Navigating in the Solar System. The Mars Polar Lander spacecraft was launched on January $3,1999 .$ On December $3,$ $1999,$ the day that Mars Polar Lander touched down on the Martian surface, the positions of the earth and Mars were given by these coordinates:
$$\begin{array}{lll}{\text { Earth }} & {0.3182 \mathrm{AU}} & {0.9329 \mathrm{AU}} & {0.0000 \mathrm{AU}} \\ {\text { Mars }} & {13087 \mathrm{AU}} & {-0.4423 \mathrm{AU}} & {-0.0414 \mathrm{AU}}\end{array}$$
In these coondinates, the sun is at the origin and the plane of the earth's orbit is the $x y-$ plane. The earth passes through the $+x$ -xis 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 eqnal to $1.496 \times 10^{8} \mathrm{km}$ , the average distance from the earth to the sun. (a) In a diagram, show the positions of the sun, the earth, and Mars on December 3,1999 .(b) Find the following distances in AU on December $3,1999 :$ (i) from the sun to the earth; (ii) from the sun to Mars; (iii) from 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 location at midnight on December $3,1999 .$ (When it is midnight at your location, the sun is on the opposite side of the earth from you.)