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.

Your Standard Solar Model

By combining theoretical modeling of the sun’s (unobservable) interior with observations of the energy that the sun produces, astronomers have come to an agreement on what is called a standard solar model, a mathematically-based picture of the structure of the sun. The model seeks to explain the observable properties of the sun and also describe properties of its unobservable interior. With the standard solar model, we can begin to describe some of the interior regions—regions hidden, beneath the photosphere, from direct observation. Below the photosphere is the convection zone, some 124,000 miles (200,000 km) thick. Below this is the radiation zone, 186,000 miles (300,000 km) thick, which surrounds a core with a radius of 124,000 miles (200,000 km).
The sun’s core is tremendously dense (150,000 kg/m3) and tremendously hot: some 15,000,000 K. We can’t stick a thermometer in the sun’s core, so how do we know it’s that hot? If we look at the energy emerging from the sun’s surface, we can work backward to the conditions that must prevail at the sun’s core. At this density and temperature, nuclear fusion is continuous, with particles always in violent motion. The sun’s core is a giant nuclear fusion reactor.
At the very high temperatures of the core, all matter is completely ionized—stripped of its negatively charged electrons. As a result, photons (packets of electromagnetic energy) move slowly out of the core into the next layer of the sun’s interior, the radiation zone.
Here the temperature is lower, and photons emitted from the core of the sun interact continuously with the charged particles located there, being absorbed and re-emitted. While the photons remain in the radiation zone, heating it and losing energy, some of their energy escapes into the convection zone, which in effect, boils like water on a stove so that hot gases rise to the photosphere and cool gases sink back into the convection zone. Convective cells become smaller and smaller, eventually becoming visible as granules at the solar surface. Thus, by convection, huge amounts of energy reach the surface of the sun. At the sun’s surface, a variety of processes give rise to the electromagnetic radiation that we detect from the earth. Atoms and molecules in the sun’s photosphere absorb some of the photons at particular wavelengths, giving rise to the sun’s absorption-line spectrum. Most of the radiation from a star that has the surface temperature of the sun is emitted in the visible part of the spectrum.