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<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" article-type="research-article" dtd-version="1.2" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">RUDN Journal of Engineering Research</journal-id><journal-title-group><journal-title xml:lang="en">RUDN Journal of Engineering Research</journal-title><trans-title-group xml:lang="ru"><trans-title>Вестник Российского университета дружбы народов. Серия: Инженерные исследования</trans-title></trans-title-group></journal-title-group><issn publication-format="print">2312-8143</issn><issn publication-format="electronic">2312-8151</issn><publisher><publisher-name xml:lang="en">Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">35136</article-id><article-id pub-id-type="doi">10.22363/2312-8143-2023-24-2-111-120</article-id><article-id pub-id-type="edn">CQFLUD</article-id><article-categories><subj-group subj-group-type="toc-heading" xml:lang="en"><subject>Articles</subject></subj-group><subj-group subj-group-type="toc-heading" xml:lang="ru"><subject>Статьи</subject></subj-group><subj-group subj-group-type="article-type"><subject>Research Article</subject></subj-group></article-categories><title-group><article-title xml:lang="en">Designing the low-energy lunar transfers trajectories which pass in the vicinity of the libration points of the Earth - Moon system. Part 2. Algorithm and numerical analysis</article-title><trans-title-group xml:lang="ru"><trans-title>Проектирование низкоэнергетических лунных перелетов, траектория которых проходит в окрестности точек либрации системы Земля - Луна. Часть 2. Алгоритм и численный анализ</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-0138-6190</contrib-id><contrib-id contrib-id-type="scopus">55396771600</contrib-id><contrib-id contrib-id-type="spin">3030-7494</contrib-id><name-alternatives><name xml:lang="en"><surname>Konstantinov</surname><given-names>Mikhail S.</given-names></name><name xml:lang="ru"><surname>Константинов</surname><given-names>Михаил Сергеевич</given-names></name></name-alternatives><bio xml:lang="en"><p>Doctor of Sciences (Techn.), Professor of the Space Systems and Rocket Science Department, Aerospace Institute</p></bio><bio xml:lang="ru"><p>доктор технических наук, профессор кафедры космических систем и ракетостроения, Аэрокосмический институт</p></bio><email>mkonst@bk.ru</email><xref ref-type="aff" rid="aff1"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0009-0000-1159-3292</contrib-id><name-alternatives><name xml:lang="en"><surname>Thant</surname><given-names>Aung Myo</given-names></name><name xml:lang="ru"><surname>Тант</surname><given-names>Аунг Мьо</given-names></name></name-alternatives><bio xml:lang="en"><p>PhD student, Space Systems and Rocket Science Department, Aerospace Institute</p></bio><bio xml:lang="ru"><p>аспирант, кафедра космических систем и ракетостроения, Аэрокосмический институт</p></bio><email>aungmyothant4696@gmail.com</email><xref ref-type="aff" rid="aff1"/></contrib></contrib-group><aff-alternatives id="aff1"><aff><institution xml:lang="en">Moscow Aviation Institute (National Research University)</institution></aff><aff><institution xml:lang="ru">Московский авиационный институт (национальный исследовательский университет)</institution></aff></aff-alternatives><pub-date date-type="pub" iso-8601-date="2023-06-30" publication-format="electronic"><day>30</day><month>06</month><year>2023</year></pub-date><volume>24</volume><issue>2</issue><issue-title xml:lang="en">VOL 24, NO2 (2023)</issue-title><issue-title xml:lang="ru">ТОМ 24, №2 (2023)</issue-title><fpage>111</fpage><lpage>120</lpage><history><date date-type="received" iso-8601-date="2023-07-02"><day>02</day><month>07</month><year>2023</year></date></history><permissions><copyright-statement xml:lang="en">Copyright ©; 2023, Konstantinov M.S., Thant A.M.</copyright-statement><copyright-statement xml:lang="ru">Copyright ©; 2023, Константинов М.С., Тант А.М.</copyright-statement><copyright-year>2023</copyright-year><copyright-holder xml:lang="en">Konstantinov M.S., Thant A.M.</copyright-holder><copyright-holder xml:lang="ru">Константинов М.С., Тант А.М.</copyright-holder><ali:free_to_read xmlns:ali="http://www.niso.org/schemas/ali/1.0/"/><license><ali:license_ref xmlns:ali="http://www.niso.org/schemas/ali/1.0/">https://creativecommons.org/licenses/by-nc/4.0/legalcode</ali:license_ref></license></permissions><self-uri xlink:href="https://journals.rudn.ru/engineering-researches/article/view/35136">https://journals.rudn.ru/engineering-researches/article/view/35136</self-uri><abstract xml:lang="en"><p style="text-align: justify;">An algorithm for designing a low-energy lunar flight trajectory is presented. It is based on the assumption that low-energy flight trajectories pass through the vicinity of one of the collinear libration points of the Earth - Moon system (L1 or L2). It is also assumed that at the moment of spacecraft flight in the vicinity of the libration point, the elements of the osculating geocentric orbit of the spacecraft are close to the elements of the osculating geocentric orbit of the libration point itself. The results of a numerical analysis of the obtained low-energy lunar flight trajectory are presented. It is shown that the use of such a trajectory makes it possible to reduce the deceleration impulse of the velocity during the transition to a low lunar orbit to a value of 638 m/s (in the traditional flight scheme, this impulse turns out to be more than 800 m/s). The influence of solar gravitational disturbances on the flight trajectory is analyzed. It is demonstrated that these perturbations ensure the approach of the spacecraft to the Moon with a negative selenocentric energy constant and contribute to the temporary capture of the spacecraft by the Moon. The influence of the terrestrial gravitational perturbation on the circumlunar part of the trajectory is studied. It is displayed that on the trajectory found this perturbation effectively reduces the selenocentric velocity of the spacecraft. The conditions for spacecraft flight in the vicinity of the libration point are considered.</p></abstract><trans-abstract xml:lang="ru"><p style="text-align: justify;">Приводится алгоритм проектирования низкоэнергетической траектории лунного перелета. Он основан на предположении о том, что траектории низкоэнергетического перелета проходят через окрестность одной из коллинеарных точек либрации системы Земля - Луна (L1 или L2). Предполагается, что в момент пролета космическим аппаратом окрестности точки либрации элементы оскулирующей геоцентрической орбиты космического аппарата близки к элементам оскулирующей геоцентрической орбиты самой точки либрации. Представлены результаты численного анализа полученной низкоэнергетической траектории лунного перелета. Показано, что использование такой траектории позволяет уменьшить тормозной импульс скорости при переходе на низкую окололунную орбиту до значения 638 м/с (при традиционной схеме перелета этот импульс оказывается больше 800 м/с). Проанализировано влияния солнечных гравитационных возмущений на траекторию перелета. Выявлено, что эти возмущения обеспечивают подлет космического аппарата к окрестности Луны с отрицательной селеноцентрической константой энергии и способствуют временному захвату космического аппарата Луной. Исследовано влияние земного гравитационного возмущения на окололунный участок траектории. Установлено, что на найденной траектории это возмущение эффективно уменьшает селеноцентрическую скорость космического аппарата. Рассмотрены условия пролета космического аппарата окрестности точки либрации.</p></trans-abstract><kwd-group xml:lang="en"><kwd>low-energy trajectory</kwd><kwd>flight trajectory</kwd><kwd>gravitational perturbations</kwd><kwd>velocity impulse</kwd></kwd-group><kwd-group xml:lang="ru"><kwd>низкоэнергетическая траектория</kwd><kwd>траектория полета</kwd><kwd>гравитационные возмущения</kwd><kwd>импульс скорости</kwd></kwd-group><funding-group><funding-statement xml:lang="en">The study was carried out with the support of the Russian Science Foundation, project No. 21-19-00683.</funding-statement><funding-statement xml:lang="ru">Исследование выполнено при поддержке Российского научного фонда, проект № 21-19-00683.</funding-statement></funding-group></article-meta></front><body></body><back><ref-list><ref id="B1"><label>1.</label><mixed-citation>Parker JS, Anderson RL. Low-energy lunar trajectory design. Hoboken, New Jersey: John Wiley &amp; Sons, Inc.; 2014. https://doi.org/10.1002/9781118855065</mixed-citation></ref><ref id="B2"><label>2.</label><mixed-citation>McCarthy BP, Howell KC. Cislunar transfer design exploiting periodic and quasi-periodic orbital structures in the four-body problem. 71st International Astronautical Congress, The CyberSpace Edition, October 12-14, 2020. Paris; 2020.</mixed-citation></ref><ref id="B3"><label>3.</label><mixed-citation>Scheuerle ST, McCarthy BP, Howell KC. Construction of ballistic lunar transfers leveraging dynamical systems techniques. AAS/AIAA Astrodynamics Specialist Conference, Lake Tahoe, California (Virtual), August 9-12, 2020. South Lake Tahoe, California; 2021.</mixed-citation></ref><ref id="B4"><label>4.</label><mixed-citation>McCarthy BP, Howell KC. Trajectory design using quasi-periodic orbits in the multi-body problem. Proceedings of the 29th AAS/AIAA Space Flight Mechanics Meeting, 2019. Maui; 2019.</mixed-citation></ref><ref id="B5"><label>5.</label><mixed-citation>Ivashkin VV. On the Earth-to-Moon trajectories with temporary capture of a particle by the moon. 54th International Astronautical Congress, Bremen, Germany, September 29 - October 3, 2003. Paper IAC-03-A.P.01. https://doi.org/10.2514/6.IAC-03-A.P.01</mixed-citation></ref><ref id="B6"><label>6.</label><mixed-citation>Ivashkin VV. Low energy trajectories for the Moon-to-Earth space flight. Journal of Earth System Science. 2005;114:613-618. https://doi.org/10.1007/BF02715945</mixed-citation></ref><ref id="B7"><label>7.</label><mixed-citation>Belbruno EA, Carrico JP. Calculation of weak stability boundary ballistic lunar transfer trajectories. Proceedings of the AIAAJ'AAS Astrodynamics Specialist Conference, August 14-17, 2000, Denver, Colorado. Paper AIAA 2000-4142. https://doi.org/10.2514/6.2000-4142</mixed-citation></ref><ref id="B8"><label>8.</label><mixed-citation>Belbruno EA, Miller JK. Sun-perturbed Earthto-Moon transfers with ballistic capture. Journal of Guidance, Control, and Dynamics. 1993;16(4):770-774. https://doi.org/10.2514/3.21079</mixed-citation></ref><ref id="B9"><label>9.</label><mixed-citation>Koon WS, Lo MW, Marsden JE, Ross SD. Low energy transfers to the Moon. Celestial Mechanics and Dynamical Astronomy. 2001;81(1):63-73. https://doi.org/10.1023/A:1013359120468</mixed-citation></ref><ref id="B10"><label>10.</label><mixed-citation>Miller JK. Lunar transfer trajectory design and four body problem. 13th AAS/AIAA Space Flight Mechanics Meeting at Ponce, Puerto Rico, 2003. American Astronomical Society, American Institute of Aeronautics and Astronautics; 2003.</mixed-citation></ref><ref id="B11"><label>11.</label><mixed-citation>Miller JK, Hintz GR. Weak stability boundary and trajectory design. Spaceflight Mechanics. Conference paper AAS 15-297. Williamsburgh, VA; 2015.</mixed-citation></ref><ref id="B12"><label>12.</label><mixed-citation>Konstantinov MS, Thant AM. Designing the low-energy lunar transfers trajectories which pass in the vicinity of the libration points of the Earth - Moon system. Part 1. Theory and method. RUDN Journal of Engineering Re-search. 2023;24(1):7-16. http://doi.org/10.22363/2312-8143-2023-24-1-7-16</mixed-citation></ref><ref id="B13"><label>13.</label><mixed-citation>Martins JRRA, Sturdza P, Alonso JJ. The complex-step derivative approximation. ACM Transactions on Mathematical Software. 2003;29(3):245-262. https://doi.org/10.1145/838250.838251</mixed-citation></ref><ref id="B14"><label>14.</label><mixed-citation>Konstantinov MS, Nikolichev IA, Thein M. Optimization of low thrust multi-revolution orbital transfer using the method of dual numbers. Proceedings of the 6th International Conference on Astrodynamics. Tools and Technics (ICATT-2016). Available from: https://indico.esa.int/indico/event/111/session/21/contribution/99/material/paper/0.pdf (accessed: 12.11.2022).</mixed-citation></ref><ref id="B15"><label>15.</label><mixed-citation>Petukhov VG, Yoon SW. Optimization of perturbed spacecraft trajectories using complex dual numbers. Part 1. Theory and method. Cosmic Research. 2021;59(5):401-413. https://doi.org/10.1134/S0010952521050099</mixed-citation></ref></ref-list></back></article>
