Tidal heating power estimated by this calculator
This tidal heating calculator estimates tidal dissipation power—energy converted to internal heat per unit time, in watts—for a moon or planet flexed by a nearby, more massive body. It uses a low-eccentricity, synchronous-rotation expression intended for rapid orbital and interior-response comparisons.
Use the tidal-power result to compare orbital cases, such as asking how the heat budget changes when eccentricity doubles, or to check whether an orbit could provide a meaningful internal energy source. A single wattage cannot describe every geological outcome, but ratios between otherwise comparable cases are often informative: a configuration producing 100 times as much tidal power has a much larger potential internal heat source.
Eccentricity-driven tidal heating is relevant to volcanic moons, icy satellites, possible subsurface oceans, and hypothetical exomoons. Since orbital forcing and a body’s interior response both matter, this calculator is best used as a transparent first estimate rather than a full thermal-orbital evolution model.
Tidal heating power model and formula
The tidal heating power calculation uses the time-averaged eccentricity-tide expression for dissipation power P:
P = (21/2) · (G · Mp2) · R5 · e2 · n · k2 / (Q · a6)
Here G is the gravitational constant and n is orbital mean motion: n = √(G·Mp/a3). Tidal power is especially sensitive to e2 and to orbital distance; including mean motion gives an overall fixed-mass dependence of a−15/2.
Physically, the central body’s mass controls tidal forcing, the heated body’s radius controls the amount of material participating in deformation, and eccentricity produces the changing tidal stress. The interior parameters k2 and Q represent deformability and energy loss: higher k2 raises the estimate, while higher Q lowers it.
Tidal heating power inputs and their meanings
Each tidal-heating field supplies one term in the eccentricity-driven dissipation model, so the displayed units are important.
- Planet mass (Mp, kg): mass of the central tide-raising body, such as Jupiter when estimating heating in Io. For a planet heated by its star, use the star’s mass here.
- Moon radius (R, m): radius of the tidally heated body. The R5 dependence makes this input highly consequential.
- Semi-major axis (a, m): orbital distance in meters. Tidal power declines sharply as the orbit expands.
- Eccentricity (e): orbital non-circularity. In this approximation, e = 0 removes the eccentricity-driven periodic flexing term.
- Love number (k2): a measure of how readily the body deforms under the tidal potential.
- Dissipation factor (Q): a measure of how much energy is dissipated during repeated deformation. Lower Q produces more heating in this model.
Worked tidal heating example using Io-like inputs
The default tidal-heating inputs are loosely Io-like: Mp ≈ 1.898×1027 kg, R ≈ 1.821×106 m, a ≈ 4.22×108 m, e ≈ 0.0041, k2 ≈ 0.3, and Q ≈ 100. They provide a useful starting point for exploring how a close, slightly eccentric orbit can support substantial internal dissipation.
For a direct tidal-heating sensitivity check, halve e while leaving the other fields unchanged. Since the expression contains e2, the computed power becomes one quarter of its previous value. Increasing a has an even stronger suppressing effect because distance enters both the tidal term and mean motion.
Changing Q isolates the assumed interior loss behavior. Doubling Q while holding every other input fixed halves the calculated tidal power, which is why uncertain material properties can dominate uncertainty in an absolute estimate.
How to interpret the tidal heating power result
Interpret the tidal heating result as a modelled global power budget, not as a guaranteed prediction of surface activity or melt distribution.
- Mild (below 1010 W): a comparatively small global tidal power budget that can still matter locally if heat is concentrated.
- Moderate (1010–1014 W): potentially important for the thermal evolution of smaller rocky or icy bodies.
- Extreme (above 1014 W): a large internal power source that may be associated with vigorous activity or rapid thermal and orbital evolution.
To convert a tidal-power result into an average surface heat flux, divide the result by surface area, 4πR2, giving W/m2. This calculator intentionally reports raw power so it can be compared directly with other internal energy budgets.
Assumptions and limitations of the tidal heating model
This tidal heating estimate assumes a Keplerian, low-eccentricity orbit, synchronous rotation, and bulk effective values of k2 and Q. Real moons and planets may have layered interiors, frequency-dependent dissipation, obliquity tides, and spatially concentrated heating. Treat the output as an order-of-magnitude comparison under those stated assumptions.
Tidal heating can also feed back on the inputs: resonances may maintain eccentricity, while heating can alter melt fraction and thereby change k2 and Q. At high eccentricity or away from synchronous rotation, additional tidal terms can matter. These limitations do not prevent exploratory use, but they explain why detailed published models may yield different values.
Approximate tidal heating reference comparison
This tidal-heating comparison table gives broad literature-scale context only; modeled values depend on orbital state and the adopted k2/Q.
| Body | Typical P (W) | Notes |
|---|---|---|
| Io | ~1014 | Intense volcanism; resonance maintains eccentricity. |
| Europa | ~1011–1012 | May help sustain a subsurface ocean. |
| Enceladus | ~1010–1012 | Observed plume activity; simple models can underpredict without additional physics. |
| Earth’s Moon | ~109 | Low present-day heating; mostly geologically inactive. |
Practical tips for tidal heating power estimates
For useful tidal-heating comparisons, change one orbital or interior parameter at a time and keep the remaining inputs fixed. The steep radius and distance exponents make unit checks essential.
- Unit sanity: enter both radius and semi-major axis in meters. A giant-planet moon orbit of hundreds of thousands of kilometers is on the order of 108 m.
- Eccentricity range: use a dimensionless fraction, not a percentage. The low-eccentricity assumption becomes less suitable as e increases.
- k2 and Q are effective parameters: when they are uncertain, compare a plausible range of values rather than treating a single pair as a measured constant.
- Compare energy budgets carefully: computed tidal power is interior dissipation power; it is not the same quantity as stellar energy absorbed at the surface.
Additional tidal heating scenario: moving a moon outward
For tidal heating, moving the same moon outward while holding eccentricity unchanged rapidly lowers the estimate. The explicit a−6 term and mean-motion dependence n ∝ a−3/2 combine to give P ∝ a−15/2 when Mp, R, e, k2, and Q remain fixed.
An outward shift therefore reduces tidal forcing far faster than a simple inverse-square relationship. Resonances and orbital evolution are important because they can affect both the orbital distance and the eccentricity required for continued tidal dissipation.
Tidal heating questions and input troubleshooting
These checks address common issues when an eccentricity-driven tidal heating calculation returns an unexpected value.
- “I got zero or NaN.” Check that every input is numeric and that a and Q are nonzero. Invalid or blank entries can propagate through the calculation.
- “The result seems too large.” Confirm that Mp is in kilograms and that R and a are in meters. Also verify that eccentricity was entered as a fraction, such as 0.0041 rather than 0.41.
- “Can I use this for a planet heated by a star?” As a simplified comparison, use the star as the tide-raising mass and the planet as the deformed body. Spin state, obliquity, and frequency dependence may then be especially important.
- “Does this include obliquity tides?” No. The calculation includes the eccentricity-driven synchronous term only.
- “What if the body is not synchronous?” A non-synchronous body can dissipate through different terms, so this formula is not generally sufficient for that case.
Tidal Drift Mini-Game
Keep an exomoon alive by steering its orbit so tidal heat stays inside a habitable band. Every correction teaches how eccentricity, distance, and dissipation reshape interior power.
The mini-game is optional. It does not change the calculator result; it simply uses your latest inputs to set a target heating level and then challenges you to keep the system stable. If you prefer not to play, you can ignore this section and focus on the computed watts above.
- Heating is highly sensitive to e and a, which makes a good “balance” mechanic.
- It reinforces the idea that small orbital changes can cause large thermal consequences.
- It helps build intuition for nonlinear scaling: a small push can overshoot, while gentle corrections are often best.
- Guide a glowing moon across a heating track; drift too hot and the crust fractures, too cold and the interior freezes.
- Resonance gusts push you off course; timely counter-burns earn points.
- The teal band represents a “stable” heating window around the target power derived from your calculator inputs.
- Pointer: click/drag on the canvas to nudge the system.
- Keyboard: A/D (←/→) to adjust, Space to pause/resume.
- Accessibility note: the overlay is keyboard-focusable; press Enter or Space to start.
- Canvas loop uses delta timing and respects reduced-motion preferences.
- Difficulty scales from your latest calculator inputs; best runs are saved locally.
- Because the game is a visualization, it uses a simplified “drift” model rather than integrating a full orbital solution.
Overlay shows Click to Play. Stay in the teal band to score; survive as long as you can.
Documenting tidal heating scenarios
Record a tidal heating result with Copy Result when comparing orbital cases in lab notes, mission concepts, or a spreadsheet. For a reproducible estimate, retain the full input set (Mp, R, a, e, k2, Q) with the reported power.
For tidal-heating uncertainty ranges, compare lower-e, higher-Q inputs with a baseline and with higher-e, lower-Q inputs. Because this model is nonlinear in orbital distance and eccentricity, a bracket of scenarios is usually more revealing than one nominal value.
When sharing a tidal-power estimate, state that it assumes synchronous rotation, a low-eccentricity orbit, and constant effective k2/Q. That context identifies the result as a comparative model output rather than a direct measurement.