<p>In terms of mean angular velocity, the distribution of rotation rates among main sequence stars is well known. Allen (1963) compiled data on mass, radius, and equatorial velocity, which was then expanded upon by <a href="http://adsabs.harvard.edu/abs/1965Obs....85..166M" rel="nofollow noreferrer">McNally (1965)</a>, who focused on angular velocity and angular momentum. It became clear that angular velocity increases from low rates for spectral types of G and below before rising to a peak around type A stars and then slowly decreasing.</p>
<p>Equatorial velocity continues increasing to mid-B type stars, before slowly decreasing, but because of the increased radii of O and B type main sequence stars, the peak in angular velocity occurs before this. As part of <a href="http://catdir.loc.gov/catdir/samples/cam032/99016740.pdf" rel="nofollow noreferrer">Jean-Louis Tassoul's <em>Stellar Rotation</em></a> notes, many O type stars have rotational periods similar to that of the G-type stars like the Sun!</p>
<p>The distribution is not smooth and uniform (McNally noticed a strange discontinuity in angular momentum per unit mass right for A0 and A5 stars; see his Figure 2); <a href="https://arxiv.org/abs/astro-ph/0303631" rel="nofollow noreferrer">Barnes (2003)</a> observed two distinct populations in open clusters, consisting of slower rotators (the I sequence) and faster rotators (the C sequence). Stars may migrate from one sequence to another as they evolve. Interestingly enough, stars on the I sequence lose angular momentum <span class="math-container">$J$</span> faster than stars on the C sequence:
<span class="math-container">$$\frac{\mathrm{d}J}{\mathrm{d}t}\propto-\omega^n,\quad\text{where}\begin{cases}
    n=3\text{ on the I sequence}\\
    n=1\text{ on the C sequence}\\
  \end{cases}$$</span>
Here, of course, <span class="math-container">$\omega$</span> is angular velocity. These results obey <a href="http://adsabs.harvard.edu/abs/1972ApJ...171..565S" rel="nofollow noreferrer">Skumanich's law</a>.</p>
<p>Oblateness can be determined from mass, radius, and angular velocity <a href="https://en.wikipedia.org/wiki/Equatorial_bulge" rel="nofollow noreferrer">as</a>
<span class="math-container">$$f=\frac{5\omega^2R^3}{4GM}$$</span>
Using this and McNally's data, some quick calculations get me the following table:
|Spectral type|<span class="math-container">$f/f(O5)$</span>|
|--|-------|
|O5 | 1    |
|B0 | 1.28 |
|B5 | 1.84 |
|A0 | 1.67 |
|A5 | 1.35 |
|F0 | 0.482|
|F5 | 0.0387|
|G0 | 0.000314|</p>

CC BY-SA 3.0
2016-04-30T06:39:35.913Z
I realize my question perhaps is not answerable in the form it was asked. But this is awesome anyway.
SE - stop firing the good guys
CC BY-SA 3.0
2016-04-30T17:03:36.450Z
What has this got to do with how smooth a star is?
ProfRob
CC BY-SA 3.0
2016-04-30T21:52:29.083Z
@Hohmannfan Does it still satisfy the criteria of the question? I didn't discuss irregularities at all.
HDE 226868
CC BY-SA 3.0
2016-04-30T22:08:11.740Z
@HDE226868 There was quite some discussion going on if my question was meaningful excluding angular velocity, so in the end, I decided to change that. (Your answer was what finally convinced me.) But if you have got some impossible-to-obtain stellar data, feel free to include it :)
SE - stop firing the good guys
CC BY-SA 4.0
2016-04-29T22:45:10.767Z
2020-12-15T00:35:58.733Z
2020-12-15T00:35:58.733Z
HDE 226868
<p>The Earth is a very smooth sphere, and the Sun even more so, with only minor fluctuations. I am wondering: are larger stars even rounder? Intuitively, that seems self evident, but I am not so sure. For instance, the hydrostatic equilibrium causes larger stars to be much less dense than red dwarfs. So what is the most important factor for how round a star is, a higher mass, or less activity? The most prominent cause of irregularities is of course the rotation rate of the star, which is pretty much independent of size. Ignoring that, do larger stars have smaller a deviations from the ellipsoid relative to their size?</p>

<p><strong>Edit</strong></p>

<p>As it seems like the "other than rotation rate" criterion is not really meaningfull, I now terminate it.</p>

CC BY-SA 3.0
2016-04-29T19:04:40.353Z
What do you mean by rounder?
ProfRob
CC BY-SA 3.0
2016-04-29T19:10:59.063Z
Indeed, more round is smaller irregularities relative to size.
SE - stop firing the good guys
CC BY-SA 3.0
2016-04-29T19:21:45.000Z
I have tried to edit the question now, to make it more clear I am asking about irregularities, not oblateness.
SE - stop firing the good guys
CC BY-SA 3.0
2016-04-29T19:25:35.453Z
I thought larger stars tended to have higher angular velocity than smaller stars?
called2voyage
CC BY-SA 3.0
2016-04-29T21:24:17.350Z
I have no idea how you would define an irregularity for a gaseous spheroid. In any case, we have no images or measurements of this for stars other than the Sun.
ProfRob
CC BY-SA 3.0
2016-04-29T22:22:46.167Z
[I would challenge your assumption that the Sun is rounder than the Earth](https://en.wikipedia.org/wiki/Solar_prominence).
rob
<star><hydrostatic-equilibrium><stellar-structure>
Are larger stars rounder?
SE - stop firing the good guys
# Are larger stars rounder?

(Asked by: username_0 on 2016-04-29)

The Earth is a very smooth sphere, and the Sun even more so, with only minor fluctuations. I am wondering: are larger stars even rounder? Intuitively, that seems self evident, but I am not so sure. For instance, the hydrostatic equilibrium causes larger stars to be much less dense than red dwarfs. So what is the most important factor for how round a star is, a higher mass, or less activity? The most prominent cause of irregularities is of course the rotation rate of the star, which is pretty much independent of size. Ignoring that, do larger stars have smaller a deviations from the ellipsoid relative to their size?

**Edit**

As it seems like the "other than rotation rate" criterion is not really meaningfull, I now terminate it.

**Tags:** star, hydrostatic-equilibrium, stellar-structure
> **Comments:**
> **username_1**: What do you mean by rounder?  
> **username_0**: Indeed, more round is smaller irregularities relative to size.  
> **username_0**: I have tried to edit the question now, to make it more clear I am asking about irregularities, not oblateness.  
> **username_2**: I thought larger stars tended to have higher angular velocity than smaller stars?  
> **username_1**: I have no idea how you would define an irregularity for a gaseous spheroid. In any case, we have no images or measurements of this for stars other than the Sun.  
> **username_3**: [I would challenge your assumption that the Sun is rounder than the Earth](https://en.wikipedia.org/wiki/Solar_prominence).  

---

## 1 answers

### Answer by username_4 on 2016-04-29. Score: 4 | Accepted answer

In terms of mean angular velocity, the distribution of rotation rates among main sequence stars is well known. Allen (1963) compiled data on mass, radius, and equatorial velocity, which was then expanded upon by [McNally (1965)](http://adsabs.harvard.edu/abs/1965Obs....85..166M), who focused on angular velocity and angular momentum. It became clear that angular velocity increases from low rates for spectral types of G and below before rising to a peak around type A stars and then slowly decreasing.

Equatorial velocity continues increasing to mid-B type stars, before slowly decreasing, but because of the increased radii of O and B type main sequence stars, the peak in angular velocity occurs before this. As part of [Jean-Louis Tassoul's *Stellar Rotation*](http://catdir.loc.gov/catdir/samples/cam032/99016740.pdf) notes, many O type stars have rotational periods similar to that of the G-type stars like the Sun!

The distribution is not smooth and uniform (McNally noticed a strange discontinuity in angular momentum per unit mass right for A0 and A5 stars; see his Figure 2); [Barnes (2003)](https://arxiv.org/abs/astro-ph/0303631) observed two distinct populations in open clusters, consisting of slower rotators (the I sequence) and faster rotators (the C sequence). Stars may migrate from one sequence to another as they evolve. Interestingly enough, stars on the I sequence lose angular momentum $J$ faster than stars on the C sequence:
$$\frac{\mathrm{d}J}{\mathrm{d}t}\propto-\omega^n,\quad\text{where}\begin{cases}
n=3\text{ on the I sequence}\\
n=1\text{ on the C sequence}\\
\end{cases}$$
Here, of course, $\omega$ is angular velocity. These results obey [Skumanich's law](http://adsabs.harvard.edu/abs/1972ApJ...171..565S).

Oblateness can be determined from mass, radius, and angular velocity [as](https://en.wikipedia.org/wiki/Equatorial_bulge)
$$f=\frac{5\omega^2R^3}{4GM}$$
Using this and McNally's data, some quick calculations get me the following table:
|Spectral type|$f/f(O5)$|
|--|-------|
|O5 | 1 |
|B0 | 1.28 |
|B5 | 1.84 |
|A0 | 1.67 |
|A5 | 1.35 |
|F0 | 0.482|
|F5 | 0.0387|
|G0 | 0.000314|

> **Comments:**
> **username_0**: I realize my question perhaps is not answerable in the form it was asked. But this is awesome anyway.  
> **username_1**: What has this got to do with how smooth a star is?  
> **username_4**: @Hohmannfan Does it still satisfy the criteria of the question? I didn't discuss irregularities at all.  
> **username_0**: @HDE226868 There was quite some discussion going on if my question was meaningful excluding angular velocity, so in the end, I decided to change that. (Your answer was what finally convinced me.) But if you have got some impossible-to-obtain stellar data, feel free to include it :)  

---

