The answer for your question is true not false
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Answer:
Two stars (a and b) can have the same luminosity, but different surface area and temperature if the following condition is met:
(T_a^4)(R_a^2) = (T_b^4)(R_b^2)
Explanation:
The luminosity of a star is the total energy that produces in one second. It depends on the size of the star and its surface temperature.
L = σ(T^4)(4πR^2)
L is the luminosity f the star, T is the temperature of the surface of the star and R is its radius.
Two stars can have the same luminosity if the relation between the radius and the surface temperature is maintained.
To see this lets suposed you have 2 stars, a and b, and the luminosities of each one of them:
L_a = σ(T_a^4)(4πR_a^2)
L_b = σ(T_b^4)(4πR_b^2)
you can assume that L_a and L_b are equal:
σ(T_a^4)(4πR_a^2) = σ(T_b^4)(4πR_b^2)
Now, you can cancel the constants:
(T_a^4)(R_a^2) = (T_b^4)(R_b^2)
as long as this relation between a and b is true, then the luminosity can be the same.
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