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| patched conic approximation | |
|---|---|
| Name | Patched conic approximation |
| Field | Astrodynamics |
patched conic approximation is a heuristic method used in astrodynamics for approximating spacecraft trajectories by partitioning an interplanetary trajectory into segments governed by the gravity of a single dominant body. It reduces the three-body or n-body problem to a sequence of two-body problems, enabling analytic and semi-analytic planning of transfers and flybys for missions such as those by NASA, European Space Agency, Roscosmos, JAXA, and private firms like SpaceX. The approximation underpins preliminary designs for trajectories to targets including Mars, Venus, Jupiter, Saturn, Mercury (planet), and minor bodies like Comet Halley, Comet 67P/Churyumov–Gerasimenko, and numerous Near-Earth object missions.
The patched conic approximation emerged to simplify trajectory design for projects associated with institutions such as Jet Propulsion Laboratory, Goddard Space Flight Center, and European Space Agency (ESA) mission teams tasked with planning flights to Moon, Mars Exploration Program, and outer planet probes like Voyager program and Galileo (spacecraft). By treating regions of dominance for bodies including Sun, Earth, Moon, and other planets, engineers from organizations like Aerospace Corporation and agencies like Russian Federal Space Agency applied the technique in early mission concepts such as Mariner program, Pioneer program, and the Apollo program trajectory analyses. The method interfaces with concepts used by figures and institutions like Konstantin Tsiolkovsky, Yuri Kondratyuk, and teams at MIT and Caltech.
Patched conics draws on classical mechanics developed by scholars such as Isaac Newton, Joseph-Louis Lagrange, and Pierre-Simon Laplace, and leverages two-body solutions popularized in treatments by Johannes Kepler and later formalized by mathematicians at institutions like University of Cambridge and University of Göttingen. The approach approximates the restricted three-body problem as sequential two-body problems by introducing spheres of influence or Hill spheres defined by formulas credited to researchers associated with George William Hill and concepts refined at Royal Society. It connects to methods used in perturbation theory by contributors from Princeton University and Harvard University, and aligns with numerical strategies developed at Los Alamos National Laboratory and Sandia National Laboratories for mission analysis.
The patched conic procedure partitions space into regions—typically the heliocentric region near Sun and planetocentric regions near bodies like Earth or Jupiter—using radii such as the sphere of influence often attributed to formulas discussed in works from Jet Propulsion Laboratory and textbooks from Cornell University faculty. Within each region, the trajectory is computed as a Keplerian conic (ellipse, parabola, hyperbola) about the dominant body using constants from sources like International Astronomical Union. At region boundaries, the conic arcs are "patched" by matching position and velocity vectors, a procedure employed by analysts at NASA Ames Research Center and taught in courses at Stanford University and Massachusetts Institute of Technology. The method uses orbital elements (semi-major axis, eccentricity, inclination) and relies on transformations explored by scholars at California Institute of Technology and University of Michigan; delta-v budgeting and Lambert problem solutions are often computed following algorithms refined by teams at Ball Aerospace and Northrop Grumman.
Mission designers at NASA Jet Propulsion Laboratory and European Space Agency routinely apply the patched conic approximation to design gravity-assist sequences for missions such as Voyager 2, Cassini–Huygens, Galileo (spacecraft), Messenger (spacecraft), and interplanetary sample-return concepts explored by Canadian Space Agency and JAXA. It aids preliminary selection of launch windows, resonant returns used by teams at University of Arizona and University of Colorado Boulder, and interplanetary trajectory libraries maintained by NASA Ames Research Center. Commercial entities like Blue Origin and SpaceX have used similar simplifications in feasibility studies for cislunar transportation between Low Earth orbit and Lunar Gateway, while planetary protection planning for missions coordinated with Committee on Space Research uses patched-conic-based estimates for impact probabilities.
Despite widespread use by organizations including European Space Agency and NASA, the patched conic approximation neglects multi-body perturbations central to the restricted three-body dynamics studied by researchers at CITA and Max Planck Institute for Gravitational Physics. Errors arise near Lagrange points investigated by Joseph-Louis Lagrange and in regions with significant third-body perturbations as encountered in missions to Jupiter and Saturn, where models from Jet Propulsion Laboratory show discrepancies compared to n-body integrations used by teams at Goddard Space Flight Center. Quantitative error analyses performed by researchers from University of California, Berkeley and Imperial College London compare patched-conic predictions to high-fidelity propagation using integrators from NASA Ames Research Center and codebases developed at Los Alamos National Laboratory, revealing limitations for long-duration, low-thrust, or chaotic trajectories studied in works at Princeton University and Caltech.
Historically, patched conics underpinned early interplanetary mission planning at Jet Propulsion Laboratory for the Mariner program and were central to trajectory designs for the Apollo program lunar transfers developed by engineers at MIT flight dynamics groups and by personnel at NASA Manned Spacecraft Center. The technique informed gravity-assist routing in the Voyager program orchestrated by Edward C. Stone and teams at Jet Propulsion Laboratory, and guided the deployment of probes like Cassini–Huygens and Galileo (spacecraft) with planning input from European Space Agency and Ames Research Center. Contemporary applications remain pedagogically important in curricula at Massachusetts Institute of Technology, Stanford University, and California Institute of Technology, even as mission teams at NASA Jet Propulsion Laboratory, European Space Agency, and research groups at University of Texas at Austin augment patched-conic estimates with n-body simulations and optimal control methods originating from work at IBM Research and Sandia National Laboratories.