{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:02:51Z","timestamp":1760241771821,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,8,15]],"date-time":"2018-08-15T00:00:00Z","timestamp":1534291200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/EMS\/50022\/2013"],"award-info":[{"award-number":["UID\/EMS\/50022\/2013"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Aerospace"],"abstract":"<jats:p>The calculation of rocket trajectories is most often performed using purely numerical methods that account for all relevant parameters and provide the required results. There is a complementary need for analytical methods that make more explicit the effect of the various rocket and atmospheric parameters of the trajectory and can be used as test cases with unlimited accuracy. The available analytical methods take into account (i) variable rocket mass due to propellant consumption. The present paper includes four new analytical methods taking into account besides (i) also (ii) nonlinear aerodynamic forces proportional to the square of the velocity and (iii) exponential dependence of the mass density with altitude for an isothermal atmospheric layer. The four new methods can be used in \u201chybrid analytical-numerical\u201d approach in which: (i) the atmosphere is divided into isothermal rather than homogeneous layers for greater physical fidelity; and (ii) in each layer, an exact analytical solution of the equations of motion with greater mathematical accuracy than a numerical approximation is used. This should allow a more accurate calculation of rocket trajectories while discretizing the atmosphere into a smaller number of layers. The paper therefore concentrates on four analytical methods of calculation of rocket trajectories in an isothermal atmospheric layers using new exact solutions of the equations of motion beyond those currently available in the literature. The four methods are developed first for the simpler case of a vertical climb and will be subsequently extended to the practically more relevant case of a gravity turn.<\/jats:p>","DOI":"10.3390\/aerospace5030088","type":"journal-article","created":{"date-parts":[[2018,8,15]],"date-time":"2018-08-15T10:40:07Z","timestamp":1534329607000},"page":"88","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On Four New Methods of Analytical Calculation of Rocket Trajectories"],"prefix":"10.3390","volume":"5","author":[{"given":"Lu\u00eds","family":"Campos","sequence":"first","affiliation":[{"name":"CCTAE, IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2183-6221","authenticated-orcid":false,"given":"Paulo","family":"Gil","sequence":"additional","affiliation":[{"name":"CCTAE, IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1063\/1.1722828","article-title":"Universal Gravity Turn Trajectories","volume":"28","author":"Culler","year":"1957","journal-title":"J. Appl. Phys."},{"key":"ref_2","unstructured":"Thomson, W.T. (1986). Introduction to Space Dynamics, Dover Publications. [2nd ed.]."},{"key":"ref_3","unstructured":"Miele, A. (1962). Flight Mechanics, Addison-Wesley."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1308","DOI":"10.2514\/3.28652","article-title":"Gravity turn trajectories through the atmosphere","volume":"3","author":"Connor","year":"1966","journal-title":"J. Spacecr. Rocket."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"538","DOI":"10.2514\/2.4914","article-title":"Semi-Analytical Formulas for Launcher Performance Evaluation","volume":"25","author":"Sotto","year":"2002","journal-title":"J. Guid. Control Dyn."},{"key":"ref_6","unstructured":"Rutherford, D.E. (1967). Classical Mechanics, Oliver and Boyd. [2nd ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1007\/BF00937371","article-title":"Method of particular solutions for linear, two-point boundary-value problems","volume":"2","author":"Miele","year":"1968","journal-title":"J. Optim. Theory Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/BF00927913","article-title":"Sequential gradient-restoration algorithm for optimal control problems","volume":"5","author":"Miele","year":"1970","journal-title":"J. Optim. Theory Appl."},{"key":"ref_9","unstructured":"Von Mises, R. (1959). Theory of Flight, Dover."},{"key":"ref_10","unstructured":"Perkins, C.D., and Hage, R.E. (1949). Airplane Performance, Stability and Control, Wiley."},{"key":"ref_11","unstructured":"Milne-Thomson, L.M. (1958). Theoretical Aerodynamics, Dover."},{"key":"ref_12","unstructured":"Campos, L.M.B.C. (2010). Complex Analysis with Applications to Flows and Fields, CRC Press."},{"key":"ref_13","unstructured":"Campos, L.M.B.C. (2003). Mec\u00e2nica Aplicada, Escolar Editora."},{"key":"ref_14","unstructured":"(2018, August 10). Ariane 5. Available online: https:\/\/en.wikipedia.org\/wiki\/Ariane_5."},{"key":"ref_15","unstructured":"(2018, August 10). Ariane 5\u2014VA226\u2014Launch Profile. Available online: https:\/\/spaceflight101.com\/ariane-5-va226\/ariane-5-va226-launch-profile\/."}],"container-title":["Aerospace"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2226-4310\/5\/3\/88\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:18:55Z","timestamp":1760195935000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2226-4310\/5\/3\/88"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,15]]},"references-count":15,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,9]]}},"alternative-id":["aerospace5030088"],"URL":"https:\/\/doi.org\/10.3390\/aerospace5030088","relation":{},"ISSN":["2226-4310"],"issn-type":[{"type":"electronic","value":"2226-4310"}],"subject":[],"published":{"date-parts":[[2018,8,15]]}}}