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The science
Softened gravity
Each particle feels a = -GM r / (r2 + ε2)3/2, the field of a Plummer sphere with softening radius ε. The potential is Φ(r) = -GM / √(r2 + ε2), finite at the centre. Far away it reduces to Newton's -GM/r. The circular speed is vc = √(GM r2 / (r2 + ε2)3/2) and the escape speed is √(-2 Φ).
Conservation and loss
Without drag and turbulence both the energy and the angular momentum of each particle are conserved (the page uses the symplectic leapfrog integrator). Gas drag, a = -γ v, removes energy so orbits shrink and particles fall into the core, as in an accretion flow; turbulence adds random kicks that heat the disc.
Honest simplifications: test particles that do not attract each other, a fixed softened core instead of a point mass, drag proportional to velocity with no gas dynamics or pressure, and a respawn at the outer edge when a particle is captured or ejected.
What to try
- Use Pure orbits and watch the total energy graph stay flat.
- Use Radial infall: particles with almost no swirl plunge through the soft core.
- Raise the drag: the mean radius drops and the capture counter grows.
- Increase softening and compare the potential well depth.