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Local Gravitational Instabilities Modeled as a Dynamical System

Presentation #406.03 in the session “Rings, Disks, and Migration”.

Published onJun 01, 2021
Local Gravitational Instabilities Modeled as a Dynamical System

The traditional theory for describing local gravitational instabilities in unperturbed particle disks is formulated in Fourier space as a Volterra integral equation for the perturbed surface density, with the ratio of the radial and azimuthal wave numbers acting as the independent variable (Julian and Toomre 1966; Fuchs 2001). This formulation is inconvenient for describing gravitational instabilities in a perturbed disk, such as a density wave or satellite wake, where the background dynamics is time-dependent. A more useful and versatile formulation can be obtained by first transforming to a locally sheared coordinate system before transforming to Fourier space. If we further focus on the radial and azimuthal displacements from the guiding center of the particle orbits, then we find a linear system of two differential equations in time with time-dependent coefficients. In this formulation, it is rather straightforward to add a time-dependent background state that can represent a density wave or satellite wake. This dynamical system is readily solved using standard numerical algorithms in Matlab. The perturbed surface density is recovered by calculating the Jacobian determinant of the transformation from guiding center variables to local variables and by performing a numerical inverse Fourier transform in the two spatial variables. A primary objective of this work is to determine the how the size and pitch angle of local gravitational instabilities vary with the amplitude and phase of the background density wave. The goal is to understand the structures observed by the Cassini mission in strongly perturbed regions of Saturn’s rings.

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