Читать книгу The Doppler Method for the Detection of Exoplanets - Professor Artie Hatzes - Страница 50

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. 2015).

The K0 III star β Gem was shown to host a giant planet with a mass of 3M_Jup in a 589 day orbit (Hatzes et al. 2006; Reffert et al. 2006). Figure 3.14 shows RV measurements taken from the McDonald and Tautenburg Observatories phased to the orbital period of the planet. The star has a mass of 1.9M⊙ (Hatzes & Zechmeister 2007; Hatzes et al. 2012). Because the star is cool and slowly rotating, the rms scatter about the orbit is 13 m s⁻¹, or a factor of 20 less than for WASP-33, a main-sequence star of comparable mass.

Figure 3.14. RV measurements of β Gem phased to the 585 day orbital period of the planet (Hatzes et al. 2006). This star is a K0 III giant star with a mass of 1.9 M⊙. The red curve is the orbital solution. The rms scatter about the orbit is 13 m s⁻¹.

However K giant stars may avoid one problem (better RV measurement precision), but it is replaced by others. First, determining the mass of the host star is more problematic. On the main sequence, there is a tight relationship between spectral type and stellar mass. So if you measure the effective temperature of the star, you have a good handle on the stellar mass. For K giant stars, one must rely on evolutionary tracks, so the stellar mass is model dependent. Unfortunately, the main sequence covers a wide range of masses (≈1−4M⊙) with evolutionary tracks that converge to the same region of the color–magnitude diagram. Placing the star on the correct track requires an accurate effective temperature, luminosity, metal abundance, and of course, good stellar models.

The second problem is that K giant stars show relatively high-amplitude RV variations due to stellar oscillations, and these create excess “noise,” which can hinder the detection of low-mass companions (see Chapter 10). However, as the saying goes, “one person’s noise is another person’s signal,” and we can exploit the second problem (more “noise” in the RV measurement) to address the first (poorly known stellar mass). One can use the properties of these stellar oscillations to derive the mass of the host star as was done in the case of β Gem (Hatzes & Zechmeister 2007; Hatzes et al. 2012). In particular, the Kepler space mission has provided us with a wealth of data on stellar oscillations in K giant stars, and these can provide a good sample of giant stars with well-determined masses that are suitable for planet searches (Hrudková et al. 2017) In Chapter 10, we will return to the subject of stellar oscillations as a noise component to RV measurements.