Stellar Doppler Shift and Binary Systems

00:00 6 marks

Light from a distant star is analysed. A particular spectral line corresponding to the emission from hydrogen atoms has a wavelength \(\lambda_0\) = 656.3 nm when measured in a laboratory on Earth. The same spectral line observed in the light from the distant star has a wavelength \(\lambda\) = 658.1 nm.

Part a

3 minutes 2 marks

Question

Calculate the speed of the star relative to Earth. [2]

Hint Hint

Hint

Use the non-relativistic Doppler shift formula for light: \(\Delta \lambda / \lambda_0 \approx v/c\). Remember \(c = 3.00 \times 10^8 \text{ m/s}\).

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\(v \approx 8.2 \times 10^5 \text{ m/s}\)

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Part b

1 minutes 1 marks

Question

State whether the star is moving towards or away from Earth, justifying your answer. [1]

Hint Hint

Hint

Compare the observed wavelength (\(\lambda\)) to the rest wavelength (\(\lambda_0\)). Is it longer (redshift) or shorter (blueshift)? What does this imply about the direction of motion?

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Away from Earth because the observed wavelength is longer than the rest wavelength (redshift).

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Part c

5 minutes 3 marks

Question

The star is part of a binary system, orbiting a common centre of mass with another star. Explain how observations of the Doppler shift of the spectral lines from both stars over time can be used to deduce properties of the binary system. [3]

Hint Hint

Hint

Think about how the orbital motion causes the velocity relative to Earth to change. How does this changing velocity affect the observed wavelengths? What properties can be calculated from these periodic changes?

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Observing the periodic Doppler shift over time reveals the orbital period. The maximum shifts give the radial velocities. Using these with Kepler’s laws allows estimation of mass ratio, total mass, and separation.

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