<blockquote>
<p>This is for instance the case for the Saturn moons Enceladus and Dione.</p>
</blockquote>
<p>This is not the case for Enceladus and Dione. Enceladus's orbit about Saturn has an eccentricity of 0.0047, which while low, is more than enough to result in tidal heating. Dione does play a role in the tidal heating of Enceladus, but that role is secondary. Dione is what keeps Enceladus's eccentricity non-zero via Enceladus's 2:1 mean motion orbital resonance with Dione.</p>

CC BY-SA 4.0
2020-07-23T19:17:57.140Z
This removes some of the motivation (different mechanism for example chosen), but it doesn't really address what I'm asking for.
SE - stop firing the good guys
CC BY-SA 4.0
2020-07-23T19:19:58.780Z
What you are asking for effectively does not exist.
David Hammen
CC BY-SA 4.0
2020-07-23T19:21:43.830Z
Well, that would be the answer then, if you have some way to show it.
SE - stop firing the good guys
CC BY-SA 4.0
2020-07-23T17:25:19.390Z
2020-07-23T17:25:19.390Z
David Hammen
<p>You have asked a very reasonable question.</p>
<p>Although moon-moon tidal heating is <em>usually</em> very weak, it may in principle be of interest in tightly packed systems. A detailed theory of this effect (motivated by TRAPPIST-1) was developed in 2021 by <a href="https://www.mff.cuni.cz/en/faculty/organizational-structure/people?hdl=8148" rel="nofollow noreferrer">Michaela Walterova</a> in her outstanding Ph.D. thesis. Also see page 39 in her <a href="https://geo.mff.cuni.cz/theses/2021-Walterova-PhD-slides.pdf" rel="nofollow noreferrer">defence slides</a>.</p>
<p>I begged Michaela to publish the solution, but she keeps procrastinating.</p>

CC BY-SA 4.0
2024-03-23T01:30:00.017Z
2024-03-23T01:38:51.270Z
2024-03-23T01:38:51.270Z
Michael_1812
<p>Between a moon and the primary, <a href="https://en.wikipedia.org/wiki/Tidal_heating" rel="nofollow noreferrer">the equation for tidal heating</a> is:</p>
<p><span class="math-container">$$\dot E_\mathit{Tidal} = - Im(k_2) \frac{21}{2} \frac{GM_h^2 R^5 n e^2}{a^6}$$</span></p>
<p>But how does one calculate the tidal heating between moons?</p>
<p><strong>Simplifying assumptions I'm fine with:</strong></p>
<ul>
<li>The moons are coplanar, <span class="math-container">$I_\mathit{affected} = I_\mathit{perturbing} = 0$</span></li>
<li>Both orbits have no eccentricity, <span class="math-container">$e_\mathit{affected} = e_\mathit{perturbing} = 0$</span> (which also implies no heating from the primary)</li>
<li>The affected moon is tidally locked to the primary</li>
<li>The perturbing moon can be treated as a point mass</li>
</ul>
<p>While I'm not able to come up with any formula, I suspect the following properties hold:</p>
<ul>
<li>The tidal heating is still proportional to <span class="math-container">$Im(k_2)$</span>, as this seems to only be an internal property of the moon.</li>
<li>It's still proportional to <span class="math-container">$R^5$</span></li>
<li>Since tidal forces are inversely proportional to distance cubed, I think the overall heating is proportional to <span class="math-container">${(a_\mathit{affected} - a_\mathit{perturbing})^{-3}}$</span>, due to most heating happening while they are in close proximity.</li>
<li>It's inversely proportional to the relative synodic period of the two moons.</li>
</ul>

CC BY-SA 4.0
2020-07-23T14:40:56.683Z
Why do you think the formula only applies when  mass of primary >> mass of moon?
Carl Witthoft
CC BY-SA 4.0
2020-07-23T14:50:35.043Z
@CarlWitthoft I don't see I have made such an assumption, and I can't see such an assumption in the linked article either. If you are suggesting adapting the equation by treating the second moon as the "primary", I'm not sure that's valid since the tidal locking condition no longer holds.
SE - stop firing the good guys
CC BY-SA 4.0
2020-07-23T17:14:46.047Z
OK, why wouldn't tidal locking apply?
Carl Witthoft
CC BY-SA 4.0
2020-07-23T19:15:57.043Z
Re *Both orbits have no eccentricity,* $e_\text{affected}=e_\text{perturbing}=0$ *(which also implies no heating from the primary)* This is not true. Enceladus has an eccentricity of 0.0047, greater than that of the 0.0041 eccentricity of Io's orbit about Jupiter. Just as Jupiter causes significant tidal heating in Io, Saturn causes significant tidal heating in Enceladus.
David Hammen
CC BY-SA 4.0
2020-07-23T19:20:34.477Z
@DavidHammen $e = 0$ is the model I'm looking for, I'm not claiming Enceladus has zero eccentricity. May I also point out that Dione is not a point mass.
SE - stop firing the good guys
<tidal-forces>
What's the equation for moon-moon tidal heating?
SE - stop firing the good guys
# What's the equation for moon-moon tidal heating?

(Asked by: username_0 on 2020-07-23)

Between a moon and the primary, [the equation for tidal heating](https://en.wikipedia.org/wiki/Tidal_heating) is:

$$\dot E\_\mathit{Tidal} = - Im(k\_2) \frac{21}{2} \frac{GM\_h^2 R^5 n e^2}{a^6}$$

But how does one calculate the tidal heating between moons?

**Simplifying assumptions I'm fine with:**

* The moons are coplanar, $I\_\mathit{affected} = I\_\mathit{perturbing} = 0$
* Both orbits have no eccentricity, $e\_\mathit{affected} = e\_\mathit{perturbing} = 0$ (which also implies no heating from the primary)
* The affected moon is tidally locked to the primary
* The perturbing moon can be treated as a point mass

While I'm not able to come up with any formula, I suspect the following properties hold:

* The tidal heating is still proportional to $Im(k\_2)$, as this seems to only be an internal property of the moon.
* It's still proportional to $R^5$
* Since tidal forces are inversely proportional to distance cubed, I think the overall heating is proportional to ${(a\_\mathit{affected} - a\_\mathit{perturbing})^{-3}}$, due to most heating happening while they are in close proximity.
* It's inversely proportional to the relative synodic period of the two moons.

**Tags:** tidal-forces
> **Comments:**
> **username_1**: Why do you think the formula only applies when  mass of primary >> mass of moon?  
> **username_0**: @CarlWitthoft I don't see I have made such an assumption, and I can't see such an assumption in the linked article either. If you are suggesting adapting the equation by treating the second moon as the "primary", I'm not sure that's valid since the tidal locking condition no longer holds.  
> **username_1**: OK, why wouldn't tidal locking apply?  
> **username_2**: Re *Both orbits have no eccentricity,* $e_\text{affected}=e_\text{perturbing}=0$ *(which also implies no heating from the primary)* This is not true. Enceladus has an eccentricity of 0.0047, greater than that of the 0.0041 eccentricity of Io's orbit about Jupiter. Just as Jupiter causes significant tidal heating in Io, Saturn causes significant tidal heating in Enceladus.  
> **username_0**: @DavidHammen $e = 0$ is the model I'm looking for, I'm not claiming Enceladus has zero eccentricity. May I also point out that Dione is not a point mass.  

---

## 2 answers

### Answer by username_2 on 2020-07-23. Score: 2

> This is for instance the case for the Saturn moons Enceladus and Dione.

This is not the case for Enceladus and Dione. Enceladus's orbit about Saturn has an eccentricity of 0.0047, which while low, is more than enough to result in tidal heating. Dione does play a role in the tidal heating of Enceladus, but that role is secondary. Dione is what keeps Enceladus's eccentricity non-zero via Enceladus's 2:1 mean motion orbital resonance with Dione.

> **Comments:**
> **username_0**: This removes some of the motivation (different mechanism for example chosen), but it doesn't really address what I'm asking for.  
> **username_2**: What you are asking for effectively does not exist.  
> **username_0**: Well, that would be the answer then, if you have some way to show it.  

---

### Answer by username_3 on 2024-03-23. Score: 2

You have asked a very reasonable question.

Although moon-moon tidal heating is *usually* very weak, it may in principle be of interest in tightly packed systems. A detailed theory of this effect (motivated by TRAPPIST-1) was developed in 2021 by [Michaela Walterova](https://www.mff.cuni.cz/en/faculty/organizational-structure/people?hdl=8148) in her outstanding Ph.D. thesis. Also see page 39 in her [defence slides](https://geo.mff.cuni.cz/theses/2021-Walterova-PhD-slides.pdf).

I begged Michaela to publish the solution, but she keeps procrastinating.


---

