### 4.6 Modeling the radiometric Doppler observable

The Pioneer spacecraft were navigated using radiometric Doppler data. The Doppler observable is defined as the difference between the number of cycles received by a receiving station and the number of cycles produced by a (fixed or ramped) known reference frequency, during a specific count interval.

The expected value of the Doppler observable can be calculated accurately if the trajectory of the transmitting station (e.g., a spacecraft) and receiving station (e.g., a ground-based tracking station) are known accurately, along with information about the transmission medium along the route of the received signal.

The trajectory of the spacecraft is determined using the spacecraft’s initial position and velocity according to Section 4.1, in conjunction with a model of nongravitational forces, as detailed in Section 4.4.

The signal propagation delay due to the gravitational field of solar system can be calculated using Equation (4.4). Afterwards, from the known arrival times of the first and last cycle during a Doppler count interval, the times of their transmission can be obtained. Given the known frequency of the transmitter, one can then calculate the actual number of cycles that were transmitted during this interval. Comparing the two figures gives the difference known as the Doppler residual.

The model parameters, which include an estimate of the spacecraft’s initial state vector and other factors, can be adjusted to achieve a “best fit” between model and observation. A commonly used method to achieve a best fit is the use of a least squares estimator. The “solve-for” parameters can include orbital parameters of solar system bodies; the visible light and infrared radiometric properties of the spacecraft; or properties of the Earth’s atmosphere or the interplanetary medium.

While a spacecraft is in flight, the revised model can be used to make navigational predictions and provide guidance, as depicted in Figure 4.1. In the case of the historical Doppler data of Pioneer 10 and 11, the purpose is not to navigate a live spacecraft, but to provide a model of as large a segment of the spacecraft’s trajectory as possible, using a consistent set of parameters and minimizing the model residual. Several, independent efforts to analyze the trajectories of the two spacecraft have demonstrated that this can be accomplished using multi-year spans of data with a root-mean-square model residual of no more than a few mHz.