Reconstruct an orbit from two turning radii
At an apsis of a noncircular orbit, radial velocity vanishes and speed is tangential. With specific angular momentum h, that speed is h/r. The same total energy must therefore equal h²/(2rp²) − μ/rp at pericenter and h²/(2ra²) − μ/ra at apocenter.
Learning outcome: Derive energy and angular momentum from both apsides, check both energy substitutions, and identify the circular limiting case.
Starting recap
Integrate mass moments about named axes, derive torque and angular-momentum constraints, and test orbital energy, angular and numerical consistency against independent evidence.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
For rp = ra = 3 m and μ = 1, the orbit is circular: eccentricity zero, speed 1/sqrt(3) m/s, energy −1/6 m²/s² and period 2π sqrt(27) s.
Open the experimentPredict
For rp = 1 and ra = 3 in the supplied μ = 1 orbit, pericenter/apocenter speed ratio is:
Experiment
Derive energy and angular momentum from both apsides, check both energy substitutions, and identify the circular limiting case.
Reconstruct an ellipse from two turning radii
Worked teaching model · synthetic values · Ideal test-particle Newtonian orbit with fixed mu=1. Selected radii satisfy rp<=ra, including a circular equality. Radii are from the force center, not ellipse center. No real mission design or body surfaces.
Reset keeps your prediction, captures and practice history.
Ellipse with force center at origin
Spatial axes use the same pixels per physical unit, preserving geometry. 1: geometric orbit, not equally timed markers (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Ellipse with force center at origin
geometric orbit, not equally timed markers
| east coordinate (m) | north coordinate (m) |
|---|---|
| 1 | 0 |
| 0.9938347 | 0.1358951 |
| 0.9753767 | 0.2709524 |
| 0.9447398 | 0.4043392 |
| 0.902113 | 0.5352331 |
| 0.8477591 | 0.6628271 |
| 0.782013 | 0.7863346 |
| 0.7052803 | 0.9049941 |
| 0.618034 | 1.018074 |
| 0.5208119 | 1.124877 |
| 0.4142136 | 1.224745 |
| 0.2988961 | 1.317062 |
| 0.1755705 | 1.401259 |
| 0.04499713 | 1.476816 |
| -0.092019 | 1.543269 |
| -0.2346331 | 1.600206 |
| -0.381966 | 1.647278 |
| -0.5331093 | 1.684194 |
| -0.6871311 | 1.710726 |
| -0.8430818 | 1.726711 |
| -1 | 1.732051 |
| -1.156918 | 1.726711 |
| -1.312869 | 1.710726 |
| -1.466891 | 1.684194 |
| -1.618034 | 1.647278 |
| -1.765367 | 1.600206 |
| -1.907981 | 1.543269 |
| -2.044997 | 1.476816 |
| -2.175571 | 1.401259 |
| -2.298896 | 1.317062 |
| -2.414214 | 1.224745 |
| -2.520812 | 1.124877 |
| -2.618034 | 1.018074 |
| -2.70528 | 0.9049941 |
| -2.782013 | 0.7863346 |
| -2.847759 | 0.6628271 |
| -2.902113 | 0.5352331 |
| -2.94474 | 0.4043392 |
| -2.975377 | 0.2709524 |
| -2.993835 | 0.1358951 |
| -3 | 2.12115e-16 |
| -2.993835 | -0.1358951 |
| -2.975377 | -0.2709524 |
| -2.94474 | -0.4043392 |
| -2.902113 | -0.5352331 |
| -2.847759 | -0.6628271 |
| -2.782013 | -0.7863346 |
| -2.70528 | -0.9049941 |
| -2.618034 | -1.018074 |
| -2.520812 | -1.124877 |
| -2.414214 | -1.224745 |
| -2.298896 | -1.317062 |
| -2.175571 | -1.401259 |
| -2.044997 | -1.476816 |
| -1.907981 | -1.543269 |
| -1.765367 | -1.600206 |
| -1.618034 | -1.647278 |
| -1.466891 | -1.684194 |
| -1.312869 | -1.710726 |
| -1.156918 | -1.726711 |
| -1 | -1.732051 |
| -0.8430818 | -1.726711 |
| -0.6871311 | -1.710726 |
| -0.5331093 | -1.684194 |
| -0.381966 | -1.647278 |
| -0.2346331 | -1.600206 |
| -0.092019 | -1.543269 |
| 0.04499713 | -1.476816 |
| 0.1755705 | -1.401259 |
| 0.2988961 | -1.317062 |
| 0.4142136 | -1.224745 |
| 0.5208119 | -1.124877 |
| 0.618034 | -1.018074 |
| 0.7052803 | -0.9049941 |
| 0.782013 | -0.7863346 |
| 0.8477591 | -0.6628271 |
| 0.902113 | -0.5352331 |
| 0.9447398 | -0.4043392 |
| 0.9753767 | -0.2709524 |
| 0.9938347 | -0.1358951 |
| 1 | -4.24230e-16 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Semimajor axis | 2 | m |
| Eccentricity | 0.5 | ratio |
| Specific angular momentum | 1.224745 | m²/s |
| Specific energy | -0.25 | m²/s² |
| Pericenter speed | 1.224745 | m/s |
| Apocenter speed | 0.4082483 | m/s |
| Orbital period | 17.77153 | s |
The same energy and angular momentum must hold at both turning radii.
At apsides radial speed is zero and tangential speed h/r. Equate h²/(2rp²)−1/rp and h²/(2ra²)−1/ra to obtain h²=2rp ra/(rp+ra), with the circular case obtained by continuity. Energy is−1/(rp+ra), semimajor axis is their half sum, and period2pi a^(3/2). The ellipse parameter in the plot is not elapsed time.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The same energy and angular momentum must hold at both turning radii.
At apsides radial speed is zero and tangential speed h/r. Equate h²/(2rp²)−1/rp and h²/(2ra²)−1/ra to obtain h²=2rp ra/(rp+ra), with the circular case obtained by continuity. Energy is−1/(rp+ra), semimajor axis is their half sum, and period2pi a^(3/2). The ellipse parameter in the plot is not elapsed time.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Why can equal spacing around an ellipse fail to represent equal time steps?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.