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The required time separation is $\tau \ll t_{\sec }$, where $t_{\sec }$ is the evolution timescale of $f_0$, together with $t_{\mathrm{peak}}\gg \tau$ and an observation time well after the pulse. The first condition permits freezing $f_0(\mathbf{J},t^{\prime })$ at its pre-pulse value; its neglected change would enter beyond the retained second order. The second makes the original initial time lie in a negligible tail of the Gaussian function, permitting replacement of the lower time limits by $-\infty$. Short duration alone would not justify that replacement for a pulse centered at the initial time.

Integrate the continuity equation in c over the whole pulse. For fixed $\mathbf{J}$, the time factor is the ordered Gaussian response integralA common translation by $t_{\mathrm{peak}}$ removes the peak time. Pair the positive and negative azimuthal harmonics: $(-m,-k_R)$ is paired with $(m,k_R)$. Both $k_{\phi }$ and $\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0$ change sign, so their product does not; ${\omega }_{-\mathbf{k}}=-{\omega }_{\mathbf{k}}$ makes the two time factors complex conjugates. Hence the pair contributes $2\mathrm{Re}\mathcal{I}$.

For completeness, the real part of the integrand is symmetric under $t\leftrightarrow t^{\prime }$. The triangular time domain therefore gives half of the full plane, and the Fourier transform of a Gaussian function givesThis derives the supplied identity rather than requiring a separate time-ordering calculation.

DefineUsing the paired harmonics and $\Delta F_0=-{\partial }_{J_{\phi }}\int Q_{0}dt$ now yieldsBy the stated assumption, ${\omega }_{m{k}_R}$ is independent of $J_R$, so its exponential can be taken outside the radial integral. Put $\Gamma =(\sqrt{2}\tau {)}^{-1}$ and useThereforeThe outer derivative acts on the Gaussian resonance broadening factor as well as on the radial integral. The result describes quasilinear diffusion in action space weighted by the squared pulse amplitude, with a detuning width $\Gamma$. A shorter pulse has broader frequency selectivity, while a longer pulse selects narrower orbital resonances. If the action-space boundary flux vanishes, integrating $\Delta F_0$ over $J_{\phi }$ gives zero, as required by conservation of mass.

For $k_R=0$, the detuning is ${\omega }_{m0}=m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})$. The Gaussian resonance broadening factor selects the corotation resonance, where the star's azimuthal frequency matches the pattern speed. The derivative driving the flux is just $m\partial f_0/\partial J_{\phi }$. In fact this contribution alone has the formA single Fourier series harmonic changes the action variables in the direction $(m,k_R)$, because $\dot{\mathbf{J}}=-i\mathbf{k}\delta {\varphi }_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{\theta }}$ plus its conjugate. For this harmonic, $\Delta J_R=0$: stars change angular momentum and shift their epicyclic guiding center without a leading-order increase of random radial motion. This is stellar radial migration driven by the corotation resonance. The resonance condition explains the absence of leading heating: $\Delta E\simeq {\Omega }_{\phi }\Delta J_{\phi }={\Omega }_{\mathrm{p}}\Delta J_{\phi }$ at the corotation resonance.

The second-order, angle-averaged evolution of a galactic distribution function caused by weak fluctuating gravitational perturbations. In action-angle variables, the first-order response to a Fourier series harmonic feeds a second-order flux proportional to $\mathbf{k}(\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0)$ and to the perturbation's power. Orbital resonances select where transport is strongest; a finite pulse produces Gaussian resonance broadening when its time envelope is a Gaussian function. The general flux retains time memory until a further short-memory or pulse-integrated approximation is made.