High School Explaining the Color Variations of Main Sequence Stars

[jordankonisky]

If all main sequence stars are engaged only in hydrogen fusion, why don't they all exhibit the same color?

 

[Vanadium 50]

Why do you think they should?

 

[jordankonisky]

Hmm, I thought that I was asking the question. Seriously though, since all main sequence stars generate thermal energy via the same hydrogen fusion reactions and it is the thermal energy that determines a star's surface temperature, why should a larger star display a different color (as dictated by the Wien relationship) than a smaller star? I must be missing something basic here.

 

[Bandersnatch]

More massive stars fuse hydrogen more rapidly, hence have higher temperatures, hence have the corresponding colour.

 

[Chronos]

The visual color of a star depends on it surface temperature which also depends on its mass. While composition appears to affect the apparent color displayed by burning objects on Earth this is because these objects do not behave as perfect black bodies - a term coined by the physicist kirchhoff in the 19th century to describe a body in thermal equilibrium that does not reflect or otherwise re-emit any light that falls upon it. While, strictly speaking, no such thing as a perfect blackbody actually exists in nature, the difference is negligibly small for virtually all stars as well as most other souces of thermal radiation.

 

[Vanadium 50]

So write doen the Wien relationship, solve for T,and look at the right hand side. Which terms are the same for all stars and which ones are different?

 

[Ken G]

The question can be answered fairy simply. Main sequence stars are fusing hydrogen in their core via a mechanism that closely regulates the core temperature, so to within about a factor of 2, they all have the same core temperature-- and that's your question. But this does not require that they should have the same surface temperature. The logic is, the surface temperature is set by the luminosity and the radius of the star, via the formula L = kR²T_s⁴, for k a constant, which solves for the surface temperature T_s = (L/kR²)^1/4. So this shows that you need to know R and L. The L is generally set by radiative diffusion in the interior of the star, which depends on mass essentially because the mass is the "stuff" the light has to diffuse through, and this leads to a relation like L is proportional to M raised to the power 3.5 or so, depending on some opacity details. The point is, M determines L. So now you only need R, and this depends on the history of contraction required to get the core to fusion temperature. That can be determined by using the fact that the average energy per particle necessary for fusion must be about equal to the potential energy per particle, set by M/R. So that sets R-- it is the R needed, given M, to get fusion in the core, so roughly R is proportional to M. Put it all together, and you find T_s is proportional to M to a power of about 0.4, roughly. That says a star 10 times more massive, or 1/10 as massive, than the Sun should have a surface T that is about 3 times higher, or 1/3 as high. That's about right.