Doppler Shift

• Wavelength Compression: As a source moves toward an observer, each successive wave crest is emitted from a position closer to the observer than the previous crest. This reduces the distance between crests, effectively shortening the wavelength.

• Wavelength Expansion: Conversely, as a source recedes, each crest is emitted from a further distance, increasing the gap between successive wavefronts and resulting in a longer observed wavelength.

• Frequency Relation: Because the speed of light $c$ is constant, the relationship $c = f\lambda$ dictates that an increase in wavelength must result in a proportional decrease in frequency, and vice versa.

• The Doppler Equation: The relative velocity of a celestial body can be calculated using the ratio of the change in wavelength to the reference wavelength:

$\frac{\Delta \lambda}{\lambda_0} = \frac{v}{c}$

• Identifying $\Delta \lambda$: The change in wavelength is defined as the difference between the observed wavelength and the reference wavelength: $\Delta \lambda = \lambda - \lambda_0$.

• Step-by-Step Velocity Calculation:

1. Identify the reference wavelength $\lambda_0$ from laboratory standards.
2. Measure the observed wavelength $\lambda$ from the object's spectrum.
3. Calculate the difference $\Delta \lambda$.
4. Rearrange the formula to solve for velocity: $v = c \cdot (\Delta \lambda / \lambda_0)$.

• Sign Convention: A positive $\Delta \lambda$ (redshift) indicates the object is moving away, while a negative $\Delta \lambda$ (blueshift) indicates the object is approaching.

• Redshift vs. Blueshift: These terms describe the direction of the spectral shift in relation to the visible spectrum.

• Sound vs. Light Analogies: While Doppler shift in sound affects pitch (frequency), Doppler shift in light affects color. For example, a receding ambulance's siren drops in pitch, just as a receding galaxy's light shifts toward the red spectrum.

• Variable Identification: Students often confuse the observed wavelength $\lambda$ with the reference wavelength $\lambda_0$. Always remember that $\lambda_0$ is the 'natural' or 'lab' value, while $\lambda$ is the 'shifted' value seen through a telescope.

• Unit Consistency: Ensure that $\Delta \lambda$ and $\lambda_0$ are in the same units (typically nanometers or meters) so the ratio is dimensionless. The final velocity $v$ will have the same units as the speed of light $c$.

• Magnitude Check: Since $c$ is extremely large ($3 \times 10^8$ m/s), even small shifts in wavelength correspond to massive celestial velocities. If your calculated velocity is faster than light, re-check your algebra.

• Spectral Line Patterns: In exams, look for patterns of dark absorption lines. The entire pattern shifts as a unit; identifying the shift of one known line allows you to calculate the shift for the whole system.

• 'Change in Speed' Error: A common misconception is that the Doppler shift changes the speed of the wave. In reality, the speed of light $c$ remains constant; only the wavelength and frequency are altered by the motion.

• Absolute vs. Relative Wavelength: Some students mistakenly use the change $\Delta \lambda$ as the denominator in the equation. The ratio must always be compared against the original reference wavelength $\lambda_0$.

• Frequency vs. Wavelength Inversion: Forgetting that wavelength and frequency are inversely related leads to errors. Redshift is an increase in wavelength but a decrease in frequency.