Even for the laser satellites, whose satellite forms are
uncomplicated, proper treatment of the direct solar radiation pressure
force as the satellite enters and exits the Earth's shadow is
non-trivial [ Tapley et al., 1993]. A conical shadow model (containing
both the umbra and penumbra) and the effects of lunar shadowing
are important modeling considerations. Numerical integration treatments
of the boundary properties of the forcing also present challenges
[ Fuelner, 1990]. Rowlands et al., [1994 in press] have adopted a
multi-rate numerical integration method which allows this type of
force component to be sampled in greater detail while not requiring
frequent evaluation of the whole force model within orbit integrations.
3.2.1.2. Planetary reradiation: Albedo/infrared:
Albedo and infrared emissions from the Earth also provide radiative
flux which acts on a spacecraft's exposed surfaces in a fashion similar
to that of direct solar radiation pressure. This force is
commonly evaluated by dividing the Earth into a grid and taking the
vector sum of the spot-to-satellite accelerations. These models
are presently quite simplistic and do not account for the large
temporal variation due to the changing albedo of the Earth's land
masses arising from snow and vegetative covering, and the distribution
of clouds. Martin and Rubincam [1993] have performed a detailed
calculation of the specular and diffuse albedo re-radiation based
on measured global albedo time histories. They evaluated the net effect
of more detailed modeling in order to assess the performance of the
standard models adopted in most orbit determination codes [ Knocke et
al., 1988; Borderies and Langaretti, 1990]. The Martin and Rubincam
study indicated that while the time history of the albedo forcing
is complex, when integrated over the Lageos orbit, it produces effects
which are similar to that obtained with existing albedo models. The
current approach which approximates the visible planetary surface
integral by the summation of the contributions of a highly
generalized gridded surface is still favored, but efforts continue
to improve the temporal characterization of the Earth's reradiation.
3.2.1.3. Thermal imbalances: Accelerations arising
from thermal gradients across the spacecraft surfaces must also
be considered for precise orbit determination. A surface emits
infrared radiation at levels proportional to the emissive properties of
the material and to the fourth power of its temperature. The
surface temperature varies according to its exposure to both external
(i.e. solar radiation) and internal heat sources (i.e.
electronics). Additionally, many spacecraft use both passive and
active thermal control systems to regulate the temperature of the
electronic components. Often this implies that heat is
directionally expelled from the spacecraft, causing an
acceleration. Modeling the spacecraft's overall thermal behavior
is problematic given that the knowledge of the past history of
each surface's temperature and exposure to heat sources is required.
For example, duty cycles for onboard equipment are not generally
systematic or easily predicted.
Thermal models of the spacecraft are normally computed as a part of the design process using finite element analysis software. Antreasian and Rosborough [1992] used this technique to develop a thermal imbalance force model history for the T/P spacecraft. However, this method is far too computationally intensive for nominal orbit determination methods. Therefore, Marshall and Luthcke [1994a] developed a simplified approach based upon the results of this finite element analysis. Unfortunately, this technique is satellite dependent and the results of the thermal finite element analysis are not typically available for orbit determination. Furthermore, it is difficult to realistically determine the temperature of a given surface on orbit from satellite tracking data.
One possible solution to this dilemma might be the incorporation
of telemetered temperature data into the thermal model. This approach
also has its drawbacks in that the temperature readings are only a
measure of the local characteristics and may not be representative of
the whole surface. Also, many of the measurements are taken internal to
the spacecraft and therefore, do not play a direct role in the
thermal model. Clearly this is a difficult problem and will require
further study and innovative methodologies in the future.
3.2.1.4. Thermal modeling of Lageos-1: The
ultra-stable orbit provided by Lageos-1 has allowed the study of
orbital dynamics to unprecedented levels of accuracy. Given
this satellite's spherical shape and high density coupled with the
extensive global SLR data set which has been acquired since 1976,
many previously unobservable dynamical effects have now been detected.
It has been observed that even high altitude laser satellites like
Lageos-1 (orbiting at nearly an Earth's radius in altitude) experience
drag which decays their orbits at the sub-millimeter/day level. While
this drag effect is small, the resulting orbital perturbations are
clearly seen [ Rubincam, 1990]. The source of this drag has been
extensively studied. A Yarkovsky thermal drag model for Lageos was
first proposed by Rubincam [1987a, 1988]. It alone has been able to
account for most of the average along track acceleration seen in
Lageos' orbit evolution. The Yarkovsky thermal drag arises from
the hemispheric temperature imbalance between the satellite's Earth
facing and opposite hemispheres due to the heating characteristics of
the retroreflectors by the Earth's reradiated IR. This effect coupled
with that of neutral density and charged particle drag accounts for
Lageos' overall orbital decay [ Rubincam, 1990].
The Yarkovsky-Schach photon thrust, which is induced by the Sun and the satellite's recoil during Earth's shadow passages, has been proposed to account for the large variations seen in Lageos' along track acceleration during eclipse seasons [ Afonso et al., 1989; Scharroo et al., 1991]. This effect is a function of Lageos' spin orientation. Direct heating from the sun produces ``summer'' and ``winter'' hemispheres on Lageos which introduces a thermal imbalance. This imbalance migrates slowly in time as the spin axis orientation changes.
Anisotropic reflectivity, which arises from the asymmetry in the satellite's reflectivity and Lageos' recoil during Earth shadow passages, has also been proposed [ Rubincam et al., 1987b]. This effect arises from the difference in optical properties of the two physical hemispheres of the satellite which were prepared separately and bolted together to form the Lageos sphere. Scharroo et al., [1991] estimated that the northern hemisphere of the satellite is 1.5% more reflective than the southern one to explain the satellite's behavior during eclipses. Ries et al., [1991] speculated that the non-symmetric distribution of the Germanium/infrared corner cubes may be a contributing factor. Analysis of Lageos-2 SLR data, which benefits from good knowledge of spin axis orientation, are generally confirming these models but no definitive explanation for the cause of the asymmetric reflectivity is yet available.
Since the modeling of these thermal effects depends in many cases on knowledge of Lageos' spin axis orientation, this too has received considerable study. Currie [1994] has used direct optical glint observations of the satellite to estimate both the spin rate and orientation of Lageos. Ries et al., [1993b] and Robbins et al., [1994] have attempted to use the observed orbit evolution to develop time histories of Lageos-1's spin orientation.
Recently, Mallama and Rubincam [1994] have made an assessment of whether the solar radiation pressure modeling for Lageos requires consideration of the absorption and refraction of the sunlight by the Earth's atmosphere. They also assessed if volcanic dust and aerosols, unevenly distributed within the atmosphere immediately after a major eruption, can cause a discernible effect.