# How might planet size affect volcanic activity?

I'm working on a project involving the evolution of life on different hypothetical habitable planets. In imagining different atmospheres on planets of different masses, I'm wondering how a planet's size might affect volcanic activity. Would a more massive rocky world have a thicker crust? And would the thickness of the crust affect the power of volcanoes? I would assume a planet with lower gravity would have a crust that breaks more easily and the volcanoes would release their magma and gases more frequently but with less force. Does this make sense? Also, I'm excluding moons in our solar system as examples in answering this question (f/e Io has intense volcanism because of extreme tidal forces).

## Rayleigh number

Copying from a thing I did on Astronomy SE on this very subject:

The competition between forcing by thermal buoyancy and damping by viscosity and thermal diffusion is characterized by a dimensionless ratio called the Rayleigh number.

In other words, a Rayleigh number represents the ratio between how much hot mass in a convection current rises and how much the viscosity and thermal conductivity of the mass stop it from rising and bleed away its heat, respectively. In this case, the mass in question is the mantle of a terrestrial planet, and the convection current in question is mantle convection.

ππ (Rayleigh number) needs to exceed a certain value, called the critical Rayleigh number [πππ], in order to excite convective flow. The value of πππ is typically on the order of 1,000, with the exact value depending on the thermal and mechanical properties of the horizontal boundaries, (e.g., whether the boundary is rigid or open to the air or space, see Chandrasekhar, 1961).

Therefore, it can be concluded that, if, within a body's mantle, the ratio of "stuff-forcing-its-way-up" to "stuff-being-held-down" is β€ ~1,000, said body won't experience convective flow in its mantle. As far as I know, convective flow in the mantle is necessary for a body to have plate tectonics; this is supported by the fact that the Earth and Venus, with the highest Rayleigh numbers calculated in Mantle Convection in Terrestrial Planets, are fairly volcanically active, whereas Mercury, with the lowest one, is a dead rock, and Mars, with the second-lowest one, features stagnant-lid tectonics, if I recall correctly.

Assuming their material properties are similar to Earthβs, we can estimate the Rayleigh numbers for the mantles of other terrestrial planets: $${10}^4$$ for Mercury, $${10}^7$$ for Venus, and $${10}^6$$ for Mars. With the exception of Mercury, whose $$Ra$$ is at most an order of magnitude above critical, the mantle of the rocky planets in the solar system appear to be cooling predominantly by convection.

The equation for finding a Rayleigh number is: $$Ra = \frac{ \rho g \alpha \Delta Td^3 }{\upsilon \kappa}$$ where:

• $$\rho$$ = density of mantle
• g = planet's gravity
• $$\alpha$$ = thermal expansivity of mantle
• $$\Delta T$$ = difference in temperature between top and bottom boundaries of mantle
• d = thickness of mantle
• $$\upsilon$$ = viscosity of mantle
• $$\kappa$$ = thermal diffusivity of mantle

Per the one answer to my Astronomy SE question, the largest possible Rayleigh number is within the ballpark of $$3 x {10}^8$$. The Rayleigh number Mantle Convection in Terrestrial Planets calculates for the Earth is $${10}^7$$, which gets you your answer: even the most massive terrestrial planet is only going to have $$3 x {10}^8$$ / $${10}^7$$ = 30 times more tectonic activity than the Earth.

However, consider that "30x more volcanic activity" is a lot. Io, for instance, is likely less than 30 times more tectonically active than the Earth, and it's the closest thing to Mustafar in our solar system.

Without you providing specific statistics on the specific planets in question (i.e. the ones for that Rayleigh number equation) I would say that, for the purposes of your project, you can have tectonically active planets with a Rayleigh number between $${10}^3$$ (defined in that paper as when tectonic activity halts) and $$3x{10}^8$$ (picture of hell). Note, however, that if you have something below $${10}^7$$ (i.e. Earth and Venus), you're probably going to have stagnant-lid tectonics like Mars, which aren't suitable for life as we know it but may be useful for what you're working on. I'm unsure about higher numbers.

• +1! Impressive for such a quick answer! Sep 6, 2023 at 18:04
• Thank you so much for this! So basically, higher mass planets are likely to have more volcanic activity? Sep 6, 2023 at 18:39
• @Elhammo Several factors. Mass, yes, but radius is also relevant, and by extension that makes density and gravity relevant. The temperature difference between the top/bottom of the mantle is also important β if there isn't one, there are no convection currents carrying heated magma to the surface. This temperature difference depends on the mantle thickness; thicker mantles mean more room for a high temperature difference. If you want, use figures for Earth for thermal expansivity, diffusivity, and viscosity; those are the hardest to determine as they rely on the mantle's chemical composition. Sep 6, 2023 at 19:12
• @Elhammo Essentially, massive planets with higher gravity/density (i.e. more mass relative to radius), thicker, denser, less viscous, and less thermally conductive mantles, and hotter insides will have higher Rayleigh numbers, whereas the opposite β less massive planets with less density, thinner mantles, and more freely-flowing, liquid magma β will rapidly expend their internal heat budget early on in their formation via heavy Kilauea-esque volcanic activity, killing off plate tectonics before your lifeforms have a chance to evolve. That's kind of what happened with Mars. Sep 6, 2023 at 19:16
• Thanks so much! That all makes a lot of sense. Ok, I also have a related but different question that it seems like you might have a good answer for: I have a smaller world (not tiny, but like 75% the mass of earth) that is the moon of a super Jovian planet. Let's say it has enough tidal heating to have greater tectonic activity than earth despite being smaller but no so much that it's uninhabitable. Do you think it would have more frequent but smaller volcanic eruptions, due to a thinner crust? Sep 6, 2023 at 20:55