More on Orion

Based on an initial simulation it seems that the Apollo 16 Lunar Module Orion, abandoned in lunar orbit in 1972, smashed into the surface of the Moon about 5 weeks later. Would it be possible to locate the impact crater? Let’s get the best estimate we can for the initial orbit, run a randomized series of simulations, to try to narrow the search.

In previous investigations I have relied on the Mission Reports to provide the initial orbit state. For Apollo 16, the record is much richer. In particular, there are orbital state vectors (i.e., position and velocity information) for each of the hundreds of photographs taken by the “Metric Camera” experiment, which was in the science bay of the Command Module. In particular, this file has all the state information for every photo taken during the mission. 

A subset of these is of special interest…those from the “Rev 60” and “Rev 63” mapping passes. This refers to the fact that the photos were all taken during the 60th or 63rd revolution of the Apollo 16 CSM around the Moon. By this time John Young and Charlie Duke had returned from the lunar surface, and rendezvous and docking took place during Rev 53.  The LM was jettisoned during Rev 62, so the Rev 60 and 63 mapping passes bracket the last known position of Orion.

I grabbed a set of the Rev 60 data points and adjusted a simulation so that I fit them as closely as I could. Though trial and error I found that I could match them to within +/- 20 meters of altitude. Based on the documentation, this original state vector data was fitted in 1972 to a solution for the orbit that was generated using the best gravity model and ephemeris available at that time. I’m using a better gravity model and a modern ephemeris, so I’m not surprised that I don’t match up exactly. A residual error of +/- 20 meters seems pretty good.

Then I did the same thing for Rev 63 data points to get another estimate of the CSM orbit after the LM was jettisoned. With these estimates for the orbit, I can propagate to the moment of jettison and compare to the values published in the Mission report. (The simulator can run “backwards”, so it can reverse propagate from Rev 63 just as easily as it propagates forward from Rev 60.) The results are shown in Figure 1.

Figure 1: Comparing Orion's initial state, from 3 different sources.

Not bad! The positions are all within 0.5 km of each other, and the velocities match up well. The velocity difference when working backwards from Rev 63 is slower than the others, which is expected…it accounts for the ~2 ft/sec separation maneuver performed after Orion was jettisoned. 

The next step is to compensate the horizontal flight path angle, i.e., the HFPA. Orion was jettisoned “upwards”, away from the Moon, at about 2 f.p.s. and so this slightly increases the angle that it is moving relative to the local horizontal direction. It works out to be about 0.021 degrees, so adding this to the HFPA gives 0.42 degrees.

Now we can generate random sets of initial conditions that are similar to these and see where the impacts occur. I adopted the "Rev 60" state as the “nominal” case, and then varied randomly around those values. I generated a total of 350 randomized parameter sets and ran them all. Each run takes about 8 minutes to simulate, and I have a python script that automatically runs then one after another, so the whole batch completes in about 2 days. Each run produces a ground track file, and I post-process all of these as a batch to extract the impact locations.

What are the results? All 350 simulated impacts occurred between May 28th, and June 2nd of 1972. That's a fairly tight cluster. Looking deeper, there are some things that were expected, and other things I found surprising. Figure 2 is a plot of the impact date versus impact longitude for all 350 trials. The first thing to notice is the overall trend. Impacts that occur earlier are farther to the East, and the later ones are progressively farther West. This is to be expected, because the Moon is rotating Westward under the orbit as it destabilizes. (The impacts are also focused in a narrow band of latitudes, from 8.5 to 10 °N, which is under the plane of the orbit.)

Figure 2: Impact Longitude versus Date. Later impacts occur farther West.

Then notice the next trend…the impact longitudes occur in bands. For instance, notice that near the end of May 29th, the impacts are mostly around 104 °E, and then during May 30th they shift and are clustered around 99 °E. This clustering also isn’t surprising…it’s a consequence of the lunar terrain. The impacts always occur on the highest nearby terrain as the orbit drops lower and lower, so these impact clusters correspond to tall features of the Moon’s surface. Figure 3 below shows the latitude and longitude of the impacts (as red dots) superimposed on a map of the Moon, and it’s obvious that the dots are clustered around craters and mountainous areas. (At this scale it's hard to see that there are a lot of overlapping dots.)

Figure 3: Impact locations superimposed on a map of the Moon.

One thing I did find surprising is that the impact times are also quantized…they tend to be concentrated into buckets separated by about two hours. A lot of the individual impacts are overlapping on the plot in Figure 1 above, so it’s easier to see this time quantization if we zoom in on a few impacts like in Figure 4 below. Now you can see that there are 6 trials that resulted in impacts around 20:00 hours (10 PM) on May 29th, then 3 others around 22:00, then another 4 around Midnight, and so forth. On the one hand, the two-hour separation makes sense. If the spacecraft barely misses a mountain on one pass, it is likely to be lower, and therefore to impact that mountain on the next pass, which will come two hours later. (The time to complete one revolution is about two hours.) But all the trials have slightly different orbits and slightly different periods, so after 5 weeks orbiting the Moon all those virtual spacecraft in all those trials should have spread out, like race cars around an oval track in a long race.  

Figure 4: Details of impacts occurring around midnight on May 29th, showing how the impacts are occurring in clusters separated by about 2 hours.

It turns out that they do spread out, but the ones that are in similar orbits, with similar orbital periods, tend to impact around the same time. So, the impact times are sorted by orbital period. I would not have guessed this, but in hindsight it also makes sense. It turns out that the impact times depend on the energy of the initial orbit. Lower energy results in earlier impact, and vice versa. The energy of the orbit also determines the orbital period. Therefore, impacts that occur around the same time, which started with similar energy, also have the same period, resulting in the clustering of the impact times. Figure 5 below shows a plot of initial orbit altitude versus impact time, and there is a clear relationship. (The altitude plotted is the average of the initial apolune and perilune altitudes, which directly ties to the orbital energy.)

Figure 5: The time of impact correlates to the initial energy of the orbit, which also relates to the orbital period. This explains the clustering of the impact times.

What have we learned so far about Orion? It seems clear that Orion must have impacted the Moon sometime around the end of May of 1972, about 5 weeks after it was jettisoned. Longitude of the impact is probably between 70 and 120 °E. The impact location is likely to be on a mountain or crater wall, along a narrow track between 8 and 11 °N Latitude, and there seem to be just a handful of locations where the impacts are concentrated. Could we actually locate the impact crater? Although we have narrowed down the search considerably, we are still talking about huge areas of the Moon. (One degree of longitude along the Moon's equator is about 20 miles.) In my next post we'll dig into some other sources of data to see if there are any clues that could be helpful to narrow the search even more.

What About Orion?

As I write this, the 50th anniversary of the Apollo 16 mission is a few months away. Like all of the later Apollo missions, the plan for the ascent stage of the Lunar Module “Orion”, after it returned the astronauts to orbit, was to intentionally crash it into the lunar surface. This would generate seismic waves that would reveal the inner structure of the Moon by way of the seismometers left on the surface by the astronauts. These intentional impacts were done for the Apollo 12, 14, 15, and 17 missions. But something went wrong during Apollo 16, and the LM “Orion” stage did not hold its attitude (orientation) after it was jettisoned. With no way to control which way it was pointing, there was no safe way to command its engine to fire. It was abandoned in orbit, left to drift under the forces of lunar gravity, and no one knows what happened to it. Let’s turn our attention to Orion and see if we can shed some light on what became of this vessel.

If you’ve read my previous posts, you know the drill by now. We can get the initial conditions for the stage orbit from the Mission Report. We use GMAT and a high-fidelity gravity map to simulate the spacecraft. We record the “perilune”, the lowest point, for each revolution, and watch to see how close the stage comes to the surface over time. The orbital period is about two hours, as usual for low lunar orbits, so we get about 12 perilune points per day. GMAT simulates one year of spacecraft time in about an hour (on my computer) so we get a month of simulated orbit data every 5 minutes or so. The simulator doesn’t know about terrain, but we can tell the simulator to stop once the perilune point is a few km below the mean radius…at a point where an impact would surely already have occurred.

Figure 1: Simulated Perilune Altitude for Orion

Using the “nominal” initial conditions from the Mission Report, I get a script like this one. And the resulting perilune sequence is shown in Figure 1. It looks like Orion didn’t remain in orbit very long. By the end of May 1972, the spacecraft is already zero km above the Moon’s mean radius. A day or two later the perilune has dropped to 5 km below mean radius. Whack!

If you read my earlier post about “PFS-2” the Apollo 16 subsatellite, you might notice something interesting. The perilune sequence looks very similar, and PFS-2 is known to have also impacted the Moon near the end of May. Did I use the wrong script? No, actually the similar result is not surprising, when you consider the release of each object. Figure 2 shows an excerpt from the Mission Report, and details the sequence. After Orion was jettisoned, the astronauts performed a small “separation burn”, changing their velocity by just 2 feet per second. Then an hour later they released PFS-2. So, Orion was drifting along nearby when PFS-2 was jettisoned, and the two objects were in very similar orbits. They were in almost the same plane, and at almost the same altitude and speed. In hindsight it is not surprising that both orbits destabilize in a similar fashion.

Figure2: This excerpt from the Mission Report shows how the PFS-2 subsatellite was launched soon after Orion was jettisoned. Both events are during the 62nd revolution around the Moon, and just one small "CSM separation" maneuver was performed in between. Thus Orion and PFS-2 were in very similar orbits. Similar impact dates should not be surprising.

Where does Orion impact the Moon in this nominal case? Using the analysis method I describe here, Orion strikes the Moon at 77.5E, 8.16N at 8:14 on June 1st, 1972. Again, that is very similar to the impact time and point reported by NASA for PFS-2. The time is day or so later, and the impact is farther West. This simulation runs fairly quickly, completing in under 10 minutes on my computer. That means it will be quick to run a lot of variations of the initial conditions, so as to understand how much uncertainty there is in the result. I'll post those results later.

There is another important source of data we can look to for clues about Orion: those seismometers left on the surface that I mentioned above. In May of 1972 NASA had 4 stations operating, at the landing sites of Apollo 12, 14, 15, and 16. (The station at the Apollo 11 site stopped working late in August of 1969.) The data from all 4 stations was monitored and recorded continuously. The impact of Orion, weighing 5,000 pounds and moving at a mile per second, should have registered as a meteoroid strike, so we’ll be able to look for seismic events that might help to pinpoint the time of impact.

And then, in addition to the Mission Report, there are other clues we can use to help understand Orion’s initial orbit. During the later Apollo Missions, including Apollo 16, an extensive set of photographs were taken from lunar orbit by Panoramic and Mapping Cameras operating in the science bay of the Service Module. Each of these photos has an associated blob of data about the location of the spacecraft when it was taken. Hopefully this data can be used to further refine the initial conditions. And then if we get very lucky, these photographs might contain a “before” view of the area where Orion hit the surface, making it easier to identify any “new” crater.

Okay, so we have a large spacecraft that struck the Moon within weeks of its last sighting. We have a number of great data sources that we can search for clues. We have photos of the surface taken before the impact, and of course we have the high-resolution images captured by LRO and other lunar satellites. Perhaps we can identify the impact crater of Orion in time for the 50th anniversary of the mission. The game is on!

Impact Analysis

 

The simulation environment I am using, GMAT, doesn’t know anything about the Moon’s mountains or valleys. When it calculates the altitude of an orbit, that altitude is relative to the “mean radius” of the Moon, somewhat comparable to “sea level” on Earth. So far, in the simulations I have run for the Eagle and Snoopy, that hasn’t mattered, because the orbits of both objects remain high above the surface. But to estimate the location of a possible impact, as for the Apollo 16 subsatellite, we have to do a deeper analysis. In this post I’ll explain how that works. 

Let's start with the big picture. An orbit has become more eccentric…the high point of the orbit has moved higher, and the low point of the orbit has moved lower. On Earth, as the spacecraft began to skim the atmosphere, it would slow down, then burn up, and the heaviest bits might make it all the way to the surface. Since the Moon has no atmosphere, the spacecraft, at the low point of its orbit, can zip past the lunar surface at very low altitude, moving more than a mile per second, and if it misses the surface, even barely, it can continue around for another pass. 

Figure 1: The path of a spacecraft passing low over the Moon's surface

If the Moon were a smooth ball the impact might occur at the lowest approach. However, the Moon is not so smooth. In fact, it is quite rough, with jagged surface features thrown up by countless meteoroid impacts, undiminished by wind or rain. If a spacecraft comes streaking by and strikes the surface, that impact will likely be on a piece of high terrain, perhaps the side of a mountain or a crater wall. To work out the location, we’ll need a good elevation model of the Moon’s terrain. Fortunately, these are freely available, with resolution as high as 512 points per degree, which works out to one elevation point for every 100 feet or so.

We also need the simulator to give us a full “ground track” record, instead of just the lowest point of each orbit. The ground track is a list of latitude, longitude, and altitude points and typically a point is recorded for every 10-20 seconds of simulated time, as in the example shown below. Given that the spacecraft is traveling at about one mile per second, there is a lot of ground between each point. It’s not enough detail to locate the impact exactly, but it shows us the places where the spacecraft is close to the surface…places where we can zoom in for a closer look. As we go through this exercise, we will be zooming in on the points highlighted in green.

Figure 2: Ground track file excerpt. The terrain and "AGL" altitude data (in km) was added by post-processing the output from the simulator.

The first thing we need to do is to “post-process” the ground track file, looking up the height of the surface for each point in the file. For this method I am greatly indebted to a space enthusiast named Daniel Estevez, who ran simulations to try to estimate the impact location of a lunar satellite and posted his results here. I use a modified version of his method, wherein I run two passes on the ground track file.

Step one is to go through the ground track, one line at a time. From the latitude and longitude, we can look up the nearest terrain altitude point in the elevation model. Once we know the spacecraft altitude and the terrain height, we can calculate its “Above Ground Level” altitude. I write out a new copy of the ground track file with the extra terrain altitude and “AGL” data points added to each line, like the yellow values shown in the figure above. To limit the size of the new file I discard any point where the spacecraft is more than 5 km above the surface.

Here is a plot of a ground track file showing two low revolutions of a spacecraft over an area of the Moon. One thing to notice right away, even though this is covering 30 degrees of longitude, or a distance of about 900 km, the "zero" altitude point is flat, and the satellite trajectory curves upward, away from the Moon. This is just to make it easier to plot out the data. I promise the Moon is NOT flat, and the spacecraft is always curving towards the center of the Moon, as in Figure 1 above. The flattening of the Moon for this chart doesn't affect our ability to find the impact point. Another thing to keep in mind is that the vertical scale is greatly exaggerated. This chart is 900 km wide and just 8 km high.

Figure 3: A "flattened" plot of low passes of a spacecraft over the Moon. The area in the green box depicts the values highlighted in green from Figure 2.

Figure 3 allows us to see the areas where the spacecraft is coming close to the surface of the Moon. In particular, the points in the small green box highlight the closest approach visible at this resolution, and the closest point shows a separation of 449 meters above the ground. Given that these two data points are separated by a distance of over 20 km, we need to zoom in and take a closer look.

Figure 4: A "zoomed in" look at the region of the green box from Figure 3. We see that the spacecraft altitude is lower than the terrain altitude at 98.97°. Kaboom!

The figure above shows the zoomed in view. The two red dots are the points from the green box of Figure 3, the same data points highlighted in the ground track file in Figure 2.  What we have done is to make a straight line between these points, and break that line into 100 shorter segments, like the dotted blue line. (This process is called linear interpolation.) For each blue dot we can again look up the terrain altitude and compare it to the spacecraft altitude. Sure enough, in that intervening 20 km between the two red points there was a mountain, about 800 meters tall, and the spacecraft (moving from right to left in this example) strikes it near its top, at around 98.97°. 

How do we know there wasn't another mountain lurking somewhere further to the East? We don't! We have to check. In my code, posted here, I check any time I find a point that is within 3 km of the ground. (I also tried higher thresholds, but 3 km seems to be sufficient to catch all the lurking mountains in my tests so far.)

This is how I have used GMAT to estimate the impact location of the Apollo 16 Particles and Fields Subsatellite, the notorious (in some circles) PFS-2. Now that I have this tool working, I am interested to use it to investigate another Apollo impact. Yes, there is another spacecraft from the Apollo era whose final resting place is unknown...the ascent stage of the Apollo 16 Lunar Module "Orion". As I write these words the 50th anniversary of that mission is fast approaching. Stand by for further updates.

A Reality Check

 What about that Apollo 16 subsatellite?

In this blog I have shown that two Apollo spacecraft were left in orbits that are stable over decades. That’s really surprising and unexpected. Some people have asked if I can simulate an object that is known to have decayed out of orbit, as a reality check, to show that these simulations aren’t out of whack. That’s what we’ll do in this post.

One very notable case is the Apollo 16 Particles and Fields Subsatellite, otherwise known as PFS-2, which decayed out of lunar orbit in 1972 after only 5 weeks in orbit. Weighing just 36 kg, it was jettisoned from the Apollo 16 Service Module not long before the crew left lunar orbit to return to Earth. Originally it was planned to raise the orbit of Apollo 16, so that PFS-2 would remain in orbit for a year. Due to problems during the mission, that orbit change was skipped, and the expected orbital lifetime of PFS-2 was cut down to a few months. PFS-2 was equipped with a transmitter so that it could be tracked, and its data could be sent back to Earth. Only 34 days after it was jettisoned, the transmissions ceased, and PFS-2 impacted the Moon. 

Let’s run a simulation of PFS-2 and see what happens. As with previous simulations we can get the initial conditions from the Mission Report. The Figure 1 shows the data from the report. I believe the report is showing the state of the Command-Service Module (CSM) rather than the PFS-2, but it should be close enough to see if we are in the right ballpark. After converting the parameters to metric and getting them into the right coordinate frame, I get a GMAT script like this one, posted on GitHub.

Figure 1: Showing the initial conditions of PFS-2 from the Mission Report.

For starters, we’ll just record the low point in each revolution, as we have done previously. Figure 2 shows how it looks over time. We see the minimum altitude dropping for several weeks, and then there is a reversal and it starts to rise up again around the middle of May. About a week later the orbit starts to become more eccentric, and the perilune altitude begins dropping again. Sure enough, just 5 weeks after jettison, at the end of May 1972, the low point of the orbit is below the average radius of the Moon...zero altitude...and that is a sure sign that impact has occurred. (The simulator doesn’t check for impact while it is running…it will happily simulate an object that is actually beneath the surface, so we’ll have to look in greater detail to see exactly when and where the impact occurs. I’ll explain how to do that in a future blog post.)

Figure 2: Simulated perilune altitude.

A more detailed analysis of the simulation results gives the location and time of impact, and it comes out as below. Tracking data from the satellite ended shortly after 10:31 PM on May 29th, with an estimated impact at 111° East longitude, and 10° North latitude. This simulation puts the impact about 14 hours later, and about 13 degrees further west. That's not bad! 

Figure 3: Impact times and locations reported by NASA and estimated by simulation

There is another source of information on the PFS-2 initial orbit, at this page. It describes the orbit in a different way, and the parameters don’t completely agree with those in the Mission Report. If we run again with those initial conditions, we get the results labelled “Nominal 2”. This time we get closer to the 1972 estimated impact location and time…impacting about three hours earlier and about 4° more to the west, with the latitude agreeing almost exactly.

In my view, the basic answer is “yes”, these simulations do compare well against reality. We are able to predict the impact of PFS-2 within a few hours of the actual time, and within a few degrees of the estimated location. Considering the uncertainty of the initial conditions, with two different NASA sources that don’t agree, errors of a few hours either way don’t seem too surprising. Having gone through this exercise, I have even greater confidence in the results obtained for Eagle and Snoopy.

By the way, if you are looking for a lunar sleuthing challenge, the actual impact crater of PFS-2 has never been located. This web page states that the original raw PFS-2 tracking data has been preserved, and if you were to obtain that data and fit a simulation to it, I suspect you would be able to map out a very small area where the PFS-2 impact occurred on the surface of the Moon almost 50 years ago. You might be the person to identify the final resting place of the infamous short-lived Apollo subsatellite. Good luck and happy hunting!

Feedback and Stability

The Moon’s uneven gravity field causes most lunar orbits to be unstable. Over time the orbits increase in eccentricity, which is to say that the high part of the orbit gets higher, and the low part gets lower, until the object strikes the lunar surface. In this blog I have described the orbits of two different Apollo artifacts that show long-term stability in their orbits. (The Eagle and Snoopy.) They somehow manage to evade the instability that dooms most lunar satellites. How could that be? In this post we’ll dig in deeper to try to understand what is going on in greater detail.

I’ll start by focusing on the Eagle, and then at the end we can do a similar analysis for Snoopy. To start, as a reminder, look at the way the perilune altitude varies over time in the figure below. (Remember, perilune altitude is the lowest point of each revolution.) You see a cycle that repeats as the minimum altitude dips lower then climbs higher about every 25 days. I showed in a previous post that this 25-day cycle reflects the way the orbit changes as the Moon rotates underneath. The lowest lows always occur on the near side of the Moon. The fact that the cycle completes in 25 days, while the Moon completes a full rotation in 27.32 days, means that the Eagles orbit is also precessing. (This is also sometimes called “Apsidal advance”.) In this way the long axis of the Eagle’s orbit, called the Apse Line, does a complete circuit of the Moon in about 25 days, and this drives the short-period variation.

Figure 1: Minimum altitude of the Eagle in the first year after jettison. Notice the shorter variations every 25 days, and the longer variation every 4-5 months.

What about that longer variation in the perilune altitude? Notice how every 4-5 months the minimum altitude goes higher and then lower. What’s going on there? If you look at the figure above, notice that the minimum altitude is nearly the same at point A and point B, but somehow this system “knows” that at point A the longer cycle is increasing, and at point B the longer cycle is decreasing. Somehow there is “state” information being stored in the system, so that it “remembers” where it is in the long-period cycle. Let’s dig in and look for that “state” signal.

Figure 2: Showing the time (in days) between the peaks of the first 4 complete cycles for the Eagle. Notice that the time between peaks increases as the altitudes move lower.

For starters, let's look for differences between the “low” cycles and “high” cycles. One thing to measure is the “period” of the cycle, i.e., how many days it takes to complete a cycle. We can measure the time between the highest point in each cycle. In figure 2 above, I show the time (in days) to go from one peak to the next for the first 4 complete cycles of the Eagle’s orbit back in 1969. Do you notice anything interesting? As the altitudes get higher, the times get a bit shorter. As the altitudes get lower, the times get a bit longer. We can plot these on a graph that makes the relationship easier to see, and in the figure below I show the first 14 cycles…the first year of the Eagle after jettison. If I plotted out the data for 52 years you would see that the same relationship continues to the present day. This is a persistent feature of the Eagle's orbit.

Figure 3: Cycle length and end peak altitude for 14 Eagle cycles during its first year in lunar orbit.

The next thing to notice about these cycles is how they relate to the Moon. The plot below shows perilune altitude versus the Moon’s longitude, for one year. As the Moon rotates underneath the orbit, we see 14 tracks wrapping around. Each of the blue dots represents the lowest point of one revolution, and the longitude where that low point occurs above the Moon. What’s interesting is that the lowest parts of the cycles always occur on the near side of the Moon, near 30 degrees East, while the highest parts occur on the lunar far side. (From Earth we can only see lunar longitudes between -98° and +98°.)

Figure 4: Mapping how perilune altitude varies with lunar longitude. Eccentricity of the orbit is highest when perilune occurs on the near side of the Moon.

You might also notice some “sloshing” back and forth in that pattern in Figure 4. Notice on the left part of the figure where the highest points in each cycle are marked with red dots. The dots actually form a loop. It's even more interesting to connect the successive dots, as in Figure 5 below. In this figure I’m only showing the highest points of each cycle, like the red dots above, but now I added a dotted blue line showing the sequence. You can see that over the course of a year these dots trace out a series of loops. And these loops tie back to the slower 4- to 5-month variation you see that first figure above. Now we can see the difference between points A and B in the first figure. I’ve marked them again in Figure 5. Point A occurs about 30 degrees farther to the East than point B. This longitudinal variation is how the system stores its “state” information…how it “remembers” whether the short cycles are increasing or decreasing. And just to be clear, this is another pattern that is stable over decades. On the left in Figure 5, notice how this variation looks over a 50-year period. It doesn’t expand or contract or drift away. It remains centered on this longitude.

Figure 5: These plots show the lunar longitude where the perilune cycle peaks occur. Points A and B on the left are the same ones marked in Figure 1. All the red points on the left are also marked in red in Figure 4. Data for 50 years is plotted on the right, showing the long-term stability of the pattern.

We’ve seen how the eccentricity variation of the orbit stays locked to lunar longitude over decades. How can that be? There must be some feedback mechanism that prevents it from drifting away. It’s interesting to look at the rate that the perilune longitude point changes. To do that, for every revolution, we have to measure how far Eastward the perilune point shifted and compare that to the elapsed time. If we divide the longitude change by the elapsed time, we get a measure of the rate. (The elapsed time is nearly constant…about 1 hour and 58 minutes per revolution, but it’s a spreadsheet doing the math, so why not recalculate it for each point.) I’ll call this measurement the “precession rate”. That’s kind of a misnomer…the longitude is mostly changing because of the Moon’s rotation under the orbit, which is not really precession. (This component is also constant, because the rate of the Moons rotation is constant.) But there is an additional precession in the orbit so there is some variation in this precession rate. Here it is...

Figure 6: The "Precession Rate" varies depending on perilune altitude.

What is interesting here is that the rate gets much faster as the perilune altitude gets higher. Put another way, the precession rate varies inversely with eccentricity. As the orbit becomes more eccentric, the rate slows down. As the rate slows down, the Moon's gravity begins to drive the eccentricity lower. Lower eccentricity causes the rate to speed up, and the cycle repeats. Again, and again. For decades.

OK we found an interesting pattern in the orbit of the Eagle that persists for decades and can plausibly explain its long-term stability. (“Explain” is a strong word here…I believe there are more layers to this onion.) How about the Snoopy descent stage? If we go through the same exercise with Snoopy, we see very similar patterns. Compare the figures below for Snoopy’s orbit to those above for the Eagle. There is something about these retrograde equatorial orbits that leads to long-term stability, somehow evading the unstable fate of most other lunar satellites. Pretty cool, eh?

Figure 7: These plots of Snoopy's orbit data show similar patterns to those of the Eagle. A similar feedback mechanism seems to be responsible for the long-term stability displayed by both orbits.

Start Here

I started this blog to document my search for the descent stage of "Snoopy", the Apollo 10 Lunar Module. I thought that I could simulate its orbit and maybe find the crater where it hit the moon. To my surprise, I found instead that Snoopy's orbit was stable over decades. This blog was my attempt to show what I had done, and how I had done it, and hopefully get some feedback on anything I missed. I have to say that traffic was quite light.

I then turned to the ascent stage of the Apollo 11 Eagle...no one knows what became of that little piece of history either. And again, amazingly, the simulated orbit shows a long-term stability very similar to that of the Snoopy stage. Wow! That was in September of 2020, and you might notice that the blog posts stopped around that time. Instead, I focused on writing up the results for peer review and publication, and after some fits and starts I am proud to say that the paper is published.

My intent now is to expand on the Eagle results and continue looking for other interesting objects from that era. 

You can page through the blog posts in order for a "tutorial" on the process I went through. Or skip around to whatever looks interesting.

Love it? Hate it? Don't believe it? Post your (respectful) comments.

Hope you enjoy the material!

-Roger

Has the Eagle Landed?

 No one knows what became of the Eagle. That seems wrong. 

After it carried Neil Armstrong and Buzz Aldrin back from the surface of the Moon in 1969, the ascent stage of the Apollo 11 Lunar Module "Eagle" was jettisoned into lunar orbit. The astronauts watched out the window as it drifted away. The NASA tracking network followed it for a few revolutions, until they lost the signal. Since then no one has seen or heard from the Eagle. Without question it is one of the most important machines ever created by humanity. Not knowing her fate is a terrible wrong which must be righted.

The assumption has always been that the Moon's lumpy gravity caused the Eagle's orbit to decay, and she impacted the Moon at an unknown location. In this post I will go through the last known orbital state of the Eagle, and show the results of simulating that orbit with the best gravity models available. Spoiler alert: as I found previously with "Snoopy", the orbit is quasi-stable. Lunar gravity alone may not have brought the Eagle down.

For the orbital state of the Eagle at the time it was jettisoned, we look to the Apollo 11 Mission Report. Table 7-II lists information about the spacecraft at various points in the mission, and in particular there is an entry for "Ascent stage jettison" as below.

Orbital State of the Eagle at jettison, from the Mission Report

As I have described in a previous post, I use a simulation tool developed by NASA, and gravity models derived from GRAIL data. It's fairly straightforward to plug in the values from the table and simulate the stage. There is one problem with the Mission Report, though. It's wrong! When you think back to 1969, a world where word processing does not yet exist, and data processing is cumbersome, it isn't shocking that there is a problem in the table. But if you know a bit about the Apollo 11 orbit, the error is rather glaring.

All of the Apollo missions followed orbits that were low in inclination...that is, they stayed close to the lunar equator. It means that their "Space-fixed heading angle East of South" in degrees was never far from -90 degrees. If we use the value in the table, the orbit is inclined to the lunar equator by about 8 degrees...it can't be right.

What to do? Fortunately there is another source. This paper from 1970 lists orbit data for several Apollo missions, including Apollo 11. In particular, it lists the inclination for several revolutions leading up to the moment the Eagle was cast off. By plotting out the values and extrapolating, one can find an accurate inclination at that moment...178.817 degrees. (Inclination would be 0 degrees for an orbit following the lunar equator, in the direction the Moon rotates. Because the orbit is "against" the rotation, the inclination is close to 180 degrees.) Using GMAT this inclination can be translated into a heading angle...-89.63 degrees.

Extrapolating to find the inclination at the moment of jettison

Plugging this value into GMAT, along with the other values from table 7-II leads to a simulated orbit that matches up nicely to what is known about the mission. For instance the ground track of the orbit matches up well with those depicted in the Mission Report; the apolune and perilune values agree with values reported by Public Affairs Officer during the mission; and longitude values from the simulation agree well with Aquisition Of Signal and Loss Of Signal (AOS/LOS) times reported by tracking.

So what happens to our simulated Eagle? Let's look at the first five days after jettison. The stage was initially in an orbit that was "63.3 by 56 nautical miles", according to a P.A.O. announcement a few hours after jettison. That's a nearly circular orbit that is 117.2 kilometers at the high point and 103.7 km at the low point. From there we can see that our simulated stage is pulled into a more eccentric orbit, with higher highs and lower lows over the next 5 days.

If this trend were to continue, the Eagle would have indeed impacted the moon within a few weeks. However, as I have seen in previous simulations of Snoopy, there is a pattern that takes hold, and the orbit cycles through periods of higher and lower eccentricity, completing one cycle about every 22 days. In the plot below we see that after about 10 days, the orbit begins to return to a more circular pattern, with lower highs and higher lows, until around August 13th, when it is nearly back to the original state. Then a new cycle begins and we see the minimum altitude dropping again.

For simplicity in the plots below, I will ignore the higher parts just focus on the lowest points of each orbit, following the lower envelope of the plot. If I plot out these low points (the "perilune" points) for the first year, we see that the orbit continues to oscillate throughout the year.

What is exciting about this simulation is that there is no impact! Across the first three cycles of eccentricity, the low point of the orbit drops down to within 20 km of the surface in September of 1969. Then the trend reverses, and the minimum altitude begins trending higher. We see that there is a slower cycle of highs and lows superimposed on the 22 day cycle, which repeats about every 4 months. 

These cycles are very similar to the behavior of Snoopy's descent stage, and the cycles of eccentricity are can be explained by precession of the major axis of the orbit around the Moon. For whatever reason, the orbit always reaches it's highest eccentricity when the perilune point is above the near side of the moon. For Snoopy a cycle of precession takes about 25 days, while for the Eagle, in a lower orbit, it is about 22 days.

Now the big question. What happens if we run the simulation longer? When does the stage impact the moon? The answer, very surprisingly, is NEVER! I ran the simulation out to the present, which took about a week to complete on my home laptop. Here is a plot of the perilune points of the Eagle, simulated to the present...

Simulation of Eagle to the present shows no contact with the Moon!

The cycles of high and low eccentricity are almost completely lost in this graph, but there is no secular trend...the closest approach to the surface in 1969 is about the same as the closest approach in 2020. If the simulation is to be believed, then lunar gravity did not bring the Eagle down.

I have posted the simulation script and other information on GitHub, and I welcome you do try it yourself.

 It sounds crazy, but there is some possibility that the Eagle never impacted the Moon. Wouldn't it be amazing if we could find this amazing little vessel and bring her back to Earth!!!!