Question

1 (10 pts). In 1601 the German astronomer Johannes Kepler became director of the Prague Observatory. Kepler had been helping Tycho Brache in collecting 13 years of observations on relative motion of the planet Mars. By 1609, Kepler had formulated his first two laws of planetary motion: i. Each planet moves on an ellipse with the sun at one focus. ii. For each planet, the line from the sun to the planet sweeps out equal areas is equal times. Kepler's third law relates the period of a planet's orbit to its mean distance from the sun. He observed that the square of the period of the orbit is proportional to the cube of the mean distance from the sun (that is, P2 ox D3). Directly apply least squares (use the normal equation for a power function) to fit the model y to the observational data. Compute D and dmax as bounds for cmax; what do these bounds suggest about the Chebyshev criterion? ar3/2 Period (seconds) Mean Distance (meters) Body Mercury 7.60 x 106 1.94 x 10 5.79 x 1010 Venus 1.08 x 1011 Earth 3.16 x 107 1.50 x 1011 5.94 x 107 Mars 2.28 x 101 7.79 x 101 Jupiter 3.74 x 108 Saturn 9.35 x 108 1.43 x 1012 Uranus 2.64 x 109 2.87 x 1012 Neptune 5.22 x 10 7.82x 10 4.50 x 1012 Pluto 5.91x 1012 We can debate whether we should include Pluto, since it has been recategorized as a dwarf planet. However, Uranus wasn't discovered until 1781, Neptune in 1846, and Pluto in 1930 (according to "Ask an Astronomer" at cornell.edu), so Kepler would have made his observation based upon the six planets known to him.

Answer 1

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that iven - kepea's third law relates the period of a planet's corbit Toate Mean distance for the san. He Observed that the square of the period of the arbit is proportiona ITO the Cube of the mean distand form the sun Diorerfly Apply least square To fit the model y = qx3/ 70 The Obsauational dato compute D And day as bounds for max what do these Bounds suggest About the CHe bysheu coriterion.

Given that: we can debate whetho we should pluto since it has been Recateqarind as a dwool planet . However Cranus wash't discoverd cenfil 1781 Neptune In 1846 and pluto in 1930 so Kepler would have made His Observation Based upon the Six plantes known to Him.

Body mercury period(seconds) 7.60 X 106 1.948 107 Venus Earth 3.16 x 107 mean Distance (meters) 5.79 X 100 1.08x10" 1.50X10" 2-28x lo!! 7.79 x 10! 1.43 x 1012 mars 5.94 x 107 Jupiter saturn .74x10F -35 x 108 2.64 x 109 5.22x109 2.87 X10 Uranus Neptune 4.50x 1002 12. 5.91 x 10 pluto 7.82 8109

Given that! Anuswer: T: Enc - D Taking Log Bouth Sider от т. ер L + с (997 y. A+ cx ->0) This Bouth Sides are. The In whear : 7- le17 A. leg E y = logor Now harnea 1 Ep" to ху, пA +сx — 70 Exy = aɛy to ex²

T R xy 8-118 94472 gs. 36 s4/334 X² (6-4231848 | 81. 43948]49 Хя 995 365 | ts) | 93.99 514 108- 8 194: 6 229 1+3.3 1+83 1ча gap Y=1097 x = logo +.Ч772368 19 4-053-7385 54161 бо Ҷ08 ч. 683 481366 Ф. без 43 5 84891354 5oo79650 66 8454647989 &s-o1971ч). 6.534 33 4293 5:428506 °36 3s. 4639 344s 89,42303as 8 343D 18ча с. (910 до 54 55 146 1 067 44-31714о4 | 9.894ockЧ 1863х84 61 62 чаsоsis) 52. Эs545эг b-330214) S 1.464669309 8 - #s384[43 63 3945583 11. ооч S96 -ЧЛ1 6se; • s+14 РЭЧч 20. 79710781 1.41 418) Xas 63х33 6 3 ч. ч€998 18-23197397 58 ts+1 38 ғ9 505-6585589 379-s 2094ъе | jo153 Зовь 6ol50 966-0 4445 S903 exy Ex2 Ex ху

from the 1. Go and 9 72.739 94 + 5d.157 505.656 = o.1574+ 399.5916 the solering A = -1.6092 c=1.4996 af k. 2. ankle (4) Odooe 4 T:0. 2000 47590'