Reduced Solar Quadrupole Moment Compensates for Lack of Asteroids in Long-term Solar System Integrations - Richard E. Zeebe; Ilja J. Kocken (The Astronomical Journal)
State-of-the-art long-term solar system integrations include several second order effects such as the Sun’s quadrupole moment J _2 and a contribution from asteroids (plus the Moon and general relativity). We recently showed that including 10 asteroids and a reduced J _2 in our astronomical solutions provides the best match with geologic data to −58 Myr. However, the rationale for the reduced J _2 remained ambiguous and may suggest that parameters for long-term integrations compatible with geologic observations are not fully compatible with our knowledge of the current solar system (specifically J _2 ). Here we show that a reduced J _2 compensates for a diminished asteroid population in long-term solar system integrations, which may appear surprising. We present an analysis and offer a mechanism for the long-term compensating effects of J _2 and asteroid mass in the solar system (not planetary systems in general). Our analysis suggests that “differential effects” on specific secular frequencies involved in resonant terms (i.e., ( g _4  −  g _3 ) and ( s _4  −  s _3 )), are critical in the long term, rather than short-term effects on the orbital elements of individual planetary orbits across the board. Also, our results indicate that if long-term integrations including the full asteroid population were computationally feasible, a J _2 value (within errors) compatible with our current knowledge of the solar system could be used. Attempts to improve the long-term accuracy of astronomical solutions by, e.g., tinkering with initial conditions using current/future astronomical observations are futile unless asteroid deficiencies in the solar system model are addressed.