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Stability of orbits

An orbit is stable only relative to a stated model and kind of disturbance. In ideal two-body gravity, a small change to a circular orbit usually creates a nearby ellipse, not immediate collision or escape. The radius stays bounded, but the perturbed body can gradually move out of phase with the reference orbit.

What kind of stability?

Several questions are easily confused:

An ideal conservative orbit is not asymptotically stable: without dissipation it does not settle back to the exact original circular path. A small perturbation generally produces persistent oscillation or a neighboring orbit. Drag can make altitude decay rather than restore the original orbit, while active control can deliberately hold one.

Circular reference orbit

For gravitational parameter \(\mu\) and circular radius \(r_0\), circular speed is

\[v_0=\sqrt{\frac{\mu}{r_0}}\]

and the orbital angular frequency is

\[n=\sqrt{\frac{\mu}{r_0^3}}\]

The initial velocity must be perpendicular to the radius. A speed formula alone does not specify the orbit orientation or angular-momentum sign.

Specific energy and angular momentum are

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

Both are constant in the ideal two-body problem. They determine whether the orbit is bound and which conic it follows.

Effective potential explains radial stability

Conservation of angular momentum lets the radial motion be described with an effective potential per unit mass:

\[V_{\mathrm{eff}}(r)=\frac{h^2}{2r^2}-\frac{\mu}{r}\]

The first term is associated with transverse motion and becomes large at small radius when angular momentum is nonzero. The second is gravitational potential. A circular orbit occurs at a stationary point:

\[\frac{dV_{\mathrm{eff}}}{dr}=-\frac{h^2}{r^3}+\frac{\mu}{r^2}=0\]

Thus

\[r_0=\frac{h^2}{\mu}\]

At that radius, the curvature is

\[\left.\frac{d^2V_{\mathrm{eff}}}{dr^2}\right|_{r_0}=\frac{\mu}{r_0^3}>0\]

The circular orbit is therefore a local minimum of effective potential and is stable to small radial disturbances within the ideal Newtonian two-body model. Small radial motion oscillates rather than growing without bound. For the inverse-square Kepler potential, the small radial oscillation frequency equals the circular angular frequency, consistent with closed ellipses.

import sympy as sp

r, mu, h = sp.symbols("r mu h", positive=True)
effective_potential = h**2 / (2 * r**2) - mu / r
circular_radius = h**2 / mu
curvature = sp.simplify(sp.diff(effective_potential, r, 2).subs(r, circular_radius))
print("Radial curvature:", curvature)
assert curvature.is_positive

This test holds angular momentum fixed. A perturbation that changes velocity can also change angular momentum and therefore changes the effective-potential curve itself.

Small changes in tangential speed

Begin at radius \(r_0\) with the circular tangential speed. If speed is raised slightly but remains below escape speed, specific energy stays negative. The body initially moves outward and follows an ellipse whose starting point is periapsis. If speed is lowered slightly but remains nonzero, it moves inward and the starting point is apoapsis.

Escape speed at the same position is

\[v_{\mathrm{esc}}=\sqrt{\frac{2\mu}{r_0}}=\sqrt2\,v_0\]

The finite gap between circular and escape speed explains why a small speed increase does not cause escape. A sufficiently large decrease can make the new ellipse intersect the central body's physical surface even though its mathematical point-mass orbit remains bound.

Changing direction while holding speed fixed preserves energy but changes angular momentum. Adding radial velocity can increase eccentricity and move the turning points. The perturbation must therefore be described as a vector change, not just “slightly faster.”

Why nearby bound orbits separate in phase

Kepler's third law gives

\[T=2\pi\sqrt{\frac{a^3}{\mu}}\]

where \(a\) is semimajor axis. A small energy change changes \(a\) and therefore period. After many revolutions, two individually accurate nearby orbits can be far apart in angular phase even though each remains bound and preserves its own invariants.

This secular phase difference is not energy drift and does not by itself mean radial instability. Report radius envelope and phase separately. If the question is rendezvous, phase is crucial; if the question is escape, energy sign and long-term radial behavior are more direct.

A controlled paired calculation

Create a reference state and copy it. Change one position or velocity component in the copy and record the magnitude and units. Both runs must use identical numerical policy.

import numpy as np

# Osculating Kepler elements at radius 1 in units with mu=1.
for tangential_speed in [0.99, 1.0, 1.01]:
    energy = 0.5 * tangential_speed**2 - 1
    angular_momentum = tangential_speed
    eccentricity = np.sqrt(max(0, 1 + 2 * energy * angular_momentum**2))
    print(tangential_speed, "eccentricity:", eccentricity)

At every output time save each orbit's radius, radial velocity, specific energy, angular momentum, and phase. Also save differences between the two runs. The zero-perturbation control should produce identical deterministic traces under the same execution policy.

Numerical stability is a separate question

A poor integrator can make a stable physical orbit spiral inward or outward. Ordinary forward Euler typically adds energy to conservative orbital motion. Some dissipative updates remove energy. Either artifact can be mistaken for real instability.

To separate them, compare each numerical run with a smaller-step version at matching physical times. Examine energy and angular-momentum residuals, periapsis and apoapsis, period, and phase. A physical response should converge as numerical resolution improves. An artifact generally changes systematically with step or solver.

Near-circular paths that fail to close exactly are not automatically wrong. Numerical phase error can rotate the apparent ellipse while energy remains well controlled. Name the observable and acceptable error rather than using a picture as a pass condition.

Perturbations beyond the two-body model

Real orbital stability depends on omitted influences: atmospheric drag, nonspherical gravity, third bodies, solar radiation pressure, tides, thrust, and relativistic effects. Some perturbations cause bounded periodic changes; others produce slow secular drift or resonance. A stable orbit under one force model may be unsuitable under another.

In a rotating multi-body frame, equilibrium regions and effective potentials differ from the two-body result. Multi-body systems can also be chaotic, so long-term state prediction may be limited even when energy and momentum are accurately conserved.

Parameter uncertainty belongs in the analysis. Repeat with plausible initial-state and model variations, not only roundoff-sized perturbations. A finite-time run can establish behavior only over its tested interval; it cannot prove asymptotic stability.

Evidence for a careful stability claim

State the force model, frame, reference orbit, perturbation vector, duration, metric, and classification. Verify the analytic circular speed and effective-potential minimum. Confirm that the perturbed energy remains negative if claiming a nearby bound orbit, and check whether its periapsis remains outside the body's surface.

Refine step and event detection independently. Show that invariant errors shrink or remain within a justified bound, while the measured physical response approaches a consistent result. Distinguish radial boundedness, collision avoidance, element changes, and phase separation.

The ideal Kepler circle is locally radially stable but not an attractor. That precise statement is more useful than saying simply that “the orbit is stable.”

A worked Python notebook

Hold specific angular momentum fixed at one and inspect the radial effective potential. The question for this calculation is precise: how does effective potential change when we change radial distance, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

This reduction applies to a central inverse-square field with gravitational parameter one. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.6 to 2 scaled length units. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for radial distance, expressed in scaled length units. The output is effective potential, expressed in scaled specific energy units. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = - \frac{1}{q} + \frac{1}{2 q^{2}}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1 scaled length units, the expression gives -0.5 scaled specific energy units. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("1/(2*q**2)-1/q", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "scaled length units")
print(
    "Predicted effective potential:", reference_output, "scaled specific energy units"
)

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives -0.277778 scaled specific energy units. The midpoint, at 1.3 scaled length units, gives -0.473373 scaled specific energy units. The upper endpoint gives -0.375 scaled specific energy units. Read these together with the assumptions: they describe different choices of radial distance within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.6, 1.3, 2.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(
        f"{input_value:.6g} scaled length units -> "
        f"{output_value:.6g} scaled specific energy units"
    )
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows radial distance; the vertical axis shows effective potential. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.6, 2.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Radial distance (scaled length units)")
ax.set_ylabel("Effective potential (scaled specific energy units)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does effective potential change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 scaled length units is 0 in those ratio units. Locally, the output has no first-order change as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = \frac{1}{q^{2}} - \frac{1}{q^{3}}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.007
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "scaled specific energy units")
print("Full expression:", full_prediction, "scaled specific energy units")
print(
    "Approximation difference:",
    full_prediction - linear_prediction,
    "scaled specific energy units",
)

Here the changed input is 1.007 scaled length units. The tangent estimate is -0.5 scaled specific energy units, while the full expression gives -0.499976 scaled specific energy units. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 1.4
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of radial distance inside the stated interval, predict whether effective potential will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: This reduction applies to a central inverse-square field with gravitational parameter one. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of scaled specific energy units: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(2.0) - evaluate(0.6))
for points in [101, 201, 401]:
    grid = np.linspace(0.6, 2.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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