A multi-spiral set of solutions of a 3-D Hamiltonian system

Various sets of periodic solutions of a 3-D Hamiltonian system crossing perpendicularly thez=0 plane are presented. These sets form a main multi-spiral pattern and two secondary ones which have three focal points. The main pattern is inside a stochastic region that surrounds a simple complex unstable periodic orbit, while the two secondary patterns are parts of a stochastic sea. Through these regions the stochastic region communicates with the stochastic sea.     

The edge of the Kuiper belt: the Planet X scenario

Our goal is to determine whether or not the observed sudden termination of the Edgeworth-Kuiper belt can be the result of perturbations from a hypothetical planet. We investigate the effects that such an object would produce on the primordial orbital distribution if the trans-neptunian objects, for a range of masses and orbital parameters of the hypothetical planet. In this numerical investigation, the motion of the hypothetical planet was influenced by the existing planets but not by its interaction with the disk. We find that no set of parameters produce results that match the observed data. Dynamical interaction with the disk is likely to be important so that the orbit of the hypothetical planet changes significantly during the integration interval. This is also discussed. The overall conclusion is that none of the models for the hypothetical planet that were investigated can reproduce the observed features of the Edgeworth-Kuiper belt starting from any probable primordial distribution.     

Mutual events of the uranian satellites 2006-2010

The upcoming crossing of the Sun and the Earth through the equatorial plane of the planet Uranus presents an opportunity to observe mutual eclipses and occultations of the uranian satellites. We present predictions for 321 such events from 2006 to 2010. 230 of these events are “nominal” i.e. they are predicted to occur based on the currently available ephemeris while a further 91 “grazing” events are allowable given the positional uncertainties of the satellites. Taking into account the statistical frequency of events that occur too close to the planet, during solar conjunction or are too “shallow” to observe, we conclude that about 150 events should be detectable from different longitudes around the world. We argue that a worldwide campaign of photometric observations of these events will yield, as in the case of the jovian and saturnian systems, high-precision astrometric information on the satellites toward improving their ephemerides as well as the system constants (satellite masses, uranian zonal harmonics, etc.). In addition, mathematical inversion of the lightcurves should permit, subject to the photometric quality and number of observed events, mapping of albedo variegations over the satellite hemispheres that were in darkness during the Voyager 2 encounter with the uranian system in 1985/1986.     

A three-dimensional solution for the orbit of the asteroidal satellite of 22 Kalliope

We carried out new observations of the binary asteroid 22 Kalliope (S2/2001) with the Shane 3-m telescope of the Lick observatory in October and November 2001. With a FWHM (full width at half maximum) of 0″.2, Kalliope (apparent size of about 0″.15) was not resolved but it was possible to separate the secondary from its primary whose apparent separation was of the order of 0″.7 with a magnitude difference of 3.22±0.20. As each set of observations spanned a few days of time, they are well distributed along the secondary's orbit, enabling us to accurately estimate its orbit.The satellite orbits 22 Kalliope in a prograde manner with respect to Kalliope's rotational spin (which is in a retrograde sense relative to its orbit around the Sun), on a highly inclined (i=19.8±2.0 with respect to the equator of 22 Kalliope) and moderately eccentric orbit (e=0.07±0.02) with an orbital period of 3.58±0.08 days. The semi-major axis is 1020±40 km. Using Kalliope's diameter as determined from IRAS data, the asteroid's bulk density is about 2.03±0.16 g cm⁻³, suggestive of a highly porous body with a porosity of 70% considering that the grain density of its meteoritic analog is of ∼7.4 g cm⁻³. This suggests a rubble pile, rather than solid, body. The measured nodal precession rate of the secondary's orbit seems to be much higher than expected from Kalliope's oblateness, assuming a homogeneous body (constant density). This suggests that Kalliope may be 60% more elongated or 35% larger than presently believed or/and that its internal structure is highly inhomogeneous with a denser outer shell.     

Meteor outbursts from long-period comet dust trails

Long-period comets have narrow one-revolution old dust trails that can cause meteor outbursts when encountered by Earth. To facilitate observing campaigns that will characterize and perhaps help find Earth-threatening, long-period comets from their trace of meteoric debris, we use past accounts of outbursts from 14 different showers to calculate the future dust trail positions near Earth’s orbit. We also examine known near-Earth, long-period comets and identify five potential new showers, which can be utilized to learn more about these objects. We demonstrate that it is the one-revolution trail that is responsible for meteor outbursts. A method that calculates in what year these showers are likely to return and at what hour is presented. The calculations improve on earlier approximate methods that used the Sun’s reflex motion to gauge the trail motion relative to Earth’s orbit.     

A 1 Gyr climate model for Mars: new orbital statistics and the importance of seasonally resolved polar processes

New results from a 1 Gyr integration of the martian orbit are presented along with a seasonally resolved energy balance climate model employed to illuminate the gross characteristics of the long-term atmospheric pressure evolution. We present a new analysis of the statistical variation of the martian obliquity and precession prior to and subsequent to the formation of the Tharsis uplift, and explore the long term effects on the martian climate. We find that seasonal polar cycles have a critical influence on the ability for the regolith to release CO₂ at high obliquities, and find that the atmospheric CO₂ actually decreases at high obliquities due to the cooling effect of polar deposits at latitudes where seasonal caps form. At low obliquity, the formation of massive, permanent polar caps depends critically on the values of the frost albedo, A_frost, and frost emissivity, ?_frost. Using our model with values of A_frost=0.67 and ?_frost=0.55, matched to the NASA Ames General Circulation Model (GCM) results (Haberle et al., 1993, J. Geophys. Res. 98, 3093-3123, and Haberle et al., 2003, Icarus 161, 66-89), we find that permanent caps only form at low obliquities (<13°), suggesting that any permanent deposits on the surface of Mars today may be residuals left over from a period of very low obliquity, or are the result of mechanisms not represented by this model. Thus, contrary to expectations, the martian atmospheric pressure is remarkable static over time, and decreases both at high and low obliquity. Also, from our one billion year orbital model, we present new results on the fraction of time Mars is expected to experience periods of low obliquity and high obliquity.     

Transition spectra of dynamical systems

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape P_n(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of a_k in the continued fraction expansion of the rotation number a = [a₀,a₁,... a_k,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q₀ indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands. This revised version was published online in July 2006 with corrections to the Cover Date.     

On particle trajectories in restricted 3-body problem including the effect of radiation: Sun-Jupiter system

In this work, we have simulated orbits of a particle moving in gravitational field of the Sun-Jupiter system. The effect of solar radiation pressure, including Poynting Robertson drag, on the evolution of particle orbits in phase space have been studied for different values of the parameter β ₁ (the ratio of radiation to gravitational force) and initial conditions. Characteristics of various computed trajectories have been studied using wavelet transform (WT), Fourier transform (FT) and Poincare surface of section method. We use wavelet analysis to identify transitions of a trajectory in time-frequency plane and further apply it to classify it as regular or chaotic in phase space. Unlike the Fourier transform method (FT), we observe that the wavelet transform (WT) also provides a basis to identify ‘sticky’ trajectories in the present dynamical system.