Potential softening and eccentricity dynamics in razor-thin, nearly-Keplerian discs

Area of Science:

• Astrophysics
• Celestial Mechanics

Background:

• Gravitational potential of discs influences orbital dynamics.
• Secular approximation and softened gravity are used to study disc dynamics, avoiding singularities in calculations.

Purpose of the Study:

• To evaluate softening formalisms for accurately reproducing eccentricity dynamics in disc potentials.
• To develop a general framework for computing secular disturbing functions with arbitrary softening prescriptions.
• To analyze the computational demands for accurate numerical simulations of secular disc dynamics.

Main Methods:

• Analysis of orbit-averaged potential and disturbing function using classical Laplace-Lagrange theory.
• Exploration of various softening formalisms in the literature.
• Development of a general framework for secular disturbing function computation.
• Numerical simulations of gravitationally interacting annuli.

Main Results:

• Identification of softening models that converge to expected behaviors in the zero-softening limit.
• Demonstration that accurate eccentricity dynamics requires a large number of annuli (N) for small softening parameters (ς).
• Finding that discs with sharp edges necessitate very small softening parameters (ς ≲ 10⁻³) for accurate dynamics near boundaries.

Conclusions:

• The choice of softening formalism significantly impacts the accuracy of simulated disc dynamics.
• Numerical simulations of secular disc dynamics are computationally intensive, requiring substantial resolution for precision.
• Accurate modeling of phenomena like planetary rings necessitates careful consideration of softening parameters, especially near disc edges.