We present transit observations of the WASP-2 exoplanet system by the Apache Point Survey of Transit Lightcurves of Exoplanets (APOSTLE) program. Model fitting to these data allows us to improve measurements of the hot-Jupiter exoplanet WASP-2b and its orbital parameters by a factor of ~2 over prior studies; we do not find evidence for transit depth variations. We do find reduced chi^2 values greater than 1.0 in the observed minus computed transit times.
We apply two equilibrium tide theories – a constant-phase-lag model and a constant-time-lag model – to compute the obliquity evolution of terrestrial planets orbiting in the habitable zones around LMSs. The time for the obliquity to decrease from an Earth-like obliquity of 23.5° to 5°, the “tilt erosion time”, is compared to the traditional insolation habitable zone (IHZ) in the parameter space spanned by the semi-major axis a, the eccentricity e, and the stellar mass Ms. We also compute tidal heating and equilibrium rotation caused by obliquity tides as further constraints on habitability. The Super-Earth Gl581 d and the planet candidate Gl581 g are studied as examples for these tidal processes.
We evaluate the orbital evolution and several plausible origin scenarios for the mutually inclined orbits of υ And c and d. These two planets have orbital elements that oscillate with large amplitudes and lie close to the stability boundary. This configuration, and in particular the observed mutual inclination, demands an explanation. The planetary system may be influenced by a nearby low-mass star, υ And B, which could perturb the planetary orbits, but we find it cannot modify two coplanar orbits into the observed mutual inclination of 30°. However, it could incite ejections or collisions between planetary companions that subsequently raise the mutual inclination to >30°. Our simulated systems with large mutual inclinations tend to be further from the stability boundary than υ And, but we are able to produce similar systems. We conclude that scattering is a plausible mechanism to explain the observed orbits of υ And c and d, but we cannot determine whether the scattering was caused by instabilities among the planets themselves or by perturbations from υ And B. We also develop a procedure to quantitatively compare numerous properties of the observed system to our numerical models. Although we only implement this procedure to υ And, it may be applied to any exoplanetary system.
We study the dynamical stability of planetary systems consisting of one hypothetical terrestrial-mass planet (1 or 10 M ⊕) and one massive planet (10 M ⊕-10 M jup). We consider masses and orbits that cover the range of observed planetary system architectures (including non-zero initial eccentricities), determine the stability limit through N-body simulations, and compare it to the analytic Hill stability boundary. We show that for given masses and orbits of a two-planet system, a single parameter, which can be calculated analytically, describes the Lagrange stability boundary (no ejections or exchanges) but diverges significantly from the Hill stability boundary. However, we do find that the actual boundary is fractal, and therefore we also identify a second parameter which demarcates the transition from stable to unstable evolution. We show the portions of the habitable zones (HZs) of ρ CrB, HD 164922, GJ 674, and HD 7924 that can support a terrestrial planet. These analyses clarify the stability boundaries in exoplanetary systems and demonstrate that, for most exoplanetary systems, numerical simulations of the stability of potentially habitable planets are only necessary over a narrow region of the parameter space. Finally, we also identify and provide a catalog of known systems that can host terrestrial planets in their HZs.
The habitable zones (HZs) of main-sequence stars have traditionally been defined as the range of orbits that intercept the appropriate amount of stellar flux to permit surface water on a planet. Terrestrial exoplanets discovered to orbit M stars in these zones, which are close-in due to decreased stellar luminosity, may also undergo significant tidal heating. Tidal heating may span a wide range for terrestrial exoplanets and may significantly affect conditions near the surface. For example, if heating rates on an exoplanet are near or greater than that on Io (where tides drive volcanism that resurfaces the planet at least every 1 Myr) and produce similar surface conditions, then the development of life seems unlikely. On the other hand, if the tidal heating rate is less than the minimum to initiate plate tectonics, then CO2 may not be recycled through subduction, leading to a runaway greenhouse that sterilizes the planet. These two cases represent potential boundaries to habitability and are presented along with the range of the traditional HZ for main-sequence, low-mass stars. We propose a revised HZ that incorporates both stellar insolation and tidal heating. We apply these criteria to GJ 581 d and find that it is in the traditional HZ, but its tidal heating alone may be insufficient for plate tectonics.
We present N-body simulations of planetary accretion beginning with 1 km radius planetesimals in orbit about a 1 M⊙ star at 0.4 AU. The initial disk of planetesimals contains too many bodies for any current N-body code to integrate; therefore, we model a sample patch of the disk. Although this greatly reduces the number of bodies, we still track in excess of 105 particles. We consider three initial velocity distributions and monitor the growth of the planetesimals. The masses of some particles increase by more than a factor of 100. Additionally, the escape speed of the largest particle grows considerably faster than the velocity dispersion of the particles, suggesting impending runaway growth, although no particle grows large enough to detach itself from the power law size-frequency distribution. These results are in general agreement with previous statistical and analytical results.
The known extrasolar multiple-planet systems share a surprising dynamical attribute: they cluster just beyond the Hill stability boundary. Here we show that the planet-planet scattering model, which naturally explains the observed exoplanet eccentricity distribution, can reproduce the observed distribution of dynamical configurations. We calculated how each of our scattered systems would appear over an appropriate range of viewing geometries; as Hill stability is weakly dependent on the masses, the mass-inclination degeneracy does not significantly affect our results. We consider a wide range of initial planetary mass distributions and find that some are poor fits to the observed systems. In fact, many of our scattering experiments overproduce systems very close to the stability boundary. The distribution of dynamical configurations of two-planet systems may provide better discrimination between scattering models than the distribution of eccentricity. Our results imply that, at least in their inner regions which are weakly affected by gas or planetesimal disks, planetary systems should be “packed,” with no large gaps between planets.
Three planets with minimum masses less than 10 M⊕ orbit the star HD 40307, suggesting these planets may be rocky. However, with only radial velocity data, it is impossible to determine if these planets are rocky or gaseous. Here we exploit various dynamical features of the system in order to assess the physical properties of the planets. Observations allow for circular orbits, but a numerical integration shows that the eccentricities must be at least 10–4. Also, planets b and c are so close to the star that tidal effects are significant. If planet b has tidal parameters similar to the terrestrial planets in the solar system and a remnant eccentricity larger than 10–3, then, going back in time, the system would have been unstable within the lifetime of the star (which we estimate to be 6.1 ± 1.6 Gyr). Moreover, if the eccentricities are that large and the inner planet is rocky, then its tidal heating may be an order of magnitude greater than extremely volcanic Io, on a per unit surface area basis. If planet b is not terrestrial, e.g., Neptune-like, these physical constraints would not apply. This analysis suggests the planets are not terrestrial-like, and are more like our giant planets. In either case, we find that the planets probably formed at larger radii and migrated early-on (via disk interactions) into their current orbits. This study demonstrates how the orbital and dynamical properties of exoplanet systems may be used to constrain the planets’ physical properties.
We study the dynamical stability of an additional, potentially habitable planet in the HD 47186 planetary system. Two planets are currently known in this system: a “hot Neptune” with a period of 4.08 days and a Saturn-mass planet with a period of 3.7 years. Here we consider the possibility that one or more undetected planets exist between the two known planets and possibly within the habitable zone (HZ) in this system. Given the relatively low masses of the known planets, additional planets could have masses $ \rlap{<}{\lower1.0ex\hbox{$\sim $}}10 {\,M_\oplus }$, and hence be terrestrial-like and further improving potential habitability. We perform N-body simulations to identify the stable zone between planets b and c and find that much of the inner HZ can harbor a 10 M ⊕ planet. With the current radial velocity threshold of ~1 m s–1, an additional planet should be detectable if it lies at the inner edge of the habitable zone at 0.8 AU. We also show that the stable zone could contain two additional planets of 10 M ⊕ each if their eccentricities are lower than ~0.3.
The tidal heating of hypothetical rocky (or terrestrial) extrasolar planets spans a wide range of values depending on stellar masses and initial orbits. Tidal heating may be sufficiently large (in many cases, in excess of radiogenic heating) and long-lived to drive plate tectonics, similar to the Earth’s, which may enhance the planet’s habitability. In other cases, excessive tidal heating may result in Io-like planets with violent volcanism, probably rendering them unsuitable for life. On water-rich planets, tidal heating may generate subsurface oceans analogous to Europa’s with similar prospects for habitability. Tidal heating may enhance the outgassing of volatiles, contributing to the formation and replenishment of a planet’s atmosphere. To address these issues, we model the tidal heating and evolution of hypothetical extrasolar terrestrial planets. The results presented here constrain the orbital and physical properties required for planets to be habitable.
Tides raised on a planet by the gravity of its host star can reduce the planet’s orbital semi-major axis and eccentricity. This effect is only relevant for planets orbiting very close to their host stars. The habitable zones of low-mass stars are also close in, and tides can alter the orbits of planets in these locations. We calculate the tidal evolution of hypothetical terrestrial planets around low-mass stars and show that tides can evolve planets past the inner edge of the habitable zone, sometimes in less than 1 billion years. This migration requires large eccentricities (>0.5) and low-mass stars (≲0.35 M⊙). Such migration may have important implications for the evolution of the atmosphere, internal heating, and the Gaia hypothesis. Similarly, a planet that is detected interior to the habitable zone could have been habitable in the past. We consider the past habitability of the recently discovered, ∼5 M⊕ planet, Gliese 581 c. We find that it could have been habitable for reasonable choices of orbital and physical properties as recently as 2 Gyr ago. However, when constraints derived from the additional companions are included, most parameter choices that indicate past habitability require the two inner planets of the system to have crossed their mutual 3:1 mean motion resonance. As this crossing would likely have resulted in resonance capture, which is not observed, we conclude that Gl 581 c was probably never habitable. Astrobiology 8, 557–568.
To date, two planetary systems have been discovered with close-in, terrestrial-mass planets forumla. Many more such discoveries are anticipated in the coming years with radial velocity and transit searches. Here we investigate the different mechanisms that could form ‘hot Earths’ and their observable predictions. Models include: (1) in situ accretion; (2) formation at larger orbital distance followed by inward ‘type 1’ migration; (3) formation from material being ‘shepherded’ inward by a migrating gas giant planet; (4) formation from material being shepherded by moving secular resonances during dispersal of the protoplanetary disc; (5) tidal circularization of eccentric terrestrial planets with close-in perihelion distances and (6) photoevaporative mass-loss of a close-in giant planet. Models 1–4 have been validated in previous work. We show that tidal circularization can form hot Earths, but only for relatively massive planets forumla with very close-in perihelion distances (≲0.025 au), and even then the net inward movement in orbital distance is at most only 0.1–0.15 au. For planets of less than forumla, photoevaporation can remove the planet’s envelope and leave behind the solid core on a Gyr time-scale, but only for planets inside 0.025–0.05 au. Using two quantities that are observable by current and upcoming missions, we show that these models each produce unique signatures, and can be observationally distinguished. These observables are the planetary system architecture (detectable with radial velocities, transits and transit timing) and the bulk composition of transiting close-in terrestrial planets (measured by transits via the planet’s radius).
We present results of numerical simulations that examine the dynamical stability of known planetary systems, a star with two or more planets. First we vary the initial conditions of each system on the basis of observational data. We then determine regions of phase space that produce stable planetary configurations. For each system we perform 1000 ~ 106 yr integrations. We examine υ And, HD 83443, GJ 876, HD 82943, 47 UMa, HD 168443, and the solar system. We find that the resonant systems, two planets in a first-order mean motion resonance (HD 82943 and GJ 876) have very narrow zones of stability. The interacting systems, not in first-order resonance, but able to perturb each other (υ And, 47 UMa, and the solar system), have broad stable regions. The separated systems, two planets beyond 10 : 1 resonance (we examine only HD 83443 and HD 168443) are fully stable. We find that the best fits to the interacting and resonant systems place them very close to unstable regions. The boundary in phase space between stability and instability depends strongly on the eccentricities and (if applicable) the proximity of the system to perfect resonance. Furthermore, we also find that the longitudes of periastron circulate in chaotic systems but librate in regular systems. In addition to 106 yr integrations, we also examined stability on ~108 yr timescales. For each system we ran ~10 long-term simulations, and find that the Keplerian fits to these systems all contain configurations that are regular on this timescale.
Because of the high eccentricities (~0.3) of two of the possible planets about the star υ Andromeda, the stability of the system requires careful study. We present results of 1000 numerical simulations which explore the orbital parameter space as constrained by the observations. The orbital parameters of each planet are chosen from a Gaussian error distribution, and the resulting configuration is integrated for 1 Myr. We find that 84% of these integrations are stable. Configurations in which the eccentricity of the third planet is lesssim0.3 are always stable, but when the eccentricity is gsim0.45, the system is always unstable, typically producing a close encounter between the second and third planets. A similar exercise with the gas giants in our solar system sampled with the same error distribution was performed. Approximately 81% of these simulations were stable for 106 yr.