{"paper":{"title":"Why do launch trajectories end downwards","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Max Cerf","submitted_at":"2017-06-04T08:58:00Z","abstract_excerpt":"The problem of finding the optimal thrust profile of a launcher upper stage is analyzed. The engine is assumed to be continuously thrusting, following either a linear or a bilevel parametric profile, until reaching the targeted coplanar orbit. The minimum-fuel problem is analytically solved using the maximum principle. A closed-loop solution for the thrust direction is derived and the final point is found to be necessarily at an apside, reached downwards in the case of a perigee injection. The optimal control problem reduces to a nonlinear problem with only the thrust profile parameters as unk"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.01044","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}