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target="_blank" rel="nofollow" href="#ulink_c035de27-58ec-5401-9030-f58bed590fc7">1992). These were discovered using timing variations.

      2As of this writing, this paper was submitted but not yet accepted.

      3In recognition of this discovery Mayer and Queloz were awarded the 2019 Nobel Prize in physics.

      

      The Doppler Method for the Detection of Exoplanets

      A P Hatzes

      Chapter 2

      The Instruments for Doppler Measurements

      The measurement of stellar radial velocities (RVs) requires spectrographs that break the light into its component wavelengths and detectors to record the resulting spectrum. In this chapter, we give a brief overview of the essential equipment needed for stellar RV measurements, namely spectrographs and detectors. Entire books are required to do the subject justice, and an excellent source is the text Spectroscopic Instrumentation: Fundamentals and Guidelines for Astronomers by Eversberg & Vollman (2015). Rather, in this chapter, we will cover just the basics principles of instrumentation needed for RV measurements.

      The left panel of Figure 2.1 shows the classic layout of a high-resolution spectrograph. Light from the telescope comes to a focus at the slit and then diverges. A collimator having the same focal ratio1 as the telescope then converts this diverging beam into a parallel one that strikes a dispersing element. This is generally a reflection grating (see below) that breaks the light up into its component wavelengths. The dispersed light is then focused onto the detector by a camera, which is either a reflective, Schmidt-type camera (mirror plus corrective lens), or a refractive, lens-based system. Reflective cameras are generally used for spectrographs with slits, while lens cameras are the choice for spectrographs fed by a fiber optic.

image

      Figure 2.1. The layout of a classic spectrograph (left) and a white-pupil design spectrograph (right). Spectrographs using fine-ruled gratings do not have the cross-dispersing element.

      The cross-disperser that is shown is an optical element that disperses the light in the direction perpendicular to the grating dispersion in order to separate the spectral orders. This will be discussed in more detail below. Classic spectrographs from 40 years ago typically used finely ruled gratings at low spectral order. Filters had to be inserted into the light path to block out light from unwanted orders. Modern spectrographs use echelle gratings at high orders coupled with a cross-disperser. So, before the advent of cross-dispersing elements, the layout looks the same, except for the cross-dispersing element.

      It is important to note that a spectrograph is merely a camera (you can consider it a telescope as it is also bringing a parallel wavefront, like starlight, to a focus). The only difference is the presence of the echelle grating to disperse the light, and in this case, the cross-dispersing element. Remove these and what you would see at the detector is a white-light image of your entrance slit. Reinsert the grating, and the spectrograph now produces a dispersed image of your slit at the detector.

      Many modern echelle spectrographs are designed after the white-pupil concept (Baranne 1972) shown in the right panel of Figure 2.1. In this design, an additional optical element is used to produce an intermediate focus between the grating and the cross-disperser at a position where the various spectral orders created by the grating are not yet separated. Here, there is a superposition of all wavelengths, and an intermediate white-light image of the slit is formed.

      Naively, one may think that adding an additional optical element is unwise as it reduces the efficiency of your spectrograph, due to the extra optical element, but there are two good reasons to do this. First, because of the intermediate focus, all subsequent apertures, like the camera, can be made smaller. As a general rule, smaller optics translates into reduced costs. Furthermore, smaller optics can make for a more compact spectrograph, which is easier to stabilize thermally and mechanically. As we shall see, this is important for precise RV measurements.

      Second, a spatial filter at the intermediate focus can reduce stray light, which is not desired in your spectrograph. Overall, the advantages of the white-pupil design outweighs the disadvantage of the small loss of light due to the extra optical element.

      The key part of any spectrograph is a dispersing element that breaks the light up into its component wavelengths. For high-resolution astronomical spectrographs, this is almost always a reflecting grating, a schematic that is shown Figure 2.2. The grating is ruled with a groove spacing, σ. Each groove, or facet, has a tilt at the so-called blaze angle, ϕ, with respect to the grating normal. This blaze angle diffracts most of the light into higher orders, m, rather than the m = 0 order, which is white light with no wavelength information.

image

      Figure 2.2. Schematic of an echelle grating. Each groove facet has a width σ and is blazed at an angle θB with respect to the grating normal (dashed line). Light strikes the grating at an angle α and is diffracted at an angle β, both measured with respect to the grating normal.

      Light hitting the grating at an angle α is diffracted at an angle βb. and satisfies the grating equation:

      Note that at a given λ, the right-hand side of Equation (2.1) is ∝m/σ. This means that the grating equation has the same solution for small m and small σ (finely grooved), or alternatively, for large m and large σ (coarsely grooved).

      One can compute the angular dispersion dλ/dβ by taking the derivative of the grating equation:

      dβdλ=mσcosβ.(2.2)

      Thus, a higher dispersion can be achieved by using higher spectral orders.

      The grating equation (Equation (2.1)) can be used to eliminate the spectral order number and obtain

      If we chose the blaze angle, θB such that α=β=θB, we get

      dβdλ=2λtanθB.(2.4)

      In other words, large angular dispersions require large blaze angles. The blaze angles of echelle gratings are typically 63.4° or 75.9°. These are often called “R2 grating” or “R4 grating,” due to the fact that the tangent of 63.4° or 75.9° is 2 and 4, respectively. Because of the tangent, the angular dispersion is a steeply increasing function of the blaze angle. Note that the angular dispersion of an R4 grating is a factor of two larger than that of an R4. Increasing the blaze angle by just another 7° would result in another factor of two increase in the dispersive power.

      Now let us consider the case of a constant diffraction angle, β=βc. The right side of Equation (2.1) is now a constant, and the associated central wavelength of an order, λc, is inversely proportional to the order number m:

      λc(m)=σm(sinα+sinβc).(2.5)

      If we differentiate with respect to m, we get the change in central wavelength with respect to m,

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