The ω\omega-limit set in a positively invariant compact region and a new description of the Lorenz attractor

The $\omega$-limit set in a compact positively invariant region $R \subset \mathbb{R}^n$ has been identified for $n=1$, 2, and 3, with examples in each case. It has been shown that the $\omega$-limit set becomes more complex as $n$ increases from 1 to 3, and we expect this to also be true for $n>3$. Our example for $n=3$ is the Lorenz equations, for which we have shown that its $\omega$-limit set is a {\em twisted torus

Long-term impact risk for (101955) 1999 RQ36

The potentially hazardous asteroid (101955) 1999 RQ36 has the possibility of collision with the Earth in the latter half of the 22nd century, well beyond the traditional 100-year time horizon for routine impact monitoring. The probabilities accumulate to a total impact probability of approximately 10E-3, with a pair of closely related routes to impact in 2182 comprising more than half of the total. The analysis of impact possibilities so far in the future is strongly dependent on the action of the Yarkovsky effect, which raises new challenges in the careful assessment of longer term impact hazards. Even for asteroids with very precisely determined orbits, a future close approach to Earth can scatter the possible trajectories to the point that the problem becomes like that of a newly discovered asteroid with a weakly determined orbit. If the scattering takes place late enough so that the target plane uncertainty is dominated by Yarkovsky accelerations then the thermal properties of the asteroid,which are typically unknown, play a major role in the impact assessment. In contrast, if the strong planetary interaction takes place sooner, while the Yarkovsky dispersion is still relatively small compared to that derived from the measurements, then precise modeling of the nongravitational acceleration may be unnecessary.Comment: Reviewed figures and some text change

Infrared Lightcurves of Near Earth Objects

We present lightcurves and derive periods and amplitudes for a subset of 38 near earth objects (NEOs) observed at 4.5 microns with the IRAC camera on the the Spitzer Space Telescope, many of them having no previously reported rotation periods. This subset was chosen from about 1800 IRAC NEO observations as having obvious periodicity and significant amplitude. For objects where the period observed did not sample the full rotational period, we derived lower limits to these parameters based on sinusoidal fits. Lightcurve durations ranged from 42 to 544 minutes, with derived periods from 16 to 400 minutes. We discuss the effects of lightcurve variations on the thermal modeling used to derive diameters and albedos from Spitzer photometry. We find that both diameters and albedos derived from the lightcurve maxima and minima agree with our previously published results, even for extreme objects, showing the conservative nature of the thermal model uncertainties. We also evaluate the NEO rotation rates, sizes, and their cohesive strengths.Comment: 16 pages, 4 figures, 3 tables, to appear in the Astrophysical Journal Supplement Serie

The Impact Trajectory of Asteroid 2008 TC3

Asteroid 2008 TC3 was the rst asteroid ever discovered before reaching Earth. By using the almost 900 astrometric observations acquired prior to impact we estimate the trajectory of 2008 TC3 and the ground-track of the impact location as a function of altitude. For a reference altitude of 100 km the impact location 3- formal uncertainty is a 1.4 km 0.15 km ellipse with a semimajor axis azimuth of 105. We analyze the contribution of modeling errors and nd that the second-order zonal harmonics of the Earth gravity eld moves the ground-track by more than 1 km and the location along the ground-track by more than 2 km. Non-zonal and higher order harmonics only change the impact prediction by less than 20 m. The contribution of the atmospheric drag to the trajectory of 2008 TC3 is at the numerical integration error level, a few meters, down to an altitude of 50 km. Integrating forward to lower altitudes and ignoring the break-up of 2008 TC3, the atmospheric drag causes an along-track error that can be as large as a few kilometers at sea level. The locations of the recovered meteorites is consistent with the computed ground-track

Projecting Asteroid Impact Corridors onto the Earth

No abstract availabl