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To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.icarus.2007.01.016 Byline: J.E. Chambers Keywords: Moon; Terrestrial planets; Cratering; Planetary dynamics; Celestial mechanics Abstract: The stability of an additional planet between the orbit of Mars and the asteroid belt is examined in the context of the Planet V hypothesis. In this model, the Solar System initially contained a fifth terrestrial planet, 'Planet V,' which was removed after [approximately equal to]700 Myr, a possible trigger for the late heavy bombardment on the inner planets. The model is investigated using 96 N-body integrations of the 8 major planets with an additional body between Mars and the asteroid belt. In more than 1/4 of simulations, Planet V survives for 1000 Myr. In most other cases, Planet V collides with the Sun or hits another planet after several hundred Myr, leaving 4 surviving terrestrial planets. In 24/96 simulations, Planet V is lost by ejection or collision with the Sun while the other four terrestrial planets survive without undergoing a collision. In 18 cases, Planet V is removed at least 200 Myr after the beginning of the simulation. The endstate depends sensitively on the mass of Planet V. Collision with the Sun is likely when Planet V's mass is 0.25 Mars masses or less. When Planet V is more massive than this, collisions involving it and/or other terrestrial planets become commonplace. In unstable systems, the times of first encounter and first collision/ejection depend on the initial aphelion distance of Mars. Reducing Mars's aphelion distance increases these times and also increases the fraction of systems surviving for 1000 Myr. When Mars's current orbit is used, the stability of Planet V increases when these two planets are widely separated initially. Planet V's aphelion distance Q typically begins to cross the asteroid belt within a few tens to a few hundred Myr, and its orbit last leaves the belt several hundred Myr later in most cases. The total time spent with Q2.1AU is typically less than 200 Myr. Author Affiliation: Carnegie Institution of Washington, 5241 Broad Branch Road, NW, Washington, DC 20015, USA Article History: Received 30 September 2003; Revised 5 September 2006

Several recent studies have noted that the orbital evolution of many comets is influenced by mean-motion resonances with Jupiter. However, the distribution and relative importance of these resonances and the orbital characteristics of the comets affected have not been addressed to date. Here I show analytically that cometary orbits with periods greater than a critical value, [P.sub.c], (which depends upon the orbital inclination) are prevented from undergoing librations about a mean-motion resonance. Conversely, numerical integrations indicate that resonances play an important role in the dynamics of comets with P [less than] [P.sub.c]. The inclination-averaged value of [P.sub.c] approximately coincides with the traditional and arbitrary dividing line between Halley-type and long-period comets, which explains why many of the former are currently observed to be in resonance, whereas the latter are not. Thus, we now have a dynamical justification for separating comets into those of short and long period.

A system of two small planets orbiting the Sun on low-eccentricity, low-inclination orbits is stable with respect to close encounters if the initial semi-major axis difference, [Delta], measured in mutual Hill radii, [R.sub.H], exceeds 2 [square root of 3], due to conservation of energy and angular momentum. We investigate the stability of systems of more than two planets using numerical integrations. We find that systems with [Delta] < 10 are always unstable, with the time, t, of first close encounter given approximately by log t = b[Delta] + c, where b and c are constants. It is likely that systems with [Delta] > 10 are also unstable. The slope b depends weakly on the number of planets, but is independent of planetary mass, m, if we measure [Delta] in units that are proportional to [m.sup.1/4] rather than the usual [R.sub.H] [proportional to] [m.sup.1/3]. Instability in multi-planet systems arises because energy and angular momentum are no longer conserved within each two-planet subsystem due to perturbations by the additional planet(s). These results suggest that planetary embryos will not become isolated prior to the final stage of terrestrial-planet formation simply due to a failure to achieve close encounters. Other factors leading to isolation cannot be ruled out at this stage.

A study of the orbit of comet Swift-Tuttle shows that the comet will move very close to earth in the year 4479 AD. The chances of a collision are one in 10 to the sixth power. For the next 20000 years from 1995, the orbital descending node will remain within 0.1 AU of Earth's orbit. The movement of the orbital nodes will be affected by resonant librations, which increase the orbit's Lyapunov time by a factor of two.

We simulate the late stages of terrestrial-planet formation using N-body integrations, in three dimensions, of disks of up to 56 initially isolated, nearly coplanar planetary embryos, plus Jupiter and Saturn. Gravitational perturbations between embryos increase their eccentricities, e, until their orbits become crossing, allowing collisions to occur. Further interactions produce large-amplitude oscillations in e and the inclination, i, with periods of [approximately]105 years. These oscillations are caused by secular resonances between embryos and prevent objects from becoming re-isolated during the simulations. The largest objects tend to maintain smaller e and i than low-mass bodies, suggesting some equipartition of random orbital energy, but accretion proceeds by orderly growth. The simulations typically produce two large planets interior to 2 AU, whose time-averaged e and i are significantly larger than Earth and Venus. The accretion rate falls off rapidly with heliocentric distance, and embryos in the 'Mars zone' (1.2 < a < 2 AU) are usually scattered inward and accreted by 'Earth' or 'Venus,' or scattered outward and removed by resonances, before they can accrete one another. The asteroid belt (a [greater than] 2 AU) is efficiently cleared as objects scatter one another into resonances, where they are lost via encounters with Jupiter or collisions with the Sun, leaving, at most, one surviving object. Accretional evolution is complete after 3 x [10.sup.8] years in all simulations that include Jupiter and Saturn. The number and spacing of the final planets, in our simulations, is determined by the embryos' eccentricities, and the amplitude of secular oscillations in e, prior to the last few collision events. Key Words: planetary formation; terrestrial planets; planetary dynamics; extra-solar planetary systems.