Page 64 - analysis-and-interpretation-of-astronomical-sp

Analysis and Interpretation of Astronomical Spectra 64

15.4 The Relativistic Doppler Effect for Electromagnetic Waves

From the view of observer , at very high radial velocities an additional effect from the
Special Theory of Relativity STR becomes noticeable: detects that the system time of the
spaceship has significantly slowed down and therefore the period duration of the gener-
ated light signal is accordingly shortened from to . According to STR applies:

From the view of observer combines here the classical- with the relativistic Doppler-Effect:

1. Classical Doppler Effect: The observed duration of one oscillation period is the sum of
the period duration plus the extension, or shortening, of the run time due to the radial ve-
locity . (sect. 15.2).

2. Relativistic Doppler Effect: The combination with the relativistic Doppler Effect can be
obtained by replacing T in with T ' from

solved for

For reasons of proportionality and due to the following applies:

inserted into finally yields the well-known Relativistic Doppler Formula for cal-

culating the radial velocity .

This applies to kinematically induced radial velocities. Their extended application to the ap-
parent "cosmological escape velocity", due to the expansion of the "space-time lattice" is
under debate (see chap. 15.8).

15.5 The Measurement of the Doppler Shift

For radial velocity measurements ( ) in most cases, the Doppler shift is determined accord-
ing to sect. 15.3. The difference is calculated between a specific spectral line and their
well-known rest wavelength of a stationary laboratory spectrum.

The calculation of is finally enabled by formula {16} {18} or {20}. For the accurate deter-
mination of , a calibration lamp spectrum must be recorded, immediately before and/or
after the object spectrum with a totally unchanged spectrograph setup. The analysis soft-
ware (eg Vspec) allows in a calibrated spectral profile the -measurement with so called
Gaussian fits. For this purpose, clearly identified lines must be selected which are not too
strongly deformed by blends. A detailed description of the procedure can be found in [30].

In the red region of the spectrum and with high-resolution spectrographs, a very accurate
calibration can also be achieved by use of the atmospheric H2O lines [30]. In addition, this
is also enabled by the naturally- and artificially generated emissions of the night sky spec-
trum [33].