Middle Planet Research Articles

Secular Dynamics of Compact Three-planet Systems

The secular Laplace–Lagrange orbital solution, decomposing eccentricities into a set of uniformly precessing eigenmodes, is a classical result that is typically solved numerically. However, in the limit where orbits are closely spaced, several simplifications make it possible to make analytical progress. We derive simple expressions for the eccentricity eigenmodes in a coplanar three-planet system where the middle planet is much less massive than its neighbors, and we show that these approximate the true eigenmodes of more general systems with three massive planets in various limits. These results provide intuition for the secular dynamics of real systems, and have applications for understanding the stability boundary for compact multiplanet systems.

Sandwiched planet formation: restricting the mass of a middle planet

ABSTRACT We conduct gas and dust hydrodynamical simulations of protoplanetary discs with one and two embedded planets to determine the impact that a second planet located further out in the disc has on the potential for subsequent planet formation in the region locally exterior to the inner planet. We show how the presence of a second planet has a strong influence on the collection of solid material near the inner planet, particularly when the outer planet is massive enough to generate a maximum in the disc’s pressure profile. This effect in general acts to reduce the amount of material that can collect in a pressure bump generated by the inner planet. When viewing the inner pressure bump as a location for potential subsequent planet formation of a third planet, we therefore expect that the mass of such a planet will be smaller than it would be in the case without the outer planet, resulting in a small planet being sandwiched between its neighbours – this is in contrast to the expected trend of increasing planet mass with radial distance from the host star. We show that several planetary systems have been observed that do not show this trend but instead have a smaller planet sandwiched in between two more massive planets. We present the idea that such an architecture could be the result of the subsequent formation of a middle planet after its two neighbours formed at some earlier stage.

Three young planets around the K-dwarf K2-198: high-energy environment, evaporation history, and expected future

ABSTRACT Planets orbiting young stars are thought to experience atmospheric evaporation as a result of the host stars’ high-magnetic activity. We study the evaporation history and expected future of the three known transiting exoplanets in the young multiplanet system K2-198. Based on spectroscopic and photometric measurements, we estimate an age of the K-dwarf host star between 200 and 500 Myr, and calculate the high-energy environment of these planets using eROSITA X-ray measurements. We find that the innermost planet K2-198c has likely lost its primordial envelope within the first few 10s of Myr regardless of the age at which the star drops out of the saturated X-ray regime. For the two outer planets, a range of initial envelope mass fractions is possible, depending on the not-yet-measured planetary mass and the stars’ spin-down history. Regarding the future of the system, we find that the outermost planet K2-198b is stable against photoevaporation for a wide range of planetary masses, while the middle planet K2-198d is only able to retain an atmosphere for a mass range between ∼7 and 18 M⊕. Lower mass planets are too susceptible to mass-loss, and a very thin present-day envelope for higher mass planets is easily lost with the estimated mass-loss rates. Our results support the idea that all three planets started out above the radius valley in the (sub-)Neptune regime and were then transformed into their current states by atmospheric evaporation, but also stress the importance of measuring planetary masses for (young) multiplanet systems before conducting more detailed photoevaporation simulations.

The shared evaporation history of three sub-Neptunes spanning the radius–period valley of a hyades star

ABSTRACT We model the evaporation histories of the three planets around K2-136, a K-dwarf in the Hyades open cluster with an age of 700 Myr. The star hosts three transiting planets, with radii of 1.0, 3.0, and 1.5 Earth radii, where the middle planet lies above the radius–period valley and the inner and outer planets are below. We use an XMM–Newton observation to measure the XUV radiation environment of the planets, finding that the X-ray activity of K2-136 is lower than predicted by models but typical of similar Hyades members. We estimate the internal structure of each planet, and model their evaporation histories using a range of structure and atmospheric escape formulations. While the precise X-ray irradiation history of the system may be uncertain, we exploit the fact that the three planets must have shared the same history. We find that the Earth-sized K2-136b is most likely rocky, with any primordial gaseous envelope being lost within a few Myr. The sub-Neptune, K2-136c, has an envelope contributing 1–1.7 per cent of its mass that is stable against evaporation thanks to the high mass of its rocky core, whilst the super-Earth, K2-136d, must have a mass at the upper end of the allowed range in order to retain any of its envelope. Our results are consistent with all three planets beginning as sub-Neptunes that have since been sculpted by atmospheric evaporation to their current states, stripping the envelope from planet b and removing most from planet d whilst preserving planet c above the radius-period valley.

Detection of two additional circumbinary planets around Kepler-451

ABSTRACT We announce the detection of two new planetary-mass companions around Kepler-451 binary system in addition to the one detected previously based on eclipse timing variation analysis. We found that an inner planet with 43 d period with a minimum mass of 1.76 Mjup and an outer one with a ∼1800 d orbital period with a minimum mass of 1.61 Mjup can explain the periodic variations in the residuals of the one-planet fit of the eclipse timings. We updated the orbital period of the middle planet as 406 d, and determined its eccentricity as 0.33. The newly discovered outer planet is also on an eccentric orbit (0.29), while the innermost planet was assumed to have a circular orbit. All three Jovian planets have similar masses, and our dynamical stability test yields that the system is stable.

A Five-Planet Resonant Chain: Reevaluation of the Kepler-80 System

Abstract Since the launch of the Kepler space telescope in 2009 and the subsequent K2 mission, hundreds of multiplanet systems have been discovered. The study of such systems, both as individual systems and as a population, leads to a better understanding of planetary formation and evolution. Kepler-80, a K dwarf hosting six super-Earths, was the first system known to have four planets in a chain of resonances, a repeated geometric configuration. Transiting planets in resonant chains can enable us to estimate not only the planets’ orbits and sizes but also their masses. Since the original resonance analysis and TTV fitting of Kepler-80, a new planet has been discovered whose signal likely altered the measured masses of the other planets. Here, we determine masses and orbits for all six planets hosted by Kepler-80 by direct forward photodynamical modeling of the light curve of this system. We then explore the resonant behavior of the system. We find that the four middle planets are in a resonant chain, but that the outermost planet only dynamically interacts in ∼14% of our solutions. We also find that the system and its dynamic behavior are consistent with in situ formation and compare our results to two other resonant chain systems, Kepler-60 and TRAPPIST-1.

Semianalytical model for planetary resonances

Context. Planetary resonances are a common dynamical mechanism acting on planetary systems. However, no general model for describing their properties exists, particularly for commensurabilities of any order and arbitrary eccentricity and inclination values. Aims. We present a semianalytical model that describes the resonance strength, width, location and stability of fixed points, and periods of small-amplitude librations. The model is valid for any two gravitationally interacting massive bodies, and is thus applicable to planets around single or binary stars. Methods. Using a theoretical framework in the Poincaré and Jacobi reference system, we developed a semianalytical method that employs a numerical evaluation of the averaged resonant disturbing function. Validations of the model are presented that compare its predictions with dynamical maps for real and fictitious systems. Results. The model describes many dynamical features of planetary resonances very well. Notwithstanding the good agreement found in all cases, a small deviation is noted in the location of the resonance centers for circumbinary systems. As a consequence of its application to the HD 31527 system, we found that the updated best-fit solution leads to a high-eccentricity stable libration between the middle and outer planets inside the 16/3 mean-motion resonance (MMR). This is the first planetary system whose long-term dynamics appears dominated by such a high-order commensurability. In the case of circumbinary planets, the overlap of N/1 mean-motion resonances coincides very well with the size of the global chaotic region close to the binary, as well as its dependence on the mutual inclination.

Near mean motion resonance of terrestrial planet pair induced by giant planet: application to Kepler-68 system

ABSTRACT In this work, we investigate configuration formation of two inner terrestrial planets near mean motion resonance (MMR) induced by the perturbation of a distant gas giant for the Kepler-68 system, by conducting thousands of numerical simulations. The results show that the formation of terrestrial planets is relevant to the speed of type I migration, the mass of planets, and the existence of giant planet. The mass and eccentricity of the giant planet may play a crucial role in shaping the final configuration of the system. The inner planet pair can be trapped in 5:3 or 7:4 MMRs if the giant planet revolves the central star with an eccentric orbit, which is similar to the observed configuration of Kepler-68. Moreover, we find that the eccentricity of the middle planet can be excited to roughly 0.2 if the giant planet is more massive than 5 MJ; otherwise, the terrestrial planets are inclined to remain in near-circular orbits. Our study may provide a likely formation scenario for the planetary systems that harbour several terrestrial planets near MMRs inside and one gas giant exterior to them.

Discovery of a Third Transiting Planet in the Kepler-47 Circumbinary System

Abstract Of the nine confirmed transiting circumbinary planet systems, only Kepler-47 is known to contain more than one planet. Kepler-47 b (the “inner planet”) has an orbital period of 49.5 days and a radius of about 3 R ⊕. Kepler-47 c (the “outer planet”) has an orbital period of 303.2 days and a radius of about 4.7 R ⊕. Here we report the discovery of a third planet, Kepler-47 d (the “middle planet”), which has an orbital period of 187.4 days and a radius of about 7 R ⊕. The presence of the middle planet allows us to place much better constraints on the masses of all three planets, where the 1σ ranges are less than 26 M ⊕, between 7–43 M ⊕, and between 2–5 M ⊕ for the inner, middle, and outer planets, respectively. The middle and outer planets have low bulk densities, with g cm−3 and ρ outer < 0.26 g cm−3 at the 1σ level. The two outer planets are “tightly packed,” assuming the nominal masses, meaning no other planet could stably orbit between them. All of the orbits have low eccentricities and are nearly coplanar, disfavoring violent scattering scenarios and suggesting gentle migration in the protoplanetary disk.

K2-155: A Bright Metal-poor M Dwarf with Three Transiting Super-Earths

Abstract We report on the discovery of three transiting super-Earths around K2-155 (EPIC 210897587), a relatively bright early M dwarf (V = 12.81 mag) observed during Campaign 13 of the NASA K2 mission. To characterize the system and validate the planet candidates, we conducted speckle imaging and high-dispersion optical spectroscopy, including radial velocity measurements. Based on the K2 light curve and the spectroscopic characterization of the host star, the planet sizes and orbital periods are and 6.34365 ± 0.00028 days for the inner planet; and 13.85402 ± 0.00088 days for the middle planet; and and 40.6835 ± 0.0031 days for the outer planet. The outer planet (K2-155d) is near the habitable zone, with an insolation 1.67 ± 0.38 times that of the Earth. The planet’s radius falls within the range between that of smaller rocky planets and larger gas-rich planets. To assess the habitability of this planet, we present a series of three-dimensional global climate simulations, assuming that K2-155d is tidally locked and has an Earth-like composition and atmosphere. We find that the planet can maintain a moderate surface temperature if the insolation proves to be smaller than ∼1.5 times that of the Earth. Doppler mass measurements, transit spectroscopy, and other follow-up observations should be rewarding, as K2-155 is one of the optically brightest M dwarfs known to harbor transiting planets.

Assessing the Habitability of the TRAPPIST-1 System Using a 3D Climate Model

Abstract The TRAPPIST-1 system provides an extraordinary opportunity to study multiple terrestrial extrasolar planets and their atmospheres. Here, we use the National Center for Atmospheric Research Community Atmosphere Model version 4 to study the possible climate and habitability of the planets in the TRAPPIST-1 system. We assume the worlds are ocean-covered, with atmospheres composed of N2, CO2, and H2O, and with orbital and geophysical properties defined from observation. Model results indicate that the inner three planets (b, c, and d) presently reside interior to the inner edge of the traditional liquid water habitable zone. Thus, if water ever existed on the inner planets, they would have undergone a runaway greenhouse and lost their water to space, leaving them dry today. Conversely, the outer three planets (f, g, and h) fall beyond the maximum CO2 greenhouse outer edge of the habitable zone. Model results indicate that the outer planets cannot be warmed, despite having as much as 30 bar CO2 atmospheres, instead entering a snowball state. The middle planet (e) represents the best chance for a presently habitable ocean-covered world in the TRAPPIST-1 system. Planet e can maintain at least some habitable surface area with 0–2 bar CO2, depending on the background N2 content. Near-present-day Earth surface temperatures can be maintained for an ocean-covered planet e with either 1 bar N2 + 0.4 bar CO2, or a 1.3 bar pure CO2 atmosphere.

The mass of the Mars-sized exoplanet Kepler-138 b from transit timing.

Extrasolar planets that pass in front of their host star (transit) cause a temporary decrease in the apparent brightness of the star, providing a direct measure of the planet's size and orbital period. In some systems with multiple transiting planets, the times of the transits are measurably affected by the gravitational interactions between neighbouring planets. In favourable cases, the departures from Keplerian orbits (that is, unaffected by gravitational effects) implied by the observed transit times permit the planetary masses to be measured, which is key to determining their bulk densities. Characterizing rocky planets is particularly difficult, because they are generally smaller and less massive than gaseous planets. Therefore, few exoplanets near the size of Earth have had their masses measured. Here we report the sizes and masses of three planets orbiting Kepler-138, a star much fainter and cooler than the Sun. We determine that the mass of the Mars-sized inner planet, Kepler-138 b, is 0.066(+0.059)(-0.037) Earth masses. Its density is 2.6(+2.4)(-1.5) grams per cubic centimetre. The middle and outer planets are both slightly larger than Earth. The middle planet's density (6.2(+5.8)(-3.4) grams per cubic centimetre) is similar to that of Earth, and the outer planet is less than half as dense at 2.1(+2.2)(-1.2) grams per cubic centimetre, implying that it contains a greater portion of low-density components such as water and hydrogen.

The HU Aqr planetary system hypothesis revisited

We study the mid-egress eclipse timing data gathered for the cataclysmic binary HU Aquarii during the years 1993-2014. The (O-C) residuals were previously attributed to a single ~7 Jupiter mass companion in ~5 au orbit or to a stable 2-planet system with an unconstrained outermost orbit. We present 22 new observations gathered between June, 2011 and July, 2014 with four instruments around the world. They reveal a systematic deviation of ~60 - 120 seconds from the older ephemeris. We re-analyse the whole set of the timing data available. Our results provide an erratum to the previous HU Aqr planetary models, indicating that the hypothesis for a third and fourth body in this system is uncertain. The dynamical stability criterion and a particular geometry of orbits rule out coplanar 2-planet configurations. A putative HU Aqr planetary system may be more complex, e.g., highly non-coplanar. Indeed, we found examples of 3-planet configurations with the middle planet in a retrograde orbit, which are stable for at least 1Gyr, and consistent with the observations. The (O-C) may be also driven by oscillations of the gravitational quadrupole moment of the secondary, as predicted by the Lanza et al. modification of the Applegate mechanism. Further systematic, long-term monitoring of HU Aqr is required to interpret the (O-C) residuals.

PLANET HUNTERS. VII. DISCOVERY OF A NEW LOW-MASS, LOW-DENSITY PLANET (PH3 C) ORBITING KEPLER-289 WITH MASS MEASUREMENTS OF TWO ADDITIONAL PLANETS (PH3 B AND D)

We report the discovery of one newly confirmed planet ($P=66.06$ days, $R_{\rm{P}}=2.68\pm0.17R_\oplus$) and mass determinations of two previously validated Kepler planets, Kepler-289 b ($P=34.55$ days, $R_{\rm{P}}=2.15\pm0.10R_\oplus$) and Kepler-289-c ($P=125.85$ days, $R_{\rm{P}}=11.59\pm0.10R_\oplus$), through their transit timing variations (TTVs). We also exclude the possibility that these three planets reside in a $1:2:4$ Laplace resonance. The outer planet has very deep ($\sim1.3%$), high signal-to-noise transits, which puts extremely tight constraints on its host star's stellar properties via Kepler's Third Law. The star PH3 is a young ($\sim1$ Gyr as determined by isochrones and gyrochronology), Sun-like star with $M_*=1.08\pm0.02M_\odot$, $R_*=1.00\pm0.02R_\odot$, and $T_{\rm{eff}}=5990\pm38$ K. The middle planet's large TTV amplitude ($\sim5$ hours) resulted either in non-detections or inaccurate detections in previous searches. A strong chopping signal, a shorter period sinusoid in the TTVs, allows us to break the mass-eccentricity degeneracy and uniquely determine the masses of the inner, middle, and outer planets to be $M=7.3\pm6.8M_\oplus$, $4.0\pm0.9M_\oplus$, and $M=132\pm17M_\oplus$, which we designate PH3 b, c, and d, respectively. Furthermore, the middle planet, PH3 c, has a relatively low density, $\rho=1.2\pm0.3$ g/cm$^3$ for a planet of its mass, requiring a substantial H/He atmosphere of $2.1^{+0.8}_{-0.3}%$ by mass, and joins a growing population of low-mass, low-density planets.

SPATIALLY RESOLVED SPECTROSCOPY OF THE EXOPLANET HR 8799 c

HR 8799 is a multi-planet system detected in direct imaging, with three companions known so far. Here, we present spatially resolved VLT/NACO 3.88--4.10 micron spectroscopy of the middle planet, HR 8799 c, which has an estimated mass of ~10 Mjup, temperature of ~1100 K and projected separation of 38 AU. The spectrum shows some differences in the continuum from existing theoretical models, particularly longwards of 4 microns, implying that detailed cloud structure or non-equilibrium conditions may play an important role in the physics of young exoplanetary atmospheres.

A THIRD GIANT PLANET ORBITING HIP 14810

We present new precision radial velocities and a three-planet Keplerian orbit fit for the V = 8.5, G5 V star HIP 14810. We began observing this star at Keck Observatory as part of the N2K Planet Search Project. Wright et al. announced the inner two planets to this system, and subsequent observations have revealed the outer planet and the proper orbital solution for the middle planet. The planets have minimum masses of 3.9, 1.3, and 0.6 M_(Jup) and orbital periods of 6.67, 147.7, and 952 day, respectively. We have numerically integrated the family of orbital solutions consistent with the data and find that they are stable for at least 10^6 yr. Our photometric search shows that the inner planet does not transit.

Planet–planet scattering in the upsilon Andromedae system

Doppler spectroscopy has detected 152 planets around nearby stars. A major puzzle is why many of their orbits are highly eccentric; all planets in our Solar System are on nearly circular orbits, as is expected if they formed by accretion processes in a protostellar disk. Several mechanisms have been proposed to generate large eccentricities after planet formation, but so far there has been little observational evidence to support any particular model. Here we report that the current orbital configuration of the three giant planets around upsilon Andromedae (upsilon And) probably results from a close dynamical interaction with another planet, now lost from the system. The planets started on nearly circular orbits, but chaotic evolution caused the outer planet (upsilon And d) to be perturbed suddenly into a higher-eccentricity orbit. The coupled evolution of the system then causes slow periodic variations in the eccentricity of the middle planet (upsilon And c). Indeed, we show that upsilon And c periodically returns to a very nearly circular state every 6,700 years.