Astronomy Principles and Practice Fourth Edition.pdf

228 The radiation laws
Figure 15.9. A source S at a distance d from the observer O.
Suppose that light is being emitted by atoms in a source, S, with a frequency, ν, and that the waves
travel to an observer, O, at a distance, d, from the source (see figure 15.9). After ‘switching-on’ the
source, the first wave arrives at O after a time d/c. During this time the source has emitted νd/c waves
and the apparent wavelength is obviously
Distance occupied
λ =
Number of waves = d
νd/c = c ν .
Suppose that the source has velocity, V , away from the observer. During the same time interval as
before (d/c), the source moved a distance equal to V × d/c. Thus, the waves emitted during the same
interval now occupy a distance equal to
d + Vd
c .
The apparent wavelength, λ ′ , of the radiation is, therefore, given by
λ ′ = d + Vd/c
νd/c
= c ν + V ν
= λ + V λ
c . (15.24)
Therefore,
λ ′ − λ = λ = V λ
c
so giving
z = λ
λ = V c . (15.25)
The difference, λ, between the observed wavelength and the wavelength that would have been
observed from a stationary source is called the Doppler shift. It is positive (λ ′ >λ)for an object
which is receding from the observer and negative for an object which is approaching the observer.
Thus, lines present in a spectrum of the light from a moving source are shifted towards the red for a
receding object and are shifted towards the blue for an approaching object.
Equation (15.25) is valid only for V ≪ c. For velocities near that of light, such as for some
galaxies and quasars, the concepts of relativity theory need to be applied giving a Doppler shift which
is nonlinear in V . Under this circumstance, the apparent wavelength is written as
[
λ ′ = γλ 1 + V ]
(15.26)
c
where γ is the Lorentz factor:
[ ( ) ]
V
2 − 1
2
γ = 1 − .
c