= Solution
For $k_R=\pm1$, the selected <orbital resonances> satisfy
$$
m(\Omega_\varphi-\Omega_{\mathrm p})\pm\Omega_R=0.
$$
For near-circular <stellar orbits>, $\Omega_R$ is the <radial epicyclic frequency>. Thus $k_R=-1$ is the <inner Lindblad resonance>, $m(\Omega_\varphi-\Omega_{\mathrm p})=\Omega_R$, and $k_R=+1$ is the <outer Lindblad resonance>, $m(\Omega_\varphi-\Omega_{\mathrm p})=-\Omega_R$.
The flux is driven by $m\partial_{J_\varphi}f_0\pm\partial_{J_R}f_0$, and each resonant harmonic exchanges both <angular momentum> and <radial action> in the ratio
$$
\boxed{\Delta J_R=\frac{k_R}{m}\Delta J_\varphi.}
$$
These terms can therefore produce <galactic disk heating> as well as <angular momentum transport>. In the usual absorbing response, stars losing <angular momentum> at the <inner Lindblad resonance> gain <radial action>, and stars gaining <angular momentum> at the <outer Lindblad resonance> also gain <radial action>. The actual direction of flux follows from the displayed derivatives of $f_0$; it is not fixed by the resonance condition alone. A finite pulse replaces exact <Lindblad resonance> selection by a band of detunings of width of order $1/\tau$.
Back to article page
