SCIENCE · INVESTIGATE AND EXPLAIN

Separate an interior’s volume, mass and inertia

Bulk density alone does not reveal a unique planetary interior. Rock, metal, volatile material, temperature and pressure can produce overlapping mass–radius relations. Differentiation redistributes material when sufficiently mobile dense components sink and lighter components rise; a layered interior is an inference constrained by gravity, rotation, composition and thermal history, not a direct photograph of the core.

Learning outcome: Change core radius and mantle density separately. Retain all three fractions and the radial weighting of inertia; explain what additional measurement could discriminate two mass–radius-compatible interiors.

Starting recap

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

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Try

For f=0.4 and mantle density 3, core volume fraction is 0.064, mean density 3.32 g/cm³ and core mass fraction 0.512/3.32≈0.154217. Normalized inertia is 0.4×3.0512/3.32≈0.367614, below a uniform sphere.

Open the experiment

Predict

Does a core occupying 40% of radius occupy 40% of volume or mass?

Your prediction and reason

Experiment

Change core radius and mantle density separately. Retain all three fractions and the radial weighting of inertia; explain what additional measurement could discriminate two mass–radius-compatible interiors.

Separate core volume, mass and rotational inertia

Worked teaching model · synthetic values · Original incompressible two-layer sphere: core density 8 g/cm³, outer radius normalized 1, sharp spherical boundary. Core volume fraction f³; mean density rho_m+(8−rho_m)f³. Normalized inertia C/(MR²)=(2/5)[rho_m+(8−rho_m)f⁵]/mean density. No pressure/compression, measured core size or unique composition inference.

Core radius divided by planet radius: 0.4 fraction Mantle density: 3 g/cm³

Reset keeps your prediction, captures and practice history.

A dense small core can hold a larger mass fraction than its volume fraction.

Bulk density and normalized inertia are distinct integrals over interior structure. Multiple compositions and density profiles can share a bulk density; pressure, temperature and gravity data must constrain a real inference.

Original conditional model ledger
Quantity	Value	Unit
Core volume fraction	0.06400000	fraction
Bulk mean density	3.320000	g/cm³
Core mass fraction	0.1542169	fraction
Mantle mass fraction	0.8457831	fraction
Normalized axial moment of inertia	0.3676145	C/(MR²)

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

A dense small core can hold a larger mass fraction than its volume fraction.

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 two-layer sphere has normalized outer radius one, core density 8 g/cm³ and mantle density ρm. If core radius fraction is f, its volume fraction is f³ and mean density is ρm+(8−ρm)f³. Core mass fraction is 8f³ divided by mean density. Radius fraction, volume fraction and mass fraction therefore cannot be interchanged.

Rotation supplies another weighting: material farther from the axis contributes more strongly to moment of inertia. Integrating spherical shells gives C/(MR²)=(2/5)[ρm+(8−ρm)f⁵]/[ρm+(8−ρm)f³]. A uniform sphere has ratio 0.4; concentrating mass inward lowers it under these assumptions. The fifth power in the numerator is not a typographical replacement for the third-power volume term.

Terrestrial planets are predominantly rock and metal, whereas gas and ice giants contain much larger volatile inventories and strongly compressed interiors. Small moons, asteroids and comets have different gravity and thermal histories. Our incompressible sharp-boundary sphere omits compression, gradients, irregular shape and rotation-induced distortion, so its fitted core fraction would not be a unique real composition.

3. Work the supplied case

4. Predict and investigate

Does a core occupying 40% of radius occupy 40% of volume or mass?

Apply the evidence to a changed case

Use f=0.6 and mantle density 3. Find volume fraction, mean density and core mass fraction.

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

Compare a worked response

Volume fraction is 0.216; mean density is 4.08 g/cm³; core mass fraction is 1.728/4.08≈0.423529. None equals the radius fraction 0.6.

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.

Use f=0.6 and mantle density 3. Find volume fraction, mean density and core mass fraction.

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: Audit an actual spectrum before treating a curve as an inference Next: Compare internal production with surface heat loss

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