Integrated tidal tensor of a straight-line flyby ($\int Tdt=\frac{2G{m}_p}{v{p}^2}\mathrm{diag}(1,-1,0)$)

For a perturber following $(p,0,vt)$, the impulse approximation integrates its tidal tensor to $2G{m}_p\mathrm{diag}(1,-1,0)/(v{p}^2)$. A fixed target position therefore receives $\Delta \mathbf{v}=2G{m}_p(x,-y,0)/(v{p}^2)$. The longitudinal kick vanishes after integrating the symmetric trajectory.

If the initial stellar velocities are uncorrelated with the impulse approximation kicks, mean specific heating is $|\Delta \mathbf{v}{|}^2/2$. Spherical position averaging gives total heating $4G^2{m}_p^2{m}_g⟨r^2⟩/(3v^2{p}^4)$, where the mass-weighted mean square radius must be finite.