A Solar System Survey of Forced Librations

Introduction: Physical librations in longitude are forced periodic variations of a body’s rotation rate. If the torque producing the librations can be calculated, then observations of the phase and amplitude of librations can provide information on mass distribution, and effective strength of the body. In the near future prospects for observing physical librations look quite promising. Radio interferometric observations of Venus and Mercury [1] may yield sufficiently accurate rotational observations that librations there may be visible. Range measurements from Earth to networks of landed instrument packages on Mars [2] are likely to yield librational data there as well. We compute expected libration amplitudes from physical and orbital parameters of a set of planets and satellites partially motivated by a desire to identify candidates for future observations. Solar system bodies occupy one of three general rotation states: non-resonant states, resonant states, and the synchronous resonant state. Analytical treatments of forced librations were initially motivated by the Moon. Lunar librations were predicted by Newton, first detected telescopically by Bessel [3], and definitively resolved through lunar laser ranging [4] which has led to quite thorough analysis of of librations for the synchronous case[5, 6, 7, 8]. The synchronous resonant state is commonly observed among satellites. The only known body to exist in a non-synchronous resonance is Mercury [9] which exists in a 3:2 resonance, completing three rotations for every two revolutions about the sun. The analysis of Goldreich and Peale [10] has lead to improved understanding of the general case of half integer resonance states. The dynamics of forced librations in non-resonant rotators has received less attention. While there are few cases in which non-resonant forced librations have been observed, Earth is an important exception, and current observing techniques may have the capacity to detect them on Venus. A comprehensive observing program spanning a range of solar system bodies can address an array of geophysical issues involving interior mass distribution of planets, satellites, and asteroids. Calculations of expected librations can supply amplitude estimates helpful in identifying the likelihood of detecting librations observationally. We present a survey of expected libration amplitudes for a subset of solar system bodies, identifying those bodies with amplitudes likely to be detectable, and commenting on spin state, and radial structure implications. Theory: The equation of librational motion in the absence of tidal torques can be written as