That was one of my first thoughts...making it with cotton, only problem is that you need to paint it. I thought that the cotton wouldn't look as good as it is without the paint. The toothpicks as struts is a great idea.
With the ball idea I have....
I figured if I cut the expanding ball in half, flip it upside down so it was a closed bowl shape and then when I get to the planet, flip it over and expand it so it covers the planet.