Propagate orbital observations without hiding a clock gain
For a circular test-particle orbit, gravitational acceleration μ/r² equals v²/r and v = 2πr/T. Combining them gives μ = 4π²r³/T². An inferred parameter therefore depends strongly on both radius and the clock used for period.
Learning outcome: Derive the circular inference, evaluate its monotone endpoint bounds and compare a repeated shared-clock record with genuinely independent calibration.
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.
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Try
With μtrue = 1 and an exact gain of 1, the nominal inference returns 1. With gain 1.1 it returns 1/1.21 ≈ 0.826446 m³/s², even if repeated indicated periods agree perfectly.
Open the experimentPredict
For μ = 4π²r³/T², which interval corner gives its lower bound?
Experiment
Derive the circular inference, evaluate its monotone endpoint bounds and compare a repeated shared-clock record with genuinely independent calibration.
Propagate radius and clock assumptions through orbital inference
Worked teaching model · synthetic values · Supplied circular radius2 m and true period2pi sqrt(8/mu). Readout gain multiplies period. Inferred mu uses reported radius2±.02 m and indicated period±.1 s as deterministic observation bounds. They do not include an unknown systematic gain unless explicitly corrected.
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Parameter inferred under hypothesized clock correction
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: nominal parameter times gain squared (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Parameter inferred under hypothesized clock correction
nominal parameter times gain squared
| assumed clock gain | inferred gravitational parameter (m³/s²) |
|---|---|
| 0.9 | 0.7346939 |
| 0.9025 | 0.7387812 |
| 0.905 | 0.7428798 |
| 0.9075 | 0.7469898 |
| 0.91 | 0.7511111 |
| 0.9125 | 0.7552438 |
| 0.915 | 0.7593878 |
| 0.9175 | 0.7635431 |
| 0.92 | 0.7677098 |
| 0.9225 | 0.7718878 |
| 0.925 | 0.7760771 |
| 0.9275 | 0.7802778 |
| 0.93 | 0.7844898 |
| 0.9325 | 0.7887132 |
| 0.935 | 0.7929478 |
| 0.9375 | 0.7971939 |
| 0.94 | 0.8014512 |
| 0.9425 | 0.80572 |
| 0.945 | 0.81 |
| 0.9475 | 0.8142914 |
| 0.95 | 0.8185941 |
| 0.9525 | 0.8229082 |
| 0.955 | 0.8272336 |
| 0.9575 | 0.8315703 |
| 0.96 | 0.8359184 |
| 0.9625 | 0.8402778 |
| 0.965 | 0.8446485 |
| 0.9675 | 0.8490306 |
| 0.97 | 0.853424 |
| 0.9725 | 0.8578288 |
| 0.975 | 0.8622449 |
| 0.9775 | 0.8666723 |
| 0.98 | 0.8711111 |
| 0.9825 | 0.8755612 |
| 0.985 | 0.8800227 |
| 0.9875 | 0.8844955 |
| 0.99 | 0.8889796 |
| 0.9925 | 0.8934751 |
| 0.995 | 0.8979819 |
| 0.9975 | 0.9025 |
| 1 | 0.9070295 |
| 1.0025 | 0.9115703 |
| 1.005 | 0.9161224 |
| 1.0075 | 0.9206859 |
| 1.01 | 0.9252608 |
| 1.0125 | 0.9298469 |
| 1.015 | 0.9344444 |
| 1.0175 | 0.9390533 |
| 1.02 | 0.9436735 |
| 1.0225 | 0.948305 |
| 1.025 | 0.9529478 |
| 1.0275 | 0.957602 |
| 1.03 | 0.9622676 |
| 1.0325 | 0.9669444 |
| 1.035 | 0.9716327 |
| 1.0375 | 0.9763322 |
| 1.04 | 0.9810431 |
| 1.0425 | 0.9857653 |
| 1.045 | 0.9904989 |
| 1.0475 | 0.9952438 |
| 1.05 | 1 |
| 1.0525 | 1.004768 |
| 1.055 | 1.009546 |
| 1.0575 | 1.014337 |
| 1.06 | 1.019138 |
| 1.0625 | 1.023951 |
| 1.065 | 1.028776 |
| 1.0675 | 1.033611 |
| 1.07 | 1.038458 |
| 1.0725 | 1.043316 |
| 1.075 | 1.048186 |
| 1.0775 | 1.053067 |
| 1.08 | 1.057959 |
| 1.0825 | 1.062863 |
| 1.085 | 1.067778 |
| 1.0875 | 1.072704 |
| 1.09 | 1.077642 |
| 1.0925 | 1.082591 |
| 1.095 | 1.087551 |
| 1.0975 | 1.092523 |
| 1.1 | 1.097506 |
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 |
|---|---|---|
| True circular period | 17.77153 | s |
| Indicated period | 18.66011 | s |
| Uncorrected inferred gravitational parameter | 0.9070295 | m³/s² |
| Observation-bound inferred lower parameter | 0.8707322 | m³/s² |
| Observation-bound inferred upper parameter | 0.9446106 | m³/s² |
| Actual parameter included by stated observation bounds | 0 | flag |
Narrow random-free observation bounds do not cover an omitted systematic clock scale by declaration.
Circular balance gives mu=4pi²r³/T². It increases with radius and decreases with period, so interval endpoints pair low radius with high period and vice versa. The disclosed clock gain causes nominal inference mu/gain². Correcting that gain requires independent timing evidence; refitting the same period cannot supply it.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Narrow random-free observation bounds do not cover an omitted systematic clock scale by declaration.
Circular balance gives mu=4pi²r³/T². It increases with radius and decreases with period, so interval endpoints pair low radius with high period and vice versa. The disclosed clock gain causes nominal inference mu/gain². Correcting that gain requires independent timing evidence; refitting the same period cannot supply it.
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Why can an interval be narrow and still exclude the actual parameter?
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