How to calculate the expected rotational speed of a star

Question

I am working on fictional stars, and I want to have as many points about them as I can, and most things I found relatively easy, except for rotational speed.

I am not sure exactly how to find at what speed a star should be rotating in relation to its mass and spectral class. I found a small list of a few ones on wikipedia, but it doesn't help that much as it doesn't include the exact class I am looking for at the moment.

So, how is it that one can estimate what rotational velocity a star should have given the parameters. The parametres I know include Mass, Stellar Class, Radius, Density, Volume, Luminosity, Expected Maximum Age, Current Age (technically it changes as my world takes place over a long period, but the difference is in like, thousands of years, not billions) and Temperature.

I am aware massive outliers can exist, like Achernar, but I just need help finding like, the average, expected range so that I can use believable values.

Your help is much appreciated. Equations and citations would be beneficial, which is why I used the Hard Science tag.

Edit:

As requested by Neil Iyer, here is a random one of my stars so you can try answering that. I still do request a generalized answer, though.

• Class: F2.2V

  Mass: 1.35

  Radius: 1.187

  Density: 0.808

  Luminosity: 3.322

  Temperature: 7158 K

  Age: 3.6 Gyr

Mass, Radius, Density, and Luminosity are based on the Sun.

Edit 2:

One more specification, I am looking for equatorial velocity.

Answer 1

Unfortunately, you cannot.*

Angular momentum is an independant parameter of the system, like mass, or age. You cannot derive what it should be from other independant parameters, unlike a dependant parameter like temperature or radius. A star can have any angular velocity*, just like it could be at any age, or have any mass, within limits. It all depends on what the initial angular momentum of the cloud of gas it came from. While there may be trends based on mass and age, they do not effect the actual value. For example, while older stars will tend to be slower than younger stars, there's no reason to expect a young star not to have no rotation whatsoever.

*While the actual value of an independant paramaeter is not calculatable and must be observed, the minimum and maximum values can be. A star of a certain mass will have a maximum age, and one of a certain luminosity will have a maximum mass. Taking the link posted in a comment by JBH as a source, the break-up velocity of a star is given by:

$v_{max} = (\frac{2}{3}\frac{GM}{r_p})^\frac{1}{2}$

Where $r_p$ is the polar radius. A star can have any equatorial velocity between 0 and this value. For your example star, the value is 380 km/s, or 188 times the sun's equatorial velocity.

I would say pick a value between these extremes, but as you might have guessed it is not a uniform distribution, and most stars are no where near the extremes. But any given star could be anywhere in the range. This is because, to reiterate, the angular momentum a star starts with is not dependant on the mass it starts with (unlike, for example, the temperature, radius, and luminosity, which all can be calculated from age and mass). It is one of the starting values which must be plugged into the model.

What you can do is take observations of many star rotations for a given set of parameters (age and mass, spectral class, etc.) and fit a function to this distribution, and take the peak of this function as your average, but I wouldn't expect a very tight distribution around this average.

Either way the answer is to use values from observation. As slarty pointed out, here is a table of average values based on spectral class that you can use as a first approximation: The distribution of angular momentum among main sequence stars

Based on that, a good approximate value may be 50 km/s, about ~25 solar velocities.