924 Elementary Problems and Answers in Solar System Astronomy

Answers to Chapter 3. Sun, Earth, and Moon

ANSWERS TO CHAPTER 3 Sun, Earth, and Moon 214. G, the universal constant of proportionality in Newton's law of gravitation. 215. By measuring the attractive force between two known masses m! and m2 separated by a distance r and using Newton's law of gravitation F - G m}m2 - r2 ' one can find the value of G. Then one can measure the acceleration g of a freely falling body in vacuo at the surface of the Earth. Finally, the relationship g = GM/r2, where M is the mass of the Earth and r its radius, determines M in terms of known values of g, r, and G. 216. 4,000 miles. 166 Sun, Earth, and Moon / 167 217. An idealized method is to make simultaneous measurements of the meridian transit altitude of a star, the Sun, or a planet at two stations at the same longitude but separated along the surface of the Earth by a known North-South distance s. If the difference of the two observed altitudes is ~h radians, then the radius of the Earth is given by r = sf~h, as can be confirmed by a simple diagram. There are obvious variations of the above idealized scheme to yield the same result. 218. 1.852 km (or 1.000 nautical mile). 219. 19.6 km. The navigator's convenient version of the given formula is d = 1.1v'H where d is in nautical miles and H in feet. 220. (d) 210 miles. 221. (a) 1,700 ft. 222. 1,304,000. 223. (a) Study of the relative abundances of various radiogenic elements in granites. 224. (b) 4 billion. 225. (a) 4 X 109 years. 226. (b) 4.5 billion years ago. 227. 3.9 X 109 years. 228. 4.5 X 109 years. 229. 0.25 g. 230. The Earth is composed mostly of rocky material (and some iron) under the high pressure of its self-gravitation. 168 / Sun, Earth, and Moon 231. Its mean density 5.52 g cm-3 is substantially less than even the uncompressed density of iron, namely 7.8 g cm-3 , and much less than its compressed value. 232. (b) Its rotation. 233. 68 degrees Kelvin. 234. The heat released by the decay of radioactive substances and the residual (gravitational) heat from its original formation by the accretion of dispersed material. 235. Volcanic eruptions of molten lava, hot water from geysers, and measured temperatures in deep oil wells. 236. Because of variable refraction along the observer's line of sight through the atmosphere. 237. (a) Scattering. 238. (a) Sunlight scattered in the atmosphere. 239. (c) Refraction. 240. (d) The apparent altitude of a star to be greater than its true (geometric) altitude. 241. (c) 3.0. 242. 11.9 meters. The density of liquid air is about 0.87 g cm-3 • 243. (d) Nitrogen. 244. (a) Ozone is 03 or triatomic oxygen. (b) 25 km. (c) It absorbs actinic ultraviolet light from sunlight, which would otherwise cause severe sunburn and possibly cancer on exposed human skin; cataracts in eyes; and other adverse effects on biological material. Sun, Earth, and Moon / 169 245. The minimum orbital period of any free-fiying satellite of the Earth is 84.5 minutes, in an orbit of radius equal to the radius of the Earth (neglecting atmospheric drag). Any higher altitude orbit has a greater period. In principle, a continuously propelled aircraft with "upside down" wings to cause a negative aerodynamic lift could fiy a great circle path around the Earth in the upper atmosphere in less than 84.5 minutes. 246. 490 meters (1,600 feet, nearly three times the height of the Washington Monument). 247. (c) Electrical currents in its interior. 248. (b) The solar wind. 249. (a) Would not exist if the Earth were not magnetized. 250. A durably trapped electrically charged particle travels at constant energy in a spiral path around a magnetic line of force, with a typical period of the order of a millisecond, and oscillates back and forth between "mirror points" in opposite hemispheres with a typical period of a second. The particle's spiral path drifts around the planet in longitude with a typical period of an hour. The foregoing exemplary periods are for electrons in the Earth's magnetosphere. They are different for protons and other ions and for other planets. 251. Perpendicular to the plane of the Earth's orbit...