Curves in space

Things are different out there.

Our astronomer has recently been advising an aspiring science-fiction author on his science.  The latter freely admits to a sparse background in science and math, but wants very much to get his details correct.  This has often meant going through the same subject matter as our tutoring consultant’s Physics students; indeed, our tutor is making a collection of the calculations our astronomer has made, to set them as interesting problems for such students.  Several of them are about orbits.

The picture of planets orbiting the Sun, following a nearly circular path and held in place by the Sun’s gravity, is generally known.  The fact that as you go farther from the Sun the planet takes longer to go around is also pretty familiar; less common is the realization that this comes from both the orbit being bigger and the planet moving more slowly.

But strange things happen when you change orbits.  Suppose you’re in a nice circular orbit, at the proper speed to keep that distance from the Sun.  You point your spaceship’s nose in the direction you’re going and fire your thrusters.  Of course you speed up, right?  But you also move into a higher orbit, farther from the Sun, and by trading off kinetic for potential energy wind up slowing down.  Similarly, if you fire retro-rockets to slow yourself down, you fall into a lower orbit and wind up speeding up.

Suppose, then, you’re on Earth and you want to get to Mars.  You can’t just point yourself at where it sits in the sky and go.  Once off this planet you fire your thrusters to get you into a higher orbit, actually an ellipse whose lowest point touches Earth’s orbit and highest point touches Mars’; when you get there you’ll need to fire them again to match speeds with that planet.  Except that Mars probably won’t be there.  You have to leave Earth at just the right time for your elliptical orbit to get you to Mars, and not just Mars’ orbit.  For Earth and Mars, these opportunities come about 26 months apart, so you’ll see flocks of Mars-probes setting off more or less together, every two years and a bit.  (This is a description of the Hohmann Transfer Orbit, worked out in the 1920s, decades before there was any prospect of actually doing it.)

Suppose, now, we’re outside our space station in Earth orbit, doing some maintenance or repair work while in our space suits.  It’s not going well, and in exasperation your partner flings her crescent wrench away, directly at the blue-white planet below.  It’s long fall, but of course it’s lost, and will be deducted from her pay next month.  Well, no.  The wrench is doing several kilometers per second sideways as it orbits along with the station; throwing it down just modifies its orbit slightly, and after a certain time it will meet up again with station, astronaut and all.

So space travel is not like sea travel, or anything we have experience with on the surface of our planet.  It’s different out there.