Temperature of blackbody (star)

AI Thread Summary
Two stars radiating the same total energy per second have a cooler star with a surface temperature T and a diameter three times that of the hotter star. The relationship between their temperatures is derived from the area and power equations, leading to the conclusion that the temperature of the hotter star is T2 = T1/sqrt(3). The area of the cooler star is nine times that of the hotter star due to its larger diameter. The discussion emphasizes the importance of correctly applying the Stefan-Boltzmann law, which states that power is proportional to area times temperature to the fourth power. The final temperature relationship and area ratio are confirmed through collaborative problem-solving.
pat666
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Homework Statement


b) Two stars, both of which behave like black bodies, radiate the same total energy per second. The cooler star has a surface temperature, T, and 3.0 times the diameter of the hotter star.
i) What is the temperature of the hotter star in terms of T?
ii) What is the ratio of the peak intensity wavelength of the hotter star to that of the cooler star?

Homework Equations


The Attempt at a Solution


Calling the cooler star 1 and the hotter 2
A1=4\pir12
A2=4\pi(3r1)2
P1=P2
P=\epsilon\sigmaAT4
so P\proptoAT and that I think means T \propto 1/A
Here is where I get stuck, not sure if anything I am doing is right (basically just playing to see what happens) and not sure where to go now?
 
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pat666 said:

The Attempt at a Solution


Calling the cooler star 1 and the hotter 2
A1=4\pir12
A2=4\pi(3r1)2
P1=P2
P=\epsilon\sigmaAT4
so P\proptoAT and that I think means T \propto 1/A
Here is where I get stuck, not sure if anything I am doing is right (basically just playing to see what happens) and not sure where to go now?
You're pretty much on track, except that your T has lost it's 4 exponent.
P α A T 4​
 
So A_2 is 12 times the size of A_1 and T is proportional to 1/A^4
T=12T_1?
 
pat666 said:
So A_2 is 12 times the size of A_1
No, it isn't. How did you calculate that? What is A2/A1, given the correct expressions for A1 and A2 that you wrote earlier?
and T is proportional to 1/A^4
No, you need to use the fact that A·T4 is a constant to get the proportionality relation.
T=12T_1?
No.
 
A2/A1=9
P is proportional to AT^4
still not sure what to do here?
 
pat666 said:
A2/A1=9
Yes.
P is proportional to AT^4
still not sure what to do here?
Yes, and since P is the same for both stars,
A1 T14 = A2 T24​
Solve for T2
 
I get T2=1/sqrt(3)T1? since the indices on T cancel out.
 
Looks good, that's what I get too.

(I assume you mean T1/sqrt(3), and not 1/[T1*sqrt(3)] )
 
yeah I meant the 1st one. Thanks for helping me to the answer.
 

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