Equal-mass tidal-capture threshold ($pv<[32G^2{m}_g^2⟨r^2⟩/3{]}^{1/4}$)

Two equal galaxies each receive the tidal impulse heating of a spherical galaxy, while their relative orbital energy at infinity is $m_g{v}^2/4$. Transfer of more internal energy than that orbital energy gives $pv<[32G^2{m}_g^2⟨r^2⟩/3{]}^{1/4}$. Slow encounters require adiabatic shielding of tidal encounters, so the impulsive inequality does not apply there.