# What is the Roche limit for these two planets? [duplicate]

I want may planets to be as close as possible, but I really don't want this to happen.

So, both of these planets are earth-sized, and they have a moon orbiting the two. These planets are also tidally locked, and forming a binary system together. How close can these planets be to each other while still being roughly spherical? They have to be about as spherical as earth.

I want to just say that this question is not a duplicate because I still want the planets to be spherical, and not be stretched far enough to be able to share an atmosphere.

This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

## marked as duplicate by Serban Tanasa, Hohmannfan, Jim2B, bowlturner, HDE 226868♦Jan 30 '16 at 18:05

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• A resource you may find useful, if you don't mind a bit of a simplified explanation. YouTube: Artifexian - Worldbuilding Videos (Chronological Order) – platypus-rising Jan 30 '16 at 4:37
• You'll have trouble keeping the moon stable. – JDługosz Jan 30 '16 at 6:09
• I know, it's pretty far away, and it was recently captured. – Xandar The Zenon Jan 30 '16 at 15:29
• Oops, I accidentally clicked reopen. I meant to do it on a different question. Curse you multiple tabs! – Xandar The Zenon Feb 6 '16 at 0:50
• @XandarTheZenon it's actually still a duplicate, since the linked duplicate has an answer that gives you everything you need to calculate the roche limit in the form of fancy formulas (and thus, answer the question "What is the roche limit". – Aify Feb 6 '16 at 2:58

## 1 Answer

There's an answer here for the Roche limit (somewhere between touching each other and 2800 km depending on composition).

The oblateness of the Earth is caused by centrifugal forces induced by rotation about its axis. If you want an Earth-like planet to have Earth-like oblateness, it needs to have an Earth-like rotational period, about 24 hours.

Now, you want the two planets to be tidally locked. This means they're orbiting each other at the same speed they're rotating about their axes. Since we want axial rotation to be 24 hours, we want orbital period to be 24 hours.

From this random Google result, we get the derivation of the orbital period of binary stars. The assumptions they make about the stars should hold for our Earth-like planets, and gravity is gravity.

$2\pi\sqrt{{(M+m)^2 r^3\over GM^3}}=T$

From above and a quick Google search, we know $T=24h=86400s$, $M=m=5.972\cdot 10^{24} kg$, and $G=6.67408\cdot 10^{-11}{m^3\over kg\cdot s^2}$. We just need to solve for radius.

$\sqrt{{(2M)^2 r^3\over GM^3}}={T\over 2\pi}$ -- subst $M+m=2M$, div sides by $2\pi$

${4M^2 r^3\over GM^3}={T^2\over 4\pi^2}$ -- square sides, subst $(2M)^2=2^2M^2=4M^2$

$r^3={T^2GM^3\over 16\pi^2M^2}$ -- div sides by $4M^2$, mult sides by $GM^3$

$r^3={T^2GM\over 16\pi^2}$ -- reduce ${M^3\over M^2}=M$

$r=\sqrt[3]{T^2GM\over 16\pi^2}$ -- cube root sides

$r=\sqrt[3]{86400^2s^2\cdot 6.67408\cdot 10^{-11}{m^3\over kg\cdot s^2}\cdot 5.972\cdot 10^{24}kg\over 16\pi^2}$ -- sub known values

$r=\sqrt[3]{1.88417\cdot 10^{22}{s^2m^3kg\over s^2kg}}$ -- simplify numerical part, collect all units

$r=\sqrt[3]{1.88417\cdot 10^{22}}\sqrt[3]{m^3}$ -- $s^2$ and $kg$ cancel out, separate number and units

$r=2.66097\cdot 10^7 m$ -- simplify

$r=26,609,700m=26,610km$ -- convert to km

$26,610 km \gg 2800 km$, so you shouldn't have any trouble with the planets breaking apart.

As long as the moon isn't ridiculously close, it shouldn't affect the outcome. The given Roche limit seems unsettlingly small to me, but appears to be valid from the links given on the other page.

• So it looks like my two thousand mile goal was met. Hooray! Now for some interspecies wars, and the unintentional annihilation of both planets by smooshing. (The moon is a harsh mistress) – Xandar The Zenon Jan 30 '16 at 4:59
• @XandarTheZenon Sounds like a fun story. :) – Draco18s Feb 5 '16 at 21:23
• @XandarTheZenon You might want to glance over at First Cycle by Piper and the Rocheworld series. – user487 Feb 6 '16 at 2:34
• @MichaelT Don't worry, Rocheworld is on my list. – Xandar The Zenon Feb 6 '16 at 2:41
• @XandarTheZenon The thing is Forward was a hard science fiction author (and physicist). Piper was on the softer side and the two books are worlds apart in their treatment of the subject. – user487 Feb 6 '16 at 2:47