Tidal Heating Power Calculator

Tidal heating is the conversion of orbital and rotational energy into internal heat through friction inside a deforming body. A satellite on a perfectly circular orbit around its primary feels a tidal bulge that is fixed in shape and orientation in the rotating frame, so no work is done and no heat is produced. Eccentricity breaks that symmetry: the tide-raising force is stronger at periapsis than at apoapsis and the bulge also librates back and forth in longitude, so the body is kneaded once per orbit. Dissipation therefore scales with the square of eccentricity, which is why the term e squared appears and why even values as small as 0.0041 matter. The mean motion n equals 2 pi divided by the orbital period, and the fifth-power dependence on both radius and mean motion makes the result extremely sensitive to a large body on a short-period orbit. The ratio k2 over Q lumps together two rheological properties: the degree two Love number k2, which measures how readily the body deforms under a tidal potential, and the quality factor Q, which measures how lossy that deformation is. Low Q means high dissipation, characteristic of warm partially molten rock; rigid cold ice and gas giant interiors have high Q. A default of 0.015 is a reasonable order-of-magnitude choice for a silicate moon. Applied to Io, with a radius of 1,821 km and a 1.77-day period, the result is on the order of 100 TW, equivalent to a surface heat flux near 2 W/m2 and more than twenty times Earth mean geothermal flux of about 0.087 W/m2. Europa sits nearer 0.1 to 1 TW and Enceladus around 10 GW. The main limitations are that k2 and Q are treated as fixed when melting actually changes them, that the body is assumed homogeneous and synchronously rotating, and that the expansion holds only for small eccentricity.

The theory of tidal friction was developed by George Howard Darwin, second son of Charles Darwin, in a series of papers from 1879 onward on the tidal evolution of the Earth-Moon system. Darwin introduced the framework of a lagging tidal bulge and the phase-lag parameter that later became the quality factor Q, and showed that tidal dissipation drives orbital expansion and spin-down over geological time. His formalism remained the standard treatment of satellite tidal evolution for most of the following century. The application to volcanically active moons produced one of the most striking predictions in planetary science. Stanton Peale, Patrick Cassen and Ray Reynolds published a paper titled Melting of Io by Tidal Dissipation in Science on 2 March 1979, arguing that the forced eccentricity Io maintains through the 4:2:1 Laplace resonance with Europa and Ganymede should generate enough heat to melt a substantial fraction of its interior and drive widespread volcanism. Voyager 1 made its closest approach to Jupiter three days later, and on 8 March 1979 navigation engineer Linda Morabito identified a volcanic plume rising above Io limb in a Voyager image, confirming the prediction almost immediately. The Galileo orbiter from 1995 to 2003 mapped Io heat flow in detail and found the induced magnetic signature indicating a subsurface ocean at Europa, while Cassini discovery of the Enceladus plumes in 2005 and its later gravity measurements extended tidal heating arguments to the Saturn system. The same physics is now applied to close-in exoplanets and their moons.