Observations of the absorption and emission lines of the spectra of distant galaxies provide a way of estimating its astronomical distance. It has been found that for many stars and galaxies, these spectral lines do not occur at their 'correct' frequencies. The pattern of lines is correct, but they are displaced, usually towards the red (longer wavelength) end of the spectrum. That is, the spectrum shows a red-shift.
The explanation for the red-shift lies in the motion of the star or galaxy. If it is moving away from the observer at a significant speed, then all its radiation will be red-shifted. Say, for example, there are two stars, each emitting a pulse of four waves. Star A is stationary, with a wave pulse of length 4 lambda in space. Star B is moving away from the obsever. Wave 1 is emitted while it is close to the observer. By the time it emits wave 4, it has moved a distance further away. So the four wabes occupy a length greater than 4 lambda in space.
In this way, the red-shift of spectral lines can be used to determine the velocity of a moving astronomical object. If a star moves ar 10% the speed of light away from the observer, its spectral lines will show a 10% increase in their wavelengths.
As you might have noticed, the pitch of the siren of an ambulance changes as it approaches and passes you. The frequency increases as it approaches you and decreases as it recedes. This is an example of the Doppler effect, as is the red-shift of light from moving stars as discussed above. The Doppler effect, among other things, can be used to determine the orbital period of binary stars and the recessional velocity of a distant star or galaxy.
Instead of moving away like Distance Red Shifting, Gravitational Red Shifting is caused by slowing down light waves. According to General Relativity, Gravity affects both space and time. Around a black hole, extreme gravity causes time to dilated or slowed down to the outside observer. As the waves reach us more slowly, the light that we see changes color.
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