{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:05:43Z","timestamp":1760231143491,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,27]],"date-time":"2022-08-27T00:00:00Z","timestamp":1661558400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Council for Scientific and Technological Development (CNPq)","award":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"],"award-info":[{"award-number":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"]}]},{"name":"S\u00e3o Paulo Research Foundation (FAPESP)","award":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"],"award-info":[{"award-number":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"]}]},{"name":"Coordination for the Improvement of Higher Education Personnel (CAPES)","award":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"],"award-info":[{"award-number":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"]}]},{"name":"Spanish Ministry of Economy and groups E24_20R from the Aragon Government and European Social Fund","award":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"],"award-info":[{"award-number":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"]}]},{"name":"RUDN University Strategic Academic Leadership Program","award":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"],"award-info":[{"award-number":["305210\/2018-1","300923\/2017-1","406841\/2016-0","301338\/2016-7","2019\/15180-0","2016\/24561-0","2016\/23542-1","ESP2017-87113-R"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The study of orbital maneuvers in space missions is a very important problem in astrodynamics. One of the options is the use of a \u201cgravity assisted\u201d maneuver, which is a technique where a spacecraft passes close to a celestial body and uses the gravity of this body to change its trajectory. This approach trajectory has symmetry with respect to the periapsis line when observed from a reference frame fixed in the approached body. There is also a more complex maneuver, when the passage by the celestial body is combined with the application of propulsion, either to give extra energy to the spacecraft or to help to satisfy other constraints required by the mission, like passing by a giving point or region in space. The main object of this study is to measure the efficiency of the application of a continuous thrust combined with the \u201cgravity assisted\u201d maneuver. The effect of this combination is analyzed using maps that give the energy variation of the spacecraft as function of the parameters related to the maneuver. This analysis is made from the point of view of the variation of energy of the spacecraft with respect to the main body of the system. The continuous thrust is applied in the different regions of the trajectory to evaluate the effects of the locations of the thrusting arcs in the variations of energy. The effects of the variations of the direction of the thrust are also studied. This type of combined maneuver may be used to give extra energy to the spacecraft and keeping control of the trajectory during the close approach to better position the spacecraft to complete the mission.<\/jats:p>","DOI":"10.3390\/sym14091786","type":"journal-article","created":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T02:09:36Z","timestamp":1661911776000},"page":"1786","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Low Thrust Propelled Close Approach Maneuvers"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6162-9078","authenticated-orcid":false,"given":"Alessandra","family":"Ferreira","sequence":"first","affiliation":[{"name":"Department of Mathematics, Universidade Estadual Paulista\u2014UNESP, 333 Ariberto Pereira da Cunha Ave, Guaratinguet\u00e1 12516-410, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Antonio","family":"Elipe","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1ticas y Aplicaciones, Universidad de Zaragoza-UNIZAR, 12 Pedro Cerbuna Street, 50009 Zaragoza, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rodolpho","family":"De Moraes","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universidade Estadual Paulista\u2014UNESP, 333 Ariberto Pereira da Cunha Ave, Guaratinguet\u00e1 12516-410, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ant\u00f4nio","family":"Prado","sequence":"additional","affiliation":[{"name":"Instituto Nacional de Pesquisas Espaciais\u2014INPE, Divis\u00e3o de P\u00f3s-Gradua\u00e7\u00e3o, 1758 Avenida dos Astronautas Ave, S\u00e3o Jos\u00e9 dos Campos 12227-010, SP, Brazil"},{"name":"Academy of Engineering, RUDN University, Miklukho-Maklaya Street 6, 117198 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Othon","family":"Winter","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universidade Estadual Paulista\u2014UNESP, 333 Ariberto Pereira da Cunha Ave, Guaratinguet\u00e1 12516-410, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vivian","family":"Gomes","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universidade Estadual Paulista\u2014UNESP, 333 Ariberto Pereira da Cunha Ave, Guaratinguet\u00e1 12516-410, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/BF00200846","article-title":"Voyager mission description","volume":"21","author":"Kohlhase","year":"1977","journal-title":"Space Sci. Rev."},{"key":"ref_2","unstructured":"Nock, K.T., and Upholf, C.W. (1979, January 25\u201327). Satellite aided orbit capture. Proceedings of the AAS\/AIAA, Provincetown, MA, USA."},{"key":"ref_3","first-page":"235","article-title":"Trajectories and orbital maneuvers for the ISEE-3\/ICE comet mission","volume":"33","author":"Farquhar","year":"1985","journal-title":"J. Astronaut. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Schmitt, R.N., Prado, A.F.B.A., Sukhanov, A., and Gomes, V.M. (2022). Swing-By Applications and Estimation of the Van Allen Belts\u2019 Radiation Exposure for a Spacecraft in a Low Thrust Transfer to the Moon. Symmetry, 14.","DOI":"10.3390\/sym14030617"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Broucke, R.A. (1988, January 15\u201317). The Celestial Mechanics of Gravity Assist. Proceedings of the AIAA\/AAS Astrodynamics Conference, Minneapolis, MN, USA.","DOI":"10.2514\/6.1988-4220"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/0094-5765(94)90059-0","article-title":"Optimal atmospheric trajectory for aero-gravity-assist","volume":"32","author":"Lohar","year":"1994","journal-title":"Acta Astronaut."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"637","DOI":"10.2514\/2.4451","article-title":"Optimal Low-Thrust Scape Trajectories Using Gravity Assist","volume":"22","author":"Casalino","year":"1999","journal-title":"J. Guid. Control Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"156","DOI":"10.2514\/2.4362","article-title":"Simple Strategy for Powered Swing-By","volume":"22","author":"Casalino","year":"1999","journal-title":"J. Guid. Control Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.asr.2015.04.016","article-title":"A numerical study of powered Swing-Bys around the Moon","volume":"56","author":"Ferreira","year":"2015","journal-title":"Adv. Space Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.actaastro.2017.02.018","article-title":"Method of interplanetary trajectory optimization for the spacecraft with low thrust and swing-bys","volume":"136","author":"Konstantinov","year":"2017","journal-title":"Acta Astronaut."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1007\/s11071-017-3485-2","article-title":"A numerical mapping of energy gains in a powered Swing-By maneuver","volume":"89","author":"Ferreira","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2071","DOI":"10.1016\/j.asr.2017.01.033","article-title":"Effects of the eccentricity of the primaries in powered Swing-By maneuvers","volume":"59","author":"Ferreira","year":"2017","journal-title":"Adv. Space Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4621","DOI":"10.1093\/mnras\/sty2634","article-title":"Powered swing-by in the elliptic restricted three-body problem","volume":"481","author":"Qi","year":"2018","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"370","DOI":"10.23919\/JSEE.2020.000014","article-title":"Tracking the maneuvering spacecraft propelled by swing propulsion of constant magnitude","volume":"31","author":"Zhai","year":"2020","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"He, B., Pengbin, M., and Hengnian, L. (2021). Properties of the Transfers from LEO to the Retrograde-GEO using Lunar Swing-by in a Three-Body Model. Res. Sq.","DOI":"10.21203\/rs.3.rs-464236\/v1"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"18813","DOI":"10.1038\/s41598-021-98231-1","article-title":"Properties of the lunar gravity assisted transfers from LEO to the retrograde-GEO","volume":"11","author":"He","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"151","DOI":"10.2514\/1.G005360","article-title":"Optimal Powered Aerogravity-Assist Trajectories","volume":"44","author":"Qi","year":"2021","journal-title":"J. Guid. Control Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1016\/j.asr.2021.10.025","article-title":"A trajectory design and optimization framework for transfers from the Earth to the Earth-Moon triangular L4 point","volume":"69","author":"Ren","year":"2021","journal-title":"Adv. Space Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"994","DOI":"10.1016\/j.asr.2011.11.036","article-title":"Low-altitude, near-polar and near-circular orbits around Europa","volume":"49","author":"Carvalho","year":"2012","journal-title":"Adv. Space Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"291","DOI":"10.2514\/1.5686","article-title":"Dynamic behavior of an orbiter around Europa","volume":"28","author":"Martin","year":"2005","journal-title":"J. Guid. Control Dyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s40295-021-00269-w","article-title":"Design of High-Inclination Artificial Frozen Orbits around Europa","volume":"68","author":"Huang","year":"2021","journal-title":"J. Astronaut. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3552","DOI":"10.1093\/mnras\/stw2619","article-title":"The dynamical environment of asteroid 21 Lutetia according to different internal models","volume":"464","author":"Aljbaae","year":"2017","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Colangeli, L. (2004). Astrophysics and Space Science Library, Kluwer. The New Rosetta Targets.","DOI":"10.1007\/978-1-4020-2573-0"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"111","DOI":"10.2514\/1.G004358","article-title":"Powered Swing-By with Continuous Thrust","volume":"43","author":"Qi","year":"2020","journal-title":"J. Guid. Control Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1504","DOI":"10.2514\/1.28694","article-title":"Improved Edelbaum\u2019s approach to optimize low earth\/geostationary orbits low-thrust transfers","volume":"30","author":"Casalino","year":"2007","journal-title":"J. Guid. Control Dyn."},{"key":"ref_26","first-page":"111","article-title":"A numerical-analytical approach based on canonical transformations for computing optimal low-thrust transfers","volume":"54","author":"Fernandes","year":"2018","journal-title":"Rev. Mex. Astron. Astr."},{"key":"ref_27","unstructured":"JPL NASA (2020, March 30). Jet Propulsion Laboratory\u2014California of Institute Technology. Mission to Asteroid 9969 Braille, Comet Borrelly: Deep Space 1. Site Manager: Veronica McGregor, Available online: https:\/\/www.jpl.nasa.gov\/missions\/deep-space-1-ds1\/."},{"key":"ref_28","unstructured":"ESA (2020, March 30). The Europe Space Agency.SMART-1. Available online: https:\/\/www.esa.int\/Enabling_Support\/Operations\/SMART-1."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"380","DOI":"10.2514\/2.3973","article-title":"Design and Optimization of Low-Thrust Trajectories with Gravity Assists","volume":"40","author":"McConaghy","year":"2003","journal-title":"J. Spacecr. Rocket."},{"key":"ref_30","unstructured":"Pascale, P., Vasile, M., and Casotto, S. (2004, January 11\u201315). Preliminary Analysis of Low-Thrust Gravity Assist Trajectories by an Inverse Method and a Global Optimization Technique. Proceedings of the 18th International Symposium on Space Flight Dynamics (ESA SP-548), Munich, Germany."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"776","DOI":"10.2514\/2.3650","article-title":"Low-thrust Trajectories to Jupiter via Gravity Assists from Venus, Earth and Mars","volume":"37","author":"Okutsu","year":"2000","journal-title":"J. Spacecr. Rocket."},{"key":"ref_32","first-page":"15","article-title":"Design of low-thrust multi-gravity assist trajectories to Europa","volume":"62","author":"Vasile","year":"2009","journal-title":"J. Brit. Inter. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1134\/S0010952510050114","article-title":"The Aster project: Flight to a near-Earth asteroid","volume":"48","author":"Sukhanov","year":"2010","journal-title":"Cosm. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"151","DOI":"10.21042\/AMNS.2018.1.00011","article-title":"Design of Gravity Assist trajectory from Earth to Jupiter","volume":"3","author":"Ammar","year":"2018","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.actaastro.2018.10.012","article-title":"Set-oriented design of interplanetary low-thrust trajectories using Earth Gravity Assist","volume":"156","author":"Tamakoshi","year":"2019","journal-title":"Acta Astronaut."},{"key":"ref_36","unstructured":"Ozimek, M.T., Riley, J.F., and Arrieta, J. (2019, January 13\u201317). The Low-Thrust Interplanetary Explorer: A Medium-Fidelity Algorithm for Multi-Gravity Assist Low-Thrust Trajectory Optimization. Proceedings of the AAS\/AIAA 29th Space Flight Mechanics Meeting, Ka\u2019anapali, HI, USA. AIAA Paper 19-348."},{"key":"ref_37","unstructured":"Marec, J.P. (1979). Optimal Space Trajectories, Elsevier."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1786\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:16:28Z","timestamp":1760141788000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1786"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,27]]},"references-count":37,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091786"],"URL":"https:\/\/doi.org\/10.3390\/sym14091786","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,27]]}}}