Stability of orbits

An ideal circular orbit is a balance between inward gravity and the turning required by tangential velocity. Small state changes move it into a different orbit rather than necessarily destroying it. Stability is therefore a finite-time, model-specific perturbation question, not a promise that radius stays exactly constant or that two rendered trails stay close in phase.

Think like a programmer

Run a reference state and a paired state with exactly one declared perturbation, using the same integrator, step schedule, frame, force model, output times, and event policy. Compare radius envelope, orbital elements or turning events, specific energy, angular momentum, phase, and invariant residuals. Keep physical perturbation response separate from the difference between each run and its refined/reference numerical solution.

Model checklist

Inputs: Reference orbit state, one-component perturbation and magnitude, force model, frame, integrator/step, duration, output/event times, radius/phase metric, invariant tolerances, and refinement schedule.
State: Paired trajectories, perturbation difference metrics, radius extrema/events, specific energy, angular momentum, phase, invariant residuals, and reference/refined comparisons.
Rule: Advance both initial-value problems under identical numerical policy except for the named perturbation; compare diagnostics at common physical times; repeat at reduced step.
Output: Bounded/escaping/colliding/undetermined finite-time classification, physical-response curve, numerical-error curve, invariant-ledger evidence, and provenance.
Check: Zero perturbation yields identical deterministic traces; small tangential speed increase from circular remains bound in the stated ideal range; phase differences are not mislabeled as energy drift; reduced step improves numerical invariant/residual metrics; drag/thrust/third-body changes are recorded as model changes rather than solver error.

\[v_{circ}=\sqrt{\frac{\mu}{r}},\qquad \varepsilon=\frac{v^2}{2}-\frac{\mu}{r},\qquad h=\mathbf r\times\mathbf v\]

Loading the interactive visual. The lesson text and model remain available while it starts.

For an ideal inverse-square two-body model, a sufficiently small tangential speed increase from circular creates a different bound conic rather than instant escape. Radius then oscillates between turning points while energy and angular momentum remain constant in the analytic model. A numerical spiral can imitate physical inward/outward evolution; distinguish it by refining step and checking invariant residuals and the direction of systematic drift.

Phase is a separate observable. Two correct nearby bound orbits may gradually occupy different angles at the same time even if both preserve energy and angular momentum. Define whether the question is radial boundedness, event timing, element variation, or trajectory separation; no single trail distance answers all of them.

Try this experiment

Prediction: A small speed increase from circular produces an ellipse rather than instant escape.

Choose a speed ratio just above one and predict the first radial event. Run reference and perturbed states to the same final time, recording radius extrema, energy, angular momentum, and phase separately. Repeat with a smaller step. Identify an integrator-created spiral by its invariant drift and lack of convergence, not merely by a path that does not close exactly.

Where this model breaks

Stability depends on the model. Perturbing an ideal two-body system differs from perturbing a satellite with drag, thrust, resonance, oblateness, and third bodies. Finite-time numerical evidence cannot prove asymptotic stability; collisions, escape thresholds, chaotic multi-body regimes, parameter uncertainty, and measurement error need model-specific classifications.

Summary

Test orbital stability with controlled one-component perturbations and multiple diagnostics. Separate radial boundedness, phase response, energy/angular-momentum behavior, and finite-time numerical drift through fixed-time refinement.

Glossary

Perturbation: deliberate small change in initial state.
Invariant: modeled quantity expected to remain constant.
Phase error: timing or angle error along an otherwise similar path.
Radius envelope: observed minimum/maximum radial range over a declared interval.
Finite-time classification: outcome statement limited to stated model and observation duration.

Self-check

What does a slight speed change from circular create in the ideal model?
How do you distinguish numerical drift?
Why track phase separately from energy and angular momentum?
Which quantities and event rules should be compared at fixed physical times?

Sources

V. Szebehely, Theory of Orbits.
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Stability of Orbits, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Gravitation, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Stability of Orbits as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Stability of Orbits into a test

Use controlled orbital perturbations and invariant checks to separate physical stability from solver drift.

Name the inputs and units that the gravitation model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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