SpaceMath: The Goldilocks Planets -- Not Too Hot or Cold
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SpaceMath: The Goldilocks Planets -- Not Too Hot or Cold

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Science Education Standards: Earth and Space Science -- Content Standard D: Earth in the Solar System

• The Earth is the third planet from the sun in a system that includes the Moon, the sun, seven other planets and their moons, and smaller objects, such as asteroids and comets. The sun, an average star, is the central and largest body in the solar system.

Short Description: Students use a table of the planets discovered by the Kepler satellite, and estimate the number of planets in our Milky Way galaxy that are about the same size as Earth and located in their Habitable Zones. They estimate the average temperature of the planets, and study their tabulated properties using histograms.

Source: Space Math (GSFC)

Once you have discovered a planet, you need to figure out whether liquid water might be present. In our solar system, Mercury and Venus are so close to the sun that water cannot remain in liquid form. It vaporizes! For planets beyond Mars, the sun is so far away that water will turn to ice. Only in what astronomers call the Habitable Zone (shown in green in the figure above) will a planet have a chance for being at the right temperature for liquid water to exist in large quantities (oceans) on its surface!

The Table on the following page lists the 54 planets that were discovered by NASA's Kepler Observatory in 2010. These planets come in many sizes as you can see by their radii. The planet radii are given in terms of the Earth, where '1.0' means a planet has a radius of exactly 1 Earth radius (1.0 Re) or 6,378 kilometers. The distance to each planet's star is given in multiples of our Earth-Sun distance, called an Astronomical Unit, so that '1.0 AU' means exactly 150 million kilometers.

Problem 1 - For a planet discovered in its Habitable Zone, and to the nearest whole number, what percentage of planets are less than 4 times the radius of Earth?

Problem 2 - About what is the average temperature of the planets for which R <4.0 Re?

Problem 3 - About what is the average temperature of the planets for which R >4.0 Re?

Problem 4 - Create two histograms of the number of planets in each distance zone between 0.1 and 1.0 AU using bins that are 0.1 AU wide. Histogram-1: for the planets with R > 4.0 Re. Histogram-2 for planets with R < 4.0 Re. Can you tell whether the smaller planets favor different parts of the Habitable Zone than the larger planets?

Problem 5 - If you were searching for Earth-like planets in our Milky Way galaxy, which contains 40 billion stars like the ones studies in the Kepler survey, how many do you think you might find in our Milky Way that are at about the same distance as Earth from its star, about the same size as Earth, and about the same temperature (270 - 290 K) if 157,453 stars were searched for the Kepler survey?

Problem 1 - For a planet discovered in its Habitable Zone, and to the nearest whole number, what percentage of planets are less than 4 times the radius of Earth? Answer: There are 28 planets for which R < 4.0 re, so P = 100% x (28/54) = 52%

Problem 2 - About what is the average temperature of the planets for which R <4.0 Re? Answer; Students will identify the 28 planets in the table that have R < 4.0, and then average the planet's temperatures in Column 6. Answer: 317 K.

Problem 3 - About what is the average temperature of the planets for which R >4.0 Re? Students will identify the 26 planets in the table that have R > 4.0, and then average the planet's temperatures in Column 6. Answer: 306 K.

Problem 4 - Create two histograms of the number of planets in each distance zone between 0.1 and 1.0 AU using bins that are 0.1 AU wide. Histogram-1: for the planets with R > 4.0 Re. Histogram-2 for planets with R < 4.0 Re. Can you tell whether the smaller planets favor different parts of the Habitable Zone than the larger planets? Answer; They tend to be found slightly closer to their stars, which is why in Problem 2 their average temperatures were slightly hotter than the larger planets

Problem 5 - If you were searching for Earth-like planets in our Milky Way galaxy, which contains 40 billion stars like the ones studies in the Kepler survey, how many do you think you might find in our Milky Way that are at about the same distance as Earth from its star, about the same size as Earth, and about the same temperature (270 - 290 K) if 157,453 stars were searched for the Kepler survey?

Answer: Students may come up with a number of different strategies and estimates. For example, they might create Venn Diagrams for the data in the table that meet the criteria given in the problem. Then, from the number of planets in the intersection, find their proportion in the full sample of 54 planets, then multiply this by the ratio of 40 billion to 157,453. Estimates near 1 million are in the right range.

 # Planet Name (KOI) Orbit Period (days) Distance To Star (AU) Planet Radius (Re) Planet Temp. (K) Star Temp. (K) 1 683.01 278 0.84 4.1 239 5,624 2 1582.01 186 0.63 4.4 240 5,384 3 1026.01 94 0.33 1.8 242 3,802 4 1503.01 150 0.54 2.7 242 5,356 5 1099.01 162 0.57 3.7 244 5,665 6 854.01 56 0.22 1.9 248 3,743 7 433.02 328 0.94 13.4 249 5,237 8 1486.01 255 0.80 8.4 256 5,688 9 701.03 122 0.45 1.7 262 4,869 10 351.01 332 0.97 8.5 266 6,103 11 902.01 84 0.32 5.7 270 4,312 12 211.01 372 1.05 9.6 273 6,072 13 1423.01 124 0.47 4.3 274 5,288 14 1429.01 206 0.69 4.2 276 5,595 15 1361.01 60 0.24 2.2 279 4,050 16 87.01 290 0.88 2.4 282 5,606 17 139.01 225 0.74 5.7 288 5,921 18 268.01 110 0.41 1.8 295 4,808 19 1472.01 85 0.37 3.6 295 5,455 20 536.01 162 0.59 3.0 296 5,614 21 806.01 143 0.53 9.0 296 5,206 22 1375.01 321 0.96 17.9 300 6,169 23 812.03 46 0.21 2.1 301 4,097 24 865.01 119 0.47 5.9 306 5,560 25 351.02 210 0.71 6.0 309 6,103 26 51.01 10 0.06 4.8 314 3,240 27 1596.02 105 0.42 3.4 316 4,656 28 416.02 88 0.38 2.8 317 5,083 29 622.01 155 0.57 9.3 327 5,171 30 555.02 86 0.38 2.3 331 5,218 31 1574.01 115 0.47 5.8 331 5,537 32 326.01 9 0.05 0.9 332 3,240 33 70.03 78 0.35 2.0 333 5,342 34 1261.01 133 0.52 6.3 335 5,760 35 1527.01 193 0.67 4.8 337 5,470 36 1328.01 81 0.36 4.8 338 5,425 37 564.02 128 0.51 5.0 340 5,686 38 1478.01 76 0.35 3.7 341 5,441 39 1355.01 52 0.27 2.8 342 5,529 40 372.01 126 0.50 8.4 344 5,638 41 711.03 125 0.49 2.6 345 5,488 42 448.02 44 0.21 3.8 346 4,264 43 415.01 167 0.61 7.7 352 5,823 44 947.01 29 0.15 2.7 353 3,829 45 174.01 56 0.27 2.5 355 4,654 46 401.02 160 0.59 6.6 357 5,264 47 1564.01 53 0.28 3.1 360 5,709 48 157.05 118 0.48 3.2 361 5,675 49 365.01 82 0.37 2.3 363 5,389 50 374.01 173 0.63 3.3 365 5,829 51 952.03 23 0.12 2.4 365 3,911 52 817.01 24 0.13 2.1 370 3,905 53 847.01 81 0.37 5.1 372 5,469 54 1159.01 65 0.30 5.3 372 4,886