Impulse approximation

In a rapid encounter, the impulse approximation holds stellar positions fixed while integrating an external acceleration over time. A stellar velocity kick is $\Delta \mathbf{v}=\int {\mathbf{a}}_{\mathrm{tidal}}(t)dt$. If the encounter time $p/v$ is much shorter than the orbital time ${\Omega }^{-1}$, this gives reliable leading tidal heating for a weak, distant encounter. Slow perturbations require adiabatic shielding of tidal encounters instead.

Integrate the transverse force along a complete straight path: ${\int }_{-\infty }^{\infty }(1+s^2{)}^{-3/2}ds=2$. This gives the transverse kick. Conservation of relative speed gives longitudinal speed reduction $\Delta v_{\perp }^2/(2V)$ at second order. As an independent check, exact weak hyperbolic scattering has angle $2\arctan \eta$, where $\eta =GM/(b{V}^2)$, and reduction $V(1-\cos (2\arctan \eta ))=2V{\eta }^2/(1+{\eta }^2)$. The small-$\eta$ coefficient agrees. Keeping only one half of the encounter in the transverse kick makes this quadratic reduction four times too small.

For a straight shearing-sheet encounter, $V=S|x|$, and the outer and inner longitudinal changes have opposite signs. The complete-flyby gravitational impulse gives $\chi =2$. Some simplified prescribed kick models use a different coefficient; carrying it explicitly makes the corresponding one-sided impulse torque on a disk and migration rates unambiguous. This coefficient does not assert an exact solution of the full rotating Hill encounter.

Multiply encounter mass flux $\Sigma (x)Sxdx$ by specific angular-momentum gain $r_0\Delta v_y$. The net disk torque replaces $\Sigma (x)$ by $\Sigma (x)-\Sigma (-x)$, so a symmetric mass density gives zero net torque. For constant surface density, Keplerian shear and the full-flyby value $\chi =2$, the one-sided coefficient is $8/27$ in $q^2\Sigma r_0^4{\Omega }^2(r_0/H{)}^3$, matching the primary coplanar impulse normalization.

A local odd mass density slope $\Sigma (x)-\Sigma (-x)=2{\Sigma }_0\beta x/r_0$ changes the cancelling one-sided torques into this net torque. A local outer cutoff $L$ adds the factor $1-(H/L{)}^2$. Positive slope transfers net angular momentum from satellite to disk. An infinite linear mass density law is only a formal local extension, since it cannot remain nonnegative on both sides.

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.

Adiabatic shielding suppresses the heating of an orbit when an external tidal tensor varies slowly compared with its orbital frequency. In a nonresonant slow encounter, adiabatic invariance of an orbital action lets the orbit adjust approximately reversibly, so a frozen-position impulse approximation overestimates the energy transfer. An orbital resonance can invalidate the simple slow-variation argument.