{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Stability of orbits\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/stability-of-orbits). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "An orbit is stable only relative to a stated model and kind of disturbance. In ideal\n", "two-body gravity, a small change to a circular orbit usually creates a nearby ellipse,\n", "not immediate collision or escape. The radius stays bounded, but the perturbed body can\n", "gradually move out of phase with the reference orbit.\n", "\n", "## What kind of stability?\n", "\n", "Several questions are easily confused:\n", "\n", "- Does a small radial displacement produce a restoring radial tendency?\n", "- Does the orbit remain bound rather than escape or collide?\n", "- Do nearby trajectories remain close at equal times?\n", "- Do the orbital elements remain near their initial values?\n", "- Does a numerical calculation preserve the intended continuous orbit?\n", "\n", "An ideal conservative orbit is not asymptotically stable: without dissipation it does not\n", "settle back to the exact original circular path. A small perturbation generally produces\n", "persistent oscillation or a neighboring orbit. Drag can make altitude decay rather than\n", "restore the original orbit, while active control can deliberately hold one.\n", "\n", "## Circular reference orbit\n", "\n", "For gravitational parameter\n", "$\\mu$ and circular radius\n", "$r_0$, circular speed is\n", "\n", "\n", "$$\n", "v_0=\\sqrt{\\frac{\\mu}{r_0}}\n", "$$\n", "\n", "\n", "and the orbital angular frequency is\n", "\n", "\n", "$$\n", "n=\\sqrt{\\frac{\\mu}{r_0^3}}\n", "$$\n", "\n", "\n", "The initial velocity must be perpendicular to the radius. A speed formula alone does not\n", "specify the orbit orientation or angular-momentum sign.\n", "\n", "Specific energy and angular momentum are\n", "\n", "\n", "$$\n", "\\varepsilon=\\frac{v^2}{2}-\\frac{\\mu}{r}\\qquad\\mathbf h=\\mathbf r\\times\\mathbf v\n", "$$\n", "\n", "\n", "Both are constant in the ideal two-body problem. They determine whether the orbit is\n", "bound and which conic it follows.\n", "\n", "## Effective potential explains radial stability\n", "\n", "Conservation of angular momentum lets the radial motion be described with an effective\n", "potential per unit mass:\n", "\n", "\n", "$$\n", "V_{\\mathrm{eff}}(r)=\\frac{h^2}{2r^2}-\\frac{\\mu}{r}\n", "$$\n", "\n", "\n", "The first term is associated with transverse motion and becomes large at small radius\n", "when angular momentum is nonzero. The second is gravitational potential. A circular orbit\n", "occurs at a stationary point:\n", "\n", "\n", "$$\n", "\\frac{dV_{\\mathrm{eff}}}{dr}=-\\frac{h^2}{r^3}+\\frac{\\mu}{r^2}=0\n", "$$\n", "\n", "\n", "Thus\n", "\n", "\n", "$$\n", "r_0=\\frac{h^2}{\\mu}\n", "$$\n", "\n", "\n", "At that radius, the curvature is\n", "\n", "\n", "$$\n", "\\left.\\frac{d^2V_{\\mathrm{eff}}}{dr^2}\\right|_{r_0}=\\frac{\\mu}{r_0^3}>0\n", "$$\n", "\n", "\n", "The circular orbit is therefore a local minimum of effective potential and is stable to\n", "small radial disturbances within the ideal Newtonian two-body model. Small radial motion\n", "oscillates rather than growing without bound. For the inverse-square Kepler potential,\n", "the small radial oscillation frequency equals the circular angular frequency, consistent\n", "with closed ellipses." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import sympy as sp\n", "\n", "r, mu, h = sp.symbols(\"r mu h\", positive=True)\n", "effective_potential = h**2 / (2 * r**2) - mu / r\n", "circular_radius = h**2 / mu\n", "curvature = sp.simplify(sp.diff(effective_potential, r, 2).subs(r, circular_radius))\n", "print(\"Radial curvature:\", curvature)\n", "assert curvature.is_positive\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "This test holds angular momentum fixed. A perturbation that changes velocity can also\n", "change angular momentum and therefore changes the effective-potential curve itself.\n", "\n", "## Small changes in tangential speed\n", "\n", "Begin at radius $r_0$ with the circular\n", "tangential speed. If speed is raised slightly but remains below escape speed, specific\n", "energy stays negative. The body initially moves outward and follows an ellipse whose\n", "starting point is periapsis. If speed is lowered slightly but remains nonzero, it moves\n", "inward and the starting point is apoapsis.\n", "\n", "Escape speed at the same position is\n", "\n", "\n", "$$\n", "v_{\\mathrm{esc}}=\\sqrt{\\frac{2\\mu}{r_0}}=\\sqrt2\\,v_0\n", "$$\n", "\n", "\n", "The finite gap between circular and escape speed explains why a small speed increase\n", "does not cause escape. A sufficiently large decrease can make the new ellipse intersect\n", "the central body's physical surface even though its mathematical point-mass orbit remains\n", "bound.\n", "\n", "Changing direction while holding speed fixed preserves energy but changes angular\n", "momentum. Adding radial velocity can increase eccentricity and move the turning points.\n", "The perturbation must therefore be described as a vector change, not just “slightly\n", "faster.”\n", "\n", "## Why nearby bound orbits separate in phase\n", "\n", "Kepler's third law gives\n", "\n", "\n", "$$\n", "T=2\\pi\\sqrt{\\frac{a^3}{\\mu}}\n", "$$\n", "\n", "\n", "where $a$ is semimajor axis. A small\n", "energy change changes $a$ and therefore\n", "period. After many revolutions, two individually accurate nearby orbits can be far apart\n", "in angular phase even though each remains bound and preserves its own invariants.\n", "\n", "This secular phase difference is not energy drift and does not by itself mean radial\n", "instability. Report radius envelope and phase separately. If the question is rendezvous,\n", "phase is crucial; if the question is escape, energy sign and long-term radial behavior\n", "are more direct.\n", "\n", "## A controlled paired calculation\n", "\n", "Create a reference state and copy it. Change one position or velocity component in the\n", "copy and record the magnitude and units. Both runs must use identical numerical policy." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "import numpy as np\n", "\n", "# Osculating Kepler elements at radius 1 in units with mu=1.\n", "for tangential_speed in [0.99, 1.0, 1.01]:\n", " energy = 0.5 * tangential_speed**2 - 1\n", " angular_momentum = tangential_speed\n", " eccentricity = np.sqrt(max(0, 1 + 2 * energy * angular_momentum**2))\n", " print(tangential_speed, \"eccentricity:\", eccentricity)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "At every output time save each orbit's radius, radial velocity, specific energy, angular\n", "momentum, and phase. Also save differences between the two runs. The zero-perturbation\n", "control should produce identical deterministic traces under the same execution policy.\n", "\n", "## Numerical stability is a separate question\n", "\n", "A poor integrator can make a stable physical orbit spiral inward or outward. Ordinary\n", "forward Euler typically adds energy to conservative orbital motion. Some dissipative\n", "updates remove energy. Either artifact can be mistaken for real instability.\n", "\n", "To separate them, compare each numerical run with a smaller-step version at matching\n", "physical times. Examine energy and angular-momentum residuals, periapsis and apoapsis,\n", "period, and phase. A physical response should converge as numerical resolution improves.\n", "An artifact generally changes systematically with step or solver.\n", "\n", "Near-circular paths that fail to close exactly are not automatically wrong. Numerical\n", "phase error can rotate the apparent ellipse while energy remains well controlled. Name\n", "the observable and acceptable error rather than using a picture as a pass condition.\n", "\n", "## Perturbations beyond the two-body model\n", "\n", "Real orbital stability depends on omitted influences: atmospheric drag, nonspherical\n", "gravity, third bodies, solar radiation pressure, tides, thrust, and relativistic effects.\n", "Some perturbations cause bounded periodic changes; others produce slow secular drift or\n", "resonance. A stable orbit under one force model may be unsuitable under another.\n", "\n", "In a rotating multi-body frame, equilibrium regions and effective potentials differ from\n", "the two-body result. Multi-body systems can also be chaotic, so long-term state prediction\n", "may be limited even when energy and momentum are accurately conserved.\n", "\n", "Parameter uncertainty belongs in the analysis. Repeat with plausible initial-state and\n", "model variations, not only roundoff-sized perturbations. A finite-time run can establish\n", "behavior only over its tested interval; it cannot prove asymptotic stability.\n", "\n", "## Evidence for a careful stability claim\n", "\n", "State the force model, frame, reference orbit, perturbation vector, duration, metric, and\n", "classification. Verify the analytic circular speed and effective-potential minimum.\n", "Confirm that the perturbed energy remains negative if claiming a nearby bound orbit, and\n", "check whether its periapsis remains outside the body's surface.\n", "\n", "Refine step and event detection independently. Show that invariant errors shrink or\n", "remain within a justified bound, while the measured physical response approaches a\n", "consistent result. Distinguish radial boundedness, collision avoidance, element changes,\n", "and phase separation.\n", "\n", "The ideal Kepler circle is locally radially stable but not an attractor. That precise\n", "statement is more useful than saying simply that “the orbit is stable.”\n", "\n", "## A worked Python notebook\n", "\n", "Hold specific angular momentum fixed at one and inspect the radial effective potential. The question for this calculation is precise: how does **effective potential**\n", "change when we change **radial distance**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "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\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0.6 to 2 scaled length units. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for radial distance, expressed in scaled length units. The output is\n", "effective potential, expressed in scaled specific energy units. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = - \\frac{1}{q} + \\frac{1}{2 q^{2}}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 1 scaled length units, the expression gives -0.5 scaled specific energy units. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"1/(2*q**2)-1/q\", locals={\"q\": q})\n", "reference_input = 1.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"scaled length units\")\n", "print(\n", " \"Predicted effective potential:\", reference_output, \"scaled specific energy units\"\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "-0.277778 scaled specific energy units. The midpoint, at 1.3 scaled length units, gives\n", "-0.473373 scaled specific energy units. The upper endpoint gives -0.375 scaled specific energy units.\n", "Read these together with the assumptions: they describe different choices of radial distance\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.6, 1.3, 2.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(\n", " f\"{input_value:.6g} scaled length units -> \"\n", " f\"{output_value:.6g} scaled specific energy units\"\n", " )\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows radial distance; the vertical axis shows effective potential. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "input_grid = np.linspace(0.6, 2.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Radial distance (scaled length units)\")\n", "ax.set_ylabel(\"Effective potential (scaled specific energy units)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "effective potential change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1 scaled length units is\n", "0 in those ratio units. Locally, the output has no first-order change as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = \\frac{1}{q^{2}} - \\frac{1}{q^{3}}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 1.007\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"scaled specific energy units\")\n", "print(\"Full expression:\", full_prediction, \"scaled specific energy units\")\n", "print(\n", " \"Approximation difference:\",\n", " full_prediction - linear_prediction,\n", " \"scaled specific energy units\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 1.007 scaled length units. The tangent estimate is\n", "-0.5 scaled specific energy units, while the full expression gives\n", "-0.499976 scaled specific energy units. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_span = 1.4\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "radial distance inside the stated interval, predict whether effective potential will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "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\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of scaled specific energy units:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(2.0) - evaluate(0.6))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.6, 2.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 1*, Section 13.3, \"Gravitational Potential Energy and Total Energy\"](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy).\n", "- [OpenStax, *University Physics Volume 1*, Section 13.4, \"Satellite Orbits and Energy\"](https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy).\n", "- [OpenStax, *University Physics Volume 1*, Section 13.5, \"Kepler's Laws of Planetary Motion\"](https://openstax.org/books/university-physics-volume-1/pages/13-5-keplers-laws-of-planetary-motion).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 25, Sections 25.3-25.4](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter25new/).\n", "- [Benjamin Crowell, *Light and Matter*, orbital energy and angular-momentum discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Physics LibreTexts, \"Satellite Orbits and Energy\"](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/13%3A_Gravitation/13.05%3A_Satellite_Orbits_and_Energy)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "stability-of-orbits.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/stability-of-orbits", "source_sha256": "ac41c735662c69ecf2d0f912bf8c4b425dc7bc24fcf1bae4cb844ef0f45f8771" } }, "nbformat": 4, "nbformat_minor": 5 }