Solved: Navigating in the Solar System. The Mars Polar

QUESTION:

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: x y z Earth 0.3182 AU 0.9329 AU 0.0000 AU Mars 1.3087 AU -0.4423 AU -0.0414 AU With these coordinates, the sun is at the origin and the earths 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 * 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.)