project six -- the laskar problem

"These laws which thus regulate the eccentricities and inclinations of the planetary orbits, combined with the invariability of the mean distances secure the permanence of the solar system throughout an indefinatie lapse of ages, and offer to us an impressive indication of the Supreme Intelligence."

-Robert Grant, 1852, from A History of Physical Astronomy

The problem of the stability of the solar system has a 400 year history, and has still not been definitively solved.

An interesting development in this long-standing problem was presented by Jaques Laskar (of the Bureau de Longitudes of Paris) in 1996. Laskar performed the following experiment: Using a high-order secular theory for the interactions between the planets, he is able to quickly compute the effects of planet-planet interactions over long time periods. Back in the mid-nineties, he was able to use this method (which is approximate) to integrate the solar system through timespans of billions of years. Laskar took the current solar system ephemeris, and produced four different versions of the solar system which differed only in a shift of the earth in its orbit by less than 1 cm. Because the solar system is chaotic, even tiny differences in the initial condition lead to large divergances in the motion after many Lyapunov times. Using his high-order secular theory, Laskar integrated each of his four solar system realizations for a period of 500 million years. After the integrations were completed, he went through each realization and indentified the point in the integrations in which the planet Mercury achieved its largest eccentricity. He then used this maximum eccentricity point as the basis for the initial conditions of four further integrations, each differing by only a tiny change in their intial conditions. He repeated this process recursively through fourteen cycles, and was able to cause Mercury's orbit to leave the Solar system.

Our goal in this project is to reproduce this calculation using a full integration of the equations of motion rather than a secular theory, and to investigate the consequences of this experiment, both for our own solar system's global stability, and for stability integrations of other planetary systems.


(1) Laskar's review article describing his numerical experiments. .pdf file is here.

(2) John Chambers' mercury.f code. This code is an easy-to-use package for high-speed integration, and provides a significant performance upgrade to the integrator.f routine that we use in class.

(3) An essay that I've written that discusses the details of the history of the problem of the stability of the solar system.