Adiabatic Invariants in Planetary Motion

In our simulation of Jupiter's effect on Earth's orbit, if we abruptly increase Jupiter's mass by around a factor of 100, the Earth flies out of the solar system.

What happens if we gradually increase Jupiter's mass? (Say, Jupiter is used as a cosmic garbage dump, and gets heavier over a period of thousands or tens of thousands of years.) Try gradually increasing the Jovian mass; does the Earth seem more stable? (To do this in a fairly controlled way, you can click on the little arrow to the right of the slider. Change the increment by clicking on the vertical bar just above the arrow.)

This started as Earth's orbit with Jupiter massless, and slowly increased until Jupiter was 50,000 Earth masses. Had the increase been sudden, the Earth would have departed at a lower mass.

Changing things adiabatically (slowly) produces very different results from changing them abruptly.

Questions

  1. Adiabatic invariants are quantities which do not change as the parameters in the system change. I'm guessing that there are some adiabatic invariants for this problem. Can you find them (either in the literature, by numerical investigation, or by analytical work)?

Jupiter:

How to Get Jupiter

Jupiter is available for Windows 95, Windows NT, Macintosh, and several Unix platforms (the IBM RS6000, Sun Sparc, Dec Alpha (courtesy Kamal Bhattacharya), Linux, and the PowerPC running AIX4.1). The files are available without charge by anonymous FTP (ftp.lassp.cornell.edu) or via the World Wide Web.
Last modified: May 19, 1996

James P. Sethna, sethna@lassp.cornell.edu.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).