A three million year integration of the earth's orbit

Abstract

We have integrated the equations of motion ofthe nine planets and the Earth’s spin axis for 3.05 million years into the past. The equations include the dominant relativistic corrections and corrections for the quadrupole moment of the Earth-Moon system; solar mass loss is neglected but the principal consequence ofthis neglect is an unimportant nonuniformity in the overall timescale ofless than one part per million. The initial conditions are taken from the JPL DE 102 ephemeris. We believe that the heliocentric direction to the Earth and the direction of the Earth’s pole are correct to within about 0.03 radians at the end of the integration; errors in the other planets are larger, because of larger integration errors (for Mercury) or less accurate initial conditions (for the outer planets). All the planetary orbits appear to be regular on this timescale (i.e., there is no evidence of exponential divergence in adjacent trajectories). We have computed the orbital elements of all the planets and the Earth’s spin direction at 0.1 yr intervals and have removed short-term variations using a low-pass filter, in order to isolate the longterm variations (period> 2000 yr). The smoothed elements are available at 500 yr intervals. They can be used to check or to replace the results of secular perturbation theory and as input to geophysical models that test the Milankovich hypothesis that climate variations are caused by changes in the Earth’s orbit.