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Analytical tools are extremely hard to find for non-linear gravitationalcollpase. Only a few simple but powerful tools exist so far. Two examples arethe spherical collapse model (SCM) and stable clustering hypothesis (SCH). Wepresent a new analytical tool, a two-body collapse model (TBCM), that plays thesame fundamental role as harmonic oscillator in dynamics. For convenience, TBCMis formulated for gravity with any potential exponent $n$ in a staticbackground with a fixed damping ($n$=-1 for Newtonian gravity). The competitionbetween gravity, expanding background (or damping), and angular momentumclassifies two-body collapse into: 1) free fall collapse, where free fall timeis greater if same system starts to collapse at earlier time; 2) equilibriumcollapse that persists longer in time, whose perturbative solutions lead topower-law evolution of system energy and momentum. Two critical values$\beta_{s1}=1$ and $\beta_{s2}=1/3\pi$ are identified that quantifies thecompetition between damping and gravity. Value $\beta_{s2}$ only exists fordiscrete values of potential exponent $n=(2-6m)/(1+3m)=$ -1,-10/7... forinteger $m$. Critical density ratio ($\Delta_c=18\pi^2$) is obtained for $n$=-1that 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 ofextremely fast mass accretion. TBCM is able to demonstrate SCH, i.e. meanpairwise velocity (first moment) $\langle\Delta u\rangle=-Hr$. A generalizedSCH is developed for higher order moments $\langle\Deltau^{2m+1}\rangle=-(2m+1)\langle\Delta u^{2m}\rangle Hr$ that is validated byN-body simulation. Energy evolution in TBCM is independent of particle mass andenergy equipartition does not apply. TBCM can be considered as a non-radialSCM. Both models predict the same critical density ratio, while TBCM containsmuch richer information.

Two-body collapse model for self-gravitating flow of dark matter and generalized stable clustering hypothesis for pairwise velocity