I've been scratching my head over this for a while now. Seems pretty simple, but I just can't get anywhere! So, this is a problem out of pater & Lissauer's Planetary Sciences book (prob 2.10). Starts off by describing how Neptune was discovered by observing anomalies in Uranus's orbit. Here's the question: Estimate the maximum displacement in the position of Uranus caused by the gravitational effect of Neptune as Uranus catches up and passes this slowly moving planet. Quote answer in km along Uranus's orbital path (and in sec of arc against the sky as observed from Earth. Assumptions: U & N are in ciruclar, coplanear orbits (initially, unperturbed) neglect effects of other planets and affect of Uranus on Neptune. assume that the potentail energy released as Uranus gets closer to neptune increases URanus's semimajor axis and thus slows Uranus down. assume that Uranus's average semimajor axis during the interval under consideration is halfway between its semimajor ais a the beginning and the end of the interval Use the synodic (relative) period of the pair of planets rather than just Uranus's orbital period This is a crude estimate (ie could integrate along U's path, but this can be solved using simple algebra and conservation laws). So in the most basic statement I guess it would be: what is delta r for Uranus given a stationary Neptune (what is the change in distance between where it should be with no Neptune, and what it actually is with Neptune tugging on it and expanding it's orbit in the vicinity of Neptune). ANY help, leads, hints etc would be greatly appreciated!