Several authors have done detailed studies on the theoretical uncertainties from RV measurements (Beatty & Gaudi 2015; Bouchy et al. 2001; Bottom et al. 2013). It is not our intention to go into a deep mathematical investigation into the uncertainties of RV measurements for two reasons. First, it is more important to have a general feel as to how a spectrograph or a star of given characteristics will influence your RV precision. These will be done through simple numerical simulations. Second, as we shall soon see, systematic instrumental errors and intrinsic stellar variability will introduce errors in your RV measurement that will completely overwhelm the theoretical errors due to photon statistics. You rarely get perfect results.
3.1 Instrumental Characteristics
The fundamental characteristics of a spectrograph that can influence the RV precision are
the wavelength coverage,
the resolving power of your spectrograph, and
the S/N of your data.
An increased wavelength coverage usually results in more spectral lines to measure your Doppler shift. All other factors constant (i.e., the properties of the star), the spectral resolution determines the recorded width of your spectral line and how easy it is to measure its centroid. The S/N naturally depends on the brightness of the star, the exposure time, and the observing conditions. If all these factors are equal, then the S/N also depends on the efficiency of the spectrograph. We therefore include it among the instrumental characteristics.
3.1.1 Wavelength Coverage
Every spectral line gives you a Doppler measurement of the star with a certain error. Use a second line and your measurement error should decrease by the square root of two. Indeed, a simple simulation using spectral lines that all have the same strength confirms that if you achieve an RV error of σ for a single line, the total RV error decreases by σRV = σ(N), where N is the number of spectral lines (left panel of Figure 3.1). So naively, if we have a wavelength coverage of Δλ (not to be confused with spectral resolution, δλ) then we expect the RV error to scale as
σRV∝(Δλ)−0.5.(3.1)
Figure 3.1. (Left) Simulations (points) showing the normalized RV error σRV as a function of the number of spectral lines, Nlines, used in the RV measurement. The fit (line) shows that σRV∝ Nlines−0.5. (Right) The normalized RV uncertainty as a function of wavelength coverage for real data using the spectrum of a Sun-like star. The line represents a fit with σRV∝ Nlines−0.55.
Reality is a bit more complicated. The right panel of Figure 3.1 shows the behavior for real RV measurements on a solar-type star taken with the Tautenburg Coudé Echelle spectrograph (TCES). The method employed was the iodine absorption cell method, which will be covered in more detail in Chapters 4 and 6. The RV was calculated using an increasingly larger wavelength region.
The value of σRV indeed decreases with wavelength coverage, Δλ, but for a short wavelength coverage, it is markedly worse than the predicted behavior. However, it quickly falls in line with the predicted behavior. Beyond a wavelength coverage of about 1000 Å, the measurement error is flat, that is, increasing the wavelength coverage results in no substantial reduction in the measurement error. A fit to the data results in σ∝(Δλ)−0.55, close to the theoretical expectation.
There are three reasons for these deviations with the Δλ−0.5 law in Figure 3.1. First, an increase in wavelength coverage is not always followed by an increase in the number of useful spectral lines for a Doppler measurement. Depending on the effective temperature of the star, there can be spectral regions where the number density of lines is sparse. This is especially true of early-type stars. Furthermore, not all spectral lines have the same strength, and as we shall shortly see, this will affect the RV precision.
Second, once your wavelength coverage extends beyond about 6000 Å, telluric lines start to become prevalent (see Chapter 12). These features are not tied to the Doppler motion of the star, and they only serve to decrease the RV error precision.
Finally, as we shall soon see, the molecular iodine lines that were used to compute the relative RV shifts in Figure 3.1 become weak beyond about 5800 Å, and this results in a degradation of the RV precision. This, along with the presence of telluric lines, largely explains the flattening of the RV precision.
In practice, when performing precise RV measurements on a star, it is useful to explore how the measurement error behaves as one uses various spectral regions for the Doppler calculation. It is best to avoid those regions where there is no noticeable improvement in the error, or at least weight these accordingly. However, when performing RV measurements for solar-type stars in the optical region, we can expect σRV∝(Δλ)−0.5.
3.1.2 Signal-to-noise Ratio
In the absence of other sources of noise, the error in your spectral data, σp, is determined purely by the number of photons you detect, Np, and according to photon statistics, this is simply σp=Np. The photon noise is the theoretical limit to the RV precision you can achieve for a given exposure. If you do this, then you have a perfect spectrograph and have done an excellent job of eliminating systematic errors, or at least to a level where they are not important.
Figure 3.2 shows the result of simulations where increasing levels of noise were added to a spectral line and the relative Doppler shift was calculated with respect to a noise-free line. The RV uncertainty follows the simple power law
σ∝(S/N)−1,(3.2)
Figure 3.2. Simulations (points) showing the normalized RV measurement error as a function of signal-to-noise ratio, S/N. The error has been normalized to unity for S/N = 1000. The behavior shows that σRV∝(S/N)−1.
At face value, it would seem that simply increasing the S/N is a quick way of achieving a lower RV uncertainty. However, as with most things in life, an increase in quality comes at a price, and in this case, it is paid with exposure time. Suppose you achieve an RV error of σ = 10 m s−1 for S/N = 50 using a 15 minute exposure. Decreasing this error to 1 m s−1 requires a 10-fold increase in the exposure time to about 2.5 hr. This would not be an effective strategy if you want to survey a large number of stars, or if you have limited telescope resources. For bright stars, it is wise to limit your the S/N ratio to no higher than about 200. Above this, there is relatively little gain in the RV precision. Furthermore, using a higher S/N to achieve RV precisions of 1 m s−1 is probably an inefficient use of telescope time because other factors, such as systematic errors or intrinsic stellar variability, will determine your precision before you hit the photon noise limit.
3.1.3 Resolving Power
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