Legend Superior Planet's Orbit Earth's Orbit SC Inferior Planet's Orbit SC - Superior Conjunction SC IC - Inferior Conjunction GEE - Greatest Eastern Elongation Sun GWE - Greatest Western Elongation EQ - Eastern Quadrature GEE IC GWE O - Opposition WQ - Western Quadrature EQ Earth WQ O Figure 1 Part I. Planetary Configuration on 1/01/2000, 12/31/2000, and 9/22/2000 1. The semi-major axis (a), eccentricity (e), and the ecliptic longitude of the perihelion (Op) for the Earth, Venus, and Jupiter have been provided in Table 1. 2. The ecliptic longitudes of the Earth, Venus, and Jupiter on 1/01/2000, 12/31/2000, and 9/22/2000 have been provided in Table 1. Table 1 Ecliptic Longitude Planet a e Op 1/01/2000 12/31/2000 9/22/2000 Venus 0.72 AU 0.0068 131 181 37' 450 14' 249 35' Earth 1.00 AU 0.0167 103 100 26' 99 10' 359 17' Jupiter 5.20 AU 0.0484 150 36 04' 69 09' 60 16' 3. On a sheet of unlined paper, draw to scale the orbits of the Earth and Venus. Adopt a scale in which 1 AU = 10 cm, and approximate the orbits with offset circles. a. Draw a circle representing the orbit of the Earth. N 2A37 b. Mark a point to the right of the center of the circle at the distance equal to c = ea = 0.0167x10 cm = 0.167 cm. C. Draw a line through the center of the Earth's orbit and the Sun. This line represents the major axis of Earth's orbit. d. Given the perihelion of the Earth's orbit has a heliocentric longitude of 102, draw the y-line. e. Given the perihelion angle of Venus, measure this angle along the orbit of the Earth and make a mark (a point) on the orbit of the Earth. f. Draw a line through this point and the Sun, extending the line beyond the Sun (this line represents the extension of the major axis of Venus). g. Draw a circle centered on a point offset from the Sun (along the major axis of Venus) by a distance ea. The radius of the circle is the semi-major axis converted to cm. 4. Mark the positions of the Earth and Venus on their respective orbits for January 1, 2000, and then note the date next to each planet. 5. Mark the positions of the Earth and Venus on their respective orbits for December 31, 2000 and then note the date next to each planet. You will notice that the Earth is at almost the same point at which it was on January 1. Title this drawing "Venus and Earth." 6. On the other side of the sheet, draw to scale the orbits of the Earth and Jupiter. Adopt a scale in which 1 AU = 2.0 cm. 7. Repeat steps 3, 4, and 5 for these planets using the new scale. Title this drawing 'Jupiter and Earth." 8. Based on your drawings, which planets are visible at 6:00 a.m., 6:00 p.m., and midnight on December 31, 2000? Write your results in Table 2. Table 2 Time Visible Planets 6:00 a.m. 6:00 p.m. Midnight 9. Measure the elongation angle of each planet on December 31, 2000 (do not forget to specify if the elongation angle is E or W). Enter your results in Table 3. Table 3 Planet Elongation Angle Venus Jupiter38 10. From your drawings, determine the angle the orbital radius swept between 01/01/2000 and 12/31/2000. Note: This angle is larger than 360 for Venus. Do you know why? List the angles in Table 4. 11. For each planet, calculate how many degrees the planet's orbital radius sweeps per day. Enter the values in Table 4. 12. For each planet, calculate the sidereal period of revolution in Earth's years. Note: In one sidereal period of revolution, the orbital radius sweeps 360. Also, one Earth year is 365.25 days. Enter the values in Table 4. Table 4 Planet Angle Swept Angle Swept Per Day, n Sidereal Period Venus Jupiter Part II. Planetary Conjunctions and Oppositions Knowing the daily motion of a planet (n), (i.e. the angle swept by the orbital radius per day) one can calculate the dates of superior and inferior conjunctions as well as oppositions of the planet with respect to the Earth. The ecliptic longitude (EL) of a planet can be written as: EL = ELotnd, where ELo is the ecliptic longitude of the planet on September 22, n is the planet's daily motion, and d is the number of days from the Autumnal Equinox (September 22). As an example, let's find out if Saturn is at opposition in year 2000. If it is, on what date does this occur? The daily motion for the Earth, n, is 0.986 per day; for Saturn it is 0.033. On September 22, the ecliptic ongitude of Saturn is about 55 (for the Earth, this angle is 0). At opposition, the heliocentric longitudes of the Earth and Saturn are the same, so: 0.986/day x d + 0= 55 + 0.033/day x d. Solving ford, we get roughly 58 days. This means that the opposition for the Earth and Saturn should occur 58 days after September 22, or November 19. This agrees well with the actual date of this opposition (November 19!). If we want to find when Saturn is at conjunction, we note that, in this case, the ecliptic longitudes of the Earth and Saturn differ by 180, so: 0.986/day x d + 180 = 0.033/day x d + 55. Solving for d, we get roughly -131 days. This means that conjunction occurred 131 days before September 22, or on May 14. This agrees well with the actual date of this conjunction (May 10). 1. Determine if Jupiter is ever at opposition or superior conjunction in the year 2000. a. Opposition (yeso) If yes, when (mm/dd) b. Conjunction (yeso) If yes, when (mm/dd) 2. Determine if Venus is ever at superior or inferior conjunction in the year 2000. a. Superior conjunction (yeso) If yes, when (mm/dd) b. Inferior conjunction (yeso) If yes, when (mm/dd) 4 PHYS 202A