On Apr 27, 2019, at 1:57 AM, Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia: Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter <https://en.wikipedia.org/wiki/Jupiter>'s moons Ganymede <https://en.wikipedia.org/wiki/Ganymede_(moon)>, Europa <https://en.wikipedia.org/wiki/Europa_(moon)> and Io <https://en.wikipedia.org/wiki/Io_(moon)>, and the 2:3 resonance between Pluto <https://en.wikipedia.org/wiki/Pluto> and Neptune <https://en.wikipedia.org/wiki/Neptune>.
In[234]:= MinorPlanetData["Pluto"][EntityProperty["MinorPlanet", "OrbitPeriod"]]
Out[234]= Quantity[247.92065, "JulianYears"]
In[235]:= PlanetData["Neptune"][EntityProperty["Planet", "OrbitPeriod"]]
Out[235]= Quantity[164.79132, "JulianYears"]
In[236]:= ContinuedFraction[%%/%]
Out[236]= {1, 1, 1, 55, 1, 1, 1, 7}
That's a fairly lousy approximation to 2:3. What's going on? —rwg
In the case of the Jovian moons the average orbital frequencies are very precisely modeled as w_G = w_0 + w_p w_E = 2w_0 + w_p w_I = 4w_0 + w_p where w_p << w_0. What’s going on is that after the approximate common period T_0 = 2pi/w_0 the system of three moons has “rigidly” precessed by the small angle T_0 w_p. Why they like to do that is complicated. Something similar may be going on with Neptune/Pluto. -Veit