Satellite networks are key enablers to many applications, including world-wide sensing and communications. Unlike their terrestrial counterparts, satellites are able to provide coverage in remote and hard-to-reach areas, including areas with regional conflicts. However, they are also susceptible to multiple security threats and potential failures. In addition to commonly used security techniques, it is essential to have algorithms that assess the trustworthiness of satellites as they operate, without limiting the satellites' abilities to perform their intended tasks. In this paper we focus on trust assessment methods that analyze the behavior of satellites to detect attacks and identify failed or compromised nodes in constellation networks. In this work, we (1) present a satellite threat model and enumerate possible attacks, (2) compare several existing trust assessment models when applied to low earth orbit satellite constellations, and (3) propose Trust via Asynchronous Updates (TAU), a novel trust algorithm model that is applicable to all modern satellite constellation networks. Model TAU uses finite state machines and asynchronous updates to track node behavior. Our custom simulator evaluates the performance of our algorithm in comparison to several previously proposed trust models. We consider two well-known attacks, the kinetic and black hole attacks, and show that the proposed Model TAU accurately identifies malicious satellites, with low false positive rate, in time comparable to previously proposed trust models while achieving lower computational complexity and communication overhead.