Читать книгу The Doppler Method for the Detection of Exoplanets - Professor Artie Hatzes - Страница 43

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, N_p, 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.