Power-law relationship between the semi-major axis and rotation period ratio in an eccentric system: Its physical meaning and various applications

Power-law relationship between the semi-major axis and rotation period ratio in an eccentric system: Its physical meaning and various applications | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Power-law relationship between the semi-major axis and rotation period ratio in an eccentric system: Its physical meaning and various applications Yongfeng Dai This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3911288/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study aims to address long-term questions about orbital motion, introduces a synchronous orbit radius ratio (cube root of the product of the mass ratio and square of the rotation period ratio; secondary/primary) and investigates the relationship between the semi-major axes and synchronous orbit radius ratios of known systems with the same mass ratio orders of magnitude, the same spin–orbit angles and eccentric or circular orbits, including binaries, star–planet systems and planet–moon systems. The results of this work show that in a system with an eccentric orbit (e > 0.01) and a mass ratio order with a magnitude higher than −7.5217, the semi-major axis exhibits a power-law distribution relationship with the product of the mass ratio and square of the rotation period ratio. Furthermore, under spin–orbit alignment conditions, the power-law exponent exhibits a positive linear relationship with the mass ratio order of magnitude and is greater than zero and less than or equal to one. Moreover, the power-law exponent for a spin–orbit misaligned system correlates negatively with its spin–orbit angle. This study proposes that in a local physical space of a gravitational field rotating along with the host celestial body, there exists a frame that replaces gravity and exhibits a free-fall acceleration towards the host and an acceleration from an initial zero velocity to the current velocity of the gravitational field based on Einstein's equivalence principle. This frame can be divided into two subframes: one with the acceleration towards the host and the other with the acceleration in the direction of gravitational field movement; additionally, the latter can be replaced by the drag force. The analysis of Newton's law of gravity using such a frame not only supports the proposal of gravitational field movement but also obtains an expression for drag force. In a system, when the movement of the secondary body reaches a circular orbital velocity under the action of the drag force caused by the gravitational field rotation of the primary body, the secondary body is no longer subjected to the drag force. Moreover, considering the mass centre of the system as the axis under the action of the reaction force, the primary body moves in a circular orbit. However, as the drag force acts on the primary body owing to the gravitational field rotation of the secondary body, the velocity is constrained by the mass ratio unless their masses are similar. This power-law describes a relation between the orbital velocity forming the semi-major axis caused by the drag force due to the gravitational field rotation of the secondary body and the ratio of the two drag forces when they appear within the same system and the primary and secondary bodies rotate in the same direction. Simultaneously, the drag force drives the eccentric orbital precession in the apogee region; if the spin–orbit misalignment exists, the orbital plane precession is driven by the drag force caused by the gravitational field rotation of the primary body in the direction of its rotation. Physical sciences/Astronomy and planetary science/Astronomy and astrophysics/Exoplanets Physical sciences/Astronomy and planetary science/Astronomy and astrophysics/General relativity and gravity Physical sciences/Astronomy and planetary science/Planetary science/Early solar system Physical sciences/Astronomy and planetary science/Planetary science/Giant planets Physical sciences/Astronomy and planetary science/Planetary science/Inner planets Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3911288","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":270241480,"identity":"ced29bd9-b42d-4df4-9332-064dc6c99938","order_by":0,"name":"Yongfeng Dai","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0001-5330-1294","institution":"Lishuling Institute of Physical Space Research","correspondingAuthor":true,"prefix":"","firstName":"Yongfeng","middleName":"","lastName":"Dai","suffix":""}],"badges":[],"createdAt":"2024-01-30 17:12:22","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3911288/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3911288/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":50491895,"identity":"b9551ea8-f78f-4f86-b67c-95b35c1e95d1","added_by":"auto","created_at":"2024-02-01 10:33:47","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":985916,"visible":true,"origin":"","legend":"Article File","description":"","filename":"document.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3911288/v1_covered_91e22d8a-8239-4f2c-aa93-c65613302b13.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Power-law relationship between the semi-major axis and rotation period ratio in an eccentric system: Its physical meaning and various applications","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-3911288/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3911288/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This study aims to address long-term questions about orbital motion, introduces a synchronous orbit radius ratio (cube root of the product of the mass ratio and square of the rotation period ratio; secondary/primary) and investigates the relationship between the semi-major axes and synchronous orbit radius ratios of known systems with the same mass ratio orders of magnitude, the same spin–orbit angles and eccentric or circular orbits, including binaries, star–planet systems and planet–moon systems. The results of this work show that in a system with an eccentric orbit (e \u003e 0.01) and a mass ratio order with a magnitude higher than −7.5217, the semi-major axis exhibits a power-law distribution relationship with the product of the mass ratio and square of the rotation period ratio. Furthermore, under spin–orbit alignment conditions, the power-law exponent exhibits a positive linear relationship with the mass ratio order of magnitude and is greater than zero and less than or equal to one. Moreover, the power-law exponent for a spin–orbit misaligned system correlates negatively with its spin–orbit angle. This study proposes that in a local physical space of a gravitational field rotating along with the host celestial body, there exists a frame that replaces gravity and exhibits a free-fall acceleration towards the host and an acceleration from an initial zero velocity to the current velocity of the gravitational field based on Einstein's equivalence principle. This frame can be divided into two subframes: one with the acceleration towards the host and the other with the acceleration in the direction of gravitational field movement; additionally, the latter can be replaced by the drag force. The analysis of Newton's law of gravity using such a frame not only supports the proposal of gravitational field movement but also obtains an expression for drag force. In a system, when the movement of the secondary body reaches a circular orbital velocity under the action of the drag force caused by the gravitational field rotation of the primary body, the secondary body is no longer subjected to the drag force. Moreover, considering the mass centre of the system as the axis under the action of the reaction force, the primary body moves in a circular orbit. However, as the drag force acts on the primary body owing to the gravitational field rotation of the secondary body, the velocity is constrained by the mass ratio unless their masses are similar. This power-law describes a relation between the orbital velocity forming the semi-major axis caused by the drag force due to the gravitational field rotation of the secondary body and the ratio of the two drag forces when they appear within the same system and the primary and secondary bodies rotate in the same direction. Simultaneously, the drag force drives the eccentric orbital precession in the apogee region; if the spin–orbit misalignment exists, the orbital plane precession is driven by the drag force caused by the gravitational field rotation of the primary body in the direction of its rotation.","manuscriptTitle":"Power-law relationship between the semi-major axis and rotation period ratio in an eccentric system: Its physical meaning and various applications","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-01 10:25:37","doi":"10.21203/rs.3.rs-3911288/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"7bf9ddf6-75fe-461a-af5a-8b1278b98ad5","owner":[],"postedDate":"February 1st, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":28482576,"name":"Physical sciences/Astronomy and planetary science/Astronomy and astrophysics/Exoplanets"},{"id":28482577,"name":"Physical sciences/Astronomy and planetary science/Astronomy and astrophysics/General relativity and gravity"},{"id":28482578,"name":"Physical sciences/Astronomy and planetary science/Planetary science/Early solar system"},{"id":28482579,"name":"Physical sciences/Astronomy and planetary science/Planetary science/Giant planets"},{"id":28482580,"name":"Physical sciences/Astronomy and planetary science/Planetary science/Inner planets"}],"tags":[],"updatedAt":"2024-02-01T10:25:37+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-01 10:25:37","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3911288","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3911288","identity":"rs-3911288","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}