The Doppler Method for the Detection of Exoplanets. Professor Artie Hatzes

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      ¹We take the typical wavelength coverage of an optical spectrograph used for RV measurements.

      

      The Doppler Method for the Detection of Exoplanets

      A P Hatzes

      Chapter 4

      Simultaneous Wavelength Calibration

      You are an eager exoplanet hunter, and you want to find exoplanets using the radial velocity (RV) method. Taking everything you have learned from the previous chapter, you design a spectrograph with enough resolution, wavelength coverage, etc. to achieve an RV precision of, say 10 m s⁻¹. You prepare a target list of suitable stars, i.e., bright, late-type, and slowly rotating stars. You start making measurements, and after a time, you realize that the scatter of your measurements, even for stable stars, is far worse than your estimated uncertainty. What went wrong? Most likely, instrumental shifts have introduced an unwanted and large source of errors.

      A CCD detector only records the intensity of light as a function of pixel location. To measure a Doppler shift, you need to know the intensity of light as a function of wavelength. Thus, you have to put a wavelength scale on your spectrum using a suitable calibration source. A good wavelength calibrator should have a high density of spectral features with measured wavelengths well spread across the spectral range of your Doppler measurements. The more features you have, the better the mapping between pixel location and wavelength will be.

      But that is not the whole story. We have seen in Chapter 3 that a Doppler shift is a tiny displacement on your detector. It does not take much of a mechanical shift to mimic a shift of the spectral line. The problem is that you most likely observed the calibration source at a different time than when you made your stellar observation. Furthermore, the light from the hollow cathode lamp always goes through a different optical path than your starlight, and this might introduce a systematic error. If you are getting large uncertainties in your Doppler measurements, much higher than is predicted by photon statistics, then the likely cause is instrumental shifts.

      If you want to minimize the effects of instrumental shifts on your Doppler measurement, it is essential that you observe your wavelength calibration at the same time as your stellar observation. Unless your spectrograph is extremely stable, there is a good chance that something has moved—optical elements, the detector, etc.—between the time you observed your star and the time you observed your calibrator.

      In this chapter, we examine the various methods of simultaneous wavelength calibration, both historic and modern, that have been employed to minimize the effects of instrumental shifts on the RV measurement.

4.1 Instrumental Shifts

      Instrumental shifts occur because the traditional way of performing wavelength calibration is to observe a calibration source either before or after your science observation. At optical wavelengths, this is typically a thorium–argon (Th–Ar) hollow cathode lamp (see below). In the data reduction process, you identify emission lines with wavelengths that have been measured in the laboratory. A fit to the pixel versus wavelengths of these lines using a high-order polynomial provides the mapping between pixel and wavelength space. This function is then applied to the stellar spectrum.

      The problem is that the stellar spectrum is taken at a different time to that of the calibration source. Mechanical shifts of the spectrograph and detector or temperature and pressure changes in the spectrograph room can occur between the time you take your calibration and that of the stellar observations. Table 3.1 shows that for an R = 100,000 spectrograph, a Doppler shift of 1 m s⁻¹ will cause a physical shift at 10⁻⁵ cm at the detector, or about one-fifth of the wavelength of the incoming light. It does not take much of an instrumental shift of the spectrograph or detector to obliterate this signal. These instrumental shifts can dominate the measurement error due to simple photon statistics.

      Let’s look into the instrumental shifts for modern spectrographs using measurements taken with the Tautenburg Coudé Echelle spectrograph (TCES). The TCES is an example of a modern, well-designed echelle spectrograph with good stability for standard spectroscopic work. The spectrograph has a resolving power of R