Planets orbit the star now. You're probably looking at the line showing their vector for that.

 Paladin. The lines you speak of a white and slightly curved and coming right out of each planets gravity well? 

These lines indicate the direction a planet is traveling which is why they are slightly curved. Because usually planets orbit a star (sun) in a circle. Additionally the longer the line is the faster the planet is orbiting.

There is a button in the top right of the screen that you press to see where those planets will end up over the next period of time. When you press it you will see they follow the basic curve of the white line.

I recently had a moon orbit a planet instead of the sun. Which meant the white line I don't think was curved enough to show it's very sharp orbit. I'd have to look again next time though. I don't think I noticed much of a difference in curve. It may have been a little sharper.