You observe a spectral line of hydrogen at a wavelength of \(502.3 \mathrm{nm}\) in a distant galaxy. The rest wavelength of this line is \(486.1 \mathrm{nm} .\) What is the radial velocity of this galaxy?

When studying celestial objects like stars and galaxies, one important motion to measure is their radial velocity. Radial velocity refers to the speed at which an object moves toward or away from an observer. In astronomy, this is often done by observing the Doppler Effect in the object's light. If a star or galaxy is moving away, its light is redshifted, meaning the light waves stretch out and appear more red. If moving closer, it's blueshifted, appearing more blue. Radial velocity is calculated using the difference between the observed wavelength of a spectral line and its actual, or rest, wavelength.

Spectral lines are unique elements' signatures in the spectra of stars and galaxies. They appear as distinct lines in the spectrum, corresponding to specific wavelengths of light absorbed or emitted by elements. These lines help identify the composition of celestial objects and are key to studying their movement. Each chemical element produces a set of characteristic spectral lines at specific wavelengths. By analyzing these lines, astronomers can determine various properties of stars, such as their temperature, composition, and movement. For instance, when a spectral line of hydrogen at rest is observed to be at a different wavelength, it implies movement relative to the observer.

The difference in the observed and rest wavelengths of a spectral line is known as the wavelength shift. This shift is crucial for calculating an object's radial velocity. When a star or galaxy moves, the wavelength of light it emits changes due to the Doppler Effect. If the object is moving away, the observed wavelength increases (redshift). If approaching, it decreases (blueshift). The amount of shift helps astronomers gauge the speed of the object. For example, if an observed hydrogen line is at 502.3 nm compared to its rest wavelength of 486.1 nm, the wavelength shift indicates that the object is moving away from us.

The speed of light in a vacuum, denoted as 'c,' is a fundamental constant in physics, valued at approximately 3 × 10^8 m/s. This constant plays a key role in many astronomical calculations, including determining radial velocity. When an object's spectral line is shifted, the speed of light helps relate this shift to the object's velocity. For instance, the formula for calculating radial velocity involves dividing the wavelength difference by the rest wavelength, then multiplying by the speed of light. This relationship allows us to quantify the velocity of a galaxy or star based on the observed spectral line shift.