3. Planetary Heat & Energy Transport

Learning objectives: By the end of this lecture, you will be able to identify the main energy sources driving planetary evolution, distinguish between heat transport mechanisms, and apply scaling laws for convection and thermal evolution.

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3.1. Energy sources

A planet’s geological and geophysical evolution is fundamentally governed by its thermal history: how much heat it starts with, how it generates new heat, and how efficiently it can lose that heat to space. Without internal heat, there would be no volcanism, no magnetic field, no plate tectonics, and no atmospheric outgassing. Understanding where planetary heat comes from is therefore the starting point for understanding planetary evolution.

There are four principal sources of heat in planets:

3.1.1. Accretional heating

During planet formation (Lecture 2), gravitational potential energy is converted into kinetic energy as material falls onto the growing body, and that kinetic energy is converted into heat upon impact. The total gravitational energy released in assembling a uniform sphere of mass \(M\) and radius \(R\) from dispersed material is:

(3.1)\[ E_{\mathrm{acc}} \sim \frac{3}{5} \frac{G M^2}{R} \]

For Earth (\(M = 5.97 \times 10^{24}\) kg, \(R = 6.37 \times 10^6\) m), this gives \(E_{\mathrm{acc}} \approx 2.2 \times 10^{32}\) J. If all this energy were retained as heat and distributed uniformly, it would raise the temperature by:

\[ \Delta T = \frac{E_{\mathrm{acc}}}{M c_p} \approx \frac{2.2 \times 10^{32}}{5.97 \times 10^{24} \times 1200} \approx 30{,}000 \text{ K} \]

where \(c_p \approx 1200\) J kg\(^{-1}\) K\(^{-1}\) is a representative specific heat for silicate rock (values between 1200 and 1250 J kg\(^{-1}\) K\(^{-1}\) are in common use; worksheet 2 adopts 1250). This is enough to melt Earth many times over. In practice, not all energy is retained: some is radiated back to space during accretion, especially in the later stages when the surface is molten and radiates efficiently. Nevertheless, accretional heating is more than sufficient to produce a fully molten planet (a magma ocean) by the end of formation [Lichtenberg et al., 2023].

Fig. 3.1 shows how the accretional energy and the resulting temperature rise depend on the size of the body.

Fig. 3.1 Accretional heating against body size for a uniform sphere of Earth’s mean density. (a) The gravitational energy released in assembly, \(E_{\mathrm{acc}} = \frac{3}{5} G M^2/R\), which scales as \(R^5\); Earth’s \(2.2 \times 10^{32}\) J is marked. (b) The temperature rise \(\Delta T = E_{\mathrm{acc}}/(M c_p)\) with \(c_p = 1200\) J kg\(^{-1}\) K\(^{-1}\) if all of the energy were retained, which scales as \(R^2\): about \(30\,000\) K for Earth, but only tens of kelvins for radii of 100 to 300 km. Course-original figure.

3.1.2. Gravitational differentiation

When a planet melts (from accretional heating and other sources), dense metallic iron sinks to the centre while lighter silicates rise to form the mantle. This core formation process releases additional gravitational potential energy as the denser material moves to a lower gravitational potential. For Earth, the energy released by core–mantle differentiation is estimated at:

\[ E_{\mathrm{diff}} \approx 2 \times 10^{31} \text{ J} \]

This is about 10% of the accretional energy, but is released relatively quickly during the differentiation process, further heating the interior and helping to sustain the early magma ocean. Gravitational differentiation is a one-time energy source: once the core has formed, this energy has been spent.

Fig. 3.2 sketches the process.

Fig. 3.2 Gravitational differentiation in schematic cross-section. (a) The body after accretion, with metal droplets dispersed in silicate rock. (b) After core formation, with the dense metal at the centre and the silicate mantle around it. The sinking of the metal lowers the gravitational potential energy of the body, and the difference, about \(2 \times 10^{31}\) J for Earth, is released as heat. Not to scale. Course-original figure.

3.1.3. Radioactive decay

The decay of naturally occurring radioactive isotopes provides a sustained heat source that continues to power planetary interiors billions of years after formation. There are two categories:

Long-lived isotopes have half-lives comparable to or longer than the age of the solar system. The four most important for planetary heat production are:

Isotope	Half-life (Gyr)	Heat production (W kg\(^{-1}\) of isotope)	Concentration in bulk silicate Earth
\({}^{238}\mathrm{U}\)	4.47	\(9.5 \times 10^{-5}\)	20 ppb
\({}^{235}\mathrm{U}\)	0.704	\(5.7 \times 10^{-4}\)	0.14 ppb
\({}^{232}\mathrm{Th}\)	14.0	\(2.6 \times 10^{-5}\)	80 ppb
\({}^{40}\mathrm{K}\)	1.25	\(2.9 \times 10^{-5}\)	28 ppb

Concentrations are in parts per billion by mass (1 ppb \(= 10^{-9}\)). These isotopes are lithophile (“rock-loving”) elements that concentrate in the silicate mantle and crust rather than the metallic core. Together, they produce Earth’s present-day radiogenic heat of approximately 20 TW (terawatts) [Jaupart et al., 2015]. Because the shorter-lived isotopes (\({}^{235}\mathrm{U}\), \({}^{40}\mathrm{K}\)) were more abundant in the past, total radiogenic heat production was almost 5 times higher at the time of Earth’s formation (4.5 Gyr ago) than it is today, a factor of 4.7 with the abundances in the table (Fig. 3.3).

Short-lived isotopes had half-lives of only a few million years and are now extinct, but were present when the solar system formed. The most important is \({}^{26}\mathrm{Al}\) (half-life 0.72 Myr), which was produced by nearby stellar nucleosynthesis and incorporated into the solar nebula. In the first few million years, \({}^{26}\mathrm{Al}\) dominated the total heating rate by almost four orders of magnitude (Fig. 3.3): it could melt planetesimals as small as a few kilometres in radius, initiating core formation and volatile loss even before planet-sized bodies had assembled (Fig. 3.4) [Lichtenberg et al., 2023]. The short-lived isotope \({}^{60}\mathrm{Fe}\) (half-life 2.6 Myr) also contributed, though to a lesser extent.

Fig. 3.3 Specific radiogenic heating rate of planetary material from the formation of the CAIs (calcium-aluminium-rich inclusions, the first solids to condense in the solar nebula, defining \(t = 0\)) to the present. Dashed curves: the short-lived isotopes \({}^{26}\mathrm{Al}\) (orange) and \({}^{60}\mathrm{Fe}\) (brown) in primitive chondritic rock, from their canonical initial abundances (\({}^{26}\mathrm{Al}/{}^{27}\mathrm{Al} = 5.25 \times 10^{-5}\), \({}^{60}\mathrm{Fe}/{}^{56}\mathrm{Fe} = 10^{-8}\)) [Lichtenberg et al., 2023], with half-lives from [Castillo-Rogez et al., 2009, Rugel et al., 2009]. Solid curves: the four long-lived isotopes in bulk-silicate-Earth rock, computed from the parameters in the table above, with their sum in black. \({}^{26}\mathrm{Al}\) dominates the first few Myr by almost four orders of magnitude, the window in which planetesimals melt; after \(\sim 10\) Myr the long-lived isotopes take over, and their summed output falls by a factor of 4.7 between Earth’s formation and today (92 TW to 19 TW for the bulk silicate Earth). Course-original figure.

Fig. 3.4 Decay of the principal short-lived radionuclides \({}^{26}\mathrm{Al}\) (\(t_{1/2}=0.72\) Myr; [Castillo-Rogez et al., 2009]) and \({}^{60}\mathrm{Fe}\) (\(t_{1/2}=2.62\) Myr; [Rugel et al., 2009]) over the first 10 Myr of solar-system history. Despite half-lives of only a few million years, these isotopes were the dominant heat source in the first few Myr and could melt planetesimals as small as a few kilometres in radius. Course-original figure.

3.1.4. Tidal heating

When a body on a non-circular orbit is tidally deformed by a nearby massive companion, the periodic flexing of its interior dissipates energy as heat. This is a significant heat source for certain moons in the outer solar system, particularly Io and Enceladus. We will discuss tidal heating in detail in section 7 of this lecture.

3.1.5. Earth’s heat budget

The total heat flowing out of Earth’s interior is approximately 47 TW [Davies and Davies, 2010]. The approximate breakdown is:

Source	Contribution
Radiogenic heating (mantle + crust)	~20 TW
Primordial (secular cooling)	~17 TW
Core cooling and solidification	~10 TW
Tidal dissipation (from the Moon)	~0.1 TW
Total	~47 TW

Fig. 3.5 Earth’s surface heat-flux budget (~47 TW), decomposed into radiogenic (~20 TW), primordial / secular cooling (~17 TW), core cooling and solidification (~10 TW), and lunar tidal dissipation (~0.1 TW). Tidal contribution is too small to render at this scale. After [Davies and Davies, 2010]. Course-original figure.

Roughly 40% of Earth’s current heat loss is powered by ongoing radioactive decay, with the rest being the slow release of primordial heat stored since formation (Fig. 3.5). The ratio of internal heat production \(H\) to total surface heat loss \(Q\) is called the Urey ratio, \(\mathrm{Ur} = H/Q \approx 0.4\) for the present-day Earth; a value below 1 means the planet loses more heat than it generates and therefore cools. This means Earth is still cooling: its interior temperature is gradually decreasing over geological time. The relative importance of these sources varies strongly across the solar system: small bodies like the Moon have lost most of their primordial heat long ago, while a tidally heated moon like Io has a heat budget dominated by tidal dissipation rather than radiogenic or primordial heat.

3.2. Heat transport mechanisms

Heat moves through planetary interiors and escapes to space via three fundamental mechanisms: conduction, convection, and radiation. The dominance of each depends on the material properties and physical conditions at each depth.

Fig. 3.6 compares the three mechanisms schematically.

Fig. 3.6 The three heat transport mechanisms in schematic form. (a) Conduction: heat passes through a material down the temperature gradient, \(\vec{q} = -k \nabla T\), without bulk motion. (b) Convection: hot fluid rises from the heated base and cold fluid sinks from the cooled top. (c) Radiation: a hot surface emits photons with a flux \(F = \epsilon \sigma T^4\). Schematic. Course-original figure.

3.2.1. Conduction

In conduction, heat is transferred through a material by direct molecular or atomic collisions, without bulk motion of the material itself. In solids (such as rock or ice), thermal energy propagates as lattice vibrations (phonons); in metals, free electrons also carry heat. The rate of heat flow by conduction is described by Fourier’s law:

(3.2)\[ \vec{q} = -k \nabla T \]

where \(\vec{q}\) is the heat flux (W m\(^{-2}\)), \(k\) is the thermal conductivity (W m\(^{-1}\) K\(^{-1}\)), and \(\nabla T\) is the temperature gradient. The negative sign indicates that heat flows from hot to cold regions.

A closely related quantity is the thermal diffusivity, which controls how fast temperature disturbances propagate through a material:

(3.3)\[ \kappa = \frac{k}{\rho c_p} \]

where \(\rho\) is the density and \(c_p\) is the specific heat capacity. Typical values for silicate rock are \(k \approx 3\)–\(5\) W m\(^{-1}\) K\(^{-1}\) and \(\kappa \approx 10^{-6}\) m\(^2\) s\(^{-1}\).

Fig. 3.7 illustrates Fourier’s law and compares the conductivities of planetary materials.

Fig. 3.7 Fourier’s law and the thermal conductivity of planetary materials. (a) A slab with a cold surface and a hot interior: with \(z\) measured upward, the heat flux \(q = -k\,\mathrm{d}T/\mathrm{d}z\) points up, down the temperature gradient. (b) Thermal conductivities on a logarithmic axis: air 0.025, water ice 2.2, upper-mantle silicate rock 3 to 5, lower-mantle bridgmanite 5 to 10 and solid iron 80 W m\(^{-1}\) K\(^{-1}\). Course-original figure.

Conduction is inherently a diffusive process, slow and inefficient over large distances. As we will show in the blackboard derivation, the time for heat to conduct through a distance \(L\) scales as \(L^2/\kappa\), which becomes enormous for planet-sized bodies.

3.2.2. Convection

In convection, heat is transported by the bulk motion of fluid. When a fluid layer is heated from below, hot material near the base becomes less dense (due to thermal expansion) and rises, while cool, denser material near the top sinks (Fig. 3.8). This overturning circulation (thermal convection) is far more efficient at transporting heat than conduction.

The concept of solid-state convection in planetary mantles often requires a shift in intuition. Mantle rock is entirely solid, not a liquid magma. However, at the high temperatures and pressures deep within a planetary interior, the silicate crystal lattice yields through the slow migration of atomic defects. Over geological timescales of millions of years, this microscopic solid-state creep integrates into macroscopic flow, allowing the solid mantle to behave as a highly viscous fluid. Viscosity measures a fluid’s resistance to flow: a low-viscosity fluid such as water flows freely, while a high-viscosity fluid such as cold honey resists deformation and flows only slowly under an applied force. The continuum mechanics equations that govern the overturning of liquid water apply equally to the solid mantle, but the effective viscosity is roughly \(20\) orders of magnitude higher. This enormous viscosity scales the flow velocity down from centimetres per second to centimetres per year, permitting solid rock to convect efficiently without ever crossing its melting point.

Convection is the dominant heat transport mechanism in:

• Earth’s mantle (solid-state convection of slowly creeping rock, with velocities of ~cm/year)

• Earth’s outer core (liquid metal convection, driving the geodynamo, the convection-powered generator of Earth’s magnetic field; Lecture 4)

• Giant planet interiors (convecting metallic hydrogen and other fluids)

• Planetary atmospheres (rising thermals, Hadley cells)

The efficiency of convection compared to conduction is quantified by the Nusselt number (discussed in section 4).

3.2.3. Radiation

Heat can also be transported by electromagnetic radiation. The radiative heat flux from a surface at temperature \(T\) is given by the Stefan–Boltzmann law:

(3.4)\[ F = \epsilon \sigma T^4 \]

where \(\sigma = 5.67 \times 10^{-8}\) W m\(^{-2}\) K\(^{-4}\) is the Stefan–Boltzmann constant and \(\epsilon\) is the emissivity (close to 1 for most planetary surfaces). Radiation is the dominant mechanism for:

• Heat loss from planetary surfaces to space; the planetary energy balance built on this is the subject of Lecture 5

• Energy transport through optically thin atmospheres

• Cooling of magma ocean surfaces in the early stages of planetary evolution

Inside solid planetary interiors, radiation is generally less important than conduction and convection because rocks are opaque at infrared wavelengths. However, at very high temperatures (\(\gtrsim 2000\) K), radiative heat transfer through partially transparent silicates can become significant.

Fig. 3.8 Schematic of Rayleigh-Bénard convection. A fluid layer of depth \(d\) is heated from below (\(T_h\), hot bottom plate) and cooled from above (\(T_c\), cold top plate). Hot fluid rises in plumes through a thin hot thermal boundary layer (TBL) at the base; cold fluid sinks through a thin cold TBL at the top. Lateral flow along the boundary layers closes the circulation. Convection sets in once the Rayleigh number exceeds the critical value \(\mathrm{Ra}_c\) (Eq. (3.7)). Course-original figure.

3.2.4. Comparison

The key distinction is one of timescale. Conduction is a diffusive process that becomes prohibitively slow for large bodies. Convection is an advective process (the moving material itself carries the heat) that can transport heat over large distances much more rapidly. This difference explains why small bodies (asteroids, small moons) cool primarily by conduction, while larger bodies (terrestrial planets, giant planets) must convect to lose their internal heat.

3.3. Blackboard derivation: The conductive cooling timescale

Blackboard derivation: The conductive cooling timescale

Goal: Derive the characteristic timescale for heat to conduct through a body of size \(L\), and show that conduction alone cannot cool planet-sized bodies, motivating the need for convection.

Setup.

The starting point is the heat diffusion equation (also called the heat equation), which describes how temperature evolves in a conducting medium:

(3.5)\[ \pdv{T}{t} = \kappa \nabla^2 T \]

where \(T\) is temperature, \(t\) is time, \(\kappa\) is the thermal diffusivity (Eq. (3.3)), and \(\nabla^2 T\) is the Laplacian of the temperature field. In one dimension:

\[ \pdv{T}{t} = \kappa \pdv{^2 T}{x^2} \]

This equation says that the rate of temperature change at any point is proportional to the curvature of the temperature profile: regions where the temperature profile is concave up (\(\pdv{^2 T}{x^2} > 0\)) heat up, and regions where it is concave down cool down.

Fig. 3.9 shows the setup: a body of radius \(R\) with a hot interior loses heat by conduction through its outer layers, and the only length in the problem is \(L \sim R\).

Fig. 3.9 Setup sketch of the board derivation of the conductive cooling timescale: a body of size \(L \sim R\) with a hot interior loses heat through a conductive outer shell; the orange arrows mark the outward heat flux \(q\). Course-original figure.

Derivation.

We use dimensional analysis to extract the characteristic timescale. The heat equation relates:

• A time derivative: \(\pdv{T}{t} \sim T/\tau\) (where \(\tau\) is the timescale we seek)

• A spatial derivative: \(\pdv{^2 T}{x^2} \sim T/L^2\) (where \(L\) is the lengthscale)

Substituting into the heat equation:

\[ \frac{T}{\tau} \sim \kappa \frac{T}{L^2} \]

The temperature \(T\) cancels, giving:

(3.6)\[ \boxed{\tau_{\mathrm{cond}} \sim \frac{L^2}{\kappa}} \]

This is the conductive cooling timescale. It tells us how long it takes for a thermal disturbance to propagate a distance \(L\) by conduction alone. The key feature is the \(L^2\) dependence: doubling the size of a body increases its conductive cooling time by a factor of four.

Application.

Using \(\kappa \approx 10^{-6}\) m\(^2\) s\(^{-1}\) (typical for silicate rock), we can estimate the conductive cooling timescale for bodies of different sizes:

Body	Length \(L\)	\(\tau_{\mathrm{cond}} = L^2/\kappa\)	In familiar units
1 m boulder	1 m	\(10^6\) s	~12 days
100 km asteroid	\(10^5\) m	\(10^{16}\) s	~300 Myr
Moon (\(R = 1740\) km)	\(1.7 \times 10^6\) m	\(3 \times 10^{18}\) s	~100 Gyr
Earth (\(R = 6370\) km)	\(6.4 \times 10^6\) m	\(4 \times 10^{19}\) s	~1300 Gyr

A small boulder cools in days, consistent with everyday experience. A 100 km asteroid cools in hundreds of millions of years, which is long but shorter than the age of the solar system, so small asteroids should have cooled and solidified, consistent with the meteorite record. But for the Moon and Earth, the conductive cooling time far exceeds the age of the universe (\(\sim 14\) Gyr). Earth cannot cool by conduction alone. This \(L^2\)-dependence is plotted in Fig. 3.10.

Note: This result tells us something profound: since Earth is losing heat at a rate of ~47 TW [Davies and Davies, 2010], there must be a more efficient transport mechanism operating in its interior. That mechanism is convection. The question of when convection occurs (and how vigorous it is) leads directly to the Rayleigh number.

Fig. 3.10 Conductive cooling timescale \(\tau_{\mathrm{cond}}=L^2/\kappa\) as a function of body lengthscale, for \(\kappa=10^{-6}\) m\(^2\) s\(^{-1}\) (silicate rock). Reference horizontal lines mark the ages of the Solar System (4.57 Gyr) and the universe (13.8 Gyr). Bodies plotted to the right of these lines (Moon, Mars, Earth) cannot have cooled appreciably by conduction alone, motivating the need for convection. Boulders cool in days; ~100 km asteroids cool in a few hundred Myr. Course-original figure.

A specific application of the heat-equation solution that we will return to repeatedly is the half-space cooling model for oceanic lithosphere (Fig. 3.11). A “half-space” refers to a mathematical domain bounded by a single flat surface (here, the seafloor) that extends infinitely downward. In this model, a hot half-space with its surface temperature held fixed cools by conduction, growing a thermal boundary layer whose thickness scales as \(\sqrt{\kappa t}\).

This analytical solution is of major historical importance. In the late nineteenth century, Lord Kelvin used this conductive cooling model to estimate the age of the Earth. Assuming the planet began as a molten sphere and cooled strictly by conduction, he calculated an age of 20 to 400 million years. This result ignited a fierce debate. Kelvin’s timescale was much longer than the few thousand years asserted by religious interpretations, but far too short for geologists and biologists (such as Charles Lyell and Charles Darwin), who recognised that the rock record and evolution required hundreds of millions or billions of years. Kelvin’s mathematics were correct, but his physical premises were flawed: he knew nothing of radiogenic heating (which continuously adds energy) or solid-state mantle convection (which transports heat far more efficiently than conduction).

Fig. 3.11 Half-space cooling solution for the temperature profile beneath cooling oceanic lithosphere. Temperature \(T(z,t)=T_s+(T_m-T_s)\,\mathrm{erf}\!\big(z/2\sqrt{\kappa t}\big)\) is shown for ages of 1 to 200 Myr (\(T_s=273\) K, \(T_m=1600\) K, \(\kappa=10^{-6}\) m\(^2\) s\(^{-1}\)); \(\mathrm{erf}\) is the error function, which rises smoothly from 0 at the surface to 1 at depth. The thermal lithosphere thickens as \(\sqrt{t}\), in agreement with seafloor bathymetry (ocean depth) and heat-flow observations on young oceanic crust [Turcotte and Schubert, 2002]. Course-original figure.

3.4. Rayleigh number and convective vigour

Whether a fluid layer convects or conducts depends on the competition between two forces: buoyancy (which drives convection) and viscous resistance (which opposes it). The balance between these forces is captured by a single dimensionless number.

3.4.1. Onset of convection

Consider a horizontal fluid layer of depth \(d\), heated from below with a temperature difference \(\Delta T\) between the bottom and top boundaries. The hot fluid at the base is less dense than the cool fluid at the top, an inherently unstable configuration. But viscosity and thermal diffusion resist the onset of overturning. Convection begins only when the destabilising buoyancy force exceeds the stabilising effects of viscosity and thermal diffusion.

3.4.2. The Rayleigh number

The Rayleigh number Ra quantifies this balance:

(3.7)\[ \mathrm{Ra} = \frac{\alpha \rho g \Delta T \, d^3}{\kappa \eta} \]

where:

• \(\alpha\) is the thermal expansion coefficient (K\(^{-1}\)): how much the fluid expands when heated

• \(\rho\) is the density (kg m\(^{-3}\))

• \(g\) is gravitational acceleration (m s\(^{-2}\))

• \(\Delta T\) is the temperature difference across the layer (K)

• \(d\) is the layer depth (m)

• \(\kappa\) is the thermal diffusivity (m\(^2\) s\(^{-1}\))

• \(\eta\) is the dynamic viscosity (Pa s)

The numerator (\(\alpha \rho g \Delta T \, d^3\)) represents the buoyancy force driving convection. The denominator (\(\kappa \eta\)) represents the combined dissipative forces resisting it.

3.4.3. Critical Rayleigh number

Convection occurs when Ra exceeds a critical value. For Rayleigh-Bénard convection with rigid boundaries (the relevant limit for a planetary mantle):

(3.8)\[ \mathrm{Ra}_c \approx 1708 \]

(the exact value depends on the boundary conditions; \(\mathrm{Ra}_c = 657.5\) for free-slip boundaries, where fluid slides parallel to the surface without friction, and \(\mathrm{Ra}_c = 1707.8\) for rigid boundaries, where fluid sticks to the wall with zero velocity [Turcotte and Schubert, 2002]). When \(\mathrm{Ra} < \mathrm{Ra}_c\), heat is transported by conduction alone. When \(\mathrm{Ra} \gg \mathrm{Ra}_c\), vigorous convection dominates.

Whether the fluid is stable or convecting depends on both the Rayleigh number and the horizontal size of the circulation. This size is expressed as a non-dimensional horizontal wavenumber \(k = 2\pi d/\lambda\), where \(\lambda\) is the horizontal wavelength of a convection cell and \(d\) is the fluid depth. The threshold for convection across different wavenumbers forms a marginal-stability curve (Fig. 3.12). The fluid first becomes unstable at the absolute minimum of this curve, which defines both the critical Rayleigh number \(\mathrm{Ra}_c\) and the preferred cell size \(k_c\).

Fig. 3.12 Marginal stability curve for Rayleigh-Bénard convection between rigid horizontal boundaries (schematic, normalised to the canonical critical values). Convection grows for parameter combinations above the curve; conduction is the only steady transport mode below it. The minimum of the curve is the critical Rayleigh number \(\mathrm{Ra}_c\approx 1708\), attained at non-dimensional wavenumber \(k_c\approx 3.12\) [Turcotte and Schubert, 2002]. Course-original figure.

For Earth’s mantle, typical parameters are \(\alpha \sim 2 \times 10^{-5}\) K\(^{-1}\), \(\rho \sim 4000\) kg m\(^{-3}\), \(g \sim 10\) m s\(^{-2}\), \(\Delta T \sim 2500\) K, \(d \sim 3 \times 10^6\) m, \(\kappa \sim 10^{-6}\) m\(^2\) s\(^{-1}\), and \(\eta \sim 10^{21}\) Pa s. This gives:

\[ \mathrm{Ra}_{\oplus} \sim \frac{2 \times 10^{-5} \times 4000 \times 10 \times 2500 \times (3 \times 10^6)^3}{10^{-6} \times 10^{21}} \sim 10^7 \text{–} 10^8 \]

This is many orders of magnitude above \(\mathrm{Ra}_c\), confirming that Earth’s mantle convects vigorously [Schubert et al., 2001].

3.4.4. The Nusselt number

The Nusselt number Nu measures how much more efficiently convection transports heat compared to conduction alone:

(3.9)\[ \mathrm{Nu} = \frac{q_{\mathrm{total}}}{q_{\mathrm{cond}}} \]

where \(q_{\mathrm{total}}\) is the actual heat flux and \(q_{\mathrm{cond}} = k \Delta T / d\) is the heat flux that would occur by conduction alone across the same temperature difference. When \(\mathrm{Nu} = 1\), all heat is transported by conduction. When \(\mathrm{Nu} \gg 1\), convection dominates.

A classic result from convection theory relates the Nusselt number to the Rayleigh number through a power law:

(3.10)\[ \mathrm{Nu} \propto \mathrm{Ra}^\beta \]

where \(\beta \approx 1/3\) for simple convection in a layer heated from below [Turcotte and Schubert, 2002]. For Earth’s mantle, \(\mathrm{Ra} \sim 10^7\) to \(10^8\), well above \(\mathrm{Ra}_c \approx 1700\). Adopting the \(\mathrm{Nu} = (\mathrm{Ra}/\mathrm{Ra}_c)^{1/3}\) scaling that vanishes at marginal stability:

\[ \mathrm{Nu} \sim \left(\frac{10^7}{1700}\right)^{1/3} \approx 18 \quad \text{to} \quad \left(\frac{10^8}{1700}\right)^{1/3} \approx 39 \]

This means convection transports heat roughly 20 to 40 times more efficiently than conduction would across the same temperature drop (Fig. 3.13), confirming the essential role of convection in planetary thermal evolution.

Fig. 3.13 Schematic Nusselt-Rayleigh scaling. Below the critical Rayleigh number \(\mathrm{Ra}_c\approx 1708\) heat is conducted only (\(\mathrm{Nu}=1\)); above it the boundary-layer scaling \(\mathrm{Nu}\propto\mathrm{Ra}^{1/3}\) applies (Eq. (3.10)). Earth’s mantle, with \(\mathrm{Ra}\sim 10^7\text{--}10^8\), transports heat \(\sim 20\) to \(40\) times more efficiently than conduction alone under this idealised isoviscous scaling. Course-original figure.

3.5. Thermal boundary layers and mantle convection

3.5.1. The boundary layer structure

A convecting planetary interior does not have a uniform temperature gradient. Instead, it develops a characteristic three-layer structure [Schubert et al., 2001]:

1. Cold thermal boundary layer (TBL) at the top: a thin conductive layer where temperature drops steeply from the hot interior to the cold surface. In Earth, this is the lithosphere, the rigid outer shell of the planet, approximately 100 km thick beneath oceans and up to 200 km beneath continents.

2. Well-mixed convecting interior: the bulk of the mantle, where temperatures vary only slowly with depth, following an adiabatic gradient: the small temperature increase with depth caused by compression alone, with no heat exchanged between rising or sinking material and its surroundings. Vigorous convective stirring keeps this region nearly isothermal on large scales.

3. Hot thermal boundary layer at the base: a thin layer above the core–mantle boundary where temperature rises steeply into the hot core. In Earth, this corresponds to the D’’ layer (pronounced “D double prime”, the lowermost ~200 km of the mantle), where the temperature increases from the mantle adiabat (~2500 K) to the core–mantle boundary temperature (~4000 K).

The temperature drop across the whole mantle is accommodated almost entirely in these two thin boundary layers, while the interior between them is nearly isothermal (Fig. 3.14).

Fig. 3.14 Schematic temperature profile through Earth’s mantle. The cold thermal boundary layer (lithosphere, ~100 km) is conductive and contains a steep gradient from the surface to the well-mixed convecting mantle. Through the bulk mantle the profile follows a near-adiabatic gradient (\(\sim 0.5\) K km\(^{-1}\)). The hot D’’ boundary layer (~200 km) bridges the temperature jump from the mantle adiabat to the molten outer core (\(T_\mathrm{CMB}\sim 4000\) K). Schematic after [Schubert et al., 2001]. Course-original figure.

3.5.2. Plate tectonics: the mobile lid regime

The cold thermal boundary layer at Earth’s surface behaves in a remarkable way: rather than remaining as a stagnant conductive lid, it breaks into rigid plates that move laterally, subduct back into the mantle at convergent boundaries, and are recycled. This is plate tectonics, the surface expression of mantle convection in a mobile lid regime.

Plate tectonics is the most efficient way for a rocky planet to lose heat, because subducting plates carry cold surface material deep into the mantle, while hot material rises at mid-ocean ridges and volcanic hotspots (Fig. 3.15). This vigorous recycling is why Earth’s surface heat flow is as high as it is. The surface geology of plate boundaries is treated in Lecture 7, and the contrast with Venus’s episodic resurfacing in Lecture 9.

Fig. 3.15 Cross-section through the upper mantle and lithosphere illustrating the principal plate-tectonic settings: divergent oceanic ridges (mantle upwelling, new crust formed), convergent boundaries (oceanic plate subducts beneath island arc or continent, generating arc volcanism), transform boundaries, and a hot-spot plume rising from depth. The lithosphere is the cold thermal boundary layer of mantle convection; its mobile expression on Earth is plate tectonics. Credit: USGS, public domain.

3.5.3. The stagnant lid regime

Most other rocky bodies in the solar system (including Venus, Mars, Mercury, and the Moon; Lecture 9, Lecture 10) appear to operate in a stagnant lid regime, where the cold thermal boundary layer forms a single, thick, immobile shell. Without plate tectonics, heat can only escape through the lid by conduction (and volcanism, where magma punches through the lid). This is less efficient than plate tectonics, so stagnant lid planets cool more slowly and may retain more internal heat. The full taxonomy of plausible tectonic modes (Fig. 3.16) is broader than the simple binary, including sluggish-lid and episodic regimes intermediate between the active and stagnant end-members [Lenardic, 2018].

Fig. 3.16 Tectonic modes of terrestrial planets, beyond the simple mobile-versus-stagnant binary. An active lid moves as plate tectonics or as distributed deformation; a sluggish lid shows ridge-only spreading or regional to global mobility driven by sub-lithospheric convection; episodic (transient) modes switch between states globally or locally, leaving non-recyclable “continental” regions; a stagnant lid is cold (thermal stress only) or hot (heat pipes with vertical cycling, or plumes with lower-crust recycling). Reproduced from [Lenardic, 2018], Fig. 2.

This spectrum is not just a classification scheme: numerical mantle convection models reproduce it from a single set of physics. Depending on the strength of the lithosphere and the thermal state of the interior, the same simulations settle into mobile, episodic, sluggish, stagnant, or magmatism-dominated (“squishy”) lid regimes (Fig. 3.17; [Lyu et al., 2025]).

Fig. 3.17 Six tectonic lid regimes emerging in numerical mantle convection simulations of terrestrial planets. Each annulus shows one simulation snapshot, with temperature on the left half and basalt fraction on the right half: mobile, stagnant, episodic, sluggish, plutonic-squishy, and episodic-squishy lid. Reproduced from [Lyu et al., 2025], Fig. 1 (CC BY 4.0).

Why Earth has plate tectonics while other rocky planets do not is one of the major unsolved problems in geophysics. Factors likely include:

• The presence of water in the mantle, which weakens rocks and enables the formation of narrow shear zones (plate boundaries)

• The size of the planet: a larger planet has more vigorous convection and higher stresses in the lithosphere

• The thermal history: the balance between internal heating and surface cooling

3.5.4. Whole-mantle or layered convection?

A long-standing question for Earth is whether the mantle convects as a single layer or as two. The mantle transition zone ends at a depth of 660 km with a phase transition in olivine that resists vertical flow: it deflects some subducting slabs and could, in principle, separate the mantle into two independently convecting shells. In layered convection, separate circulation systems would operate above and below 660 km, and the upper and lower mantle would exchange little material. In whole-mantle convection, convection cells span the full mantle depth: slabs sink through the 660 km boundary into the lower mantle, and plumes rise from the core-mantle boundary to the surface (Fig. 3.18).

Seismic tomography has settled the question in favour of whole-mantle convection. Images of the mantle beneath North America show the Farallon slab (the remnant of an ancient oceanic plate that subducted beneath the continent) descending far below 660 km toward the core-mantle boundary, and global tomographic models trace plume conduits rooted near the core-mantle boundary beneath major hotspots [French and Romanowicz, 2015]. The 660 km phase transition acts as a partial barrier that slows and deforms slabs, but it does not stop them: the mantle convects as one layer.

Fig. 3.18 Layered versus whole-mantle convection. In layered convection (left), the phase transition at 660 km depth would act as an impermeable boundary, with separate circulation systems in the upper and lower mantle. In whole-mantle convection (right), the 660 km boundary is only a partial barrier: convection cells span the full mantle depth, subducting slabs sink through the transition zone to the deep mantle, and plumes rise from the core-mantle boundary (CMB) to feed surface volcanism. Seismic tomography supports the whole-mantle picture. Course-original figure.

3.5.5. Mantle plumes

In addition to the broad convective circulation, hot upwellings can develop from the hot thermal boundary layer at the base of the mantle. These mantle plumes are narrow columns of anomalously hot material that rise through the mantle and produce localised volcanism at the surface (so-called hotspots, e.g., Hawaii, Iceland, Yellowstone; Fig. 3.20). Plumes are a direct consequence of the thermal boundary layer structure: just as the cold TBL can become gravitationally unstable and sink (subduction), the hot TBL can become unstable and rise. Whole-mantle seismic tomography directly images broad plume conduits rooted near the core-mantle boundary beneath several major hotspots [French and Romanowicz, 2015]. A cutaway view of the layered Earth that ties the mantle-convection picture to the inner-core/outer-core/mantle/crust geometry is shown in Fig. 3.21.

Plumes and plates represent two distinct modes of mantle upwelling (Fig. 3.19). Upwelling beneath a mid-ocean ridge is passive: the diverging plates pull apart, and ambient-temperature mantle rises broadly to fill the gap, melting by decompression (the drop in pressure lowers the melting point below the rock’s actual temperature). A plume, in contrast, is an active upwelling: a narrow conduit of material some 200 to 300 K hotter than its surroundings, driven upward by its own buoyancy and rooted in the deep mantle. Because a plume is anchored in the deep mantle while the plate above it moves, a long-lived plume produces an age-progressive chain of volcanoes such as the Hawaiian-Emperor chain. When a large plume head first arrives beneath the lithosphere, it can generate a Large Igneous Province (LIP): a region covered by exceptionally voluminous eruptions of basalt (the dark volcanic rock produced when mantle rock melts), far larger than any historical volcanic activity. Examples include the Deccan Traps in India, the Siberian Traps, and the Ontong-Java Plateau in the western Pacific. The eruption ages of several LIPs coincide with mass extinction events in the fossil record, most prominently the Siberian Traps with the end-Permian extinction [Courtillot and Renne, 2003].

Fig. 3.19 Two modes of mantle upwelling. Beneath a mid-ocean ridge (left), upwelling is passive: diverging plates draw a broad column of ambient-temperature mantle upward, which melts by decompression at shallow depth. In a mantle plume (right), upwelling is active: a narrow conduit of anomalously hot material (\(\Delta T \approx 200\) to \(300\) K) rises buoyantly from the deep mantle, feeding a hotspot volcano; the moving plate carries older, extinct volcanoes away from the plume. Course-original figure.

Fig. 3.20 Global distribution of plate boundaries (divergent: thin lines; convergent: hatched; transform: dotted) with selected major hotspots (orange dots) including Hawaii, Iceland, Yellowstone, Galápagos, Azores, and Afar. Hotspots cluster within plate interiors as well as on ridges and reflect the surface expression of long-lived deep-mantle plumes. Credit: USGS, public domain.

Fig. 3.21 Cutaway view of Earth’s interior, showing the layered structure from crust to inner core. The mantle convects in broad circulation patterns; the cold surface boundary layer (lithosphere) breaks into tectonic plates, while the hot boundary layer at the core–mantle boundary spawns mantle plumes. Credit: Wikimedia Commons, CC BY-SA 3.0.

3.6. Surface heat flow

The heat flowing out of a planet’s surface is one of the most directly measurable quantities in planetary geophysics, and it provides a fundamental constraint on the thermal state of the interior.

3.6.1. Earth’s surface heat flow

Earth’s average surface heat flux is approximately \(90\) mW m\(^{-2}\), totalling about 47 TW when integrated over the entire surface area [Davies and Davies, 2010]. This heat flow is not uniform:

• Oceanic regions account for about 70% of the total heat loss, despite covering about 60% of the surface. The high oceanic heat flow reflects the thinness of oceanic lithosphere and the creation of new, hot crust at mid-ocean ridges.

• Continental regions account for about 30% of the total heat loss. Continental heat flow is lower on average because the thick continental lithosphere is an effective insulator, and because continents preferentially accumulate heat-producing elements (U, Th, K) in the crust.

The spatial pattern of heat flow (highest at mid-ocean ridges, decreasing with the age of the ocean floor, lowest in old continental shields, the ancient stable cores of the continents; Fig. 3.22) is a direct expression of plate tectonics and confirms that convection controls Earth’s heat loss. The age-dependence is captured by the half-space cooling prediction \(q\propto t^{-1/2}\) (Fig. 3.23), which describes the data well over young oceanic crust and converges to a finite asymptote at old ages.

Fig. 3.22 Global surface heat-flow map of Earth (mW m\(^{-2}\)). The map combines a half-space cooling model in regions of young oceanic crust with the global compilation of \(\sim 70{,}000\) borehole heat-flow measurements from [Lucazeau, 2019]. Mid-ocean ridges (orange) are the dominant heat-loss feature; cratonic continental shields (dark blue) are the coldest. Total integrated heat loss is \(\sim 47\) TW [Davies and Davies, 2010]; Lucazeau et al. (2019) report 40–42 TW from direct measurements alone. Credit: Fabio Crameri, Wikimedia Commons, CC BY-SA 4.0, with the underlying dataset from [Lucazeau, 2019].

Fig. 3.23 Surface heat flux \(q\) as a function of seafloor age. The half-space cooling model predicts \(q(t)=k(T_m-T_s)/\sqrt{\pi\kappa t}\propto t^{-1/2}\) (blue curve, \(k=3.3\) W m\(^{-1}\) K\(^{-1}\), \(\Delta T=1327\) K, \(\kappa=10^{-6}\) m\(^2\) s\(^{-1}\)); the plate-cooling model instead converges to a finite asymptote (~48 mW m\(^{-2}\), dashed red) at old ages once the lithosphere reaches its equilibrium thickness [Stein and Stein, 1992]. Grey points show the global compilation of marine heat-flow measurements of Richards et al. [2018], binned in 2.5 Myr age windows (filled circles: bin median with interquartile range; open circles: bins containing a single measurement). Over young crust the compilation retains only measurements at thickly sedimented sites away from seamounts, where hydrothermal circulation (seawater circulating through the hot, permeable young crust) is suppressed, and corrects for the thermal effect of rapid sedimentation; these filtered data follow the conductive \(t^{-1/2}\) prediction, while raw unfiltered surveys over young crust fall well below it because hydrothermal circulation carries part of the heat. At old ages the measurements level off near the plate-cooling asymptote. Course-original figure.

3.6.2. Comparison across the solar system

Body	Surface heat flux (mW m\(^{-2}\))	Total heat loss (TW)	Dominant source
Earth	~90	~47	Radiogenic + primordial
Moon	~10–15	~0.3	Primordial (largely cooled)
Mars	~15–25 (estimated)	~2–4	Radiogenic + primordial
Io	~2000–2500	~105	Tidal dissipation

The Moon and Mars have much lower surface heat flow than Earth, which reflects their smaller sizes (faster cooling through their conductive cooling timescale), the absence of plate tectonics (stagnant lid), and their smaller inventories of heat-producing elements. Io stands out dramatically: its surface heat flux is 20–30 times higher than Earth’s, entirely driven by tidal heating. NASA’s InSight mission (Fig. 3.24) recently provided the first direct seismological constraints on Mars’s interior, including the radius of its core (Lecture 10).

Fig. 3.24 NASA InSight mission concept showing how marsquake travel times constrain Mars’s interior structure. The lander’s seismometer (SEIS) records the arrivals of P- and S-waves (compressional and shear seismic waves) from quakes and from impactors, which combined with InSight’s heat-flow probe (HP\(^3\)) and rotation-rate measurements yielded the first direct seismic constraints on the radius of the Martian core, the thickness of the crust, and the structure of the mantle [Khan et al., 2021, Stähler et al., 2021]; the 2023 reanalyses place the metallic-core radius at \(\sim\)1650–1675 km beneath a molten silicate layer [Khan et al., 2023, Samuel et al., 2023]. Credit: NASA/JPL-Caltech, public domain.

3.6.3. Connection to tectonic regime

A planet’s surface heat flow is intimately linked to its tectonic regime:

• Mobile lid (plate tectonics): Efficient heat loss through subduction and seafloor spreading → high surface heat flux

• Stagnant lid: Heat loss limited to conduction through a thick lid plus episodic volcanism → lower surface heat flux

• Heat-pipe volcanism: On bodies like early Earth or Io, vigorous volcanism can transport heat by erupting hot magma at the surface, which then cools and is buried by subsequent flows, an alternative to plate tectonics for efficient heat loss [Turcotte and Schubert, 2002]

Fig. 3.25 compares the three heat-loss modes.

Fig. 3.25 Heat loss under three tectonic regimes, in schematic cross-section. (a) Mobile lid: hot mantle rises at ridges and the cold plate sinks at subduction zones, so the surface heat flux is high. (b) Stagnant lid: heat leaves by conduction through a thick lid, with episodic volcanism, so the surface heat flux is lower. (c) Heat pipe: erupted magma cools at the surface and is buried by later flows, which carries heat out efficiently without plates, as on Io. Not to scale. Course-original figure.

3.7. Tidal dissipation

We introduced tidal forces in Lecture 2 as a consequence of the gradient in gravitational acceleration across an extended body. Here we explore the consequences: when tidal forces are combined with orbital eccentricity, they become a sustained source of internal heating that drives some of the most spectacular geological activity in the solar system.

3.7.1. How tidal heating works

On a circular orbit, a tidally locked moon maintains a constant distance from its planet, and the tidal bulge points steadily toward the planet: there is no flexing and no heat generation. But if the orbit has even a small eccentricity, the distance varies periodically: at periapsis (the closest point of the orbit) the tidal force is stronger and the moon moves faster than it rotates, and at apoapsis (the farthest point) it is weaker and the moon rotates faster than it orbits. The result is a continually varying tidal deformation: the body is rhythmically squeezed and stretched every orbit (Fig. 3.26). The energy dissipated in this cyclic deformation is converted to heat.

Fig. 3.26 Origin of tidal heating on an eccentric orbit. Left: at periapsis (\(r=a(1-e)\)) the moon is closest to the planet and the raised tidal bulge is largest. Right: at apoapsis (\(r=a(1+e)\)) the bulge is smallest. The cyclic flexing of the body’s shape on each orbit dissipates orbital energy as heat at a rate \(\dot E_\mathrm{tidal}\propto e^2/Q\), where \(Q\) is the tidal quality factor (orbital geometry exaggerated for clarity). Course-original figure.

The tidal heating rate depends on:

• Orbital eccentricity \(e\): zero eccentricity means zero tidal heating

• Orbital distance \(a\): tidal forces fall off steeply with distance (heating \(\propto a^{-15/2}\) for a given eccentricity, combining the explicit \(a^{-6}\) in Eq. (3.11) with the Keplerian scaling \(n \propto a^{-3/2}\) of the mean motion)

• Internal dissipation, parameterised by the tidal quality factor \(Q\) (lower \(Q\) means more dissipation per deformation cycle)

• Body rigidity: a stiffer body deforms less per cycle but may dissipate more depending on its rheology (how the material deforms and flows under stress)

These dependencies are collected in the standard expression for the tidal heating rate of a synchronously rotating satellite on an orbit of small eccentricity [Peale et al., 1979]:

(3.11)\[ \dot E_\mathrm{tidal} = \frac{21}{2}\,\frac{k_2}{Q}\,\frac{G M_p^2 R^5 n\, e^2}{a^6} \]

where \(M_p\) is the mass of the planet, \(R\) the radius of the satellite, \(n = \sqrt{G M_p/a^3}\) the orbital mean motion, and \(k_2\) the tidal Love number, which measures how strongly the satellite’s shape responds to the tidal potential (larger \(k_2\) means a larger tidal bulge for the same forcing). Every dependence discussed above is visible in this formula: the heating vanishes as \(e^2 \to 0\), scales inversely with \(Q\), grows steeply with satellite radius as \(R^5\), and falls off with distance as \(a^{-6} \times n \propto a^{-15/2}\).

3.7.2. Io: the most volcanically active body in the solar system

Jupiter’s innermost Galilean moon Io (radius 1821 km, similar in size to our Moon) is the most extreme example of tidal heating in the solar system; the giant-planet moon systems are surveyed in Lecture 11. Despite having no significant radiogenic heat sources at present, Io produces a staggering ~\(10^{14}\) W of tidal heat [Spencer et al., 2000], roughly twice Earth’s total heat output, from a body with only 1.5% of Earth’s mass.

This was predicted before it was observed. In a landmark 1979 paper, Peale, Cassen, and Reynolds calculated that tidal dissipation in Io, maintained by the Laplace resonance with Europa and Ganymede (Lecture 2), should be sufficient to melt Io’s interior [Peale et al., 1979]. Just days after the paper was published, the Voyager 1 spacecraft flew past Io and discovered active volcanic eruptions, a stunning confirmation of theory.

The Laplace resonance is essential: it forces Io’s orbital eccentricity to remain at \(e \approx 0.004\) (Fig. 3.27). Without the resonance, tidal forces would circularise Io’s orbit on a timescale of \(\sim 10^7\) yr and tidal heating would cease. The resonance continuously pumps the eccentricity, maintaining a steady heat source.

Fig. 3.27 The Laplace resonance: the inner three Galilean satellites Io, Europa, and Ganymede orbit Jupiter in a 4 : 2 : 1 mean-motion ratio. The resonant configuration forces Io’s orbital eccentricity to a small but non-zero value (\(e\approx 0.004\)). This forced eccentricity sustains the tidal heating that powers Io’s volcanism; in the absence of the resonance, Io’s orbit would circularise on a timescale of \(\sim 10^7\) yr and the heating would cease. Course-original figure.

The consequences for Io are dramatic (Fig. 3.28, Fig. 3.29, Fig. 3.30):

• Surface heat flux of ~2–2.5 W m\(^{-2}\), about 20–30 times higher than Earth’s

• 343 active volcanic centres catalogued from Galileo, New Horizons, and ground-based monitoring [Davies et al., 2024]

• The entire surface is resurfaced by lava flows on a ~Myr timescale, so there are essentially no impact craters

• Internal structure consistent with a partially molten mantle (asthenosphere) [Keane et al., 2023]

Fig. 3.28 Galileo full-disk colour composite of Io, showing the global distribution of volcanic centres, lava flows, and yellow-red sulfur deposits. Davies et al. [2024] catalogue 343 active volcanic centres from combined Galileo, New Horizons, and ground-based monitoring; the entire surface is resurfaced by lava on a \(\sim 1\) Myr timescale, leaving essentially no impact craters. Credit: NASA/JPL/University of Arizona, public domain.

Fig. 3.29 The Tvashtar volcanic plume on Io, captured by the New Horizons spacecraft during its Jupiter flyby in February 2007. The plume rises ~300 km above Io’s surface, driven by the eruption of silicate magma heated by tidal dissipation. Io’s extreme volcanism is powered entirely by tidal heating from the Laplace resonance with Europa and Ganymede. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Southwest Research Institute, public domain.

Fig. 3.30 Global map of Io’s 343 catalogued volcanic hot spots, in Mollweide (equal-area) projection centred on the anti-Jovian point, from nearly 30 years of combined Galileo, Juno, New Horizons, and ground-based monitoring. Marker size and colour indicate each hot spot’s estimated power output, increasing from blue to yellow; prominent volcanoes are named, with Loki Patera the most powerful at \(6000\) to \(12{,}000\) GW. The catalogued hot spots emit \(58 \pm 1\) TW, approximately 55% of Io’s total endogenic heat output of \(105 \pm 12\) TW; the remainder is attributed to buried and subsurface volcanic activity. The emission pattern is consistent with tidal-dissipation heating driven by the Laplace-resonance forcing of Io’s orbital eccentricity. Reproduced from Davies et al. [2024], Fig. 1.

3.7.3. Enceladus: a tiny moon with a big secret

Saturn’s small moon Enceladus (radius 252 km) provides another dramatic example of tidal heating. In 2005, the Cassini spacecraft discovered jets of water ice and vapour erupting from four parallel fractures (dubbed “tiger stripes”) near the south pole [Porco et al., 2006]. Subsequent far-infrared measurements with Cassini’s Composite Infrared Spectrometer (CIRS) showed that the south polar terrain emits \(\sim 16\) GW of thermal power (\(15.8 \pm 3.1\) GW; Howett et al. [2011]), revising upward an earlier \(\sim 6\) GW estimate, and far more than can be explained by radioactive decay alone for such a small body.

The heat source is tidal dissipation, driven by Enceladus’s 2:1 orbital resonance with the larger moon Dione. The heating is sufficient to maintain a global subsurface ocean of liquid water beneath an ice shell \(\sim\)15–25 km thick [Nimmo and Pappalardo, 2016]. The erupted material from the tiger stripes (Fig. 3.31, Fig. 3.32) feeds Saturn’s E ring and provides direct samples of the subsurface ocean: analysis by Cassini’s instruments revealed the presence of salts, silica nanoparticles (indicating hydrothermal activity on the ocean floor), and even complex organic molecules. The thermal-emission map in Fig. 3.33 localises the heat output along the four “tiger stripe” fractures.

Fig. 3.31 Full-disk false-color mosaic of the anti-Saturn hemisphere of Enceladus. The prominent “tiger stripe” fractures are visible in blue near the south pole. Credit: NASA/JPL/Space Science Institute, public domain.

Fig. 3.32 Geysers of water ice erupt from the “tiger stripe” fractures near the south pole of Enceladus, imaged by the Cassini spacecraft. The jets are driven by tidal heating that maintains a subsurface liquid water ocean. The erupted material feeds Saturn’s E ring and provides evidence for hydrothermal activity on the ocean floor. Credit: NASA/JPL/Space Science Institute, public domain.

Fig. 3.33 High-resolution Cassini CIRS thermal scan (colour overlay) along a segment of one tiger-stripe fracture in Enceladus’s south polar terrain, draped on a visible-light mosaic. The warmest emission hugs the fracture and falls off within a few kilometres to either side, localising the heat loss to the fractures themselves. Integrated over the full south polar terrain, the thermal output is \(15.8 \pm 3.1\) GW [Howett et al., 2011], revising upward the earlier \(\sim 6\) GW estimate and far above any plausible radiogenic budget for a body of this size. Credit: NASA/JPL/GSFC/SwRI/SSI, public domain.

3.7.4. Europa and Titan: subsurface oceans

Tidal heating is also implicated in maintaining subsurface oceans on other outer solar system moons (Lecture 11) [Nimmo and Pappalardo, 2016]:

• Europa (Jupiter): The Laplace resonance forces Europa’s eccentricity, generating tidal heat (though less than Io’s, because Europa is further from Jupiter). Magnetic field measurements by the Galileo spacecraft, combined with gravity data, strongly suggest a global liquid water ocean ~100 km deep beneath an ice shell of ~15–25 km (Fig. 3.34).

Fig. 3.34 Inferred interior structure of Europa (NASA/JPL artist’s cutaway). From the centre outward: an iron-nickel core, a rocky silicate mantle, a global liquid-water ocean (\(\sim 100\) km deep, blue), and an outer ice shell (\(\sim 15\text{--}25\) km thick). The ocean is maintained against freezing by tidal dissipation in the ice shell and silicate interior, forced by the Laplace resonance with Io and Ganymede. Credit: NASA/JPL, public domain.

• Titan (Saturn): Saturn’s largest moon shows evidence for a subsurface water ocean from measurements of its rotational state and tidal response by Cassini. Titan’s ocean may be maintained by a combination of tidal heating and radiogenic heat in its rocky core.

These “ocean worlds” are among the most promising targets in the search for extraterrestrial life, because they possess liquid water, chemical energy sources (from rock–water interactions at the ocean floor), and the organic building blocks needed for biology. We will return to this topic in Lecture 14.

3.8. Recent advances

NASA’s InSight mission (2018–2022) provided the first direct seismological measurements of another planet, fundamentally advancing our understanding of Mars’s thermal state. Seismic wave travel times revealed Mars’s core size, mantle structure, and crustal thickness [Khan et al., 2021, Stähler et al., 2021]; reanalyses place the metallic-core radius at \(\sim\)1650–1675 km beneath a molten silicate layer (Khan et al. [2023]: \(150 \pm 15\) km thick) [Khan et al., 2023, Samuel et al., 2023] (see also Lecture 8). The mission’s heat flow probe (HP\(^3\)) was unable to penetrate to the required depth due to unexpected soil properties, but the attempt provided valuable constraints on the thermal conductivity of the regolith (the fragmented surface soil) and on near-surface heat flow.

Tidal heating models for icy moons have been refined using updated rheological models and orbital evolution calculations [Nimmo and Pappalardo, 2016]. Recent work suggests that tidal dissipation in Enceladus may be concentrated in its ice shell rather than its rocky core, affecting predictions for the longevity and temperature of its subsurface ocean. For Europa, estimates of ice shell thickness have been narrowed to \(\sim\)15–25 km using multiple independent constraints, with implications for the Europa Clipper mission [Howell and Pappalardo, 2020] (see also Lecture 14).

Updated analyses of chondritic meteorites continue to refine our understanding of the radiogenic heating budget available during early solar system evolution [Lichtenberg et al., 2023], with implications for the thermal history of planetesimals and the onset of differentiation (Lecture 4).

3.9. Looking ahead to Lecture 4

This lecture followed the heat: where it comes from and how it escapes. Lecture 4 follows the chemistry that the heat enables. A molten planet separates into a metallic core and a silicate mantle, and the elements sort themselves between the two: siderophile (“iron-loving”) elements follow the metal down, while lithophile elements stay behind in the silicates. The Hf-W chronometer turns this sorting into a clock for how fast core formation happened. A convecting metallic core, finally, generates the magnetic field that shields a planet’s atmosphere. The heat sources assembled here set the pace for all of it: differentiation needs melting, and the dynamo needs a cooling core.

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