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

      Latham et al. (1989) found a possible giant planet with a minimum mass of 11 M_Jup orbiting HD 114762 with an orbital period of 83.8 days (Figure 1.6). With an eccentricity of e = 0.38, HD 114762 b was the prototype of the so-called massive eccentric planets. These are massive planets (M ≈ 10 MJup) in eccentric (e ≳ 0.3) orbits. The RV measurements were made with “traditional” methods, i.e., without simultaneous wavelength calibration (see Chapter 4), so the measurement error is σ ≈ 400 m s⁻¹. This is comparable to the RV amplitude of ≈600 m s⁻¹ and demonstrates that with sufficient measurements, one can detect a planet with RV amplitude comparable to the measurement error. However, the true nature of the companion is unknown until the orbital inclination is measured. Preliminary results from the GAIA astrometric space mission indicate an orbital inclination of 6.2 degrees which yields a companion mass of 107−2730 MJup, or in the M dwarf star regime (Kiefer 2019)².

      Figure 1.6. The RV variations of HD 114762 due to a planetary candidate companion with an orbital period of 83.8 days and a minimum mass of 11 MJup (Latham et al. 1989). The rms scatter about the orbital solution (red curve) is 412 m s⁻¹ and was typical for high-quality RV measurements before the use of simultaneous wavelength calibration.

      Inspired by the early work of Walker et al. (1989), who found RV variations in a sample of K giant stars, Hatzes & Cochran (1993) monitored several of these stars and found long-period RV variations in α Tau, α Boo, and β Gem. They hypothesized that these could be due to giant-planet companions. Indeed, it was later shown that early RV measurements for β Gem were due to a giant planet with M = 2.7 MJup in a 590 day orbit (Hatzes et al. 2006; Reffert et al. 2006).

1.5 The 51 Peg Revolution

      The discovery of 51 Peg b (Mayor & Queloz 1995) clearly marked an explosion of the field³. This giant planet (M ≈ 0.5 MJup) in a 4.2 day orbit shocked astronomers, except maybe for the ghost of Otto Struve. It also demonstrated that RV surveys were probing the wrong parameter space (orbital distances of 5 au rather than 0.05 au)—the dangers of planning surveys based on one example, our solar system. The RV amplitude of ≈50 m s⁻¹ (Figure 1.7) clearly benefited from the increased precision of Doppler measurements. Figure 1.8 shows the discovery rate of exoplanets found using the Doppler method. The sharp increase after the discovery of 51 Peg is for three reasons. First, once astronomers realized that giant planets could occur in short period orbits, they changed their observing strategies so that these short-period planets could be discovered rather quickly. Second, using the Doppler method to detect exoplanets became quite fashionable, with many groups “jumping on the bandwagon.” Prior to 1995, there were only a handful of groups using precise stellar RVs to search for exoplanets. Now, the number of such groups is in the dozens. Currently, approximately a hundred exoplanets per year are discovered with the Doppler method. Note that RV measurements also play a key role in the confirmation and mass determination of discoveries made by the transit method.

      Figure 1.7. The discovery of 51 Peg b. The RV variations have an amplitude of ≈50 m s⁻¹ and are phased to the orbital period of 4.2 days. The typical measurement error is ≈15 m s⁻¹. (Adapted by permission from Macmillan Publishers Ltd: Mayor & Queloz 1995.)

      Figure 1.8. The rate of exoplanet discovered planets using the Doppler method (transit discoveries are not included; data from http://www.exoplanet.eu).

      Finally, the increased detection rate of exoplanets using Doppler measurement also benefited from the dramatic increase in RV precision over the past 30 years. The top panel of Figure 1.9 shows precise RV measurements of γ Cep (Walker et al. 1992). These show a scatter of about 15 m s⁻¹—the measurement error they could achieve in the mid-1980s. The lower panel shows modern RV measurements of Proxima Centauri showing the variations of the Earth-mass companion in an 11.8 day orbit (Anglada-Escudé et al. 2016). These have a scatter of a mere 1 m s⁻¹. Note that the scale of the y-axis in this lower panel is the size of the error bar in the top panel. An important aspect of this book is to show how this dramatic increase was achieved.

      Figure 1.9. (Top) RV measurements of γ Cep phased to the orbital motion using Doppler measurements taken in the mid-1980s. (Bottom) RV measurements of Proxima Centauri phased to the orbital motion as measured in 2016. The box size for this panel is the same as the measurement error for the earlier measurements.

      The parameter space in the mass versus semimajor axis of exoplanets discovered with the Doppler method is shown in Figure 1.10. These are only planets discovered through RV measurements and not those from transit discoveries, although Doppler measurements were important for confirming the nature of these discoveries and measuring the companion mass.

      Figure 1.10. The detection parameter space for planets found using the Doppler method. This does not include transit discoveries. The method only measures the mass multiplied by the sine of the orbital inclination, i (M sin i).

1.6 The Doppler Method

      This book is written primarily for astronomers who wish to use the Doppler method for the detection of exoplanets. Given the ubiquity of Doppler measurements in astronomy, its utility is not restricted to this narrow field. Measurement precision is only one aspect of the process.

      The redshift z of an object is defined as

      z=λmeasλemit−1,(1.3)

      where λmeas is the measured wavelength of the spectral feature and λemit is the emitted wavelength.

      We are interested in the nonrelativistic Doppler shift, and if the change in wavelength is Δλ = λmeas−λemit, then the Doppler shift can be converted to an RV by

      So, in principle, the method is quite simple. You measure a position of a spectral feature, compare it to its rest wavelength, and convert that to a Doppler shift in velocity using Equation (1.4). Once you have sufficient velocity measurements, you can fit a Keplerian orbit and derive the companion mass. Simple in theory, but as we shall soon see, challenging in practice.

      The Doppler method we will deal with in this book is strictly a relative, as opposed to absolute, velocity measurement of stars. To detect companions to a star, we