<p>Comments under this answer to <em>What orbital period would produce one New Moon (and one Full Moon) each year? What other effects would this produce?</em> link to</p>
<ol>
<li><a href="https://ui.adsabs.harvard.edu/abs/2006MNRAS.373.1227D/abstract" rel="nofollow noreferrer">Stable satellites around extrasolar giant planets</a> which gives a<sub>E</sub> ~ 0.4895 (1.0000 - 1.0305e<sub>P</sub> - 0.2738e<sub>sat</sub>). So for circular obits the limit is 0.4895 R<sub>Hill</sub>. This is based on dynamics simulations.</li>
<li><a href="https://academic.oup.com/mnras/article/392/1/181/1071655#ss9" rel="nofollow noreferrer">Transit timing effects due to an exomoon</a> (<a href="https://academic.oup.com/mnras/article/392/1/181/1071655#ss3-1" rel="nofollow noreferrer">section 3.1</a> and <a href="https://academic.oup.com/mnras/article/392/1/181/1071655#ss9" rel="nofollow noreferrer">Appendix B</a>) which states (with <span class="math-container">$\chi$</span> as some fraction of the Hill sphere radius:</li>
</ol>
<p><span class="math-container">$$\frac{P_{moon}}{P_{planet}} = \sqrt{ \frac{\chi^3}{3} }$$</span></p>
<p><a href="https://astronomy.stackexchange.com/a/39413/7982">This answer</a> cites <a href="https://arxiv.org/abs/1209.5323" rel="nofollow noreferrer">Exomoon habitability constrained by illumination and tidal heating</a> that links to item 2, and seems to say that this limit results in the conclusion that stability requires the period of the Moon to be <em>no longer than</em> 1/9 (one ninth) that of the planet's period.</p>
<blockquote>
<p>The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P<sub>p</sub>/9, P<sub>p</sub> being the planet’s orbital period about the star (Kipping 2009a).</p>
</blockquote>
<p>I don't understand how the factor of 1/9 arrises here, nor how these two different stability criteria can be brought into alignment.</p>
<p>For example, if I put 0.4895 from the first item into the equation from the 2nd item, I get 1/5 not 1/9, that's a small disagreement considering the topic of stability is a bit mushy, but I do not see at all how 1/9 arrises without using item 1. How did they get 1/9?</p>

CC BY-SA 4.0
2020-10-17T08:58:38.220Z
+1 for a nicely presented (and asked question)!
Infinity Milestone
CC BY-SA 4.0
2020-10-17T15:33:56.103Z
@Uhoh  That's a good question.  It is always good to know that there are rival solitions to a problem.  But you phrased part of it incorrectly.  You state referring to my answer that "...stability requires the period of the Moon to be 1/9 (one ninth) that of the planet's period."  When actually the sources I cited say stability requires the orbital period to be no more than one ninth that of the planet.
M. A. Golding
CC BY-SA 4.0
2020-10-17T15:52:34.960Z
@M.A.Golding I've made an edit that I think fixes it thanks! In the future please feel free to just click edit and fix problems like that if you like. I'm signing off for the night so if it needs more adjustment you or anyone are welcome to do so. *Thanks!*
uhoh
<exoplanet><orbital-mechanics><mathematics>
Help understanding why these two sub-Hill sphere moon stability limits are so different?
uhoh
# Help understanding why these two sub-Hill sphere moon stability limits are so different?

(Asked by: username_0 on 2020-10-17)

Comments under this answer to *What orbital period would produce one New Moon (and one Full Moon) each year? What other effects would this produce?* link to

1. [Stable satellites around extrasolar giant planets](https://ui.adsabs.harvard.edu/abs/2006MNRAS.373.1227D/abstract) which gives aE ~ 0.4895 (1.0000 - 1.0305eP - 0.2738esat). So for circular obits the limit is 0.4895 RHill. This is based on dynamics simulations.
2. [Transit timing effects due to an exomoon](https://academic.oup.com/mnras/article/392/1/181/1071655#ss9) ([section 3.1](https://academic.oup.com/mnras/article/392/1/181/1071655#ss3-1) and [Appendix B](https://academic.oup.com/mnras/article/392/1/181/1071655#ss9)) which states (with $\chi$ as some fraction of the Hill sphere radius:

$$\frac{P\_{moon}}{P\_{planet}} = \sqrt{ \frac{\chi^3}{3} }$$

[This answer](https://astronomy.stackexchange.com/a/39413/7982) cites [Exomoon habitability constrained by illumination and tidal heating](https://arxiv.org/abs/1209.5323) that links to item 2, and seems to say that this limit results in the conclusion that stability requires the period of the Moon to be *no longer than* 1/9 (one ninth) that of the planet's period.

> The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about Pp/9, Pp being the planet’s orbital period about the star (Kipping 2009a).

I don't understand how the factor of 1/9 arrises here, nor how these two different stability criteria can be brought into alignment.

For example, if I put 0.4895 from the first item into the equation from the 2nd item, I get 1/5 not 1/9, that's a small disagreement considering the topic of stability is a bit mushy, but I do not see at all how 1/9 arrises without using item 1. How did they get 1/9?

**Tags:** exoplanet, orbital-mechanics, mathematics

> **Comments:**
> **username_1**: +1 for a nicely presented (and asked question)!  
> **username_2**: @Uhoh  That's a good question.  It is always good to know that there are rival solitions to a problem.  But you phrased part of it incorrectly.  You state referring to my answer that "...stability requires the period of the Moon to be 1/9 (one ninth) that of the planet's period."  When actually the sources I cited say stability requires the orbital period to be no more than one ninth that of the planet.  
> **username_0**: @M.A.Golding I've made an edit that I think fixes it thanks! In the future please feel free to just click edit and fix problems like that if you like. I'm signing off for the night so if it needs more adjustment you or anyone are welcome to do so. *Thanks!*  

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## 0 answers

