1. From the Siren to the Cosmos: What Does Doppler Measure?

In acoustics we hear the pitch rise as an ambulance approaches and fall as it moves away. In relativity, the Doppler effect applies to light, not sound—and the classical formula no longer suffices.

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2. Longitudinal Doppler: Frequency and Wavelength

For a source and observer receding from or approaching each other along the line of sight:   $$ \frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{em}}} = \sqrt{\frac{1 + \beta}{1 - \beta}}, \quad \beta = \frac{v}{c}. $$   In terms of frequencies:   $$ \nu_{\mathrm{obs}} = \nu_{\mathrm{em}}\, \sqrt{\frac{1 - \beta}{1 + \beta}}. $$

• If \(v > 0\) (receding): redshift (\(\lambda_{\mathrm{obs}} > \lambda_{\mathrm{em}}\)).
• If \(v < 0\) (approaching): blueshift (\(\lambda_{\mathrm{obs}} < \lambda_{\mathrm{em}}\)).

Astronomical example:   A planet orbiting its star at 30 km/s induces a shift of about 0.01 nm in an iron absorption line—detectable with modern spectrographs.

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3. Light Aberration: Apparent Angle Change

Aberration changes the apparent angle \(\theta\) between a light ray and the direction of motion. If \(\theta\) is the source’s emission angle and \(\theta'\) the observer’s reception angle:   $$ \cos\theta' = \frac{\cos\theta - \beta}       {1 - \beta\,\cos\theta}. $$

• For \(\theta = 90^\circ\): stars appear “dragged” toward the direction of Earth’s motion, by up to \(\approx 20.5''\).

Historical note:   In 1727, James Bradley, while searching for stellar parallax, discovered aberration—first proof that Earth moves.