The rigid body obliquity history of Mars
Authors: Bills Bruce G. 
Published: February 2008

The primary objective of this study is to explore the range of uncertainty in the obliquity history of Mars associated with the present uncertainty in the axial precession rate. The obliquity, or angular separation between the spin axis and the orbit normal, is the most important parameter for determining the seasonal and latitudinal pattern of insolation. Thus significant variations in obliquity are a likely source of major climatic variations on Mars. The present obliquity is well known, and the torques acting to reorient the spin axis of Mars can be readily calculated for time spans of order 107 years into the past (or future). The primary limitation to reconstructing the obliquity history is uncertainty in the mean moment of inertia of Mars, which governs its response to the applied torques. The range of axial precession rates corresponding to recent theoretical estimates of the moment of inertia is 8.29–8.77 arc sec/yr, but even the most recent observational limits are still much broader: 8–12 arc sec/yr. Nominal estimates of the axial precession rate suggest that resonant amplification of a number of small terms in the orbital inclination series will lead to significant variations in the obliquity of Mars, a behavior much different from the Earth. The major variations are on a 105 year time scale, with significant amplitude modulation on a 106 year time scale. Because of this resonant amplification, estimates of the obliquity history depend very sensitively on assumed values for the axial precession rate. Three different analytic techniques are applied to the obliquity problem. Both linear perturbation analysis and direct numerical integration of the governing differential equations can be used to obtain an obliquity time series, once a model value is selected for the moment of inertia. The linear solution, because of its simplicity, provides useful insight into the amplification of near‐resonant forcing, but fails in the case of exact resonance. The secular orbital theory of Laskar, which is complete to fifth order in eccentricity and inclination, has been used to integrate numerically a suite of 501 different obliquity histories, each spanning the 20‐m.y. interval centered on the present. Each of the computed histories corresponds to a different axial precession rate. Another approach, which utilizes the adiabatic invariance of action variables associated with the spin evolution, gives additional insight into the obliquity behavior expected during passage through resonance but does not yield an actual obliquity time series. On approach to resonance, the amplitude of obliquity oscillations increases dramatically, and on passage through resonance, the mean value about which the oscillations occur changes abruptly. Clearly, passage through a resonance involves either a change in the resonant frequency, which is determined by the internal mass distribution of Mars, or a change in the effective forcing frequency, due either to actual changes in the orbital configurations of the perturbing planets or merely to interference effects among closely spaced forcing frequencies. It has previously been hypothesized that the formation of Tharsis and/or core differentiation may have driven the spin axis precession rate through resonance with one of the orbital inclination eigenfrequencies early in the history of Mars, giving rise to very large obliquity oscillations. A major result of the present analysis is the observation that, within the plausible range of present axial precession rates, a very wide range of obliquity histories are possible, including some which involve resonance passages within the relatively recent past. As a result, obliquities as high as 51.4°, or as low as 0.2°, may have occurred within the last 10‐m.y.