Radius, Luminosity, Temperature: A Key Relationship

We don’t have to give up on measuring the sizes of stars, however. We just have to be more clever. What astronomers do is determine the temperature and mass of a star, which can be done using a star’s color and spectrum. Then, using numerical models of how stars hold together, they derive the quantity that they are interested in (radius, for example). It is akin to looking out over a parking lot and seeing a Cadillac. Now, you may not know its size, but you know (consulting a chart) that this model of Cadillac is 18.5 feet long. You can see clearly that it is indeed this particular model of Cadillac, so you know its length, even though you didn’t actually measure it with a ruler.
Stefan’s law states that a star’s luminosity (its wattage, or the rate at which it emits energy into space) is proportional to the fourth power to the star’s surface temperature. This relationship can be extended further. A star’s luminosity is not only related to its temperature, but to its surface area. Heat the head of a pin to 400 degrees F and a large metal plate to the same temperature. Which will radiate more heat? Obviously, the object with the larger surface area. Given the same surface temperature, a larger body will always radiate more energy than a smaller one.
This relationship can be expressed in this way: A star’s luminosity is proportional to the square of its radius (that’s the surface area term) times its surface temperature to the fourth power (luminosity ×radius2 ×temperature4). Thus, if we know a star’s luminosity and temperature (which can be measured by available astronomical instruments), we can calculate its radius. How do we measure a star’s luminosity and temperature? Let’s see.