SCIENCE · INVESTIGATE AND EXPLAIN

Close the torque and work boundary of an orbital change

Conservation applies to a defined system. An orbiting body subject to an external torque can gain orbital angular momentum; that does not violate conservation when the driver is included. Similarly, orbital energy can increase through work. A mechanism such as interaction with another body or a disk needs a complete exchange ledger rather than the assertion that every orbit must conserve its own quantities while being driven.

Learning outcome: Vary final radius at fixed mass, then mass at fixed endpoints. Close ΔE and ΔL, compare their ratio with endpoint angular rates and identify the driver that a complete physical conservation argument would need to include.

Starting recap

Retain relative versus barycentric geometry, time versus angle, fitted versus instantaneous velocity and complete energy/angular-momentum boundaries.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

For m=2 and R=4, L changes from 2 to 4, so ΔL=2. E changes from −1 to −0.25, so ΔE=0.75. The mean work/torque-transfer ratio is 0.375 per time, between endpoint rates 1 and 0.125. Multiplying the final rate by ΔL would give only 0.25 energy units and miss the changing rate.

Open the experiment

Predict

Can a body move outward to a slower circular orbit while gaining total orbital energy?

Your prediction and reason

Experiment

Vary final radius at fixed mass, then mass at fixed endpoints. Close ΔE and ΔL, compare their ratio with endpoint angular rates and identify the driver that a complete physical conservation argument would need to include.

Account for external torque and work between circular states

Worked teaching model · synthetic values · Original quasi-static sequence of circular states around fixed μ=1, initial radius 1. L=m√r and E=−m/(2r); external torque/work changes orbital L and E. Along this ideal sequence dE/dL=Ω=r^(−3/2). The driver belongs in a complete conservation boundary; no physical migration mechanism is assumed.

Final circular radius: 4 normalized length Orbiting test mass: 2 mass units

Reset keeps your prediction, captures and practice history.

An outward circular-state change requires both angular-momentum gain and less-negative orbital energy.

The orbit alone is not an isolated system while a driver exerts torque and work. Along the ideal circular sequence, integrate Ω dL rather than multiplying the entire transfer by the final angular rate. A real tidal, disk or spacecraft mechanism requires its own energy and angular-momentum ledger.

Original conditional quantities and explicit units
Quantity	Value	Unit
Initial orbital angular momentum	2	mass length²/time
Final orbital angular momentum	4	mass length²/time
Required angular-momentum gain	2	mass length²/time
Initial orbital energy	-1	energy
Final orbital energy	-0.2500000	energy
Required orbital energy gain	0.7500000	energy
Mean work per angular-momentum transfer	0.3750000	per time
Initial circular angular rate	1	per time
Final circular angular rate	0.1250000	per time
Mistaken final-rate times angular-momentum gain	0.2500000	energy

These original models are not measured orbit records or operational mission instructions. Retain the frame, assumptions and missing evidence before interpreting a real system.

An outward circular-state change requires both angular-momentum gain and less-negative orbital energy.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the orbital evidence

2. Connect geometry, force and conservation

The original comparison follows a hypothetical slow sequence of circular states around a fixed central μ=1, from radius 1 to R. For orbiting test mass m, angular momentum is L=m√r and orbital energy is E=−m/(2r). An outward change raises L and makes E less negative. Final orbital speed decreases, but the total orbital energy increases because the potential-energy change dominates.

Differentiating the circular-state relations gives dE/dL=r^(−3/2)=Ω, the local angular rate. Because Ω changes along the sequence, total work is the integral of Ω dL, not the final Ω multiplied by all transferred angular momentum. The ratio ΔE/ΔL is a transfer-weighted mean rate between the endpoint rates. This check distinguishes a local differential relation from a finite-change shortcut.

The model does not specify how a real body remains circular while migrating, nor does it supply the driver’s efficiency or dissipated energy. Its external work and torque must be provided by an additional physical process. In a real tidal exchange, for example, spin, orbit and heat can all matter. The original ledger establishes required orbital changes, not a prediction that a specified migration will occur.

3. Work the original case

4. Predict and investigate

Can a body move outward to a slower circular orbit while gaining total orbital energy?

Apply the evidence to a changed case

Use m=1 and R=4. Find ΔL, ΔE and their ratio; explain which results scale with mass.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

ΔL=1 and ΔE=0.375, giving the same ratio 0.375 per time. Both transfers scale linearly with mass, but their ratio and endpoint angular rates do not. The source of torque/work remains outside the orbit-only ledger.

Source and boundaries

Primary mechanism reference. Checked 30 October 2026. Original prose, numerical cases and diagrams; source context is not a claim that these fictional records were observed. No actual mission operation or instrument aiming is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Use m=1 and R=4. Find ΔL, ΔE and their ratio; explain which results scale with mass.

Your explanation and limitations

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.

Previous: Subtract common acceleration before interpreting a tidal effect Next: Separate electromagnetic frequency from photon arrival rate

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.

Choose a science backup (maximum 36 MB)

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.

Export scope

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.

Choose the manifest and all parts together

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.