The Doppler Effect of Light (DP IB Physics: HL): Revision Note

A stationary source of light is found to have a spectral line of wavelength 438 nm.  The same line from a distant star that is moving away from us has a wavelength of 608 nm.

Calculate the speed at which the star is travelling away from Earth.

Answer:

Step 1: List the known quantities

• Unshifted wavelength, λ = 438 nm

• Shifted wavelength, λ₀ = 608 nm

• Change in wavelength, Δλ = (608 – 438) nm = 170 nm 

• Speed of light, c = 3.0 × 10⁸ m s^–1

Step 2: Write down the Doppler equation and rearrange for velocity v

$\frac{∆\lambda }{\lambda }=\frac{v}{c}$

$v=\frac{c∆\lambda }{\lambda }$

Step 3: Substitute values to calculate v

$v=\frac{(3.0 \times {10}^8)\times 170}{438}$ = 1.16 × 10⁸ m s^–1

The stars in a distant galaxy can be seen to orbit about a galactic centre. The galaxy can be observed 'edge-on' from the Earth.

Light emitted from a star on the left-hand side of the galaxy is measured to have a wavelength of 656.44 nm. The same spectral line from a star on the right-hand side is measured to have a wavelength of 656.12 nm.

The wavelength of the same spectral line measured on Earth is 656.28 nm.

(a) State and explain which side of the galaxy is moving towards the Earth.

(b) Calculate the rotational speed of the galaxy. 

Answer:

(a)

• The light from the right-hand side (656.12 nm) is observed to be at a shorter wavelength than the reference line (656.28 nm)

• Therefore, the right-hand side has been blue-shifted and must be moving towards the Earth

(b)

Step 1: List the known quantities

• Observed wavelength on LHS, ${\lambda }_{LHS}$ = 656.44 nm

• Observed wavelength on RHS, ${\lambda }_{RHS}$ = 656.12 nm

• Reference wavelength, λ = 656.28 nm

• Speed of light, c = 3.0 × 10⁸ m s⁻¹

Step 2: Calculate the average change in wavelength

$∆\lambda =\frac{{\lambda }_{LHS}-{\lambda }_{RHS}}{2}=\frac{656.44-656.12}{2}$

$∆\lambda$= 0.16 nm

Step 3: Write down the Doppler equation and rearrange for velocity v

$\frac{∆\lambda }{\lambda }=\frac{v}{c}$

$v=\frac{c∆\lambda }{\lambda }$

Step 4: Substitute values into the velocity equation

$v=\frac{\left(3\times {10}^8\right)\times 0.16}{656.28}$

Rotational speed:  $v=7.31\times {10}^4\mathrm{m}{\mathrm{s}}^{-1}=73.1\mathrm{km}{\mathrm{s}}^{-1}$

You need to know that in the visible light spectrum red light has the longest wavelength and the smallest frequency compared to blue light which has a shorter wavelength and higher frequency.

The second worked example didn't change the wavelengths from nm into m, since it doesn't matter in the equation as the units will cancel out.