The paper presents results of the 14-year studies (1966-1980) with dogs irradiated by presumable doses of the galactic cosmic rays (GCR) and solar cosmic rays (SCR) that might affect cosmonauts during a flight to Mars. Along with data of chronic experiments with small laboratory animals exposed to a broad variety of dose rates, our data formed a basis for construction of mathematical models of processes of radiation damage to living organisms, and growth of the probability of mammal death or life span reduction caused by repeated acute and chronic irradiation by different dose rates. Described are present-date estimates of equivalent doses from GCR and SCR behind various shields, and radiation risks for cosmonauts of different age on a mission to Mars. Also compared are values of total radiation risks over the length of life, risks of cancer and lifetime reduction after the planetary mission derived by the mathematical modeling and calculation.