DATA ?Navigating in the Solar System. The Mars Polar Lander spacecraft was launched on January 3, 1999. On December 3, 1999, the day ?Mars Polar Lander impacted the Martian surface at high velocity and probably disintegrated, the positions of the earth and Mars were given by these coordinates: With these coordinates, the sun is at the origin and the earth’s orbit is in the ?xy -plane. The earth passes through the +x-axis once a year on the autumnal equinox, the first day of autumn in the northern hemisphere (on or about September 22). One AU, or ?astronomical unit?, is equal to 1.496 X 108 km, the average distance from the earth to the sun. (a) Draw the positions of the sun, the earth, and Mars on December 3, 1999. (b) Find these distances in AU on December 3, 1999: from (i) the sun to the earth; (ii) the sun to Mars; (iii) the earth to Mars. (c) As seen from the earth, what was the angle between the direction to the sun and the direction to Mars on December 3, 1999? (d) Explain whether Mars was visible from your current location at midnight on December 3, 1999. (When it is midnight, the sun is on the opposite side of the earth from you.)
Solution 100CP The coordinates of object A are (X ,Y ) and object B coordinates are (X ,Y ) , the A A B B vector RABfrom A to B has x component X BX anA y component Y -Y B A a) The positions of the sun, the earth, and Mars on December 3, 1999. Diagram clearly shows sun at origin and earth is orbiting around sun in xy plane b ) i)The distance(in AU) between sun and earth on December 3, 1999, 2 2 (0.3182) + (0.9329) =0.9857 ii)The distance(in AU) between sun and mars on December 3, 1999, (1.3087) + (0.4423) + ( 0.0414) =1.3820 iii)The distance(in AU) between earth and mars on December 3, 1999, 2 2 2 2 2 (0.3182) + (1.3087) + ( 0.9329) + ( 0.4423) + (0.0414) =1.695 c)the angle between the direction to the sun and the direction to Mars on December 3, 1999 is obtained from the dot product. = arccos( (0.3182)(1.3(0.9857)(1.695)0.4423)(0.9329)+(0) =54.6 0 d) Mars was not visible from your current location at midnight on December 3, 1999., because the sun -mars angle is less than 90
