The Doppler Method for the Detection of Exoplanets. Professor Artie Hatzes

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 T_eff, 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 T_eff 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 (T_eff = 8100 K) rotating with a sin i = 90 km s⁻¹, which is a challenge for RV work. To complicate matters, it is a δ-Scuti star, and the stellar oscillations add an additional noise component (see Chapter 10).

      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 and the measurements have an rms scatter of 245 m s⁻¹ about the orbital curve. By taking many measurements, one can take binned averages to reduce the scatter to 23 m s⁻¹ (squares in Figure 3.13).

      Figure 3.13. RV measurements of WASP-33 phased to the orbital period of the transiting planet (Lehmann et al. 2015). WASP-33 is an A5 star with T_eff = 8100 K that exhibits δ-Scuti pulsations. The star has a mass of 1.5 M⊙ and rotates with sin i = 90 km s⁻¹. The individual measurements have an rms scatter of 245 m s⁻¹ while the phase-binned averages (blue squares) have an rms scatter of 23 m s⁻¹. The red curve is the orbital solution.

      Let’s see if the measurement errors are consistent with the predictions from our scaling relationships. The RV data for WASP-33 were taken with the TCES. This spectrograph has a resolving power of R≈60,000 and can achieve an RV precision of 3 m s⁻¹ on a slowly rotating solar-type star with data having S/N of 100. Figure 3.7 and Equation (3.3) indicate that at this resolving power, a star rotating at 90 km s⁻¹ should have an RV uncertainty a factor of 30 times larger than for our slowly rotating “reference” star stemming just from the larger rotational velocity. An A5 type star has approximately 6.5 times fewer spectral lines (Figure 3.10) than a solar-type star. This gives another factor of 2.5 increase in the measurement uncertainty. The RV measurements of WASP-33 have roughly the same S/N as our reference star, so we expect an RV error of ≈230 m s⁻¹, comparable to the actual rms scatter. This means that the RV measurements for WASP-33 have an error fairly close to the expected uncertainty due to photon statistics.

      A way to circumvent the low RV precision for early-type stars is to use evolved giant stars as proxies for investigating the frequency of planets around stars more massive than the Sun. As an intermediate-mass star (M = 1.5–3 M⊙) evolves off the main sequence, it expands and becomes cooler, thus it shows more spectral lines. More importantly, its rotation rate slows. A 2M⊙ K giant star has an effective temperature Teff≈4000K and rotates at a few km s⁻¹, thus it is highly amenable to precise RV measurements. This fact has inspired a large number of surveys for planets around evolved intermediate-mass stars (Setiawan et al. 2003; Reffert et al. 2006; Johnson et al. 2007; Döllinger et al. 2007; Sato et al. 2008; Niedzielski et al. 2009; Han et al. 2010; Wittenmyer et al. 2011; Quirrenbach et al.