原创 Fly me to the Moon (Part 5)

Physicists know that many problems can be treated as though the dynamical event happened instantaneously. Examples might include the impact of a hammer and nail, two billiard balls colliding, or a baseball bat hitting the ball. Deep down, we know that there is a complex interaction involve. Materials get deformed, forces get exerted, and velocities get changed. But we don't need to know the details or the time history of the collision. From our perspective, the scale of time is so short that it might as well be zero.

We call such collisions impulsive. From our perspective, the velocity used to have some value v. Now it has a new one, v+Δv. The velocity has changed by Δv. This model fits beautifully with the behavior of a rocket. When you watch a Space Shuttle take off, it certainly seems that the buildup of velocity is anything but instantaneous. But at a burn time of 8 minutes or so, it's still a very short time compared to the 90 minutes of the orbital period. So an impulsive Δv is not a bad approximation to the real rocket dynamics. What's more, Δv is the natural result value from Tsiolkovsky's Ideal Rocket Equation. The result doesn't depend on burn time; only on initial and burnout mass, and the energy density of the propellant.

How does Δv analysis help design a lunar trajectory? Well, try this. If we ignore the Moon completely, the trajectory must be a two-body orbit—an ellipse—which has a perigee of the Earth's radius, plus enough altitude to be outside the atmosphere. The apogee must be big enough to reach the Moon, some 384,400 km away.

For this orbit, the velocity at perigee must be about 11 km/s, implying a Δv over the orbital velocity of 3.1 km/s. Despite the primitive method of its calculation, you can take that number to the bank. At the other end, at apogee, the velocity will be 188 m/s (symmetry, remember?). There's your first estimate of the transfer orbit, and it didn't take an expensive simulation to get it.

We can do similar things for all the other needed maneuvers, namely LOI, de-orbit, and landing. Again, we recognize that the numbers that come out of a Δv analysis are first-guess estimates, but they turn out to be surprisingly good. And I think I know why.

When you look at figure 1, it's easy to see that the effect of the Moon's gravity is powerful—enough to bend the motion from counter-clockwise to clockwise. Because the spacecraft still has some radial velocity, its trip time from the point where it "feels" the Moon's presence to perilune is reasonably short.

However, the lower energy orbit shown in figure 3 is different. Relative to the Moon, it loops far up the y-axis, and approaches the Moon from that axis, as we've discussed. When the spacecraft gets near the Moon's orbit, the Moon is still far away from it. So far that it doesn't alter the trajectory much. And when it does, it only accelerates the spacecraft down along the -y axis, without changing its direction.

Patched Conic

Remember, the trajectory near the Earth looks very much like an Earth-centered ellipse, while near the Moon it looks like a Moon-centered hyperbola. The problem lies in stitching the two portions together in a mathematically meaningful way. The Patched Conic method gives us that way.

Because the mass of the Earth is greater—81 times greater—than the mass of the Moon, its gravity dominates, not only near the Earth but also far away from it. The Moon's gravity dominates only when the spacecraft is within a surface called the Sphere Of Influence (SOI). Laplace gave us a rigorous definition of this sphere. It's not really spherical, and its shape is complicated, but for practical purposes we can think of it as a sphere about 66,000 km in diameter. In the patched-conic method, we switch from Earth-centric to Moon-centric motion at the point where the trajectory pierces the SOI. To do this, we translate both the position and velocity vectors from Earth-relative to Moon-relative motion. The vector addition of the Moon's velocity drives the velocity hyperbolic.

The folks doing interplanetary trajectories routinely use the patched-conic method. In an interplanetary trajectory, the spacecraft spends all but a tiny fraction of its life moving under the influence of the Sun. The sphere of influence of a planet is tiny, on a planetary scale. That's because the mass of the planet, as great as it is, is still negligible compared to that of the Sun.

The method is a lot more questionable in the case of lunar trajectories, because the mass ratio between the Earth and Moon—about 1/81—is much larger. Even so, the method can be used for preliminary planning. I haven't used the patched conic method much, either during the Apollo days or now. But I'm still a big fan. I think it's a very clever idea, and should be considered when high accuracy isn't needed.

Wrapping up

I find the problem of lunar trajectories a very interesting one, on a couple of levels.

First, there's nothing new about it. Newton's laws of motion haven't changed since the 16th century. Neither has the state of the art in solving the two-body problem, the N-body problem, the three-body problem, or the restricted three-body problem. Our understanding of the classical RTBP hasn't changed; the Lagrange points are still out there.

The only thing that's changed, really, is the tools we have to solve the problem numerically. The math techniques for numerical integration haven't changed, but the way we apply them have changed so profoundly, it's just not possible to over-emphasize the matter. Newton, Lagrange, and their colleagues used pen and paper. Also, I presume, WhiteOut—or its 16th century equivalent—by the gallon.

In 1959 we used slide rules, mechanical desk calculators (adding machines on steroids), and one of the first practical, large-scale computers on the planet.

Today, we have computers on every desk. Very fast ones. Not only can they do the same calculations Lagrange did, they can do them in seconds, perhaps milliseconds. I calculated once that my 2.4GHz, dual-core computer could have calculated all the trajectories I did for Apollo, 10,000 times over, in the time it takes this computer to boot Windows. Why does it take so long to boot? We can talk about that another day.

Second, there's such a huge range between the complexity and difficulty of the "exact" model, and the practical methods we can use to deal with it. To model the Earth-Moon system exactly, we first need extremely accurate data for the paths the planets follow—their ephemerides. We also need accurate models for the gravitational fields of the Earth and Moon, for light pressure and solar wind, and all manner of other effects. Including relativity.

On the other hand, we can get decent, back-of-the-envelope estimates about the trajectory with much cruder methods and models, ranging from the RTBP to the patched-conic method to the crudest of rules of thumb like the "angular momentum" requirement that drives the symmetry of the circumlunar trajectory.

In short, the more we work the problem, the better we understand it, and the more we can see how to apply approximate methods to get in the ball park. I suppose this range of methods, of varying degrees of fidelity, exist for other challenging problems, but from my perspective, the range between high accuracy methods and rules of thumbs is profound.

As we continue to work the problem, our understanding of it continues to grow.

And that, dear readers, is how I spent MY summer vacation last year. We hope you liked the Show-n-Tell.

Endnotes:

 1. "Trajectory Considerations for Circumlunar Missions," with W. H. Michael, presented at Inst. Aerospace Sci. 29th Annual Meeting, New York, January 1961. IAS Paper #61-35.