Red light emitted by hydrogen atoms at rest in the laboratory has wavelength \(656 \mathrm{nm}\). Light emitted in the same process on a distant galaxy is received at Earth with wavelength \(708 \mathrm{nm}\) Describe the galaxy's motion relative to Earth.

Imagine looking at a celestial body like a distant galaxy and finding that the light it emits seems slightly different than what we expect. Redshift occurs when the light from an object in space is stretched out, making it appear more red than it usually would. In the context of our exercise, the light from a hydrogen atom that is normally at a wavelength of 656 nm was detected at 708 nm.

This stretching of light indicates that the galaxy is moving away from us. The higher wavelength or 'redder' appearance serves as evidence of the galaxy's recession. This phenomenon, as observed in our universe, was key in determining that the universe is expanding, with galaxies steadily moving away from each other.

The wavelength of light is the distance between consecutive points of the same phase on a wave, such as two peaks or troughs. It determines the color of the light as perceived by the human eye or detected by instruments. Typically, light wavelengths are measured in nanometers (nm), with visible light spanning from about 380 nm (violet) to 750 nm (red).

The significance of wavelength is evident in our exercise, where the observed light from a distant galaxy has a longer wavelength when observed from Earth compared to its original emission. This change suggests that the light has traveled a great distance and has been affected by the motion of the galaxy relative to Earth.

The Doppler effect is a phenomenon observed when there is relative motion between a source of waves and an observer. It affects all kinds of waves, including sound and light. When the source moves towards the observer, the waves are compressed, leading to a higher frequency or a 'blueshift' in the case of light. Conversely, when the source moves away, the waves are stretched, resulting in a lower frequency or a 'redshift'.

The formula \( \frac{\Delta \lambda}{\lambda_{0}} = \frac{v}{c} \) captures this effect for light waves, where \( \Delta \lambda \) is the change in wavelength, \( \lambda_{0} \) is the original wavelength, \( v \) is the velocity of the source, and \( c \) is the speed of light. The exercise applies this formula to find the velocity of the distant galaxy moving away from Earth.

Astronomical spectroscopy is a powerful tool used by scientists to analyze the light from stars, galaxies, and other celestial objects. It involves spreading out the light into its various components or wavelengths, creating a spectrum. By studying these spectra, astronomers can determine many properties of celestial objects, such as their chemical composition, temperature, density, mass, distance, luminosity, and motion.

In our exercise, spectroscopy enables us to compare the wavelengths of light emitted by hydrogen in a laboratory setting to the wavelengths of the same light coming from a distant galaxy. The differences in these wavelengths lead to insights about the galaxy's motion, and it's this technique that underpins the discovery of redshift and the contributions to our understanding of the expanding universe.