<p>(This is more an answer to this part: &quot;If any of these are physically impossible, let me know&quot;.)</p>
<p>As you mention volcanism, plate tectonics, mountains... I make the assumption that you consider rocky planets only. If so, it is pretty straightforward to calculate the theoretical maximum Rayleigh number of a rocky planet. The rocks being essentially the same as those on Earth/Mars/Venus, parameters like <span class="math-container">$\alpha$</span>, <span class="math-container">$\upsilon$</span>, <span class="math-container">$\kappa$</span> and <span class="math-container">$\rho$</span> will be similar. That leaves us <span class="math-container">$g$</span>, <span class="math-container">$\Delta T$</span> and <span class="math-container">$d$</span> to work with. They all work the same way: the bigger the planet, the bigger they are, and the bigger <span class="math-container">$Ra$</span> gets.</p>
<p>Now, according to the following graph, the largest rocky super-Earth (near the &quot;67 % Fe&quot; dashed grey line) has a mass of about 7 <span class="math-container">$M_E$</span> and a radius of about 2 <span class="math-container">$R_E$</span> (a bit lower, but let's say 2 for the sake of simplicity):</p>
<p><a href="https://i.sstatic.net/GWZGA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GWZGA.png" alt="enter image description here" /></a>
<em>Scatter plot of super-Earth characteristics in context showing the mass, <span class="math-container">$m$</span>, and radius, <span class="math-container">$r$</span>, for transiting Super-Earths (black dots), public domain</em></p>
<p>This translates in:</p>
<ul>
<li><span class="math-container">$g = 17.26 \approx$</span> 2 times that of the Earth</li>
<li><span class="math-container">$d^3 =$</span> 8 times that of the Earth</li>
<li><span class="math-container">$\Delta T$</span> is trickier, but is fairly linear in Earth's mantle, so extrapolating the trend yields ~6000 K, about 2 times that of the Earth</li>
</ul>
<p>This back-of-the-envelope calculation shows that the largest <em>rocky</em> super-Earth can have a <span class="math-container">$Ra$</span> of about <span class="math-container">$3 \times 10^8$</span>, no more.</p>
<p>As to how such a vigorous mantle convection would translate in terms of surface processes, I agree with you: probably more volcanism and plate movements. The tectonic regime could also be different: some people think that in the Archean (when the Earth likely had a higher <span class="math-container">$Ra$</span>) the dynamics was more vertical, as opposed to the modern horizontal tectonic regime.</p>

CC BY-SA 4.0
2023-01-12T15:18:40.090Z
I think this is a very good heuristic for finding the largest possible Rayleigh number. I'll note, however: larger rocky bodies have been hypothesized. For instance, Kepler-277b may have 87.4 M⊕ and 2.92 R⊕, and Kepler-145b may have 37.1 M⊕ and 2.65 R⊕. However, these figures equate to densities roughly comparable to depleted uranium and lead, respectively, which seem overly large, so your predictions likely hold true for bodies *known* to be rocky. Honestly, that's an entirely different question I should ask Astronomy SE. Also, thanks for the edit; these symbols can be tricky to work with.
KEY_ABRADE
CC BY-SA 4.0
2023-01-12T18:48:28.693Z
Did the math. If mass/radius figures are accurate, Kepler-277b has ~10.25⊕ g, 24.9⊕ d^3, and 2.92⊕ delta-T, for a Rayleigh number of ~7.45E9, and Kepler-145B has ~5.28⊕ g, 18.61⊕ d^3, and 2.65⊕ delta-T, for a Rayleigh number of ~2.604E9. Do you think such high numbers would equate to proportionately greater more volcanic and tectonic activity, though? At a certain point, it seems like it would be difficult for a planet to get more tectonically active and volcanic.
KEY_ABRADE
CC BY-SA 4.0
2023-01-13T08:30:00.167Z
Are you sure of your calculations? According to Wikipedia, Kepler-277b has a very large iron core, of 2.435 $R_{\oplus}$. That leaves only 0.485 $R_{\oplus}$ of rocky, convective layer (i.e., mantle), or ~3000 km, like Earth. So $d^3$ and $\Delta T$ would be similar, leaving just a factor of 10 to account for $g$, yielding a $Ra$ of $10^8$.
Jean-Marie Prival
CC BY-SA 4.0
2023-01-13T08:31:51.440Z
And I agree, there is probably a limit, at some point things would probably stop being solid, so no more plate tectonics or volcanism, just a big magma ocean world.
Jean-Marie Prival
CC BY-SA 4.0
2023-01-12T09:54:39.263Z
2023-01-12T09:54:39.263Z
Jean-Marie Prival
<p>What's a Rayleigh number? Well, it doesn't have anything to do with <a href="https://en.wikipedia.org/wiki/Rayleigh_scattering" rel="nofollow noreferrer">Rayleigh scattering</a>. <a href="https://en.wikipedia.org/wiki/Rayleigh_number" rel="nofollow noreferrer">Here's a Wikipedia page on this concept</a>, but I believe Elvira Mulyukova's and David Bercovici's <em><a href="https://oxfordre.com/planetaryscience/display/10.1093/acrefore/9780190647926.001.0001/acrefore-9780190647926-e-109;jsessionid=E0978952D39C68EC05DEED914229758C" rel="nofollow noreferrer">Mantle Convection in Terrestrial Planets</a></em> put it better:</p>
<blockquote>
<p>The competition between forcing by thermal buoyancy and damping by
viscosity and thermal diffusion is characterized by a dimensionless
ratio called the Rayleigh number.</p>
</blockquote>
<p>In other words, a <a href="https://en.wikipedia.org/wiki/Rayleigh_number" rel="nofollow noreferrer">Rayleigh number</a> represents the ratio between how much hot mass in a <a href="https://en.wikipedia.org/wiki/Convection" rel="nofollow noreferrer">convection current</a> rises and how much the <a href="https://en.wikipedia.org/wiki/Viscosity" rel="nofollow noreferrer">viscosity</a> and <a href="https://en.wikipedia.org/wiki/Thermal_conductivity" rel="nofollow noreferrer">thermal conductivity</a> 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 <a href="https://en.wikipedia.org/wiki/Mantle_convection" rel="nofollow noreferrer">mantle convection</a>.</p>
<p><em>Mantle Convection in Terrestrial Planets</em> goes on to state the following:</p>
<blockquote>
<p><strong><span class="math-container">$Ra$</span></strong> [Rayleigh number] needs to exceed a certain value, called the
critical Rayleigh number <span class="math-container">$Ra_c$</span>, in order to excite
convective flow. The value of <span class="math-container">$Ra_c$</span> 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 <a href="https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/hydrodynamic-and-hydromagnetic-stability-by-s-chandrasekhar-clarendon-press-oxford-university-press-1961-652-pp-5-5s/DC77FFB6F40A90109063FEAE444692BB" rel="nofollow noreferrer">Chandrasekhar, 1961</a>).</p>
</blockquote>
<p>Therefore, it can be concluded that, if, within a body's mantle, the ratio of &quot;stuff-forcing-its-way-up&quot; to &quot;stuff-being-held-down&quot; is ≤ ~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 here, 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.</p>
<p>Continuing from <em>Mantle Convection in Terrestrial Planets</em>:</p>
<blockquote>
<p>Assuming their material properties are similar to Earth’s, we can
estimate the Rayleigh numbers for the mantles of other terrestrial
planets: <span class="math-container">${10}^4$</span> for Mercury, <span class="math-container">${10}^7$</span> for Venus, and <span class="math-container">${10}^6$</span> for Mars. With the
exception of Mercury, whose <span class="math-container">$Ra$</span> 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.</p>
</blockquote>
<p>Mulyukova and Bercovici calculated these Rayleigh numbers using the following values:</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th>Property</th>
<th>Mercury</th>
<th>Venus</th>
<th>Earth</th>
<th>Mars</th>
</tr>
</thead>
<tbody>
<tr>
<td><strong>Density (kilograms per cubic meter)</strong></td>
<td>3500</td>
<td>4000</td>
<td>4000</td>
<td>3500</td>
</tr>
<tr>
<td><strong>Surface temperature (degrees Kelvin)</strong></td>
<td>440</td>
<td>730</td>
<td>285</td>
<td>220</td>
</tr>
<tr>
<td><strong>Core-mantle boundary temperature (degrees Kelvin)</strong></td>
<td>3000</td>
<td>3500</td>
<td>3500</td>
<td>3000</td>
</tr>
<tr>
<td><strong>Mantle thickness (kilometers)</strong></td>
<td>400</td>
<td>2900</td>
<td>2900</td>
<td>2000</td>
</tr>
<tr>
<td><strong>Gravity (meters per second squared)</strong></td>
<td>3.7</td>
<td>10</td>
<td>10</td>
<td>3.5</td>
</tr>
</tbody>
</table>
</div>
<p>The gravity values (if not others) are slightly inaccurate and could use slight refining (for instance, 9.80665 <span class="math-container">$m/s^2$</span> rather than 10 <span class="math-container">$m/s^2$</span> for the Earth's gravity), but that's irrelevant to this question.</p>
<p>For the purposes of this question, the equation for finding a Rayleigh number is: <span class="math-container">$Ra = \frac{ \rho g \alpha \Delta Td^3 }{\upsilon \kappa}$</span> where:</p>
<ul>
<li><span class="math-container">$\rho$</span> = density of mantle</li>
<li>g = planet's gravity</li>
<li><span class="math-container">$\alpha$</span> = <a href="https://en.wikipedia.org/wiki/Thermal_expansion" rel="nofollow noreferrer">thermal expansivity</a> of mantle</li>
<li><span class="math-container">$\Delta T$</span> = difference in temperature between top and bottom boundaries of mantle</li>
<li><em>d</em> = thickness of mantle</li>
<li><span class="math-container">$\upsilon$</span> = viscosity of mantle</li>
<li><span class="math-container">$\kappa$</span> = <a href="https://en.wikipedia.org/wiki/Thermal_diffusivity" rel="nofollow noreferrer">thermal diffusivity</a> of mantle</li>
</ul>
<p>My question is: <strong>what would the geology of a planet with a high (specifically, <span class="math-container">$Ra \geq 10^8$</span>) Rayleigh number look like?</strong> The few planets for which I can find calculated Rayleigh numbers seem to fit a trend in which higher Rayleigh numbers correlate to more volcanism, more mountain-forming, more tectonic plates moving about, etc. Obviously, a sample size of 4 doesn't allow itself to accurate extrapolations, but that's why you're here: I'm asking this in the hope that someone more informed and/or qualified can provide further insight.</p>
<p>Let's use four hypothetical planets here:</p>
<ul>
<li><span class="math-container">$Ra = 10^8$</span></li>
<li><span class="math-container">$Ra = 10^9$</span></li>
<li><span class="math-container">$Ra = 10^{10}$</span></li>
<li><span class="math-container">$Ra = 10^{11}$</span></li>
</ul>
<p>If any of these are physically impossible, let me know.</p>
<p>Even before this is answered, I think I can draw several conclusions myself. Such a planet would cool relatively faster than planets with low Rayleigh numbers (<em>how</em> much more quickly, and what the relationship between Rayleigh number and heat loss is, I don't know) as the convection currents within its mantle would carry heat away from its core more quickly. I think that this means that, if the Rayleigh number-to-surface-area ratio (and, therefore, the Rayleigh number-to-planetary-radius ratio) is greater than that of planets with equal surface areas and ratios, the <a href="https://en.wikipedia.org/wiki/Geothermal_gradient" rel="nofollow noreferrer">geothermal gradient</a> would likely be steeper; more heat would be soaking out of the ground per unit of surface area. I also imagine such planets would see more volcanism, more mountain-forming, more tectonic plates moving about, etc.</p>
<p>I'm fairly sure this question fits Astronomy SE better than it fits Physics SE. If you think otherwise, let me know.</p>

CC BY-SA 4.0
2023-01-12T10:01:02.297Z
Kind of related: https://astronomy.stackexchange.com/q/40769/34513
Jean-Marie Prival
CC BY-SA 4.0
2023-01-12T18:51:03.903Z
@Jean-MariePrival As a matter of fact, the answer to that post is the very thing which put me onto the concept of Rayleigh numbers in the first place.
KEY_ABRADE
<geology><terrestrial-planets>
Geological/tectonic/thermal/etc. implications of Rayleigh numbers ≥ 100,000,000
KEY_ABRADE
# Geological/tectonic/thermal/etc. implications of Rayleigh numbers ≥ 100,000,000

(Asked by: username_0 on 2023-01-12)

What's a Rayleigh number? Well, it doesn't have anything to do with [Rayleigh scattering](https://en.wikipedia.org/wiki/Rayleigh_scattering). [Here's a Wikipedia page on this concept](https://en.wikipedia.org/wiki/Rayleigh_number), but I believe Elvira Mulyukova's and David Bercovici's *[Mantle Convection in Terrestrial Planets](https://oxfordre.com/planetaryscience/display/10.1093/acrefore/9780190647926.001.0001/acrefore-9780190647926-e-109;jsessionid=E0978952D39C68EC05DEED914229758C)* put it better:

> 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](https://en.wikipedia.org/wiki/Rayleigh_number) represents the ratio between how much hot mass in a [convection current](https://en.wikipedia.org/wiki/Convection) rises and how much the [viscosity](https://en.wikipedia.org/wiki/Viscosity) and [thermal conductivity](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Mantle_convection).

*Mantle Convection in Terrestrial Planets* goes on to state the following:

> **$Ra$** [Rayleigh number] needs to exceed a certain value, called the
> critical Rayleigh number $Ra\_c$, in order to excite
> convective flow. The value of $Ra\_c$ 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](https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/hydrodynamic-and-hydromagnetic-stability-by-s-chandrasekhar-clarendon-press-oxford-university-press-1961-652-pp-5-5s/DC77FFB6F40A90109063FEAE444692BB)).

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 ≤ ~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 here, 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.

Continuing from *Mantle Convection in Terrestrial Planets*:

> Assuming their material properties are similar to Earth’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.

Mulyukova and Bercovici calculated these Rayleigh numbers using the following values:

| Property | Mercury | Venus | Earth | Mars |
| --- | --- | --- | --- | --- |
| **Density (kilograms per cubic meter)** | 3500 | 4000 | 4000 | 3500 |
| **Surface temperature (degrees Kelvin)** | 440 | 730 | 285 | 220 |
| **Core-mantle boundary temperature (degrees Kelvin)** | 3000 | 3500 | 3500 | 3000 |
| **Mantle thickness (kilometers)** | 400 | 2900 | 2900 | 2000 |
| **Gravity (meters per second squared)** | 3.7 | 10 | 10 | 3.5 |

The gravity values (if not others) are slightly inaccurate and could use slight refining (for instance, 9.80665 $m/s^2$ rather than 10 $m/s^2$ for the Earth's gravity), but that's irrelevant to this question.

For the purposes of this question, 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](https://en.wikipedia.org/wiki/Thermal_expansion) 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](https://en.wikipedia.org/wiki/Thermal_diffusivity) of mantle

My question is: **what would the geology of a planet with a high (specifically, $Ra \geq 10^8$) Rayleigh number look like?** The few planets for which I can find calculated Rayleigh numbers seem to fit a trend in which higher Rayleigh numbers correlate to more volcanism, more mountain-forming, more tectonic plates moving about, etc. Obviously, a sample size of 4 doesn't allow itself to accurate extrapolations, but that's why you're here: I'm asking this in the hope that someone more informed and/or qualified can provide further insight.

Let's use four hypothetical planets here:

* $Ra = 10^8$
* $Ra = 10^9$
* $Ra = 10^{10}$
* $Ra = 10^{11}$

If any of these are physically impossible, let me know.

Even before this is answered, I think I can draw several conclusions myself. Such a planet would cool relatively faster than planets with low Rayleigh numbers (*how* much more quickly, and what the relationship between Rayleigh number and heat loss is, I don't know) as the convection currents within its mantle would carry heat away from its core more quickly. I think that this means that, if the Rayleigh number-to-surface-area ratio (and, therefore, the Rayleigh number-to-planetary-radius ratio) is greater than that of planets with equal surface areas and ratios, the [geothermal gradient](https://en.wikipedia.org/wiki/Geothermal_gradient) would likely be steeper; more heat would be soaking out of the ground per unit of surface area. I also imagine such planets would see more volcanism, more mountain-forming, more tectonic plates moving about, etc.

I'm fairly sure this question fits Astronomy SE better than it fits Physics SE. If you think otherwise, let me know.

**Tags:** geology, terrestrial-planets

> **Comments:**
> **username_1**: Kind of related: https://astronomy.stackexchange.com/q/40769/34513  
> **username_0**: @username_1 As a matter of fact, the answer to that post is the very thing which put me onto the concept of Rayleigh numbers in the first place.  

---

## 1 answers

### Answer by username_1 on 2023-01-12. Score: 4 | Accepted answer

(This is more an answer to this part: "If any of these are physically impossible, let me know".)

As you mention volcanism, plate tectonics, mountains... I make the assumption that you consider rocky planets only. If so, it is pretty straightforward to calculate the theoretical maximum Rayleigh number of a rocky planet. The rocks being essentially the same as those on Earth/Mars/Venus, parameters like $\alpha$, $\upsilon$, $\kappa$ and $\rho$ will be similar. That leaves us $g$, $\Delta T$ and $d$ to work with. They all work the same way: the bigger the planet, the bigger they are, and the bigger $Ra$ gets.

Now, according to the following graph, the largest rocky super-Earth (near the "67 % Fe" dashed grey line) has a mass of about 7 $M\_E$ and a radius of about 2 $R\_E$ (a bit lower, but let's say 2 for the sake of simplicity):

[![enter image description here](https://i.sstatic.net/GWZGA.png)](https://i.sstatic.net/GWZGA.png)
*Scatter plot of super-Earth characteristics in context showing the mass, $m$, and radius, $r$, for transiting Super-Earths (black dots), public domain*

This translates in:

* $g = 17.26 \approx$ 2 times that of the Earth
* $d^3 =$ 8 times that of the Earth
* $\Delta T$ is trickier, but is fairly linear in Earth's mantle, so extrapolating the trend yields ~6000 K, about 2 times that of the Earth

This back-of-the-envelope calculation shows that the largest *rocky* super-Earth can have a $Ra$ of about $3 \times 10^8$, no more.

As to how such a vigorous mantle convection would translate in terms of surface processes, I agree with you: probably more volcanism and plate movements. The tectonic regime could also be different: some people think that in the Archean (when the Earth likely had a higher $Ra$) the dynamics was more vertical, as opposed to the modern horizontal tectonic regime.

> **Comments:**
> **username_0**: I think this is a very good heuristic for finding the largest possible Rayleigh number. I'll note, however: larger rocky bodies have been hypothesized. For instance, Kepler-277b may have 87.4 M⊕ and 2.92 R⊕, and Kepler-145b may have 37.1 M⊕ and 2.65 R⊕. However, these figures equate to densities roughly comparable to depleted uranium and lead, respectively, which seem overly large, so your predictions likely hold true for bodies *known* to be rocky. Honestly, that's an entirely different question I should ask Astronomy SE. Also, thanks for the edit; these symbols can be tricky to work with.  
> **username_0**: Did the math. If mass/radius figures are accurate, Kepler-277b has ~10.25⊕ g, 24.9⊕ d^3, and 2.92⊕ delta-T, for a Rayleigh number of ~7.45E9, and Kepler-145B has ~5.28⊕ g, 18.61⊕ d^3, and 2.65⊕ delta-T, for a Rayleigh number of ~2.604E9. Do you think such high numbers would equate to proportionately greater more volcanic and tectonic activity, though? At a certain point, it seems like it would be difficult for a planet to get more tectonically active and volcanic.  
> **username_1**: Are you sure of your calculations? According to Wikipedia, Kepler-277b has a very large iron core, of 2.435 $R_{\oplus}$. That leaves only 0.485 $R_{\oplus}$ of rocky, convective layer (i.e., mantle), or ~3000 km, like Earth. So $d^3$ and $\Delta T$ would be similar, leaving just a factor of 10 to account for $g$, yielding a $Ra$ of $10^8$.  
> **username_1**: And I agree, there is probably a limit, at some point things would probably stop being solid, so no more plate tectonics or volcanism, just a big magma ocean world.  

---

