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

narrower and the wavelength coverage larger such that the blaze function can become an issue in data reductions.

      Figure 2.7 shows an extracted spectral order from an echelle spectrograph that has a strong blaze function. This should be removed for RV measurements. The blaze function can have an influence on your RV measurement in two ways. First, if you use the cross-correlation method (see Chapter 5), the blaze should be removed (flattened) from all extracted spectral orders before calculating RVs. As we shall soon see, the cross-correlation method matches two signals, and in this case it will simply be matching the blaze functions rather than the location of spectral lines.

      Figure 2.7. A spectral order from an echelle spectrograph that has not been corrected for the blaze function. The blaze function is the interference pattern of a single groove and has the shape of a sinc function.

      The second way the blaze function can influence the RVs is by altering the photon statistics after removal of the blaze. Note in Figure 2.7 that the number of counts at the edges of the spectrum are about half that at the center of the spectrum. This means that the signal-to-noise ratio (S/N) is a function of position along the spectrum, being a factor of 1.4 lower at the edges. If you normalize the spectrum in order to remove the blaze, the count rate in the continuum is no longer related to the true photon count. This will alter your statistics if you want to weight your RV measurement according to the S/N ratio.

      Typically, the blaze function is not removed via the normal flat-fielding process (see below), so one needs to take an extra step in the reduction process to remove it. There are two approaches to doing this. One way to remove the blaze function is by dividing it with a low-order polynomial fit to the continuum points in your spectrum. This can be tricky as it depends on the choice of continuum points. As you see in Figure 2.7, it is sometimes difficult to identify the true continuum. This is particularly true for late-type stars with an abundance of spectral lines.

      Alternatively, you can divide the blaze function with the spectrum of a rapidly rotating hot star with no spectral features. This has the advantage in that a hot star spectrum was taken through the same optical path as the science target. The disadvantage is that you are introducing additional noise, and for high-precision RV work, you not want to avoid increasing the noise level if possible. Also, there may still be stellar lines present in the hot star spectrum even though rapid rotation will make these shallow. These will alter the shape of the continuum.

2.1.5 Scattered Light

      Scattered light comes from photons that are redirected into the optical path. Because the light has not properly traversed the optical path of the spectrograph, it is recorded on the detector at a position that has no relationship to its original wavelength. In short, it is light appearing on your detector where it should not be. It primarily comes from dust on the grating, optics, light leaks, etc., as well as light reflected off (in an undesired direction) optical surfaces.

      Figure 2.8 shows a cut perpendicular to spectral orders of an unreduced stellar spectrum. The peaks represent the spectral orders where most of the light should be, but one can see that the intensity in between the orders does not reach zero (the bias level has been subtracted; see below), which would be the case if there were no scattered light.

      Figure 2.8. A cut along CCD detector columns of an unreduced echelle spectrum. The peaks are the spectral orders. Note the nonzero intensity in the interorders due to scattered light.

      Improper scattered-light subtraction will alter the true depths of the spectral lines. This is definitely important if you want to perform stellar abundance studies, which depend on the line strength, but it can also have an influence on the RV measurement. This is primarily through slight changes in the line depth (see Chapter 3), but also because of the slight mismatch between the stellar observation and the template used to calculate the RV (see Chapter 5).

      Proper scattered-light subtraction requires a good separation between orders, which should be considered when designing a spectrograph.

2.2 Fourier Transform Spectrometers

      Most stellar RV measurements are made with spectrometers using an echelle grating as a dispersing element. Another form of spectrometer is a Fourier transform spectrometer (FTS). Unlike a dispersing spectrometer which measures the intensity over a narrow range of wavelengths at a time, an FTS simultaneously collects high spectral resolution data over a wide spectral range. The FTS is not directly used for RV measurements, but it plays an important role in providing a reference spectrum for the gas absorption cell method (Chapter 6).

      An FTS is a Michelson interferometer with a movable mirror (Figure 2.9). Light from a source is split into two beams using a beam splitter. One beam is reflected off a fixed mirror while the other is reflected off a movable mirror, which can vary the optical path length. The combined beams interfere and are recorded at the detector. By measuring the temporal coherence of the light at a different optical path difference, one can convert the time domain signal into a spatial coordinate.

      Figure 2.9. The Fourier transform spectrometer. The light source goes through a beam splitter, which divides the light equally into two light paths, L₁ and L₂. The optical path length L₁ remains fixed, but a translating movable mirror changes the optical path length L₂. The combined interference beam is recorded at the detector. Moving the translating mirror over a fixed range produces an interferogram in frequency space. A Fourier transform converts this into a spectra in the spatial domain.

      If d is the optical path difference of the combined beams at the detector, then for a monochromatic beam of frequency ν0(=λ−1), the intensity is

      I(d)=I021+cos(2πν0d).(2.17)

      For a polychromatic source, S(ν), only at d = 0 will all the light waves add constructively to produce a signal. The output signal is thus

      I(d)=1/2∫0∞S(ν)1+cos(2πν)ddν.(2.18)

      If I¯(ν)=2I(d)−I(0), then

      I¯(ν)=∫0∞S(ν)cos(2πνd)dν.(2.19)

      The function I¯ is called the interferogram, and it is merely the Fourier transform of the source, which can be recovered by the Fourier integral theorem:

      S(ν)=∫0∞I¯(d)cos(2πνd)dν.(2.20)

      Hence, as the name FTS implies, it is a device that records the Fourier transform of your spectrum.

      The resolution of an FTS is determined by the maximum distance that the translating mirror moves during the observation. Thus, it is relatively easy for an FTS to achieve a very high resolving power, typically up to R=500,000−1,000,000. If you want lower resolution, simply move the translating mirror a shorter distance.

      There are no reported instances of an FTS being used for precise RV measurements. These instruments, however, play an important role in the absorption cell method (see Chapter 4). The top panel of Figure 2.10 shows a portion of the spectrum of molecular iodine, I₂, recorded using an echelle spectrograph with reasonably high resolution, R = 67,000. The lower panel shows the same wavelength region recorded with the McMath FTS (see Deming & Plymate 1994). The resolving power is R ≈ 1,000,000, and one can see that the broad and shallow spectral features seen at lower resolving power is really a myriad of very narrow and deep lines.

      Figure 2.10. (Top) A portion of a spectrum of molecular iodine taken using an echelle