Tidal Heating Power Calculator

What this calculator does

This page estimates tidal heating power (energy dissipated as heat per unit time, in watts) inside a moon or exoplanet that is being flexed by a nearby massive body. The model is the standard low-eccentricity, synchronous-rotation approximation used for quick comparisons and back-of-the-envelope checks.

Use it to compare scenarios (for example, “How much does heating change if eccentricity doubles?”) or to sanity-check whether a proposed orbit could plausibly power volcanism, cryovolcanism, or a subsurface ocean. The output is a single number in watts, but it is often most useful as a relative indicator: if one configuration produces 100× more power than another, it is very likely to be more geologically active, even if the absolute value is uncertain.

Tidal heating is a key ingredient in many planetary science stories: it helps explain Io’s extreme volcanism, it is a candidate energy source for Europa and Enceladus, and it is frequently discussed for hypothetical exomoons and compact multi-planet systems. Because the physics depends on both orbital dynamics and interior response, simple calculators like this one are best treated as a first pass.

Model and formula

The script implements a commonly used expression for the time-averaged tidal dissipation power P:

P = (63/4) · (G · M_p²) · R⁵ · e² · n · k₂ / (Q · a⁶)

where G is the gravitational constant and n is the mean motion: n = √(G·M_p/a³). The strongest sensitivities are typically e² and a⁻⁶ (plus the additional dependence through n).

In plain language: the central body’s mass sets the strength of the tide, the moon’s radius sets how much material is being flexed, the orbital distance sets how strong the tide is at the moon, and the eccentricity sets how much the tide varies over an orbit. The parameters k₂ and Q summarize the interior response: how easily the body deforms and how much of that deformation is converted into heat.

Inputs (what each field means)

• Planet mass (M_p, kg): mass of the central body raising tides (e.g., Jupiter for Io). If you are modeling a planet heated by its star, M_p would be the star’s mass.
• Moon radius (R, m): radius of the tidally heated body. Heating scales as R⁵, so size matters a lot.
• Semi-major axis (a, m): orbital distance. Heating drops very steeply with distance, so be careful with units (meters, not kilometers).
• Eccentricity (e): how non-circular the orbit is. e = 0 gives (nearly) no periodic flexing in this approximation.
• Love number (k₂): how readily the body deforms under tides (depends on rigidity and internal structure). Typical values range from ~0.01 (very rigid) to ~0.5 (more deformable).
• Dissipation factor (Q): how “lossy” the material response is. Lower Q means more heat per cycle. Q can vary widely with temperature, melt fraction, and forcing frequency.

Worked example (Io-like values)

The default values are loosely Io-like: M_p ≈ 1.898×10²⁷ kg, R ≈ 1.821×10⁶ m, a ≈ 4.22×10⁸ m, e ≈ 0.0041, k₂ ≈ 0.3, Q ≈ 100. With these inputs, the calculator returns a power on the order of 10¹⁴ W, which is consistent with Io being extremely volcanically active.

Try a quick sensitivity check: keep everything the same but halve e. Because heating scales with e², the power should drop by about a factor of 4. If you instead increase a by 10%, the power should drop dramatically because of the steep distance dependence.

Another useful check is to change Q. If you double Q (for example, from 100 to 200) while keeping everything else fixed, the predicted power halves. This is why uncertainty in Q often dominates uncertainty in the final number: Q is not a simple constant for real materials, and it can change as the interior warms or partially melts.

How to interpret the result

• Mild (below 10¹⁰ W): likely small global effect; could still matter locally depending on heat transport, porosity, and whether heating is concentrated in a shell.
• Moderate (10¹⁰–10¹⁴ W): potentially geologically important, especially for smaller or icy bodies; may help maintain liquid water beneath an ice shell.
• Extreme (above 10¹⁴ W): often associated with strong volcanism/cryovolcanism or rapid orbital/thermal evolution; may imply significant resurfacing or interior melt.

If you want to compare to surface heat flux, you can divide power by surface area (4πR²) to get an average flux in W/m². This is not done automatically here because many users prefer to keep the raw power for energy-budget comparisons, but it is a common next step when thinking about ice shell thickness or volcanic output.

Assumptions and limitations

This is a simplified estimator. It assumes a Keplerian orbit with small eccentricity, synchronous rotation, and an effective (bulk) k₂ and Q. Real bodies can have layered interiors, frequency-dependent dissipation, obliquity tides, and localized heating (e.g., Enceladus). Treat the output as an order-of-magnitude estimate and use it to compare scenarios consistently.

Additional caveats that often matter in research contexts: (1) eccentricity is frequently maintained by resonances, so e is not always a free parameter; (2) the same heating that you compute can change the interior structure, which changes k₂ and Q, creating feedback; (3) for very high eccentricity or non-synchronous rotation, other terms become important. None of these invalidate the calculator for quick exploration, but they do explain why published values can differ between models.

Reference comparison (very approximate)

The table below is a qualitative guide to typical tidal heating magnitudes discussed in the literature. Values vary by model and by the assumed k₂/Q.

Practical tips for using the calculator

For the most reliable comparisons, change one parameter at a time and keep a short record of what you changed. Because the formula is steep in distance and radius, it is easy to accidentally introduce a unit error that overwhelms the physics. A few quick checks can prevent confusion.

• Unit sanity: a typical giant-planet moon orbit is hundreds of thousands of kilometers, which is hundreds of millions of meters (10⁸ m). If you enter 4.22e5 instead of 4.22e8, heating will be wildly overstated.
• Eccentricity range: e must be between 0 and 1 for bound Keplerian orbits. For many moons, e is a few thousandths to a few hundredths; values like 0.2 are possible in some exoplanet contexts but may violate the “small e” assumption.
• k₂ and Q are effective parameters: if you do not know them, use the calculator to explore plausible ranges (for example, k₂ from 0.05 to 0.5 and Q from 10 to 500) and report a band of outcomes.
• Compare to other energy sources: radiogenic heating in small rocky bodies is often ~10⁹–10¹² W depending on size and composition; stellar insolation affects surface temperature but not necessarily interior power.

Additional worked scenario: “move the moon outward”

Suppose you keep the same moon and planet but move the orbit outward while leaving eccentricity unchanged. Because the power contains a factor of a⁻⁶ and also depends on mean motion n ∝ a^−3/2, the combined scaling is approximately P ∝ a^−15/2 for fixed M_p, R, e, k₂, and Q. That means a modest outward migration can shut down heating quickly.

As a rough illustration: increasing a by 20% (multiplying by 1.2) reduces P by about 1.2^−7.5, which is roughly a factor of 3 to 4 decrease. This is why resonances and orbital evolution are so important: they can keep a moon close enough (and eccentric enough) for heating to remain significant over long times.

Common questions and troubleshooting

The calculator is intentionally simple, but a few recurring questions come up when people first explore tidal dissipation.

• “I got zero or NaN.” Check that all inputs are numbers and that a is not zero. Extremely small a can also cause overflow-like behavior in the exponentials.
• “The result seems too large.” Verify that R and a are in meters and that Mp is in kilograms. Also check whether you entered e as a percent (e.g., 0.41) instead of a fraction (0.0041).
• “Can I use this for a planet heated by a star?” Yes as a toy model: treat the star as the tide-raiser (Mp) and the planet as the deformed body (R, k2, Q). Be aware that stellar tides can involve additional effects (spin state, obliquity, and frequency dependence).
• “Does this include obliquity tides?” No. Obliquity tides can be important for some bodies; this page focuses on the eccentricity-driven term for clarity.
• “What if the body is not synchronous?” The expression used here is most appropriate for synchronous rotation. Non-synchronous cases can dissipate differently and may require a different model.