With the rapid evolution of space-borne capabilities, space edge computing (SEC) is becoming a new computation paradigm for future integrated space and terrestrial networks. Satellite edges adopt advanced on-board hardware, which not only enables new opportunities to perform complex intelligent tasks in orbit, but also involves new challenges due to the additional energy consumption in power-constrained space environment. In this paper, we present PHOENIX, an energy-efficient task scheduling framework for emerging SEC networks. PHOENIX exploits a key insight that in the SEC network, there always exist a number of sunlit edges which are illuminated during the entire orbital period and have sufficient energy supplement from the sun. PHOENIX accomplishes energy-efficient in-orbit computing by judiciously offloading space tasks to "sunlight-sufficient" edges or to the ground. Specifically, PHOENIX first formulates the SEC battery energy optimizing (SBEO) problem which aims at minimizing the average battery energy consumption while satisfying various task completion constraints. Then PHOENIX incorporates a sunlight-aware scheduling mechanism to solve the SBEO problem and schedule SEC tasks efficiently. Finally, we implement a PHOENIX prototype and build an SEC testbed. Extensive data-driven evaluations demonstrate that as compared to other state-of-the-art solutions, PHOENIX can effectively reduce up to 54.8% SEC battery energy consumption and prolong battery lifetime to 2.9$\times$ while still completing tasks on time.