4. Light from a distant star is analysed. A spectral line corresponding to hydrogen has a wavelength of 656.3 nm when measured in a laboratory on Earth. The same spectral line observed in the light from the star has a wavelength of 658.1 nm.
(a) Calculate the speed of the star relative to Earth.
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(b) State whether the star is moving towards or away from Earth, justifying your answer.
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(c) The star is part of a binary system, orbiting a common centre of mass with another star. Outline how observations of the Doppler shift of the spectral lines from both stars over time can be used to determine properties of the binary system.
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(d) Explain **one** limitation of using the non-relativistic Doppler formula calculated in (a) for astronomical objects.
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Question
Calculate the speed of the star relative to Earth.
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Hint
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Use the non-relativistic Doppler shift formula for light: \(\Delta\lambda / \lambda \approx v / c\). Calculate \(\Delta\lambda\) first.
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8.2 × 10⁵ m s⁻¹
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Question
State whether the star is moving towards or away from Earth, justifying your answer.
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Hint
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Compare the observed wavelength to the rest wavelength. Is it longer (redshift) or shorter (blueshift)?
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Away from Earth, because the observed wavelength (658.1 nm) is longer than the rest wavelength (656.3 nm), indicating redshift.
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The star is part of a binary system … determine properties of the binary system.
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Think about how velocity changes affect observed wavelengths. What properties can be deduced?
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Periodic Doppler shifts give the orbital period. The amplitude gives line-of-sight speeds. Using Kepler’s laws and these speeds and period yields stellar masses and separation.
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Explain **one** limitation … non-relativistic Doppler formula.
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Consider when v << c assumption fails.
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The approximation Δλ/λ ≈ v/c requires v ≪ c. At high speeds (e.g., distant galaxies, near black holes), relativistic effects like time dilation become significant, so the non-relativistic formula is invalid.
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