The availability of vast quantities of data through electronic archives has transformed astronomical research. It has also enabled the creation of new products, models and simulations, often from distributed input data and models, that are themselves made electronically available. These products will only provide maximal long-term value to astronomers when accompanied by records of their provenance; that is, records of the data and processes used in the creation of such products. We use the creation of image mosaics with the Montage grid-enabled mosaic engine to emphasize the necessity of provenance management and to understand the science requirements that higher-level products impose on provenance management technologies. We describe experiments with one technology, the "Provenance Aware Service Oriented Architecture" (PASOA), that stores provenance information at each step in the computation of a mosaic. The results inform the technical specifications of provenance management systems, including the need for extensible systems built on common standards. Finally, we describe examples of provenance management technology emerging from the fields of geophysics and oceanography that have applicability to astronomy applications.