Thermodynamic Bounds and Mass-Trade Criteria for Heat Rejection in Orbital Data Centers

Orbital data centers must reject waste heat by far-field thermal radiation, and the resulting radiator area requirements are widely regarded as the binding engineering constraint on the concept. We derive, within a gray-body effective-sink radiator model, a set of exact thermodynamic results governing this constraint. First, we prove that the Carnot efficiency evaluated at the environmental sink temperature is unattainable by any heat engine with a material cold reservoir at finite radiator area and positive throughput, and we characterize both limiting routes (infinite area at fixed throughput; vanishing throughput at fixed area). Second, we show that radiator area per unit work output is minimized, along the reversible lower envelope, at a cold-side temperature T꜀* = (3/4)·Tₕ with a 25% efficiency ceiling, and we give an exact implicit characterization of the nonzero-sink optimum, T꜀*/Tₕ = (3 + q⁴)/4 with q = Tₛᵉᶠᶠ/T꜀*, whose fractional shift above (3/4)·Tₕ is exactly q⁴/3. Third, we prove that converting waste heat to work before rejection multiplies required radiator area by at least (Tₕ/T꜀)³, with equality only for reversible conversion at zero sink temperature. Fourth, we show no cyclic system can sustain its compute load solely from its own waste heat. Finally, we state reduced-order mass-trade criteria for radiator-type selection, heat-pump inclusion, and topping-cycle inclusion. All quantitative claims are verified by an accompanying machine-executable test suite (Python and Wolfram Language). The central design implication is that orbital thermal management is a temperature-architecture problem: within the model, the temperature at which heat finally leaves the system dominates fixed-temperature efficiency optimizations. Derivations were developed and adversarially audited through a multi-model AI workflow (Claude Fable 5, GPT 5.5 with Wolfram verification, Perplexity.)