Tidal Heating Power Calculator

Explanation

Tidal heating arises when a satellite experiences varying gravitational forces during its orbit, flexing its interior and converting orbital energy into heat. The calculator above evaluates the widely used expression for the time-averaged tidal power in a synchronously rotating body on a low-eccentricity orbit. The basic scaling emerges from the constant time lag model in which the tidal bulge raised by a central planet lags behind the line connecting the centers because of internal friction. The difference between the instantaneous direction of the bulge and the direction of the tidal potential leads to a torque, and the work done by that torque is dissipated as heat. The rate of energy conversion scales as the square of the tidal potential and hence the square of the eccentricity, while being inversely proportional to the dissipation function Q, which parametrizes how effectively the body can deform and relax.

To understand the pieces of the formula, consider the parameters that enter. The gravitational constant G is universal. The planet mass M_p and the orbital semi-major axis a set the strength of the tides and the orbital angular frequency n=$\mathrm{\backslash sqrt\{GM\_p/a^3\}}$. The moon's radius R determines the lever arm for the tidal forces, entering as the fifth power because both the tidal potential and the resulting volume scale with R. The eccentricity e measures how much the distance to the planet changes over an orbit; a perfectly circular orbit has e=0 and therefore no periodic flexing. The Love number k₂ encapsulates the body's rigidity and internal structure: larger values indicate that the body deforms more readily under tidal stresses, enhancing the dissipation. The dissipation factor Q represents the efficiency of energy loss per tidal cycle. Low Q values correspond to high internal friction and therefore strong heating, while a perfectly elastic body would have Q tending to infinity and negligible heating. Plugging these pieces into the theoretical model yields the expression implemented in the script above: $P=\frac{63}{4}\backslash frac\{(GM\_p)^2R^5n\; e^2k\_2\}\{Q\; a^6\}$.

The numerical example preloaded in the calculator corresponds loosely to Jupiter's moon Io, the most volcanically active body in the Solar System. With M_p=1.898×10²⁷ kg, R=1.821×10⁶ m, a=4.22×10⁸ m, e=0.0041, k₂≈0.3, and Q≈100, the result is of order 10¹⁴ W, consistent with detailed geophysical models. This extreme heating drives the resurfacing of Io every few million years and powers intense sulfur dioxide plumes. In contrast, Earth's Moon has e≈0.054 but k₂/Q so small that its tidal heating is only about 10⁹ W, barely affecting its geology. The quadratic dependence on eccentricity means that even modest perturbations from other moons can drastically alter heating, a key concept in the resonant Laplace configuration that maintains Io's orbit.

Why include k₂ and Q separately? In some literature the combination k₂/Q is treated as a single tidal dissipation factor, but distinguishing them allows users to explore the influence of interior structure. For icy moons, k₂ can range from 0.1 to 0.5 depending on the presence of subsurface oceans. Q can vary from ~10 for partially molten silicate interiors to >1000 for rigid, cold bodies. Exoplanet studies often treat k₂/Q as a free parameter to gauge the tidal circularization timescales and heating rates, yet data from ring seismology and libration measurements in our Solar System provide empirical anchors. By adjusting these parameters, the calculator can model hypothetical bodies with strange rheologies, helping researchers gauge the potential for cryovolcanism or subsurface oceans.

Tidal heating is not simply an exotic curiosity. It plays a pivotal role in the habitability of moons and exoplanets, providing long-lived energy sources independent of stellar radiation. For instance, Europa's tidal heating could maintain a global ocean beneath its icy crust, raising the tantalizing possibility of extraterrestrial life. In exoplanetary science, tidal dissipation influences orbital evolution, migration, and the ultimate fate of close-in bodies. Hot Jupiters may experience tidal inflation or even Roche-lobe overflow if the internal heating becomes extreme. For super-Earths, a delicate balance between radiogenic and tidal heating could drive plate tectonics, a process thought to be essential for maintaining long-term climate stability.

The formula used here stems from the constant time lag model, which assumes a fixed phase lag between the tidal potential and the body's deformation. Alternative models, such as the constant Q model or frequency-dependent viscoelastic treatments, yield similar scaling but with different numerical coefficients. The choice depends on the rheology: Maxwell or Andrade models capture the complex response of real planetary materials. Nevertheless, the simplified expression retains the essential physics and suffices for order-of-magnitude estimates, which are often all that is needed in the early stages of mission planning or theoretical investigations.

Users should be aware of the assumptions baked into the calculator. The orbit is assumed to be Keplerian with small eccentricity, the rotation is synchronous (no obliquity tides), and the interior is homogeneous. In reality, many moons exhibit higher-order tidal terms due to orbital inclination, internal layering, or non-synchronous rotation. For example, Enceladus's south polar heat flux appears to exceed predictions from simple models, possibly due to localized softening or resonant effects. Additionally, the dissipation factor Q may depend on the tidal frequency, temperature, and amplitude, rendering the heating rate time-dependent. Such complexities lie beyond the scope of the simple formula but can be explored by adjusting parameters to mimic effective values.

Consider compiling example values into a small reference table for comparison.

This table illustrates how dramatically tidal heating can vary. The thresholds between “mild” and “extreme” activity are not sharply defined, but Io’s value serves as a useful benchmark. Any body exceeding ~10¹³ W is likely to exhibit global volcanism or cryovolcanism, while those below 10¹⁰ W may have only localized or ancient tectonics. Between these extremes lies a realm of possibilities that exoplanet missions are just beginning to probe.

Historically, the concept of tidal heating traces back to George Darwin’s 19th century investigations of Earth’s Moon. He showed that tidal friction causes the Moon to spiral outward while Earth’s rotation slows. Later, in the mid-20th century, scientists recognized that Jupiter’s Galilean moons are locked in orbital resonances that continually pump up their eccentricities, preventing the orbits from circularizing and thereby sustaining ongoing heating. Spacecraft observations by Voyager, Galileo, and Cassini confirmed the theoretical predictions by revealing active worlds shaped by tides. Today, tidal heating is incorporated into models of planetary system evolution, exomoon detection strategies, and astrobiological assessments.

Mathematically, the energy dissipation rate can also be derived using the concept of tidal torque. The torque due to tidal bulge misalignment is $\backslash tau\; =\; -\backslash frac\{3\}\{2\}\; k\_2\; \backslash frac\{GM\_p^2\; R^5\}\{a^6\; Q\}\; \backslash text\{sign\}(\backslash omega\; -\; n)$. The work done per unit time is then τ times the difference between the spin rate ω and the orbital mean motion n. In synchronous rotation, small residuals remain due to eccentricity, leading to nonzero power. By using energy balance arguments, one recovers the expression for P used here, emphasizing how angular momentum transfer and heat generation are intertwined.

Finally, the calculator’s simple categorization of “mild,” “moderate,” and “extreme” heating is meant purely as a qualitative guide. Actual habitability or geological vigor depends on how heat is transported to the surface, which involves convection, conduction, and advection through melts or cracks. A body with extreme internal heating might still have a cold surface if a thick lithosphere insulates its interior. Conversely, moderate heating concentrated along faults or plumes could produce striking localized features. Users are encouraged to experiment with parameter ranges reflective of diverse planetary compositions to appreciate the rich diversity of outcomes in tidal environments.