# The Doppler Shift

If an object is moving toward us or away from us, its light will appear shifted in wavelength:
• Moving towards us: blueshift
• Moving away: redshift
(Think of the analogy of the train whistle.)

This doppler shift is given by

where c is the speed of light. If the motion is nonrelativistic, ie vr<<c, this boils down simply to

Things to notice:

• The velocity in question is line of sight radial velocity.
• As defined, if vr is positive the object is moving away.

## The Motions of Stars

Remember, Doppler shift only gives us a star's radial velocity. There is also a tangential component to the star's motion.

How do we determine this tangential component? Watch a star for a long time (years). Then measure how far it moves in the sky. The rate at which it moves is called its proper motion.  For example, Vega's proper motion is 0.345"/yr.

Then remember that proper motion (mu) is an angular motion across the sky. It is related to the true tangential velocity by

More things to note:
• Proper motion must be converted into radians/time for that equation to work.
• We can't solve for the tangential motion until we know the distance to the star.
• At a given velocity, distant stars have smaller proper motions.
Once we have a radial velocity, a proper motion, and a distance, we can solve for the true space motion of the star:
Most stars near the sun have space motions of 30 km/s or less.