An Integrated Analysis with Predictions on the Architecture of the τ Ceti Planetary System, Including a Habitable Zone Planet

Under the assumption that planets are found with equal period ratios between them, the DYNAMITE predictions agree very well with the combined picture from the analyses by T13 and F17. Specifically, DYNAMITE predicts four planet candidates (PxP–1 through PxP–4, see Table 2) with orbital periods at the relative likelihood maxima of 12.0, 31.4, 89.7, and 322 days. We point out that the planet candidates b, c, and d reported by T13 (which were not part of the DYNAMITE input) have very similar periods (b = 14.0 days, c = 35.4 days, and d = 94.1 days) to the predicted planets PxP–1, PxP–2, and PxP–3. Combining the four planets from F17 with the three unconfirmed (but now supported) candidates and the prediction of PxP–4 (which has no current equivalent candidate in the observational literature), the τ Ceti system becomes strongly dynamically packed. The relative likelihood in log-period space for the period ratio prescription is shown in the top of Figure 2.

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Under the assumption that the planets are clustered in period space, DYNAMITE also finds a very similar configuration to that described previously. Specifically, there are still four relative likelihood maxima for the predicted planets (again named PxP–1 through PxP–4 here) at periods of 14, 34, 67, and 468 days. These predictions are even closer (in orbital period) for PxP–1 and PxP–2 to planet candidates b and c, but farther away for PxP–3 to planet candidate d. The cluster assumption means DYNAMITE finds planet f in a second cluster in period space, away from the main cluster containing the other planets. Thus, DYNAMITE predicts the planet as close to the center of that cluster (and therefore as close to planet f) as possible (i.e., dynamically stable). The relative likelihood for the predicted planets in log-period space, under the clustered periods prescription, is shown in the bottom of Figure 2.

On the basis of the DYNAMITE predictions we will now explore the possible ranges in mass and radius for the planets, planet candidates, and predicted planets in the τ Ceti system. We will follow three steps: first, we derive the likely mass distributions of the planets. Second, combining these with planet radius–mass relationships and planet occurrence rates (expressed in planet radii) we derive the likely distributions of planet radii and masses in the system. Third—Section 4.4—we will use the derived M–R probability distributions to identify the possible natures of the planets, also considering the impact of key assumptions on the results. In the discussions in this and the following subsection, we also consider the uncertainties in the measurements as well as the sensitivity of the final results to key assumptions. The most important assumption is the adopted M–R relationship; therefore, we present and contrast results assuming three different M–R relationships, which are representative to the state of the art.

For our first step—deriving the probability distribution functions of the planet masses—we start our analysis from the $m\sin (i)$ values observed for the confirmed planets and combine these with a constraint on the orbital inclinations to derive the likely true masses of the planets. The planetary orbital inclinations are not directly measured, but—as explained in Section 3—there are very strong astrophysical reasons to assume that the planetary orbits are overall well-aligned with the debris disk, for which good geometrical constraints exist from spatially resolved ALMA observations. With this constraint on the orbital inclinations, we find that all planets and planet candidates in the inner system likely fall between 3 and 7 ${M}_{\oplus }$ in mass. Table 2 shows the derived planet mass ranges and inclinations.

A key assumption in the second step of our analysis is the M–R relationship adopted for smaller planets, of which multiple somewhat different variants have been proposed in the literature (e.g., Bashi et al. 2017; Chen & Kipping 2017; Ning et al. 2018; Otegi et al. 2020). Instead of adopting any single M–R relationship, we explore three different relationships, one from Ning et al. (2018) and two from Otegi et al. (2020). Therefore, our exploration of multiple M–R relationships quantifies the impact of this choice on the results. As one of our three relationships to explore, we chose the nonparametric relationship from Ning et al. (2018) as it provides a probabilistic measure of the mass and radius. This relationship conditions its predictions on the known mass/radius data we have gathered from exoplanet systems, where super-Earths/sub-Neptunes have the highest occurrence and therefore would be the most predictive. However, this conditionality also means there is not a 1:1 relationship between radius and mass, and it also greatly increases the computation time required for deriving mass distributions. Furthermore, as our second and third M–R relationships to explore, we adopted the two power-law relationships from Otegi et al. (2020) as these match data up to 120 ${M}_{\oplus }$ very well. Based on these, we explore the difference between sub-Neptunes and super-Earths in a region of M–R space where these two planet populations overlap. This combined approach, therefore, also provides an assessment of the possible range of natures for these worlds.

We applied each M–R relationship to the known planet masses to find their radii, before—following Dietrich & Apai (2020)—we fit them to the Log-normal distribution for planet radius to find the best-fit distribution. We find that the known planet radii are close enough to each other to be statistically considered a single cluster in planet radius by DYNAMITE. The predicted planet radius distribution found from this best-fit of the four planets, for each of the three M–R relationships adopted, is shown in the left side of Figure 3. We then reversed the process and applied each M–R relationship on the radius distribution to get a distribution in planet masses, shown on the right side of Figure 3.

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In this section we will build on the planet M–R probability distributions derived in Section 4.3 to explore the possible natures of the planets, i.e., by which broad categories they are most likely to fit (e.g., Earth/super-Earth/sub-Neptune/ice giants/evaporated cores). For our results on the natures of the planets, we simply refer to planet candidates b, c, and d also as planets, as the derived natures would only be meaningful if the planets exist at all. In the following, we split the planets into two groups based on our mass predictions.

Planets b, g, and h. When using the nonparametric M–R relationship from Ning et al. (2018), these three planets would have likely been sub-Neptunes, with predicted radii above $2{R}_{\oplus }$ and a density consistent with a gaseous-volatile envelope. However, as these planets have expected masses of 5 ${M}_{\oplus }$, when using both realizations of the power-law relationships from Otegi et al. (2020), these planets would be rocky planets, with radii less than 1.5 ${R}_{\oplus }$. Therefore, we find that these worlds are either sub-Neptunes or rocky worlds, depending on which M–R relationships better capture reality.

Planets c–f. As with the previous planets, we find that the nonparametric M–R relationship from Ning et al. (2018) describes these four planets as sub-Neptunes. The planets have expected masses between 5.4 and 7 $M\oplus$, and therefore fall in the overlap region for the Otegi et al. (2020) power-law relationships. If predominantly rocky, these planets would have radii between 1.7 and 1.8 ${R}_{\oplus }$ and would have approximately terrestrial density. These radii are close in size to the Fulton gap, a relative paucity of planets (Fulton et al. 2017), which may also separate primarily rocky worlds from those with significant gaseous envelopes. However, the data are also consistent with a different nature: under the volatile M–R relationship from Otegi et al. (2020), these planets would have planet radii between 2 and 2.4 ${R}_{\oplus }$, and thus densities and nature comparable to the ice giants in our solar system.

When comparing the irradiation of the planets in the τ Ceti system to photoevaporation models (e.g., Owen & Wu 2017; Carrera et al. 2018), we find that the three innermost planets b, g, and c would likely have suffered from a significant degree of atmospheric loss due to photoevaporation. Therefore, if these planets had any gaseous envelopes, it is likely that these atmospheres would have evaporated and left behind a rocky core (Pascucci et al. 2019). However, for planet h and outwards, the radiative flux received from τ Ceti would not be strong enough to strip the envelope from the planet. Thus it is likely that if these planets had volatile-rich atmospheres, they would still retain them to this day.

Based on RV measurements, T13 reported five planets in the τ Ceti system (labeled τ Ceti b–f). However, follow-up analysis by F17 found four planets in the system. They discovered new planets at 20 and 49 days (labeled τ Ceti g and h) and confirmed the presence of planets e and f at 163 and 636 days, but were not able to confirm the presence of planet candidates b–d at 14, 35, and 94 days. They found that the 14 day period signal becomes much less significant in later data sets when the 20 day signal is subtracted, and is, thus, attributed to stellar activity. Furthermore, the 35.4 day period signal was found to be much weaker than previously thought and close to the rotation period of the star. Finally, the 94 day period signal was noticeable but found in only a few of the data sets.

To assess the likelihood of planet candidates b–d in the system, we show that if planets were to be added within the orbit of planet e (P ≲ 162 days), the periods for τ Ceti b–d are near the local maxima of the relative likelihood in period space. This is true for both exoplanet period distributions; the clustered periods prescription is closer to the known signals for b and c, while the period ratio prescription is closer for d. Therefore, the predictions for planet candidates PxP–1, PxP–2, and PxP–3 provide contextual, statistical evidence that support the existence of planet candidates τ Ceti b–d, as does the dynamical stability analysis of the τ Ceti system. When only considering planets g, h, e, and f in the system, the average separation between each planet in units of mutual Hill radii is ∼31.4 (assuming all planets orbit near the disk inclination). The peak in separation seen in population statistics is ∼20 mutual Hill radii, with larger numbers (i.e., close to 40) indicating the presence of missing planets (e.g., Gilbert & Fabrycky 2020). Adding in τ Ceti b–d and PxP–4 with the period ratio description pushes the average separation down to ∼15.6 Hill radii.

Both flavors of the orbital period distributions explored in our analysis result in the robust prediction that a planet in the habitable zone of τ Ceti is very likely. This prediction is, as discussed below, consistent with available data. T13 identify a possible ∼315 day signal and performed extensive Keplerian modeling of the system both with and without that planet. They concluded that it was more likely an alias of the 168 day signal. F17 also found a relatively strong signal at ∼318 days and concurred that it was an alias of their 162 day signal. However, due to the gap of factor ∼4 in period space between planets e and f, DYNAMITE (and dynamically packing, in general) predicts another planet candidate, PxP–4, in the period range between those two known planets.

With the four-planet architecture found by F17, under the clustered periods assumption, our analysis finds that planet f is separated from the other planets in period space and is in its own cluster. Therefore, we predict another planet as close to the center of that cluster (i.e., the period of planet f) as dynamically possible, near an orbital period of 468 days. In contrast, under the period ratio assumption, our analysis places the new planet with symmetric period ratios between itself and planets e and f, at a period of 322 days. Given the mass of τ Ceti as $0.783\,{M}_{\odot }$, an orbital period of 322 days corresponds to an orbital semimajor axis of 0.848 au, and an orbital period of 468 days corresponds to orbital semimajor axis of 1.09 au. The conservative habitable zone for τ Ceti (effective temperature ${T}_{\mathrm{eff}}=5344$ K), as defined by Kopparapu et al. (2013) using their moist greenhouse and maximum greenhouse limits, lies between stellocentric distances of 0.703 and 1.26 au. Thus, PxP–4 would comfortably lie inside the habitable zone of τ Ceti.

As before, we explore the potential nature of PxP–4 by contrasting it to the exoplanet size–density trends. As explained below, we find that it may be a super-Earth, but could also be a volatile-rich sub-Neptune. We calculated the mass of the PxP–4 planet candidate by translating its predicted planet radius distribution into a mass distribution via our M–R relationship. This provided the predicted mass of 3.33 ${M}_{\oplus }$ for the power-law relationships from Otegi et al. (2020) and 6.30 ${M}_{\oplus }$ for the nonparametric M–R relationship from Ning et al. (2018). Planets of the former mass (3.33 ${M}_{\oplus }$) would likely always be rocky, as the lower limit for the volatile-rich population from Otegi et al. (2020) where anything below 5 ${M}_{\oplus }$ is considered rocky. Under the rocky assumption, 1 ${M}_{\oplus }$ lies at the 7th percentile of the planet mass distribution for the τ Ceti system. Planets with the latter mass (${M}_{p}\approx 6\mbox{--}7\,{M}_{\oplus }$) would have planet radii ∼1.8 ${R}_{\oplus }$ if they were rocky, but could also likely contain a gaseous envelope and have radii of $2.1\mbox{--}2.4\,{R}_{\oplus }$.

Our prediction of a habitable zone planet, PxP–4, in the τ Ceti system raises the question of whether this planet could be detected. In this section we explore the observational signatures expected from this planet and discuss what measurements may clarify whether it is a habitable zone super-Earth or a—presumably uninhabitable—sub-Neptune. Specifically, we will discuss the possibility of detecting PxP–4 via transits, RV, and direct imaging. Our analysis assumes that the inclination of PxP–4 is aligned with the disk and other planets in the system to within a few degrees. This assumption would make it very unlikely that PxP–4 is detectable via transit observations, leaving RV and direct imaging as the viable options for the next decades of observations.

To estimate the RV semi-amplitude, we assume that PxP–4's orbital eccentricity is low, as would be expected in a high-multiplicity system (e.g., Limbach & Turner 2015; Van Eylen et al. 2019). Given its value for $m\sin i$ of $3.3\mbox{--}6.3\,{M}_{\oplus }$, the RV semi-amplitude of PxP–4 would be ∼0.2–0.4 m s⁻¹, which is similar in magnitude to the signals detected from τ Ceti g and h. This value is at or just above the noise limit for τ Ceti on the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) spectrograph, as characterized by F17. The Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO) spectrograph (Pepe et al. 2010) would likely be able to reach the required precision to detect PxP–4, as it is able to reach ∼0.25 m s⁻¹ for Proxima Centauri (Suárez Mascareño et al. 2020), a fainter and more active star.

We will now explore τ Ceti 's PxP–4 as a potential target for direct imaging. The relative proximity of τ Ceti to the solar system makes this system one of the premier targets for next-generation direct-imaging systems, both from space and from the ground. A direct-imaging detection of a potentially habitable PxP–4 would be revolutionary, as it could be exploited for a broad variety of follow-up efforts to characterize the planet in depth and to address a multitude of new science questions (for a community report, see Apai et al. 2017). At a distance of only 3.65 pc, the orbital semimajor axis of 0.848 au of PxP–4 corresponds to an angular separation of 250 mas, which is on par to recently imaged super-Jupiter exoplanets (e.g., Lagrange et al. 2010; Macintosh et al. 2015). Unlike the hot young super-Jupiters imaged with extreme adaptive systems, however, PxP–4 is neither young, hot, or massive, and thus poses an orders-of-magnitude greater contrast challenge than anything imaged as of now.

While imaging PxP–4 with current state-of-the-art facilities is not possible, the feat may be within grasp of future facilities. Thermal emission from PxP–4 should peak close to ${\lambda }_{\mathrm{th}}=10\,\mu {\rm{m}}$ due to the equilibrium temperature of this habitable zone planet. Current efforts for detecting a habitable zone Earth-sized planet around α Centauri through a 100 h-long integration with the Very Large Telescope's VISIR instrument show it is possible (e.g., Kasper et al. 2019). Although τ Ceti is at a distance 2.4 times greater than α Centauri, PxP–4 may be a factor of a few larger than Earth, somewhat countering the greater distance. Thus, while the intensity of PxP–4 may be detectable with the sensitivity of present-day, ultra-deep thermal infrared images (see also Wagner 2020), its relatively small projected separation (250 mas) falls within the contrast-limited region of such images, where sensitivity is greatly reduced (K. Wagner et al. 2020, in preparation). The next-generation large ground-based telescopes, such as the Giant Magellan Telescope (GMT),⁴ the Thirty Meter Telescope (TMT), or the European Extremely Large Telescope (E-ELT), should deliver increased thermal infrared sensitivity and a factor of several smaller inner working angles (e.g., Quanz 2015; Mazin et al. 2019; Wang et al. 2019), making direct detection of τ Ceti PxP–4 a promising possibility.

PxP–4 may also be detected in the future via reflected (scattered) light imaging. The brightness of the planet in scattered light depends on the physical separation from the host star, the Bond albedo, the scattering phase angle, and the illumination phase, which complicate the detection, identification, and interpretation of directly imaged habitable planets (e.g., Bixel & Apai 2020). It is likely that PxP–4 would be ∼10⁻⁹ fainter than τ Ceti, requiring space-based, ultra-high-contrast imaging or interferometry to detect. Mission concepts that would be capable of detecting PxP–4 have been proposed such as the Large UltraViolet Optical InfraRed Surveyor (LUVOIR; The LUVOIR Team 2019) the Habitable Exoplanet Observatory (HabEx; Gaudi et al. 2020), and the Large Interferometer For Exoplanets (LIFE; Quanz 2019), and may become operational within the next two decades.

Thus, we conclude that PxP–4 is already likely detectable indirectly, via the RV modulations it imprints on its host star, and it will be directly detectable by the end of the current decade in thermal emission (with the extremely large ground-based telescopes), as well as an ideal target for space-based high-contrast imaging and spectroscopy in about two decades.

τ Ceti has long been a candidate to search for nearby life, and we find that the probability of having a rocky planet in the habitable zone is relatively high. The probability of additional planets to be sub-Neptunes with a gaseous/volatile envelope versus rocky super-Earths ranges from roughly equal to leaning toward rocky planets; across the M–R relationships 50%–75% of injected planets had a planet mass below 5 ${M}_{\oplus }$. However, the probability of finding specifically an Earth-like planet (0.5 ${M}_{\oplus }\lt {M}_{p}\ \lt$ 1.5 ${M}_{\oplus }$) is low. The nonparametric M–R relationship from Ning et al. (2018) had no significant probability of finding a planet with a mass between 0.5 and 1.5 ${M}_{\oplus }$, whereas the rocky and volatile populations of the power-law relationships from Otegi et al. (2020) had only ∼12% and ∼3%, respectively, in that mass range.

A super-Earth planet in the habitable zone around τ Ceti would have an RV semi-amplitude of 0.4 m s⁻¹, which is above the noise floor of current spectrographs like HARPS and ESPRESSO. However, even across a decade of observations, this signal would still be difficult to find, especially due to the alias caused by Earth's own orbital period. With even better extreme-precision RV (EPRV) measurements pushing down toward 1 cm s⁻¹ precision, the ability to observe a signal from rocky planets in the habitable zone around τ Ceti will become much easier, and EPRV data would even be able to find an Earth-like planet (RV semi-amplitude $\lesssim 0.1$ m s⁻¹) around τ Ceti.

The period ratio prescription has low probability of adding in an additional planet after inserting PxP–1-3 at the values for the known planet candidates b–d and adding PxP–4 at its relative likelihood maxima. These four predicted planets dynamically pack the system with relative period ratios for each pair of planets close to the Kepler statistical mean. The clustered periods prescription, after adding in PxP–1-3 at the values for planet candidates b–d and PxP–4 at its relative likelihood maxima, also allows for an additional planet candidate to be placed at ∼270 days, corresponding to an orbital semimajor axis of 0.754 au. This hypothetical planet, too, would be within the habitable zone assuming an inner limit of 0.703 au. This additional prediction from the clustered periods prescription is favorable as it fits the system parameters well. The integrated likelihood of the second injection is higher than the first injection, and the overall average planet separation in units of mutual Hill radii for the nine planet system prediction via the clustered periods prescription is 15.5. This is very similar to the eight-planet system prediction via the period ratio prescription of 15.6. The new injection test, after adding in the eight current candidates, is shown in Figure 4.

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