<p>I think all this means is that up to some critical rotation rate, the equilibrium shape of a rotating, self-gravitating fluid is an oblate spheroid. That is, this shape defines a unique global minimum of energy.</p>

<p>Above this threshold there are two possible equilibrium solutions. Apparently one is an oblate spheroid, whereas the other is a triaxial ellipsoid. These will define <em>local</em> energy minima, but the triaxial ellipsoid is the global minimum.</p>

<p>The gory mathematical details are no doubt contained in Chandrasekhar's textbook <a href="http://ukads.nottingham.ac.uk/abs/1969efe..book.....C" rel="nofollow noreferrer">on equilibrium ellipsoidal figures</a>. Pages 3-5 of <a href="https://arxiv.org/abs/1409.3858" rel="nofollow noreferrer">Iurato (2014)</a> appear to summarise the historical development of this bifurcation in the equilibrium solutions.</p>

CC BY-SA 3.0
2016-11-28T01:03:50.100Z
OK that makes sense, and this is best followed up by a trip to the library. Iurato (2014) certainly sets the stage from the historical perspective, and it's possible that numerical simulations may show the minimum energy figures are not necessarily pure ellipsoids. Thank you again for another helpful answer!
uhoh
CC BY-SA 3.0
2016-11-27T20:09:52.727Z
2016-11-27T20:18:13.770Z
2016-11-27T20:18:13.770Z
ProfRob
<p>An <a href="https://astronomy.stackexchange.com/q/19186/7982">intriguing comment</a> below <a href="https://astronomy.stackexchange.com/questions/19186/#comment31073_19186">this question</a> links to <a href="https://www.quora.com/How-and-why-are-planets-spherical-What-makes-them-round#!n=96" rel="nofollow noreferrer">this Quora question</a> and points to the answer by Robert Walker. </p>

<p>The sentence that particularly intrigued me is:</p>

<blockquote>
  <p>It turns out that there are two possible solutions, as the spin rate increases. You can get an oblate spheroid, or a triaxial ellipsoid - the solution "bifurcates". But the triaxial ellipsoid is the most stable of the two as Jacobi found out in his paper published in 1834.  Figures of Equilibrium - Historical Account* - Chandrasekar</p>
</blockquote>

<p>It seems to refer both to an 1834 paper by Jacobi and it's reference in the 1964 paper <em>Ellipsoidal figures of equilibrium—an historical account</em> by S. Chandrasekhar, Communications on Pure and Applied Mathematics, 20 (2), May 1967, 251–265. While the latter is paywalled, there seems to be a readable version <a href="https://books.google.com.tw/books?id=h_9pDQAAQBAJ&amp;lpg=PA1145&amp;ots=w9oKES1OYs&amp;dq=Figures%20of%20Equilibrium%20-%20Historical%20Account&amp;pg=PA1138#v=onepage&amp;q=Figures%20of%20Equilibrium%20-%20Historical%20Account&amp;f=false" rel="nofollow noreferrer">here in google</a> in the book <em>A Quest for Perspectives: Selected Works of S. Chandrasekhar : with Commentary By Subrahmanyan Chandrasekhar</em>, Kameshwar C. Wali, Volume 1. Imperial College Press, 2001.</p>

<p>Mathematical bifurcation is a fascinating topic and there is even an <a href="http://www.worldscientific.com/worldscinet/ijbc" rel="nofollow noreferrer">entire mathematics journal</a> dedicated to the subject.</p>

<p>In this case, <em>where exactly in the calculation of hydrostatic equillibrium of rotating bodies does this bifurcation</em> (mentioned in the Quora answer) <em>occur?</em> I'm wondering if it simply means that for a given volume and density, below a certain rotation rate the hydrostatic equilibrium shape is an oblate sphere, but above it, the hydrostatic equilibrium shape will be a triaxial ellipsoid? Or is the bifurcation part of the evolution of the shape over time, involving viscocity or external effects?</p>

<p>If the triaxial ellipsoide is always(?) the most stable, then is a oblate spheroid actually an intermediate shape, and not really the shape of hydrostatic equillibrium?</p>

CC BY-SA 3.0
2016-11-27T19:00:59.727Z
Doesn't it just mean that for a given mass there are two local minima in the energy density?
ProfRob
CC BY-SA 3.0
2016-11-27T19:10:35.780Z
@RobJeffries I don't know for sure. For example when we talk about bifurcation of orbits in a 3-body problem, two trajectories with almost identical starting conditions will appear to suddenly separate (bifurcate) at a point in time as the orbits evolve. Bifurcation means more than there being two minima. It may mean that there is some rotation rate above-which one minimum changes to two minima, I don't know.
uhoh
CC BY-SA 3.0
2016-11-27T19:12:01.900Z
Oh, it definitely refers to there being more than one possible solution. i.e.  there is no unique shape that is stable above some critical rotation threshold.
ProfRob
CC BY-SA 3.0
2016-11-27T19:25:35.400Z
@RobJeffries can you support this with something that *shows the mathematical bifurcation?* I'm not asking about uniqueness, I am asking *specifically about this bifurcation*. In what space or along what dimension or axis does a minimum bifurcate?
uhoh
CC BY-SA 3.0
2016-11-28T03:39:04.217Z
"bifurcates" just means "goes one way or another".  here it means "the whole situation" goes either one way or another; it heads towards one or the other solution-paradigms.
Fattie
CC BY-SA 3.0
2016-11-28T04:22:26.837Z
@JoeBlow no, not in astronomy, physics nor mathematics it doesn't. The "x just means y" dumbing-down of terms is not helpful in stackexchange. If you think the Quora answer has misused the term you should comment there instead, but please don't muddy the waters here.
uhoh
CC BY-SA 3.0
2016-11-28T04:41:13.450Z
Hi @Uhoh dude. You're wrong. The sentence means ***the solution*** bifurcates.  Exact quote, I added italics and the "A/B":  "You can get (A) an oblate spheroid, or (B) a triaxial ellipsoid - *the solution* "bifurcates"".
Fattie
CC BY-SA 3.0
2016-11-28T04:47:25.293Z
You could be confusing "bifurcating an object" (meaning "chopping it in two" - eg, "the Nile bifurcates", to quote the OED) with the Quora writer simply using the word in his sentence to say the *solution* bifurcates.  Just BTW there was no reason whatsoever for the Quora writer to put bifurcates in quotes, in the extract shown.
Fattie
CC BY-SA 3.0
2016-11-28T05:28:41.250Z
@JoeBlow I think the use in the Quora answer has the same meaning as it does when Chandrasekhar, Iurato and RobJeffries use it. Hopefully I can find a plot that actually shows the bifurcation in question - if so I'll post it somehow.
uhoh
<planetary-formation><mathematics>
What is meant by the bifurcation of hydrostatic equilibrium shapes in planetary formation?
uhoh
# What is meant by the bifurcation of hydrostatic equilibrium shapes in planetary formation?

(Asked by: username_0 on 2016-11-27)

An [intriguing comment](https://astronomy.stackexchange.com/q/19186/7982) below [this question](https://astronomy.stackexchange.com/questions/19186/#comment31073_19186) links to [this Quora question](https://www.quora.com/How-and-why-are-planets-spherical-What-makes-them-round#!n=96) and points to the answer by Robert Walker.

The sentence that particularly intrigued me is:

> It turns out that there are two possible solutions, as the spin rate increases. You can get an oblate spheroid, or a triaxial ellipsoid - the solution "bifurcates". But the triaxial ellipsoid is the most stable of the two as Jacobi found out in his paper published in 1834. Figures of Equilibrium - Historical Account\* - Chandrasekar

It seems to refer both to an 1834 paper by Jacobi and it's reference in the 1964 paper *Ellipsoidal figures of equilibrium—an historical account* by S. Chandrasekhar, Communications on Pure and Applied Mathematics, 20 (2), May 1967, 251–265. While the latter is paywalled, there seems to be a readable version [here in google](https://books.google.com.tw/books?id=h_9pDQAAQBAJ&lpg=PA1145&ots=w9oKES1OYs&dq=Figures%20of%20Equilibrium%20-%20Historical%20Account&pg=PA1138#v=onepage&q=Figures%20of%20Equilibrium%20-%20Historical%20Account&f=false) in the book *A Quest for Perspectives: Selected Works of S. Chandrasekhar : with Commentary By Subrahmanyan Chandrasekhar*, Kameshwar C. Wali, Volume 1. Imperial College Press, 2001.

Mathematical bifurcation is a fascinating topic and there is even an [entire mathematics journal](http://www.worldscientific.com/worldscinet/ijbc) dedicated to the subject.

In this case, *where exactly in the calculation of hydrostatic equillibrium of rotating bodies does this bifurcation* (mentioned in the Quora answer) *occur?* I'm wondering if it simply means that for a given volume and density, below a certain rotation rate the hydrostatic equilibrium shape is an oblate sphere, but above it, the hydrostatic equilibrium shape will be a triaxial ellipsoid? Or is the bifurcation part of the evolution of the shape over time, involving viscocity or external effects?

If the triaxial ellipsoide is always(?) the most stable, then is a oblate spheroid actually an intermediate shape, and not really the shape of hydrostatic equillibrium?

**Tags:** planetary-formation, mathematics
> **Comments:**
> **username_1**: Doesn't it just mean that for a given mass there are two local minima in the energy density?  
> **username_0**: @RobJeffries I don't know for sure. For example when we talk about bifurcation of orbits in a 3-body problem, two trajectories with almost identical starting conditions will appear to suddenly separate (bifurcate) at a point in time as the orbits evolve. Bifurcation means more than there being two minima. It may mean that there is some rotation rate above-which one minimum changes to two minima, I don't know.  
> **username_1**: Oh, it definitely refers to there being more than one possible solution. i.e.  there is no unique shape that is stable above some critical rotation threshold.  
> **username_0**: @RobJeffries can you support this with something that *shows the mathematical bifurcation?* I'm not asking about uniqueness, I am asking *specifically about this bifurcation*. In what space or along what dimension or axis does a minimum bifurcate?  
> **username_2**: "bifurcates" just means "goes one way or another".  here it means "the whole situation" goes either one way or another; it heads towards one or the other solution-paradigms.  
> **username_0**: @JoeBlow no, not in astronomy, physics nor mathematics it doesn't. The "x just means y" dumbing-down of terms is not helpful in stackexchange. If you think the Quora answer has misused the term you should comment there instead, but please don't muddy the waters here.  
> **username_2**: Hi @Uhoh dude. You're wrong. The sentence means ***the solution*** bifurcates.  Exact quote, I added italics and the "A/B":  "You can get (A) an oblate spheroid, or (B) a triaxial ellipsoid - *the solution* "bifurcates"".  
> **username_2**: You could be confusing "bifurcating an object" (meaning "chopping it in two" - eg, "the Nile bifurcates", to quote the OED) with the Quora writer simply using the word in his sentence to say the *solution* bifurcates.  Just BTW there was no reason whatsoever for the Quora writer to put bifurcates in quotes, in the extract shown.  
> **username_0**: @JoeBlow I think the use in the Quora answer has the same meaning as it does when Chandrasekhar, Iurato and RobJeffries use it. Hopefully I can find a plot that actually shows the bifurcation in question - if so I'll post it somehow.  

---

## 1 answers

### Answer by username_1 on 2016-11-27. Score: 3 | Accepted answer

I think all this means is that up to some critical rotation rate, the equilibrium shape of a rotating, self-gravitating fluid is an oblate spheroid. That is, this shape defines a unique global minimum of energy.

Above this threshold there are two possible equilibrium solutions. Apparently one is an oblate spheroid, whereas the other is a triaxial ellipsoid. These will define *local* energy minima, but the triaxial ellipsoid is the global minimum.

The gory mathematical details are no doubt contained in Chandrasekhar's textbook [on equilibrium ellipsoidal figures](http://ukads.nottingham.ac.uk/abs/1969efe..book.....C). Pages 3-5 of [Iurato (2014)](https://arxiv.org/abs/1409.3858) appear to summarise the historical development of this bifurcation in the equilibrium solutions.

> **Comments:**
> **username_0**: OK that makes sense, and this is best followed up by a trip to the library. Iurato (2014) certainly sets the stage from the historical perspective, and it's possible that numerical simulations may show the minimum energy figures are not necessarily pure ellipsoids. Thank you again for another helpful answer!  

---

