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This thesis presents fundamental work that explains two mysteries concerning the trajectory of interplanetary spacecraft. For the first problem, the so-called Pioneer anomaly, a wholly new and innovative method was developed for computing all contributions to the acceleration due to onboard thermal sources. Through a careful analysis of all parts of the spacecraft Pioneer 10 and 11, the application of this methodology has yielded the observed anomalous acceleration. This marks a major achievement, given that this problem remained unsolved for more than a decade. For the second anomaly, the flyby anomaly, a tiny glitch in the velocity of spacecraft that perform gravity assisting maneuvers on Earth, no definitive answer is put forward; however a quite promising strategy for examining the problem is provided and a new mission is proposed. The proposal largely consists in using the Galileo Navigational Satellite System to track approaching spacecraft, and in considering a small test body that approaches Earth from a highly elliptic trajectory.

Physics. --- Extraterrestrial Physics, Space Sciences. --- Theoretical, Mathematical and Computational Physics. --- Aerospace Technology and Astronautics. --- Astrophysics. --- Astronautics. --- Physique --- Astrophysique --- Astronautique --- Space trajectories. --- Space vehicles -- Propulsion systems. --- Trajectory optimization -- Mathematical models. --- Astronomy & Astrophysics --- Physical Sciences & Mathematics --- Astrophysics --- Trajectory optimization --- Linear programming. --- Mathematical models. --- Optimization, Trajectory --- Space vehicles --- Trajectories --- Space sciences. --- Aerospace engineering. --- Aerodynamics --- Space trajectories --- Astrodynamics --- Space flight --- Trajectories (Mechanics) --- Production scheduling --- Programming (Mathematics) --- Space Sciences (including Extraterrestrial Physics, Space Exploration and Astronautics). --- Space sciences --- Aeronautics --- Astronomical physics --- Astronomy --- Cosmic physics --- Physics --- Mathematical physics. --- Aeronautical engineering --- Astronautics --- Engineering --- Physical mathematics --- Science and space --- Space research --- Cosmology --- Science --- Mathematics