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This work seeks to predict secondary radiation levels induced in spacecraft structures by space protons. The radiations analyzed are secondary neutrons from spallation and evaporation reactions and gamma and beta rays from the decay of induced radioactivity, as sources of interfering background to spaceborne measurements of galactic and planetary gamma rays below 10 MeV. The spacecraft considered are the Multi-Mission Spacecraft (MMS) and the Space Shuttle, modeled as spherical shells. The proton environment is that of the South Atlantic Anomaly, as well as cosmic ray protons. The induced radioactivity is analyzed in terms of its interference with various gamma-ray lines of astrophysical interest, as well as its contribution to several spectral regions of the gamma-ray continuum. The buildup of the line and continuum radioactivity background is predicted for a period of nearly 9 months in orbit (~4100 orbits). In addition, background contributions from cosmic ray electron bremsstrahlung and earth gamma-ray albedo are estimated.