Orbits as numerical experiments

An orbit is the trajectory generated by gravity from an initial position and velocity. It is not an ellipse drawn in advance. A numerical orbit lets us explore initial conditions that have no convenient closed-form solution, but it also introduces errors that can imitate precession, spiraling, escape, or collision.

The ideal central-force problem

For a small body orbiting a much larger fixed spherical body, relative position \(\mathbf r\) obeys

\[\ddot{\mathbf r}=-\mu\frac{\mathbf r}{|\mathbf r|^3}\]

where \(\mu=GM\). The initial position and velocity determine the future path. They must use compatible units and the same reference frame.

If both bodies have important mass, the same equation describes their relative separation when

\[\mu=G(m_1+m_2)\]

and both bodies orbit their common center of mass. Holding one body fixed is accurate only when its motion is negligible for the question.

The ideal model assumes point or spherical bodies, no atmosphere, no thrust, no other gravitation, and Newtonian weak-field dynamics. A numerical experiment should state which of these assumptions it uses.

Circular orbit as a reference case

For a circular orbit of radius \(r\), gravity supplies exactly the required centripetal acceleration:

\[\frac{v_{\mathrm c}^2}{r}=\frac{\mu}{r^2}\]

Therefore

\[v_{\mathrm c}=\sqrt{\frac{\mu}{r}}\]

The velocity must also be perpendicular to the radius. The formula supplies a speed, not its direction. Reversing the tangential direction reverses angular momentum but produces the same ideal circular geometry.

The orbital period is

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

This is a strong numerical fixture: after one analytic period, a sufficiently accurate calculation should return close to its starting state and should not show systematic energy or angular-momentum drift.

Energy classifies ideal two-body motion

Specific mechanical energy means energy per unit orbiting mass:

\[\varepsilon=\frac{|\mathbf v|^2}{2}-\frac{\mu}{r}\]

With potential energy set to zero at infinity, negative specific energy describes bound elliptic motion, zero describes the parabolic escape boundary, and positive describes an unbound hyperbola. A circle is the special ellipse with constant radius.

For a bound Kepler orbit with semimajor axis \(a\),

\[\varepsilon=-\frac{\mu}{2a}\]

and the period generalizes to

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

Energy classification is valid for the stated ideal model. A softened force, drag, thrust, or time-varying multi-body field changes the conservation statement.

Angular momentum and areal speed

Specific angular momentum is

\[\mathbf h=\mathbf r\times\mathbf v\]

The central force has zero torque about the center, so \(\mathbf h\) remains constant. The motion stays in the plane perpendicular to this vector. Its magnitude also gives the swept area per time:

\[\frac{dA}{dt}=\frac{|\mathbf h|}{2}\]

This is Kepler's equal-area law. Sampling equal time intervals and comparing swept triangle areas is a useful geometric check, though direct angular-momentum evaluation is usually more accurate.

Advance the differential equation

Velocity Verlet is a useful explicit method for position-dependent acceleration. It is time-reversible for this conservative problem and generally controls long-run energy behavior better than ordinary forward Euler. No finite-step method is exact.

import numpy as np


# Scaled units: gravitational parameter mu = 1, initial radius = 1.
def gravity(position):
    return -position / np.linalg.norm(position) ** 3


def advance(position, velocity, dt):
    old_a = gravity(position)
    new_position = position + velocity * dt + 0.5 * old_a * dt**2
    new_velocity = velocity + 0.5 * (old_a + gravity(new_position)) * dt
    return new_position, new_velocity


print(advance(np.array([1.0, 0.0]), np.array([0.0, 0.8]), 0.01))

Ordinary forward Euler updates position and velocity using only the old state. In an orbit it commonly produces systematic energy growth and an outward spiral. That spiral is a numerical artifact, not a prediction of Newtonian gravity.

Adaptive high-order solvers can be excellent for event accuracy and close approaches, while symplectic methods are valuable for long conservative integrations. Solver choice depends on the question. In all cases, repeat the calculation at smaller tolerances or time steps.

A complete experiment loop

Store diagnostics at fixed physical output times so runs with different internal steps can be compared directly. Rendering frequency must not determine the physics step.

position, velocity = np.array([1.0, 0.0]), np.array([0.0, 0.8])
history = []
dt = 0.005
for step in range(2001):
    radius = np.linalg.norm(position)
    energy = 0.5 * np.dot(velocity, velocity) - 1 / radius
    radial_speed = np.dot(position, velocity) / radius
    history.append([step * dt, radius, energy, radial_speed])
    position, velocity = advance(position, velocity, dt)
history = np.array(history)
print("Energy range:", np.ptp(history[:, 2]))

All acceleration evaluations within one step must use the state required by the chosen integrator. Mixing components updated at different times defines a different, often lower-quality method.

Predicting changes in initial speed

Begin at a circular position with tangential circular speed. If the tangential speed is reduced but remains nonzero, gravity is initially stronger than the centripetal requirement for that speed, and the body moves inward toward a future periapsis. The starting point is an apoapsis of the new ellipse.

If tangential speed is increased above circular but remains below escape speed, the body moves outward first and the starting point is a periapsis. Escape speed at radius \(r\) is

\[v_{\mathrm{esc}}=\sqrt{\frac{2\mu}{r}}=\sqrt{2}\,v_{\mathrm c}\]

Exactly reaching this speed is the ideal parabolic boundary only when the velocity and model are interpreted through total energy. Adding a radial velocity component changes energy and the location of turning points.

Detecting orbital events

Radial velocity is

\[v_r=\frac{\mathbf r\cdot\mathbf v}{r}\]

Periapsis and apoapsis occur where this quantity changes sign. A coarse output schedule will not normally land exactly on the event. Detect a sign-changing bracket during integration and interpolate or solve within that bracket using a documented event rule.

for previous, current in zip(history[:-1], history[1:]):
    if previous[3] < 0 <= current[3]:
        fraction = -previous[3] / (current[3] - previous[3])
        event_time = previous[0] + fraction * (current[0] - previous[0])
        print(f"Estimated periapsis time: {event_time:.4f} scaled time units")

Event times and radii should converge when the step is reduced. Trail vertices alone can miss the true extremes.

Singularities, surfaces, and collisions

The point-mass acceleration diverges at zero radius. A planet has a surface and an extended mass distribution, so a practical model may terminate when radius reaches the body radius. A star or compact body needs its own collision or interior model. Softening the force avoids divergence but changes orbital precession and close-approach dynamics. Record softening as physics input, not a hidden numerical safeguard.

Near periapsis an eccentric orbit moves fastest and acceleration changes most rapidly. A fixed step adequate near apoapsis may be inadequate there. Monitor the smallest radius and test with a smaller step or adaptive policy.

Evidence that the orbit is credible

Test the acceleration separately before integrating. At the initial circular state, verify perpendicular velocity, circular speed, inward acceleration, and analytic period. During the run, track relative changes in energy and angular momentum. Bounded oscillation of numerical energy may be acceptable for a symplectic method; systematic drift that shrinks under refinement is still numerical error and must be reported.

Compare runs at the same physical times, not at the same step index. Refine time step and event tolerance independently. Check radius extrema, phase, period, energy, and angular momentum because agreement in one observable does not guarantee agreement in the others.

A convincing numerical orbit is therefore a reproducible initial-value calculation with declared units, frame, force law, solver, singularity policy, and convergence evidence. The picture is useful, but the saved physical diagnostics carry the scientific claim.

A worked Python notebook

Choose initial velocities for circular orbits in units where the gravitational parameter is one. The question for this calculation is precise: how does circular speed change when we change radius, 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.

A numerical orbit must also maintain the direction of velocity perpendicular to the radius. 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 0.5 to 3 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 radius, expressed in scaled length units. The output is circular speed, expressed in scaled speed 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.

\[f(q) = \sqrt{\frac{1}{q}}\]

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 1 scaled speed 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("sqrt(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 circular speed:", reference_output, "scaled speed 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 1.41421 scaled speed units. The midpoint, at 1.75 scaled length units, gives 0.755929 scaled speed units. The upper endpoint gives 0.57735 scaled speed units. Read these together with the assumptions: they describe different choices of radius 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([0.5, 1.75, 3.0])
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 speed 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 radius; the vertical axis shows circular speed. 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(0.5, 3.0, 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("Radius (scaled length units)")
ax.set_ylabel("Circular speed (scaled speed 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 circular speed 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 -0.5 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

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

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 = 1.0125
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 speed units")
print("Full expression:", full_prediction, "scaled speed units")
print(
    "Approximation difference:",
    full_prediction - linear_prediction,
    "scaled speed units",
)

Here the changed input is 1.0125 scaled length units. The tangent estimate is 0.99375 scaled speed units, while the full expression gives 0.993808 scaled speed 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 = 2.5
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 radius inside the stated interval, predict whether circular speed 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: A numerical orbit must also maintain the direction of velocity perpendicular to the radius. 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 speed 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(3.0) - evaluate(0.5))
for points in [101, 201, 401]:
    grid = np.linspace(0.5, 3.0, 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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