Читать книгу The Doppler Method for the Detection of Exoplanets - Professor Artie Hatzes - Страница 44

Spectral resolution is the spectrograph characteristic that largely determines the basic RV precision you can achieve. Clearly, if you have more sampling points across a stellar line profile, the more accurately you can determine the centroid and thus Doppler shift of the line. Most spectrographs are designed to satisfy the Nyquist criterion, i.e., two detector pixels cover the spectral resolution, δλ. So, if a spectrograph has a resolving power of R = 100,000, this will produce a dispersion of 0.055 Å per pixel, or a velocity resolution of 3 km s⁻¹ at a wavelength of 5500 Å.

Table 3.1 lists the expected velocity resolution for spectrographs over a wide range of resolving powers, R=δλ/λ. Also listed is the expected shift in millimeter at the detector assuming CCD pixels of 15μm size a wavelength of 5000 Å and a Doppler shift of 1 m s⁻¹. If you can measure the position of a spectral line to a fixed fraction of a CCD pixel, it make sense to have a higher resolving power. Clearly, one should avoid low-resolution spectrographs for precision RV measurements. For example, if you want to get an RV precision of 1 m s⁻¹ with an R = 1000 spectrograph, this would result in a shift at the detector of only 6.7 × 10⁻⁶ pixels, or equivalently only 10⁻⁴ μm (= 1 Å!).

Table 3.1. The Doppler Shift of 1 m s⁻¹ for Different Resolving Powers

Resolving Power	Dispersion (Å/pixel)	Velocity Resolution (m s−1 pixel−1)	Shift in Pixels	Shift at Detector (mm)
1000	2.5	150,000	6.7 × 10−6	10−7
5000	0.5	30,000	3.3 × 10−5	5.0 × 10−7
10,000	0.25	15,000	6.7 × 10−5	1.0 × 10−6
25,000	0.10	6000	1.7 × 10−4	2.5 × 10−6
50,000	0.05	3000	3.3 × 10−4	5.0 × 10−6
100,000	0.025	1500	6.7 × 10−4	1.0 × 10−5
200,000	0.0125	750	1.4 × 10−3	2.0 × 10−5
500,000	0.005	300	3.3 × 10−3	5.0 × 10−5

Several works have investigated the dependence of the RV uncertainty on spectral resolution, or resolving power. Bouchy et al. (2001) found that σRV∝R−1. This result is consistent with simple simulations using a single spectral line generated at different resolutions of spectral lines generated using model atmospheres (Figure 3.3). Bottom et al. (2013) found that the RV uncertainty behaved as σRV∝R−1.2. Earlier work by Hatzes & Cochran (1992) reported a steeper variation of the uncertainty with resolving power, σRV∝R−1.5, which is consistent with what Beatty & Gaudi (2015) reported. Why the discrepancies?

Figure 3.3. (Red circles) Simulations of the normalized RV measurement error (σRV as a function of resolving power R, normalized for the error at R = 120,000). These simulations used synthetic spectral lines with a rotational velocity sin i = 1 km s⁻¹. The red dashed line shows that σ∝R−1. (Blue squares) Simulations of the RV measurement error as a function of R, but this time using a δ-function as the “spectral line,” i.e., a feature that is unresolved at R = 100,000. The broadening of this feature is dictated by the instrumental profile. In this case, σ∝R−1.8.

One possibility is how the Doppler shifts were calculated. The simulations of Hatzes & Cochran (1992) mimicked the data that were taken with an iodine gas absorption cell. The method will be discussed at length in the next chapter, but basically Doppler shifts are calculated with respect to molecular iodine absorption lines which are unresolved even at a resolving power R = 100,000. The shape of these lines are thus dominated by the instrumental profile. Figure 3.4 summarizes the different power dependencies of the RV uncertainty on resolving power found by various investigations. Although Beatty & Gaudi (2015) reported an R−1.5 dependence in the RV uncertainty, a close inspection of their figures shows that the dependence follows the red line shown in the figure.

Figure 3.4. The behavior of the RV uncertainty as a function of spectral resolving power from various studies: BG15 (red line): Beatty & Gaudi (2015), BPQ01: Bouchy et al. (2001); B13: Bottom et al. (2013); HC93: Hatzes & Cochran (1992).

Interestingly, when one computes the relative shift of an unresolved profile (δ-function) that is convolved with a Gaussian instrumental profile, the RV uncertainty has a much steeper dependence on R, namely σRV∝R−1.8 (Figure 3.3). This is because the uncertainty depends on the line depth (see below), and for lower resolution, this decreases more rapidly for lower resolution than for resolved lines.

From the various studies, we expect the RV uncertainty to follow a power law, σRV∝R−α with α = 1–1.5. For a good approximation of the dependence of σRV on R, it is sufficient to take the average of the extreme values for α, namely σRV∝R−1.2. In subsequent discussions, we will adopt this as the dependence of the RV uncertainty with spectral resolving power.

If you were building a high RV precision spectrograph, you might naively assume that it should have the highest resolving power possible. However, there are trade-offs to consider. First, high-resolution spectrographs are much more expensive to build. Everything is bigger: collimator optics, gratings, cameras, etc. Bigger implies more expensive. If you are designing a spectrograph, budget constraints may be the factor that dictates resolving power.

Second, at higher resolving powers, you are dispersing the light over more CCD pixels. This will decrease the count rate and thus S/N of your spectrum for a given star and exposure time. Double your resolving power, and you have just decreased the S/N and thus RV precision by a factor of 1.4.

Finally, at higher resolving power for a fixed-sized detector, you have a decreased wavelength coverage, and this scales as Δλ∝R−1. If you have a CCD detector with a certain size and you install it on a spectrograph with twice the resolving power, you will have decreased the wavelength coverage by a factor of 2. The RV uncertainty for a fixed S/N has just increased by the square root of 2, due to the lost wavelength coverage. You can of course try to compensate for this by using more detectors in a mosaic configuration, but again, that comes with increased cost for the spectrograph.

It is easy to calculate how the RV changes if the same detector were moved to a spectrograph with higher resolving power, i.e., for a “fixed-size detector.” In this case, what you gain in precision from the increased resolving power is partially offset by the loss in precision due to the smaller wavelength coverage. The uncertainty due to resolving power scales as σR∝R−α. The uncertainty with wavelength coverage, Δλ, scales as σΔλ ∝(Δλ)−1/2. The wavelength coverage scales as R−1, so substituting into the previous expression, one gets σΔλ∝R1/2. So, for the case of the fixed-sized detector, the total uncertainty is given by the product of the two, namely σTotal∝R1/2−α.

Figure 3.5 shows the actual RV error determined from solar spectra (day sky observations) using an iodine absorption cell at resolving powers of R = 2300, 15,000, and 200,000. These were taken with the same detector and spectrograph, but different gratings to provide different resolutions. The solid red curve is the function σ∝R−1, while the dotted curve is σ∝R−1/2. At first glance this seems to support σ∝R−1. Therefore, (1/2−α)=−1, which implies that α = 3/2 as opposed to unity. Keep in mind the caveat that these data were taken with the iodine absorption cell.

Figure 3.5. (Points) The radial velocity error taken with a spectrograph at different resolving powers. This is the actual data taken of the day sky all with the same S/N values. The solid red line shows a σ∝R−1 fit. The dashed black line shows a σ∝R−1/2 fit. The detector size is fixed for all data, thus the wavelength coverage is increasing with decreasing resolving power.