 Interior eccentricity-type mean-motion resonance

For an eccentric inner body and circular outer perturber near the reduced $(m+n):m$ period ratio, the leading resonant argument is $\varphi =(m+n){\lambda }_{\mathrm{out}}-m{\lambda }_{\mathrm{in}}-n{\varpi }_{\mathrm{in}}$. Its disturbing function amplitude is of order $e_{\mathrm{in}}^n$. At a mean conjunction, $\varphi =n(\Lambda -{\varpi }_{\mathrm{in}})$; at actual alignment the exact expression is $(m+n)x-my$ with $x=f_{\mathrm{in}}$ and $y=M_{\mathrm{in}}(x)$.