General Solar Radiation Pressure Model for Global Positioning System Orbit Determination

T HERE are a wide variety of solar radiation pressure (SRP) models used for the GPS satellites. The most complicated numerical methods use ray-tracing methods [1] to attempt to predict the details of the physical interactions that cause the SRP accelerations. Simpler models, such as those by Fliegel and Gallini [2], use geometric primitives to try and capture the main effects at a much smaller computational cost. Thesemodels suffer from the same weakness for orbit determination in that these models are based on ground-measured data of the spacecraft. Once the spacecraft enters the space environment, however, many of the assumptions that go into these models will change in an unpredictable fashion, whichwill change the accelerations caused by the solar radiation pressure. Due to the manner in which these models are developed and used, it is nearly impossible to separate how errors in model parameters impact the spacecraft’s orbit. The most common method to improve the model solution is to apply an estimated correction factor to all parts of the spacecraft equally; however, this method cannot capture all possible model errors, as it changes the accelerations in all directions proportionally. A different type of SRP model was developed independently by Springer et al. [3] at theUniversity of Bern and byBar-Sever [4] at the Jet Propulsion Laboratory, California Institute of Technology (JPL) to model the SRP effect on the orbits of GPS satellites. Both of these models are truncated Fourier series expressed in body-fixed coordinates, for which the coefficients can vary over the course of a year depending on the orbit plane tilt with respect to the sun–Earth line. The University of Bern model uses the difference between the spacecraft and sun’s respective arguments of latitude as the Fourier series argument, while the JPL model uses the sun–spacecraft–Earth angle (see Sec. IV.A for a full explanation of the JPLmodel). In either case, the coefficients were determined byminimizing the residuals of a large number of orbit fits. In other words, the SRP model was determined by observing the orbital effects of SRP without focusing on the physical process that is causing these effects. The model previously derived by the authors [5,6] is similar to the preceding models [2–4], in that it uses a Fourier series expression for the SRP force. The main difference is that the authors’ model is expressed in terms of mean anomaly in the rotating (or local-vertical/ local-horizontal) frame, which allows for an analytical determination of the secular effects on the orbit in terms of the Fourier coefficients. Using the authors’ SRP model has three main advantages over the existingmodels. First, the authors’model is derived in such away that the Fourier series coefficients can be tied to the physics of the SRP interaction with the spacecraft model. Second, an analytical framework has been developed that gives the periodic and secular orbital effects of the rotating frame coefficients. Finally, the authors’ model has a generic formulation that can be applied to any spacecraft, and not only GPS. In this Note, our model [5,6] is applied to the GPS Block IIR-M spacecraft. The model is reviewed in Sec. II. Section III reviews the previous expression for the secular changes to an orbit due to SRP. Section IV explains how the rotating frame coefficients can be determined from the JPL model [4]. In this Note, we show the values of the rotating coefficients computed from the JPL model. These coefficients were used in several estimation schemes, which are discussed in Sec.V. The results of these estimation tests are compared to the performance of the JPL model, and they are shown to be equivalent.