Symplectic integrators separate a problem into parts that can be solved in isolation, alternately advancing these sub-problems to approximate the evolution of the complete system. Problems with a single, dominant mass can use mixed-variable symplectic (MVS) integrators that separate the problem into Keplerian motion of satellites about the primary, and satellite-satellite interactions. Here, we examine T + V algorithms, where the problem is separated into kinetic T and potential energy V terms. T + V integrators are typically less efficient than MVS algorithms. This difference is reduced by using different step sizes for primary-satellite and satellite-satellite interactions. The T + V method is improved further using fourth and sixth-order algorithms that include force gradients and symplectic correctors. We describe three sixth-order algorithms, containing two or three force evaluations per step, that are competitive with MVS in some cases. Round-off errors for T + V integrators can be reduced by several orders of magnitude, at almost no computational cost, using a simple modification that keeps track of accumulated changes in the coordinates and momenta. This makes T + V algorithms desirable for long term, high-accuracy calculations.

Of the solar system's four terrestrial planets, the origin of Mercury is perhaps the most mysterious. Modern numerical simulations designed to model the dynamics of terrestrial planet formation systematically fail to replicate Mercury, which possesses just 5% of the mass of Earth and the highest orbital eccentricity and inclination among the planets. However, Mercury's large iron-rich core and low volatile inventory stand out among the inner planets, and seem to imply a violent collisional origin. Because most algorithms used for simulating terrestrial accretion do not consider the effects of collisional fragmentation, it has been difficult to test these collisional hypotheses within the larger context of planet formation. Here, we analyze a large suite of terrestrial accretion models that account for the fragmentation of colliding bodies. We find that planets with core mass fractions boosted as a result of repeated hit-and-run collisions are produced in 90% of our simulations. While many of these planets are similar to Mercury in mass, they rarely lie on Mercury-like orbits. Furthermore, we perform an additional batch of simulations designed to specifically test the single giant impact origin scenario. We find less than a 1% probability of simultaneously replicating the Mercury-Venus dynamical spacing and the terrestrial system's degree of orbital excitation after such an event. While dynamical models have made great strides in understanding Mars' low mass, their inability to form accurate Mercury analogs remains a glaring problem.

We describe an analytic model for an evolving protoplanetary disk driven by viscosity and a disk wind. The disk is heated by stellar irradiation and energy generated by viscosity. The evolution is controlled by three parameters: (i) the inflow velocity toward the central star at a reference distance and temperature, (ii) the fraction of this inflow caused by the disk wind, and (iii) the mass-loss rate via the wind relative to the inward flux in the disk. The model gives the disk midplane temperature and surface density as a function of time and distance from the star. It is intended to provide an efficient way to calculate conditions in a protoplanetary disk for use in simulations of planet formation. In the model, disks dominated by viscosity spread radially while losing mass onto the star. Radial spreading is the main factor reducing the surface density in the inner disk. The disk mass remains substantial at late times. Temperatures in the inner region are high at early times due to strong viscous heating. Disks dominated by a wind undergo much less radial spreading and weaker viscous heating. These disks have a much lower mass at late times than purely viscous disks. When mass loss via a wind is significant, the surface density gradient in the inner disk becomes shallower, and the slope can become positive in extreme cases.

How plants respond physiologically to leaf warming and low water availability may determine how they will perform under future climate change. In 2015-2016, an unprecedented drought occurred across Amazonia with record-breaking high temperatures and low soil moisture, offering a unique opportunity to evaluate the performances of Amazonian trees to a severe climatic event. We quantified the responses of leaf water potential, sap velocity, whole-tree hydraulic conductance (K-wt), turgor loss and xylem embolism, during and after the 2015-2016 El Nino for five canopy-tree species. Leaf/xylem safety margins (SMs), sap velocity and K-wt showed a sharp drop during warm periods. SMs were negatively correlated with vapour pressure deficit, but had no significant relationship with soil water storage. Based on our calculations of canopy stomatal and xylem resistances, the decrease in sap velocity and K-wt was due to a combination of xylem cavitation and stomatal closure. Our results suggest that warm droughts greatly amplify the degree of trees' physiological stress and can lead to mortality. Given the extreme nature of the 2015-2016 El Nino and that temperatures are predicted to increase, this work can serve as a case study of the possible impact climate warming can have on tropical trees.

Variations in the axial tilt, or obliquity, of terrestrial planets can affect their climates and therefore their habitability. Kepler-62f is a 1.4 R-circle plus planet orbiting within the habitable zone of its K2 dwarf host star. We perform N-body simulations that monitor the evolution of obliquity of Kepler-62f for 10-million-year timescales to explore the effects on model assumptions, such as the masses of the Kepler-62 planets and the possibility of outer bodies. Significant obliquity variation occurs when the rotational precession frequency overlaps with one or more of the secular orbital frequencies, but most variations are limited to less than or similar to 10 degrees. Moderate variations (similar to 10-20 degrees) can occur over a broader range of initial obliquities when the relative nodal longitude (Delta ohm) overlaps with the frequency and phase of a given secular mode. However, we find that adding outer gas giants on long-period orbits (similar to 1000 days) can produce large (similar to 60 degrees) variations in obliquity if Kepler-62f has a very rapid (4 h) rotation period. The possibility of giant planets on long-period orbits impacts the climate and habitability of Kepler-62f through variations in the latitudinal surface flux, where large variations can occur on million year timescales.

Conventionally, a habitable planet is one that can support liquid water on its surface. Habitability depends on temperature, which is set by insolation and the greenhouse effect, due mainly to CO(2)and water vapor. The CO(2)level is increased by volcanic outgassing and decreased by continental and seafloor weathering. Here, I examine the climate evolution of Earth-like planets using a globally averaged climate model that includes both weathering types. Climate is sensitive to the relative contributions of continental and seafloor weathering, even when the total weathering rate is fixed. Climate also depends strongly on the dependence of seafloor weathering on CO(2)partial pressure. Both these factors are uncertain. Earth-like planets have two equilibrium climate states: (i) an ice-free state where outgassing is balanced by both weathering types, and (ii) an ice-covered state where outgassing is balanced by seafloor weathering alone. The second of these has not been explored in detail before. For some planets, neither state exists, and the climate cycles between ice-covered and ice-free states. For some other planets, both equilibria exist, and the climate depends on the initial conditions. Insolation increases over time due to stellar evolution, so a planet usually encounters the ice-covered equilibrium first. Such a planet will remain ice covered, even if the ice-free state appears subsequently, unless the climate receives a large perturbation. The ice-covered equilibrium state covers a large fraction of phase space for Earth-like planets. Many planets conventionally assigned to a star's habitable zone may be rendered uninhabitable as a result.

An episode of dynamical instability is thought to have sculpted the orbital structure of the outer solar system. When modeling this instability, a key constraint comes from Jupiter's fifth eccentric mode (quantified by its amplitude M-55), which is an important driver of the solar system's secular evolution. Starting from commonly-assumed near-circular orbits, the present-day giant planets' architecture lies at the limit of numerically generated systems, and M-55 is rarely excited to its true value. Here we perform a dynamical analysis of a large batch of artificially triggered instabilities, and test a variety of configurations for the giant planets' primordial orbits. In addition to more standard setups, and motivated by the results of modern hydrodynamical simulations of the giant planets' evolution within the primordial gaseous disk, we consider the possibility that Jupiter and Saturn emerged from the nebular gas locked in 2:1 resonance with non-zero eccentricities. We show that, in such a scenario, the modern Jupiter-Saturn system represents a typical simulation outcome, and M-55 is commonly matched. Furthermore, we show that Uranus and Neptune's final orbits are determined by a combination of the mass in the primordial Kuiper belt and that of an ejected ice giant.

The absence of planets interior to Mercury continues to puzzle terrestrial-planet formation models, particularly when contrasted with the relatively high derived occurrence rates of short-period planets around Sun-like stars. Recent work proposed that the majority of systems hosting hot super-Earths attain their orbital architectures through an epoch of dynamical instability after forming in quasi-stable, tightly packed configurations. Isotopic evidence seems to suggest that the formation of objects in the super-Earth-mass regime is unlikely to have occurred in the solar system as the terrestrial-forming disk is thought to have been significantly mass deprived starting around 2 Myr after the formation of calcium-aluminum-rich inclusions-a consequence of either Jupiter's growth or an intrinsic disk feature. Nevertheless, terrestrial-planet formation models and high-resolution investigations of planetesimal dynamics in the gas-disk phase occasionally find that quasi-stable protoplanets with mass comparable to that of Mars emerge in the vicinity of Mercury's modern orbit. In this paper, we investigate whether it is possible for a primordial configuration of such objects to be cataclysmically destroyed in a manner that leaves Mercury behind as the sole survivor without disturbing the other terrestrial worlds. We use numerical simulations to show that this scenario is plausible. In many cases, the surviving Mercury analog experiences a series of erosive impacts, thereby boosting its Fe/Si ratio. A caveat of our proposed genesis scenario for Mercury is that Venus typically experiences at least one late giant impact.

The formation of gas-giant planets within the lifetime of a protoplanetary disk is challenging especially far from a star. A promising model for the rapid formation of giant-planet cores is pebble accretion in which gas drag during encounters leads to high accretion rates. Most models of pebble accretion consider disks with a monotonic, radial pressure profile. This causes a continuous inward flux of pebbles and inefficient growth. Here we examine planet formation in a disk with multiple, intrinsic pressure bumps. In the outer disk, pebbles become trapped near these bumps allowing rapid growth under suitable conditions. In the inner disk, pebble traps may not exist because the inward gas advection velocity is too high. Pebbles here are rapidly removed. In the outer disk, growth is very sensitive to the initial planet mass and the strength of turbulence. This is because turbulent density fluctuations raise planetary eccentricities, increasing the planet-pebble relative velocity. Planetary seeds above a distance-dependent critical mass grow to a Jupiter mass in 0.5-3 Myr out to at least 60 au in a 0.03 solar-mass disk. Smaller bodies remain near their initial mass, leading to a sharp dichotomy in growth outcomes. For turbulent alpha = 1e-4, the critical masses are 1e-4M (circle plus) and 1e-3M (circle plus) at 9 and 75 au, respectively. Pressure bumps in disks may explain the large mass difference between the giant planets and Kuiper Belt objects, and also the existence of wide-orbit planets in some systems.

Modern terrestrial-planet formation models are highly successful at consistently generating planets with masses and orbits analogous to those of Earth and Venus. In stark contrast to classic theoretical predictions and inferred demographics of multiplanet systems of rocky exoplanets, the mass (greater than or similar to 10) and orbital period (greater than or similar to 2) ratios between Venus and Earth and the neighboring Mercury and Mars are not common outcomes in numerically generated systems. While viable solutions to the small-Mars problem are abundant in the literature, Mercury's peculiar origin remains rather mysterious. In this paper, we investigate the possibility that Mercury formed in a mass-depleted, inner region of the terrestrial disk (a < 0.5 au). This regime is often ignored in terrestrial-planet formation models because of the high computational cost of resolving hundreds of short-period objects over similar to 100 Myr timescales. By testing multiple disk profiles and mass distributions, we identify several promising sets of initial conditions that lead to remarkably successful analog systems. In particular, our most successful simulations consider moderate total masses of Mercury-forming material (0.1-0.25 Earth masses). While larger initial masses tend to yield disproportionate Mercury analogs, smaller values often inhibit the planets' formation as the entire region of material is easily accreted by Venus. Additionally, we find that shallow surface density profiles and larger inventories of small planetesimals moderately improve the likelihood of adequately reproducing Mercury.