High Precision Analytical Solar Radiation Pressure Modelling for GNSS Spacecraft

In global navigation satellite systems (GNSS) a fundamental operational component is the calculation of the orbits of the system spacecraft. This requires understanding and modelling the forces that act on the spacecraft. Solar radiation pressure (SRP) is the force caused by the impact of solar photons on the spacecraft surface. For GNSS spacecraft this is a significant force. If SRP is not included in the force model, then the calculated position of the spacecraft can be in error by between one and two hundred metres after one 12-hour orbit. SRP can be modelled using either analytical or empirical methods, or by some combination of the two.
Historically, analytical SRP modelling has been somewhat neglected and high precision orbit estimation has relied upon empirical methods to account for SRP.
Even so, most of these empirical methods start the estimation process with an a priori analytical model. The success of this empirical approach relies upon having many
observations of the range between the system spacecraft and ground-based tracking stations, and works well within the context of the International Global Positioning System Service (IGS) network, which provides the necessary data volume. However, empirical methods do not work as well in operational GNSS, as these typically have a relatively small number of tracking stations. Moreover, empirical methods cannot be applied at the GNSS design stage, where knowledge of the system dynamics plays a key role.
Existing methods for calculating analytical SRP models can only be used with relatively simple spacecraft structures, and lack flexibility as tools for analysis. In this study a new method is developed for calculating analytical SRP models that can cope with a high level of complexity in the spacecraft structure. The method is based upon simulating the solar photon flux with a pixel array. Using the method, models are calculated and tested for the Russian GLONASS IIv spacecraft. This particular spacecraft was used as the testbed because, at the time the study was being conducted,
an international scientific campaign - called IGEX-98, the International GLONASS Experiment - was being carried out to analyse the Russian system. Developing force
models for the spacecraft was one of the campaign goals, and the IGEX-98 steering committee accepted a proposal to use SRP models for GLONASS from this study.
A detailed description is given of all the mathematics and physics that was used to develop the modelling technique. The method by which the models can be calculated and applied in practical orbit determination is also provided.
In order to test the performance of the SRP models computed for the GLONASS spacecraft using the new method, comparisons were made between two kinds of trajectory. The first kind was calculated by numerical integration of the
spacecraft's second order differential equation of motion, where this force model included the custom SRP models developed in the thesis. The second kind of trajectory, which is used as a 'truth' model in the study, was a precise orbit computed by the University of Berne using IGS range data and an empirical SRP model. Such precise orbits are the best estimates available of the true trajectories, as they are derived from the simultaneous estimation of multiple receiver tracking station network positions and spacecraft force model parameters. The repeatability of the
Berne orbit is circa 0.75m. The RMS differences between the two trajectories over one twelve-hour orbit (an arc length of circa 160,000km) were 0.7m in height, 1.3m across track and 3.5m along track. This shows that the trajectory derived from the force model alone is very close to the precise orbit. The time-varying pattern of the differences between the two trajectories strongly indicates that the residual mismodelling of the forces acting on the spacecraft is due to thermal re-radiation effects.
Further tests of the method were also conducted using satellite laser ranging (SLR) data to calculate arc lengths of 400 days, again using SRP models from the study. This enabled the calculation of model scale factors and additional empirical terms. The average SRP model scale factor was circa 1.01, which implies that the average error in the a priori SRP models calculated for the GLONASS IIv spacecraft is at the 1% level. This is consistent with an error budget based on an assessment of the accuracy of the source data supplied by the Russian authorities. The magnitude and parameterisation of the SLR empirical terms again strongly suggest that most of the remaining mis-modelling is caused by thermal effects.
An analysis is given of the effect on the a priori SRP model of unmodelled, SRP-related forces acting along the spacecraft Y-axis. This is the so-called Y-bias. It
is shown that whilst Y-bias effects are important in orbit determination, they are less critical in the process of calculating the a priori SRP model. A discussion is provided
on how the new method can be adapted to improve the modelling and understanding of thermal re-radiation and Y-bias effects, and also on what benefits might accrue
from such studies.
The new method is an improvement over existing techniques as it enables the calculation of high precision SRP models that can be applied in the design, operation and scientific analysis of GNSS.
A UK patent application has been made in respect of the new method.

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