Resonant motion of Echo I: a case study of SRP and J2 coupling

Resonant motion of Echo I: a case study of SRP and J2 coupling | 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 Research Article Resonant motion of Echo I: a case study of SRP and J 2 coupling Catherine Massé, Inna Sharf This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7238808/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Mar, 2026 Read the published version in The Journal of the Astronautical Sciences → Version 1 posted You are reading this latest preprint version Abstract In this paper, we explore the resonant behaviour exhibited by the historical NASA’s Echo I satellite. It is well known that the coupling between the SRP and the geopotential (second zonal harmonic 𝑱2) perturbations leads to a resonance effect in eccentricity under certain orbital conditions. Here, we revisit the orbital motion of Echo I in light of the recently published phase space theory for spherical spacecraft subject to the SRP-𝑱2 resonance. We first demonstrate that our modelling is in very good agreement with the recorded Two Line Element (TLE) data of the spacecraft. We then derive the first ( 𝒋 = 1) resonance model associated with the motion of Echo I by using the Hamiltonian formulation of the satellite dynamics, which we use to interpret its observed behaviour. Finally, we extrapolate the resonance model of Echo I for two imagined extreme scenarios: the motion near a stable equilibrium, and the motion under the strongest resonance possible. The results provide a new understanding of the orbital motion of Echo I and offer new insights into the SRP-𝑱2 resonance phenomenon. Resonance Echo I SRP J2 Orbital motion Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 Mar, 2026 Read the published version in The Journal of the Astronautical Sciences → 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-7238808","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":498901274,"identity":"86f002c0-e951-4ff2-a38e-3f905566929b","order_by":0,"name":"Catherine Massé","email":"data:image/png;base64,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","orcid":"","institution":"McGill University","correspondingAuthor":true,"prefix":"","firstName":"Catherine","middleName":"","lastName":"Massé","suffix":""},{"id":498901275,"identity":"5b54648d-0ee0-4808-9f10-82b1dc4d17af","order_by":1,"name":"Inna Sharf","email":"","orcid":"","institution":"McGill University","correspondingAuthor":false,"prefix":"","firstName":"Inna","middleName":"","lastName":"Sharf","suffix":""}],"badges":[],"createdAt":"2025-07-29 04:23:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7238808/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7238808/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s40295-026-00573-3","type":"published","date":"2026-03-03T15:57:34+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":104250637,"identity":"c935b30c-31d3-4864-9917-25c15e66c492","added_by":"auto","created_at":"2026-03-09 16:03:00","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3107350,"visible":true,"origin":"","legend":"","description":"","filename":"ResonantmotionofEchoI.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7238808/v1_covered_9d82b590-9133-4b7b-804d-1d634b981666.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eResonant motion of Echo I: a case study of SRP and J\u003csub\u003e2\u003c/sub\u003e coupling\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":true,"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":"Resonance, Echo I, SRP, J2, Orbital motion","lastPublishedDoi":"10.21203/rs.3.rs-7238808/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7238808/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we explore the resonant behaviour exhibited by the historical NASA’s Echo I satellite. It is well known that the coupling between the SRP and the geopotential (second zonal harmonic 𝑱2) perturbations leads to a resonance effect in eccentricity under certain orbital conditions. Here, we revisit the orbital motion of Echo I in light of the recently published phase space theory for spherical spacecraft subject to the SRP-𝑱2 resonance. We first demonstrate that our modelling is in very good agreement with the recorded Two Line Element (TLE) data of the spacecraft. We then derive the first ( 𝒋 = 1) resonance model associated with the motion of Echo I by using the Hamiltonian formulation of the satellite dynamics, which we use to interpret its observed behaviour. Finally, we extrapolate the resonance model of Echo I for two imagined extreme scenarios: the motion near a stable equilibrium, and the motion under the strongest resonance possible. 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