it takes many years to see the change of arms of a galaxy. so how was the rotation of arms measured?
Spectroscopy. You check the red shift of stars in the various arms of a galaxy.
but it is said that stars in the arms are moving at fast rate. the arms are nothing but collection of stars. so, how can arms have a different speed and if so, how can it be measured?
Define fast. Compared to what? The Sun orbits the galaxy at about 251 km/s.
While each individual star will have it's own specific velocity at any point in time, in general most stars will have roughly the same velocity as they move around the galaxy. This is measurable through spectroscopy like Chronos said.
They are indeed going faster than general relativity predicts, hence the assumption that there is hidden (dark) matter.
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