Stefan's Law gives the relationship between the flux of energy through a surface and the temperature of the surface. The energy goes as the fourth power of Temperature. This is a very strong relationship between energy and temperature. A very small increase in temperature results in a very large increase in power.
Stefan's law is used to derive the relationship between luminosity and temperature. Luminosity can be thought of as the energy output of a star. Notice that it does not depend on the distance to the star. The apparent brightness is how bright the star looks to us, and it does depend on how far away the star is. The luminosity is intrinsic to the star itself.
Image credit: khadley.com
When there is some relative velocity between the source of light and an observer, it affects the frequency of the light seen by the observer. This is known as the Doppler effect. If the light source is moving toward the observer, in a sense, it is moving into its own signal, and the observed frequency increases. If the source is moving away from the observer, the opposite is true, and the observed frequency decreases. Similar increase and decrease of the observed frequency occurs when the observer moves toward/away from the light source.
Please visit the astronomy interactive site here for animations illustrating the Doppler effect.
The Doppler effect can be used to estimate the velocity of an astrophysical object. If the absorption spectrum is known for an object at rest to an observer, the apparent shift of the frequencies can be used to establish the relative speed between the object and observer. These images show absorption spectra for galaxies moving away from our Milky Way galaxy. We will learn more about absorption spectra in the next topic, spectroscopy.
You are probably more familiar with the Doppler effect as it applies to sound waves. When an object producing a sound is coming toward you, the pitch of the sound will be higher, which corresponds to a higher frequency of the sound wave. Similarly, when an object producing a sound is moving away, the observed frequency is lower.
The same is true whether the source is moving toward/away from the observer, or the observer is moving toward/away from the source. The relative motion is what matters here.
Image by Kathy Hadley
The apparent brightness can be measured from Earth. If the luminosity of a star or a galaxy is known or can be estimated, it can be compared to the apparent brightness to give an estimate to the distance of that object. Certain objects are very useful in this respect, if they are very bright and have known luminosities. These objects are known as standard candles. For example, if you have several lamps that have 100 Watt light bulbs, and place them at various distances away, the fact that you know they are shining at 100 Watts allows you to figure out how far away they are. Type Ia supernovas are often used as standard candles, since we believe we know quite a bit about their intrinsic brightness, and because they are so bright that they can be seen very far away.