<p>If you're thinking about how close planets can be, you should probably consider each planet's <a href="https://en.wikipedia.org/wiki/Hill_sphere" rel="noreferrer">Hill sphere</a>, the region in which it can retain satellites. <a href="https://arxiv.org/abs/1302.7190" rel="noreferrer">Fang &amp; Margot (2013)</a> did an analysis of <em>Kepler</em> data and found that planets had mean values of $\Delta = 21.7$, where $\Delta$ is a parameter given for two adjacent planets by
$$\Delta=\frac{a_2-a_1}{R_{H1,2}}$$
where the $a$s are the semi-major axes and $R_{H1,2}$ is the mutual Hill radius.</p>

<p>One system the authors consider is <a href="https://en.wikipedia.org/wiki/Kepler-11" rel="noreferrer">Kepler-11</a>, which has 6 planets, all with semi-major axes $\leq0.466\text{ AU}$ and with only one semi-major axis greater than $0.25\text{ AU}$. The smallest $\Delta$ there is approximately $5.7$, although all the other $\Delta$s are quite small. <a href="https://en.wikipedia.org/wiki/Kepler-36" rel="noreferrer">Kepler-36</a>, with only two planets, still has a $\Delta$ of $4.7$.</p>

<p>According to <a href="http://www.nature.com/nature/journal/v542/n7642/full/nature21360.html" rel="noreferrer">the <em>Nature</em> paper</a> about TRAPPIST-1, all seven planets have semi-major axes within $\sim0.063\text{ AU}$. They have mean $\Delta$s of $10.5\pm1.9$ - not much different from the Kepler-11 planets, because they have smaller Hill spheres. They may be closer together, but they can be much closer together without having stability problems. Additionally, they are in a "near-<a href="https://en.wikipedia.org/wiki/Orbital_resonance" rel="noreferrer">resonant</a>" configuration.</p>

<p>How close planets can be depends strongly on their masses, then, which in turn determines their mutual Hill radii, which determines stability.</p>

<hr>

<p>All that said, the authors believe that the TRAPPIST-1 planets may have migrated in from further out, thus entering the resonances. Without more information, we can't know whether this is the case, but if so, it is not, then, an example of planets forming close to each other.</p>

CC BY-SA 3.0
2017-02-22T23:43:36.127Z
To say 'oh because they have smaller masses, their $\Delta_{max}$ is smaller', is completely inconsistent with their mutual Hill-radii already being dependent on the masses and encapsulating all interesting physics. The point is, either there is some $\Delta_{max}$ for all planets, or there's a part of the physics that we don't understand.
AtmosphericPrisonEscape
CC BY-SA 3.0
2017-02-23T00:13:59.153Z
@AtmosphericPrisonEscape I never said that that was the case. Where specifically in my answer are you looking?
HDE 226868
CC BY-SA 3.0
2017-02-23T00:27:34.533Z
You say "They have mean Δs of 10.5±1.9 - not much different from the Kepler-11 planets, because they have smaller Hill spheres." while comparing this value to $\Delta \approx 21.7$, so it seemed you're trying to explain the smaller critical value by some scaling with the mass, which wouldn't make sense... I apologize if I've misunderstood the message.
AtmosphericPrisonEscape
CC BY-SA 3.0
2017-02-23T00:30:40.737Z
@AtmosphericPrisonEscape Their semi-major axes and differences in semi-major axes are also smaller. Therefore, $R_{H1,2}$ and $a_2-a_1$ are both smaller than with the Kepler-11 planets, so there's not necessarily a large chance in $\Delta$.
HDE 226868
CC BY-SA 3.0
2017-02-23T14:05:10.597Z
I think you left out a "not" in the sentence "...they can [not] be much closer together without... stability problems"   I might add that we haven't come up with a name (so far as I know) for a putative cluster of equally-sized bodies orbiting a star as a group, with the usual unpredictable intra-group orbits (3-body problem).
Carl Witthoft
CC BY-SA 3.0
2017-02-23T14:21:23.540Z
@CarlWitthoft I meant that they can be closer together than the more massive planets around Kepler-11 - so no, that's not a typo.
HDE 226868
CC BY-SA 3.0
2017-02-23T18:26:00.637Z
Maybe replace "but" with "can," then?
Carl Witthoft
CC BY-SA 3.0
2017-02-22T21:22:28.377Z
2017-02-22T21:38:37.787Z
2017-02-22T21:38:37.787Z
HDE 226868
<p>With the NASA announcement today regarding the discovery of a system containing seven earth-sized planets (3 within the habitable zone), I wondered about the seemingly crowded conditions.  What principles guide the formation of planets and the distances between them?  Are there laws governing this that are well established?  How close have we observed two planets form?</p>

CC BY-SA 3.0
2017-02-22T21:45:19.623Z
As you mention laws, I find myself compelled to say that they can form as close as they like to each other until one of them evolves lawyers and zoning laws ruin the whole thing.  I apologize for any offense this causes.  Unless it's to lawyers, of course. :-)
StephenG - Help Ukraine
CC BY-SA 3.0
2017-02-22T22:24:30.607Z
FYI, the paper suggests the planets formed farther out and migrated inwards as indicated by their resonances, meaning their formation processes likely weren't able to interrupt each other as they formed and they only became so crowded afterwards.
zephyr
<planet><exoplanet><planetary-formation>
How close can planets form to one another?
SDH
# How close can planets form to one another?

(Asked by: username_0 on 2017-02-22)

With the NASA announcement today regarding the discovery of a system containing seven earth-sized planets (3 within the habitable zone), I wondered about the seemingly crowded conditions. What principles guide the formation of planets and the distances between them? Are there laws governing this that are well established? How close have we observed two planets form?

**Tags:** planet, exoplanet, planetary-formation
> **Comments:**
> **username_1**: As you mention laws, I find myself compelled to say that they can form as close as they like to each other until one of them evolves lawyers and zoning laws ruin the whole thing.  I apologize for any offense this causes.  Unless it's to lawyers, of course. :-)  
> **username_2**: FYI, the paper suggests the planets formed farther out and migrated inwards as indicated by their resonances, meaning their formation processes likely weren't able to interrupt each other as they formed and they only became so crowded afterwards.  

---

## 1 answers

### Answer by username_3 on 2017-02-22. Score: 7 | Accepted answer

If you're thinking about how close planets can be, you should probably consider each planet's [Hill sphere](https://en.wikipedia.org/wiki/Hill_sphere), the region in which it can retain satellites. [Fang & Margot (2013)](https://arxiv.org/abs/1302.7190) did an analysis of *Kepler* data and found that planets had mean values of $\Delta = 21.7$, where $\Delta$ is a parameter given for two adjacent planets by
$$\Delta=\frac{a\_2-a\_1}{R\_{H1,2}}$$
where the $a$s are the semi-major axes and $R\_{H1,2}$ is the mutual Hill radius.

One system the authors consider is [Kepler-11](https://en.wikipedia.org/wiki/Kepler-11), which has 6 planets, all with semi-major axes $\leq0.466\text{ AU}$ and with only one semi-major axis greater than $0.25\text{ AU}$. The smallest $\Delta$ there is approximately $5.7$, although all the other $\Delta$s are quite small. [Kepler-36](https://en.wikipedia.org/wiki/Kepler-36), with only two planets, still has a $\Delta$ of $4.7$.

According to [the *Nature* paper](http://www.nature.com/nature/journal/v542/n7642/full/nature21360.html) about TRAPPIST-1, all seven planets have semi-major axes within $\sim0.063\text{ AU}$. They have mean $\Delta$s of $10.5\pm1.9$ - not much different from the Kepler-11 planets, because they have smaller Hill spheres. They may be closer together, but they can be much closer together without having stability problems. Additionally, they are in a "near-[resonant](https://en.wikipedia.org/wiki/Orbital_resonance)" configuration.

How close planets can be depends strongly on their masses, then, which in turn determines their mutual Hill radii, which determines stability.

---

All that said, the authors believe that the TRAPPIST-1 planets may have migrated in from further out, thus entering the resonances. Without more information, we can't know whether this is the case, but if so, it is not, then, an example of planets forming close to each other.

> **Comments:**
> **username_4**: To say 'oh because they have smaller masses, their $\Delta_{max}$ is smaller', is completely inconsistent with their mutual Hill-radii already being dependent on the masses and encapsulating all interesting physics. The point is, either there is some $\Delta_{max}$ for all planets, or there's a part of the physics that we don't understand.  
> **username_3**: @AtmosphericPrisonEscape I never said that that was the case. Where specifically in my answer are you looking?  
> **username_4**: You say "They have mean Δs of 10.5±1.9 - not much different from the Kepler-11 planets, because they have smaller Hill spheres." while comparing this value to $\Delta \approx 21.7$, so it seemed you're trying to explain the smaller critical value by some scaling with the mass, which wouldn't make sense... I apologize if I've misunderstood the message.  
> **username_3**: @AtmosphericPrisonEscape Their semi-major axes and differences in semi-major axes are also smaller. Therefore, $R_{H1,2}$ and $a_2-a_1$ are both smaller than with the Kepler-11 planets, so there's not necessarily a large chance in $\Delta$.  
> **username_5**: I think you left out a "not" in the sentence "...they can [not] be much closer together without... stability problems"   I might add that we haven't come up with a name (so far as I know) for a putative cluster of equally-sized bodies orbiting a star as a group, with the usual unpredictable intra-group orbits (3-body problem).  
> **username_3**: @CarlWitthoft I meant that they can be closer together than the more massive planets around Kepler-11 - so no, that's not a typo.  
> **username_5**: Maybe replace "but" with "can," then?  

---

