DDA 2015 – The Öpik Approximation and Giant Planet Shielding of the Inner Solar System

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

William Newman (UCLA)

Abstract

Öpik (1976) proposed that close-range gravitational interactions between planetesimal material and planets could be approximated by a two-step integration scheme: (1) while the planetesimal was outside the gravitational sphere of influence of the planet, its orbit would be described by a heliocentric Keplerian orbit; and (2) once its orbit entered the sphere of influence of the planet, its trajectory would then become a planetocentric Keplerian orbit until it exited the sphere of influence and resumed a heliocentric path. This approximation, however, was also limited by the requirement noted by Öpik that the perihelion or aphelion distance of the planetesimal differ from the orbital distance of the planet from the sun. This approximation proved to be a useful tool during early solar system dynamical investigations but this process was often employed as a numerical integration method without checking Öpik’s requirements, as well as establishing whether the orbital passage through the sphere of influence was sufficiently accurate. Öpik’s scheme was used to establish many features of solar system evolution, including the commonly-held belief that the giant planets serve as a shield preventing substantial numbers of planetesimals from entering the inner solar system. Wetherill (1994) in a pioneering work that exploited the Öpik approximation as an integration scheme estimated that present-day Jupiter could prevent 99.9% of planetesimals from entering the inner solar system. Here, we employ high precision first principles calculations of the orbits of swarms of planetesimals emerging from the Jupiter-Saturn, Saturn-Uranus, and Uranus- Neptune zones and have shown (1) the conditions necessary for Öpik’s approximation to be valid fail for a substantial fraction of the planetesimal population during their lifetimes, and (2) approximately 44% of the planetesimal swarm originating in the Jupiter-Saturn zone alone are injected into the inner Solar System while 18% ultimately become Earth-crossers.

Notes

  • Does Jupiter shield the inner solar system?
    • Impact history
    • Öpik:
      • novel scheme for solar system integrations
      • but identified a useful criterion for valid (numerical) results
      • exploits near-Keplerian orbits of inner SS
    • Öpik’s method:
      • Keplerian time step
    • Criterion: aphelion or perihelion must be different from mean distance. [um…duh]
    • Öpik’s result: 99.95% of outer solar system planetesimals could not have entered inner solar system!
  • Newman: let’s check, using an extremelyaccurate numerical integrator.
    • counted fraction of particles in each planet-planet outer zone fail Öpik’s criterion
    • found: ~20% get through
    • thus: Öpik’s criterion gets violated a lot

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