{"title":"On the Asymptotic Behavior of the Secular Perturbation Function in the Circular Restricted Three-Body Problem","authors":"P. S. Krasilnikov, A. V. Dobroslavskiy","doi":"10.1134/s0010952524600252","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The asymptotic behavior of the secular perturbation function expanded in a power series in μ, the ratio of the semimajor axes of the massless point (asteroid) and Jupiter, is studied in the restricted spatial circular three-body problem. It is assumed that <span>\\(\\mu < 1\\)</span> (internal case). A new derivation of the expansion of a secular perturbation function into a power series with coefficients expressed through Gauss and Clausen functions is described based on Parseval’s formula. For different values of μ at fixed values of the Lidov-Kozai constant, the radius of convergence of the reduced series, the areas of convergence and divergence are described in the plane of osculating elements <i>e</i>, ω. It is shown that power series is asymptotic in the sense of Poincaré in divergence regions, and that truncating the series after a 70 number of terms provides an high value approximation to a secular perturbation function. It is shown that the asymptotic properties of the series deteriorate on the nonanalyticity curves of secular perturbation function and completely disappear in a small neighborhood of <span>\\(\\mu = 1\\)</span>. The asymptotic nature of the series allows, using ordinary methods of perturbation theory, to study the evolution of Keplerian orbital elements for all values of <span>\\(\\mu \\)</span> from the interval [0, 1), excluding the case <span>\\(\\mu \\approx 1\\)</span>.</p>","PeriodicalId":56319,"journal":{"name":"Cosmic Research","volume":"2 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cosmic Research","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1134/s0010952524600252","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"ASTRONOMY & ASTROPHYSICS","Score":null,"Total":0}
引用次数: 0
Abstract
The asymptotic behavior of the secular perturbation function expanded in a power series in μ, the ratio of the semimajor axes of the massless point (asteroid) and Jupiter, is studied in the restricted spatial circular three-body problem. It is assumed that \(\mu < 1\) (internal case). A new derivation of the expansion of a secular perturbation function into a power series with coefficients expressed through Gauss and Clausen functions is described based on Parseval’s formula. For different values of μ at fixed values of the Lidov-Kozai constant, the radius of convergence of the reduced series, the areas of convergence and divergence are described in the plane of osculating elements e, ω. It is shown that power series is asymptotic in the sense of Poincaré in divergence regions, and that truncating the series after a 70 number of terms provides an high value approximation to a secular perturbation function. It is shown that the asymptotic properties of the series deteriorate on the nonanalyticity curves of secular perturbation function and completely disappear in a small neighborhood of \(\mu = 1\). The asymptotic nature of the series allows, using ordinary methods of perturbation theory, to study the evolution of Keplerian orbital elements for all values of \(\mu \) from the interval [0, 1), excluding the case \(\mu \approx 1\).
期刊介绍：
Cosmic Research publishes scientific papers covering all subjects of space science and technology, including the following: ballistics, flight dynamics of the Earth’s artificial satellites and automatic interplanetary stations; problems of transatmospheric descent; design and structure of spacecraft and scientific research instrumentation; life support systems and radiation safety of manned spacecrafts; exploration of the Earth from Space; exploration of near space; exploration of the Sun, planets, secondary planets, and interplanetary medium; exploration of stars, nebulae, interstellar medium, galaxies, and quasars from spacecraft; and various astrophysical problems related to space exploration. A chronicle of scientific events and other notices concerning the main topics of the journal are also presented.
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