It's a fairly subtle distinction in any case, but I see a difference in implied scope between
calculating trajectories that got spacecraft in and out of space
and
calculating the trajectories that got spacecraft in and out of space
Something similar happens with this pair of sentences:
Dr. Mithridates spent several hours identifying reasons that people do idiotic things while driving.
and
Dr. Mithridates spent several hours identifying the reasons that people do idiotic things while driving.
So let's consider the latter pair first. To my ear, the first version of the sentence—the one with "identifying reasons"—indicates that Dr. Mithridates identified some reasons; and the second version—the one with "identifying the reasons"—suggests that Dr. Mithridates was engaged in identifying all of the reasons. It therefore seems to me that pointing to "reasons" without a definite article makes a more modest claim than pointing to "the reasons" does.
Similarly, I read "calculating trajectories" as implying "calculating some trajectories that got spacecraft in and out of space," whereas "calculating the trajectories" comes closer to implying "calculating all of the trajectories that got spacecraft in and out of space."
With regard to your question about whether the definite article is necessary, I hope it is clear that although expressing the idea with and without the definite article can yield sentences with arguably different meanings, neither wording is inherently unfinished or otherwise faulty.