Nucleus properties of Comet 67P/Churyumov-Gerasimenko estimated from non-gravitational force modeling

By considering model comet nuclei with a wide range of sizes, prolate ellipsoidal shapes, spin axis orientations, and surface activity patterns, constraints have been placed on the nucleus properties of the primary Rosetta target, Comet 67P/Churyumov-Gerasimenko. This is done by requiring that the model bodies simultaneously reproduce the empirical nucleus rotational lightcurve, the water production rate as function of time, and non-gravitational changes (per apparition) of the orbital period (ΔP), longitude of perihelion (Δ?), and longitude of the ascending node (ΔΩ). Two different thermophysical models are used in order to calculate the water production rate and non-gravitational force vector due to nucleus outgassing of the model objects. By requiring that the nominal water production rate measurements are reproduced as well as possible, we find that the semi-major axis of the nucleus is close to 2.5 km, the nucleus axis ratio is approximately 1.4, while the spin axis argument is either 60°±15° or 240°±15°. The spin axis obliquity can only be preliminarily constrained, indicating retrograde rotation for the first argument value, and prograde rotation for the second suggested spin axis argument. A nucleus bulk density in the range 100-370 kg m⁻³ is found for the nominal ΔP, while an upper limit of 500 kg m⁻³ can be placed if the uncertainty in ΔP is considered. Both considered thermophysical models yield the same spin axis, size, shape, and density estimates. Alternatively, if calculated water production rates within an envelope around the measured data are considered, it is no longer possible to constrain the size, shape, and spin axis orientation of the nucleus, but an upper limit on the nucleus bulk density of 600 kg m⁻³ is suggested.     

Asteroid spin‐axis longitudes from the Lowell Observatory database

By analyzing brightness variation with ecliptic longitude and using the Lowell Observatory photometric database, we estimate spin‐axis longitudes for more than 350,000 asteroids. Hitherto, spin‐axis longitude estimates have been made for fewer than 200 asteroids. We investigate longitude distributions in different dynamical groups and asteroid families. We show that asteroid spin‐axis longitudes are not isotropically distributed as previously considered. We find that the spin‐axis longitude distribution for Main Belt asteroids is clearly nonrandom, with an excess of longitudes from the interval 30°–110° and a paucity between 120° and 180°. The explanation of the nonisotropic distribution is unknown at this point. Further studies have to be conducted to determine if the shape of the distribution can be explained by observational bias, selection effects, a real physical process, or other mechanism.     

Out-of-plane librations of spinning tethered satellite systems

We analyze the out-of-plane librations of a tethered satellite system that is nominally rotating in the orbit plane. To isolate the librational dynamics, the system is modeled as two point masses connected by a rigid rod with the system mass center constrained to an unperturbed circular orbit. For small out-of-plane librations, the in-plane motion is unaffected by the out-of-plane librations and a solution for the in-plane motion is determined in terms of Jacobi elliptic functions. This solution is used in the linearized equation for the out-of-plane librations, resulting in a Hill’s equation. Floquet theory is used to analyze the Hill’s equation, and we show that the out-of-plane librations are unstable for certain ranges of in-plane spin rate. For relatively high in-plane spin rates, the out-of-plane librations are stable, and the Hill’s equation can be approximated by a Mathieu’s equation. Approximate solutions to the Mathieu’s equation are determined, and we analyze the dominant characteristics of the out-of-plane librations for high in-plane spin rates. The results obtained from the analysis of the linearized equations of motion are compared to numerical simulations of the nonlinear equations of motion, as well as numerical simulations of a more realistic system model that accounts for tether flexibility. The instabilities discovered from the linear analysis are present in both the nonlinear system and the more realistic system model. The approximate solutions for the out-of-plane librations compare well to the nonlinear system for relatively high in-plane rotation rates, and also capture the significant qualitative behavior of the flexible system.     

New measures of the orientation of the linear structure in the sunward hemisphere of the coma of Comet 19P/Borrelly: apparitions of 1994, 2001, and 2008

The sky‐projected orientation (position angle) of the axis (line of maximum density or maximum brightness) of the long time‐known, linear structure (LS) in the sunward hemisphere of the coma of Comet 19P/Borrelly is measured on 45 photographs taken by different observers under different projection conditions and covering three consecutive apparitions (1994, 2001, and 2008) for a total time interval of 5174 days. The analysis of the results by a tomographic approach yields an LS axis constantly oriented towards a fixed point in the space, at Right Ascension 214°.4 ± 0°.5 and Declination –7°.0 ± 0°.5 (J2000), corresponding to an obliquity of 103°.5 ± 1° and an orbital longitude of 147°.2 ± 1°, throughout the relevant interval. Such coordinates are close to the ones found by other authors for the spatial orientation of the nucleus spin axis during the apparitions of 1994 and 2001. In the hypothesis of an LS orientation aligned with the nucleus spin axis, the new results confirm the previous ones and show that this orientation remained unchanged during the subsequent 2008 apparition (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theory of satellite orbit-orbit resonance

On the basis of the strong mathematical and physical parallels between orbit-orbit and spin-orbit resonances, the dynamics of mutual orbit perturbations between two satellites about a massive planet are examined, exploiting an approach previously adopted in the study of spin-orbit coupling. The satellites are assumed to have arbitrary mass ratio and to move in non-intersecting orbits of arbitrary size and eccentricity. Resonances are found to exist when the mean orbital periods are commensurable with respect to some rotating axis, which condition also involves the apsidal and nodal motions of both satellites. In any resonant state the satellites are effectively trapped in separate potential wells, and a single variable is found to describe the simultaneous librations of both satellites. The librations in longitude are 180° out-of-phase, with fixed amplitude ratio that depends only on their relative masses and semimajor axes. At the same time the stroboscopic longitude of conjunction also librates about the commensurate axis with the same period. The theory is applicable to Saturn's resonant pairs Titan-Hyperion and Mimas-Tethys, and in these cases our calculated libration periods are in reasonably good agreement with the observed periods.This research supported under a grant from the California Institute of Technology President's Fund and NASA Contract NAS 7-100.     

地球动力学扁率及其与岁差章动的关系

由岁差常数求得的日月岁差是天文学的重要参数之一,它和地球动力学扁率相联系。地球动力学扁率在章动理论的计算中也是一个重要的物理量。介绍了由不同的观测方法和模型给出的地球动力扁率值,并讨论了它也岁差的关系和对章动计算的影响。在刚体地球章动振幅的计算中,地球动力学扁率值起着尺度因子的作用,要改善刚体地球章动振幅的计算,需要修改目前的黄经总岁差值。非刚体地球章动的转换函数中所采用的简正模和常数都直接或间接地依赖地球动力学扁率值。在IAU1980章动理论中,计算刚体地球章动振幅所使用的地球动力学扁率值计算转换函数中简正模频率和常数所使用的地球动力学扁率值并不一致。随着观测和计算精度的提高,地球动力学扁率值的不一致将影响章动振幅的计算。在建立刚体地球章地动理论中,如何解释地球动力学扁率值的差异,如何选取地球动力学扁率值,还有待进一步的研究。     

Determination of Planetocentric Coordinates of the Center of the Illuminated Part of the Spherical Planet's Visible Disk

Exact formulas are derived for calculating the correction for phase to the central meridian longitude and the latitude of the center of the illuminated part of the visible disk of a spherical planet in projection onto the plane of the sky. The range of possible values of the phase correction to the central meridian longitude is determined. The proposed formulas are valid for any planet's orientation with respect to the Earth and Sun and allow a transition, in planetocentric coordinates, from the center of the geometric disk of a spherical planet to the center of the illuminated part of the planet's visible disk. The reduction formulas are derived for direct and inverse transitions between the two aforementioned points of the planetary disk in geocentric equatorial coordinates. The examples of special cases of illumination of visible disks of planets in specific ephemeris are given.     

Physical librations due to the third and fourth degree harmonics of the lunar gravity potential

The existence of third and fourth harmonics of the lunar gravity potential gives rise to sizable lunar physical librations. Using one recent set of potential estimates, the following effects are noted: the mean sub-Earth point is displaced from the earthward principal moment of inertia axis by 168″; the inclination of the lunar equator to the ecliptic is decreased by 14″.5; and a six year period libration in longitude, with amplitude 13″.1, is induced.     

Latitudinal librations of Mercury with a fluid core

In the framework of the space missions to Mercury, an accurate model of rotation is needed. Librations around the 3:2 spin-orbit resonance as well as latitudinal librations have to be predicted with the best possible accuracy. In this paper, we use a Hamiltonian analysis and numerical integrations to study the librations of Mercury, both in longitude and latitude. Due to the proximity of the period of the free libration in longitude to the orbital period of Jupiter, the 88-day and 11.86-year contributions dominate Mercury’s libration in longitude (with the Hermean parameters chosen). The amplitude of the libration in latitude is much smaller (under 1 arcsec) and should not be detected by the space missions. Nevertheless, we point out that this amplitude could be much larger (up to several tens of arcsec) if the free period related to the libration in latitude approaches the period of the Jupiter-Saturn Great Inequality (883 years). Given the large uncertainties on the planetary parameters, this new resonant forcing on Mercury’s libration in latitude should be borne in mind.     

The Moon’s physical librations and determination of their free modes

The Lunar Laser Ranging experiment has been active since 1969 when Apollo astronauts placed the first retroreflector on the Moon. The data accuracy of a few centimeters over recent decades, joined to a new numerically integrated ephemeris, DE421, encourages a new analysis of the lunar physical librations of that ephemeris, and especially the detection of three modes of free physical librations (longitude, latitude, and wobble modes). This analysis was performed by iterating a frequency analysis and linear least-squares fit of the wide spectrum of DE421 lunar physical librations. From this analysis we identified and estimated about 130–140 terms in the angular series of latitude librations and polar coordinates, and 89 terms in the longitude angle. In this determination, we found the non-negligible amplitudes of the three modes of free physical libration. The determined amplitudes reach 1.296′′ in longitude (after correction of two close forcing terms), 0.032′′ in latitude and 8.183′′ × 3.306′′ for the wobble, with the respective periods of 1056.13 days, 8822.88 days (referred to the moving node), and 27257.27 days. The presence of such terms despite damping suggests the existence of some source of stimulation acting in geologically recent times.     

Long-period forcing of Mercury's libration in longitude

Planetary perturbations of Mercury's orbit lead to forced librations in longitude in addition to the 88-day forced libration induced by Mercury's orbital motion. The forced librations are a combination of many periods, but 5.93 and 5.66 years dominate. These two periods result from the perturbations by Jupiter and Venus respectively, and they lead to a 125-year modulation of the libration amplitude corresponding to the beat frequency. Other periods are also identified with Jupiter and Venus perturbations as well as with those of the Earth, and these and other periods in the perturbations cause several arc second fluctuations in the libration extremes. The maxima of these extremes are about 30″ above and the minima about 7″ above the superposed ∼^60″, 88-day libration during the 125-year modulation. Knowledge of the nature of the long-period forced librations is important for the interpretation of the details of Mercury's rotation state to be obtained from radar and spacecraft observations. We show that the measurement of the 88-day libration amplitude for the purposes of determining Mercury's core properties is not compromised by the additional librations, because of the latter's small amplitude and long period. If the free libration in longitude has an amplitude that is large compared with that of the forced libration, its ∼10-year period will dominate the libration spectrum with the 88-day forced libration and the long-period librations from the orbital perturbations superposed. If the free libration has an amplitude that is comparable to those of the long-period forced libration, it will be revealed by erratic amplitude, period and phase on the likely time span of a series of observations. However, a significant free libration component is not expected because of relatively rapid damping.     

Detailed survey of the phase space around Nix and Hydra

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto–Charon system. The spatial elliptic restricted three-body problem was used as model and stability maps were created by chaos indicators. The orbital elements of the moons are in the stable domain on the semimajor axis, eccentricity and inclination spaces. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps. They do not contain the positions of the moons, confirming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericentre or longitude of node falls in a certain range. The results strongly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of pericentre or longitude of node.     

Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009

Every three years the IAU Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the directions of the poles of rotation and the prime meridians of the planets, satellites, minor planets, and comets. This report takes into account the IAU Working Group for Planetary System Nomenclature (WGPSN) and the IAU Committee on Small Body Nomenclature (CSBN) definition of dwarf planets, introduces improved values for the pole and rotation rate of Mercury, returns the rotation rate of Jupiter to a previous value, introduces improved values for the rotation of five satellites of Saturn, and adds the equatorial radius of the Sun for comparison. It also adds or updates size and shape information for the Earth, Mars?? satellites Deimos and Phobos, the four Galilean satellites of Jupiter, and 22 satellites of Saturn. Pole, rotation, and size information has been added for the asteroids (21) Lutetia, (511) Davida, and (2867) ?teins. Pole and rotation information has been added for (2) Pallas and (21) Lutetia. Pole and rotation and mean radius information has been added for (1) Ceres. Pole information has been updated for (4) Vesta. The high precision realization for the pole and rotation rate of the Moon is updated. Alternative orientation models for Mars, Jupiter, and Saturn are noted. The Working Group also reaffirms that once an observable feature at a defined longitude is chosen, a longitude definition origin should not change except under unusual circumstances. It is also noted that alternative coordinate systems may exist for various (e.g. dynamical) purposes, but specific cartographic coordinate system information continues to be recommended for each body. The Working Group elaborates on its purpose, and also announces its plans to occasionally provide limited updates to its recommendations via its website, in order to address community needs for some updates more often than every 3 years. Brief recommendations are also made to the general planetary community regarding the need for controlled products, and improved or consensus rotation models for Mars, Jupiter, and Saturn.     

Rotational modeling of Hyperion

Saturn’s moon, Hyperion, is subject to strongly-varying solid body torques from its primary and lacks a stable spin state resonant with its orbital frequency. In fact, its rotation is chaotic, with a Lyapunov timescale on the order of 100 days. In 2005, Cassini made three close passes of Hyperion at intervals of 40 and 67 days, when the moon was imaged extensively and the spin state could be measured. Curiously, the spin axis was observed at the same location within the body, within errors, during all three fly-bys—~ 30° from the long axis of the moon and rotating between 4.2 and 4.5 times faster than the synchronous rate. Our dynamical modeling predicts that the rotation axis should be precessing within the body, with a period of ~ 16 days. If the spin axis retains its orientation during all three fly-bys, then this puts a strong constraint on the in-body precessional period, and thus the moments of inertia. However, the location of the principal axes in our model are derived from the shape model of Hyperion, assuming a uniform composition. This may not be a valid assumption, as Hyperion has significant void space, as shown by its density of 544± 50  kg m⁻³ (Thomas et al. in Nature 448:50, 2007). This paper will examine both a rotation model with principal axes fixed by the shape model, and one with offsets from the shape model. We favor the latter interpretation, which produces a best-fit with principal axes offset of ~ 30° from the shape model, placing the A axis at the spin axis in 2005, but returns a lower reduced χ ² than the best-fit fixed-axes model.     

Three tenuous rings/arcs for three tiny moons

Using Cassini images, we examine the faint material along the orbits of Methone, Anthe and Pallene, three small moons that reside between the orbits of Mimas and Enceladus. A continuous ring of material covers the orbit of Pallene; it is visible at extremely high phase angles and appears to be localized vertically to within ±25 km of Pallene's inclined orbit. By contrast, the material associated with Anthe and Methone appears to lie in longitudinally confined arcs. The Methone arc extends over ∼10° in longitude around the satellite's position, while the Anthe arc reaches ∼20° in length. The extents of these arcs are consistent with their confinement by nearby corotation eccentricity resonances with Mimas. Anthe has even been observed to shift in longitude relative to its arc in the expected manner given the predicted librations of the moon.     

Analytical model of the long-period forced longitude librations of Mercury

The shaking of Mercury’s orbit by the planets forces librations in longitude in addition to those at harmonics of the orbital period that have been used to detect Mercury’s molten core. We extend the analytical formulation of Peale et al. (Peale, S.J., Margot, J.L., Yseboodt, M. [2009]. Icarus 199, 1-8) in order to provide a convenient means of determining the amplitudes and phases of the forced librations without resorting to numerical calculations. We derive an explicit relation between the amplitude of each forced libration and the moment of inertia parameter (B-A)/C_m. Far from resonance with the free libration period, the libration amplitudes are directly proportional to (B-A)/C_m. Librations with periods close to the free libration period of ∼12 years may have measurable (∼arcsec) amplitudes. If the free libration period is sufficiently close to Jupiter’s orbital period of 11.86 years, the amplitude of the forced libration at Jupiter’s period could exceed the 35 arcsec amplitude of the 88-day forced libration. We also show that the planetary perturbations of the mean anomaly and the longitude of pericenter of Mercury’s orbit completely determine the libration amplitudes.While these signatures do not affect spin rate at a detectable level (as currently measured by Earth-based radar), they have a much larger impact on rotational phase (affecting imaging, altimetry, and gravity sensors). Therefore, it may be important to consider planetary perturbations when interpreting future spacecraft observations of the librations.     

Theory of the rotation of Janus and Epimetheus

The saturnian coorbital satellites Janus and Epimetheus present a unique dynamical configuration in the Solar System, because of high-amplitude horseshoe orbits, due to a mass ratio of order unity. As a consequence, they swap their orbits every 4 years, while their orbital periods is about 0.695 days. Recently, Tiscareno et al. (Tiscareno, M.S., Thomas, P.C., Burns, J.A. [2009]. Icarus 204, 254-261) got observational informations on the shapes and the rotational states of these satellites. In particular, they detected an offset in the expected equilibrium position of Janus, and a large libration of Epimetheus.We here propose to give a three-dimensional theory of the rotation of these satellites in using these observed data, and to compare it to the observed rotations. We consider the two satellites as triaxial rigid bodies, and we perform numerical integrations of the system in assuming the free librations as damped.The periods of the three free librations we get, associated with the three dimensions, are respectively 1.267, 2.179 and 2.098 days for Janus, and 0.747, 1.804 and 5.542 days for Epimetheus. The proximity of 0.747 days to the orbital period causes a high sensitivity of the librations of Epimetheus to the moments of inertia. Our theory explains the amplitude of the librations of Janus and the error bars of the librations of Epimetheus, but not an observed offset in the orientation of Janus.     

Determining a tilt in Titan’s north-south albedo asymmetry from Cassini images

Analysis of Titan’s hemispheric brightness asymmetry from mapped Cassini images reveals an axis of symmetry that is tilted with respect to the rotational axis of the solid body. Twenty images taken from 2004 through 2007 show a mean axial offset of 3.8 ± 0.9° relative to the solid body’s pole, directed 79 ± 24° to the west of the sub-solar longitude. These values are consistent with recent measurements of an implied atmospheric spin axis determined from isothermal mapping by [Achterberg, R.K., Conrath, B.J., Gierasch, P.J., Flasar, F.M., Nixon, C.A., 2008. Icarus 197, 549-555].     

Rotational Properties of Cometary Nuclei

We review several techniques used to retrieve rotational parameters from observations. The spin period of a dozen of comets retrieved with these techniques are summarized. We describe how the spin period of comet Hale-Bopp (C/1995 O1) has been calculated with a high accuracy (11.30–11.34 h). Although several authors converged to a spin axis orientation at (α,δ) = (275 ± 15°, -55 ± 5°), detailed studies indicate that the dust jets morphology in 1996–1997 may be incompatible with this orientation. Comet 19P/Borrelly has been recently observed by the Deep Space 1 spacecraft. At the same time, its spin axis orientation and period have been determined by several authors to be respectively (α,δ) = (225 ± 15°, -10 ± 10°)and 26h. These two comets are likely to be in (or close to) a principal axis spin state. We discuss new modeling of the spin state of comet 46P/Wirtanen, the target of the Rosetta mission. The model involves a three-dimensional shape and thermal model, from which the torque of the non gravitational force is calculated at each time step. The moments of inertia are computed for each irregular shape. The results from numerical integrations show that this comet can remain in a principal axis spin state during more than 10 orbits if the spin period does not get above～6 h. If the spin period increases, its nucleus gets rapidly into excited spin states. It shows that even small and very active short-period comets are not necessarily in non principal axis spin states. In the last section, the consequences of recent observations and modeling of the rotational parameters of comet nuclei are discussed, and unsolved problems are presented.