## A. Introduction

The recent discovery (Ref. 1) of seven exoplanets orbiting the red dwarf star Trappist-1 is an exciting and potentially key development in the search for extraterrestrial life in the universe. All of the seven planets are Earth-sized, and at least three lie in the temperate “Goldilocks” zone, neither too near nor too far from their host star for surface oceans to exist. Given the mass and temperature of these planets, their atmospheres are likely to contain nitrogen, oxygen, water vapor, and carbon dioxide, just like on Earth. As an added bonus, the small size of the host star will help astronomers determine the chemical composition of each planet’s atmosphere.

The goal of this post is to show students how scientists use elementary physics and trigonometry to derive many of the physical properties of this and other extrasolar systems. We present straightforward explanations appropriate to an introductory physics course, and offer exercises suitable for a homework assignment or recitation class to help students appreciate the significance of the Trappist-1 discovery. We hope these materials will be useful to you, and that your students will enjoy and benefit from them.

## B. Background

In February 2017, an international team of astronomers, led by Michaël Gillon of Belgium, announced the discovery (Ref. 1) of seven Earth-sized exoplanets orbiting the red dwarf Trappist-1, so named for the instrument in Chile (TRAPPIST: *Tra*nsiting *P*lanets and *P*lanetesimals *S*mall *T*elescope) first employed in their work. Observations continued for one year using an array of ground-based telescopes located in Chile, Morocco, Hawaii, Spain, and South Africa, plus the orbiting Spitzer Space Telescope. The exoplanet discoveries were made by the *transit photometry* technique: when a planet eclipses its host star, i.e., as it passes between the star and Earth, it blocks a tiny fraction of starlight from reaching Earth. The resulting light curve (flux *vs*. time, see Fig. 2) and its period* P *(time between successive transits), plus estimates of the star’s mass and radius , suffice to determine the planet’s radius and the radius of its orbit . This information can then be used to estimate the surface temperature and the potential for life on the planet.

Astronomers determine stellar masses by studying binary stars. For each binary system, they measure the distance from Earth by parallax, and measure the apparent brightness of each star to calculate its luminosity . Mass is found by studying the orbital motion of the binary, as described in *PPE*, Section 8.11. By compiling data on many binaries, an empirical mass-luminosity relation can be constructed. A mass-radius relation can be constructed in a similar way, using data from eclipsing binary stars and long baseline interferometry. For now, let’s accept the mass and radius of Trappist-1 given in Ref. 1: , , where the subscript indicates the Sun. Table 1 is adopted from Ref. 1, and lists many of the properties of the Trappist-1 exosystem.

1. A commonly quoted mass-luminosity relation, suitable for some small-mass main sequence stars, is , where the subscript indicates the Sun. The measured luminosity of Trappist-1 is . Show that this leads to a value of that agrees with the value quoted in Ref. 1.

## C. Kepler’s Laws and Orbital Radii

Assume that the mass of each planet is much less than that of the host star, , and that each planet is in a near-circular orbit of radius , just like the planets of our own solar system. Kepler’s 3rd law is

(1)

where * P* is the orbital period. After calculating from its luminosity, and measuring

*, we can use Eqn. 1 to derive*

**P***.*

**a**2. Astronomers have recently (February 2017) discovered seven exoplanets orbiting the star Trappist-1, an ultracool red dwarf about 12 pc from Earth. This is an exciting discovery because at least three planets are in the “temperate zone” of the host star, with surface temperatures favorable for life. The table below lists properties of six bodies that orbit either Trappist-1 () or one other central body with mass . Fill in the missing numbers. (Hint: Use ratios rather than converting to meters, seconds and kilograms.) Can you identify* M* and the bodies in orbit about it?

## D. Analyzing the Light Curve

Fig. 2 contains a great deal of information that can be used to further characterize the Trappist-1 system. Even if and are not known *a priori*, the period * P* and the shape of the transiting planet’s light curve provide enough information to determine , the orbital radius

*and inclination*

**a****, and the density of the host star . Combining the star’s density with the appropriate mass-radius relation for red dwarfs yields and . This strategy uses Kepler’s 3rd law plus simple kinematics and trigonometry, making it amenable to first-year physics students. It is described in detail by Seager and Mallén-Ornelas in Ref. 2, and we will follow their treatment closely.**

*i*As usual, assume that the exoplanet’s mass is much less than the star’s, and that it is in a circular orbit of radius . Fig. 3a is a “side view” of the orbit, perpendicular to the line of sight from Earth to the star. The inclination angle * i *is defined so that if , observers on Earth view the orbit edge-on. The

*impact parameter*is defined by . Clearly, for a transit to occur, and since , this implies that .

**b**From Fig. 3b, the amount of light blocked by the planet is proportional to its area, so the fractional change in flux reaching Earth during a transit is

.

(2)

In practice, the straight line segments of the light curve shown in Fig. 3b are rounded by *limb-darkening* (as seen in Fig. 2): less light reaches us from areas near the edge of the stellar disc than from equal areas near the center of the disc. For simplicity, we will ignore this effect.

3. Fig. 2 displays the light curves of the seven planets orbiting Trappist-1. Which planet has the largest radius? (Ans: Trappist-1b)

Since , we may approximate the trajectory of the planet as a straight line as it moves across the face of the star. For a circular orbit, the planet’s speed is . By the Pythagorean theorem, the distance traveled by the planet in time is , so

.

(3)

4. Derive the corresponding equation for

(Ans: )

If and are not known *a priori*, then the only information available is the light curve and its period. Nevertheless, it is still possible to proceed. From Eqn. 3 and the corresponding expression for (Question 4), it is easy to show that

,

where we have used Eqn. 2 to express . Rearranging the above expression,

or

.

(4)

Combining Eqn. 4 with (Kepler’s 3rd law) yields an expression for the stellar “density”:

.

It is convenient to re-express this in terms of the solar “density”

(5)

where the pre-factor . This is the key result of this section.

For classroom purposes, we can crudely express the mass-radius relation as . Choosing , we can now calculate the mass and radius of Trappist-1 in solar mass units:

or

,

(6)

where . The validity of these results depends sensitively on the mass-radius relation chosen. Admittedly, the exponent was chosen to yield results consistent with those of Ref. 1. In the literature, is more commonly quoted.

5. According to Ref. 1, the transit duration of Trappist-1d is , the period is , and the transit depth is .

a. Use the values of and given in section B to compute and then find . Check your answer against the light curve in Fig. 2. (Ans: )

b. Find the radius of the planet’s orbit. (Ans: )

c. Find the radius of the planet. Express as a fraction of . (Ans: )

6. Researchers at Trump University recently reported the discovery of an *eighth* planet orbiting Trappist-1. In a series of tweets, team leaders provided a summary of their observations taken with the privately funded Andrew Jackson Telescope in Palm Beach, FL: , , , and . Their measurements have not yet been confirmed. If you were a program director at the National Science Foundation, would you approve a grant to the Trump exoplanet program?

## E. Too Hot, Too Cold, or Just Right?

A blackbody is an object that absorbs all of the radiation incident on its surface. (Since it reflects no light, it appears black.) But a blackbody is not just an ideal absorber of radiant energy, it is an ideal emitter of energy as well. The power * S* emitted by a blackbody per unit surface area is given by the

*Stefan-Boltzmann Law*,

,

(7)

where ** T** is the surface temperature in kelvin, and is called the Stefan-Boltzmann constant. The Sun and a cloudless Earth behave approximately as blackbodies. The power emitted by the Sun, its luminosity, is . At a distance

*from the Sun, this power is spread evenly over a sphere of area . A spherical planet of radius presents a cross-sectional area to the Sun, so if it acts as a blackbody, it intercepts and absorbs energy at a rate . This energy warms the planet’s surface to a temperature*

**r***, causing it to emit energy at a rate . When , the temperature reaches a steady-state value given by*

**T**.

(8)

7. The Earth is 1 AU () from the Sun. What is the steady-state temperature on Earth, according to Eqn. 8? (Ans: 278 K. Keep in mind, though, that this ignores the greenhouse effect, which raises the temperature significantly.)

To apply Eqn. 8 to an extrasolar system, simply replace by the luminosity of the host star .

8. The period of Trappist-1d is 4.05 days. Use the star’s mass and luminosity to calculate the steady-state surface temperature on the planet. Based only on your answer, what is the likelihood of finding liquid water on this planet? (Ans: T ≈ 300 K. However, if the planet has an Earth-like atmosphere, the greenhouse effect would raise its temperature too high to harbor liquid water on its surface. This is mentioned briefly in Ref. 1.)

9. What is the likelihood of finding intelligent life on Trappist-1i, the planet claimed by researchers at Trump University?

## References

1. Michaël Gillon et al, *Nature* **542**, 456 (2017)

2. S. Seager and G. Mallén-Ornelas, *Ap. J.* **585**, 1038 (2003)