softening.hpp File Reference

Plummer softening, defined once for everything that needs it. More...

#include <cmath>
#include "orrery/core/types.hpp"

Go to the source code of this file.

Classes
class	orrery::core::Softening
	The softening length of the Plummer kernel above. More...

Functions
Real	orrery::core::softened_inverse_distance (Real separation_squared, Softening softening) noexcept
	The factor 1 / sqrt(r^2 + eps^2) of the softened potential.
Real	orrery::core::softened_inverse_distance_cubed (Real separation_squared, Softening softening) noexcept
	The factor 1 / (r^2 + eps^2)^(3/2) of the softened acceleration.

Detailed Description

Plummer softening, defined once for everything that needs it.

A gravitational force between point masses diverges as the separation goes to zero. In a simulation that divergence is not physics but a modelling error: a simulation particle represents a great many stars rather than one, so a close approach between two of them is an artefact of the sampling and not an encounter that happened. Left alone it produces an unbounded acceleration, which no fixed timestep can integrate, and the resulting energy error is arbitrarily large.

Plummer softening replaces the point mass with the potential

Phi(r) = -G m / sqrt(r^2 + eps^2)

which is the exact potential of a Plummer sphere of scale radius eps. That is what makes this the softening to choose: it is not a fudge that keeps the denominator away from zero, it is the potential of a real mass distribution, so a softened simulation is an exact simulation of extended masses rather than an approximate one of point masses. It agrees with the point-mass result to better than a part in a thousand beyond about twenty softening lengths and is finite everywhere.

The file exists so that the force kernel and the potential energy diagnostic cannot disagree. Energy conservation is one of the two headline validation results of this project, and if the diagnostic integrated a different potential from the one the kernel differentiated, the measured drift would be a property of the mismatch rather than of the integrator. Both forms below are therefore stated together: the second is the negative gradient of the first, divided by the separation vector, and a test checks that against a numerical derivative. ADR-0008 records why they live here rather than beside the kernel that will use them.

Function Documentation

softened_inverse_distance()

Real orrery::core::softened_inverse_distance ( Real separation_squared, Softening softening )

inlinenodiscardnoexcept

The factor 1 / sqrt(r^2 + eps^2) of the softened potential.

Takes the squared separation because that is what a caller has: computing the separation itself would mean a square root that this function would immediately undo.

softened_inverse_distance_cubed()

Real orrery::core::softened_inverse_distance_cubed ( Real separation_squared, Softening softening )

inlinenodiscardnoexcept

The factor 1 / (r^2 + eps^2)^(3/2) of the softened acceleration.

The gradient of the potential above carries one more power of the softened distance in the denominator, and one factor of the separation vector in the numerator, which the caller supplies. Written as the cube of the reciprocal rather than as a second square root because the two forms must agree in the last bits: an acceleration derived from a slightly different value than the potential energy uses would show up as a drift that no integrator caused.