In this paper, we introduce and study scheduling problems, where tasks yield a processing time dependent profit. In these problems, the processing time of a task is not given as input but must be decided; the profit from a task then depends on its allocated processing time and the objective is to maximize the total profit. These problems originate from astrophysics. First, we state formally the problem and propose several profit functions, including non-continuous and non-concave ones. Next, we prove and discuss some generic properties on the structure of solutions. Then, we study the complexity of the main models, proving the NP-completeness of some, and proposing polynomial algorithms for others.