Downloading a large amount of data from a low Earth orbit satellite to a ground station can be challenging due to the limited contact window, dynamic channel quality, solar energy supply, and thermal management without an atmosphere. Considering such dynamics, this paper proposes a joint design of in-orbit computation and communication for download time minimization. We combine the non-convex thermal constraints and energy constraints into unified energy budget constraints with upper bound approximation, and computational efficiency is achieved by decomposing the resulting large-scale problem into a non-convex communication sub-problem, a convex computation sub-problem solvable with interior point method and a master problem that optimizes the energy budget allocation between computation and communication. The communication sub-problem is solved with a generalized-benders-decomposition-based algorithm that decouples downlink scheduling and power allocation based on a closed-form solution of optimal dual variables in the power allocation primal problem. And the master problem is solved with ternary search by proving the minimal download time is quasi-convex with respect to the energy budget allocation between computation and communication. Simulation results demonstrate that the proposed solution effectively reduces the download time, especially under strict energy constraints and severe channel variations.