Exam (elaborations) TEST BANK FOR An Introduction to Modern Astrophysics By Carroll B.W. and Ostlie D.A. (Solution manual)

1.1 From Fig. 1.7, Earth makes S=P˚ orbits about the Sun during the time required for another planet to make S=P orbits. If that other planet is a superior planet then Earth must make one extra trip around the Sun to overtake it, hence S P˚ D S P C 1: Similarly, for an inferior planet, that planet must make the extra trip, or S P D S P˚ C 1: Rearrangement gives Eq. (1.1). 1.2 For an inferior planet at greatest elongation, the positions of Earth (E), the planet (P), and the Sun (S) form a right triangle (∠EPS D 90ı). Thus cos.∠PES/ D EP=ES. From Fig. S1.1, the time required for a superior planet to go from opposition (pointP1) to quadrature (P2) can be combined with its sidereal period (from Eq. 1.1) to find the angle ∠P1SP2. In the same time interval Earth will have moved through the angle ∠E1SE2. Since P1, E1, and S forma straight line, the angle ∠P2SE2 D ∠E1SE2  ∠P1SP2. Now, using the right triangle at quadrature, P2S=E2S D 1= cos.∠P2SE2/. P1 E1 S P2 E2 Figure S1.1: The relationship between synodic and sidereal periods for superior planets, as discussed in Problem 1.2. 1.3 (a) PVenus D 224:7 d, PMars D 687:0 d (b) Pluto. It travels the smallest fraction of its orbit before being “lapped” by Earth. 1.4 Vernal equinox: ˛ D 0h, ı D 0ı Summer solstice: ˛ D 6h, ı D 23:5ı Autumnal equinox: ˛ D 12h, ı D 0 ı Winter solstice: ˛ D 18h, ı D 23:5ı 1 2 Chapter 1 The Celestial Sphere 1.5 (a) .90ı  42ı/C 23:5ı D 71:5ı (b) .90ı  42ı/  23:5ı D 24:5ı 1.6 (a) 90ı  L ı 90ı (b) L 66:5ı (c) Strictly speaking, only at LD˙90ı. The Sun will move along the horizon at these latitudes. 1.7 (a) Both the year 2000 and the year 2004 were leap years, so each had 366 days. Therefore, the number of days between January 1, 2000 and January 1, 2006 is 2192 days. From January 1, 2006 to July 14, 2006 there are 194 days. Finally, from noon on July 14, 2006 to 16:15 UT is 4.25 hours, or 0.177 days. Thus, July 14, 2006 at 16:15 UT is JD .177. (b) MJD 53930.677. 1.8 (a) ˛ D 9m53:55s D 2:4731ı, ı D 2ı9016:200 D 2:1545ı. From Eq. (1.8),  D 2:435ı. (b) d D r  D 1:7  1015 m D 11,400 AU. 1.9 (a) From Eqs. (1.2) and (1.3), ˛ D 0:ı D 0:m and ı D 0:ı D 2:. This gives the 2010.0 precessed coordinates as ˛ D 14h30m29:4s, ı D 62ı43025:2600. (b) From Eqs. (1.6) and (1.7), ˛ D 5:46s and ı D 7:98400. (c) Precession makes the largest contribution. 1.10 In January the Sun is at a right ascension of approximately 19h. This implies that a right ascension of roughly 7h is crossing the meridian at midnight. With about 14 hours of darkness this would imply observations of objects between right ascensions of 0 h and 14 h would be crossing the meridian during the course of the night (sunset to sunrise). 1.11 Using the identities, cos.90ı  t/ D sin t and sin.90ı  t/ D cos t , together with the small-angle approximations cos   1 and sin  1, the expression immediately reduces to sin.ı C ı/ D sin ı C  cos ı cos : Using the identity sin.a C b/ D sin a cos b C cos a sin b, the expression now becomes sin ı cos ı C cos ı sinı D sin ı C  cos ı cos : Assuming that cos ı  1 and sinı  ı, Eq. (1.7) is obtained.