# Use the Doppler Effect to Measure the Astronomical Unit

Historically, measuring the Astronomical Unit – the distance to the Sun – was an exceedingly difficult task. A method that we will explore here involves the Doppler effect.

The method is to measure the Earth’s orbital velocity about the Sun V_{orbit.} This we will do by measuring the wavelength of light from a distant star. When we do so, we will find that the wavelength has been shifted: we interpret this shift as being due to the fact that we are moving, carried along by the Earth’s motion. Using the Doppler effect formula, we can infer the magnitude of our velocity from the measured shift. Once we have done so–

- Explain why the distance to the Sun R is then given by R = (V
_{orbit})(one year)/2π.

Before we perform our measurement, however, we need to recognize that what the Doppler effect tells us is the relative velocity between us and the star – and both we and the star are moving! How can we deal with the fact that the star we observe is moving – with an unknown velocity?

Here’s an analogy: let the Earth be a bus driving down the road, and let the distant star be a car which is also driving down the road. You are in the bus. Suppose they are heading towards each other:

The relative velocity between you and the car (and relative velocity is what the Doppler effect measures) is

V _{relative} = V _{bus} + V _{car}

What you measure (V _{relative}) is composed of two velocities: the bus’s and the car’s. But what you want to measure is only the first. How can we get rid of the second?

Here’s a lucky break: after a while, that bus is going to turn around and start heading in the opposite direction:

And now the relative velocity between you and the car will be

V _{relative later} = V _{bus} – V _{car}

Suppose we add these two formulas. We get

V _{relative} + V _{relative later} = 2 V_{ bus}

So V_{ bus}, which is what we want to measure, is just half the sum of these two relative velocities!

Before we move to the astronomical problem, let us get some practice with our Earthly analogy. Suppose you are in a bus which is traveling at 65 miles / hour, and you are observing a car which is traveling at 40 miles / hour. In the above configurations, what is your velocity relative to the car

- Initially?
- After the bus has reversed direction?

Now suppose the driver of the car is honking her horn, and it is emitting sound with a wavelength of one meter. What wavelength do you in the bus hear from this horn

- Initially?
- After the bus has reversed direction?

Now let’s move on to the astronomical situation. What makes “the bus” reverse direction? The fact that after 6 months the Earth has reversed its direction as it orbits about the Sun!

Here are the observations. We observe a spectral line from the star which ought to be at a wavelength 6562 Angstroms. We find

– When we are heading towards it, the observed wavelength is 6561.13 Angstroms.

– When we are heading away from it, to observed wavelength is 6562.44 Angstroms.

- How fast is the Earth moving in its orbit?
- How fast is the star moving?
- What is the distance to the Sun?

*ƒGeorge Greenstein*

*Emeritus Professor of Astronomy*

*Amherst College*

*gsgreenstein@amherst.edu*