Final answer:
Astronomers can tell if a star is approaching or receding from Earth by observing the Doppler effect on the star's spectrum. The shift in absorption lines towards the red or blue end indicates the star's motion. They can also measure rotation speeds via line broadening in the spectrum.
Step-by-step explanation:
Astronomers can determine whether a star is approaching or receding from Earth by examining the Doppler effect as it applies to the star's light spectrum. As a star moves in relation to Earth, the wavelengths of light and the position of absorption lines in its spectrum will shift. If the star is moving away, the lines shift toward the red end of the spectrum (a redshift), whereas if the star is moving closer, they shift toward the blue end (a blueshift). This shift is due to the wavelengths of light being stretched apart or squeezed closer together, depending on the direction of the star's motion.
Additionally, the Doppler effect can be used to measure other dynamic aspects of stars, like their rotation speeds. Line broadening occurs because each side of the rotating star contributes a different Doppler shift, which results in a broadening of spectral lines. Through these observations, astronomers can understand various kinematic properties of the stars.