ASTR 1110H - Quantitative Exercises Solutions: Orbital Periods and Asteroid Sizes, Assignments of Astronomy

Download ASTR 1110H - Quantitative Exercises Solutions: Orbital Periods and Asteroid Sizes and more Assignments Astronomy in PDF only on Docsity! ASTR 1110H – Solutions to Quantitative Exercises Chapter 5 #1. Jupiter’s orbital period is 11.86 years (from the inside front cover of your text book). So the asteroids whose periods are resonances of 1/2, 2/5, 1/3, and 1/4 with that of Jupiter have periods of 11.86/2 = 5.93 years, (2 ¥ 11.86)/5 = 4.74 years, 11.86/3 = 3.95 years, and 11.86/4 = 2.97 years, respectively. Using Kepler’s 3rd law (D3 = P2) results in semi- major axes of 3.28 AU, 2.82 AU, 2.50 AU, and 2.07 AU, respectively. #2. Ceres has a diameter of 940 km (radius = 470 km). The Earth’s radius is 6,378 km and the Moon’s is 1,738 (from the inside front cover of your text book). The volumes of these three objects can be compared by cubing their radii (from Volume = 4/3 p R3): volumes of Earth vs. Ceres = (6,378/470)3 = 2500; volumes of Moon vs. Ceres = (1,738/470)3 = 50. If the asteroids had originated from the breakup of a planet the size of the Earth and all the asteroids from this hypothetical breakup were of the same size as Ceres, then you’d expect to find about 2500 of them. Since Ceres is the biggest asteroid and the sizes of asteroids decrease quickly after the first few large ones (1 Ceres, 2 Pallas, 3 Juno, 4 Vesta, etc.), the fact that there are not 2500 (or even 25) the size of Ceres indicates that the asteroids couldn’t have come from the breakup of an Earth-sized planet, nor even a Moon-sized body.