How do astronomers measure the radius of a star?
In order to find the diameter of a star you really need three pieces of information about the star 1) distance 2) brightness 3) and color
Step 1: We can figure out the total power output of the star by knowing its brightness at Earth and its distance.
Step 2: Figure out the surface temperature. The universe really gave us scientists a lucky break because stars are fairly predictable objects. What I mean is that, for most stars once we know one property of the star we can generally figure out everything else we want to know about it. This is because stars behave like "black bodies". This is a physics term which describes how an object of certain temperature glows at different wavelength (thus influencing its color). Most ordinary matter resembles a black body (has nothing to do with being black, it's just terminology). You probably already have an intuitive understanding of this from your everyday experience. For example, when you heat up an object, at first it might start to glow red. And as it gets hotter it might start to glow yellow, blue, and eventually white. Because stars approximate black bodies very well, by knowing the color of the star we can figure out what its surface temperature is quite accurately.
Step 3: Another great thing about black bodies is that, for any specific amount of surface area we can predict exactly how much light it will radiate (hotter objects are brighter). And since we already know the temperature and the total brightness of the star we can figure out its surface area, and thus it's diameter :)
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