Using a clip or a mask will achieve the effect. Clipping is like using a cookie cutter, it hides everything that falls outside the desired shape, here's how to do it:
1. Select the "curvy lines" (select + click will add to selection)
2. Combine (ctrl + K or Path -> Combine ... you can also group them but I recommend Combining) all the curvy bits into one object
3. Select the sphere/circle, and duplicate it (ctrl + D or Edit -> Duplicate). A new sphere will popup above everything else.
4. Select the duplicated sphere and the combined curvy bits and go to Objects -> Clip -> Set (The duplicated sphere must be above the curvy bits for this to work)
5. If you want, you can select the curvy bits and the original sphere and group them (ctrl + G) to make it easier to drag the planet around

That should do it. Hope I haven't made it sound too difficult...