The new space race is heating up as private companies such as SpaceX and Amazon are building large satellite constellations in low-earth orbit (LEO) to provide global broadband internet access. As the number of subscribers connected to this access network grows, it becomes necessary to investigate if and how edge computing concepts can be applied to LEO satellite networks. In this paper, we discuss the unique characteristics of the LEO edge and analyze the suitability of three organization paradigms for applications considering developer requirements. We conclude that the serverless approach is the most promising solution, opening up the field for future research.