Capture of Quasi-Satellites by a Protoplanet
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The motions of a protoplanet (red) and a planetesimal (yellow) are shown here in two different reference frames. On the left is a sun-centered reference frame rotating with the mean motion of the protoplanet. The protoplanet and planetesimal are on eccentric orbits and move faster than the mean motion when nearer to the sun and slower than the mean motion when farther from the sun. This causes them to trace out small ellipses. On the right is a planet-centered frame with fixed inertial (non-rotating) orientation of the axes. The circle around the protoplanet denotes the Hill radius, where the planet's gravity and solar gravity are approximately equal.
The simulation that produced the orbital configuration shown here included gas drag from a primordial solar nebula. This gas drag allowed the planetesimal to become trapped in 1:1 mean-motion resonance with the protoplanet --- the planet and the planetesimal both orbit the sun with the same period. Traditional objects trapped in 1:1 resonance include Jupiter's Trojan asteroids, which exist in two swarms centered on the L4 and L5 Lagrangian equilibrium points, indicated in the left animation. The resonant planetesimal shown here is trapped as a so-called quasi-satellite.
All true satellites orbit their respective planets well within their Hill radii. However, quasi-satellites stay well outside the Hill radius and do not orbit the planet like true satellites. Instead, because solar nebula gas drag allows for a permanent 1:1 resonance, this quasi-satellite always appears in the same quadrant of the sky as seen from the protoplanet. The quasi-satellite is in 1:1 resonance with the protoplanet, so it completes one circuit of the looping path in the same time it takes the planet to complete one orbit around the sun. Use the two syncronized animations to compare positions of the protoplanet and quasi-satellite.
The next slides demonstrate what happens if equilibrium between solar nebula gas drag and the resonance is altered, either by gas dissipation or growth of the protoplanet.
Growth of Protoplanet
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In these figures the mass of the protoplanet (red) increases by about 50% over the course of about 100,000 years, from 2 to 2.9 Earth-masses. This steady growth disrupts the equilibrium between solar nebula gas drag and resonant perturbations. The result is that the quasi-satellite (yellow) evolves closer to the protoplanet. The animation on the left shows the last few thousand years of the evolution. Eventually the 1:1 resonance is broken and the quasi-satellite is briefly captured as a true satellite. During this time the satellite has two close-encounters with the protoplanet, one in the retrograde direction and the other a deep prograde encounter. The close-up on the right shows the path of the planetesimal during these encounters.
For other interesting examples showing capture of quasi-satellites, go to the next slide.
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These two examples show the result of slowly dissipating the solar nebula gas density. This dissipation disrupts the equilibrium between solar nebula gas drag and resonant perturbations. The quasi-satellites evolve closer to the protoplanet, entering the frame in the lower left quadrant. Eventually the 1:1 resonance is broken. On the left, the quasi-satellite then impacts the protoplanet on the first encounter inside the Hill radius. On the right, the quasi-satellite is captured as a true satellite orbiting almost entirely within the Hill radius. Eventually this satellite also impacts the protoplanet.