## Stars and the Doppler Effect

When you take white light, and sort it by wavelength through the use of a diffraction grating (a piece of glass, water, a CD-Disc, etc), a rainbow appears. On one end, the bluer, shorter wavelengths, and on the other, the redder, longer wavelengths. The redder wavelengths, being longer, did not interact with the diffraction grating enough to be bent as severely as the other wavelengths, and therefore isn’t angled as severely. The opposite is true of the shorter, bluer wavelengths. A device that splits light into the spectrum is called a spectroscope.

Due to reasons that are outside the scope of this writing, atoms absorb and emit specific wavelengths of light depending on their electron structure, being unique to each element. Absorbing atoms in the photospheres of stars leave these absorption lines in the stellar spectrum, producing a pattern of dark lines. Below is the spectrum of the star Rigel. These spectral lines shed light on the composition of the photosphere of a star and allow us to classify them based on these spectral lines into “spectral types.” A device that measures these spectral lines is called a “spectrometer.”

A main reason spectral lines are commonly seen in prisms is because the light entering the prism is unregulated. A narrow light beam is needed to detect these spectral lines (a wider beam will cause the spectral lines to be submerged in the overlapping spectra). As such, a narrow slit is placed between the light source and the diffraction grating.

In a two-body system (for example, a star and a planet), the barycentric motion of each component will have a velocity component toward or away from us, called the radial velocity, assuming that the system is not inclined face on (i = 0° = 180°). Because the Doppler Effect works on all waves, light included, we would therefore expect to be able to detect the radial velocity of a star by paying attention to how the wavelengths of a stars light change over the orbital period of the system. For a light source receding from an observer, all wavelengths would shift to the redder end of the spectrum (called the redshift). For a light source approaching an observer, all wavelengths would shift to the bluer end of the spectrum (called the blueshift, or negative redshift). Specifically, the redshift, z, of a wavelength λ will result in that wavelength being observed as wavelength λ0, when the emitting source has a velocity v,

$\displaystyle z=\frac{\lambda-\lambda_0}{\lambda_0} = \frac{1+v_r/c}{\sqrt{1-v^2/c^2}} - 1$

Where c is the speed of light, and vr is the radial velocity of the light source, in this case the star.

This is where absorption lines become important, because they give us markers to trace the redshift of the spectrum. Watching them move across the spectrum tells us the changing radial velocity of the target star. The accuracy to which this is done is currently the main limiting factor in the search for extrasolar planets through spectroscopy. It is not difficult to detect redshifts for light sources moving at velocities of several tens of kilometres per second, but the barycentric motion of stars due to the influence of a planet is typically far less.

Eggenberger and Udry (2009) give us a nice overview of the challenges facing precise Doppler spectrometry.

• The motion of the centre of the light source at the spectrograph slit must be accounted for. Errors larger than 1 m s-1 will occur if the photocentre moves away from the slit by a few thousandths of the slit width. Photocenter motions due to telescope guiding errors, focus, seeing fluctuations, and atmospheric refraction usually amount to a tenth of the slit width, leading to errors of several kilometres per second. The usual solution to this is to use an optical fibre (occasionally with a scrambling device) to scramble the starlight from the telescope to the spectrograph. This produces a nearly uniformly illuminated disk at the spectrograph entrance.
• Changes in the environment can cause severe errors in Doppler measurements. For the CORALIE spectrograph, a temperature change of 1 K produces a total velocity drift of 90 m s-1, while a pressure change of 1 mbar produces a net velocity drift of ~300 m s-1. The favoured way to deal with this is to stabilize and control the entire spectrograph in temperature and pressure, but small wavelength shifts can’t be avoided, and often a simultaneous wavelength calibration is done so that instrumental effects can be detected and corrected.
• The motion of Earth. Earth orbits around the sun at +/- 30 km s-1 per year, and has a diurnal rotation of 1-2 km s-1. To obtain precise barycentric radial velocities, one must have precise Solar System ephemeris and needs to know the photon-weighted midpoint of each observation to better than 30 s.

Intrinsic sources of error arise from stellar phenomena. A rapidly-rotating star will blur its spectral lines due to the range of radial velocities of the rotating photosphere relative to the observer. Photospheric jitter can lead to a few metres per second of noise as well, but can be accounted for by taking a time-averaged observation, assuming the jitter is random and uncorrelated.