= Solution
{c}
Ignoring <gravitational focusing>, a <galaxy> sweeps a cylinder of volume $\pi R^2v_{\rm rel}T$. Multiplying this by the number density $N$ of target <galaxies> and averaging the relative speed gives the mean encounter count
$$
\boxed{\mu=\pi R^2\langle v_{\rm rel}\rangle NT.}
$$
For independent encounters modeled by a <Poisson process>, the probability of at least one <galaxy merger> is $1-e^{-\mu}$. The printed expression $P=\mu$ is its rare-encounter approximation $\mu\ll1$; it is not an exact probability for arbitrary $T$.
For an illustrative present-day field population take an effective merger <impact parameter> $R=20\,\mathrm{kpc}=0.02\,\mathrm{Mpc}$, $N=10^{-2}\,\mathrm{Mpc}^{-3}$, and $\langle v_{\rm rel}\rangle=200\,\mathrm{km\,s^{-1}}$. These are assumed order-of-magnitude inputs, not precise observational measurements. Taking a <Hubble time> of $T\simeq1.4\times10^{10}\,\mathrm{yr}\simeq4.4\times10^{17}\,\mathrm s$, the given distance conversion yields
$$
\langle v_{\rm rel}\rangle T\simeq
\frac{200(4.4\times10^{17})}{3\times10^{19}}\,\mathrm{Mpc}
\simeq2.9\,\mathrm{Mpc},
$$
and hence
$$
\boxed{P\simeq\mu\simeq\pi(0.02)^2(10^{-2})(2.9)
\simeq3.7\times10^{-5}.}
$$
Thus the geometric field estimate is of order $10^{-5}$ to $10^{-4}$ per <Hubble time>. It scales as $R^2N\langle v_{\rm rel}\rangle T$ and is very sensitive to environment and the adopted effective merger radius. Enhanced density in groups, <gravitational focusing>, and the evolution of the galaxy population are all omitted; this number is not a prediction of the full cosmological merger fraction.
Now use the stipulated rapid, distant encounter. The consistent rectilinear trajectory is
$$
\mathbf R(t)=\mathbf r_p(t)=(p,0,vt),\qquad
R(t)=\sqrt{p^2+v^2t^2}.
$$
With the stated <impact parameter> along $x$ and velocity along $z$, this trajectory lies in the $xz$-plane. The original PDF's reference to the $xy$-plane is a typo; the local TeX also corrupts the impact-parameter direction. The original PDF fixes that direction as $x$.
Use the positive potential $Gm_p/|\mathbf R-\mathbf r|$, whose gradient is the attractive acceleration under the question's sign convention. Its <multipole expansion> is
$$
\frac1{|\mathbf R-\mathbf r|}
=\frac1R+\frac{\mathbf r\cdot\mathbf R}{R^3}
+\frac{3(\mathbf r\cdot\mathbf R)^2-R^2r^2}{2R^5}
+O\!\left(\frac{r^3}{R^4}\right).
$$
The first term has no force; the linear term accelerates the entire galaxy and disappears in the frame following its <centre of mass>. The leading internal tidal potential is therefore
$$
\boxed{\psi(\mathbf r,t)=\frac{Gm_p}{R^3}
\left[-\frac{r^2}{2}+\frac{3(\mathbf r\cdot\mathbf R)^2}{2R^2}\right].}
$$
This is a <quadrupole approximation> valid for $r\ll p$, rather than an exact equality for every $r<p$. Its acceleration <tidal tensor> acts on $\mathbf r$ as
$$
\nabla\psi=Gm_p\left[-\frac{\mathbf r}{R^3}
+\frac{3(\mathbf r\cdot\mathbf R)\mathbf R}{R^5}\right].
$$
In the <impulse approximation>, each stellar position is held fixed during the flyby and the velocity kick is the integral of this acceleration. Put $s=vt$; terms odd in $s$ integrate to zero, while
$$
\int_{-\infty}^{\infty}\frac{ds}{(p^2+s^2)^{3/2}}=\frac2{p^2},
\qquad
\int_{-\infty}^{\infty}\frac{ds}{(p^2+s^2)^{5/2}}=\frac4{3p^4},
\qquad
\int_{-\infty}^{\infty}\frac{s^2\,ds}{(p^2+s^2)^{5/2}}=\frac2{3p^2}.
$$
The integrated diagonal coefficients of the <tidal tensor> are consequently $(2,-2,0)/(vp^2)$, giving
$$
\boxed{\Delta\mathbf v=\frac{2Gm_p}{vp^2}(x,-y,0).}
$$
The encounter stretches the galaxy along the <impact parameter>, compresses it along $y$, and gives no net leading kick along the flyby direction.
Denote the pre-encounter stellar velocity by $\mathbf u$, to distinguish it from the relative flyby speed $v$. The instantaneous change of specific <kinetic energy> is
$$
\Delta e=\mathbf u\cdot\Delta\mathbf v+\frac12|\Delta\mathbf v|^2.
$$
Uncorrelated kicks with zero mean cross term imply $\langle\mathbf u\cdot\Delta\mathbf v\rangle=0$, so the phase-averaged heating at a specified position is
$$
\boxed{\langle\Delta e\rangle=\frac{2G^2m_p^2}{v^2p^4}(x^2+y^2).}
$$
For an individual star the cross term need not vanish: the formula is an ensemble or orbital-phase average. Integrating over a spherical <galaxy> of mass $m_g$, symmetry gives $\langle x^2\rangle=\langle y^2\rangle=\langle z^2\rangle=\langle r^2\rangle/3$. Therefore the total <tidal heating> is
$$
\boxed{\Delta E_g=\frac{4G^2m_p^2m_g}{3v^2p^4}\langle r^2\rangle.}
$$
For two identical galaxies, each receives this heating with $m_p=m_g$. Adding both contributions gives
$$
\boxed{\Delta E_{\rm int}=\frac{8G^2m_g^3}{3v^2p^4}\langle r^2\rangle.}
$$
In the <centre of mass> frame, equal masses approach with speeds $v/2$ if $v$ is their relative speed at infinity. The initial orbital energy is
$$
E_{\rm orb}=2\left[\frac12m_g\left(\frac v2\right)^2\right]
=\frac12\left(\frac{m_g}{2}\right)v^2
=\boxed{\frac14m_gv^2},
$$
where $m_g/2$ is the <reduced mass>. The orbital <potential energy> vanishes at infinite separation. Within the weak-deflection approximation this asymptotic speed is also the nearly constant speed used in the flyby calculation.
<Conservation of energy> transfers the positive internal heating out of the relative orbit. <Tidal capture of galaxies> occurs in this model if $E_{\rm orb}-\Delta E_{\rm int}<0$, which gives
$$
\frac{8G^2m_g^3\langle r^2\rangle}{3v^2p^4}>\frac14m_gv^2,
\qquad
\boxed{pv<\left[\frac{32}{3}G^2m_g^2\langle r^2\rangle\right]^{1/4}.}
$$
The equality is marginal capture. A bound pair still needs subsequent evolution to coalesce, so the result is an approximate capture criterion used here as a merger criterion. Its right-hand side has dimensions of length times speed, as required.
The <impulse approximation> requires the flyby duration $\tau\sim p/v$ to be short compared with the stellar dynamical time $\Omega^{-1}$, namely $v/p\gg\Omega$. If $v/p<\Omega$, a star moves substantially during the encounter. Its orbital <adiabatic invariance of an orbital action> suppresses net heating by a slowly varying tidal field: <adiabatic shielding of tidal encounters> replaces the frozen-position calculation. The divergent $v^{-2}$ heating predicted by extrapolating the impulsive formula to slow encounters is therefore spurious. Close passages with $p$ comparable to the galaxy size or strong <gravitational focusing> also lie outside the derivation.
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