of the components. As it is the case for the RTG contribution the magnitude of the effect scales with mission time which reflects the dependence on the available electrical energy as well as the compartment temperature sensor data.
In total the emission of waste heat from RTG, Louver system and partially the MLI leads to an effective TRP acting against flight direction in the magnitude of the PA at 25 AU which slowly decreases due to generally decreasing temperatures and available powers. Such a decrease in the evaluated residuals is expected by the JPL as result of the ongoing evaluation of the full Doppler data set [57]. Both the matching magnitude of the computed effect as well as the resembling characteristics of the JPL residuals and the computed results indicate that accurate modelling of TRP is the solution to the Pioneer anomaly.
6.5 Parameter variation and error analysis 99 be seen, the TRP solution is considerably stable to variations of the input parameters which mainly results from the fact that the thermal FE solution uses the measured temperature data as boundary conditions. By this the range of possible solutions is already highly constrained which reduces the magnitude of influence of other model parameters. If, for example, a surface emissivity is varied while keeping the heat load on the surface as well as the temperature boundaries at specific places constant, the result is a variation of the temperature distribution but not a drastic change of the total emitted heat flux since the total radiated power stays constant.
Besides the influence of parameter variations additional errors might be inflicted by numerical inaccuracies in the FE solution, the ray tracing process as well as inaccuracies in the sensor readings which have been used as temperature boundaries. Inaccuracies in the FE solution are below 1 %, which has been realised through extensive global and local mesh refinement. The accuracy of the TRP computation has been demonstrated with the test cases described in section 3.6 and can be estimated to a maximum error of about 0.5 %. This low value has been realised by implementing a large number of rays per element (500000) as well as numerical integration of the radiation view factors by means of Gauss quadrature with 8 integration points per dimension. The temper-ature sensor readings aboard the Pioneer 10 spacecraft have a digital resolution of 6 bit which translates to a temperature inaccuracy of ±1.44 F. The resulting worst case inaccuracy of the boundary temperature can directly be computed to ±1.7 K. In the late stage of the mission this inaccuracy leads to a TRP variation of about 3 %. As the compartment model uses the same type of temperature sensors a corresponding error on the TRP of about 1.5 % has to be added. Note that this error is much lower for the early and intermediate mission periods, resulting from the higher mean temperature in the compartment.
In a coarse worst case scenario (where all individual errors are simply summarised) the total error in the TRP calculation evolves to about ±11.5%. With this value the presented TRP solution is considerably robust with respect to the different error sources discussed above. Further smaller variation may result from unknown effects such as non-documented differences in the geometry of flight and engineering model or the exact characteristics of the degradation of the surface materials. For Pioneer 10, most of the degradation is supposed to have occurred during Jupiter flyby. Thus it would be reasonable to assume non-degraded optical properties before the Jupiter encounter and EOL values afterwards. However, due to the sensor temperature constraints and the small range of possible variations in the optical parameters, the computed TRP characteristics will practically not change for such a degradation profile.
Part III
Thermal perturbation analysis
for Rosetta
Chapter 7
Solar radiation pressure (SRP) model
This chapter introduces a numerical method for the determination of SRP acting on a spacecraft with complex shapes. An analytical model for simple geometries is presented and the expansion into a numerical method for complex models is discussed.
7.1 Introduction to SRP modelling
Since SRP is also caused by the interaction of radiation with target surfaces the an-alytical models derived for the determination of the TRP can also be used for the determination of SRP. The main difference is that the source of radiation is not the model surface itself, but the Sun which can be considered as an external radiation source. Therefore a corresponding numerical method for the calculation of SRP can be developed based on the numerical methods developed for the TRP computation (de-scribed in section 3.4). The modelling approach for SRP analysis is based on a GMM of the spacecraft including geometry and optical surface properties where the spacecraft surface is represented by a set of quadrilateral FE surfaces. In difference to the TRP algorithm no surface temperature distribution of the spacecraft and thus no thermal FE analysis is necessary for the computation of the SRP as the effect only depends on the spacecraft shape and the state of illumination by the Sun.
In this approach the Sun is modelled as a high resolution pixel array from which individual parallel solar ray vectors originate. In a ray tracing process the interaction of the solar ray vectors with the spacecraft surfaces as well as shadowing aspects are assessed. The SRP is calculated on each individual model surface. By summing up all individual surface drags, the resulting SRP can be computed. The algorithm for the numerical computation of the SRP is realised as a set of c-functions which import text files containing GMM info in order to calculate the forces. The geometric modelling is performed with an FE preprocessor as explained in section 3.4. The individual modelling steps are discussed in the following.