Unit 2: Interiors and Heat

Lecture 03: Planetary Differentiation and Interior Structure

Cores, mantles, crusts, and how we know they are there

Learning Objectives

Explain gravitational differentiation: how a molten early planet separates into density-stratified layers.
Use the moment of inertia factor to constrain a planet's internal density distribution from spacecraft tracking data alone.
Compare terrestrial planet interior structures (core size, mantle composition) using real measured moment-of-inertia factors.
Explain how seismology (Apollo, InSight) provides an independent, direct check on gravity-based interior models.

Terms for This Lecture

differentiation

Definition: Differentiation is a self-reinforcing process: as iron sinks, it releases gravitational potential energy as heat, which keeps the interior hot enough for continued sinking, until the densest material has segregated into a core.

Why it matters here: Differentiation converts a newly assembled planet's accretional and radiogenic heat into a density-stratified core-mantle-crust structure.

Relationship: The Moon's ancient anorthositic highland crust, sampled directly by Apollo astronauts, is thought to have crystallized by flotation from a global magma ocean, a direct physical sample of primary differentiation.

Boundary or common confusion: The efficiency and timing of this process differs between planets and directly explains why Mercury has an anomalously large core (Lecture on giant-impact stripping, this section) while Mars's core appears smaller and only partially differentiated relative to its mantle.

Why Planets Differentiate

Accretion energy, radioactive decay (Lecture 04), and core formation itself release enough heat to melt (at least partially) a newly assembled terrestrial planet.
In a molten or partially molten body, denser material (iron, nickel) sinks toward the center while lighter silicates float upward, forming a core-mantle-crust structure.
This process, called differentiation, converts gravitational potential energy into heat as dense material sinks, which further promotes melting -- a runaway that likely completed within the first ∼10-100 Myr for Earth.

The Moment of Inertia Factor

For a uniform-density sphere, the moment of inertia is I = (2/5) MR²; the dimensionless factor C/MR² = 0.4 for a uniform body.
A body with a dense core concentrated toward the center has C/MR² < 0.4; Earth's measured value is 0.3307, Mars's is 0.3644, and the Moon's is 0.3931 (nearly uniform).
C/MR² is measured from precession and gravity-field harmonics (J2) obtained from spacecraft tracking, not from any direct interior probe.

Terrestrial Planet Interior Comparison

Mercury: anomalously large core (∼85% of the planet's radius by some models), likely from a giant impact that stripped early mantle material.
Venus: Earth-like bulk density and probable core size, but no measured moment of inertia factor exists yet (no orbiting Venus gravity mission has achieved the precision Earth and Mars have).
Mars: smaller, partially differentiated core inferred from InSight seismic data (radius ∼1830 km, confirmed directly by seismic core-reflected phases in 2021).

Seismology as Ground Truth

Apollo seismometers (1969-1977) detected moonquakes and used travel-time analysis to infer the Moon's crust-mantle-core structure directly.
NASA's InSight lander (2018-2022) recorded over 1300 marsquakes, including core-reflected (ScS) phases that directly measured the Martian core radius and confirmed it is at least partially liquid.
Direct seismic detection is the gold standard because it requires no assumption about bulk composition, unlike gravity-field-only inference.

Pressure and Density in Planetary Interiors

Interior pressure increases with depth according to hydrostatic equilibrium, dP/dr = -ρ(r) g(r) -- the same governing equation developed for atmospheres in Lecture 07, applied here to solid/liquid interiors.
Unlike an ideal-gas atmosphere, planetary interiors require an equation of state relating pressure to density for iron, silicates, and (in giant planets) metallic hydrogen under extreme compression.
Self-compression alone can raise a planet's density substantially above its zero-pressure material density -- important when comparing Earth's and Mercury's bulk densities, since Earth's stronger self-gravity compresses its interior more.

Core Composition: Iron Alloys and Light Elements

Earth's core is primarily iron-nickel, but seismic density measurements require ∼5-10% by mass of lighter elements (sulfur, oxygen, silicon, or hydrogen) to match observed density and sound-speed profiles.
The precise light-element budget affects a core's freezing behavior and, through compositional buoyancy during inner-core growth, the vigor of the convective dynamo (developed fully in Lecture 10).
Mercury's core is inferred to be unusually sulfur-rich, which lowers its freezing point and may help sustain an active dynamo in such a small body.

Crustal Formation and Differentiation Products

The lightest, most incompatible elements concentrate in partial melts that rise to form a planet's earliest crust.
Earth's continental crust (average density ∼2700 kg/m³) is chemically distinct from and less dense than its oceanic crust (∼3000 kg/m³) and mantle (∼3300 kg/m³), itself a product of repeated partial melting over geological time.
The Moon's ancient anorthositic highland crust, sampled directly by Apollo astronauts, is thought to have crystallized by flotation from a global magma ocean, a direct physical sample of primary differentiation.

The Core-Mantle Boundary and D″ Layer

Earth's core-mantle boundary (2891 km depth) is one of the sharpest physical/chemical discontinuities in the solar system, separating liquid iron alloy from solid silicate.
The D″ (D-double-prime) layer just above the core-mantle boundary shows anomalous seismic properties, possibly a distinct chemical reservoir or a phase transition in mantle mineralogy.
Heat flow across the core-mantle boundary drives core convection (Lecture 10) and couples core cooling to mantle convection (Lecture 04).

Interior Structure Beyond the Terrestrial Planets

Giant planets have no solid surface; "interior structure" instead refers to a continuous transition from molecular to metallic hydrogen (Lecture 11) with a possible dense core.
Icy satellites (Europa, Enceladus, Titan; Lecture 13) can differentiate into a rock/metal core, an ice mantle, and, critically, a liquid subsurface ocean layer.
Gravity data from the Galileo, Cassini, and Juno missions extend the moment-of-inertia method used for terrestrial planets to these very different interior structures.

Worked Example Setup: Comparing C/MR² Across Bodies

We compare the measured moment of inertia factors of Earth, Mars, and the Moon against the uniform-sphere reference value of 0.4.
A lower C/MR² indicates a proportionally larger, denser core; a value near 0.4 indicates a nearly uniform interior.
This comparison is entirely observational (no interior model assumed beyond the sphere reference) and previews Lab 02's quantitative moment-of-inertia core-size inversion.

Reading and Scope Note

OpenStax Astronomy 2e Section 9.1 (Overview of Earth Structure) introduces core-mantle-crust structure at an introductory, non-quantitative level.
The moment-of-inertia formalism and its use to infer interior structure from spacecraft tracking data go beyond OpenStax and follow de Pater & Lissauer, Planetary Sciences, Chapter 4.

Worked Example: Moment of Inertia Factors: Earth, Mars, and the Moon

Compare each body's measured moment of inertia factor C/MR² to the uniform-sphere reference value of 0.4.

Body	C/MR² (measured)	Deviation from uniform (0.4)	Interpretation
Earth	0.3307	-17.3%	Strongly core-concentrated (large dense iron core)
Mars	0.3644	-8.9%	Moderately core-concentrated (smaller, partially differentiated core)
Moon	0.3931	-1.7%	Nearly uniform (small or absent core)

Result: the deviation from 0.4 tracks independently measured core sizes (Earth ∼3480 km, Mars ∼1830 km by InSight seismology, Moon ∼350 km by Apollo seismology and later gravity/laser-ranging refinement) in the correct rank order, validating the moment-of-inertia method against direct seismic ground truth.

Visual Reasoning

Study how this lecture’s figure turns the derivation and worked example above into a concrete quantitative picture.

Which numbers in the figure come directly from measurement, and which are computed from this lecture’s physical law?
Where do the marked points or curves match the worked-example values above?
What would change in the figure if a key assumption in the derivation failed?

Differentiation converts a newly assembled planet's accretional and radiogenic heat into a density-stratified core-mantle-crust structure.

Summary and Next Lecture

Differentiation converts a newly assembled planet's accretional and radiogenic heat into a density-stratified core-mantle-crust structure.
The moment of inertia factor, measurable from spacecraft tracking alone, is a powerful and non-invasive probe of a planet's internal density distribution.
Seismology (Apollo, InSight) independently confirms gravity-based interior inferences, the strongest kind of scientific validation available in planetary science.

Next: Lecture 04 asks what powers a planet's interior heat budget over geological time, and why some planets stay geologically active while others do not.