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The Black Hole Explorer is a space-based very-long baseline interferometry (VLBI) mission that will seek to perform precision black hole measurements, detect the photon ring around a black hole, explore the spacetime, spin, and mass properties of black holes, and attempt to experimentally validate predictions of General Relativity. These ambitious goals are achieved through the use of cryogenic receivers offering quantum-limited sensitivities across a wide frequency coverage. The dual-band receivers at 80-106 GHz and 240-320 GHz require 20 K and 4.5 K operating temperatures, respectively. To reach this, the planned cryocooling system will include two cold stages; a 20 K stage which must lift a heat load of approximately 125 mW and a 4.5 K stage lifting 10 mW of heat load. A survey of 4 K cryocooler development for spaceflight is explored in order to baseline the cryocooling system design for BHEX and leverage existing technology in the space industry at high TRLs. Notable space missions of relevance include Planck, JEM/SMILES, Hitomi, XRISM, and the advancement of US cryocoolers in this temperature range thanks to the ACTDP/JWST. Integration of the cryocooler with the receivers and broader instrument requires careful consideration, as it influences the instrument operation and thermal challenges. The latter includes thermally linking the cold ends of each cooling stage whilst minimising heat losses and ensuring adequate passive cooling for the cryocooler warm end heat rejection. Moreover, the challenges and trade-offs in sizing the mass and reducing the power consumption are explored: varying modes of operation in conjunction with other key instrument subsystems, the receiver cold temperature requirements, which in turn influence the scientific objectives of the mission, and mitigating the mission critical risks of the system. Overall, this paper presents an overview of cooling needs, initial design considerations, a survey of 4 K spaceflight cryocooler developments and current progress, and balancing scientific requirements of the instrument with the limitations of technical cryocooling capabilities, within the framework of a small-class (SMEX) space mission aiming to achieve breakthrough goals in experimental black hole physics.
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