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

will decrease as 1/m². This means at high orders, the wavelength intervals will overlap (Figure 2.3).

      Figure 2.3. Schematic of the wavelength intervals in different orders from an echelle grating. For higher orders, the central wavelength of the order λ_m has approximately the same dispersion angle.

      Most modern, high-resolution echelle gratings operate at high spectral orders of m ∼ 100, such that these occur spatially at the same location (same β) as the detector. To eliminate the undesired light from the other orders, one has to resort to blocking filters to isolate the light from the desired wavelength range. But why waste light? The elegant solution is to use an additional optical element to disperse the light perpendicular to the grating dispersion (i.e., the so-called “cross-disperser”). The use of two-dimensional detectors such as charge-coupled devices (CCDs) means that the separated spectral orders can now all be recorded at the same time.

      The advent of echelle spectrographs, which provide a large wavelength coverage and high quantum efficiency of two-dimensional digital detectors (as opposed to photographic plates), was the main driver for the dramatic increase in RV precision in the past couple of decades. Figure 2.4 shows a spectrum of sunlight taken with an echelle spectrograph, and it nicely shows the large wavelength coverage one can achieve with an echelle spectrograph.

      Figure 2.4. A spectrum of sunlight taken with a prism cross-dispersed echelle spectrograph.

2.1.2 Cross-dispersers

      The cross-disperser element currently comes primarily in three forms: prisms, gratings, and grisms. Each provides its own wavelength dependence on the dispersive power in the cross-dispersion direction, y.

      Prisms provide a separation of spectral orders that is inversely proportional to the central wavelength of the order

      Δy∝λ−1(prism).(2.6)

      Thus, for prism cross-dispersers, the separation between spectral orders increases as one goes to bluer wavelengths. There are two advantages to using prisms. First, they have a high throughput and tend to be more efficient than other options such as gratings. Second, they provide an efficient packing of spectral orders and thus effectively use the real estate of the CCD detector. Prisms are often the choice for echelle spectrographs built for 2–3 m class telescopes (Vogt 1987; Tull et al. 1995; Kaufer & Pasquini 1998; Vogt et al. 2014)

      There are two main disadvantages to using prisms as cross-dispersers. First, the packing of orders at red wavelengths becomes too tight, making it more difficult to determine interorder scattered light (see below). Second, for large diameter telescopes prisms do not provide enough dispersive power for good order separation. One must use several prisms in a chain or, preferably, another type of cross-disperser, such as a grating.

      Grating cross-dispersers provide a separation of spectral orders that is proportional to the wavelength squared:

      Δy∝λ2(grating).(2.7)

      In contrast to prisms, gratings have a spectral order separation that increases toward redder wavelengths. They have the advantage in that they provide a much larger separation of orders which can be set by the designer by choosing a grating with the appropriate groove spacing. This makes it much easier to remove the interorder scattered light. On the other hand, gratings have an efficiency of ≈70%, which is generally less than that for prisms. Because they are more effective dispersing devices, they are the choice of cross-disperser on echelle spectrographs built for large telescopes (Vogt et al. 1994; Dekker et al. 2000; Strassmeier et al. 2015).

      Another option is a grism cross-disperser, which is a combination of prisms and gratings. This produces an order separation that should roughly be proportional to wavelength,

      Δy∝λ−1(prism)×λ2(grating)∝λ(grism).(2.8)

      The advantage of grisms is that they provide a more uniform spacing of orders from blue to red wavelengths. However, they tend to have even lower throughput than either prisms or grating cross-dispersers. The Tautenburg Echelle Spectrograph is one example of an echelle spectrograph that uses grisms as cross-dispersers. Figure 2.5 shows the spectral format of the various types of cross-dispersing elements.

      Figure 2.5. The various types of cross-dispersers. The wavelength of the central order increases toward the bottom. (Left) A grating cross-disperser produces an order separation, Δλ, that increases rapidly toward longer wavelengths (Δλ∝λ2). (Center) A prism cross-disperser produces orders that are more tightly packed at redder wavelengths (Δλ∝λ−1). A grism produces a more uniform spacing of orders, but with a linear increase toward the red (Δλ∝λ).

2.1.3 Dispersion and Spectral Resolution

      The spectral resolution is one of the most important parameters of your spectrograph for precise stellar RV measurements. It fundamentally determines the smallest shift of a spectral line that you can measure.

      The spectral resolution, δλ, is defined as the difference in wavelength of two monochromatic beams that can just be resolved by your spectrograph. The spectral resolving power, R, of a spectrograph is defined by the ratio

      R=λδλ.(2.9)

      It is a common mistake for astronomers to interchange the term “spectral resolution” with “spectral resolving power.” The two are related, but technically not the same. Spectral resolution, δλ, has the dimensions of length, typically angstroms, and the smaller it is, the easier it is to distinguish fine spectral details. On the other hand, R is a dimensionless quantity, and the larger it is, the higher the resolution (equivalent to small δλ).

      Before we derive the spectral resolution of a spectrograph, it is useful to consider the basic geometry, parameters, and angles that are involved in the telescope plus spectrograph system. Figure 2.6 shows the layout of the telescope and spectrograph. For a detailed discussion see Schroeder (1987). The important parameters are as follows:

       D : telescope diameter;

       f: telescope focal length;

       d1: collimator diameter;

       f1: collimator focal length;

       d2: camera diameter;

       f2: collimator focal length;

       A: dispersing element (e.g. the echelle grating);

       w, h: slit width and height (for fibers, the diameter);

       w′, h′: projected slit width and height at detector.

      Figure 2.6. A schematic showing the geometry and angles of a spectrograph. See text for the definition of the various elements.

      The slit width subtends an angle ϕ=w/f on the sky. At the detector, it subtends an angle of ϕ′=w/f1. For