Dynamical Constraints on Nontransiting Planets Orbiting TRAPPIST-1

Few topics in exoplanet science provoke livelier debates than whether the label "habitable" may be attached to a planet orbiting a distant star. This has been particularly pertinent since the Kepler mission whose remarkable data set has allowed various estimates of the occurrence rate of Earth-like planets around Sun-like and cooler stars (e.g., Dressing & Charbonneau 2013; Fressin et al. 2013; Gaidos 2013; Kopparapu 2013; Foreman-Mackey et al. 2014). The recently discovered TRAPPIST-1 multi-transiting system (Gillon et al. 2016, 2017) around an M8 ultracool dwarf has renewed the debate. Assuming planetary albedos of zero, TRAPPIST-1 has two planets with equilibrium temperatures between 273 and 373 K; the temperature range of liquid water on Earth at sea level, and an additional planet with an equilibrium temperature of 251 K, within just a few degrees of Earth's equilibrium temperature. All three of these planets are roughly Earth-sized and theory suggests that they are likely rocky in composition. However, there is enough uncertainty on their densities that mixtures of ice and rock are consistent with their measured sizes and masses. The fifth planet, TRAPPIST-1 f, has its density constrained well enough to favor a composition that is less dense than rock, either a mixture of rock and ice or rock with a deep gaseous atmosphere (Gillon et al. 2017; Wang et al. 2017).

While the size and equilibrium blackbody temperature of TRAPPIST-1 e, and possibly c and d are well-constrained, whether or not they could be hospitable to life depends on additional considerations including the internal processes that may be present on these planets. These include atmospheric chemistry and water retention (Bolmont et al. 2017; Wolf 2017) and possibly tidal heating (Gillon et al. 2017). Furthermore, the planets are subject to additional external factors such as the radiation environment, which is much stronger in XUV than for the Earth (O'Malley-James & Kaltenegger 2017; Wheatley et al. 2017) and varies much more dramatically with stellar flares (Vida et al. 2017).

Finally, the dynamical history of the planetary system can have important consequences for possible living conditions. At TRAPPIST-1, the long-term orbital stability of the known planets depends on their poorly constrained masses, and the influence of any undetected bodies in the system. An undetected massive planet beyond the known seven may cause secular chaos and render the long-term stability of the system unpredictable, as has been demonstrated in the long-term chaotic evolution of Mercury in the solar system (Batygin & Laughlin 2008; Laskar & Gastineau 2009; Hansen 2017).

Additional planets may well be transiting TRAPPIST-1, although the likelihood of seeing a transiting planet becomes negligible with even modest inclinations for an outer planet, given the small size of the host star. For example, a planet at 0.1 au would be unlikely to transit if its inclination relative to TRAPPIST-1 h exceeds just 03.

The existence of additional planets at TRAPPIST-1 will ultimately be determined by radial velocity spectroscopy observations (RV), astrometry, and additional transit timing data over the next few years. We have estimated the maximum TTV induced in TRAPPIST-1 h from a perturber on a circular orbit further out, using the TTVfaster analytical solution (Agol & Deck 2016). Signals decline rapidly with orbital period ratio, and a planet of several Jupiter-masses induces TTVs less than a few minutes if it orbits beyond ∼0.5 au, as shown in Figure 1. To match this constraint with TTVs would require transit timing precision of a few minutes or less.

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Massive planets at greater distances are ruled out by the non-detection of an astrometric signal for TRAPPIST-1. Boss et al. (2017) rule out a 1.6 M_Jup companion with a 5-year orbital period (≈1.3 au). Planets less massive than Jupiter and significantly more distant companions cannot be ruled out by astrometry without more time and/or greater astrometric precision.

In the meantime, some constraints on the presence of a massive planet beyond the transiting seven planets can be found with dynamical models. These can inform the design of any RV campaigns.

First, long-term integrations with a putative Jovian planet can assess the stability of the system over gigayear timescales for a range of hypothetical perturbers acting on the transiting seven planets. This can be a numerically expensive exercise due to the short orbital period of TRAPPIST-1 b, and the number of secular resonances that can effect such a crowded system. Such conclusions are sensitive to the precise masses and initial conditions chosen.

Second, a massive perturber with even a modest inclination may disturb the remarkably low mutual inclinations between the planets over the secular timescale of tens of thousands of years. Several studies have examined the effect of massive planets on the coplanarity of multi-planet systems and their cotransiting likelihood. If a perturbing planet is detected beyond cotransiting inner planets, from its mass and the likelihood of coplanarity, its inclination relative to the transiting planets may be constrained (Read et al. 2017). However, in most known systems of high multiplicity (discovered by the Kepler mission), the baseline of radial velocity spectroscopic monitoring is too short to have identified planetary companions further out. Furthermore, many of the targets are too faint, too hot, or rotating too rapidly for precise RV observations. Hence, deriving lower limits on the masses and orbital distances of putative planets beyond the known transiting planets is left to dynamical modeling.

Becker & Adams (2017) performed a population level study of high multiplicity systems, drawing from a mass–radius relation, inclination distribution, and eccentricity distribution to determine when known systems of five or more planets are not mutually transiting most of the time. In their models, the nontransiting perturber was drawn from uniform priors in semimajor axis from 1–30 au and in mass from 0 to 10 M_Jup. Overall, they found that additional companions from these priors were unlikely within 10 au. They additionally found that constraints on the presence of an outer companion at Kepler-20 was sensitive to the dynamical gap between the planets orbiting at 20 days and 78 days. The discovery of a sixth planet within this gap reduced the minimum distance to any putative outer perturber substantially.

A similar result was found by Jontof-Hutter et al. (2017) for Kepler-11, where the mutual inclinations between the inner six planets are dominated by the relative inclinations of the most widely separated planets Kepler-11 f and Kepler-11 g (orbiting at 46 and 118 days respectively). However, an additional planet of at least 3 M_⊕ in this gap would ensure a tight coplanar bond between the inner seven, and their orbital inclinations would march in step over the secular cycle. In the case of Kepler-11, transit timing variations make the presence of an unseen planet more massive than 3 M_⊕ in this gap extremely unlikely. If there are no planets in this gap, a 1 M_Jup planet inclined by 3 degrees or more from the inner six is unlikely to exist within 3 au of the star.

To some extent, the mutual inclinations of inner planets over the secular cycle due to a distant perturber can be approximated analytically. Lai & Pu (2017) derive analytical solutions for the mutual inclinations of inner planets in the limits of weak and strong secular coupling. The transiting planets at TRAPPIST-1 form the most compact assembly of planets known, with the seventh planet, TRAPPIST-1 h, at just 0.06 au from the star. Each planet is strongly coupled to its adjacent neighbors, and as an ensemble, the transiting planets experience secular variations almost as a rigid disk. Nevertheless, because the star is physically very small, mutual inclinations of just ≈04 would make the planets non-cotransiting.

To place constraints on the presence of a putative planet beyond TRAPPIST-1 h, we performed n-body simulations including the transiting planets and a putative Jovian perturber orbiting farther out. There are no significant gaps between the planets; the largest orbital period ratio between neighbors is just wide of the 5:3 commensurability. We assume throughout that there are no additional planets closer to the star than TRAPPIST-1 h. We calculated the mutual inclination between all planets and TRAPPIST-1 h at every recorded time-step over several secular cycles and calculated what fraction of the time all six planets interior to TRAPPIST-1 h transit if TRAPPIST-1 h is transiting.

We used a symplectic integrator in the HNBODY package (Rauch & Hamilton 2012) and adopted the nominal values for the masses of the star and planets and the planetary semimajor axes from Gillon et al. (2017). When that study was published, the orbital period of the outer-most planet TRAPPIST-1 h was unknown but has since been determined by observations from the K2 mission to be 18.764 days, corresponding to a semimajor axis of 0.0596 au (Luger et al. 2017). The mass of TRAPPIST-1 h is unknown, and its size has been revised by Luger et al. (2017). We assumed a mass for TRAPPIST-1 h of 0.256 M_⊕, following a mass–radius fit through the other six planets (in Earth units: M ∼ 0.7R³). Gillon et al. (2017) found that the posteriors on orbital parameters render the system unstable in a large fraction of simulations. However, Tamayo et al. (2017) found that resonant capture during disk migration stabilizes the system for a large fraction of initial configurations.

Quarles et al. (2017) used the requirement of stability (over Myr timescales) to constrain the masses of the planets. Our adopted mass for TRAPPIST-1 h is well within the uncertainties on its mass found by Quarles et al. (2017). Tidal dissipation of the eccentricities of the inner planet may enhance the stability (Tamayo et al. 2017). In our simulations, we set all initial eccentricities to zero, and all of our simulations with an eighth planet included were stable for at least 200 kyr. Wang et al. (2017) found closely consistent masses with tighter constraints, but in some cases lower than Gillon et al. (2017), using transit timing data from the K2 mission. Of note, Wang et al. (2017) find a 1σ upper limit on the mass of TRAPPIST-1 h of just 0.17 M_⊕.

In Figure 2, we plot the inclinations of the inner seven planets and a distant companion at 0.53 au over the secular cycle. While the inner seven are closely coupled and have coherent mutual inclinations with cycles just 600 years in duration, as a group, they cycle up to twice the inclination of the distant perturber over a 7 kyr period. In this system, the neighboring pairs are all in the strongly coupled limit. Treating TRAPPIST-1 g as the dominant planet of the inner seven, the group is moderately coupled to this planet. For our nominal model of a 1 M_Jup perturber inclined at 3 degrees to the inner seven at an orbital distance of 0.53 au, the analytical solution of Lai & Pu (2017, assuming the inner planets are all strongly coupled to TRAPPIST-1 g) gives a range in mutual inclinations between the inner seven of 14, slightly higher than the range seen in Figure 2. The model in Figure 2 is on the threshold of what is a plausible configuration, where the planets are cotransiting to an observer who observes TRAPPIST-1 h transiting 50% of the time. If the perturbing outer planet were closer, or more inclined, or more massive, the observed cotransiting configuration would be unlikely.

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We explored a range of masses, orbital distances, and inclinations for the putative outer planet to determine which configurations are unlikely. Figure 3 shows the results of several numerical experiments, varying the mass of the outer planet inclined at 3° over a range from 0.0802 M_Jup to 3.53 M_Jup, ruling its existence unlikely to orbital distances from 0.17 au to 0.86 au, respectively. Hence, if the inclination of an outer perturber is >3°, we can rule out the lowest mass that we consider (≈25 M_⊕) with a period of up to 0.25 years. The presence of a perturber up to 3.53 M_Jup, if inclined by at least 3°, is ruled out with an orbital distance up to 0.9 au, or an orbital period of up to 2.8 years.

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Figure 4 compares the effect of different masses and inclinations for an outer perturber on the likelihood of the observed configuration of the known planets at TRAPPIST-1. We note that a Jovian-mass planet can only be ruled out at high inclination up to 1.2 au in distance, and at shorter orbital distances for lower inclinations. We also note that the range of distances over which constraints can be deduced on any putative outer planets increased with mass.

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While we cannot rule out coplanar perturbers of any mass, the distances to which such perturbers can be ruled unlikely increased rapidly with small inclinations. For example, if a perturbing planet is inclined by just 1^◦, we rule out a Jovian-mass planet to 0.37 au.

For the case of perturbers inclined at 3°, we tested the stability of our solutions at the Ψ = 50% line, which divides configurations that are ruled unlikely from those that are plausible. Our solutions were stable for ∼10 Myr, although free eccentricity in the perturbing planet as low as 0.03 reduces this timescale to ∼5 Myr.

Figure 5 compares the distances for which a perturbing planet can be ruled unlikely at TRAPPIST-1 for a range of inclinations up to 80°, with error bars to cover the range of distances where the cotransiting configuration occurs between 40% and 60% of the time. These uncertainties are very small relative to the distances shown on the vertical axis of Figure 5 because of the steepness of the curves in Figure 4 near the 50% line.

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We derived an empirical fit to the data shown in Figure 5 to give an approximate minimum distance to a possible putative planet given its inclination, i, relative to the known planets of TRAPPIST-1.

$$\begin{eqnarray}&&a\gtrsim {{Ai}}^{2}+{Bi}+C,\end{eqnarray} \tag{ 1 }$$

where a is in au, i is in degrees, and the coefficients $A=-0.0005{\left(\tfrac{{M}_{p}}{{M}_{\mathrm{Jup}}}\right)}^{1/3}$, $B=0.04{\left(\tfrac{{M}_{p}}{{M}_{\mathrm{Jup}}}\right)}^{1/3}$, and $C=0.4{\left(\tfrac{{M}_{p}}{{M}_{\mathrm{Jup}}}\right)}^{1/3}$. This minimum distance to any undetected planet matches our numerical results to within 0.2 au over the range of masses that we have considered.

The distances at which we rule massive planets to be unlikely peaks at 45° in inclination, where torques on the orbits of the inner planets are most effective at inducing significant mutual inclinations. With the exception of our most massive putative perturber on an inclination of 10°, all of our constraints inferred in Figure 5 are limited to distances less than 1.3 au, or orbital periods less than 5 years. This is complementary to the astrometric constraints of Boss et al. (2017) for a 1.6 M_Jup planet. However, for lower masses, our constraints on the existence of inclined perturbers retreats to shorter orbital periods, while the astrometric constraint requires greater orbital periods. Hence, there is a large gap in orbital periods between the constraints of coplanarity and the constraints of astrometry where there is no information on the existence or absence of Jovian planets at TRAPPIST-1.

The coplanarity of the system imposes constraints on the outer architecture of the TRAPPIST-1 system that complement those that may be expected from TTVs, RV, and astrometry.

We summarize the orbital periods and radial velocity semi-amplitudes of these lower limits in in Table 1, for the wide range of masses and inclinations that we have considered. To provide context for the dynamical constraints we provide here, we included the expected RV amplitudes from these putative perturbers. These exceed the range in RV expected from the transiting planets (ΔRV ± 9 m s⁻¹). In Table 1, we consider two possible geometric configurations for the RV. In the first case, we assume the ascending node of the perturber in the plane of the transiting planets is in the sky-plane (at quadrature to the transit line-of-sight). Hence, sinI for the observer is cosi, where i is the inclination of the perturber relative to the transiting planets. In the second case, we assume the ascending node of the perturber is in the observer's line-of-sight to the star. In this case, sinI = 1 regardless of the inclination relative to the transiting planets. For arbitrary ascending node, the RV would be between these two estimates.

To compare these constraints with anticipated RV data on TRAPPIST-1, we estimated the precision of RV measurements for a variety of existing and proposed RV spectrographs that target M dwarfs, specifically. These estimates are based on photon-noise limited precision values calculated using the quality-factor formalism described in Bouchy et al. (2001), and they do not include any estimates of the instrumental systematic noise floor, nor of the complicated (and possibly dominant) effects of stellar activity-related noise.

We calculate the information content (referred to as Q in Bouchy et al. (2001)) of a BT-Settl (Allard et al. 2012a, 2012b) model spectrum for TRAPPIST-1 (T_eff = 2600K), broadened to vsini = 3 km s⁻¹ with the appropriate resolution and bandpass for each instrument listed in Table 2. We exclude information in regions of strong (>5%) telluric absorption. To estimate the shot noise, we scaled the model spectrum to V = 18.8 and calculated the number of photons detected by each instrument for an exposure time of 1800s, assuming full telescope aperture illumination and accounting for each instrument's predicted efficiency curves. Combining the information content with the shot noise yields the photon-limited RV precision, which is listed in Table 2, Column A. Detector read noise will also degrade the signal-to-noise of these observations; we include the effect of read noise, based on each instrument's reported detector performance and spectral format (i.e., the number of detector pixels illuminated), in Table 2 Column B. The remaining columns show the estimated RV precision (photon + read noise) for shorter exposure times of 10 and 5 minutes.

We find that in the absence of stellar activity, all putative outer planets at TRAPPIST-1 that we rule unlikely from the constraint of coplanarity are readily detectable in RV. Hence, these instruments are sensitive to lower masses and greater orbital distances than we have considered in dynamical simulations. Hence, our results may permit constraints on sinI in the event of an RV detection. Until then, our dynamical constraints will remain the strongest upper limits on the presence of nontransiting planets at TRAPPIST-1.

We thank an anonymous reviewer and Brian Weaver for comments that improved this manuscript. D.J. and V.T. acknowledge the support of the University of the Pacific and NASA Exoplanets Research Program award #NNX17AD23G. D.J. and E.F. acknowledge support of NASA XRP grant #NNX15AE21G. This study benefited from collaborations and/or information exchanged within NASA's Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA's Science Mission Directorate. This work was partially supported by funding from the Pennsylvania State University's Office of Science Engagement and Center for Exoplanets and Habitable Worlds. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University, the Eberly College of Science, and the Pennsylvania Space Grant Consortium. Portions of this research were conducted with Advanced CyberInfrastructure computational resources provided by The Institute for CyberScience at The Pennsylvania State University (http://ics.psu.edu).