How and why does the Sun vary and what are the consequences?

The Sun varies in every way we can observe it.  The Sun give off light in the infrared, visible, ultraviolet, and at x-ray energies, and it gives off magnetic field, bulk plasma (the solar wind) and energetic particles moving up to nearly the speed of light, and all of these emissions vary.  These variations occur on timescales from fractions of a second to hundreds of years and longer.

Most of these variations are related to the solar magnetic field, which is caused by the moving plasma inside the rotating Sun, which make a dynamo.

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