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