We investigate the dynamical evolution of the terrestrial planets during the planetesimal-driven migration of the giant planets. A basic assumption of this work is that giant planet migration occurred after the completion of terrestrial planet formation, such as in the models that link the former to the origin of the Late Heavy Bombardment. The divergent migration of Jupiter and Saturn causes the g5 eigenfrequency to cross resonances of the form g5=gk with k ranging from 1 to 4. Consequently these secular resonances cause large-amplitude responses in the eccentricities of the terrestrial planets. We show that the resonances g5=g_4 and g5=g3 do not pose a problem if Jupiter and Saturn have a fast approach and departure from their mutual 2:1 mean motion resonance. On the other hand, the resonance crossings g5=g2 and g5=g1 are more of a concern as they tend to yield a terrestrial system incompatible with the current one. We offer two solutions to this problem. The first uses the fact that a secular resonance crossing can also damp the amplitude of a Fourier mode if the latter is large originally. A second scenario involves a 'jumping Jupiter' in which encounters between an ice giant and Jupiter, without ejection of the former, cause the latter to migrate away from Saturn much faster than if migration is driven solely by encounters with planetesimals. In this case, the g5=g2 and g5=g1 resonances can be jumped over, or occur very briefly.
We examine how the late divergent migration of Jupiter and Saturn may have perturbed the terrestrial planets. We identify six secular resonances between the nu_5 apsidal eigenfrequency of Jupiter and Saturn and the four eigenfrequencies of the terrestrial planets (g_{1-4}). We derive analytic upper limits on the eccentricity and orbital migration timescale of Jupiter and Saturn when these resonances were encountered to avoid perturbing the eccentricities of the terrestrial planets to values larger than the observed ones. If Jupiter and Saturn migrated with eccentricities comparable to their pr…
Using numerical simulations, we show that smooth migration of the giant planets through a planetesimal disk leads to an orbital architecture that is inconsistent with the current one: the resulting eccentricities and inclinations of their orbits are too small. The crossing of mutual mean motion resonances by the planets would excite their orbital eccentricities but not their orbital inclinations. Moreover, the amplitudes of the eigenmodes characterising the current secular evolution of the eccentricities of Jupiter and Saturn would not be reproduced correctly; only one eigenmode is excited by …
We present results from a suite of N-body simulations that follow the accretion history of the terrestrial planets using a new parallel treecode that we have developed. We initially place 2000 equal size planetesimals between 0.5--4.0 AU and the collisional growth is followed until the completion of planetary accretion (> 100 Myr). All the important effect of gas in laminar disks are taken into account: the aerodynamic gas drag, the disk-planet interaction including Type I migration, and the global disk potential which causes inward migration of secular resonances as the gas dissipates. We var…
The habitability of planets in binary star systems depends not only on the radiation environment created by the two stars, but also on the perturbations to planetary orbits and rotation produced by the gravitational field of the binary and neighbouring planets. Habitable planets in binaries may therefore experience significant perturbations in orbit and spin. The direct effects of orbital resonances and secular evolution on the climate of binary planets remain largely unconsidered. We present latitudinal energy balance modelling of exoplanet climates with direct coupling to an N Body integrato…
We describe the long-term evolution of compact systems of terrestrial planets, using a set of simulations that match the statistical properties of the observed exoplanet distribution. The evolution is driven by tidal dissipation in the planetary interiors, but the systems evolve as a whole due to secular gravitational interactions. We find that, for Earth-like dissipation levels, planetary orbits can be circularised out to periods of order 100 days, an order of magnitude larger than is possible for single planets. The resulting distribution of eccentricities is a qualitative match to that infe…
With N-body simulations of a planetary system with an asteroid belt we investigate how the asteroid impact rate on the Earth is affected by the architecture of the planetary system. We find that the $ν_6$ secular resonance plays an important role in the asteroid collision rate with the Earth. Compared to exoplanetary systems, the solar system is somewhat special in its lack of a super-Earth mass planet in the inner solar system. We therefore first consider the effects of the presence of a super-Earth in the terrestrial planet region. We find a significant effect for super-Earths with a mass of…
The orbital distributions of currently observed extrasolar giant planets allow marginally stable orbits for hypothetical, terrestrial planets. In this paper, we propose that many of these systems may not have additional planets on these "stable" orbits, since past dynamical instability among giant planets could have removed them. We numerically investigate the effects of early evolution of multiple giant planets on the orbital stability of the inner, sub-Neptune-like planets which are modeled as test particles, and determine their dynamically unstable region. Previous studies have shown that t…
The non-resonant secular dynamics of compact planetary systems are modeled by a perturbing function which is usually expanded in eccentricity and absolute inclination with respect to the invariant plane. Here, the expressions are given in a vectorial form which naturally leads to an expansion in eccentricity and mutual inclination. The two approaches are equivalent in most cases, but the vectorial one is specially designed for those where a quasi-coplanar system tilts as a whole by a large amount. Moreover, the vectorial expressions of the Hamiltonian and of the equations of motion are slightl…
Investigations were carried out on the stability of the two gas giant system HD 60532 itself including additional, smaller bodies. The methods used were numerical simulations with the Lie Series integration method, and secular perturbation theory. The aims were to find stable regions for potential Earth-like planets within the system, and confirming the stability of the detected planets. As results, the given configuration can be considered stable for a period of at least 65 million years. The two planets create a large unstable section between 0.25 and 3.6 AU, where no additional body is able…
We present the results of an extensive study of the final stage of terrestrial planet formation in disks with different surface density profiles and for different orbits of Jupiter and Saturn. We carried out simulations for disk densities proportional to r^-0.5, r^-1, and r^-1.5, and also for partially depleted disks as in the recent model of Mars formation by Izidoro et al (2014). The purpose of our study is to determine how the final assembly of planets and their physical properties are affected by the total mass of the disk and its radial profile. Because of the important roles of secular r…
We revisit the problem of secular resonance sweeping during the dissipation of a protoplanetary disk and its possible role in exciting the orbits of primordial asteroids, in the light of recent models of solar system evolution. We develop an integrator that incorporates the gravitational effect of a uniformly (or not) depleting, axisymmetric disk with arbitrary surface density profile; its performance is verified by analytical calculations. The secular response of fictitious asteroids, under perturbations from Jupiter, Saturn and a decaying disk, is thoroughly studied. Note that the existence …
We consider a dynamical shake-up model to explain the low mass of Mars and the lack of planets in the asteroid belt. In our scenario, a secular resonance with Jupiter sweeps through the inner solar system as the solar nebula depletes, pitting resonant excitation against collisional damping in the Sun's protoplanetary disk. We report the outcome of extensive numerical calculations of planet formation from planetesimals in the terrestrial zone, with and without dynamical shake-up. If the Sun's gas disk within the terrestrial zone depletes in roughly a million years, then the sweeping resonance i…
In this paper, we investigate whether hypothetical Earth-like planets have high probability of remaining on stable orbits inside the habitable zones around the stars A and B of α Centauri, for lengths of time compatible with the evolution of life. We introduce a stability criterion based on the solution of the restricted three-body problem and apply it to the α Centauri system. In this way, we determine the regions of the short-term stability of the satellite-type (S-type) planetary orbits, in both planar and three-dimensional cases. We also study the long-term stability of hypothetical planet…
The distribution of heavy elements is anomalously low in the asteroid main belt region compared with elsewhere in the solar system. Observational surveys also indicate a deficit in the number of small ($ \le 50$~km size) asteroids that is two orders of magnitude lower than what is expected from the single power-law distribution that results from a collisional coagulation and fragmentation equilibrium. Here, we consider the possibility that a major fraction of the original asteroid population may have been cleared out by Jupiter's secular resonance, as it swept through the main asteroid belt du…
Asteroid impacts with the Earth may have played an essential role in the emergence of life on Earth through their creation of favorable niches for life, changes to the atmosphere and delivery of water. Consequently, we suggest two potential requirements for life in an exoplanetary system: first, that the system has an asteroid belt, and second, that there is a mechanism to drive asteroids to impact the terrestrial habitable planet. Since in the solar system, the $ν_6$ secular resonance has been shown to have been important in driving these impacts, we explore how the masses and locations of tw…
The newly formed giant planets may have migrated and crossed a number of mutual mean motion resonances (MMRs) when smaller objects (embryos) were accreting to form the terrestrial planets. We investigated the effects of the planetesimal-driven migration of Jupiter and Saturn, and the influence of their mutual 1:2 MMR crossing on terrestrial planet formation for the first time, by performing N-body simulations. These simulations considered distinct timescales of MMR crossing and planet migration. In total, 68 high-resolution simulation runs using 2000 disk planetesimals were performed, which wa…
Constructing dynamical maps from the filtered output of numerical integrations, we analyze the structure of the $ν_\odot$ secular resonance for fictitious irregular satellites in retrograde orbits. This commensurability is associated to the secular angle $θ= \varpi - \varpi_\odot$, where $\varpi$ is the longitude of pericenter of the satellite and $\varpi_\odot$ corresponds to the (fixed) planetocentric orbit of the Sun. Our study is performed in the restricted three-body problem, where the satellites are considered as massless particles around a massive planet and perturbed by the Sun. Depend…
The only discovery of Earth Trojan 2010 TK$_7$ and the subsequent launch of OSIRIS-REx motive us to investigate the stability around the triangular Lagrange points $L_4$ and $L_5$ of the Earth. In this paper we present detailed dynamical maps on the $(a_0,i_0)$ plane with the spectral number (SN) indicating the stability. Two main stability regions, separated by a chaotic region arising from the $ν_3$ and $ν_4$ secular resonances, are found at low ($i_0\leq 15^\circ$) and moderate ($24^\circ\leq {i_0}\leq 37^\circ$) inclinations respectively. The most stable orbits reside below $i_0=10^\circ$ …
The formation of the solar system's terrestrial planets has been numerically modeled in various works, and many other studies have been devoted to characterizing our modern planets' chaotic dynamical state. However, it is still not known whether our planet's fragile chaotic state is an expected outcome of terrestrial planet accretion. We use a suite of numerical simulations to present a detailed analysis and characterization of the dynamical chaos in 145 different systems produced via terrestrial planet formation in Kaib and Cowan (2015). These systems were created in the presence of a fully f…
The cratering record on the Earth and Moon shows that our planet has been exposed to high velocity impacts for much or all of its existence. Some of these craters were produced by the impact of long period comets (LPCs). These probably originated in the Oort cloud, and were put into their present orbits through gravitational perturbations arising from the Galactic tide and stellar encounters, both of which are modulated by the solar motion about the Galaxy. Here we construct dynamicalmodels of these mechanisms in order to predict the time-varying impact rate of LPCs and the angular distributio…
Dwarf stars are believed to have small protostar disk where planets may grow up. During the planet formation stage, embryos undergoing type I migration are expected to be stalled at inner edge of magnetic inactive disk ($a_{\rm crit} \sim 0.2-0.3 ~$AU). This mechanism makes the location around $a_{\rm crit}$ a sweet spot of forming planets. Especially, $a_{\rm crit}$ of dwarf stars with masses $\sim 0.5 M_\odot$ is roughly inside the habitable zone of the system. In this paper we study the formation of habitable planets due to this mechanism with a model system OGLE-06-109L. It has a $0.51 M_\…
We perform numerical simulations to study the secular orbital evolution and dynamical structure in the quintuplet planetary system 55 Cancri with the self-consistent orbital solutions by Fischer and coworkers (2008). In the simulations, we show that this system can be stable at least for $10^{8}$ yr. In addition, we extensively investigate the planetary configuration of four outer companions with one terrestrial planet in the wide region of 0.790 AU $\leq a \leq $ 5.900 AU to examine the existence of potential asteroid structure and Habitable Zones (HZs). We show that there are unstable region…
The inner solar system's modern orbital architecture provides inferences into the epoch of terrestrial planet formation; a ~100 Myr time period of planet growth via collisions with planetesimals and other proto-planets. While classic numerical simulations of this scenario adequately reproduced the correct number of terrestrial worlds, their semi-major axes and approximate formation timescales, they struggled to replicate the Earth-Mars and Venus-Mercury mass ratios. In a series of past independent investigations, we demonstrated that Mars' mass is possibly the result of Jupiter and Saturn's ea…
We present radial velocity (RV) measurements of our sample of 40 M dwarfs from our planet search programme with VLT+UVES begun in 2000. Although with our RV precision down to 2 - 2.5 m/s and timebase line of up to 7 years, we are capable of finding planets of a few Earth masses in the close-in habitable zones of M dwarfs, there is no detection of a planetary companion. To demonstrate this we present mass detection limits allowing us to exclude Jupiter-mass planets up to 1 AU for most of our sample stars. We identified 6 M dwarfs that host a brown dwarf or low-mass stellar companion. With the e…
Although warm jupiters are generally too far from their stars for tides to be important, the presence of an inner planetary companion to a warm jupiter can result in tidal evolution of the system. Insight into the process and its effects comes form classical secular theory of planetary perturbations. The lifetime of the inner planet may be shorter than the age of the system, because the warm jupiter maintains its eccentricity and hence promotes tidal migration into the star. Thus a warm jupiter observed to be alone in its system might have previously cleared away any interior planets. Before i…
Terrestrial planets are thought to be the result of a vast number of gravitational interactions and collisions between smaller bodies. We use numerical simulations to show that practically identical initial conditions result in a wide array of final planetary configurations. This is a result of the chaotic evolution of trajectories which are highly sensitive to minuscule displacements. We determine that differences between systems evolved from virtually identical initial conditions can be larger than the differences between systems evolved from very different initial conditions. This implies t…
Astronomers commonly quote the properties of celestial objects in units of parameters for the Sun, Jupiter, or the Earth. The resolution presented here was proposed by the IAU Inter-Division Working Group on Nominal Units for Stellar and Planetary Astronomy and passed by the XXIXth IAU General Assembly in Honolulu. IAU 2015 Resolution B3 adopts a set of nominal solar, terrestrial, and jovian conversion constants for stellar and (exo)planetary astronomy which are defined to be exact SI values. While the nominal constants are based on current best estimates (CBEs; which have uncertainties, are n…
The architecture of a planetary system can influence the habitability of a planet via orbital effects, particularly in the areas of stability and eccentricity. Some of these effects are readily apparent, particularly when they occur on short timescales that are easily numerically calculable. However, the appearance and evolution of life can take place on gigayear timescales, long enough that secular effects become important. These effects are difficult to investigate, as a direct integration requires significant computational time. In this paper, we apply a semi-analytic framework in conjuncti…
The evolution of the Solar System can be schematically divided into three different phases: the Solar Nebula, the Primordial Solar System and the Modern Solar System. These three periods were characterized by very different conditions, both from the point of view of the physical conditions and from that of the processes there were acting through them. Across the Solar Nebula phase, planetesimals and planetary embryos were forming and differentiating due to the decay of short-lived radionuclides. At the same time, giant planets formed their cores and accreted the nebular gas to reach their pres…
[Context]. The standard solar model fails to predict the very low lithium abundance in the Sun, which is much lower than the proto-solar nebula. This Li problem has been debated for decades, and it has been ascribed either to planet formation or to secular stellar depletion. In order to test the evolution of Li, it is important to find solar twins in a range of ages. Also, the study of stars similar to the Sun is relevant in relation to the signature of terrestrial planet formation around the Sun. [Methods]. We acquired high-resolution (R=110,000), high S/N (~300) ESO/VLT UVES spectra of sever…
GJ 581d is a potentially habitable super-Earth in the multiple system of exoplanets orbiting a nearby M dwarf. We investigate this planet's long-term dynamics, with an emphasis on its probable final rotation states acquired via tidal interaction with the host. The published radial velocities for the star are re-analysed with a benchmark planet detection algorithm, to confirm that there is no evidence for the recently proposed two additional planets (f and g). Limiting the scope to the four originally detected planets, we assess the dynamical stability of the system and find bounded chaos in th…
