In August (2016), astronomers at the European Southern Observatory in Chile announced the discovery of an Earth-sized exoplanet orbiting the red dwarf Proxima Centauri, the nearest star to our Sun (Ref. 1, 2). The planet, called Proxima Centauri b, was detected by the radial velocity method: the orbiting planet causes the star to execute a much smaller orbit of its own (as required by momentum conservation), and light from the star is Doppler-shifted due to its radial motion relative to Earth. The discovery of Proxima Centauri b is exciting because its orbit lies within the “Goldilocks,” or temperate, zone of the star, where the planet’s surface temperature would allow water to exist in liquid form – a likely prerequisite for life. Red dwarfs are the most common stars in our galaxy, so if this particular red dwarf harbors a life-supporting planet, it is likely that the Milky Way, which contains over 100 billion stars, is brimming with life.
In this post, we will use the data reported in Reference 1, plus the strategy presented in Section 8.12 of PPE, to calculate the mass and orbital radius of the exoplanet Proxima Centauri b. Our goal is to help students share the excitement of this new discovery through their understanding of introductory mechanics.
A. Calculating the exoplanet’s mass and orbital radius
In the following, we will assume , where and are the masses of the planet and star, respectively. We will also assume that their orbits are circular, which is consistent with observations, and begin by treating the simplest case where the orbits are viewed edge-on from Earth. (Later, we’ll relax this restriction.) Radial velocity measurements of Proxima Centauri collected over the past 16 years are compiled in Figure 2. Careful analysis indicates that the period of the star’s orbit is 11.186 ± .002 d, and its orbital speed is 1.38 ± 0.02 m/s.
Kepler’s 3rd law states that
where is the orbital radius of the planet. Observations of the star’s luminosity and color indicate that its mass .
1. Find the planetary orbit radius (in AU). Hint: use and to compare this orbit to Earth’s orbit about the Sun. (Ans: .
2. The orbital radius of the star can be found from its period and radial velocity , . Using , find . (Ans: )
3. The planet and star co-orbit their center of mass. Find the mass of the exoplanet. (Ans: )
If the orbit is not viewed edge-on, but is inclined to the line of sight (LOS), the measured radial speed of the star is less than the actual orbital speed used in the above analysis: , where i, the angle of inclination, is unmeasurable. See Figure 3. In this case, .
4. Prove that, in the general case, the radial velocity measurements are only sufficient to determine . Is the exoplanet mass you found in Question 3 the maximum, or minimum, mass of the planet?
5. Because red dwarfs are much cooler than the Sun, their temperate zones are much closer to them than the Earth is to the Sun. In addition, red dwarfs are less massive than the Sun. Explain why both of these factors favor the detection of small habitable exoplanets. (See Ref. 2 for a brief discussion.)
6. Proxima Centauri subtends an angle of 1.02 ± .08 mas (milliarcseconds) when viewed from Earth. (See PPE, Section 1.7) Find its radius . (Ans: )
If Proxima centauri b transits, or passes in front of, its star, as viewed from Earth, it will intercept some of the starlight and allow its radius to be determined. Additionally, if the planet has an atmosphere, it will absorb certain wavelengths of light preferentially, enabling astronomers to detect the presence of atmospheric gases such as water, oxygen, carbon dioxide, and methane. Methane (CH4), in particular, is often associated with the presence of life.
The probability of transit is found from the geometric argument outlined in Figure 4. In the figure, the horizontal dotted line is the LOS from Earth, and the vertical line represents the plane perpendicular to the LOS passing through the star’s center. For an orbit to be transiting, its normal (shown as a red line for each of the two orbits depicted) must lie within of the vertical plane, where is the star’s radius. But the normal to the orbit does not need to lie in the plane of the figure: any transiting orbit rotated by any angle (0 – 2π) about the LOS would remain a transiting orbit. The total solid angle available for transiting orbits is therefore , whereas the total solid angle associated with all possible orientations is . Therefore, the probability of a transiting orbit is . (Check: if , then the probability equals 1.) For a fuller discussion, see http://kepler.nasa.gov/Science/about/characteristicsOfTransits/ .
7. How probable is it that Proxima Centauri b transits its star? (Ans: 1.5%)
B. How Warm is Proxima Centauri b?
The following questions are only suitable for students who have studied blackbody radiation in a course of thermal physics.
8. The luminosity of a star (its total radiated power) is proportional to its surface area and the fourth power of its surface temperature: . Proxima Centauri’s surface temperature is about 3050 K, whereas the Sun’s is 5780 K. Show that . This indicates that the temperate zone lies much closer than 1 AU to the star.
9. The fraction of the star’s radiation that is intercepted by a planet is equal to the solid angle subtended by the planet divided by 4π: . The absorbed radiation warms the planet, which reaches a steady state temperature when the blackbody radiation emitted by the planet equals the radiation absorbed from the star:
, where is called the Stefan-Boltzmann constant. (Its value is not needed to answer this question.) Use this information to compare the surface temperature of Proxima Centauri b to that of the Earth, assuming both planets are perfect black bodies (or have the same albedo). (Ans: )
1. G. Anglada-Escudé et al, Nature 536, 437 (25 August, 2016)
2. A. P. Hatzes, Nature 536, 408 (25 August, 2016)