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Is there a way to model/predict the evolution and life cycle of a star based on certain initial conditions? That is, whether it will become a red giant, brown dwarf, etc. I'm basically looking for equations defining the progress of different star-types and how varying initial conditions may affect that progress.

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You could look at tools like EZ-web or interpolation formulae like those from Hurley, Pols and Tout 2000 to do so based on the initial mass and metallicity of your star. Here is a picture of the evolution of an initial 10 solar masses star and solar metallicity computed using Hurley's formulae (accessible as a fortran library, see this page for a web-interface)

a 10 solar masses star evolution within the HR diagram
(source: rubyforge.org) !

This one compares EZ results and Hurley's formulae.

EZ and Hursley's formulae comparison

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