Radial Velocity

Another way to measure properties of stars is to calculate something called the radial velocity of the star. This is just the velocity of the star as it is moving either towards us or away from us. Remember from the Doppler Effect, that as the star is moving towards us, the star will appear bluer, i.e. the spectrum moves towards the blue end of the visible light spectrum. As the star moves away from us, the spectrum moves towards the red end of the visible light spectrum. How can we use this to find the radial velocity?


As discussed in the post about spectroscopy, each element has a unique spectral pattern based on the energy levels of electrons orbiting the nucleus. By comparing a star's spectrum to the standard spectra of the elements, one can match up lines found in the stellar spectrum. By measuring the wavelength of those lines in the spectrum, and comparing them to the standard spectrum of the element, just using the Doppler equation can give you the radial velocity of the star.


This obviously only works if the star is moving away from or towards you. If the star is moving across you line of sight, astrometry is used instead and what you are measuring is called proper motion.

Doppler Effect

Have you ever sat by a road or a railroad and listened to the cars or trains coming by? As the car or train approaches, the whine or pitch of the sound rises and as it recedes away from you, the pitch goes down. This is because of something called the Doppler Effect.

The Doppler Effect works because as a wave source approaches an observer, the wavelength of the wave decreases, which in turn increases the frequency. As the source goes away from the observer, the wavelength increases, decreasing the frequency.

In the above picture, the source in this case is a star. The wavefronts are the circles surrounding the star. To the left, the observer sees the star approaching; therefore the wavefronts are closer together, meaning a shorter wavelength and higher frequency. The observer on the right sees the star moving away, and they see the wavefronts getting farther and farther apart.

For all waves, whether sound waves or light waves, the speed of the wave is related to the frequency and the wavelength of the wave. We relate them in this way:

v=λf

where:

• v is the velocity (or speed) of the wave
• λ (Greek letter lambda) is the wavelength of the wave
• f is the frequency of the wave (in units of 1/seconds). Sometimes, frequency is shown as ν (the Greek letter nu)

Since we know the wave is traveling the same speed, if the wavelength is increasing (or decreasing), the frequency in turn must decrease (or increase).

We also know that the change in the wavelength or frequency is also related to how fast the source is moving.

where:

• f' and λ' are the frequency and wavelength measured at the observer
• f and λ are the frequency and wavelength of the source
• v is the speed of the wave
• v_O is the velocity of the observer
• v_S is the velocity of the source

The velocities of the observer and the source can be positive or negative (or zero). A positive velocity means that the source (or observer or both) are moving towards each other. A negative velocity means that the source (or observer or both) are moving away from each other. If the observer is stationary (v_O is zero), you get:

 

For sources moving towards the observer, you can see the wavelength decreases and the frequency increases. When the source is moving away from the observer, the opposite happens. Change the source velocity to zero for the top two equations, and you'll see something similar.

Now, what happens if the wave is visible light? The speed of the wave becomes c and something strange happens: the wavelengths move to opposite sides of the visible spectrum.

As the wavelength of the light decreases (source moving towards you), the wave changes colors and the wave shifts towards the blue end of the spectrum. This is called blue-shift. When the wavelength increases, the wave shifts toward the red end of the spectrum. This is called red-shift.