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Analysis and Interpretation of Astronomical Spectra 63

1. In the upper, red profile the spaceship moves away by the small amount from to
, during the transmission of one entire oscillation period. Therefore the end of the

emitted oscillation period covers this additional way to the observer with a corre-

spondingly extended runtime . therefore measures a red shift, ie an extension of
the rest wavelength by . Here, the sign of the radial velocity is positive .

2. In the middle, black profile, observer measures the originally transmitted oscillation

period with the so-called rest wavelength , because the radial velocity of the space

ship is here . ,

3. In the lower, blue profile the spaceship approaches by the small amount from to
. Therefore the end of the emitted oscillation period covers this shortened way to the

observer with a correspondingly reduced runtime . therefore measures a blue-
shift, ie a reduction of the rest wavelength by . Here, the sign of the radial veloc-
ity is negative .

The amount of shift can easily be calculated from the total oscillation period [s] and
the radial velocity :

The oscillation period can be calculated from the rest wavelength and the speed of
light :

The Spectroscopic Doppler Formula with the radial velocity , can be obtained by insert of
in :

Measured wavelength of the considered spectral line
Measured shift in wavelength of a given spectral line
Rest wavelength of a spectral line, measured in a stationary system
Speed of light 300‘000 km/s

 If the spectrum is blue-shifted, the object is approaching us and becomes negative

 If the spectrum is red-shifted, the object is receding and becomes positive.

15.3 The z-Value - A Fundamental Measure of Modern Cosmology

In modern cosmology, the inconspicuous break in formula {16} has become the most

important measuring variable in extragalactic space. Now, due to the constant speed of
light, this so-called -value – applied at such extremely distant objects – became both, a

measure for the distance, as well as for the past. In a calibrated spectrum it can easily be

determined by the measured shift of a spectral line with the wavelength and their known

rest wavelength .

The -values remain also completely independent of the debated cosmological model pa-
rameters and thus are reliably comparable. According to {16}, the radial velocity of the clas-
sical Doppler Effect results to: