|MadSci Network: Astronomy|
Sunspots do emit light, but they emit much less than their surroundings. In contrast to the bright photosphere, then, they appear relatively dark on photographs.
On the Sun, the photosphere has a temperature of about 5800 degrees Kelvin. This is measured through spectroscopy.
It is possible to use a solar telescope to get a spectrum of the light coming out of the center of a sunspot. This is done by blocking out the light from the surrounding photosphere and pointing the telescope and spectrograph right at the sunspot. The temperatures in the middle of sunspots are about 4000 Kelvin.
As you wrote, material that is hot (4000 Kelvin) will certainly emit a lot of light.
But the light that is emitted goes like the 4th-power of the temperature. Thus each square meter in a sunspot will emit only about 1/4 as much light as the same area in the photosphere. That's why the spots look dark.
I actually do research on measuring rotation periods of distant stars using their sunspots. The stars are too far off to get an image of the surface, so they just look like points. But as these stars rotate, the spots come around to the front side of the star (as we see it) and the star gets a bit dimmer. Then as the star rotates, the spots go around to the back and the star brightens a tiny amount. If you watch these stars for a while, you can see a periodic variation in the brightness, which gives you the rotation period.
You can do this on the Sun, and find that the Sun rotates about every 25 days. Many young stars can rotate in just 8 hours.
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