Cut out a circular piece of cardboard about 1 foot across. Paint one side of the cardboard black using craft paint and a medium paintbrush. Allow this to dry, then stipple it with paint to create stars by dipping a clean paintbrush in white paint and lightly tapping the brush over the cardboard.
Mark the unpainted side of the cardboard with a compass to approximate the location of the planets. Create a mark in the center of the cardboard. Move four inches from the center mark, and, using the compass, create four rings around the cardboard, each about a half-inch away from the previous circle. Skip another three inches and create four more rings, each about a half-inch apart.
Punch a hole in each of the rings and the center mark using the sharp tip of a pair of scissors. Stagger the location of the holes so the planets surround the center mark and the cardboard will be evenly weighted. Punch four holes at the outer edge of the cardboard -- 12, 3, 6 and 9 if the cardboard were the face of a clock -- which are used to hang the model.
Paint nine foam balls of different sizes to represent the planets and the sun. Paint the largest ball yellow to represent the sun, with slightly smaller balls for Jupiter, Saturn, Uranus and Neptune. Choose slightly smaller balls for Earth and Mars and the smallest balls for Mercury and Venus.
Tie the end of a piece of 12-inch string around a paperclip. Feed the other end of the string through the top, unpainted side of the cardboard into the center hole and pull the string through until the paperclip rests on the cardboard. Repeat this until string has been fed through all of the planet holes.
Tie the loose end of string onto a straight pin. Feed the straight pin into the foam ball to hang the planet. Place the sun in the center and each planet according to its position in the solar system from the sun.
Tie four pieces of string, each about 2 feet long, into the four holes at the outer edge of the cardboard. Tie the loose ends of the string together on the top side of the cardboard, around a split key ring. Use the key ring to hang the model.