Resonant argument
= Resonant argument
{title2=$\phi=j\lambda-(j-1)\lambda_p-\varpi$}
= Resonant angle
{synonym}
A <resonant argument> is an integer linear combination of orbital <angles> that evolves slowly near an <orbital resonance>. For an outer body near an exterior $j:(j-1)$ <mean-motion resonance> with a nearly circular inner <planet>, one example is
$$
\phi=j\lambda-(j-1)\lambda_p-\varpi,\qquad j\ge2.
$$
Here $\lambda$ and $\lambda_p$ are <mean longitudes>, $\varpi$ is the outer body's <longitude of periapsis>, and the corresponding <mean motions> are $n$ and $n_p$. Thus $\dot\phi=jn-(j-1)n_p-\dot\varpi$ is small near the resonance, including the <apsidal precession> correction.
In <resonant-argument libration>, an unwrapped version of $\phi$ oscillates around a centre rather than repeatedly running through every angle. Slow variation at one instant alone does not establish trapping. At equal <mean longitudes>, the example gives $\phi=\lambda_p-\varpi$: its bounded range constrains where mean <conjunctions> occur relative to <periapsis>. This connects the angular diagnostic with <resonant conjunction geometry> and possible protection from close encounters.