Читать книгу The Doppler Method for the Detection of Exoplanets - Professor Artie Hatzes - Страница 21

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 the highest efficiency, the focal ratio of the collimator (f₁/d₁) should be the same as the telescope (f/D). In general, because of dispersion, the diameter of the camera is larger than that of the collimator so we can define an anamorphic magnification, r = d₁/d₂. We can also define a “focal ratio” of the camera as F₂ = f₂/d₁. The projected slit width at the detector is

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).