Space situational awareness (SSA) and defense space control capabilities are top priorities for groups that own or operate man-made spacecraft. Also, with the growing amount of space debris, there is an increase in demand for contextual understanding that necessitates the capability of collecting and processing a vast amount sensor data. Cloud computing, which features scalable and flexible storage and computing services, has been recognized as an ideal candidate that can meet the large data contextual challenges as needed by SSA. Cloud computing consists of physical service providers and middleware virtual machines together with infrastructure, platform, and software as service (IaaS, PaaS, SaaS) models. However, the typical Virtual Machine (VM) abstraction is on a per operating systems basis, which is at too low-level and limits the flexibility of a mission application architecture. In responding to this technical challenge, a novel adaptive process based cloud infrastructure for SSA applications is proposed in this paper. In addition, the details for the design rationale and a prototype is further examined. The SSA Cloud (SSAC) conceptual capability will potentially support space situation monitoring and tracking, object identification, and threat assessment. Lastly, the benefits of a more granular and flexible cloud computing resources allocation are illustrated for data processing and implementation considerations within a representative SSA system environment. We show that the container-based virtualization performs better than hypervisor-based virtualization technology in an SSA scenario.