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4 years ago in Astrophysics , Stellar Evolution By Rajeev

How can we determine the relative sizes of two stars if they have the same spectral type and apparent magnitude but are at different distances?

This is a classic problem from my stellar astrophysics textbook, but I'm getting tangled in the logic chain. I understand that same spectral type implies same temperature, and same apparent magnitude means same flux at Earth. But connecting these facts to deduce which star is larger using the Stefan-Boltzmann law and distance is where I need a step-by-step conceptual walkthrough.

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By Rinku Answered 3 years ago

Let's walk through the physics logically. Same spectral type means identical surface temperature (T). Same apparent magnitude (m) means identical observed flux (f). Flux is Luminosity (L) divided by 4πd². Since the farther star has a larger d, to yield the same f, its L must be much larger. How can it have a larger L at the same T? The Stefan-Boltzmann law, L = 4πR²σT?, gives the answer: Luminosity depends on R² and T?. With T fixed, a larger L requires a larger radius R. Therefore, the more distant star, which appears equally bright, must be the physically larger of the two.

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How can we determine the relative sizes of two stars if they have the same spectral type and apparent magnitude but are at different distances?

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