<p><strong>Diclaimer:</strong> <em>This is not (yet) an answer! To attract answers, I decided to start an answer draft which can be expanded by others.</em></p>
<h3>Cylindrical coordinates</h3>
<p>Every point in our <a href="https://en.wikipedia.org/wiki/Cylindrical_coordinate_system" rel="nofollow noreferrer">cylindrical coordinate system</a> is defined by a tuple <span class="math-container">$(r,\varphi,z)$</span> where <span class="math-container">$r$</span> is the distance from the rotational axis. We also define <span class="math-container">$Z$</span> as the height of our <a href="https://en.wikipedia.org/wiki/Solid_of_revolution" rel="nofollow noreferrer">solid of revolution</a>, i.e. <span class="math-container">$0 \leq z \leq Z$</span>. The shape of the body is defined by shape function <span class="math-container">$s(z)$</span>.</p>
<p>The volume <span class="math-container">$V$</span> of the object is then given by <span class="math-container">$$V= \pi \int_0^Z \left( s(z) \right)^2 {\rm d}z$$</span></p>
<h3>Mass conservation</h3>
<p>The mass density <span class="math-container">$\rho(r,z)$</span> does not depend on <span class="math-container">$\varphi$</span>.</p>
<p><em>to be continued</em></p>
<h1>Specific shape curves</h1>
<p>Up to now, all maths has been performed for a general shape function <span class="math-container">$s(z)$</span>, so let us now look at some specific ones</p>
<h2>Egg as rotational body</h2>
<p>For an egg with <span class="math-container">$z$</span> being the distance from the symmetry axis, we could for instance a formula by <a href="https://www.researchgate.net/publication/248562638_The_Avian_Egg_Geometrical_Description_and_Calculation_of_Parameters" rel="nofollow noreferrer">Narushin</a>:</p>
<p><span class="math-container">$$s(z) = 1.5396 \cdot \frac{B}{Z} \cdot\sqrt{ \sqrt{Z}\cdot z^{\frac{3}{2}}-z^2}$$</span></p>
<p>In this formula, <span class="math-container">$B$</span> is the maximum breadth and <span class="math-container">$Z$</span> is the height of the egg.</p>

CC BY-SA 4.0
2021-01-20T17:20:06.737Z
I thought so. Graphed it on desmos, with B and Z being variables, z being the x. Use +- for the egg.
WarpPrime
CC BY-SA 4.0
2021-01-20T13:03:49.053Z
2021-01-20T15:24:00.207Z
2021-01-20T15:24:00.207Z
B--rian
<p>I am well aware of <a href="https://en.wikipedia.org/wiki/Stellar_structure#Equations_of_stellar_structure" rel="nofollow noreferrer">one-dimensional stellar models</a>:</p>
<blockquote>
<p>The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in a steady state and that it is spherically symmetric.  It contains four basic first-order differential equations: two represent how matter and pressure vary with radius; two represent how temperature and luminosity vary with radius.</p>
</blockquote>
<p>But what if we moved from spherical symmetry to cylindrical symmetry? <strong>Did somebody already set up all equations and solve them for general rotational symmetric ellipsoid?</strong></p>
<p>What changes, if we would assume a <a href="https://en.wikipedia.org/wiki/Lemon_(geometry)" rel="nofollow noreferrer">lemon-shaped</a> or (most interestingly) an <a href="http://www.mathematische-basteleien.de/eggcurves.htm" rel="nofollow noreferrer">egg-shaped</a> star?</p>
<p>What would be the (intutive) results of such a stellar model? I am sure, somebody solved the equations already and I am just missing the appropriate search terms.</p>
<h3>References</h3>
<ul>
<li><a href="https://www.researchgate.net/publication/259604597_The_mathematics_of_egg_shape" rel="nofollow noreferrer">The mathematics of egg shape</a> gives a brief mathematical background about one of my favorite mathematical objects</li>
</ul>
<p>Cylindrical symmetry is not as hypothetical as it might sound:</p>
<ul>
<li>Ashley Strickland wrote for CNN about &quot;<a href="https://edition.cnn.com/2020/03/09/world/pulsating-star-discovery-scn/index.html" rel="nofollow noreferrer">Unusual tear-drop shaped, half-pulsating star discovered by amateur astronomers</a>&quot;</li>
<li><a href="https://de.wikipedia.org/wiki/WASP-12b" rel="nofollow noreferrer">WASP-12b</a> is is reviewed by NASA as <a href="https://exoplanets.nasa.gov/news/1375/wasp-12b-an-egg-shaped-planet/" rel="nofollow noreferrer">An egg-shaped planet</a>.</li>
</ul>
<p>The pre-print by E.C. &amp; L.V. Nolan <a href="https://arxiv.org/abs/gr-qc/0407041" rel="nofollow noreferrer">On isotropic cylindrically symmetric stellar models</a> seems to cover the topic, but is not too intuitive.</p>
<h3>Related</h3>
<ul>
<li><a href="https://astronomy.stackexchange.com/questions/27615/can-a-donut-shape-planet-or-star-be-formed">Can a donut-shape planet or star be formed?</a></li>
</ul>

CC BY-SA 4.0
2021-01-19T03:11:08.227Z
This is fascinating! If you allow for significant rotation there are some things to explore: [What exactly was Lagrange's “grave mistake” with respect to rotating bodies under hydrostatic equilibrium?](https://hsm.stackexchange.com/q/11351/3869) and [How does a Maclaurin spheroid become a Jacobi ellipsoid? What happens?](https://math.stackexchange.com/q/3514142) Both are currently unanswered.
uhoh
CC BY-SA 4.0
2021-01-19T08:30:03.727Z
Maybe helpful, if you want to do it from scratch: Some [mathematical fun with boiling eggs](https://newton.ex.ac.uk/teaching/CDHW/egg/CW061201-1.pdf) which is mainly solving the heat conduction equation.
B--rian
CC BY-SA 4.0
2021-01-19T13:09:22.827Z
In principle a cylindrical symmetry might well be that of a the collapsing nebula, at start
Alchimista
CC BY-SA 4.0
2021-01-20T00:36:01.107Z
Models of stars filling their Roche lobe would also naturally be egg-shaped. Sounds like something there should be simulations of.
Anders Sandberg
CC BY-SA 4.0
2021-01-20T21:53:41.347Z
As I mentioned [here](https://astronomy.stackexchange.com/questions/20823/shape-of-neutron-stars#comment35129_20823) it's possible that the intense magnetic field of a young magnetar can distort it into a prolate (egg / football shaped) spheroid. Also see https://arxiv.org/abs/0712.2162
PM 2Ring
<star><mathematics><modeling>
Modeling egg shaped stars
B--rian
# Modeling egg shaped stars

(Asked by: username_0 on 2021-01-18)

I am well aware of [one-dimensional stellar models](https://en.wikipedia.org/wiki/Stellar_structure#Equations_of_stellar_structure):

> The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in a steady state and that it is spherically symmetric. It contains four basic first-order differential equations: two represent how matter and pressure vary with radius; two represent how temperature and luminosity vary with radius.

But what if we moved from spherical symmetry to cylindrical symmetry? **Did somebody already set up all equations and solve them for general rotational symmetric ellipsoid?**

What changes, if we would assume a [lemon-shaped](https://en.wikipedia.org/wiki/Lemon_(geometry)) or (most interestingly) an [egg-shaped](http://www.mathematische-basteleien.de/eggcurves.htm) star?

What would be the (intutive) results of such a stellar model? I am sure, somebody solved the equations already and I am just missing the appropriate search terms.

### References

* [The mathematics of egg shape](https://www.researchgate.net/publication/259604597_The_mathematics_of_egg_shape) gives a brief mathematical background about one of my favorite mathematical objects

Cylindrical symmetry is not as hypothetical as it might sound:

* Ashley Strickland wrote for CNN about "[Unusual tear-drop shaped, half-pulsating star discovered by amateur astronomers](https://edition.cnn.com/2020/03/09/world/pulsating-star-discovery-scn/index.html)"
* [WASP-12b](https://de.wikipedia.org/wiki/WASP-12b) is is reviewed by NASA as [An egg-shaped planet](https://exoplanets.nasa.gov/news/1375/wasp-12b-an-egg-shaped-planet/).

The pre-print by E.C. & L.V. Nolan [On isotropic cylindrically symmetric stellar models](https://arxiv.org/abs/gr-qc/0407041) seems to cover the topic, but is not too intuitive.

### Related

* [Can a donut-shape planet or star be formed?](https://astronomy.stackexchange.com/questions/27615/can-a-donut-shape-planet-or-star-be-formed)

**Tags:** star, mathematics, modeling

> **Comments:**
> **username_1**: This is fascinating! If you allow for significant rotation there are some things to explore: [What exactly was Lagrange's “grave mistake” with respect to rotating bodies under hydrostatic equilibrium?](https://hsm.stackexchange.com/q/11351/3869) and [How does a Maclaurin spheroid become a Jacobi ellipsoid? What happens?](https://math.stackexchange.com/q/3514142) Both are currently unanswered.  
> **username_0**: Maybe helpful, if you want to do it from scratch: Some [mathematical fun with boiling eggs](https://newton.ex.ac.uk/teaching/CDHW/egg/CW061201-1.pdf) which is mainly solving the heat conduction equation.  
> **username_2**: In principle a cylindrical symmetry might well be that of a the collapsing nebula, at start  
> **username_3**: Models of stars filling their Roche lobe would also naturally be egg-shaped. Sounds like something there should be simulations of.  
> **username_4**: As I mentioned [here](https://astronomy.stackexchange.com/questions/20823/shape-of-neutron-stars#comment35129_20823) it's possible that the intense magnetic field of a young magnetar can distort it into a prolate (egg / football shaped) spheroid. Also see https://arxiv.org/abs/0712.2162  

---

## 1 answers

### Answer by username_0 on 2021-01-20. Score: 1

**Diclaimer:** *This is not (yet) an answer! To attract answers, I decided to start an answer draft which can be expanded by others.*

### Cylindrical coordinates

Every point in our [cylindrical coordinate system](https://en.wikipedia.org/wiki/Cylindrical_coordinate_system) is defined by a tuple $(r,\varphi,z)$ where $r$ is the distance from the rotational axis. We also define $Z$ as the height of our [solid of revolution](https://en.wikipedia.org/wiki/Solid_of_revolution), i.e. $0 \leq z \leq Z$. The shape of the body is defined by shape function $s(z)$.

The volume $V$ of the object is then given by $$V= \pi \int\_0^Z \left( s(z) \right)^2 {\rm d}z$$

### Mass conservation

The mass density $\rho(r,z)$ does not depend on $\varphi$.

*to be continued*

# Specific shape curves

Up to now, all maths has been performed for a general shape function $s(z)$, so let us now look at some specific ones

## Egg as rotational body

For an egg with $z$ being the distance from the symmetry axis, we could for instance a formula by [Narushin](https://www.researchgate.net/publication/248562638_The_Avian_Egg_Geometrical_Description_and_Calculation_of_Parameters):

$$s(z) = 1.5396 \cdot \frac{B}{Z} \cdot\sqrt{ \sqrt{Z}\cdot z^{\frac{3}{2}}-z^2}$$

In this formula, $B$ is the maximum breadth and $Z$ is the height of the egg.

> **Comments:**
> **username_5**: I thought so. Graphed it on desmos, with B and Z being variables, z being the x. Use +- for the egg.  

---

