The Doppler Effect – formulas

Lesson Objective: How to calculate the Doppler Effect  for moving sources and detectors. How to solve the Doppler equation for the speed of a galaxy based on its  Red or Blue shift.

Warm Up: {FM} Why is the Doppler Shift formula for light different than the Doppler Shift formula for sound?

What you need to know:

Review:
For Sound: fd = fs (v-vd) / (v-vs)
Derive expressions for fd where vd=0 and where vbs=0 (see pg 415).

New material:
For light: f observer = f (1 +/- v/c) for v<<c
And the “red shift” DL =Lobserved – L = +/-(v/c) L
Where D means delta and L means Lambda.

Do Practice Problems pg 456 # 16-19, you will be called on to present your work at the board.

Relativistic Doppler Effect: Now the problem is that the speed of light is a  constant and we are stuck never being able to exceed the speed of light.

Q: What is the speed of light in m/s? in miles/sec?

One of Einstein’s colleagues, Lorenz, came up with a transformation that allowed one to create  mapping between points in space-time. Using the Lorentz transformation, they derived the relativistic Doppler Effect:

nu’ = nu [  (1-v/c) / (1+v/c) ]^1/2

This is the Relativistic Doppler Effect which applies when v gets close to c.

This post from Wikipedia has the derivation of this equation – but it’s really complicated.  Diagrams 1 and 2 will give you the general idea.
The Relativistic Doppler Effect

Now calculate the non-relativistic and relativistic Doppler “Red Shifts” for a star moving away from you at;
v=0.9c
v=0.99c
v=0.999c
Assume the star emits 500nm light.

Include your results in a table and in the third column, calculate the % difference between the non-relativistic and the relativistic Doppler red shifts.

Exit Assessment: What would 500nm light look like of you were traveling faster than the speed of light?