. It was recently pointed out by Vick & Lai (2018) and Wu (2018) that, regardless of the planetary dissipative properties, a planet in a highly eccentric orbit can diffusively excite its sphericaldegree 2 fundamental mode (fmode) to nearunity amplitudes. When this happens, that nonlinear effects can set in and effectively convert mode energy to heat. As a result, the orbital energy is lost and the orbit shrinks in a short timescale. This circumvents the difficulty surrounding the uncertain dissipation of the equilibrium tide.
As shown by Wu (2018), the mode excitation can enter the diffusive regime and the mode energy can grow linearly in time when the pericenter distance reaches
(20) 
with some small corrections that depends on the mode period. Here is the density of the planet. At this point, the orbit shrinks with a rapid timescale of (Eq. [17] in Wu 2018)
(21) 
Equation (20) also yields the final orbital semimajor axis by . This is because also constitutes an impassable wall for the planet migration. The timescale for mode growth drops steeply as , any further decrease of the pericenter distance beyond that in Equation (20) brings exponentially faster decay in orbital energy. This efficiently decouples the planet from its secular perturber. When the planet’s orbit circularizes from , we obtain , or
(22) 
which corresponds to a final period days.
In conclusion, the diffusive excitation of the mode allows for rapid orbital migration when the planet reaches inwards of the value in Equation (20). This prevents any secularly migrating planets from being pushed even closer to the star and suffering the fate of tidal disruption. We end up with an USP planet that lies around the observed distances, subject to uncertainties in fmode period, planet density, etc. Subsequent tidal dissipation, possibly via equilibrium tides (likely for solid planets), may eventually circularize the orbit.
4. Numerical experiments
We explore numerically the validity of our analytical estimates in the previous section by running direct numerical integrations of a possible initial configuration of the Kepler10 system. Our calculations should be taken as a proof of concept and not as a detailed population synthesis study, which is beyond the scope of this work.
4.1. Code
All integrations were performed using the WHFAST integrator (Rein & Tamayo, 2015) in the opensource REBOUND Nbody package (Rein & Liu, 2011). We include the effects from relativistic precession and apsidal precession from tidal bulges from the REBOUNDx^{2}^{2}2https://github.com/dtamayo/reboundx library with the option grpotential and Love numbers and for the planets and the star, respectively (Tamayo et al., in prep.).
Our experiments do not include tidal dissipation and we use the maximum eccentricity as a proxy for the potential formation of an USP planet: planets reaching AU can be tidally captured to a final semimajor axis of AU ( day).
4.2. Example and orbital architecture
We shall assume that the Kepler10 system has planets beyond days and that these planets have Neptune masses, similar to the mass of Kepler10c (Weiss et al., 2016). We place these hypothetical Kepler10x planets with periods days, days, and days. Although our choice of orbital configurations is arbitrary, its general architecture is broadly consistent with the bulk of planetary systems in the Kepler sample:

the Neptunesize planets in the range of years are at least as common than their subyear period counterparts (ForemanMackey et al., 2016).
In Figure 5, we show the evolution of one possible progenitor of the Kepler10 system. Here, Kepler10b starts at AU and its eccentricity and inclination evolve to large values due to secular chaotic diffusion driven by the outer planets (e.g., Laskar, 1996; Wu & Lithwick, 2011). As expected from these secular perturbations, the semimajor axes of all the planets remain constant (indicated by the horizontal solid lines).
The planet b reaches a maximum eccentricity of after Myr, so its pericenter distance becomes and it could be tidally captured to become an USP planet ( AU). At this point (indicated by vertical dashed lines) its inclination is also largest () so if the planet were tidally captured, it would likely have a large inclination relative to the outer planets and the host star spin axis.
In our picture, secular chaos is a means to stabilize the system by reducing its overall angular momentum deficit (Wu & Lithwick, 2011). If planet Kepler10b were tidally captured at , then the orbits of the outer planets would gain angular momentum and become more circular. In fact, at (vertical dashed lines) the planets c and d have eccentricities of compared to their averages of during the rest of the evolution.
4.3. Maximum eccentricities allowed by shortrange forces
In Figure 6 we show the maximum eccentricities reached for 500 integrations with the same architecture as Figure 5 but changing the initial semimajor axis of planet b to illustrate the effect from shortrange forces and compare with our analytical estimates from §3.2.
We initialize the orbit of the innermost planet with to artificially speed up the diffusion to large eccentricities as we use a short integration timestep (0.1 day) and relatively small maximum integration time (1 Myr). Our goal is to properly resolve very large eccentricities (). The subsequent experiments do not assume initial large for the innermost planet.
We observe that the distribution of looks roughly uniform for systems with initial semimajor axes AU, while it significantly shrinks towards the initial values () for AU as expected from the shortrange forces.
From §3.2 (Eqs. [LABEL:eq:a_gr]) and (Eq. [12]) we expect that tidal bulges limit the maximum eccentricity for AU, and we observe that apart from a few exceptions, the planets are indeed above the line from the tidal bulges (green lines). For AU, relativistic precession is expected to dominate and the bulk of the experiments are roughly above this boundary.
Finally, we notice that both relativistic precession and tidal bulges allow for the formation of USP planets for AU: the line at AU is above GR and tides in this semimajor axis range.
4.4. Formation rate of USP planets
We run 500 experiments similar to those in Figure 6, but for longer timescales of Myr and drawing the eccentricities and inclinations of all planets from a Rayleigh distribution with parameter (). We set the integration timestep to 1 day^{4}^{4}4This timestep is somewhat large and barely resolves the pericenter passages when the planet becomes an USP planet ( AU). However, we checked that by decreasing the timestep to 0.5 days for up to 10 Myr we get consistent results for the fraction of USP planets. We also checked that the secular code Rings, downloadable at https://github.com/farr/Rings, gives consistent results. and stop the simulation when AU (happening typically at ), at which time we assume the planet is tidally captured (Eq. [22]). This is our fiducial set of integrations. Our goal is to get an estimate of the fraction of systems that can become USP planets and how it depends on the evolution timescale.
In Figure 7 we show the fraction of USP planets as a function of time (solid blue lines). Recall that we take the condition AU as a proxy for tidal capture. We observe that in these integrations the fraction reaches up to with most systems reaching the tidal capture threshold after Myr ( to secular cycles, Eq. [8]). The fraction of USP planets does not level off after 30 Myr () and longer integrations are required to assess whether the ensemble reaches a saturation state. We can only say at least of these Kepler10like systems can produce USP planets. We briefly study how this fraction depends on the eccentricity and inclination dispersions as well as the planetary masses.
Lowering eccentricity and inclination dispersions
We have repeated our fiducial integrations but reduced the eccentricity and inclination dispersions slightly by changing the parameter from () to (; green line) and observe that the final fraction of USP planets decreases from to . We have checked that the systems that undergo fast eccentricity diffusion and become USP planet within 30 Myr are those that have the largest initial values of AMD (or ; Eq. [4]). Thus, by slightly reducing by we reduce the expected AMD by the same factor. In order to asses whether smaller values of allow for significant formation of USP planets we need to integrate these systems for Gyr timescales (). This study is beyond the scope of our paper and it should be the topic of a separate work.
Lowering planetary masses
We have repeated our fiducial integrations but decreased the masses of all the planets by a factor of (orange lines), so the masses might be more representative of the overall Kepler sample with and . The fraction of USP planets decreases from in the fiducial simulation to . This decrease is expected because the planetary mass scale affects the timescale of the secular evolution (). Thus, the fraction of USP planets in the runs with 3 times lower masses up to 30 Myr of should be compared to the fraction in the fiducial simulation up to 10 Myr (i.e., fixed ), which corresponds to . The small difference (not statistically significant) between these fractions seems to be due to relativistic precession since it suppresses the diffusion to large eccentricities more efficiently in systems with lower planetary masses.
4.5. Inclinations of USP planets
In Figure 8 we show the inclinations of the planets that can become USP planets at the moment they first reach AU and would get tidally capture in our fiducial integrations (orange lines). These inclinations extend from to , and in one third of the cases these reach above and are substantially higher than the initial distribution with a median of (Rayleigh with ).
4.6. Nonmigrating planets: eccentricities and inclinations
In Figure 8 we compute the timeaverage inclination of the systems that do not form USP planets (blue lines). These systems have an initial distribution with a median of (Rayleigh with ) that broadens and reaches a median of as a result of the secular excitation. This implies that the secular gravitational interactions broadens the inclinations of the innermost planets significantly.
Similarly, in Figure 9 we show the timeaveraged eccentricity (red line) for the systems without USP planets and compare this with the initial distribution (black line, Rayleigh with ). As it happens with the inclinations, the eccentricities broaden significantly from an initial median of to a timeaveraged median of (the red dashed line indicates a Rayleigh distribution with for reference).
The results above are consistent with the idea that secular chaos can drive the system toward equipartition of energy of the different secular degrees of freedom (AMD equipartition), where and reach similar values for all (Wu & Lithwick, 2011). Since the innermost planet has the lowest circular angular momentum (lowest ), it gets a larger chunk of the system’s eccentricity and inclination budget. In particular, since for our example based on Kepler10, we expect a factor of increase in the timeaveraged eccentricities and inclinations, consistent with the results above.
In summary, we find that secular gravitational interactions leads to excitation of the inner planet’s eccentricities and inclinations from an initial Rayleigh distribution with to a timeaveraged distribution with . This result is consistent with secular chaos driving the system towards equipartition of Angular Momentum Deficit.
5. Discussion
We propose that most of the ultrashortperiod planets around FGK stars are migrated inward by the combined effects of secular chaos and tidal dissipation in the planets. These planets commence their migration from orbital periods beyond days.
The key ingredients for our proposal are: the presence of several planets in the system and a moderate amount of eccentricities and/or inclinations () to drive chaotic diffusion. As discussed in the introduction, this set of requirements agrees with the observed orbital architecture of planetary systems in the Kepler sample.
In what follows, we discuss various predictions from our model and comment on previous works on this subject.
5.1. Properties of USP planets from secular chaos
5.1.1 Occurrence rate of USP planets
The observed occurrence rate of USP planets around GK dwarfs is (SanchisOjeda et al., 2014). We consider whether our proposal can explain this rate.
The occurrence rate of Kepler systems among the same stars is (Zhu et al., 2018). A fraction of these are compact, highmultiple systems that are too tight to be secularly interacting, and are instead dominated by meanmotion resonances. The likely progenitors for USP planets are widelyspaced systems that only contain three or fewer planets within days. This latter is about half of the overall population (Zhu et al., 2018). For these systems, secular chaos has to produce USP planets at an efficiency of to account for most of the observed USP planets.
For our fiducial planet architecture, with , and 4 Neptunemassed planets outside the USP planet progenitor, we find that of the systems have enough AMD to both cause secular chaos and to raise the inner planet’s eccentricity toward tidal capture, within our integration time of Myrs. Due to the diffusive nature of secular chaos, it is expected that this fraction will grow with time, but it is hard to project where it will end up at a few Gyrs.^{5}^{5}5If the linear growth (with logarithmic time) as seen in Fig. 7 continues, the final ratio will be . In the meantime, our experiments show that dropping the AMD by some (from Rayleigh with to 0.08) sharply reduces the yield within 30 Myrs by a factor of , while reducing the masses for the outer companions does not sharply trim down the yield. These considerations argue that, in order for the sparsely spaced Kepler systems to produce the desired rate of USP planets, the values of eccentricity and inclination dispersion are of order , our fiducial value.
Such a dispersion, interestingly, coincides with what is currently determined. Zhu et al. (2018) estimated that sparse systems (with three planets or fewer within days) have ; and (Xie et al., 2016) argued that for these same systems, which typically appear as singletransiting planets in the Kepler database.
5.1.2 Orbital periods: why the 1day limit for USP planets?
The conventional definition of an USP planet as one that orbits with an orbital period day is, in principle, arbitrary. Here we adopt this definition by arguing that there is something physical about the 1day cut.
In our model, an USP planet is defined as a planet that gets tidally captured. The efficacy of tidal capture drops off steeply beyond a few Roche radii. For instance, tidal capture by mode excitation leads to the formation of a planet with a final period of , which occurs around day for planets of Earth density. So we expect planets inward of day have been placed there by tidal capture, while planets outward should not have experienced this process. This naturally explain why only the USP planets are dynamically detached from the companions and why planets at day orbits are less so.
All this being said, it might also be possible to migrate planets to these latter distances ( days) via secular chaos. In the case that the innermost planet does not reach the distance for tidal capture but has acquired some substantial eccentricity, tidal circulation is sufficiently efficient that its orbit will decay gradually and it is eventually freed from the influences of other planets. In fact, Figure 1 shows a handful of systems with periods in days that have distant companions (5 systems have period ratios of ) and for which higheccentricity might have operated. This possibility should be addressed with a full population synthesis including tidal dissipation.
5.1.3 Stellar obliquities
Secular chaos leads to nonlinear mixing between the eccentricity and inclination modes, resulting in large excursions in eccentricities and in inclinations.
In our experiments in Figure 8 we find that the inclinations of the USP planets relative to the initial reference plane can often reach , much larger than the initial inclination dispersion.
The inclinations reached by USP planets in these experiments are generally lower than previous experiments of secular chaos in hot Jupiter systems by Hamers et al. (2017) where planets reach a broad range in with being retrograde (similar results were found by Lithwick & Wu, 2014). We believe that the main difference between the hot Jupiter setup and ours is that the USP planets start migration much closer in, so these only need to reach in order to migrate. On the other hand, hot Jupiters need to reach and a larger maximum eccentricity might translate into larger attainable inclinations.
5.2. Other Formation Proposals
We have provided comments on some proposed scenarios for USP planet formation in the introduction section. Here, we provide more detailed assessments on a couple scenario that are similar to ours and involve both dynamical perturbations by other planetary bodies and/or tidal evolution.
5.2.1 Detaching ultrashortperiod planets by tides raised on the star
It was recently suggested by Lee & Chiang (2017) that the orbits of closein planets can be eroded from the inner edges of protoplanetary disks to ultrashortperiods by asynchronous tides raised on their slowlyspinning host stars. This could, in principle, also explain why USP planets are more widely spaced than their longerperiod counterparts. We asses this proposal here.
The timescale to shrink its orbit by asynchronous equilibrium tides acting on the host star is (e.g., Mardling & Lin 2002)
(23) 
where and are tidal quality factor and the Love number of the star. For reference, for a Sunlike star with day one obtains
(24)  
which is an order of magnitude longer than the typical ages of the host stars and typical values of the modified quality factor constrained from observations (e.g., Ogilvie, 2014)
We further quantify the extent at which tides on the star can shrink the orbits of USP planets. We integrate Equation (23) backwards in time and estimate the initial period the planets would have had in order to reach their current locations after a time :
(25) 
where is given by Equation (24) at their current location.
In Figure 10 we show the initial period for our sample of USP planets assuming (circles) and the dashed lines indicate the range . We adopt Gyr and for the systems in which there is not a mass estimate (all but two systems), we use the massradius relation for cold, terrestrialmass planets from Seager et al. 2007. The values of of