a more rapid increase in the width of the line. It is more difficult to determine the centroid of a broad line as opposed to a narrow line. Whatever gain in precision is achieved by a slightly deeper line is more than offset by the larger line width. In this case, a spectral line with an EW of 100 mÅ yields the same Doppler uncertainty as a line twice as strong. Finally, for the strongest lines, the increase in line width dominates, and the RV uncertainty actually increases with line strength. So, the largest Doppler information is found in strong, yet unsaturated, spectral lines.
3.2.3 Number Density of Spectral Lines
The Doppler precision not only depends on the wavelength of your spectrograph, but also on the number density of stellar absorption lines. The latter of course depends on the effective temperature of the star—hot stars have much fewer absorption features.
The left panel of Figure 3.10 shows the approximate number of strong spectral lines in the wavelength range 4000–7500 Å1 as a function of the effective temperature of the star. This is for main-sequence stars in the effective temperature range Teff = 3500–10,000 K. Here we define a “strong” line has having a depth deeper than 50% of the continuum value.
Figure 3.10. (Left) The number of strong lines (50% of the continuum) in the wavelength range 4000–7500 Å as a function of effective temperature, Teff, of the host star. The drop-off at lower temperatures is due to line blending at bluer wavelengths. (Right) The scale factor, F, in the RV uncertainty due to the line density as a function of Teff referenced to σ = 1 m s−1 at Teff = 5000 K. Over the temperature range 5000–10,000 K, this can be well fit by F=0.16e1.79(T/5000).
The number of spectral lines increases sharply as the effective temperature decreases, but surprisingly, this flattens out at cooler temperatures. The reason for this is that for wavelengths less than about 5000 Å, cool stars simply have too many spectral lines. Line blending suppresses the continuum, causing even strong lines to have a relatively small depth. The line blending also results in few clean, isolated lines which provide the higher Doppler information.
The right panel of Figure 3.10 shows the natural logarithm of the scaling factor for the RV uncertainty, F, as a function of effective temperature. Beyond a temperature of 5000 K, this follows a linear trend, so the scale factor for Teff<5000K can be well fit by the expression
F=0.16e1.79(Teff/5000).(3.6)
Therefore, a main-sequence star with Teff = 8000 K will have an RV uncertainty 2.8 times higher than a main-sequence star with Teff = 5000 K just from the decreased line density (same S/N and stellar rotational velocity).
3.3 RV Precision across Spectral Types
We can now put together all we have learned to get a rough estimate of the RV error as a function of Teff, or of the spectral type for main-sequence stars. Using the mean rotational velocity of stars (Table 3.2) and the mean density of lines as a function of Teff results in Figure 3.11. The horizontal dashed line marks an RV precision of 10 m s−1, the nominal value if you want to detect a Jupiter analog around a solar-type star.
Figure 3.11. The expected radial velocity error as a function of spectral type. This was created using the mean rotational velocity and approximate line density for a star in each spectral type. The horizontal line marks the nominal precision of 10 m s−1 needed to detect a Jovian-like exoplanet.
Early RV surveys for planets strove for an initial precision of approximately 10 m s−1, the nominal precision to detect Jupiter analogs. By this criterion, you should not be able to detect planets around stars of early spectral types. Indeed, up until the mid-2000s, the earliest spectral type for which a planet had been detected was about F6. This lack of precision for early stars factored into the biases in the early surveys—investigators simply avoided stars with spectral types earlier than mid-F. It was only in the mid-2000s that RV surveys began to survey more early-type stars (Galland et al. 2005; Hartmann & Hatzes 2015).
Figure 3.11 largely explains the distribution of planet discoveries as a function of spectral type (Figure 3.12). RV surveys largely ignored stars with spectral early than mid-F (Teff≈6000K) due to the poor RV precision. Stars later than about K5 (Teff<4000K) were simply too faint. The early RV surveys were largely performed on 2–3 m class telescopes (Cochran & Hatzes 1993; Butler & Marcy 1997; Queloz et al. 1998), so one could not get good S/N ratios for observations on very cool stars.
Figure 3.12. The distribution of planet detections as a function of the effective temperature of the host star. The distribution roughly coincides with the green shaded region shown in Figure 3.11.
The distribution of RV discoveries also highlights the bias of the technique when it comes to the mass of the host star. On the main sequence, there is a one-to-one mapping between effective temperature and stellar mass. About two-thirds of the host stars of RV-detected planets have effective temperatures in the range 4500–6500 K, and this translates into the narrow mass range of M = 0.7–1.2 M⊙ for the mass of the host star.
We can now put together a grand scaling relationship for the expected RV precision that combines the S/N, the spectral resolving power R, the effective temperature of the star T, and the projected stellar rotational velocity V:
σ[m/s]∝Δλ−1/2S/N−1R−1.2f(V)(0.16e1.79(T/5000)).(3.7)
The function f(V) is given by Equation (3.3).
3.3.1 Radial Velocities of High-mass Stars
Although early-type stars are now well suited for precise RV measurements, that does not mean they are useless for exoplanet studies. Low precision can be compensated by taking more measurements. This is demonstrated by the case of WASP-33. This star hosts a transiting planet in a 1.2 day orbit (Christian et al. 2006; Collier Cameron et al. 2010). It is an A5 main-sequence star (Teff = 8100 K) rotating with a
Figure 3.13 shows RV measurements of WASP-33 phased to the orbital period (Lehmann et al. 2015). These measurements have been filtered for the δ-Scuti pulsations. One can clearly see that the orbital variation
