Low-Earth-Orbit (LEO) satellite constellations are becoming integral to 6G infrastructure, but increasing in-orbit computation accelerates battery degradation and raises sustainability concerns. Meanwhile, renewable-heavy regions worldwide experience persistent energy curtailment due to transmission bottlenecks, leaving substantial clean energy stranded near generation sites. We identify a satellite-grid co-design opportunity: adaptively offloading task-critical data from satellite to data centers co-located with renewable power plants. However, realizing this vision requires jointly considering intermittent and capacity-limited communication windows, as well as time-varying electricity budgets. In this paper, we propose SQSO, a Sustainable and QoS-aware Satellite Offloading framework that models per-interval task offloading as a constrained optimization over dynamic topology and electricity prices. Under this framework, we design $\text{AO}^2$, an adaptive offloading orchestration algorithm to solve the formulated optimization problem. Using Starlink-scale simulations and real-world electricity price traces, $\text{AO}^2$ reduces energy consumption by up to 76.03% and battery life consumption by up to 76.85% compared to state-of-the-art schemes, while also lowering task delay. This work highlights that sustainable scaling of LEO constellations requires co-design of space networking and renewable energy infrastructure, while our solution promotes renewable-aware task offloading and cross-domain collaboration for space-energy integration in the 6G era.

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