gen-define-the-doppler-shift-in-spectra-and-give-one-short-worke-6d99

Define the Doppler shift in spectra and give one short worked example.

# Doppler Shift in Spectra ## Definition The **Doppler shift** is the change in observed wavelength (or frequency) of light from a source due to relative motion between the source and observer. - **Redshift**: Wavelength increases (shifts toward red) when source moves *away* - **Blueshift**: Wavelength decreases (shifts toward blue) when source moves *toward* ## Formula For non-relativistic speeds (v << c): $$\frac{\Delta \lambda}{\lambda_0} = \frac{v}{c}$$ where: - Δλ = change in wavelength (λ_observed - λ_0) - λ₀ = rest wavelength (emitted) - v = radial velocity (positive = receding) - c = speed of light (3 × 10⁸ m/s) ## Worked Example A hydrogen Hα spectral line (rest wavelength λ₀ = 656.3 nm) is observed from a distant galaxy at λ_observed = 658.9 nm. **Find the galaxy's velocity:** $$\Delta \lambda = 658.9 - 656.3 = 2.6 \text{ nm}$$ $$v = c \times \frac{\Delta \lambda}{\lambda_0} = 3 \times 10^8 \times \frac{2.6}{656.3}$$ $$v = 1.19 \times 10^6 \text{ m/s} = 1190 \text{ km/s}$$ The galaxy is **receding** at ~1190 km/s (redshifted, so moving away).