Relationship of temperature and radius.

[darkar]

Here's the question,

Initially, a main sequence star has a surface temperature of 25000 K and a radius 10 times that of the sun. Its temperature drop to 5000 K as it becomes a supergiant. What is its radius in term of solar radii?
a) 25
b) 50
c) 100
d) 250
e) 500

Well, I tried to do it using luminosity , L=4πR²σT⁴. Since the luminosity is not given, i use the Morgan-Keenan spectral classification to get the luminosity of the star at each phase, that's is at temperature of 25000 K has luminosity of 20,000 and for 5000 K has luminosity of 0.4. I got my final answer as 1.18 solar radius, which mean my answer is wrong. How should i approach the problem? and is that the spectral classification is only for main star sequence? and my friend use the relationship of temperature^4 is inversely proportional to 1/(radius)^2 and he got the answer for 250. BUt doesn't that mean that the luminosity is the same, is this true?

Thanks!

 

[Astronuc]

If the total energy output is considered the same, then one can establish an equivalence, much like your friend did.

r₁²*T₁⁴ = r₂²*T₂⁴

or r₁*T₁² = r₂*T₂²

or r₂ = r₁*(T₁/T₂)²

So what does this say about the luminosities?

 

[darkar]

I see... so the luminosities in that two phases are the same.

Thanks!