



Appendix B: The Doppler Effect (click on equations to view enlarged)
The observed frequency ν′ of light emitted from a source moving with velocity u as depicted in Fig. B1 is given by [4]
, (B1)
where ν is the frequency of the light in the rest frame of the source, and c is the speed of light, 299,792 km/s.
If the velocity is purely radial, then Eq. B1 reduces to
, (B2)
where β^{2} = (cu)/(c+u). Note that in Eq. B2 we have adopted the convention that u > 0 indicates a receding source. For our application, we assume the velocity is purely radial (θ= 0° or 180°) and use Eq. 2.[5] The magnitude of the Doppler effect for some typical situations is given in Table B1.
Eq. B2 gives the Lorentz transformation for a monochromatic frequency ν. However, for a continuous spectrum, we cannot simply scale all frequencies.[6] We must also apply the Lorentz transformation to the (necessarily finite) aperture collecting the radiation. An aperture subtending solid angle Ω in the rest frame of the source will appear to have solid angle
(B3)
when receding with velocity u. Suppose now that the source has a restframe radiance of L^{P}(ν). An aperture of size Ω in its rest frame will receive a photon flux of N(ν) = ΩL^{P}(ν) If the aperture is receding with velocity u, the photon flux received from the receding source will be
If the source is a blackbody at temperature T, (Eq. 4), we have
If we interpret this radiation as coming from a stationary blackbody, that is, L'(ν) = N'(ν)/Ω', then the effective temperature is
, (B4)
Thus the spectrum of a receding
blackbody appears identical to a cooler, stationary blackbody. [4] J. D. Jackson, Classical Electrodynamics, 2^{nd} ed., pp 522 [5] The tangential effect is small in most realistic cases anyway. For example, 1000 cm^{1} light (10 mm) from a source moving at 100 km/s perpendicular to the line of sight is shifted only 56×10^{6} cm^{1}. [6] T. P. Gill, “The Doppler Effect”, Logos Press, Inc., 1965 
Gascell Simulator 