Analysis of the radio-metric tracking data from the Pioneer 10 and 11 spacecraft at distances between 20 and 70 astronomical units from the Sun has consistently indicated the presence of an anomalous, small, and constant Doppler frequency drift. The drift is a blueshift, uniformly changing at the rate of . The signal also can be interpreted as a constant acceleration of each spacecraft of directed toward the Sun. This interpretation has become known as the Pioneer anomaly. We provide a problem set based on the detailed investigation of this anomaly, the nature of which remains unexplained.
To be precise, the data analyzed in Ref. 2 was taken between 3 January 1987 and 22 July 1998 for Pioneer 10 (when the craft was 40 to 70.5 AU distant from the Sun), and that from Pioneer 11 was obtained between 5 January 1987 and 1 October 1990 (22.4 to 31.7 AU).
To determine the state of the spacecraft at any given time, the Jet Propulsion Laboratory’s orbit determination program uses a batch-sequential filtering and smoothing algorithm with process noise (Ref. 2). For this algorithm, any small anomalous forces may be treated as stochastic parameters affecting the spacecraft's trajectory. As such, these parameters are also responsible for the stochastic noise in the observational data. To better characterize these noise sources, the data interval is split into a number of constant or variable size intervals, called “batches,” and assumptions are made for statistical properties of these noise factors. The mean values and the second moments of the unknown parameters are estimated within the batch. In batch sequential filtering, the reference trajectory is updated after each batch to reflect the best estimate of the true trajectory. (For more details consult Ref. 2 and references therein.)
We use the symbol to identify biases in the measured value of the Pioneer anomaly produced by various sources of acceleration noise. The uncertainty in the contribution of these noise sources to the anomalous acceleration is conventionally denoted by . See an example of this usage in Eq. (21).