2 Section A: Water on Extra-Solar Planets? 2.1 Introduction A recent publication in the journal Nature described the discovery of seven Earth-like planets around a \"red dwarf\" star called TRAPPIST-1, which is a star that has 8.02 % of the mass and 11.7% of the radius of the sun. The Nature article mainly described the orbits of the seven planets, and briefly stated that \"..the seven planets have equilibrium temperatures low enough to make possible the presence of liquid water on their surfaces.\" News media immediately speculated about the existence of life on one or more of these planets. You do not need the Nature article to answer the following questions. Question 1 [5 points] Table 1 holds some important data for the Sun, Mercury and Earth, TRAPPIST-1 and its seven planets. Notice that the seven TRAPPIST-1 planets orbit their star at distances that are much closer than Mercury orbits our Sun. We know that the surface temperature of Mercury is 440 K, which does not permit the existence of liquid water. Explain why the presence of liquid water on (some) of TRAPPIST-1 planets is nevertheless a possibility. 2.2 Temperatures In Module 2 we derived a simple model for the surface temperature (Ts,rf) of a planet: (1 - A)S'pianet] % Tsur = 1 = [ 1) where o, is the Stefan-Boltzmann constant (5.6705 x 1078 kgs2K %), A is the planet's albedo and Spjanet is the solar constant for each planet (under \"irradiation\" in Table 1). This simple model was also used in the Nature paper. Question 2 [5 points| Assuming an Earth-like albedo of 0.306, calculate the surface tempera- ture of each of the seven planets. It is highly recommended to use a spreadsheet and equations to calculate these answers. You are doing it wrong if you get temperatures lower than 100 kelvin (K), or more than 500 K. Question 3 [5 points] Discuss why the surface temperature model (Eq. 1) does not work very well for Earth and Venus. Question 4 [5 points]| Discuss why setting A = 0 (as is done in the Nature manuscript) might yield more realistic surface temperature estimates. Question 5 [5 points] Based on your calculations, discuss which planet(s) orbiting TRAPPIST- 1 could support liquid water. As a hint towards your answer, it might be helpful to to compute the temperature for Earth using Eq. 1 and compare that value with the ones you got in Question 2. Table 1: Data important for the TRAPPIST-1 system. AU: Astronomical unit or 149.6 x 106 km, Mo: one solar mass (2 x 1030 kg), Searth: the solar constant at Earth (1361 W/m2). 4.25 x Searth means that this planet receives 4.24 times the solar constant in radiation. Body Mass Distance Temperature Irradiation (Splanet) Mo (AU) (K) W/m Sun 1 5772 K Sun-Earth 1 1 X Searth Sun-Mercury 0.387 6.674 X Searth TRAPPIST-1 0.0802 2559 K Planet 1b 0.01111 4.25 X Searth Planet 1c 0.01521 2.27 x Searth Planet 1d 0.02144 1.143 X Searth Planet le 0.02817 0.662 X Searth Planet If 0.0371 0.382 X Searth Planet 1g 0.0451 0.258 X Searth Planet 1h 0.063 0.131 X Searth