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News of the discovery of a potentially habitable planet around the star 40 Eridani A (about 16 light-years away) is interesting for reasons including a 1991 letter described below in Sky & Telescope's recent article Super-Earth Discovered in (Fictional) Vulcan System, part of which is shown below.

Question: Is there any way to estimate a low mass cut-off for the Dharma survey for this particular star? Could there also be a 1 or 2 Earth mass planet there as well, within a habitable zone, as-yet undetected? Or does data there exclude this possibility?

The paper about the recent discovery: https://arxiv.org/abs/1807.07098

Almost three decades ago, Gene Roddenberry (producer of the Star Trek universe) wrote a letter to Sky & Telescope, along with Harvard-Smithsonian Center for Astrophysics astronomers Sallie Baliunas, Robert Donahue, and George Nassiopoulos. In their Letter to the Editor, they argued that 40 Eridani A — an orange-ish star 16 light-years away — would make the ideal home for Vulcan, the home planet of Science Officer Mr. Spock.

Now, a new discovery puts a little more science into that science fiction assertion.

A Long-ago Letter

In the July 1991 issue, the three astronomers and one movie-maker made the case for what star should be considered Vulcan's home:

The star around which Vulcan orbits was never identified in the original series or in any of the feature films based on it and so has never been officially established. But two candidates have been suggested in related literature.

Two Star Trek books named the star 40 Eridani A as Vulcan's sun, while another publication named Epsilon Eridani instead. However, Roddenberry and the astronomers made an argument for 40 Eridani A:

We prefer the identification of 40 Eridani as Vulcan's Sun because of what we have learned about both stars at Mount Wilson. . . . The HK [Project] observations suggest that 40 Eridani is 4 billion years old, about the same age as the Sun. In contrast Epsilon Eridani is barely 1 billion years old.

Based on the history of life on Earth, life on any planet around Epsilon Eridani would not have had time to evolve beyond the level of bacteria. On the other hand, an intelligent civilization could have evolved over the aeons on a planet circling 40 Eridani. So the latter is the more likely Vulcan sun. . . . Presumably Vulcan orbits the primary star, an orange main-sequence dwarf of spectral type K1. . . . Two companion stars — a 9th magnitude white dwarf and an 11th magnitude red dwarf — orbit each other about 400 astronomical units from the primary. They would gleam brilliantly in the Vulcan sky.

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In general, finding the low mass cutoff of a particular RV survey is normally quite complicated as there are a lot of selection effects on many different variables (such as planet period, planet eccentricity, host star type, signal-to-noise, stellar activity level etc) that determine whether you can recover a planet signal. In the case of this paper, this would be particularly true as they are combining datasets from several instruments, each with different coverage, sensitivity and systematic noise properties. The way this is normally handled is through a "bootstrap" technique where you inject planet signals of known parameters and see what your detection code finds and study the recovery percentage as a function of the varied parameters.

Using the online Habitable Zone calculator for a Teff=5072 K, Stellar luminosity=0.47 * L_Sun (luminosity of the Sun) matching the host star, produces 0.678 AU as the inner edge of a conservative habitable zone. If we use this, the star's mass from the paper (0.78 M_Sun) and the 1 M_Earth mass you would like to find, then making use of the formulas given on the Exoplanet Archive website, gives us a period of 230.9 days. Plugging this into an equation for the radial velocity semi-amplitude (also on the Exoplanet Archive), produces an amplitude of about 12 cm/sec. This is assuming an edge-on orbit, so the sin i term from the inclination is 1; given the lack of detected transits in the paper, the inclination is likely lower than 90 degrees, which would lower the RV amplitude further.

This RV amplitude is far too small to be detected with current instruments. The best instrument we have right now is HARPS on the ESO 3.6m telescope, whose data is included in the paper. If I am understanding their modeling correctly, they get error bars of 181 cm/s on the HARPS data for this star in their analysis, 15x the expected signal. Also note that the calculated RV signal is somewhat best case (unless it is more massive); decreasing the inclination from edge-on (likely) and moving the planet further out in the habitable zone from the inner edge I calculated (also likely) will both decrease the RV amplitude further. Even for future instruments such as ESO's ESPRESSO which will go the 8.4m VLT and reach 10 cm/s precision, this would be a challenging measurement.

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