原创 Fly me to the Moon (Part 4)

Read the previous part here.

This is not an unreasonable approach, as long as you don't mind the 90 W part. A vertical descent is what we used for Surveyor and other unmanned missions. It simplifies the automated landing quite a bit, and you can change that landing longitude by making only the slightest tweaks to the outbound trajectory.

The vertical descent is simple to automate, but it's not very efficient. When you're landing vertically, your rocket motor expends a lot of fuel fighting gravity. With a grazing, nearly horizontal approach, centrifugal force gives you some help countering gravity. In orbit, after all, it counters gravity completely.

So instead of landing vertically, it's better to go into orbit first. You can do this by tweaking the TLI a tiny bit, so the spacecraft passes behind the Moon. Better yet, pass in front of the Moon, and save more fuel yet. Or even give the trajectory a small inclination (less than 1 degree), pass above or below the Moon, and go into a polar orbit. Going into orbit lets us go to pretty much any landing site we choose. The price we pay is a more difficult job for the software that controls the automated landing.

KISS

Simplify as much as possible, but no more. —Einstein

Keep It Simple, Sidney. —Crenshaw (among others)

I presume you've figured out, by now, that the geometry of a lunar mission is not a simple one. What I must tell you now is that it's even more complicated than you thought. How complicated is it, you ask? Like figure 4 complicated.

FIGURE 4—ORBIT PLANES

This figure shows three of the important planes involved: the Equator, the Ecliptic, and the plane of the Moon's orbit. There's a fourth plane that I didn't even have the courage to try to draw: The plane of the trajectory, which can be (and usually is) different from all three that I've shown.

But the situation is even worse than you think. All the planes are moving. Our equatorial plane precesses around the Ecliptic with a period of 26,000 years. Well, Ok, I guess it's not that fast. But the Moon's orbit also precesses about the Ecliptic, with a period of only 18.7 years. Since Project Apollo, it's been around twice, changing the inclination of the Moon's orbit to the Equator by 5 degrees. A solution that was valid three years ago is worthless today.

Finally, the plane of the Moon's orbit isn't a plane at all. The solar perturbation includes out-of-plane components, with periods of one month and one year. Neither is the orbit plane a plane.

Why am I telling you all this? Because I want you to understand that the problem is complicated. What's more, the more accurate you want the result to be, the more complicated it gets. "Exact" models for the motion of the Earth and Moon can run as large as the 1400-term series of the Hill-Brown lunar theory. Then there's the 10,000 or so terms of the Earth's gravitational field, a similar number for the Moon, and terms describing the nutation and wander of the Earth's poles. Want light pressure and solar wind? Better put those in, also.

Computer simulations exist that model all of these effects, and will give you precision solutions, accurate to a gnat's hair. NASA has one. Analytical Graphics, Inc. (AGI) has a very excellent (and expensive) one called Satellite Tool Kit (STK). AGI has graciously agreed to supply a license to each of the Google Lunar X-Prize teams.

The problem with these high-precision models is that, as noted above, all the "constants" are really changing with time. Change the launch date, and you've changed them all. This may be fine—even essential—for the nominal trajectory that you need when the spacecraft and its booster are sitting on the launch pad. But it's a terrible choice for parametric studies during the design phase. That's why I tend to invoke the KISS philosophy, and look for useful approximations.

Do they exist? You bet. Over the years, we've developed quite a number of approximate methods, rules of thumb, back-of-the-envelope calculations, etc., that give us an understanding of the situation without having to run high-precision simulations.

I'll go even further than that. Novices use high-precision solutions, because they don't have a feel for which effects matter, and which ones don't. Experts (like Einstein) can use back-of-the-envelope calculations because they know what are the dominant effects.

Want an example? Consider that return-from-the-Moon thing. You want to reenter the Earth's atmosphere with a grazing entry? From my discussion on angular momentum, we know that you must leave the vicinity of the Moon with 188 m/s. Use any other value, and you won't get back.

Before I wrap this column, I'd like to describe some of the simplifications that we use.

Impulsive delta-Vs

Years ago, my son and I played around with model rockets of the Estes ilk. What surprised me was when we fired the rocket motor, we didn't get the kind of "whoosh" we were expecting. The rocket just made a very short "Pht," and it was gone. I think the motor had burned out even before the rocket had cleared its launch tower. The rest of the flight was all coast. Sort of like the circumlunar trajectory.

Real rockets make the same kind of "Pht," only longer and louder. Though longer, they're usually still quite short compared to the orbital period or the mission time.