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Tracking the Minimum Energy Function of Disassociated N-Body Systems

Presentation #103.01 in the session “Advances in Simulations of Exoplanet Evolution”.

Published onJun 01, 2021
Tracking the Minimum Energy Function of Disassociated N-Body Systems

The amended potential / minimum energy function of a system combines the total angular momentum, current mass distribution and potential energy of a system into one function that serves as a sharp lower bound on the system energy for a given level of angular momentum [1]. This function can also be used to find and evaluate relative equilibria of N-body systems of rigid bodies and determine their stability [2]. The fact that this function easily generalizes to the gravitational and surface interactions of rigid body systems makes it a useful tool for probing the stable configurations and possible transitions within rubble pile systems as a function of increasing angular momentum. The function also provides for precise computation of the necessary energy for a rubble pile system to disrupt and lose components [3, 4]. However, once a component or group of components undergo a mutual escape the function no longer serves a useful purpose, and it takes on an extreme, conservative lower bound value that provides no significant insight. Thus, in previous analysis it has not been useful for tracking components of a system after they undergo gravitational disruption.

Here we develop a form of the Jacobi equations for an N-rigid body system that provides a remedy for this situation. We show that if the system is written in Jacobi coordinates, there is a clear way in which the minimum energy function can be decomposed to maintain its utility across an arbitrary partition of a system and its associated disruption. To show this, we prove an inequality that leverages the Jacobi coordinate formulation of the N rigid body problem, and maintains its sharp lower limit behavior for all elements of a system after it has shed components.

The presentation will provide a motivation for the use of the minimum energy function and show what its limitations are for a system that undergoes disruption. Then we will prove the necessary inequality which allows the minimum energy function to be decomposed into multiple contributions, each of which serves as a minimum energy function for a collection of the rigid bodies and, when summed together, act as the overall minimum energy function of the system. Finally, we will show how this decomposition can be used to better understand and track how a disrupting system sheds components and loses energy and angular momentum to the gravitational disruption process.

[1] D.J. Scheeres. 2012. “Minimum Energy Configurations in the N-Body Problem and the Celestial Mechanics of Granular Systems,” Celestial Mechanics and Dynamical Astronomy 113: 291-320.

[2] D.J. Scheeres. 2016. “Relative Equilibria in the Spherical, Finite Density 3-Body Problem,” Journal of Nonlinear Science 26: 1445-1482. DOI 10.1007/s00332-016-9309-6

[3] D.J. Scheeres. 2017. “Constraints on Bounded Motion and Mutual Escape for the Full 3-Body Problem,” Celestial Mechanics and Dynamical Astronomy 128(2-3): 131-148.

[4] D.J. Scheeres. 2020. “Disassociation energies for the finite density N-body problem,” Celestial Mechanics and Dynamical Astronomy 132:4.

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