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

The RV measurements have a scatter of 6.9 km s⁻¹, typical of the RV precision of that era.

      Figure 1.2. Orbital motion of HD 36954 (curve) calculated from radial velocity measurements taken from 1932–1935 using the 36 inch refractor at Lick Observatory (Neubauer 1936). The scatter about the orbital solution is 6.9 km s⁻¹.

      The first astronomer to recognize that stellar RV measurements could be used to detect exoplanets was Otto Struve. He was a Russian-born astronomer who did most of his astronomical work in the United States. Struve served as director of the Yerkes, McDonald, and National Radio Astronomy Observatories. As a director, he could recognize talent, having hired Subrahmanyan Chandrasekhar and Gerhard Herzberg, two future Nobel Prize winners.

      Struve was also a visionary. His remarkable paper “Proposal for a Project of High-precision Stellar Radial Velocity Work” (Struve 1952) was the first to propose using Doppler measurements to search for exoplanets. The discovery of 51 Peg b in 1995—a giant planet in a 4.2 day orbit—was foreseen by Struve. In his paper, p. 200, he argued that “we know that stellar companions can exist at very small distances. It is not unreasonable that a planet might exist at a distance of 1/50 of an astronomical unit. Such short-period planets could be detected by precise radial velocity measurements.” His predictive powers did not stop there. He goes on to say that “there would, of course, also be eclipses … and the loss of light in stellar magnitudes is about 0.02.” Struve not only foresaw the possibility of short-period Jupiter-mass planets, but the use of the transit method to characterize the density. His proposal did not result in the “powerful” spectrograph he advocated, which only shows that science has its own “prophets” who are often ignored. The discovery of exoplanets still had to wait another half century.

1.3 Toward Precise Stellar Radial Velocity Measurements

      With 150 years of stellar RV measurements and even proposals from the mid-20th century to build spectrographs capable of such precise measurements, why did it take until the end of the 20th century to discover the first exoplanets? The short answer: a lack of precision.

      The Doppler shift of a star due to the presence of planetary companions is small. We can use Kepler’s third law to get an estimate of the RV precision needed to detect the reflex motion of star due to the presence of a planetary companion:

      P2=4π2a3G(Ms+Mp),(1.1)

      where Ms is the mass of the star, Mp is the mass of the planet, P the orbital period, and a the semimajor axis.

      For planets, we are in the regime where Ms ≫ Mp. If we assume circular orbits and the fact that Mp × ap = Ms × as, where as and ap are the semimajor axes of the star and planet, respectively, it is trivial to derive

V[ms−1]=28.4P1yr−1/3MpsiniMJupMsM⊙−2/3,(1.2)

      where i is the inclination of the orbital axis to the line of sight. Remember, the Doppler method only measures velocities along the line of sight.

      The reflex motion of a 1 M_⊙ star due to various planets at different orbital radii calculated with Equation (1.2) is shown Figure 1.3 and a Jupiter analog (1 M_Jup orbiting at a distance of 5.2 au) will induce a 11.2 m s⁻¹ reflex motion of a Sun-like host star with an orbital period of 12 years. To detect such a planet, you would need an RV measurement precision of at least 10 m s⁻¹, which would have to be maintained for over a decade.

      Figure 1.3. The amplitude of the barycentric radial velocity variations for a 1 solar mass star orbited by an Earth, Neptune, Saturn, or Jupiter at various orbital distances.

      Moving this planet to the semimajor axis of a “Struve planet” (0.02 au) would result in a reflex motion 10 times higher. This eases your required measurement precision to a more comfortable 100 m s⁻¹ maintained over several days. If you are bold and you want to detect the first Earth analog (planet at 1 au), you would need a more challenging measurement precision of better than 10 cm s⁻¹. For a “lava” Earth-mass planet orbiting at 0.05 au from the star, the stellar Doppler amplitude would be a more reasonable 1 m s⁻¹, a precision, as we will soon see, that is achieved by modern instruments. This figure shows that to detect planets with the RV method, one needs exquisite precision coupled with long-term stability.

      The early exoplanet surveys had a search strategy that was driven by the only example of a planetary system—our own. We thus expected giant planets to all lie at approximately 5 au from the star. The meant we required an instrument capable of achieving an RV measurement precision of at least 10 m s⁻¹ and with a high premium on long-term stability.

      Figure 1.4 demonstrates just why it took 150 years to discover exoplanets with the Doppler method. It shows the approximate RV precision as a function of time. The decrease in measurement error follows a power-law fit (solid line). In the 1960s, we could measure a star moving at the speed of an SR-71 military aircraft, or about a km s⁻¹. Currently, we can measure a stellar RV under 1 m s⁻¹, or the speed of a very leisure walk or a rapidly crawling baby. If the power law holds, we should achieve an RV precision of 10 cm s⁻¹ by the mid-2020s. The “magic” precision of 10 m s⁻¹, shown by horizontal dashed line, was only achieved in the mid-1980s. Coincidentally, this was about the time the first exoplanets were discovered.

      Figure 1.4. The evolution of the radial velocity measurement error as a function of time. The horizontal line marks the reflex motion of a solar mass star with a Jupiter analog.

1.4 The Early Hints of Exoplanets

      Although the discovery of 51 Peg b (Mayor & Queloz 1995) is considered as the discovery of the first exoplanet around a Sun-like star, there were hints of discoveries before 1995¹. Campbell et al. (1988; hereafter CWY) monitored the brighter component of the spectroscopic binary γ Cep using a hydrogen fluoride absorption cell (see Chapter 4). The top panel of Figure 1.5 shows the RV measurements of γ Cep A from CWY. The long-term trend due to the binary motion is obvious, but one can see extra “wiggles.” Removing the linear trend shows clear variations, due to a possible planetary companion. CWY commented that these variations represented a possible third body in the system with a period of ≈3 yr that might be planetary in nature. Unfortunately, Walker et al. (1992) later attributed these variations to rotational modulation, largely because theoretical work could not produce giant planets in short period orbits (G. A. H. W. Walker, 2013, private communication). Hatzes et al. (2003) later demonstrated that these residual RV variations were indeed due to a 1.7 MJup giant planet in a 2.48 yr orbit.

      Figure 1.5. (Top) Early RV measurements of γ Cep A made in the 1980s using an H–F absorption cell (Campbell et al. 1988). The trend is due to orbital motion of the binary companion. (Bottom) RV measurements after removing the long-term trend. The RV motion due to the planetary companion can easily be seen (red curve: orbital solution