Sources and movement of heat within planets
Primordial heat
The general term for the heat imparted to a planetary body by the processes of its formation and differentiation. It has two major components:
- Accretional heat: Addressed in previous lecture, this is the heat generated by the conversion of the kinetic energy of impacting bodies to thermal energy. It is concentrated at the surface.
- Gravitational release: The gravitational potential of dense materials is converted to heat during differentiation. As iron, for example, "falls" to the center of the differentiating body, its movement gives rise to friction that releases heat according to the formula:
Energy E = - G M m / r
where G is the gravitational constant, M and m are mass, and r is distance from the center.
Thus, once the heat of accretion gets differentiation going, it causes a positive feedback with the heat of gravitational release, releasing more heat.
Radiogenic heat
Historical Note - the search for Earth's true age:
Thermodynamics: Sir William Thomson, Lord Kelvin, during the late 19th century, assumed that the Earth had originally been molten then, using average melting point of rocks and the laws of thermodynamics, determined that the Earth would completely solidify within 20 million years. Both uniformitarians and evolutionists were uncomfortable, since their notions required a much older Earth, but the quantitative rigor of Thomson's approach made his the most prestigious estimate of his day.
Lord Kelvin could not have known that new heat is generated inside the Earth by radioactive decay (nuclear fission), because the process had not been discovered.
The discovery of radioactivity: Ironically, radioactive decay, which frustrated Kelvin's purpose, ended up providing the true key to the absolute dating of rocks.
- Antoine Henri Becquerel (1852-1908): Discovered natural radioactivity (1896). In the following years, a large number of radioactive isotopes and their daughter products became known.
- Pierre (1859-1906) and Marie (1867-1934) Curie: Discovered that the radioactive element radium continuously releases newly generated heat - radiogenic heat. With this discovery, it became clear that the decay of radioactive substances provided a continuous source of new heat that Thomson hadn't accounted for. The Earth might, indeed, be much older than his calculations indicated. But how old?
Definitions:
An ongoing source of heat is the decay of the nuclei of radioactive isotopes. First, definitions:
- Radioactivity: The spontaneous decay of the nucleus of one element into that of another. Radioactive decay is unpredictable, however nuclei of a given radioactive substance have a specific probability of decaying in any given moment.
- Radioactive isotope: (A.K.A. Radioisotope) Some isotopes are naturally stable - they will not undergo radioactive decay. Others are inherently radioactive. A given radioactive substance will decay into a specific daughter product that may, itself, be radioactive or stable.
Note: To denote isotopes, we write the atomic mass in superscript above and before the chemical element symbol. Thus, 12C, 13C, and 14C are isotopes of carbon. (Only 14C is radioactive.)
The key point is that radioactive decay is a continuing source of new heat: For example, 26Al -> 26Mg + γ. In this case energy is emitted as high-energy gamma radiation.
- How much heat does it generate and how has this amount varied over time?
- Where is the heating concentrated?
Half-lives: To answer the first question above, we need to look at the time factor of radioactive decay. The fact that a given radioactive substance's nuclei have the same stochastic chance of decaying in a given interval means that when we observe huge numbers of nuclei in a sample, we find that half of them will decay in a specific time period - the half-life. Thus, we can use the methods of radiometric dating to determine the age of a sample where we can determine the relative quantity of a radioactive substance and its daughter product. This requires a closed system - i.e. that materials have not percolated into or leaked out of the sample after it formed.
Answer: It took a generation for the wrinkles to be ironed out of the techniques for radiometric dating, but that answer is that the Solar System and its planetary bodies seem to have formed ~ 4.571 ga.
We care because the many radioactive isotopes present at the beginning of the Solar System will have decayed at different rates. It is convenient to think of these as:
- Short-lived radioisotopes: Those with half lives measured in millions of years (abbreviated Ma) or fewer. Typically we don't expect to encounter these in the modern Solar System because only undetectable traces of them will remain from the time of their formation in supernovae or previous stars prior to the formation of the Solar System. One familiar example is 14C, which we see in the modern world because it is continuously formed in the upper atmosphere by the interaction of stable carbon and solar radiation.
Some short-lived radioisotopes thought to have been common in the early Solar System and their half-lives:
10Be 1.5 Ma 26Al 0.73 Ma 36Cl 0.30 Ma 41Ca 0.10 Ma 53Mn 3.7 Ma 60Fe 1.5 Ma 107Pd 6.5 Ma 129I 16 Ma Short-lived radioisotopes are thought to have been a major source of heat driving planetary differentiation in the early Solar System. 26Al, thought to have been present in significant quantities, is especially important.
- Long-lived radioisotopes: Those with half-lives measured in hundreds of millions or billions of years (abbreviated Ga). Present in the modern world in detectable quantities.
Some long-lived radioisotopes and their half-lives:
40K 1.3 Ga 232Th 13.9 Ga 235U 0.71 Ga 238U 4.46 Ga
Where is radiogenic heat concentrated? Most long-lived radioisotopes are lithophiles. As a result, radiogenic heat arises in the mantle and crust. In fact, heavy atoms like those of thorium or uranium are incompatible that is, their large size inhibits their ability to assume a close packing in high-pressure crystals, so they tend to be concentrated in the crust. We can expect more radiogenic heat to have been produced in a planet with a relatively thick mantle and crust, like Mars, than in one with very thin ones, like Mercury.
Tidal heat
One last ongoing source of planetary heat comes from tidal forces. We have discussed the nature of tides already, but not their effect on objects that experience them. In a nutshell:
Whenever a tidal bulge is raised, frictional heat is generated. If a large bulge is being raised in solid material, considerable frictional heating results.
Io - the extreme example:
- The strength of Jupiter's gravitational pull is proportional to the inverse square of its distance. Because Io orbits very close to Jupiter, this means that its Jupiter-facing side is pulled noticeably more strongly than the side facing away.
- Remember, Io's rotation is synchronous with its orbit. If this orbit were circular, the tidal bulge would be a permanent, unchanging feature.
- But Io's orbit is actually strongly eccentric. That means that when it is at its greatest distance (apogee) it is being stretched less strongly than when it is at its closest approach (perogee). In response to these changes in gravitational stretching, Io changes shape, experiencing a rock tide of roughly 100 m with every orbit. As tides go, this is huge (Compare with Earth's lunar rock tide of ~38 cm).
- Io's orbital period is less than two days. Thus, its tides are not only enormous but rapid.
- The result is friction resulting in tremendous heating. This is the force that drives Io's extensive volcanic activity.
- Adding color to this, Io is locked in a 2:1 orbital resonance with Europa and a 4:1 resonance with Ganymede, meaning that those bodies also rhythmically contribute their tidal forces.
Orbital resonances: Left alone, eccentric orbits tend slowly to evolve into more nearly circular ones over time. In some cases, orbital resonances - regular interactions between orbiting objects develop. This typically happens when the ratios of two object's orbital periods makes a simple fraction. One effect of such resonances is to maintain the eccentricities of the participating bodies' orbits. The orbits of Ganymede, Europa, and Io form a 1:2:4 resonance with the result that the eccentricities of Io's and Europa's orbits are maintained.
Because it:
- Is closest to Jupiter
- Has the shortest orbital period
- Has great orbital eccentricity
Io is the Solar System champ for tidal heating, but it can have significant effects elsewhere. All moons on this list show signs of significant geologic activity.
- Europa with its thin ice and liquid oceans is also tidally heated. Participates in a 1:2 resonance with Io and a 2:1 resonance with Ganymede
- Enceladus, which participates in a 2:1 resonance with Dione is like a slightly tamer version of Io, showing continuous cryovolcanism.
- Ganymede, participates in a 1:4 resonance with Io and a 1:2 resonance with Europa. Because its eccentricity is currently low, it does not experience significant tidal heating, but it probably did in the past.
- Miranda, although not currently in a resonant orbit, is thought to have once participated in a 1:3 resonance with Umbriel.
Solar Heating - NOT!
One heat source we haven't considered is the sun. Solar radiation is a significant contributor of heat to the planets' surfaces and atmosphere, but not to their interiors. When we think of a planet's intrinsic internal heat, we can ignore the sun.
What happens to the heat?
Black body radiation: Remember our hypothetical non-reflective body that radiated electromagnetic energy at different wavelengths depending on its temperature? All objects with any thermal energy radiate in this way. For something as hot as the sun, the majority is in visible wavelengths. Planets radiate in the lower-energy realm of the infrared. Thus, orbiting infrared observatories are powerful tool in the search for exoplanets. That's all tame compared to the accretion disk of a black hole that may blaze away in X-rays (we won't even consider gamma-ray bursts). Nevertheless, all bodies radiate away heat and cool down over time. For planets, the rate at which this happens depends on:
- The amount of new heat generated by radiogenic and tidal heating.
- The planet's size.
Surface to volume ratio: Size matters because surface area and volume scale up at different rates.
- Surface area: A = 4πr2
- Volume: V = 4/3πr3
- A sphere with a 1 km radius has a surface area of 12.57 km2 and a volume of 4.19 km3. It's surface to volume ratio is 3 km-1.
- If we double the radius, we get A = 50.27 km2, V = 33.51 km3and A/V of 1.5 km-1.
- Global mean Heat Flow for the Earth is 87±2 mW/m2
- Global mean Heat flow for the Moon is 22-31 mW/m2
Moving heat around:
Many other factors are involved, however. How is heat conveyed to the surface? There are three basic mechanisms:- Conduction
- Convection
- Advection
Convection: This is the process by which material circulates through a region that is unevenly heated. In a tea kettle, for instance:
- Water is heated at the bottom.
- It rises.
- Surface water radiates its heat into the air and cools.
- Cooler water sinks into the space evacuated by the rising warmer water and begins to warm, while the warmer water reaches the surface and cools.
- The process repeats, yielding a top to bottom circulation of water.
Convection cells in coffee
The tops of convection cells (units of convective circulation) can often be seen in cups of tea or black coffee. Condensing water vapor marks to top of rising columns of warm water. Dark lines separating them marks the location os sinking cooler water.) That we see similar features in the nitrogen ice cap of Pluto's Sputnik Planum demonstrates that it is temperature contrast, not absolute temperature, that drives this process.
Thermodynamics tell us that convection definitely occurs inside Earth's mantle. Suppose the Earth had cooled from conduction only. Thermodynamic calculations show that a given parcel of heat could have moved only 400 km in 5 Ga by conduction alone. That would mean that below 400 km, everything must be molten. The ability of S waves to propagate at much greater depths shows us that this is not true. Convection must be at work, also. In fact, plots of changes of temperature with depth in Earth show sudden discontinuities in temperature at the Moho and the CMB. Arguably this is because at those boundaries, conduction is the only means of moving heat.
The notion of convection in the solid mantle seems counterintuitive, but it is real. (See simulation and seismic data above). Remember ductile deformation? Many solids deform ductilely. At the temperatures and pressures of most of the mantle, silicates deform ductilely as well just more slowly.
Advection: The transport of heat by the simple upward movement of hot material passing from a solid to a liquid state.
Key concepts and vocabulary:
- Accretionary heat
- Gravitational release
- Io
- Europa
- Eceladus
- Ganymede
- Miranda