Using a recently developed model of gravity that differs from both the Newton and Einstein models, close agreement is achieved between the anomalous accelerations of the Pioneer and Cassini mission vehicles and the predictions of the model. This conjecture supports the notion of a temporal-inertial (TI) field that is the principle mediator of the gravitational and inertial interactions. The TI field is subject to gravity and, in response to the acceleration of gravity, transmits its own acceleration to massive particles and objects comprising massive particles. No assertion is made relating properties of the TI field to those of the Higgs field. The flux model posits that in a gravitational field, the velocity of the TI field combines with that of gravitons emitted by the gravitational body to increase the flux of gravitons relative to the TI field. Thus the response of the TI field to gravity adds to the force of gravity. The flux model may offer an explanation for the anomalous accelerations experienced by the Pioneer and Cassini spacecraft.