Security threats introduced due to the vulnerability of the transmission medium may hinder the proliferation of Ka band multibeam satellite systems for civil and military data applications. This paper sets the analytical framework and then studies physical layer security techniques for fixed legitimate receivers dispersed throughout multiple beams, each possibly surrounded by multiple (passive) eavesdroppers. The design objective is to minimize via transmit beamforming the costly total transmit power on board the satellite, while satisfying individual intended users' secrecy rate constraints. Assuming state-of-the-art satellite channel models, when perfect channel state information (CSI) about the eavesdroppers is available at the satellite, a partial zero-forcing approach is proposed for obtaining a low-complexity sub-optimal solution. For the optimal solution, an iterative algorithm combining semi-definite programming relaxation and the gradient-based method is devised by studying the convexity of the problem. Furthermore, the use of artificial noise as an additional degree-of-freedom for protection against eavesdroppers is explored. When only partial CSI about the eavesdroppers is available, we study the problem of minimizing the eavesdroppers' received signal to interference-plus-noise ratios. Simulation results demonstrate substantial performance improvements over existing approaches.