Unit 2: Interiors and Heat

Lecture 04: Planetary Heat Budgets, Radiogenic Heating, and Convection

Why some planets are still hot after 4.5 billion years

Learning Objectives

Identify the sources of a planet's internal heat: accretional heat, differentiation heat, radiogenic decay, and (for some giant planets) ongoing gravitational contraction.
Derive the exponential decay law for radiogenic heat production and apply it to Earth's major heat-producing isotopes.
Explain the Rayleigh number criterion for the onset of solid-state mantle convection.
Compare surface heat flow across terrestrial planets and connect it to their present-day geological activity.

Terms for This Lecture

radiogenic heating

Definition: Tidal heating is flagged here, ahead of its full quantitative treatment in Lecture 13, specifically to prevent an overgeneralization of this lecture's surface-to-volume cooling argument: Io is smaller than the Moon yet is the most volcanically active body in the solar system, precisely because it receives an enormous non-radiogenic heat input from tidal flexing that the simple size-scaling argument does not capture.

Why it matters here: Radiogenic heat, decaying exponentially with each isotope's own half-life, is Earth's dominant present-day internal heat source, though secular cooling contributes comparably.

Relationship: The Rayleigh Number and Radiogenic Heating Combined

Boundary or common confusion: The Rayleigh number is the same dimensionless number used in fluid dynamics and atmospheric science generally (it also governs, for example, convection in a pot of heated water or in a star's envelope); its application here to a nominally "solid" mantle is possible only because geological timescales (millions of years) are long enough for silicate rock to behave as an extremely viscous fluid, an idea first proposed to explain continental drift and now confirmed by both seismic tomography (imaging actual mantle plumes) and geodetic measurement of present-day plate motions.

Sources of Planetary Interior Heat

Accretional heat: kinetic energy of infalling planetesimals converted to heat during formation, largely radiated away or retained depending on burial depth and accretion rate.
Differentiation heat: gravitational potential energy released as dense material (iron) sinks to form a core (Lecture 03), a one-time but substantial heat source.
Radiogenic heat: ongoing heat production from the decay of long-lived radioactive isotopes (⁴⁴⁴K, ²³⁵U, ²³⁾U, ²³⁆Th) distributed through the mantle and crust.
Giant planets add a further source: slow gravitational (Kelvin-Helmholtz) contraction, and for Saturn, possible helium rain releasing gravitational energy (Lecture 11).

Radiogenic Heat Production: The Decay Law

Each radioactive isotope's heat production rate decays exponentially: H(t) = H₀ e⁻λt, with decay constant λ = ln2 / t₁⁄₂.
Earth's present-day radiogenic heat budget is dominated by four isotopes with half-lives ranging from 1.25 Gyr (⁴⁴K) to 14.0 Gyr (²³²Th).
Because ⁴⁴K has the shortest half-life of the four major heat producers, Earth's radiogenic heat production was substantially higher in the past and has been declining ever since formation.

Present-Day Radiogenic Heat Production Rates

⁴⁴K (t₁⁄₂ = 1.25 Gyr): contributes roughly 40% of Earth's present bulk silicate Earth radiogenic heat production, despite modest abundance, because of its shorter half-life.
²³⁶U (t₁⁄₂ = 4.47 Gyr) and ²³²Th (t₁⁄₂ = 14.0 Gyr): together contribute most of the remainder, with Th typically contributing somewhat more heat than U owing to greater cosmochemical abundance.
Total present-day radiogenic heat production for the bulk silicate Earth is estimated at roughly 20 TW, compared to a total observed surface heat flow of roughly 47 TW -- implying a substantial non-radiogenic (secular cooling) contribution as well.

The Rayleigh Number and the Onset of Convection

The Rayleigh number Ra = (ρ g α ΔT d³) / (κ μ) compares the destabilizing effect of thermal buoyancy to the stabilizing effects of viscous dissipation and thermal diffusion.
Convection begins once Ra exceeds a critical value Ra_c ∼ 1000-2000 (depending on boundary conditions); solid-state mantle convection occurs because Earth's mantle, though nominally solid, deforms plastically over geological timescales at very low effective viscosity (∼10²¹ Pa·s).
Earth's mantle Rayleigh number is estimated at 10⁶-10⁷, vastly super-critical, meaning vigorous convection rather than marginal stability is expected -- consistent with plate tectonics.

Mantle Convection Regimes: Stagnant Lid vs. Mobile Lid

Mobile-lid convection (Earth's plate tectonics): the cold surface layer participates directly in convection, recycling into the mantle at subduction zones.
Stagnant-lid convection (Venus, Mars, Mercury, the Moon, most other rocky bodies): the cold surface layer is too strong (or too dry) to break into mobile plates and instead sits as a single immobile shell atop a convecting interior.
Earth appears to be the only terrestrial planet with active plate tectonics today, making it observationally anomalous rather than the default expectation.

Heat Flow Measurements Across the Terrestrial Planets

Earth: mean surface heat flow ∼91-92 mW/m², with strong contrast between young oceanic ridges (>200 mW/m²) and old continental cratons (∼40-60 mW/m²).
Moon: Apollo heat-flow probes measured 16-21 mW/m², an order of magnitude below Earth, consistent with its small size, thick stagnant lid, and largely extinct interior activity.
Mars: InSight's heat-flow probe (HP³) failed to achieve its planned burial depth, but independent geophysical estimates suggest ∼20-24 mW/m², intermediate between Earth and the Moon.

Why Small Bodies Cool Faster: Surface-to-Volume Scaling

A body's total heat generation scales with its volume (∝ R³), while its capacity to lose heat scales with its surface area (∝ R²).
The ratio of heat loss to heat generation therefore scales as R²/R³ = 1/R: smaller bodies lose heat proportionally faster and cool on shorter timescales.
This single scaling argument explains why the Moon and Mercury are geologically quiescent today while Earth (and, differently, Venus) remain active.

The Rayleigh Number and Radiogenic Heating Combined

A planet's convective vigor depends jointly on its radiogenic heat production (this lecture's exponential decay law) and its size (this lecture's surface-to-volume scaling).
Mars, intermediate in size between the Moon and Earth, is a natural test case: it appears to have lost most active mantle convection and any global magnetic dynamo (Lecture 10) roughly 4 Gyr ago, consistent with faster cooling than Earth but slower than the Moon.
This combined argument previews Lab 02's quantitative heat-budget calculation for Mars versus Earth.

Tidal Heating as a Non-Radiogenic Heat Source

For satellites in eccentric or resonant orbits (Io, Europa, Enceladus), periodic tidal flexing dissipates orbital and rotational energy as internal heat.
Tidal heating can dominate over radiogenic heating in small icy or rocky satellites, as this course shows quantitatively for Io in Lecture 13.
This heat source has no analog in isolated planets and depends on orbital dynamics (eccentricity, orbital resonance) rather than only on a body's size and radioactive inventory.

Reading and Scope Note

OpenStax Astronomy 2e Section 9.1-9.2 (Earth's interior and plate tectonics) introduces heat flow and mantle convection at an introductory level.
The Rayleigh-number formalism and the quantitative radiogenic decay-law treatment go beyond OpenStax and follow Turcotte & Schubert, Geodynamics, Chapters 4 and 6.

Worked Example: Earth's Present-Day Radiogenic Heat Production

Model each isotope's heat production as H(t) = H₀ e⁻λt with λ = ln2/t₁⁄₂, using standard bulk-silicate-Earth present-day heat production rates (per kilogram of isotope) and Earth mantle abundances.

Half-lives used: ⁴⁴K = 1.25 Gyr (λ = 0.555 Gyr⁻&sup9;), ²³⁶U = 4.47 Gyr (λ = 0.155 Gyr⁻&sup9;), ²³²Th = 14.0 Gyr (λ = 0.050 Gyr⁻&sup9;).

At t = 4.5 Gyr after formation, the surviving fraction of each isotope's initial inventory is e⁻λt: ⁴⁴K retains 8.25%, ²³⁶U retains 49.8%, and ²³²Th retains 80.0% of its original abundance.

Result: ⁴⁴K has decayed by more than a factor of 8 over Earth history, while ²³²Th has barely decayed at all -- exactly why ⁴⁴K, despite being a minor heat contributor today, dominated Earth's radiogenic heat budget much more strongly in the distant past, and why the present-day total (∼20 TW) is well below the ∼60-80 TW radiogenic heat production Earth likely had shortly after formation.

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?

Radiogenic heat, decaying exponentially with each isotope's own half-life, is Earth's dominant present-day internal heat source, though secular cooling contributes comparably.

Summary and Next Lecture

Radiogenic heat, decaying exponentially with each isotope's own half-life, is Earth's dominant present-day internal heat source, though secular cooling contributes comparably.
The Rayleigh number formalizes when a planetary interior convects; Earth's mantle is vastly super-critical, consistent with observed plate tectonics.
Surface-to-volume scaling (heat loss ∝ 1/R relative to heat generation) explains why small bodies cool and become geologically quiescent faster than large ones, though tidal heating is an important exception developed later.

Next: Lecture 05 examines the surface expression of this convecting interior: plate tectonics on Earth and its absence, so far, everywhere else in the solar system.