# How to calculate the radius of a main sequence star based on mass?

What would I need in addition to the mass to figure out the radius of a main sequence star?

• You really should show research effort in your question. – Shing Sep 12 '15 at 2:09
• Research effort... Okay, I'm not an astrophysicist. So despite all the googling I've done, I actually have found a grand total of nothing, or at least nothing I could understand. I've learned that this is definitely not going to be something that has a simple equation, so I think I'll just be trying to fake it with data from other main sequence stars. – senox13 Sep 12 '15 at 20:40
• Diagram and table that answer your question. – user10851 Sep 13 '15 at 17:22
• @ChrisWhite That approximates an answer to the title, but not the question. Since the (interesting +1) question was what, in addition to mass, is needed to figure out the radius of a main sequence star. Unless you contend that nothing is? – Rob Jeffries Sep 13 '15 at 18:19
• @Shing (And those who upvoted your comment) I think you too have misread the question or are being incredibly harsh. Simple google searches and the like will yield rough and ready approximations to a main-sequence mass-radius relation - as pointed to by Chris - but these are not what the question asks for. Even undergraduate texts would not discuss this question and the topic of the rotation-dependence I discuss below is cutting-edge research. – Rob Jeffries Sep 13 '15 at 18:28

Well, you would need an accurate stellar evolution model and you would need the age, chemical composition and probably the rotation rate too.

The mass-radius relationship is an area of intense research. Masses and radii are only determined for stars in eclipsing binary systems. These could define an empirical mass-radius relationship, but stellar evolution models tell us that the radius also depends on age. eg pre main sequence stars are larger, and even whilst still on the main sequence, stars gradually get larger by tens of percent over the course of their main sequence lifetime. Practically, this means that knowing the age would be important for stars of mass of about 80% of the Sun or larger. Lower mass main sequence stars would not have departed significantly from their initial main sequence radii even if they were as old as the Galaxy.

The radius will also depend on the chemical composition - both the overall metallicity and the fractional helium abundance are important, through their effects on atmospheric opacities and mean atomic masses. This is an effect at the level of 10-20%.

Lastly, it is becoming apparent that rotation probably plays an important role. In stars less massive than the Sun it appears that rotation-induced magnetic activity inflates stars (for poorly understood reasons). In higher mass stars rotation induces interior mixing that alters internal composition gradients and changes the evolution (as well as centrifugal distortion). Again, these are phenomena that affect radii at the level of 10% or so.

If you want something rough and ready, which at least allows you to account for the effects of age and metallicity, you could interpolate the results tables of radius versus mass that are easily obtained from this website, that reports calculations from the models of Siess, Dufour & Forestini (2000).

• Damn, I was afraid of that. Okay, follow up question: what would be the most convincing way to fake it? This is for a game, so it really doesn't have to be that accurate, just believable. – senox13 Sep 12 '15 at 0:07
• @user2925591 I am available for a small consultation fee. It depends how accurate you want to be. Basically you have to decide on the factors above and then use the results (all published and available) of stellar evolution models to calculate a radius. – Rob Jeffries Sep 12 '15 at 8:06
• Are you serious? No one's going to pay you a "consulting fee" for data that's specifically designed to be inaccurate. And besides, it's a game I'm making for fun, not a triple A title. Thanks for the help you did give though. But not so much for trying to scam me. – senox13 Sep 12 '15 at 18:26
• @user2925591 No. I wasn't serious, as you might gather from the (currently) nearly 600 fee-free answers I have already provided on this site. As for the "scam" -I can't work out what you mean. You asked for very specific, detailed information to be used in what sounded like a commercial enterprise. I am a professional astrophysicist. There's no scam. Anyway - see my edit; gratis. – Rob Jeffries Sep 12 '15 at 18:45
• Okay, I think the misunderstanding here might be on me. I wasn't clear enough on what this was for initially. But telling someone that you'll help them find publicly available information, but only for a price, is about as close to a scam as you can get, and oddly enough, I didn't stalk your account to find out how good of answers you give. But as I said, I really should have been clearer on what I was looking for. Just a simple, made up equation to simulate a concept that the best of astrophysicists don't fully understand. Nothing extremely complicated. – senox13 Sep 12 '15 at 20:33