Our previous analyses of radio Doppler and ranging data from distant spacecraft in the solar system indicated that an apparent anomalous acceleration is acting on Pioneer 10 and 11, with a magnitude aP ∼ 8 × 10 −8 cm/s 2 , directed towards the Sun. Much effort has been expended looking for possible systematic origins of the residuals, but none has been found. A detailed investigation of effects both external to and internal to the spacecraft, as well as those due to modeling and computational techniques, is provided. We also discuss the methods, theoretical models, and experimental techniques used to detect and study small forces acting on interplanetary spacecraft. These include the methods of radio Doppler data collection, data editing, and data reduction.There is now further data for the Pioneer 10 orbit determination.

Radio metric data from the Pioneer 10/11, Galileo, and Ulysses spacecraft indicate an apparent anomalous, constant, acceleration acting on the spacecraft with a magnitude ∼ 8.5 × 10 −8 cm/s 2 , directed towards the Sun. Two independent codes and physical strategies have been used to analyze the data. A number of potential causes have been ruled out. We discuss future kinematic tests and possible origins of the signal.

The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clariaed in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable p and its "conjugate" angular momentum L, is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables (sin cp and cos e) to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given.A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator 4 (conjugate to the number operator N) exists is shown to be erroneous. However, cosine and sine operators C and S exist and are the appropriate phase variables. A Poisson bracket argument using action-angle (rather I, cos P, sin p) variables is used to deduce C and S. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-di6'erence" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections. " Mention is made of possible physical applications to superQuid systems.

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