A comet moves on an orbit of eccentricity about the Sun and is integrated by velocity…
PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Mechanics & Dynamical Systems · Simulating classical dynamical systems
Computational Physics, Ch An orbit that must not drift
Problem
A comet moves on an orbit of eccentricity e=0.75 about the Sun and is integrated by velocity Verlet with a fixed step Δt, in units where GM⊙=4π2AU3yr−2. The energy trace is flat for most of the orbit and shows a spike at each perihelion passage that recovers by aphelion; the depth of this spike, measured as a fraction of ∣E0∣, is known to fall as Δt2. At Δt=2×10−3yr the measured spike depth is 3.6×10−2 of ∣E0∣.
((a)) Use the Δt2 scaling to predict the spike depth at Δt=1×10−3yr and at Δt=5×10−4yr.
((b)) A verification run requires the spike depth to be no more than 1×10−4 of ∣E0∣. Estimate, to one significant figure, the largest step Δt that satisfies this requirement.