An orbit determination procedure first determines the spacecraft’s initial position and velocity in a data interval. For each data interval, we then estimate the magnitudes of the orientation maneuvers, if any. The analysis uses models that include the effects of planetary perturbations, radiation pressure, the interplanetary media, general relativity, and bias and drift in the Doppler and range (if available). Planetary coordinates and solar system masses are obtained using JPL’s Export Planetary Ephemeris DEnnn, where DE stands for the Development Ephemeris and nnn is the current number. (Earlier in the study, DE200 and DE405 were used, presently DE412 is available.)

Current versions of ODP implement computations in the J2000.0 epoch. Past versions used B1950.0. (See [361] for details on the conversion of positions and proper motions between these two epochs.)

Standard ODP modeling includes a number of solid-Earth effects, namely precession, nutation, sidereal rotation, polar motion, tidal effects, and tectonic plates drift (see discussion in [27]). Model values of the tidal deceleration, nonuniformity of rotation, polar motion, Love numbers, and Chandler wobble are obtained observationally, by means of lunar and satellite laser ranging (LLR, SLR) techniques and very long baseline interferometry (VLBI). Currently this information is provided by way of the International Celestial Reference Frame (ICRF). JPL’s Earth Orientation Parameters (EOP) is a major source contributor to the ICRF.

Since the previous analysis [24, 27], physical models for the Earth’s interior and the planetary ephemeris have greatly improved. This is due to progress in GPS, SLR, LLR and VLBI techniques, Doppler spacecraft tracking, and new radio science data processing algorithms. ODP models have been updated using these latest Earth models (adopted by the IERS) and also are using the latest planetary ephemeris. This allows for a better characterization of not only the constant part of any anomalous acceleration, but also of the annual and diurnal terms detected in the Pioneer 10 and 11 Doppler residuals [27, 274, 390].

During the last few decades, the algorithms of orbital analysis have been extended to incorporate a Kalman-filter estimation procedure that is based on the concept of “process noise” (i.e., random, nonsystematic forces, or random-walk effects). This was motivated by the need to respond to the significant improvement in observational accuracy and, therefore, to the increasing sensitivity to numerous small perturbing factors of a stochastic nature that are responsible for observational noise. This approach is well justified when one needs to make accurate predictions of the spacecraft’s future behavior using only the spacecraft’s past hardware and electronics state history as well as the dynamic environmental conditions in the distant craft’s vicinity. Modern navigational software often uses Kalman-filter estimation since it more easily allows determination of the temporal noise history than does the weighted least-squares estimation.

ODP also enables the use of batch-sequential filtering and a smoothing algorithm with process noise [27]. Though the name may imply otherwise, batch-sequential processing does not involve processing the data in batches. Instead, in this approach any small anomalous forces may be treated as stochastic parameters affecting the spacecraft trajectory. As such, these parameters are also responsible for the stochastic noise in the observational data. To characterize these noise sources, we split the data interval into a number of constant or variable size batches with respect to the stochastic parameters, and make assumptions on the possible statistical properties of the noise factors. We then estimate the mean values of the unknown parameters within the batch and their second statistical moments. (More details on this “batch-sequential algorithm with smoothing filter” in [138, 240]). ODP is permanently being updated to suit the needs of precision navigation; the progress in the estimation algorithms, programming languages, models of small forces and new navigation methods have strongly supported its recent upgrades.

There have been a number of new models developed that are needed for the analysis of tracking data from interplanetary spacecraft that are now an integral part of the latest generation of the JPL’s ODP. These include an update to the relativistic formulation of the planetary and spacecraft motion, relativistic light propagation and relevant radiometric observables (i.e., Doppler, range, VLBI, and Delta Differential One-way Ranging or DOR), coordinate transformation between relativistic reference frames, and several models for nongravitational forces. Details on other models and their application for the analysis of the Pioneer anomaly are in [27, 392].

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