The Museum of Natural History has a policy of "pay what you like, but you must pay something". So you don't have to pay the full admission price. Just tell them what you want to pay, and pay that. (You can't do this online, of course. You must actually go down to the museum, stand in a long line, and pay at the counter.)
This admission is good for both the museum and the Rose Center of Space Exploration - the home of the Hayden Planetarium. However, if you want to see a planetarium show, you must pay full price for that. You also must pay full price for admission to any special exhibit at the museum.