SCIENCE · INVESTIGATE AND EXPLAIN

Compare internal production with surface heat loss

Accretion and differentiation can supply early heat, radioactive decay supplies continuing internal energy, and tidal deformation can dissipate orbital or rotational energy. Conduction, convection and volcanism redistribute or remove that heat. A planet’s present heat output need not equal its present radioactive production because stored energy can change.

Learning outcome: Close generation, loss and storage, then compare size at the same source age. Identify the fixed-flux assumption that prevents this snapshot from becoming a complete thermal history.

Starting recap

Keep interior, thermal, surface-history, orbital and detection assumptions distinct before drawing a planetary inference.

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Try

At R=2 and one source half-life, h=5. Production is 160π/3 and loss 80π, giving storage −80π/3. The instantaneous temperature rate is (5−7.5)/20=−0.125 normalized temperature per time.

Open the experiment

Predict

Can outward heat loss exceed current radioactive production without violating conservation?

Your prediction and reason

Experiment

Close generation, loss and storage, then compare size at the same source age. Identify the fixed-flux assumption that prevents this snapshot from becoming a complete thermal history.

Compare volume heat production with surface cooling

Worked teaching model · synthetic values · Original instantaneous model: volumetric heating 10×2^(−n), fixed outward flux 5, volumetric heat capacity 20. Generation scales with volume, loss with area; dT/dt=(heating−15/R)/20. No actual radioactive abundance, temperature-dependent transport or integrated planetary history.

Normalized planetary radius: 2 length units Elapsed source half-lives: 1 half-lives

Reset keeps your prediction, captures and practice history.

Volume production and surface loss scale differently with body size.

An instantaneous warming or cooling rate does not establish a complete thermal history. Accretion, differentiation, radioactive decay, tides and evolving conductive/convective transport need their own evidence for actual planetary activity.

Original conditional model ledger
Quantity	Value	Unit
Volume	33.51032	length³
Surface area	50.26548	length²
Total internal generation	167.5516	energy/time
Total outward loss	251.3274	energy/time
Net heat storage rate	-83.77580	energy/time
Instantaneous temperature rate	-0.1250000	temperature/time
Volume/surface ratio	0.6666667	length

Original supplied models, not actual planet observations or a unique interior, atmosphere or discovery claim.

Volume production and surface loss scale differently with body size.

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 planetary evidence

2. Connect mechanism and inference

The original sphere has volumetric production h=10×2^(−n), outward surface flux 5 and volumetric heat capacity 20, all in compatible normalized units. Total production is h(4πR³/3), while total loss is 5(4πR²). The storage rate is production minus loss, not their ratio.

Dividing storage by the total heat capacity gives instantaneous dT/dt=(h−15/R)/20. The volume-to-area ratio R/3 makes size relevant even when composition and surface flux are held fixed. The elapsed-half-life control changes the source term but does not integrate the planet’s prior temperature or allow surface flux to respond to temperature.

An actual cooling history needs initial conditions, material transport, changing surface boundaries and potentially phase changes or tidal heating. A positive instantaneous storage rate is not evidence of runaway warming, and a small body can remain active when another source or boundary differs. The supplied comparison isolates one geometric mechanism rather than ranking real planets by size alone.

3. Work the supplied case

At R=2 and one source half-life, h=5. Production is 160π/3 and loss 80π, giving storage −80π/3. The instantaneous temperature rate is (5−7.5)/20=−0.125 normalized temperature per time.

4. Predict and investigate

Can outward heat loss exceed current radioactive production without violating conservation?

Close generation, loss and storage, then compare size at the same source age. Identify the fixed-flux assumption that prevents this snapshot from becoming a complete thermal history.

Apply the evidence to a changed case

At R=3 compare zero and two elapsed half-lives.

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

Compare a worked response

At zero, h=10 and dT/dt=(10−5)/20=0.25. At two, h=2.5 and the rate is −0.125. The sign changes because the source falls while the imposed loss per volume remains 5.

Source and boundaries

Primary source. Checked 31 October 2026. Original diagrams and supplied model records are not actual planetary observations. No direct solar viewing, instrument operation or physical experiment is requested.

Apply

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

At R=3 compare zero and two elapsed half-lives.

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: Separate an interior’s volume, mass and inertia Next: Separate atmospheric thickness from an escape claim

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