# How Apollo Left Earth (And Returned)

This is the fifth in my occasional series of posts about the orbits followed by the Apollo spacecraft as they departed from (and returned to) the Earth. It’s a companion to, and expansion of, my old post “How Apollo Got To The Moon”, informed by a more recent series of posts that culminated in my deriving a set of orbital elements for Apollo 11’s departure towards the moon.

That series started with a post entitled “Keplerian Orbital Elements”, which introduced the various parameters used to describe an orbit—these are the numbers you need to plug into a piece of orbit-plotting software, like Celestia, so that it will display the spacecraft’s trajectory for you. (It’s what I used to prepare the diagram at the head of this post.)

Then I progressed to “Finding Apollo Trajectory Data”, in which I provided links to the original Apollo documentation, and described how to pull the necessary data from those sources.

Then I digressed into “The Advent Of Atomic Time”, as a way of introducing the difference between the GMT times listed in the Apollo trajectory documents, and the Terrestrial Time (TT) we need to use in order to correctly describe the Apollo orbits.

Most recently, I offered a fairly equation-intensive post entitled “Converting Apollo State Vectors To Orbits”, in which I drew together the principles established in the first three posts and gave a worked example, deriving the orbital elements of Apollo 11’s departure from Earth.

The logical progression, at this point, would be to subject you to another blizzard of equations, showing how to use those orbital elements to calculate the orbital position of a spacecraft at any given time, how to convert those positions to a ground track, and how to transform the geographical coordinates of the ground track into geomagnetic coordinates, so as to plot a trajectory relative to the Van Allen Radiation Belts. But I’m going to skip all that for now, and instead just show you some actual results.

The reason all this stuff about ground tracks and so on is important is because the static image at the head of this post can’t tell the whole story—because while Apollo 11 moved along its orbit (the red curve in the picture), the Earth and the Van Allen Radiation Belts rotated beneath it. This produced some interesting dynamics, which I can demonstrate in a little (30-second) video showing Apollo 11’s view of Earth during the first ten hours of its flight to the Moon, speeded up 1000 times. The animation was produced in Celestia, using the orbital elements I derived at the end of my previous post on this topic.

The journey begins at the moment of Translunar Injection (TLI), when the Apollo S-IVB stage finished its second burn, having accelerated the Apollo spacecraft into an orbit that would take it to the Moon. We’re looking straight down at the night-time Pacific, which fills the screen. But very soon our viewpoint shoots into daylight, travelling west-to-east over the United States, where it then seems to loiter for a while above the Caribbean, before we see the Earth apparently, and belatedly, start to rotate in its normal fashion beneath the retreating spacecraft. So our view of the Earth tracks quickly west-to-east, pauses, and then begins to drift slowly east-to-west.

What’s going on there? The reversal in relative motion is caused by the shape of Apollo’s elliptical orbit. It starts off travelling very fast from west to east, and almost parallel to the Earth’s surface, so that it overtakes the rotating Earth—the Apollo 11 astronauts in fact saw the sun rising in the east ahead of them as their orbit carried them into daylight over the USA.

But their orbital trajectory was rising and slowing, and as their velocity decreased it also became directed more away from the Earth (towards the Moon!) rather than parallel to its surface. Like this:

Eventually, as their trajectory carried them almost directly away from the Earth, they were able to watch it make its usual 24-hour rotation behind them. But there was an intermediate stage, between the extreme situation in which they overtook the Earth’s rotation, and the “normal” view they obtained later—for a while, their velocity approximately offset the Earth’s rotation, so their viewpoint loitered for an hour or two over the Americas.

We can see how this played out using a couple of graphs, prepared from the orbital elements of Apollo 11’s departure trajectory. Below, I plot the spacecraft’s velocity and flight angle during the first six hours of its translunar trajectory. (The flight angle is the angle between the trajectory and the local horizontal. A flight angle of zero corresponds to an orbit parallel to the Earth’s surface. A flight angle of ninety degrees indicates a vertical trajectory.)

Apollo’s velocity, marked in blue and plotted against the left axis, starts at almost 11 kilometres per second, but decays steadily under the influence of Earth’s gravity. Its flight angle (red, right axis) begins with a slight upward tilt of about seven degrees, but quickly progresses towards the near-vertical (over 70 degrees) as it draws away from Earth.

Putting these two factors together, we can plot the angular velocity of the Apollo spacecraft as it moves around its orbit and compare that to the constant angular velocity of the rotating Earth:

We can see how Apollo moved faster than the Earth’s rotation for about an hour after TLI, but for the next hour was moving only a little faster or a little slower than the Earth, so that it would appear to hang in the sky over the Earth’s surface for a while.

Here, in blue, is Apollo 11’s ground track for its first six-and-a-half hours after TLI. The circles mark off intervals of one hour along the track. In green, I’ve superimposed latitude and “longitude” lines for the Earth’s geomagnetic field.*

At the time of the Apollo 11 mission, the north geomagnetic pole was situated close to the entrance to the Nares Strait, between Greenland and Ellesmere Island, and the geomagnetic field was correspondingly tilted southwards over the Americas, taking the Van Allen Belts with it. The inner VAB, which contains the bulk of the dangerous proton radiation, lies mainly between magnetic latitudes forty degrees either side of the magnetic equator—I’ve shaded that region in yellow. As previously described in my post “How Apollo Got To The Moon”, you can see that Apollo’s departure orbit passed north of that critical magnetic latitude band during its first hour, and entirely avoided the region of most intense radiation near the magnetic equator.

Armed with Apollo’s orbital elements, we can get a better view of the spacecraft’s passage through the VAB by converting its geographical coordinates and distance from the centre of the Earth into magnetic coordinates. After doing that we can plot its orbital radius and magnetic latitude and superimpose that on a diagram of the Van Allen Belts. I’ve adapted and coloured the VAB diagrams from NASA’s Bioastronautics Data Book, Second Edition (1973).

Here’s the Apollo 11 trajectory relative to the electrons trapped in the VAB:

The electrons outline the inner and outer radiation belts—the intense inner VAB is show in red and orange, the green band is a region of relatively decreased radiation, and then we have the larger but less intense outer VAB in yellow-brown. Apollo 11 traversed the entire region in about an hour, but was in minimal danger from electrons, which are easily blocked by the structure of the spacecraft.

More dangerous was the energetic and penetrating proton radiation, largely confined to the inner VAB:

I’ve marked the “danger zone” along the Apollo trajectory in red, but you can see that it traversed only the fringes of the inner VAB, avoiding the core area of high radiation. With reference to the specific diagram above, I find that Apollo 11 was within the sketched limits of the proton VAB for a total of 14 minutes, starting three minutes after TLI. It’s important, though, to realize that the Van Allen Belts are very variable structures, so we shouldn’t read too much into specific radiation counts on specific charts. But we can get the general message from this diagram that the radiation dose to the Apollo 11 crew members was limited by both the speed with which they traversed the VAB, and by using an orbit that avoided the most intense regions.

I can now go back to my ground track diagram, and show the red “danger zone” section on that:

It illustrates, from a different viewpoint, how the Apollo departure trajectory exploited the tilt of the Earth’s magnetic field so as to pop northwards out of the VAB as soon as possible.

I’ve also marked another event along the early departure orbit—the Transposition, Docking and Extraction manoeuvre (TD&E), during which the Apollo Command and Service Module turned around, docked with the Lunar Module in its stowed position atop the Saturn S-IVB stage, and extracted it. (See my link for a more detailed explanation.) The whole procedure typically took about an hour, and you can see it started quite soon after Translunar Injection—about thirty minutes later, in this case.

Why the rush to get going with that? One reason, I think, was to make sure that the Apollo spacecraft and the spent S-IVB stage started moving apart as soon as possible, to avoid the danger of collision during later spacecraft manoeuvres. But it was also handy that the whole operation could be carried out during Apollo’s “loiter” over the Caribbean, within line-of-sight radio transmission of the United States, thereby avoiding the potential problems involved in relaying radio messages through the other NASA ground stations dotted around the globe.

All this happened in reverse when Apollo returned to Earth, with a “loiter” in the ground track taking place over the southern Indian Ocean. And this time Apollo was approaching the Pacific Ocean from the south, again exploiting the tilt of the VAB, except with an even more inclined orbit—close to 40° inclination to the Earth’s equator, compared to 30° on departure. Like this:

Six hours before entering the atmosphere, the spacecraft was 80,000 km above Western Australia. Its ground track then swung out over the southern Indian Ocean and loitered for a bit, before turning back, gathering speed and diving through the inner VAB while it recrossed Australia, heading for splashdown in the Pacific.

With the specific plot used here, they spent just nine minutes shooting through the outer rim of the inner VAB, then another three minutes in free flight above the Pacific before hitting the “entry interface”—the point at which the Earth’s atmosphere began to have a significant effect on their trajectory, at an altitude of 400,000 feet.

Almost all the Apollo trajectories followed a very similar pattern—north over the VAB on the way out, south under the VAB on the way back. But there was one striking exception, the trajectory taken by Apollo 17. That’s what I’ll write about next time.

* The geomagnetic poles define the overall tilt of the Earth’s magnetic field, and are different from the magnetic “dip” poles towards which your compass needle points. Scientific American has a discussion of the difference between the two magnetic poles here. The World Data Center For Geomagnetism in Kyoto provides some nice maps showing the recent wanderings of the north and south magnetic poles and geomagnetic poles. To calculate the magnetic latitudes of the Apollo trajectories, I used a little program called GM POLE, from the National Oceanic and Atmospheric Administration, which provides the coordinates of the geomagnetic poles on any given date between 1900 and 2015.

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