A simple question of celestial mechanics is investigated: in what regions of phase space near a binary system can planets persist for long times? The planets are taken to be test particles moving in the field of an eccentric binary system. A range of values of the binary eccentricity and mass ratio is studied, and both the case of planets orbiting close to one of the stars, and that of planets outside the binary orbiting the system's center of mass, are examined. From the results, empirical expressions are developed for both 1) the largest orbit around each of the stars, and 2) the smallest orbit around the binary system as a whole, in which test particles survive the length of the integration (10^4 binary periods). The empirical expressions developed, which are roughly linear in both the mass ratio mu and the binary eccentricity e, are determined for the range 0.0 <= e <= 0.7-0.8 and 0.1 <= mu <= 0.9 in both regions, and can be used to guide searches for planets in binary systems. After considering the case of a single low-mass planet in binary systems, the stability of a mutually-interacting system of planets orbiting one star of a binary system is examined, though in less detail.Comment: 19 pages, 5 figures, 7 tables, accepted by the Astronomical Journa

Earth is continuously colliding with fragments of asteroids and comets of various sizes. The largest encounter in historical times occurred over the Tunguska river in Siberia in 1908, producing an airburst of energy equivalent to 5-15 megatons of trinitrotoluene (1 kiloton of trinitrotoluene represents an energy of 4.185 × 10(12) joules). Until recently, the next most energetic airburst events occurred over Indonesia in 2009 and near the Marshall Islands in 1994, both with energies of several tens of kilotons. Here we report an analysis of selected video records of the Chelyabinsk superbolide of 15 February 2013, with energy equivalent to 500 kilotons of trinitrotoluene, and details of its atmospheric passage. We found that its orbit was similar to the orbit of the two-kilometre-diameter asteroid 86039 (1999 NC43), to a degree of statistical significance sufficient to suggest that the two were once part of the same object. The bulk strength--the ability to resist breakage--of the Chelyabinsk asteroid, of about one megapascal, was similar to that of smaller meteoroids and corresponds to a heavily fractured single stone. The asteroid broke into small pieces between the altitudes of 45 and 30 kilometres, preventing more-serious damage on the ground. The total mass of surviving fragments larger than 100 grams was lower than expected.

We study the evolution of long-period comets by numerical integration of their orbits, a more realistic dynamical approach than the Monte Carlo and analytic methods previously used to study this problem. We follow the comets from their origin in the Oort cloud until their final escape or destruction, in a model solar system consisting of the Sun, the four giant planets and the Galactic tide. We also examine the effects of nongravitational forces as well as the gravitational forces from a hypothetical solar companion or circumsolar disk. We confirm the conclusion of Oort and other investigators that the observed distribution of long-period comet orbits does not match the expected steady-state distribution unless there is fading or some similar physical process that depletes the population of older comets. We investigate several simple fading laws. We can match the observed orbit distribution if the fraction of comets remaining observable after m apparitions is ∝m −0.6±0.1 (close to the fading law originally proposed by Whipple 1962); or if approximately 95% of comets live for only a few (∼6) returns and the remainder last indefinitely. Our results also yield statistics such as the expected perihelion distribution, distribution of aphelion directions, frequency of encounters with the giant planets and the rate of production of Halley-type comets.

It was realized in 1772 that small bodies can stably share the same orbit as a planet if they remain near 'triangular points' 60° ahead of or behind it in the orbit. Such 'Trojan asteroids' have been found co-orbiting with Jupiter, Mars and Neptune. They have not hitherto been found associated with Earth, where the viewing geometry poses difficulties for their detection, although other kinds of co-orbital asteroid (horseshoe orbiters and quasi-satellites) have been observed. Here we report an archival search of infrared data for possible Earth Trojans, producing the candidate 2010 TK(7). We subsequently made optical observations which established that 2010 TK(7) is a Trojan companion of Earth, librating around the leading Lagrange triangular point, L(4). Its orbit is stable over at least ten thousand years.

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