Initial Semimajor Axis Research Articles

The feeding zones of terrestrial planets and insights into Moon formation

The final stage of terrestrial planet formation consists of several hundred approximately lunar mass bodies accreting into a few terrestrial planets. This final stage is stochastic, making it hard to predict which parts of the original planetesimal disk contributed to each of our terrestrial planets. Here we present an extensive suite of terrestrial planet formation simulations that allows quantitative analysis of this process. Although there is a general correlation between a planet’s location and the initial semi-major axes of its constituent planetesimals, we concur with previous studies that Venus, Earth, and Mars analogs have overlapping, stochastic feeding zones. We quantify the feeding zone width, Δa, as the mass-weighted standard deviation of the initial semi-major axes of the planetary embryos and planetesimals that make up the final planet. The size of a planet’s feeding zone in our simulations does not correlate with its final mass or semi-major axis, suggesting there is no systematic trend between a planet’s mass and its volatile inventory. Instead, we find that the feeding zone of any planet more massive than 0.1M⊕ is roughly proportional to the radial extent of the initial disk from which it formed: Δa≈0.25(amax-amin), where amin and amax are the inner and outer edge of the initial planetesimal disk. These wide stochastic feeding zones have significant consequences for the origin of the Moon, since the canonical scenario predicts the Moon should be primarily composed of material from Earth’s last major impactor (Theia), yet its isotopic composition is indistinguishable from Earth. In particular, we find that the feeding zones of Theia analogs are significantly more stochastic than the planetary analogs. Depending on our assumed initial distribution of oxygen isotopes within the planetesimal disk, we find a ∼5% or less probability that the Earth and Theia will form with an isotopic difference equal to or smaller than the Earth and Moon’s. In fact we predict that every planetary mass body should be expected to have a unique isotopic signature. In addition, we find paucities of massive Theia analogs and high velocity Moon-forming collisions, two recently proposed explanations for the Moon’s isotopic composition. Our work suggests that there is still no scenario for the Moon’s origin that explains its isotopic composition with a high probability event.

THE OCCURRENCE OF WIDE-ORBIT PLANETS IN BINARY STAR SYSTEMS

The occurrence of planets in binary star systems has been investigated via a variety of techniques that sample a wide range of semi-major axes, but with a preponderance of such results applicable to planets with semi-major axes less than a few AU. We utilize a new method -- the presence or absence of heavy elements in the atmospheres of white dwarf stars -- to elucidate the frequency in main sequence binary star systems of planets with semi-major axes greater than a few AU. We consider only binaries where a putative planetary system orbits one member (no circumbinary planets). For main sequence binaries where the primary star is of spectral type A or F, data in the published literature suggests that the existence of a secondary star with a semi-major axis less than about 1000 AU suppresses the formation and/or long term stability of an extended planetary system around the primary. For these spectral types and initial semi-major axis $\geq$1000 AU, extended planetary systems appear to be as common around stars in binary systems as they are around single stars.

EXTRASOLAR BINARY PLANETS. I. FORMATION BY TIDAL CAPTURE DURING PLANET-PLANET SCATTERING

We have investigated i) the formation of gravitationally bounded pairs of gas-giant planets (which we call "binary planets") from capturing each other through planet-planet dynamical tide during their close encounters and ii) the following long-term orbital evolution due to planet-planet and planet-star {\it quasi-static} tides. For the initial evolution in phase i), we carried out N-body simulations of the systems consisting of three jupiter-mass planets taking into account the dynamical tide. The formation rate of the binary planets is as much as 10% of the systems that undergo orbital crossing and this fraction is almost independent of the initial stellarcentric semi-major axes of the planets, while ejection and merging rates sensitively depend on the semi-major axes. As a result of circularization by the planet-planet dynamical tide, typical binary separations are a few times the sum of the physical radii of the planets. After the orbital circularization, the evolution of the binary system is governed by long-term quasi-static tide. We analytically calculated the quasi-static tidal evolution in later phase ii). The binary planets first enter the spin-orbit synchronous state by the planet-planet tide. The planet-star tide removes angular momentum of the binary motion, eventually resulting in a collision between the planets. However, we found that the binary planets survive the tidal decay for main-sequence life time of solar-type stars (~10Gyrs), if the binary planets are beyond ~0.3 AU from the central stars. These results suggest that the binary planets can be detected by transit observations at >0.3AU.

Dynamical stability of the Gliese 581 exoplanetary system

Using numerical methods we investigate the dynamical stability of the Gliese 581 exoplanetary system. The system is known to harbour four planets (b-e). The existence of another planet (g) in the liquid water habitable zone of the star is debated after the latest analyses of the radial velocity (RV) measurements. We integrated the 4 and 5-planet model of Vogt et al. (AN 333, 561-575, 2012) with initial circular orbits. To characterize stability, the maximum eccentricity was used that the planets reached over the time of the integrations and the LCI and RLI to identify chaotic motion. Since circular orbits in the RV fits seem to be a too strong restriction and the true orbits might be elliptic, we investigated the stability of the planets as a function of their eccentricity. The integration of the circular 4-planet model shows that it is stable on a longer timescale for even an inclination i = 5{\deg}. A fifth planetary body in the 4-planet model could have a stable orbit between the two super-Earth sized planets c and d, and beyond the orbit of planet d, although another planet would likely only be stable on circular or near-circular orbit in the habitable zone of the star. Gliese 581 g in the 5-planet model would have a dynamically stable orbit, even for a wider range of orbital parameters, but its stability is strongly dependent on the eccentricity of planet d. The low-mass planet e, which quickly became unstable in eccentric models, remains stable in the circular 4-planet model, but the stable region around its initial semi-major axis and eccentricity is rather small. The stability of the inner planets e and c is dependent on the eccentricity of the Neptune-size planet b. The outermost planet d is far away from the adjacent planet c to considerably influence its stability, however, the existence of a planet between the two super-Earth planets c and d constrains its eccentricity.

The binary companion mass ratio distribution: an imprint of the star formation process?

We explore the effects of dynamical evolution in dense clusters on the companion mass ratio distribution (CMRD) of binary stars. Binary systems are destroyed by interactions with other stars in the cluster, lowering the total binary fraction and significantly altering the initial semi-major axis distribution. However, the shape of the CMRD is unaffected by dynamics; an equal number of systems with high mass ratios are destroyed compared to systems with low mass ratios. We might expect a weak dependence of the survivability of a binary on its mass ratio because its binding energy is proportional to both the primary and secondary mass components of the system. However, binaries are broken up by interactions in which the perturbing star has a significantly higher energy (by a factor of >10, depending on the particular binary properties) than the binding energy of the binary, or through multiple interactions in the cluster. We therefore suggest that the shape of the observed binary CMRD is an outcome of the star formation process, and should be measured in preference to the distributions of orbital parameters, such as the semi-major axis distribution.

On the orbits of low-mass companions to white dwarfs and the fates of the known exoplanets

The ultimate fates of binary companions to stars (including whether the companion survives and the final orbit of the binary) are of interest in light of an increasing number of recently discovered, low-mass companions to white dwarfs (WDs). In this Letter, we study the evolution of a two-body system wherein the orbit adjusts due to structural changes in the primary, dissipation of orbital energy via tides, and mass loss during the giant phases; previous studies have not incorporated changes in the primary's spin. For companions ranging from Jupiter's mass to ~0.3 Msun and primaries ranging from 1-3 Msun, we determine the minimum initial semimajor axis required for the companion to avoid engulfment by the primary during post-main-sequence evolution, and highlight the implications for the ultimate survival of the known exoplanets. We present regions in secondary mass and orbital period space where an engulfed companion might be expected to survive the common envelope phase (CEP), and compare with known M dwarf+WD short-period binaries. Finally, we note that engulfed Earth-like planets cannot survive a CEP. Detection of a first-generation terrestrial planet in the white dwarf habitable zone requires scattering from a several-AU orbit to a high-eccentricity orbit (with a periastron of ~Rsun) from which it is damped into a circular orbit via tidal friction, possibly rendering it an uninhabitable, charred ember.

THE MASS-LOSS-INDUCED ECCENTRIC KOZAI MECHANISM: A NEW CHANNEL FOR THE PRODUCTION OF CLOSE COMPACT OBJECT-STELLAR BINARIES

Over a broad range of initial inclinations and eccentricities an appreciable fraction of hierarchical triple star systems with similar masses are essentially unaffected by the Kozai-Lidov mechanism (KM) until the primary in the central binary evolves into a compact object. Once it does, it may be much less massive than the other components in the ternary, enabling the "eccentric Kozai mechanism (EKM):" the mutual inclination between the inner and outer binary can flip signs driving the inner binary to very high eccentricity, leading to a close binary or collision. We demonstrate this "Mass-loss Induced Eccentric Kozai" (MIEK) mechanism by considering an example system and defining an ad-hoc minimal separation between the inner two members at which tidal affects become important. For fixed initial masses and semi-major axes, but uniform distributions of eccentricity and cosine of the mutual inclination, ~10% of systems interact tidally or collide while the primary is on the MS due to the KM or EKM. Those affected by the EKM are not captured by earlier quadrupole-order secular calculations. We show that fully ~30% of systems interact tidally or collide for the first time as the primary swells to AU scales, mostly as a result of the KM. Finally, ~2% of systems interact tidally or collide for the first time after the primary sheds most of its mass and becomes a WD, mostly as a result of the MIEK mechanism. These findings motivate a more detailed study of mass-loss in triple systems and the formation of close NS/WD-MS and NS/WD-NS/WD binaries without an initial common envelope phase.

Interaction between massive planets on inclined orbits and circumstellar discs

We study the interaction between massive planets and a gas disc with a mass in the range expected for protoplanetary discs. We use SPH simulations to study the orbital evolution of a massive planet as well as the dynamical response of the disc for planet masses between 1 and $6\ \rmn{M_J}$ and the full range of initial relative orbital inclinations. Gap formation can occur for planets in inclined orbits. For given planet mass, a threshold relative orbital inclination exists under which a gap forms. At high relative inclinations, the inclination decay rate increases for increasing planet mass and decreasing initial relative inclination. For an initial semi-major axis of 5 AU and relative inclination of $i_0=80^\circ,$ the times required for the inclination to decay by $10^\circ$ is $\sim10^{6}\ \rmn{yr}$ and $\sim10^{5}\ \rmn{yr}$ for $1\ \rmn{M_J}$ and $6\ \rmn{M_J}$. Planets on inclined orbits warp the disc by an extent that is negligible for $1\ \rmn{M_J}$ but increases with increasing mass becoming quite significant for a planet of mass $6\ \rmn{M_J}$. We also find a solid body precession of both the total disc angular momentum vector and the planet orbital momentum vector about the total angular momentum vector. Our results illustrate that the influence of an inclined massive planet on a protoplanetary disc can lead to significant changes of the disc structure and orientation which can in turn affect the orbital evolution of the planet significantly.

The planet around the horizontal-branch star HIP 13044

The evolution of the orbit of the planet detected around the horizontal-branch star HIP 13044 is considered. The initial semi-major axis of the planet’s orbit is estimated. A formula for the final result of the common-envelope stage in a close binary systemis used to estimate the mass of the envelope ejected by the red giant. The relatively small mass of this envelope (∼0.1M ⊙) facilitated the survival of the planet after the partial dispersal of the envelope material and the transformation of the star into a horizontal-branch star.

Eccentric disc instability in stellar discs formed from inspiralling gas clouds in the Galactic Centre

The inspiral of a turbulent molecular cloud in the Galactic Centre may result in the formation of a small, dense and moderately eccentric gas disc around the supermassive black hole (SMBH). Such a disc is unstable to fragmentation and may lead to the formation of young massive stars in the central parsec of the Galaxy. Here we perform high-accuracy direct summation N-body simulations of a ring of massive stars (with initial semi-major axes 0.1 < a/pc < 0.4 and eccentricities 0.2 < e < 0.4), subject to the potential of the SMBH, a stellar cusp, and the parent gas disc, to study how the orbital elements of the ring evolve in time. The initial conditions for the stellar ring are drawn from the results of previous simulations of molecular cloud infall and disruption in the SMBH potential. While semi-major axes do not evolve significantly, the distribution of eccentricities spreads out very fast (~1 Myr) as a consequence of cusp precession. In particular, stellar orbits with initial eccentricity e>0.3 (e<0.3) tend to become even more (less) eccentric, resulting in a bimodal eccentricity distribution. The distribution is qualitatively consistent with that of the massive stars observed in the Galactic Centre's clockwise disc.

Trojans in Exosystems with Two Massive Planets

AbstractWe take as dynamical model for extrasolar planetary systems a central star like our Sun and two giant planets m1 and m2 like Jupiter and Saturn. We change the mass ratio μ=m2/m1 of the two large planets for a wide range of 1/16 &lt; μ &lt; 16. We also change the ratio between the initial semi-major axes (ν=a2/a1) in the range of 1.2 &lt; ν &lt; 3 to model the different architecture of extrasolar planetary systems hosting two giant planets. The results for possible Trojans (Trojan planets) in the equilateral equilibrium points of the inner planet m1 and the outer planet m2 were derived with the aid of numerical integration. It turned out that in many configurations – depending on the mass ratios μ and the semi-major axes ratio ν – giant planets may host Trojans.

2008 LC18: a potentially unstable Neptune Trojan

The recent discovery of the first Neptune Trojan at the planet's trailing (L5) Lagrange point, 2008 LC18, offers an opportunity to confirm the formation mechanism of a member of this important tracer population for the Solar system's dynamical history. We tested the stability of 2008 LC18's orbit through a detailed dynamical study, using test particles spread across the orbital uncertainties in a, e, i and {\Omega}. This showed that the wide uncertainties of the published orbit span regions of both extreme dynamical instability, with lifetimes < 100 Myr, and with significant stability (> 1 Gyr lifetimes). The stability of 2008 LC18's clones is greatly dependent on their semi-major axis and only weakly correlated with their eccentricity. Test particles on orbits with an initial semi-major axis less than 29.91 AU have dynamical half-lives shorter than 100 Myr; in contrast, particles with an initial semi-major axis greater than 29.91 AU exhibit such strong dynamical stability that almost all are retained over the 1 Gyr of our simulations. More observations of this object are necessary to improve the orbit. If 2008 LC18 is in the unstable region, then our simulations imply that it is either a temporary Trojan capture, or a representative of a slowly decaying Trojan population (like its sibling the L4 Neptunian Trojan 2001 QR322), and that it may not be primordial. Alternatively, if the orbit falls into the larger, stable region, then 2008 LC18 is a primordial member of the highly stable and highly inclined component of the Neptune Trojan population, joining 2005 TN53 and 2007 VL305. We attempted to recover 2008 LC18 using the 2.3m telescope at Siding Spring Observatory to provide this astrometry, but were unsuccessful due to the high stellar density of its current sky location near the galactic centre. The recovery of this object will require a telescope in the 8m class.

COAGULATION CALCULATIONS OF ICY PLANET FORMATION AT 15-150 AU: A CORRELATION BETWEEN THE MAXIMUM RADIUS AND THE SLOPE OF THE SIZE DISTRIBUTION FOR TRANS-NEPTUNIAN OBJECTS

We investigate whether coagulation models of planet formation can explain the observed size distributions of transneptunian objects (TNOs). Analyzing published and new calculations, we demonstrate robust relations between the size of the largest object and the slope of the size distribution for sizes 0.1 km and larger. These relations yield clear, testable predictions for TNOs and other icy objects throughout the solar system. Applying our results to existing observations, we show that a broad range of initial disk masses, planetesimal sizes, and fragmentation parameters can explain the data. Adding dynamical constraints on the initial semimajor axis of `hot' KBOs along with probable TNO formation times of 10-700 Myr restricts the viable models to those with a massive disk composed of relatively small (1-10 km) planetesimals.

Tidal evolution of planets around brown dwarfs

The tidal evolution of planets orbiting brown dwarfs (BDs) presents an interesting case study because BDs' terrestrial planet forming region is located extremely close-in. In fact, the habitable zones of BDs range from roughly 0.001 to 0.03 AU and for the lowest-mass BDs are located interior to the Roche limit. In contrast with stars, BDs spin up as they age. Thus, the corotation distance moves inward. This has important implications for the tidal evolution of planets around BDs. We used a standard equilibrium tidal model to compute the orbital evolution of a large ensemble of planet-BD systems. We tested the effect of numerous parameters such as the initial semi-major axis and eccentricity, the rotation period of the BD, the masses of both the BD and planet, and the tidal dissipation factors. We find that all planets that form at or beyond the corotation distance and with initial eccentricities smaller than \sim 0.1 are repelled from the BD. Some planets initially interior to corotation can survive if their inward tidal evolution is slower than the BD's spin evolution, but most initially close-in planets fall onto the BD. We find that the most important parameter for the tidal evolution is the initial orbital distance with respect to the corotation distance. Some planets can survive in the habitable zone for Gyr timescales, although in many cases the habitable zone moves inward past the planet's orbit in just tens to hundreds of Myr. Surviving planets can have orbital periods of less than 10 days (as small as 10 hrs), so they could be observable by transit.

Spacecraft long-duration phasing maneuver optimization using hybrid approach

Before a manned orbital rendezvous mission, the target spacecraft usually performs several maneuvers to adjust the initial phase angle of the orbital rendezvous and to coordinate the injection of the chaser. This maneuvering process is referred to as the “target phasing mission”. This target phasing presents an orbital long-duration two-point boundary value problem. Further, when the maneuver revolution numbers are used as design variables, the target phasing maneuver's optimization becomes a mixed integer nonlinear programming problem. This paper presents a new optimization method for this phasing maneuver mission, employing a hybrid approach. First, we provide an approximate phasing optimization problem that considers the phase angle influences of node drift and orbital altitude decay. This problem is then optimized using a hybrid approach that integrates branch-and-bound and sequential quadratic programming. Second, a shooting iteration method is adopted to improve the solution to the approximate problem in order to satisfy the terminal constraints of high-precision numerical integration. The proposed method is then applied to an operational target phasing maneuver problem. The results lead to four major conclusions: (1) The proposed approximate phasing optimization model presents a good approximation of the operational mission. (2) The hybrid optimization approach can solve the approximate problem effectively, and the shooting iteration used to arrive at a high-precision solution converges steadily and rapidly. (3) Compared with mixed-code genetic algorithm, the proposed method can obtain a similar result with a lower computation cost and, compared with the approximate model that does not consider node drift and orbital altitude decay, the proposed method has better convergence efficiency. (4) The terminal time of target phasing remains almost constant when the initial semi-major axis increases in a limited interval, and the transition appears only when there is a change in the terminal revolution number.

Trapping in three-planet resonances during gas-driven migration

We study the establishment of three-planet resonances -similar to the Laplace resonance in the Galilean satellites- and their effects on the mutual inclinations of the orbital planes of the planets, assuming that the latter undergo migration in a gaseous disc. In particular, we examine the resonance relations that occur, by varying the physical and initial orbital parameters of the planets (mass, initial semi-major axis and eccentricity) as well as the parameters of the migration forces (migration rate and eccentricity damping rate), which are modeled here through a simplified analytic prescription. We find that, in general, for planetary masses below 1.5 M_J, multiple-planet resonances of the form n3:n2:n1=1:2:4 and 1:3:6 are established, as the inner planets, m1 and m2, get trapped in a 1:2 resonance and the outer planet m3 subsequently is captured in a 1:2 or 1:3 resonance with m2. For mild eccentricity damping, the resonance pumps the eccentricities of all planets on a relatively short time-scale, to the point where they enter an inclination-type resonance (as in Libert & Tsiganis 2011); then mutual inclinations can grow to ~35{\deg}, thus forming a "3-D system". On the other hand, we find that trapping of m2 in a 2:3 resonance with m1 occurs very rarely, for the range of masses used here, so only two cases of capture in a respective three-planet resonance were found. Our results suggest that trapping in a three-planet resonance can be common in exoplanetary systems, provided that the planets are not very massive. Inclination pumping could then occur relatively fast, provided that eccentricity damping is not very efficient so that at least one of the inner planets acquires an orbital eccentricity higher than e=0.3.

An analytic technique for constraining the dynamical origins of multiple star systems containing merger products

We present a technique to identify the most probable dynamical formation scenario for an observed binary or triple system containing one or more merger products or, alternatively, to rule out the possibility of a dynamical origin. Our method relies on an analytic prescription for energy conservation during stellar encounters. With this, observations of the multiple star system containing the merger product(s) can be used to work backwards in order to constrain the initial orbital energies of any single, binary or triple systems that went into the encounter. The initial semi-major axes of the orbits provide an estimate for the collisional cross section and therefore the time-scale for the encounter to occur in its host cluster. We have applied our analytic prescription to observed binary and triple systems containing blue stragglers, in particular the triple system S1082 in M67 and the period distribution of the blue straggler binaries in NGC 188. We have shown that both S1082 and most of the blue straggler binaries in NGC 188 could have a dynamical origin, and that encounters involving triples could be a significant contributor to BS populations in old open clusters. In general, our results suggest that encounters involving triples could make up a significant fraction of those dynamical interactions that result in stellar mergers, in particular encounters that produce multiple star systems containing one or more blue stragglers.

The dynamics of Neptune Trojans - II. Eccentric orbits and observed objects★

In a previous paper, we have presented a global view of the stability of Neptune Trojan (NT hereafter) on inclined orbit. We discuss in this paper the dependence of stability of NT orbits on the eccentricity. High-resolution dynamical maps are constructed using the results of extensive numerical integrations of orbits initialized on the fine grids of initial semimajor axis (a0) versus eccentricity (e0). The extensions of regions of stable orbits on the (a0, e0) plane at different inclinations are shown. The maximum eccentricities of stable orbits in three most stable regions at low (0, 12deg.), medium (22,36deg.) and high (51, 59deg.) inclination, are found to be 0.10, 0.12 and 0.04, respectively. The fine structures in the dynamical maps are described. Via the frequency analysis method, the mechanisms that portray the dynamical maps are revealed. The secondary resonances, concerning the frequency of the librating resonant angle and the frequency of the quasi 2:1 mean motion resonance between Neptune and Uranus, are found deeply involved in the motion of NTs. Secular resonances are detected and they also contribute significantly to the triggering of chaos in the motion. Particularly, the effects of the secular resonance v8, v18 are clarified. We also investigate the orbital stabilities of six observed NTs by checking the orbits of hundreds clones of them generated within the observing error bars. We conclude that four of them, except 2001 QR322 and 2005 TO74, are deeply inside the stable region. The 2001 QR322 is in the close vicinity of the most significant secondary resonance. The 2005 TO74 locates close to the boundary separating stable orbits from unstable ones, and it may be influenced by a secular resonance.

THE EVOLUTION OF PRIMORDIAL BINARY OPEN STAR CLUSTERS: MERGERS, SHREDDED SECONDARIES, AND SEPARATED TWINS

The basic properties of the candidate binary cluster population in the Magellanic Clouds and Galaxy are similar. The fraction of candidate binary systems is $\sim$10% and the pair separation histogram exhibits a bimodal distribution commonly attributed to their transient nature. However, if primordial pairs cannot survive for long as recognizable bound systems, how are they ending up? Here, we use simulations to confirm that merging, extreme tidal distortion and ionization are possible depending on the initial orbital elements and mass ratio of the pair. The nature of the dominant evolutionary path largely depends on the strength of the local tidal field. Merging is observed for initially close primordial binary clusters but also for wider pairs in nearly parabolic orbits. Its characteristic timescale depends on the initial orbital semi-major axis, eccentricity, and cluster pair mass ratio, becoming shorter for closer, more eccentric equal mass pairs. Shredding or extreme tidal distortion of the less massive cluster and subsequent separation is observed in all pairs with appreciably different masses. Wide pairs steadily evolve into the separated twins state characterized by the presence of tidal bridges and separations of 200-500 pc after one Galactic orbit. In the Galaxy, the vast majority of observed binary candidates appear to be following this evolutionary path which translates into the dominant peak (25-30 pc) in the pair separation distribution. The secondary peak at smaller separations (10-15 pc) can be explained as due to close pairs in almost circular orbits and/or undergoing merging. Merged clusters exhibit both peculiar radial density and velocity dispersion profiles shaped by synchronization and gravogyro instabilities. Both simulations and observations show that, for the range of parameters studied here, long term binary cluster stability in the Galactic disk is unlikely.

Orbital stability of systems of closely-spaced planets, II: configurations with coorbital planets

We numerically investigate the stability of systems of 1 $${{\rm M}_{\oplus}}$$ planets orbiting a solar-mass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 108, 5 × 105, and 2 × 105 years for the 6- and 10-planet, 126-planet, and 210-planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t c /t 0) = b β + c, where t c is the crossing time, t 0 = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.