Calculating the radius of a star's core

[MookVanguard]

Hi, I'm trying to make a simple model of stellar evolution for a game project of mine, but I'm not sure how to predict the radius of a star's core without reference to its surface radius. I am trying to derive the surface radius from the amount of energy being released by the core. The goal is to find the properties of the star given only its total mass, the time elapsed from birth, and the initial distribution of materials in the star. I hope that's not too much to ask.

My guess is that I would have to sum the radiation pressure at the core surface and the thermal pressure from the gas, but I'm not quite sure how to go about doing this, even after reading online about it. How can I get the density with respect to radius?

Would anyone care to help?

 

[ToBePhysicist]

Greetings,
Virial Theorem?
Check this thread
Probably useful
http://www.iiap.res.in/kodsch/Lectures/SBasu/kodaikanal.pdf.

 

[MookVanguard]

Thanks, your third link helped a lot. I'll read through it and see what I can come up with.

I'm not looking to apply the virial theorem yet, as I haven't yet gotten to the stage of adding gravitational contraction into the mix.

 

[ToBePhysicist]

Thanks, your third link helped a lot. I'll read through it and see what I can come up with.

I'm not looking to apply the virial theorem yet, as I haven't yet gotten to the stage of adding gravitational contraction into the mix.

Good luck.

 

[phyzguy]

This is a well understood problem, but it's not simple. I don't know what level you want it to work at. If you want to get into the details, you need to solve the differential equations of stellar structure. I strongly recommend the classic text by Kippenhahn and Weigert "Stellar Structure and Evolution" if you want to understand what equations you need to solve and how to do it. If you want an already written program that will just give you the answers, you can download a copy of MESA at this site. It will allow you to simulate a broad range of stellar types (masses, compositions, ...), with some excellent graphing packages to let you look at the structure.

 

[MookVanguard]

This is a well understood problem, but it's not simple. I don't know what level you want it to work at. If you want to get into the details, you need to solve the differential equations of stellar structure. I strongly recommend the classic text by Kippenhahn and Weigert "Stellar Structure and Evolution" if you want to understand what equations you need to solve and how to do it. If you want an already written program that will just give you the answers, you can download a copy of MESA at this site. It will allow you to simulate a broad range of stellar types (masses, compositions, ...), with some excellent graphing packages to let you look at the structure.

That's a shame. I'm relatively new to the math involved so a lot of what I read is confusing to me. But if it requires a 1GB tool to simulate... then simulating stellar structure is probably not what I'm going to do. This is for a game that needs to generate stars on demand at a random point in their evolution, and at current speeds it can generate hundreds of stars per second. But right now I'm just railroading the evolutionary tracks, i.e. I'm converting time passed to % of lifetime, then interpolating the stat's position along an evolutionary track in the HR diagram.

I'd like to have something that's marginally more accurate, although it doesn't need to be perfect, just an approximation that physicists won't roll their eyes over.

Thanks for the tool though! It'll be great for testing what I come up with in the future.

 

[rootone]

For the purpose of making your simulation believable I'd say that your idea of emulating statistically what is seen in the HR diagram is a good bet.
You might also want to incorporate the statistics for the relative abundance of different star types - for example there are many more small red dwarf type stars than there are larger ones.

 

[MookVanguard]

For the purpose of making your simulation believable I'd say that your idea of emulating statistically what is seen in the HR diagram is a good bet.
You might also want to incorporate the statistics for the relative abundance of different star types - for example there are many more small red dwarf type stars than there are larger ones.

I'm curious, is there a decent statistical approximation for that? Some kind of distribution curve with a simple expression?

 

[rootone]

This link is probably useful for you.
http://www.atlasoftheuniverse.com/startype.html

(The diagram at the top is just a summary, scroll down a bit for the detailed stats)

 

[phyzguy]

There is a function called the Initial Mass Function (IMF) which quantifies the number of stars of a given mass for stars as they are born. This Wikipedia link gives some functional forms. Probably for your purposes the Chabrier IMF, which captures the roll-off at both large and small masses, would be adequate. Then you can just evolve them along the evolutionary tracks, which it sounds like from your earlier posts that you know how to do.

 

[MookVanguard]

There is a function called the Initial Mass Function (IMF) which quantifies the number of stars of a given mass for stars as they are born. This Wikipedia link gives some functional forms. Probably for your purposes the Chabrier IMF, which captures the roll-off at both large and small masses, would be adequate. Then you can just evolve them along the evolutionary tracks, which it sounds like from your earlier posts that you know how to do.

I somewhat know how to do it, although I've noticed stars with different metallicities and properties have different behaviors. That's why I wanted to make a simple model, so I can input metallicity as a factor.

Thanks for the link!