Answers to Chapter 5. Other Planets, Their Satellites, and Rings
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ANSWERS TO CHAPTER 5 Other Planets, Their Satellites, and Rings 623. Neptune. 624. (d) Inversely as the fourth power of its distance. 625. (10-2)(270) =2.7 degrees Kelvin. 626. The classification is on the basis of mean density. The terrestrial planets (high mean densities and solid surfaces) are Mercury, Venus, Earth, and Mars. The Jovian planets (low mean density and mainly gaseous and liquid) are Jupiter, Saturn, Uranus, and Neptune. Pluto has a low mean density but has a solid surface. Minor planets (asteroids) are terrestrial in nature. 627. (d) Mars. 628. Mercury, Venus, Mars, Jupiter, and Saturn. 629. Jupiter, Saturn, Uranus, and Neptune. 630. Venus. 207 208 / Other Planets, Their Satellites, and Rings 631. (a) Gravitational attraction of every element of the planet by every other element. 632.• For an isolated body of fluid, a sphere is the shape of minimum gravitational potential energy. (If the fluid is rotating, an oblate spheroid is the shape of minimum potential energy.) • An isolated solid body also tends toward a spherical shape because of self-gravitational attraction, but will not assume that shape if the crushing strength of the material exceeds the gravitational pressure. Consider a cylindrical body of radius a, length 4a, mass m, density p, and crushing strength C. For an approximate analysis, imagine the body to be composed of two subcylinders , each of length 2a. Then the gravitational attraction between the two is approximately (1) The crushing pressure across the dividing plane between the two sub-cylinders is 1r a2 p2G . (2) The threshold condition for crushing to occur is (3) or "';C/1rG a= . p (4) • For rocky material, a representative value of C is 1 X 109 dyne cm-2 and of p is 2.8 g cm-3. Using (4), one finds that if a exceeds 250 km, the body will tend toward sphericity, whereas if a is less than 250 km, it can retain a markedly nonspherical shape. For cometary material, the threshold value of a is perhaps an order of magnitude less than that for rocky material. This treatment yields only a crude estimate of a but does illustrate the nature of the problem. Other Planets, Their Satellites, and Rings / 209 633.• Suppose that the mountain is a solid rigid cone having a circular base of radius a, height a, mass m, and mean density p. The mass of the mountain is and its weight is 1r a3 pg 3 The weight is spread over an area 1r a2. Hence, the threshold crushing condition is (1) (2) 1ra3pg=C (3) 31r a2 where C is the crushing strength of the underlying crust of the planet. From (3) 3C (4) a=-. pg • Representative values of C and p for natural rocky materials are 1 X 109 dyne cm-2 and 3 g cm-3 , respectively. With these values and g = 982 em s-2, it is found by (2) that for the Earth a = 10 km. (5) This simple analysis yields, of course, only a crude estimate of the maximum possible height of a mountain on a planet, but it does illustrate the nature of the problem. • By (4) it is noted that the upper limit value of a is inversely proportional to g. For the Moon and the terrestrial planets Mercury, Venus, the Earth, and Mars, the surface values of g are 162, 363,860,982, and 374 em S-2 and by (4), the corresponding threshold values of a are 61, 21, 11, 10, and 26 km, respectively. The heights of the highest actual mountains on these bodies are as follows: 10, 7, 11 (Maxwell Montes), 9 (Mt. Everest), and 27 (Olympus Mons) km, respectively. 210 / Other Planets, Their Satellites, and Rings 634. y o • In the diagram, A represents New York; B, San Francisco; the arc AQ'B, a great circle between A and B; AQB, the tunnel along the X-axis with midpoint at Q; 0, the center of the Earth, radius a (6372 km); and C, the position of the car at a particular moment. • The respective latitudes and longitudes of New York and San Francisco are 40.75 degrees North, 73.99 degrees West; and 37.78 degrees North, 122.41 degrees West. The great circle, sea level distance between the two cities is 4,130 km and the straight line distance is 4,057 km. The angle AOB = 37.13 degrees , b = OQ =6,040 km, and QQ' =332 km, the depth of the tunnel at...