Relativistic Redshift. Access the Active Integrated Media Module "Relativistic Redshift" in Chapter \(2.5\) of the Universe Web site or eBook. Use this to calculate the redshift and recessional velocity of quasar in whose spectrum the \(\mathrm{H}_{\mathrm{\alpha}}\) emission line of hydrogen (unshifted wavelength \(656 \mathrm{~nm}\) ) appears at a wavelength of (a) \(937 \mathrm{~nm}\) and (b) \(5000 \mathrm{~nm}\).

When we gaze into the cosmos, the motion of celestial objects relative to us can cause a phenomenon known as the Doppler effect, where emitted light appears to change in wavelength. This effect is particularly useful in astronomy for understanding the motion of galaxies and distant quasars.

Recessional velocity is the speed at which an object is moving away from an observer, an essential concept in cosmology, particularly when discussing the expansion of the universe. In the context of our universe, the farther away a galaxy is, the faster it appears to be moving away from us, a relationship described by Hubble's Law.

In practical terms, to calculate the recessional velocity of a distant quasar, as in the exercise, we can see its reflected light—the H-alpha emission line—shift toward the red end of the spectrum. This 'redshift' helps us measure the galaxy or quasar's speed away from Earth. However, it's not just the speed of the object itself that contributes to this measurement, but also the continual expansion of space between us and the object.

Mathematically, the recessional velocity (\(v\)) is closely tied to the redshift (\(z\)) of the light we observe, and scientists often use this relationship to explore and quantify the universe's grand-scale movements.

The H-alpha emission line is a specific red wavelength of light that is characteristically emitted by hydrogen atoms, the most abundant element in the universe. When an electron in a hydrogen atom falls from the third to the second lowest energy level, it releases energy in the form of light with a wavelength of exactly 656 nanometers (nm), which we perceive as red light. This emission line is a crucial tool for astrophysicists.

In the practice of spectroscopy—splitting light into its component colors—it's the H-alpha emission that often helps astronomers identify the presence of hydrogen in stars and galaxies. However, the H-alpha line's wavelength can change based on the movement of the emitting object due to relativistic effects, such as when galaxies are receding from us at high velocities.

When the wavelength of H-alpha is seen to be longer than 656 nm, it indicates a redshift. In the context of the textbook exercise, observing the H-alpha line shifted from 656 nm to 937 nm or even to 5000 nm suggests a significant redshift, which implies a substantial recessional velocity for the quasar in question.

The concept of redshift is essential for studying the universe's expansion and the motion of astronomical objects. Redshift occurs when the wavelength of emitted light is stretched, making the light appear more red than it originally was, hence the term 'redshift'. This stretching happens because the source of the light is moving away from us.

To calculate redshift, we use the formula (\(z = (\lambda_{observed} - \lambda_{emission}) / \lambda_{emission}\)), where (\(\lambda_{observed}\)) is the wavelength of light we detect, and (\(\lambda_{emission}\)) is the original wavelength emitted by the source. In layman's terms, it's a comparison of the difference between what we observe and what was originally emitted, relative to the original emitted wavelength.

For example, when we say that the H-alpha emission line of hydrogen, which should ideally be at 656 nm, is observed at 937 nm, we plug these numbers into the redshift equation. This calculated redshift can then be translated to a recessional velocity using the speed of light and helps us understand the rate at which the object is moving away from us—an invaluable piece of the astronomical puzzle.