{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-non-radial-two-body-collapse-model-tbcm-for","title":"Two-body collapse model for self-gravitating flow of dark matter and generalized stable clustering hypothesis for pairwise velocity","arxiv_id":"2110.05784","date":"2021-10-12","proceeding":null,"authors":["Zhijie Xu"],"abstract":"Analytical tools are extremely hard to find for non-linear gravitational collpase. Only a few simple but powerful tools exist so far. Two examples are the spherical collapse model (SCM) and stable clustering hypothesis (SCH). We present a new analytical tool, a two-body collapse model (TBCM), that plays the same fundamental role as harmonic oscillator in dynamics. For convenience, TBCM is formulated for gravity with any potential exponent $n$ in a static background with a fixed damping ($n$=-1 for Newtonian gravity). The competition between gravity, expanding background (or damping), and angular momentum classifies two-body collapse into: 1) free fall collapse, where free fall time is greater if same system starts to collapse at earlier time; 2) equilibrium collapse that persists longer in time, whose perturbative solutions lead to power-law evolution of system energy and momentum. Two critical values $\\beta_{s1}=1$ and $\\beta_{s2}=1/3\\pi$ are identified that quantifies the competition between damping and gravity. Value $\\beta_{s2}$ only exists for discrete values of potential exponent $n=(2-6m)/(1+3m)=$ -1,-10/7... for integer $m$. Critical density ratio ($\\Delta_c=18\\pi^2$) is obtained for $n$=-1 that is consistent with SCM. TBCM predicts angular velocity $\\propto Hr^{-3/2}$ for two-body system of size $r$. The isothermal density is a result of extremely fast mass accretion. TBCM is able to demonstrate SCH, i.e. mean pairwise velocity (first moment) $\\langle\\Delta u\\rangle=-Hr$. A generalized SCH is developed for higher order moments $\\langle\\Delta u^{2m+1}\\rangle=-(2m+1)\\langle\\Delta u^{2m}\\rangle Hr$ that is validated by N-body simulation. Energy evolution in TBCM is independent of particle mass and energy equipartition does not apply. TBCM can be considered as a non-radial SCM. Both models predict the same critical density ratio, while TBCM contains much richer information.","url_abs":"https://arxiv.org/abs/2110.05784v2","url_pdf":"https://arxiv.org/pdf/2110.05784v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-non-radial-two-body-collapse-model-tbcm-for","repo_url":"https://github.com/ZhijieXu2022/dark_matter_flow_dataset","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}