What is the Doppler effect? Explain, with the aid of labelled diagrams, how this phenomenon occurs - Leaving Cert Physics - Question 7 - 2007

An emission line spectrum is produced when a gas is heated or energized, causing its atoms to become excited to higher energy levels. This energy can be supplied by heating the gas via electrical discharge (as in hydrogen discharge tubes) or by other means.

Steps in Spectrum Production

1. Atom Excitation: The atom absorbs energy and an electron is excited to a higher energy state.

2. Electron Transition: The electron eventually returns to its lower energy state, releasing energy in the form of electromagnetic radiation (light).

3. Spectroscopy: When observed with a spectrometer, the emitted light is dispersed and forms a spectrum, displaying specific wavelengths corresponding to the energies of the transitions.

4. Characteristics of Spectrum: The result is a series of bright lines on a dark background, corresponding to the specific frequencies of emitted radiation, characteristic of the element.

To determine if the star is approaching the Earth, we must compare the observed wavelength of the red line (720 nm) to the laboratory wavelength (656 nm).

Analysis

• The original wavelength is 656 nm, while the observed wavelength is longer at 720 nm.
• According to the Doppler effect, if the wavelength has increased, this indicates that the source is moving away from the observer (the Earth, in this case).

Conclusion

Thus, we can conclude that the star is not approaching the Earth; instead, it is receding.

To calculate the frequency of the red line:

Formula

The frequency can be calculated using the formula: $f = \frac{c}{\lambda}$ where:

• $c = 3.00 \times 10^8 m s^{-1}$ (speed of light)
• $\lambda = 720 \times 10^{-9} m$ (convert nm to m)

Calculation

Substituting the values: $f = \frac{3.00 \times 10^8}{720 \times 10^{-9}} = 4.17 \times 10^{14} Hz$

To calculate the speed of the moving star, we use the Doppler shift understanding that for a distant star moving away:

Formula

Using the Doppler effect approximation: $v = c \left( \frac{(\lambda_{observed} - \lambda_{rest})}{\lambda_{rest}} \right)$ where:

• $\lambda_{observed} = 720 \, nm$
• $\lambda_{rest} = 656 \, nm$
• $c = 3.00 \times 10^8 \: m/s$

Substitution

Substituting the values: $v = 3.00 \times 10^8 \left( \frac{720 - 656}{656} \right)$

Calculation

This gives: $v \approx 2.92 \times 10^2 \: m/s$