Deriving star formation histories: inverting Hertzsprung-Russell diagrams through a variational calculus maximum likelihood method

We introduce a new method for solving maximum likelihood problems through V ariational calculus, and apply it to the case of recovering an unknown star formation history, SFR(t), from the resulting Hertzsprung-Russell (HR) dia gram. This approach allows a totally nonparametric solution, which has the advantage of requiring no initial assumptions about SFR(t). As a full maxim um likelihood statistical model is used, and we take advantage of all the i nformation available in the HR diagram, rather than concentrating on partic ular features such as turn-off points or luminosity functions. We test the method using a series of synthetic HR diagrams produced from known SFR(t), and find it to be quite successful under noise conditions comparable to tho se present in current observations. At this point we restrict the analysis to situations in which the metallicity of the system is known, as is the ca se with the resolved population of dwarf spheroidal companions to the Milky Way or the solar neighbourhood Hipparcos data. We also include tests to qu antify the way uncertainties in the assumed metallicity, binary fraction an d initial mass function (IMF) affect our inferences.