The Doppler effect

The Doppler effect ^?  in the street:

When a sound wave is emitted, the sound heard by the listener is not the same if the emitter is stationary with respect to the listener or if it is moving. For example, a motorbike driving past the listener is a sound which approaches and then moves away. The sound is higher during the approach and deeper as the bike moves away. 

The Doppler in astronomy:

The Doppler effect ^? can be observed in the spectrum of an object (star, planet, galaxy ^? ...) which is obtained by scattering the light received from it. When the object moves away, or otherwise approaches, the spectrum shifts towards longer, or otherwise shorter wavelengths.

In the following figure, we can see a pair of absorption lines in 3 objects. The first spectrum corresponds to the Sun, which is stationary with respect to the observer: a pair of dark lines is visible in the green. In the second spectrum (that of a galaxy ^? that is moving away), the two lines are shifted into the yellow. In the third spectrum, the galaxy ^? is moving away faster and the two lines are shifted into the orange.

The Doppler effect ^? with real data:

In these real data, the arrows give the position of the dark lines in three increasingly distant galaxies ^? . The galaxies ^? very far from ours are moving away at high speed and their spectrum is heavily shifted towards the red.

Calculation of Doppler effect ^? spectral shift:

If the wavelength of light is $\lambda$ (in metres), the wave peaks are separated by $\lambda$ metres and leave the emitter at times separated by $P$ seconds, such that $\lambda = c \times P$ where $c$ is the speed of light ($3 \times 10^8$ meters per second).

If the source is stationary with respect to the observer, they receive a wave the peaks of which are separated by $P$, and the received wavelength of which is $\lambda$, equal to the emitted wavelength.

If the wave source is moving with respect to the observer at a speed $v_r$, the peaks are emitted at intervals of $P$. During the time interval between the emission of 2 peaks, the source has covered a distance $d = v_r \space P$.

If the source moves away from the observer, the second peak covers a greater distance before reaching them, a distance equal to $\lambda + d$. The observer receives a wave with a wavelength

$$\lambda’ = \lambda+d=Pc+v_{r} P = Pc (1+\frac {v_{r}} {c})$$

The received wavelength is $\lambda (1+\frac {v_{r}} {c})$

It is thus greater than $\lambda$ and the light observed will be redder than the emitted light.

Conversely, if the source moves closer to the observer, the second peak covers a shorter distance before reaching them, a distance equal to $\lambda - d$. The observer receives a wave with a wavelength

$$\lambda’ = \lambda-d=Pc-v_{r} P = Pc (1-\frac {v_{r}} {c})$$

The received wavelength is $\lambda (1-\frac {v_{r}} {c})$

It is thus smaller than λ and the observed light will be bluer than the emitted light.

The Doppler effect ^? in the hunt for exoplanets