The best current technology can measure radial velocities of about \(1 \mathrm{m} / \mathrm{s}\). Suppose you are observing a spectral line with a wavelength of 575 nanometers (nm). How large a shift in wavelength would a radial velocity of \(1 \mathrm{m} / \mathrm{s}\) produce?

Radial velocity is a term used in astrophysics and other fields to describe the speed at which an object moves towards or away from an observer. It is an important concept when studying the motion of stars, planets, and other celestial objects.
Radial velocity can be measured using the Doppler effect, which deals with changes in the frequency or wavelength of waves, such as sound or light, due to movement.
When an object moves closer to us, its light wavelengths shorten, causing what is known as a 'blue shift'. Conversely, when an object moves away from us, its light wavelengths lengthen, resulting in a 'red shift'.
Understanding radial velocity is crucial because it allows scientists to determine whether stars or galaxies are approaching or receding from us, thereby unveiling vital dynamics of celestial mechanics.

The speed of light in a vacuum is one of the most fundamental constants in nature, symbolized by the letter 'c'. Its approximate value is \( 3 \times 10^8 \) meters per second (m/s). This constant comes into play in a variety of physical formulas, including those related to the Doppler effect.
Since the speed of light is so fast, the changes in wavelength due to ordinary velocities (like the radial velocities of stars) are usually quite small. However, this very high speed makes this change detectable when measuring large distances typical in astrophysics.
For example, when calculating the Doppler shift, we often divide the velocity of the object by the speed of light to find out how much the wavelength is shifted.
Einstein's theory of relativity also heavily relies on the speed of light as it states that nothing can travel faster than light.

When light is emitted from a moving source, it experiences a shift in its wavelength. This effect is quantified by the Doppler shift formula in terms of how the wavelength changes. In our example, we started with a wavelength of 575 nanometers (nm).
Using the Doppler shift formula: \[ \frac{\triangle \text{wavelength}}{\text{wavelength}} = \frac{\text{velocity}}{c} \], we could calculate the change in wavelength. Plugging in our known values of radial velocity (1 m/s) and speed of light (\(3 \times 10^8 \) m/s), the wavelength shift can be determined.
After performing the calculations, we found a shift of approximately 1.92 x 10^-3 nm. While this might seem small, even minor shifts can provide extensive information about celestial bodies. This precision highlights the importance of nanometer (nm) measurements given their capability to capture minute variations in wavelength.