Two-body freefall time in a matter dominant expanding universe

Two-body freefall time in a matter dominant expanding universe | 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Download PDF Article Two-body freefall time in a matter dominant expanding universe Sean Xu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1517575/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 Predicting freefall time is a major problem in galactic dynamics. The relaxation time of collisionless dark matter, which heavily influences galaxy evolution, is closely related to its freefall time. Calculating freefall time using Kepler’s Third Law does not consider the effects of the universe’s expansion. This paper proposes a new formula for calculating freefall time in an expanding background. The derivation of the formula starts from the governing equations of an N-body system in an expanding background. Two-body simulations were run using equations of motion with constant damping (to represent the effects of expansion) derived from the N-body equations. More than 5,200 different two-body systems with different separation distances, mass, and damping were simulated. The simulated freefall time was compared with good agreement against the proposed formula. The results demonstrate that two-body freefall time is dependent on the time at which collapse began. The earlier it began, the longer the freefall time. A weak gravitational attraction, caused by the small mass of the system or the great distances involved, exacerbates this effect. The galaxy that started to form earlier is expected to take longer to form. A better understanding of freefall time can lead to greater insight into galaxies’ historical and future evolution. Full Text Additional Declarations No competing interests reported. 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. We do this by developing innovative software and high quality services for the global research community. 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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-1517575","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":96816688,"identity":"e39b667c-ce1a-41d0-810a-cf7fe174ae4b","order_by":0,"name":"Sean Xu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwklEQVRIiWNgGAWjYHCC9N9/KmyANGPjAWK1PJDgOZMG0tJArBbGBxK8LYfBTOK0yM9ITjCQbDhvt7b9MNCWGptogloMzhxLSDDccTt525lEoJZjabkNBLWw9yQcSDxzO9nsAFALY8Nhwlrkm/k/NhxsO5dsdv4hkVoYjjckMza2HbAzu0GsLQZnDqQxM5xJTjC7AbQlgRi/yM9IAGqpsLM3O5/+8MGHGhsiHAYFiWCVCcQqBwF7UhSPglEwCkbBCAMA3ClLAkCYq7sAAAAASUVORK5CYII=","orcid":"","institution":"Hanford High School","correspondingAuthor":true,"prefix":"","firstName":"Sean","middleName":"","lastName":"Xu","suffix":""}],"badges":[],"createdAt":"2022-04-02 21:44:06","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1517575/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1517575/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":20196273,"identity":"219846ec-8bad-4066-8c0b-d765f1908651","added_by":"auto","created_at":"2022-04-11 14:34:48","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":817844,"visible":true,"origin":"","legend":"","description":"","filename":"Freefalloftwobodiesinamatterdominantuniverseasdf.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1517575/v1_covered.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Two-body freefall time in a matter dominant expanding universe","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1517575/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"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":false,"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-1517575/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1517575/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Predicting freefall time is a major problem in galactic dynamics. The relaxation time of collisionless dark matter, which heavily influences galaxy evolution, is closely related to its freefall time. Calculating freefall time using Kepler’s Third Law does not consider the effects of the universe’s expansion. This paper proposes a new formula for calculating freefall time in an expanding background. The derivation of the formula starts from the governing equations of an N-body system in an expanding background. Two-body simulations were run using equations of motion with constant damping (to represent the effects of expansion) derived from the N-body equations. More than 5,200 different two-body systems with different separation distances, mass, and damping were simulated. The simulated freefall time was compared with good agreement against the proposed formula. The results demonstrate that two-body freefall time is dependent on the time at which collapse began. The earlier it began, the longer the freefall time. 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