How Apollo Got To The Moon

I’m posting this at 13:32 GMT on 16th July 2019—exactly fifty years after the launch of Apollo 11. It’s the last part of a loose trilogy of posts about Apollo—the first two being M*A*S*H And The Moon Landings and The Strange Shadows Of Apollo. This one’s about the rather complicated sequence of events required to get the Apollo spacecraft safely to the moon.

To get from the Earth to the moon, Apollo needed to be accelerated into a long elliptical orbit. The low point of this orbit was close to the Earth’s surface (for Apollo 11, the 190 kilometres altitude of its initial parking orbit); the high point of the ellipse had to reach out to the moon’s distance (380,000 kilometres), or even farther.

Extremely diagrammatically, it looked like this:

To be maximally fuel-efficient, the acceleration necessary to convert the low, circular parking orbit into the long, elliptical transfer orbit needs to be imparted at the lowest point of the ellipse—that is, on exactly the opposite side of the Earth from the planned destination. Since the moon is moving continuously in its orbit, the translunar trajectory actually has to “lead” the moon, and aim for where it will be when the spacecraft arrives at lunar orbit, about three days after leaving Earth.

Here’s the real elliptical transfer orbit followed by Apollo 11, drawn with the moon in the position it occupied at the time of launch (you’ll need to enlarge it to see detail):

(For reasons I’ll come back to, NASA gave the Apollo spacecraft a little extra acceleration, lengthening its translunar transfer ellipse so that it would peak well beyond the moon’s orbit.)

And here’s the situation three days later, with Apollo 11 arriving at the moon’s orbit just as the moon arrives in the right place for a rendezvous:

With the proximity of the moon at this point, lunar gravity in fact pulled the Apollo spacecraft away from the simple ellipse I’ve charted, warping its trajectory to wrap around the moon—something else I’ll come back to.

In the meantime, let’s go back to the fact that NASA needed to manoeuvre the Apollo spacecraft to a very exact position, on the opposite side of the Earth from the position the moon would occupy in three days’ time, and then accelerate it into the long elliptical orbit you can see in my diagrams. The process of accelerating from parking orbit to transfer orbit is called translunar injection, or TLI.

The point on the Earth’s surface opposite the moon at any given time is called the lunar antipode. (This is a horrible word, born of a misunderstanding of the word antipodes—I’ve written more about that topic in a previous post about words.) But, given that I don’t want to keep repeating the phrase “on the opposite side of the Earth from where the moon will be in three days’ time”, from now on I’ll use the word antipode with that meaning.

So TLI had to happen at this antipode, and NASA therefore needed to launch the Apollo lunar spacecraft into an Earth orbit that at some point passed through the antipode. Not only that, but they needed to do so using a minimum of fuel, and needed to get the spacecraft to the antipode reasonably quickly, so as to economize on consumables like air and food, thereby keeping the spacecraft’s launch weight as low as possible.

Now, the moon orbits the Earth in roughly the plane of the Earth’s orbit around the sun—the ecliptic plane. But the moon can stray 5.1º above or below the ecliptic. And the ecliptic is inclined at about 23.4º to the plane of the Earth’s equator. So the moon’s orbital plane can be inclined to the Earth’s equator at anything from 18.3º to 28.5º. This means the moon can never be overhead in the sky anywhere outside of a band between 28.5º north and south of the equator, and therefore its antipode is confined in the same way—always drifting around the Earth somewhere within, or just outside, the tropics.

The Cape Kennedy launch complex (now Cape Canaveral), lies at  28.6ºN. The most energy-efficient way to get a spacecraft into Earth orbit is to launch it due east, taking advantage of the Earth’s rotation to boost its speed. Such a trajectory puts the spacecraft into an orbit inclined at 28.6º to the equator. So a launch from Kennedy put a spacecraft into an orbit inclined relative to the plane of the moon’s orbit. The inclination might be a fractional degree, if the moon’s orbit were tilted favourably close to Kennedy; but generally it would be significantly larger than that, with the spacecraft’s orbit passing through the plane of the moon’s orbit at just two points.

As it happens, the situation at the time of the Apollo 11 mission shows all these angles between equator, ecliptic, moon’s orbit and Apollo parking orbit quite clearly, because all the tilts were roughly aligned with each other. Here’s a view from above the east Pacific at the time of Apollo 11’s launch: 13:32 GMT, 16 July 1969:

The red line is the ecliptic, the plane of Earth’s orbit around the sun. From the latitude and longitude grid I’ve laid on to the Earth, you can see how the Earth’s northern hemisphere is tilted towards the sun, enjoying northern summer. The plane of the moon’s orbit (in cyan) is carrying the moon above the ecliptic plane on the illuminated side of the Earth, so that the angle between the Apollo 11 parking orbit and the moon’s orbital plane is relatively small.

It wasn’t always like that, though. Here’s the situation at Apollo 14’s launch: 21:03 GMT, 31 January 1971. It’s southern summer, and the plane of the moon’s orbit crosses Australia, so Apollo 14’s parking orbit passes through the plane of the moon’s orbit at a fairly steep angle.

Whatever the crossing angle, NASA needed to launch the Apollo moon missions so that the spacecraft’s orbit took it through the moon’s orbital plane at the same moment the antipode drifted through that crossing point. And in order to economize on consumables, that needed to happen within the time it took to make two or three spacecraft orbits, each lasting an hour and a half. This requirement dictated that there was always a launch window for each lunar mission—any launch that didn’t take place within a very specific time frame had no chance of bringing the spacecraft and the antipode together to allow a successful TLI.

At first sight, it seems like the launch window should be vanishingly narrow, given that the parking orbit intersects the moon’s orbital plane at only two points, only one of which can be suitable for a TLI at any given time. In fact, by varying the direction in which the Saturn V launched, NASA was able to hit a fairly broad sector of the lunar orbital plane. Launching in any direction except due east was less energy-efficient, but with additional fuel the Apollo spacecraft could still be placed in orbit using launch directions 18º either side of due east. The technical name for the launch direction, as measured in the horizontal plane, is the launch azimuth. So Apollo could be launched on azimuths anywhere between 72º and 108º east of north.

You can see this range of orbital options drawn out in sinusoids on Apollo 11’s Earth Orbit Chart:

Cape Kennedy is at the extreme left edge of the chart, and all the options for launch azimuths between 72º and 108º are marked. Here’s a detail from that edge:

Notice how launches directed either north or south of east take the spacecraft to a higher latitude than Cape Kennedy’s, and therefore into a more inclined orbit—at the extremes, Apollo orbits were inclined at close to 33º.

So NASA could take aim at the antipode by adjusting the launch direction. By launching north of east, they could hit a more easterly antipode; by launching south of east, they could hit a more westerly antipode. This range of options allowed a launch window spanning about four hours. A launch early in the launch window would involve an azimuth close to 72º, as the launch vehicle was aimed at the antipode in its most extreme accessible eastern position. During the four-hour window, as the moon moved across the sky from east to west, the antipode would track across the Earth’s surface in the same direction, and the required launch azimuth would gradually increase, until the launch window closed when an azimuth of 108º was reached. NASA planned to have their launch vehicle ready to go just as the launch window opened, to give themselves maximum margin for delays. Apollo 11 launched on time, and so departed along an azimuth very close to 72º.

Here’s the Apollo 11 launch trajectory:

The huge S-IC stage (the first stage of the Saturn V) shut down and dropped away with its fuel exhausted after just 2½ minutes, falling into the western Atlantic (where one of its engines was recently retrieved from 4.3 kilometres underwater). The S-II second stage then burned for 6½ minutes before falling away in turn, dropping in a long trajectory that ended in mid-Atlantic. Meanwhile, the S-IVB third stage fired for another two minutes, shoving the Apollo spacecraft into Earth orbit before shutting down at a moment NASA calls Earth Orbit Insertion (EOI). The astronauts then had about two-and-a-half hours in orbit (completing about one-and-three-quarter revolutions around the Earth) before their scheduled rendezvous with the lunar antipode over the Pacific. This gave them time to check out the spacecraft systems and make sure everything was working properly before committing to the long translunar trajectory.

At two hours and forty-four minutes into the mission, the S-IVB engine was fired up again, and worked continuously for six minutes as Apollo 11 arced across the night-time Pacific. Here’s that trajectory with the S-IVB ignition and cutoff (TLI proper) marked, as well as the plane of the moon’s orbit and the position(s) of the antipode(s). On this occasion I’ve marked the true lunar antipode as “Antipode”, and the antipode of the moon’s position in 3 days’ time as “Antipode+3”.

See how Apollo 11 accelerated continuously through the lunar orbital plane, clipping neatly past the three-day antipode. The velocity change in those six minutes took the spacecraft from 7.8 kilometres per second (the orbital speed of the parking orbit) to the 10.8 kilometres per second necessary for the planned translunar trajectory.

I promised I’d come back to the reason NASA used extra energy to propel the spacecraft into an orbit that would take it well past the moon, if it were not captured by the moon’s gravity. In part, because it speeded the journey—Apollo took three days to reach its destination, rather than five. But the main reason was to put Apollo on to a free-return trajectory. It shaved past the eastern limb of the moon and then (held by the moon’s gravity) looped around behind it. If it had not fired its engine to slow down into lunar orbit at that point, it would have reemerged from behind the western limb of the moon and come straight back to Earth. So there was a safety feature built in, if the astronauts encountered a problem with the main engine of their spacecraft—any other arrival speed would have resulted in a free-return orbit that missed the Earth.

Another safety feature of the Apollo orbits was their inclination of around 30º to the equator, which was maintained as the spacecraft entered its transfer orbit. This meant that the spacecraft avoided most of the dangerous radiation trapped in Earth’s Van Allen Belts.

The Van Allen belts are trapped in the Earth’s magnetic field, which is tilted at about 10º relative to Earth’s rotation axis—and the tilt is almost directly towards Cape Kennedy, with the north geomagnetic pole sitting just east of Ellesmere Island in the Canadian Arctic.

This means that a spacecraft launched from Cape Kennedy, with an orbital inclination of 30º to Earth’s equator, has an inclination of about 40º to the geomagnetic equator. A departure orbit with that inclination rises up and over the Van Allen belts, passing through their fringes rather than through the middle. Of course, since the Earth rotates while the spacecraft’s orbital plane remains more or less fixed in space, it needs to depart within a few hours, otherwise it will lose the advantageous tilt of the radiation belts—but Apollo already had good reason to get going so as not to waste precious consumables.

To finish, here are a couple of diagrams I’ve prepared with Celestia, using an add-on created by user Cham. The add-on shows the Earth’s magnetic field lines, and the calculated trajectory of a few charged particles trapped in the radiation belt. I’ve used a subset of Cham‘s particle tracks, so I can show the position of the inner Van Allen Belt clearly—it’s the one that contains the high-energy protons which were of most danger to the astronauts.

Here’s Apollo 11’s departure orbit (red line) seen from above the Pacific; the plane of the moon’s orbit is also shown, in cyan. The plot is for the time of translunar injection.

And here’s a side view.

(You can ignore the lower part of the orbit, which is only there to show the full elliptical shape—Apollo 11 followed the upper, northern trajectory, starting from the vicinity of the equator.)

So that’s how Apollo got to the moon.

10 thoughts on “How Apollo Got To The Moon”

1. Very fascinating as usual. 50 years ago I’d no idea how complicated was the physics of the Apollo 11 flight. Like most people on earth I was almighty impressed. Your account is clear and good to have .
( As Armstrong’s foot was stepping down from the capsule my phone rang and a patient who obviously was not caught up in the drama of the occasion announced ” I feel awful” ) !

2. Oikofuge says:

I’m glad you found it clear. Celestial mechanics is sometimes difficult to explain without moving your hands!

3. Thank-you *so* much for this… i was pulling my hair out trying to understand how the Apollo 10 spacecraft could be at ~30N to the Equator when it took this picture https://eol.jsc.nasa.gov/SearchPhotos/photo.pl?mission=AS10&roll=34&frame=5026 at 160,000km at approx 15h30GMT on May 20 (I had to work that out too…)

After searching a ton you are the only person I’ve found who is plotting the Apollo missions in 3D… everyone else is just doing ‘plan’ views from (presumably) 90 degrees to the Ecliptic.

Do you know if anyone has done a 3D animation of any of the Apollo trajectories end-to-end?

Antony

1. Oikofuge says:

Glad it was useful.

Apollo 17 took a famous photograph which is complementary to the Apollo 10 view in your link. Apollo 17 was the only moon mission that launched at night and had its translunar injection over the Atlantic (rather than the Pacific). So it was the only Apollo mission that followed a trajectory south of the equator, rather than north, as is evident in their view of Africa.

I don’t know of any realistic 3D animations of the Apollo trajectories. It was the sort of thing people looked into when we were developing Celestia, but the trajectory data just don’t seem to be available in an accessible format. Nowadays, you can download trajectories for all sorts of spacecraft in the standard SPICE format–but it was all a bit Wild West back when Apollo was flying.

4. Many thanks for your prompt reply and addition info – interesting.

I guess I’m surprised after all the efforts people put into the 50th Anniversary last year no one did a 3D model / simulation / animation of at least Apollo 11’s total journal take-off to splash-down

Would have made a great addition to the folks who did the real-time replay of all the audio and video! 🙂

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