w′=rwf2f1=rϕDF2.(2.10)
This expression is important for matching the slit width to the detector. A typical pixel size, Δ, for a CCD detector is Δ=15μm. To adhere to the Nyquist sampling (see Chapter 7) of a 2 pixel projection of the slit requires 2Δ=rϕDF2.
Equation (2.10) represents one of the major problems facing designers of high-resolution spectrographs. To match the slit width to the detector, you either have to use a narrow slit (smaller ϕ) or build a faster camera, i.e., a smaller F2. In optical design, a faster camera translates into higher costs, or problems in its manufacture.
The problem becomes worse for large telescopes. Suppose you have a 4 m telescope at a site that has a median seeing of 1″. This requires a camera with F2≈1.5. If you build a spectrograph for a 10 m telescope at a better site (0.5″ median seeing), you require a faster camera of F2≈1.2.
What is important for RV measurements is not the angular dispersion but the linear dispersion at the detector as this determines the displacement of a spectral line in detector pixels for a given Doppler shift in wavelength. The change in the dispersion angle, δβ, can be converted to a displacement at the detector of dx=f2dβ, which, along with Equation (2.3), yields the dispersion at the detector:
dλdx=dλdxdλdβ=1f21dβ/dλ(Åmm−1).(2.11)
This dispersion is in units of Å mm−1, which can be converted to the more useful Å/pixel simply by multiplying by the CCD pixel size in mm per pixel.
Using Equation (2.11), a displacement in wavelength by δλ results in a displacement (in millimeters) at the detector:
dx=f2dβdλδλ.(2.12)
Setting this equal to the projected slit width,
f2dβdλδλ=rϕDF2,(2.13)
yields a spectral resolution of the spectrograph of
δλ=rϕADd1,(2.14)
where we have used F2 = f2/d1 and A=dβ/dλ.
Converting to spectral resolving power results in
R=λδλ=λAr1ϕd1D.(2.15)
It is instructive to reflect for a moment on this equation and the difficulties involved in designing high-resolution spectrographs for very large telescopes. The resolving power depends on the ratio of the collimator to telescope diameter, and inversely on the projected slit width. Suppose you want to build a spectrograph with a fixed resolving power, say R = 100,000, on a telescope with a large D. To do this, you either have to increase the collimator diameter (which is more expensive), or you have to use a smaller slit width, which results in loss of light. The latter defeats the purpose of using a large diameter telescope!
Table 2.1 summarizes the various combinations of telescope diameters and median seeing, and the required collimator diameter needed to achieve a resolving power of R = 100,000 using a dispersive power A = 1.7 × 10−3. One can see that as one designs larger diameter telescopes, the collimator diameter, and thus the overall size of your optical system, becomes larger. This can drive up the cost of your spectrograph dramatically and is the main reason that an R = 100,000 spectrograph may cost approximately one million Euros for a 3 m telescope, 10 million Euros for an 8 m class telescope, and 50 million Euros for a 30 m class telescope. Furthermore, a larger spectrograph means that it is more difficult to stabilize, and it will be more susceptible to instrumental shifts (see Chapter 4).
Table 2.1. Telescope (D) and Collimator (d1) Diameters for Various Image Sizes (ϕ) for R = 100,000
| D (m) | ϕ (arcsec) | d1 (cm) |
|---|---|---|
| 2 | 1.0 | 10 |
| 4 | 1.0 | 20 |
| 10 | 1.0 | 52 |
| 10 | 0.5 | 26 |
| 2 | 0.5 | 77 |
| 2 | 0.25 | 38 |
It is of interest to consider Equation (2.15) in the context of the diffraction limit of a telescope:
θ=1.22λD.(2.16)
Adaptive optics (AO) is the discipline where you correct the image for atmospheric distortions (see Principles of Adaptive Optics by Tyson 1987). The performance of an AO system is measured by how close the image comes to achieving the diffraction limit of the telescope. Although in practice this is rarely achieved, let’s imagine that we have the perfect AO system, one that indeed produces diffraction-limited images. In this case, the image quality is not determined by the atmospheric seeing, but rather by the diffraction limit of the telescope. If we substitute the diffraction limit for ϕ in Equation (2.15), we find that R∝d1. In other words, the resolving power depends only on the diameter of your collimator and is independent of the telescope diameter. In principle, an R = 1,000,000 spectrograph coupled to a telescope with the perfect AO system can be built with a collimator diameter of only 6 cm!
AO systems are often used to improve the image quality for imaging measurements, but the improvement that they provide for spectroscopic measurements is often overlooked. Even if an AO system is imperfect and does not achieve the diffraction limit, it can
improve the efficiency of your spectrograph by allowing more starlight to enter your slit or fiber.
allow you to build a spectrograph with smaller optical components. This decreases construction costs as well as enables you to better stabilize your spectrograph mechanically and thermally.
stabilize the image at your slit or fiber (see Chapter 12).
2.1.4 The Blaze Function
One rather annoying feature of spectral data taken with echelle gratings is the so-called “blaze function.” The blaze function results from the interference pattern of the single grooves of the grating. Each facet is a slit, so the interference pattern is described by a sinc function.
Recall from optical interference that a smaller slit results in a broader sinc function.2 So, finely grooved gratings will produce a much wider blaze function. For a grating with rulings of ≈1000grooves/mm like the ones employed decades ago, the groove spacing is small, so the blaze (sinc) function is rather broad and was hardly noticeable, or at least easily removed. However, for echelle gratings that have