David Parry Rubincam

Asteroid orbit evolution due to thermal drag

Thermal drag, a variant of the Yarkovsky effect, may act on small asteroids with sizes from a few meters to a few tens of meters. Yarkovsky thermal drag comes from an asteroids absorbing sunlight in the visible and reradiating it in the infrared. Since the infrared photons have momentum, by action-reaction, they kick the asteroid when they leave its surface. The reradiation, which is asymmetric in latitude over the asteroid, gives a net force along the asteroids pole. Due to the asteroids thermal inertia, averaging this force over one orbital period produces a net drag if the spin axis has a component in the orbital plane. A regolith-free basaltic asteroid 60 m in radius can shrink its semimajor axis by 2 AU (the distance from the asteroid belt to the Earth) over the age of the solar system. Regolith-free iron asteroids evolve at about half the rate of basaltic ones. These calculations ignore planetary perturbations, collisions, erosion, etc. The rate of evolution varies inversely with the asteroids radius for the size range considered here, so that smaller objects evolve faster than larger ones. The rate-radius relation fails for objects smaller than a few meters because the thermal skin depth becomes comparable to the size of the asteroid. Basaltic asteroids covered by regoliths more than a few centimeters deep evolve much more slowly than regolith-free ones. Thermal drag tends to circularize orbits. It can increase or decrease orbital inclinations. An object whose spin axis points in random directions over its lifetime displays little change in orbital inclination. Thermal drag appears to have little to do with the delivery of chondrites from the asteroid belt; the thermal drag timescale (108 years for meter-sized objects) is long compared with their cosmic ray exposure ages, and aphelia in the asteroid belt are not expected for mature thermal drag orbits. However, Yarkovsky thermal drag may act on the recently discovered near-Earth asteroids, which have radii of 10–30 m. Asteroid 1992 DA, for instance, might have its orbit shrunk by 0.1 AU in 3×107 years, removing it from an Earth-crossing orbit. The near-Earth asteroids also tend to have small to moderate orbital eccentricities, as expected for highly evolved thermal drag objects. However, the time needed to bring them in from the asteroid belt (about 109 years) is long compared with the collisional and dynamical lifetimes (both about 108 years) for Earth-crossing objects, arguing against their emplacement by thermal drag.

On the secular decrease in the semimajor axis of Lageos's orbit

The semimajor axis of the Lageos satellites orbit is decreasing secularly at the rate of 1.1 mm day−1. Ten possible mechanisms are investigated to discover which one (s), if any, might be causing the orbit to decay. Six of the mechanisms, resonance with the Earths gravitational field, gravitational radiation, the Poynting-Robertson effect, transfer of spin angular momentum to the orbital angular momentum, drag from near-earth dust, and atmospheric drag by neutral hydrogen are ruled out because they are too small or require unacceptable assumptions to account for the observed rate of decay. Three other mechanisms, the Yarkovsky effect, the Schach effect, and terrestrial radiation pressure give perturbations whose characteristic signatures do not agree with the observed secular decrease (terrestrial radiation pressure appears to be too small in any case); hence they are also ruled out. Charged particle drag with the ions at Lageoss altitude is probably the principal cause of the orbital decay. An estimate of charged particle drag based upon laboratory experiments and satellite measurements of ion number densities accounts for 60 percent of the observed rate of decrease in the semimajor axis, assuming a satellite potential of −1V. This figure is in good agreement with other estimates based on charge drag theory. A satellite potential of −1.5V will explain the entire decay rate. Atmospheric drag from neutral hydrogen appears to be the next largest effect, explaining about 10 percent of the observed orbital decay rate.

Yarkovsky thermal drag on small asteroids and Mars-Earth delivery

There is an optimal size for the delivery of small asteroids from Mars to the Earth by Yarkovsky thermal drag. Basaltic asteroids with radii of about 6 m take on the average 185 million years (Myr) for their semimajor axes to shrink by 0.52 AU, assuming circular orbits and ignoring planetary perturbations and collisions. All other sizes take longer. Bigger objects are slower because they are more massive, and smaller objects are slower because they are more isothermal. These results are based on treating the asteroids as spheres and solving the heat conduction equation using spherical Bessel functions. The small near-Earth asteroids show a concentration of sizes in the thermal drag range; thus some of them may come from Mars as survivors of gravitational mechanisms which eliminate them on the 10 Myr timescale. The possible role of thermal drag in Mars-Earth delivery will remain speculative until it is included in numerical integrations of the orbits of small asteroids.

Constraints on density models from radial moments: Applications to Earth, Moon, and Mars

Few objective constraints exist on radial density variations in planetary interiors. Even if the mean density and mean moment of inertia were known with perfect accuracy, they would only provide integral constraints on the density models. However, among the family of monotonic radial density models with a given mean density and mean moment of inertia, the simple two-layer piecewise constant models have useful extremal properties. The model has only three parameters, an inner density, an outer density, and a transition radius. Once the radial moment constraints are applied, there is only one remaining degree of freedom. If the outer region density is somehow specified, then the inner region density of the two-layer model provides a firm lower bound on central density of monotonic models. Likewise, if an upper bound on central density can be provided, then the two-layer model provides a lower bound on outer region density. An envelope of acceptable density models can be generated by scanning the transition depth from the center to the surface. Any monotonic model with specified mean density and mean inertial moment must lie within that envelope. Resulting extremal density envelopes for the Earth, Moon, and Mars are compared to published radial density profiles. For the Moon, the moment constraints are quite restrictive. For the Earth, which is more centrally condensed, the allowed envelope of density profiles is rather broad. Mars is an intermediate case. Present geodetic and astrometric observations only constrain the Martian mean moment of inertia to lie somewhere in the range 0.325 ≤ I/MR2 ≤ 0.365. However, our analysis shows that any value within that range can be accommodated without invoking geochemically implausible density minima or maxima.

General relativity and satellite orbits: The motion of a test particle in the Schwarzschild metric

The motion of a satellite with negligible mass in the Schwarzschild metric is treated as a problem in Newtonian physics. The relativistic equations of motion are formally identical with those of the Newtonian case of a particle moving in the ordinary inverse-square law field acted upon by a disturbing function which varies asr−3. Accordingly, the relativistic motion is treated with the methods of celestial mechanics. The disturbing function is expressed in terms of the Keplerian elements of the orbit and substituted into Lagranges planetary equations. Integration of the equations shows that a typical Earth satellite with small orbital eccentricity is displaced by about 17 cm from its unperturbed position after a single orbit, while the periodic displacement over the orbit reaches a maximum of about 3 cm. Application of the equations to the planet Mercury gives the advance of the perihelion and a total displacement of about 85 km after one orbit, with a maximum periodic displacement of about 13 km.

LAGEOS I once‐per‐revolution force due to solar heating

Photon thrust from the solar heating of the LAGEOS I satellite appears to explain much of the eccentricity variations seen in the satellites orbital elements. We invoke a thermal model of LAGEOS I in which the photon thrust from solar heating is directed along the satellites spin axis and functionally depends only on the cosine of the angle between the Suns position and the spin axis. We calibrated the amplitude of the force from the 1980–1983 equivalent along-track acceleration derived from the observed orbital perturbations; during this time the spin axis position is assumed to be known and to be that at orbit injection. The photon thrust from this simple thermal model, plus later spin axis positions obtained from Sun glint data (which show LAGEOS I to be precessing), give reasonable agreement with the observed along-track acceleration in the time period 1988–1995. Thus much of the eccentricity variations seem to be due to thermal thrust and do not have a geophysical origin (atmospheric tides) as has been proposed. However, our solar heating model does not appear to explain the highest peaks and deepest troughs seen in the along-track acceleration, indicating the need for a better thermal model and consideration of other forces, such as that due to anisotropic reflection.

Effects of Earth albedo on the LAGEOS I satellite

For several years, Earth albedo has been one of the leading candidates for explaining the anomalous along-track acceleration experienced by the LAGEOS satellites, with reasonable reflection models having been shown to give acceleration peaks comparable to those observed. The effects of Earth albedo on LAGEOS I have been studied for the period March 1985 to June 1989, during which time measurements of reflected radiation from the Earth were made by one to three Earth Radiation Budget Experiment /ERBE) satellites. Data taken during this experiment were processed into hourly radiation exitances for the entire Earth divided into 2.5° × 2.5° blocks. These ERBE data were used, along with detailed models of reflectance characteristics of the Earth, to calculate the accelerations which reflected radiation would induce on LAGEOS I. The along-track component of the acceleration was averaged over 10 revolutions in order to average out the short-period effects. The results showed that the albedo effect on along-track acceleration did not exceed 0.5 pm/s2, or about 20% of the anomalous acceleration. Even this acceleration showed no correlation in phase with the observed acceleration. The albedo accelerations were also used to calculate the effects on the LAGEOS Keplerian orbital elements. The effects on the LAGEOS node and inclination excitations were significant, having amplitudes of several milliarc seconds /mas) per year. The largest effects were found to be on the eccentricity excitation function, having amplitudes at the 50–100 mas/yr level. The patterns also correlate well with the observed eccentricity vector excitations, but with a difference in sign, suggesting that some parameters estimated with LAGEOS data, such as ocean tide parameters, may have been corrupted by albedo effects.

Has climate changed the Earth's tilt?

Have the ice ages secularly altered the obliquity (axial tilt) of the Earth over geologic time? The waxing and waning of ice caps in response to obliquity oscillations plus mantle adjustment to the weight of the caps alter the Earths dynamical flattening. This affects the lunar and solar torques on the Earth so as to secularly change the Earths axial tilt. This mechanism is dubbed “climate friction,” since analogous to tidal friction, it arises from lags in the Earths response to cyclic external forcing. But the existence of two processes, ice cap waxing and waning and mantle viscous flow, can lead to either an increase or decrease in the obliquity. Evidence indicates that the growth and decay of the ice caps greatly lag the orbital forcing; this causes the axial tilt to increase with time. But the ice cap effects are partly canceled by viscous compensation in the mantle. Low mantle viscosities (about 1021 Pa·s) lead to rapid compensation and have only a slight effect on obliquity. High viscosities (about 1022 Pa·s) slow the compensation enough so that there could be significant secular change in the obliquity over geologic time, perhaps explaining all of the present tilt of 23.5°. However, in addition to the current uncertainty as to the effective viscosity of the mantle, knowledge of past ice ages is incomplete, so that the amount of obliquity increase presently remains unknown.

Polar motion and earth tides from laser tracking

The tracking of near-Earth satellites with laser systems permits the determination of the variation of latitude of the tracking station and the variation in the rotation of the Earth. The present-day capability of a single station is approximately 75 cm in latitude averaged over 6h and 0.8 ms in the length of day. When the Laser Geodynamics Satellite (Lageos) is launched, a network of laser stations is projected to be able to achieve better than 10 cm in each coordinate from less than one day of tracking. The perturbations of near-Earth satellites by solid Earth and ocean tides are now measurable and can provide new information about the Earth and oceans. The orbit perturbations have long periods (days, months) and the analysis of orbital changes are providing estimates of the amplitudes and phases of the major tidal components.

Terrestrial atmospheric effects on satellite eclipses with application to the acceleration of LAGEOS

We examine spatial variability of attenuation in the Earths atmosphere as a cause of asymmetrical eclipses and consequent acceleration of LAGEOS, i.e., the solar radiation pressure on LAGEOS due to the Earths penumbra. Measurements of atmospheric attenuation derived from the satellite-borne Stratospheric Aerosol and Gas Experiment after the eruption of Mount Pinatubo were used to simulate the largest expected aerosol content of the atmosphere. In our experiment one hemisphere was loaded with volcanic aerosols, while the other was not. The difference between attenuation in the two hemispheres sets a maximum reasonable limit to the size of eclipse asymmetry. This condition would accelerate LAGEOS only about 0.2 picometers per second squared (pm s−2 or 10−12 m s−2) and indicates that eclipse asymmetry can only account for about 40–50% of the remaining unmodeled residuals. This is slightly less than the penumbral acceleration found by Vokrouhlicky et al. (1994).