  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.
