= Solution
\b[An encounter between <galaxies> or clusters mainly couples their collective gravitational fields; it need not involve physical impacts between individual <stars>.] Direct stellar collision rates depend on number density, relative speed and <gravitational focusing>. For two <stars> the <collision cross-section> is approximately
$$
\sigma_{\rm coll}=\pi(R_1+R_2)^2
\left[1+\frac{2G(m_1+m_2)}
{(R_1+R_2)v_\infty^2}\right],
\qquad
t_{\rm coll}\sim(n_\star\sigma_{\rm coll}v_\infty)^{-1}.
$$
In an ordinary <galaxy>, the small geometrical stellar filling factor makes such impacts rare even while the <galaxy> undergoes a major encounter. Dense stellar cores are an important exception, not a reason to treat all <galaxies> as collisional fluids.
<Tidal forces> depend on the gradient of the perturber's field. For a distant fast encounter of mass $M_p$, impact parameter $b$ and speed $V$, the impulse scale for a <star> at displacement $r$ is
$$
\Delta v\sim\frac{GM_p r}{b^2V}.
$$
A more specific result illustrates the geometry. Put the straight-line perturber trajectory in the $(x,y)$ plane along $y$, with closest approach at $x=b$. In the tidal and impulse limits,
$$
\Delta\mathbf v=\frac{2GM_p}{b^2V}(x,0,-z),
\qquad
\boxed{\Delta E=
\frac{4G^2M_p^2M\langle r^2\rangle}{3b^4V^2}}
$$
for an initially spherical, nonstreaming target with randomly sampled internal phases. The cross term averages to zero, and $\langle x^2+z^2\rangle=2\langle r^2\rangle/3$. Heating, expansion and <tidal stripping> preferentially affect weakly bound outer <stars>. A slow encounter can be adiabatically shielded rather than impulsive; a penetrating or strongly resonant encounter needs more than this distant straight-line approximation.
Disc rotation and orbital orientation matter. Prograde passages can drive stronger coherent distortions than corresponding retrograde passages. Tidal bridges, long tails, asymmetric arms and shells can record the encounter. Restricted gravitational calculations already reproduced prominent bridge and tail structures without requiring individual stellar collisions; full self-consistent models additionally follow the gravity of the responding discs and haloes.<Galactic Bridges and Tails by Alar Toomre and Juri Toomre>[1972 abstract and §II, close tidal encounters of initially rotating test-particle discs]{ref}
A first high-speed passage need not bind the pair. Capture or merging requires orbital energy and <angular momentum> to be redistributed into internal motions, escaping material, gas radiation or the surrounding halo. Dynamical friction transfers orbital energy to a gravitational wake and can bring a bound satellite inward. In a major merger, the potential changes on orbital times and <violent relaxation> redistributes stellar energies; <phase mixing> then smooths fine orbital structure. These are collective collisionless processes, distinct from slow <two-body relaxation>.
Gas behaves differently from <stars>: gas streams can shock, dissipate energy, lose <angular momentum> and fuel a central starburst or active nucleus. Gas-poor mergers are correspondingly more nearly dissipationless. Outcomes depend on mass ratio, orbital <angular momentum>, relative velocity, halo structure and gas fraction: they range from a modest flyby and tidal debris to disruption of a satellite or a merged remnant. Slow encounters in a group are generally more favorable to merging than otherwise comparable very fast unbound passages, but no single speed or mass-ratio threshold follows without an explicit structural model.
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