Superconducting electric motors are an enabling technology for the goal to electrify aircraft propulsion. A major challenge in achieving this goal is the cooling of superconducting rotor coils. Previous rotor cooling systems have used cryogenic thermosyphons or gaseous helium circuits connecting the rotor to a refrigeration plant, commonly using Gifford McMahon cryocoolers. To enable the transfer of fluid from the stationary to the rotating side, sometimes ferrofluidic seals are used, which are best suited to slow shaft speeds. For higher speeds, axisymmetric Stirling and pulse tube cryocoolers mounted on the rotor have been proposed for direct conductive cooling of the rotor. However, the suitability of pulse tube cryocoolers for commercial aircraft propulsion is questionable, as their efficiency depends on orientation during operation. In contrast, Stirling cryocoolers provide higher efficiency than pulse tube cryocoolers with orientation independence, particularly at low temperatures. They therefore seem to be a promising option for superconducting rotor cooling. However, commercially available Stirling cryocoolers have not been designed for high-speed rotation. This paper presents a new alphatype Stirling cryocooler that is specifically designed for high-speed rotation and superconducting rotor cooling. An overall axisymmetric design is proposed in which both pistons are supported by non-contact gas bearings and driven by stationary linear motors which are outside the rotating gas circuit, using the magnetic field to bridge the stationary-rotating interface.