  The Black Hole Explorer (BHEX) is a space-based very-long baseline
interferometry (VLBI) mission aimed at precision black hole measurements,
detecting the photon ring around black holes, exploring spacetime, spin, and
mass properties, and validating predictions of General Relativity. These
objectives are achieved using cryogenic receivers with quantum-limited
sensitivities across a broad frequency range. Dual-band receivers at 80-106 GHz
and 240-320 GHz require operating temperatures of 20 K and 4.5 K, respectively.
A cryocooling system with two cold stages will be employed: a 20 K stage
handling a 125 mW heat load and a 4.5 K stage handling a 10 mW heat load.
  To design the cryocooling system, the mission leverages existing space
industry technology at high Technology Readiness Levels (TRLs), informed by
missions such as Planck, JEM/SMILES, Hitomi, and XRISM, and advancements from
the ACTDP/JWST program. Integrating the cryocooler with the receivers and
broader instrument involves careful consideration of thermal challenges,
including linking the cold ends of each cooling stage to minimize heat losses
and ensuring adequate passive cooling for the cryocooler warm end heat
rejection.
  Key challenges and trade-offs include sizing the mass and reducing power
consumption while meeting the receiver cold temperature requirements, which
impact the scientific objectives. This paper addresses efforts to balance the
scientific requirements with the limitations of technical cryocooling
capabilities within the framework of a small-class (SMEX) space mission,
presenting an overview of cooling needs, initial design considerations, a
survey of 4 K spaceflight cryocooler developments, and trade-offs.
