Impact Phase

 As I mentioned in my previous post, I have generated over 350 parameter sets that strike the Moon within one hour of the May 29th seismic event that seems to record the final impact of the ascent stage of the Apollo 16 Lunar Module Orion. Are all of these useful? What time difference should disqualify a simulation? If the simulation misses by 5 minutes, is that OK? If it misses by a full hour, is it meaningless? To answer this question, we need to dig deeper into the data, to try to understand what’s behind the time shifts. 

Let’s start with Figure 1, which is showing the distribution of impact times. Each vertical bar represents one 4-minute period, and the height of the bar shows how many of the simulations strike the Moon within that period. The red bar near the middle of the graph is the target time, around 21:14 on May 29th, 1972. The height of that bar is 30, meaning that 30 of the simulations impact within that 4-minute window. Overall, this set is centered about 6 minutes later on 21:20, and the impacts cover a total time span of 90 minutes. 

Figure1: Distribution of impact times from my simulation set. The red bar is at the time of the recorded impact event, and 30 of the simulations strike the Moon within that 4-minute window.

If you read my earlier post about “nudging” the orbits closer to a target impact time, you might be wondering…why not continue nudging until all the impacts occur at exactly the right time? The reason is that nudging stops working once you get the impact within one orbital period of the target. Huh? Let’s say the period is exactly 2 hours. (It’s close to that.) And let’s say we have a case where the impact is early by 2 hours and 20 minutes. We nudge the VMAG parameter a bit higher and re-run the simulation. Sure enough, this time Orion skims over the impact point, and zooms around for an extra revolution…then slams into the Moon two hours later on its next pass. Now, instead of being 2:20 early, it’s just 20 minutes early. Unfortunately, further nudging barely affects the 20-minute miss. In fact, another nudge might push the impact out by another 2 hours, making things worse.

We need to work with the impact times we have. Maybe we can understand what’s driving those offsets in the impact times? Let's call it impact phase. Once again (as has happened over and over during this investigation) the answer becomes obvious once the data is viewed in the right way. Take a look at Figure 2, which compares the “miss” time, in minutes, versus the initial orbit period in seconds. Aha! Notice that a one second increase in the orbit period shifts the impact time by about 7.2 minutes…or about 430 seconds. If Orion struck the Moon on May 29th, it would have been on about the 435th revolution around the Moon since jettison. So, a one second increase in the period means that 435 revolutions later, it has fallen behind by 435 seconds. And that is exactly what we see in Figure 2. Slower orbits mean later impacts, and vice versa. Impact phase is controlled by the orbit period.

Figure 2: Time offsets from the target vary linearly with the orbital period. It makes perfect sense!

(Another very interesting thing about Figure 2 is that it gives us a great way to validate that the May 29th seismic event was actually Orion. If we knew exactly the orbital period of Orion, we could confirm that it lines up with the observed impact time. I have looked for sources that could confirm Orion’s orbital period without success. Do you have any references? Please leave a comment.)

But now let’s get back to the original question. If a given simulation misses the impact time, can it still be useful to predict the impact location? One way to answer this is to take a given parameter set, and tweak it so as to vary its period, shifting the impact time, and then take a look at how the impact location moves around. Dramatic position shifts would mean we should ignore those simulations that aren’t close to the right time. Modest shifts would indicate that the exact time of impact is not so critical.

Figure 3: A sweep of one parameter set, to vary the impact time. Although the impacts are spread over a range of 2-plus hours, the impact locations are all within 1.4 km of each other. 

One way to change the orbital period is to raise the orbit, so that is what I’ve done in a set of simulations shown in Figure 3. You see that the RMAG parameter (the distance from the center of the Moon) is gradually raised to 1849.2 km. The VMAG parameter has been “nudged” to bring the impacts within one orbit of the target time, while all other parameters are held constant. Notice the impact locations. They remain in a tight pattern centered around 104.3 °E, near 10 °N. These locations are all within 1.4 km of each other, despite the fact that the impact times go from 84 minutes before the target to 53 minutes after. 

From this, I conclude that I need not be too concerned about impact time shifts of less than one orbit. The full database of impact locations seems to be useful as an indicator of where Orion’s remains can be found.

Closing The Loop

In my last post I showed that there was a lunar seismic event on May 29th, 1972 that seems likely to have been caused by the impact of the Apollo 16 Lunar Module Orion. Knowing the time of this event is an extremely valuable clue to help locate the point of impact. I showed in an earlier post that the impact locations of a randomized set of simulations varied with time, shifting gradually westward on the Moon for later impact times. If we have a target impact time, we can run more trials, and look for impacts at the right time. Then we can check the area where these occur…and hopefully the area is small enough to make a visual search practical. 

There is a problem with this "shotgun" strategy though. If we generate the trials randomly the impact times with be spread out over days. Only a very small fraction of the runs will happen to hit the Moon around the desired time. Even though each simulation completes in just 8 minutes, it would take thousands of trials to build up a good database of timely impacts. We need a better way.

In looking at the data from the first trials, there is a pattern which might offer a way forward. Each simulation starts with six numbers that represent the initial state of Orion’s orbit. The numbers represent the position (latitude, longitude, and altitude) and velocity (speed and direction) of the spacecraft when it was jettisoned. To generate a set of simulations, each of these six parameters is perturbed randomly. The hope is that the random variations will make up for any errors of precision Orion’s initial state. Hopefully all the variation covers the true initial state. In looking over the six input variables, and comparing to the results, an interesting pattern emerges. 

Figure 1: Impact Date versus initial VMAG

Figure 1 is a plot of the speed parameter, VMAG, versus the resulting simulated impact date. Each dot in the figure represents one simulation run.  What is interesting is the trend in the data, as summarized by the dotted trend line. Simulations resulting in an earlier impact had, on average, smaller VMAG values, and those resulting in later impact had larger VMAG values. According to the trend line, on average a change in VMAG of just 0.000694 km/sec resulted in a one-day change in the impact. That’s 69 cm/sec, for a parameter that is in the range of 1.6 km/second. So a very slight nudge, less that 0.05%, leads to in a one-day shift in the impact date.

Therefore, the strategy is to first run a randomized set of simulations. Then calculate the number of days that the resulting impact “missed” the desired target impact time. Multiply this error by 69 cm/second, and add this “nudge” factor to the initial VMAG value, to generate a new VMAG value. Then re-run the simulation with the new VMAG value. By using the outcome to feed back into the initial conditions, we are "closing the loop". Let's give it a try. 

Figure 2: Impacts converging on the desired time after several rounds of VMAG nudging

Figure 2 shows the results using this technique. On the left side, after the initial run, the impact times range from 2 days early to 4 days late compared to the target time. For each case, I calculate the “nudge” value, update VMAG, and run the simulations again. After one round of this nudging, the results are much more concentrated around the target date. Everything is within +/-1 day of the target. Now I can repeat the process a second time, again adjusting the VMAG values by ~69 cm/sec per day of error. Since the errors are fractions of a day, the nudges are proportionally smaller. After two rounds, I have a large set of simulations which are impacting around the desired time. Voila...it’s working well!

Isn’t this cheating? We started by generating random variations, but now we are selectively adjusting one of those values. VMAG is no longer random. That is true, but there is still value in this technique. Take a look at the progression of VMAG values in the trials, shown in Figure 3. (These are sorted from lowest to highest initial values.) The values for VMAG vary randomly with a total of 6 meters/sec variation around the nominal value. (This is a very generous variation, given that NASA said in 1972 that the doppler tracking enabled them to measure the spacecraft speed to within 0.5 feet/sec.) After two rounds of nudging, the variation in VMAG is less than 3 meters/sec. Focusing on the desired impact date has compressed the variation of VMAG, but we are still testing a generous set of its possible values. And we still have fully random variation of the other five parameters.

Figure 3: Initial VMAG values (blue) and final values (red) after nudging

OK, we are able to focus the impact times, but what about the impact locations? As hoped, as the impacts begin to cluster more tightly around the desired time on May 29th, they also begin to cluster more tightly on the surface of the Moon. As I showed in a previous post, initially the impacts are spread across a wide band of longitudes, from 62° E to 126° E. That band is over 1000 miles wide! Figure 4 shows the results after nudging towards the target impact time. The impacts are now clustered around a few prominent terrain features near 10° N and 105° E. Although this is still quite a large area, we are making progress. The total search area is greatly reduced, especially since the impacts are concentrated primarily along crater rims. I'm beginning to have hope that Orion's final impact crater might be found. 

Figure 4: Locations for simulated impacts that occur near the time of the May 29th seismic event

Perhaps there are other clues we can use to further constrain the impact area...next time. Meanwhile, happy hunting!

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!

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

The Staging Disaster (That Never Happened)

Among people who celebrate the Apollo program, there is a group that can recite the details of a near disaster during Apollo 10. They talk about the Lunar Module spinning out of control, seconds from impacting the moon. A chaotic staging in the middle of engine firings. It's a dramatic story, no doubt. Fortunately, for the men on board at the time, the story is largely false. The dramatic events actually never occurred, at least not in the way they have come to be described.

A still from the 16mm film taken as the descent stage moves away during staging. Notice that the lunar horizon is "upside down".

I can't blame people for believing in this mythology. Take a look at this excerpt of a video that has been shown on the Discovery Channel. We see the LM apparently in the midst of a rehearsal for landing. The descent engine is running. Then suddenly we see the stages separate and the ascent stage goes in a rapid, wild tumble. These are powerful images.

I am sure that the Discovery Channel showed this animation over and over to attract attention to their show. I don't know if the people who produced it knew it wasn't real, but for sure, if they had spent a few hours researching the mission records, they would have known it was a false telling of the tale. 

Just to be clear, something did go wrong at staging. At the moment it occurred, in May of 1969, no one knew exactly what was happening, and there is no doubt it was a frightening incident. Both Tom Stafford and Gene Cernan, the astronauts aboard, deserve great credit for quickly recovering and keeping their mission on track. In this post I will look at some of the fictionalized accounts online, and then go back to the original records to piece together what really happened that day. The exaggerations that have been popularized in recent years don't honor the true courage and flying skills of these brave men.

First lets look at more of the false accounts.

Here is a YouTube video from "Seeker", which claims that "the spacecraft could have slammed into the Moon." In fact the astronauts were in a stable orbit, and slamming into the surface was simply not something that would have happened in any scenario. 

 

Sadly, even the Draper Corporation has joined in the fiction-fest. This company descended from the MIT instrumentation lab that designed the guidance and wrote the software for the Moon landings! But this web site, which Draper sponsors, joins the chorus of bogus descriptions of the staging incident. They claim that the crew was "seconds from crashing into the moon when they successfully regained control." Fortunately for the Apollo crews, Draper's standards for reality-checking were tighter back in 1969. No doubt Charles Stark Draper is spinning in his grave faster than an out-of-control LEM.

Here is one more misleading account of the event, and again one would hope for better fact-checking from the source...Smithsonian Magazine. While I appreciate their effort to draw attention to an overlooked Apollo mission, the claim that the staging gyration took place "after the ignition of the ascent engine" is simply false, as a few minutes of online research will quickly prove.

Surprisingly, it seems that the over-dramatized accounts of the staging incident tie back to someone who was there - Gene Cernan, the Apollo 10 Lunar Module Pilot. Back in1969 his descriptions of the events related well to the facts of the mission. By 2009, 40 years later, his sense of drama appears to have enhanced his memories, and the tale he tells has grown. He recounts "seeing the lunar horizon go by 8 times in 15 seconds". No wonder the event mythology has become so exaggerated. Eyewitness testimony! While I have great respect for Gene and all of his contributions to the space program, in this case we unfortunately need to doubt his words.

Let's go back to May of 1969 and uncover what really happened. There are a number of great sources to help understand the event, including film taken out the LM window, on-board audio recordings, telemetry data, the crew debrief, and the "anomaly" investigation report.

For starters, one fun video from the time is the live coverage that was being broadcast by CBS. They were "all in" on the Apollo program, and invested heavily in animations to complement the audio they were getting from NASA. Famously, there was no 7-second delay in those days, and we clearly hear Gene cussing on live television as things go south in the LM. In the video I linked above, Cernan recounts how his wife admonished him about his "salty" language when he got back to Houston. However even from the live coverage, it's clear something went wrong during staging, resulting in a "wild gyration".

Cernan's mic was "live", so his words went straight to Houston, and on to CBS, which is why he was tagged as the "salty" one. Meanwhile, an on-board tape recorder captured audio from both Cernan and Stafford. From the transcript of the tape produced by NASA, we see that Stafford (labelled "CDR" in the transcript) had a few choice words of his own. Fortunately for all concerned these words were not broadcast live. 

This excerpt from the LM transcript captures the tense moments of the staging incident.

To understand what really happened, there is one truly outstanding source - the Apollo Flight Journal. Robin Wheeler of the AFJ put together a video that combines the film, the on-board audio, and telemetry data, aligned to a mission timer. Anyone interested in the true story of this incident needs to watch this video. When I watch it I have to really admire the crew as they quickly and effectively dealt with a very unexpected situation. Clearly they are a great team, with years of experience training and flying together. Listen to the great communication as they keep each other apprised as events unfold.

From the video it's clear that the LM did not tumble or spin in the dramatic fashion of the Discovery Channel animation. The lunar horizon does not go by "8 times in 15 seconds" as Gene claimed. What we see in the video matches up much more closely to what Tom and Gene described in the crew debrief interview they gave when they got home. In that interview Tom states that "We got the vehicle under control after about, I'd estimate, a 360-degree maneuver." Gene's account is more conservative, saying that "We could have maneuvered 30 degrees or we could have maneuvered 90 degrees. All I know is that it was fairly slow." Then he added "whether we did a 360-degree maneuver is difficult for me to say." The account Gene gives years later, is very different, and is probably one of the the main reasons that the staging mythology grew. 

This chart shows the spin rates at the time of staging. The peak in the Z-axis (roll) was around 26 degrees per second...about 13 RPM.  Source

It's important to understand that neither the descent engine or the ascent engine was fired during the staging maneuver. The plan was to use the small "RCS" jets to speed up by 2 feet per second, kick off the stage, then slow back down to the original speed. This is basically what happened, but the unexpected attitude changes meant that the stage was kicked off more upwards than backwards. Robin Wheeler at the Apollo Flight Journal undertook a detailed analysis of the film and prepared this image showing the planned and actual staging attitudes.

The confusion about the engine firings probably arises because the LM was designed to be able to perform staging "hot", during descent. This "Fire In The Hole" capability had been tested with the first unmanned LM flight during Apollo 5. For Apollo 10, no such risk was taken with men aboard. The descent engine had been deactivated after the Phasing burn, two hours before. The ascent engine would be fired for the first time 10 minutes after staging, as the crew began their rendezvous with "Charlie Brown".

This table from the Mission Report shows the Phasing Burn that preceded staging and the Insertion Burn that occurred about 10 minutes after staging. We also see that the LM was in a 190-by-12 nautical mile orbit...with no risk of impacting the surface

Another point of confusion about the staging mishap concerns the altitude above the lunar surface when it occurred. Prior to the Phasing Burn, the LM had been in an orbit that took it to within 8.5 nautical miles of the surface. However, at the time of staging, one can see from the Mission Report that they were at an altitude of 31.4 nautical miles. They were never in danger of impacting the Moon.

Table 6-II from the Mission Report, showing the altitude of 31 miles at the time of staging

All of the original documents point to the staging event as much less dramatic than the one so widely depicted online these days. That bothers me. Although the mission was a success overall, there were plenty of challenges, and plenty of surprises for the talented crew to overcome, and they met them all in grand fashion. Thanks to the fantastic job by Gene, Tom, and John, Apollo 10 was an amazing success, which paved the way for the first landing on the Moon less than two months later. There is no need to exaggerate the mission to celebrate its triumphs.

Propellants

Snoopy's descent stage was jettisoned into lunar orbit with more than 8 tons of hypergolic propellants still aboard. If the stage orbit was long lived, what happened to all this fuel? This might be the thing that brought the stage out of orbit, so in this post I will examine the possible outcomes.

Descent Stage Cutaway View  Source

The mission was a rehearsal for the Apollo 11, without the landing. They undocked, then did the Descent Orbit Insertion (DOI) burn to drop the orbit closer to the surface, as Apollo 11 would. Then they did an additional "phasing burn" to get the right alignment with the command module for the rendezvous. One orbit later the stage was jettisoned. These two short maneuvers used only a small fraction of the fuel, leaving the tanks at about 96% of their capacity

This excerpt from the Mission Report shows the quantities of propellants loaded and consumed

The stage was not designed for long life, so no one knows exactly what might have happened. There are some facts in various NASA reports that offer clues, so we'll map out the clues and then make some guesses about the eventual outcome.

In a previous post I showed a simplified model of the Descent Propulsion System, or DPS. It was a simple, reliable system with tanks fed by pressurized helium. The tanks of fuel and oxidizer led to the combustion chamber, so when a valve was opened, the propellants mixed, ignited, and burned, creating thrust. In the earlier post I described how the main "supercritical" helium tank likely vented to space within a few days after staging. Then what?

Here is a complete schematic of the DPS plumbing. There are actually two helium tanks, plus various valves, burst disks, and so forth.

Source

In order to analyze what happened, its useful to simplify things, as I show below. The "Squib valves" were sealed during flight and then opened using small explosive charges when the engine was activated. Once opened, they never close, so it's cleaner to show them as open pipes. One set of squib valves was used to vent the propellants on the lunar surface, so for Apollo 10 these remained sealed. I believe they can be ignored. (I don't think it's possible that they could activate themselves.) So I eliminated those as well. The Supercritical Helium tank no doubt vented within a few days after staging, so I also eliminated that from the diagram, showing instead an opening to the vacuum of space. Here is the simplified schematic:

What do we know about this system when it was cast off? We know that the fuel and oxidizer tanks were pressurized to 247 PSI at 70 degrees. We know the tanks were still 96% full. The small Helium tank in the schematic is the high pressure "start bottle". (It was used to initially pressurize the tanks and start pushing fuel through the heat exchanger.) We know this tank had a slow leak. We know that the burst disks were rated to open between 260 and 275 PSI. We know the quad check valves might have had leakage rates up to 100 standard cubic centimeters per hour. We know that even without the main helium pressurization system, the DPS could operate from existing tank pressures, in "blowdown" mode, generating significant thrust.

We also know that the stage was slowly tumbling in orbit. The dramatic film taken during staging captures the unplanned attitude excursions, and a post-mission guidance report shows the rates at the moment of staging in this chart:

At staging, Snoopy's tail had yaw, pitch, and roll rates of -9, -4, and +7 degrees per second. So that's one full yaw rotation every 40 seconds, one pitch rotation every 90 seconds, and one roll every 52 seconds.

So what happened? I think there are four possibilities. 1) Slow leaks might have allowed the tanks to depressurize, until the stage reached a stable state. 2) The tank pressures might have increased until the burst disks failed, allowing things to vent to space. 3) Propellants might have leaked back through the check valves, into common helium plumbing, which might have led to combustion or even a catastrophic explosion. 4) Something might have caused the throttle to open, allowing the engine to start generating thrust. Let's take these one by one, in reverse order.

If something caused the throttle to open up, residual pressure in the tanks would have allowed for significant thrust in "blowdown" mode. For example, as Apollo 13 was headed back to Earth, it was noted that the DPS could provide an 800 f.p.s. velocity change to the full LM-CM-SM stack in this mode. In the case of Snoopy, the volume of gas in the tanks was lower, reducing the possible burn time. However the stage was much lighter than the Apollo 13 stack. If the throttle opened up, it might have knocked the stage out of orbit.

Detailed view of the throttle assembly (source)

I really doubt this happened, because several things would need to fail for it to occur. One of the two actuator isolation valves would have to fail hard open, allowing pressurized fuel to flow into the valve actuator, forcing the main shutoff valves open. Then the "thrust control actuator" would have needed to fail into the open state. Designing a thrust control valve that could fail open doesn't sound like something that would have passed muster during Apollo. If you have deeper insights, please leave a comment. In my view, a spontaneous blowdown burn is unlikely.

How about reverse leakage through the check valves? This could create big problems because it would allow both fuel and oxidizer to flow into the common helium feed plumbing. A high profile explosion during a ground test of the Crew Dragon spacecraft in April of 2019 was attributed to oxidizer leaking into helium pipes. NASA documents do state that early check valves had a higher leak rate than originally intended. So could this leakage cause the stage to explode?

For the check vales to leak, there must be reduced pressure in the helium feed. The helium system was still being fed by the small ambient "start bottle", so it probably remained at 247 PSI for some time. During the flight, the ambient bottle pressure dropped 35 PSI in 97 hours, apparently due to a leak that developed during launch. At this rate, the bottle pressure would have dropped to 247 PSI in about one year. After the other helium system vented, the leak rate might have increased.

So once the pressure was low enough for leakage to occur, does that result in an explosion? Honestly I don't know how to evaluate this possibility. If the leakage was in gaseous form, any reaction would probably be low-power. If liquids leaked, and eventually flowed together, the reaction would be more violent, perhaps even powerful enough to blow out the plumbing, or even trigger the complete destruction of the stage. Leave a note in the comments if you have insight into this possibility.

How about venting through the burst disks? These disks were designed to open up if the pressure in a tank reached 270 PSI, give or take. The tank pressures could have reached this level if they heated up, since pressure increases with temperature. The temperature in the tanks was 70 F after the second burn, and a rise of 43 degrees would have raised the pressure up to the nominal burst pressure. Could this have occurred? It seems unlikely. Orbiting the moon every two hours, half of each orbit in searing sunlight and half in freezing darkness, it seems that the heavy tanks would slowly reach thermal equilibrium. The surface of the moon in sunlight actually is quite hot, and radiates a lot of heat out into space, adding to the direct heat from the sun. But it doesn't seem that this would be enough to raise the temperature of those tanks to 110 degrees. Perhaps the hot/cold temperature cycles could lead to failure of the burst disks at a lower pressure? Again, I don't know how to evaluate this.

If the burst disks did open up, I don't think the orbit would have been significantly affected. Due to the "thrust neutralizers" and the tumbling of the stage, the net thrust would have been low. I believe this case would be similar to what occurred when the helium tank vented.

Finally, what about slow leaks that allowed the tanks to depressurize? For the fuel side, at least, this could have occurred through the isolation valves and pilot valves of the shutoff valve assembly. For the oxidizer this path doesn't exist. That sets up the possibility that the ox tanks, still pressurized, could have leaked back through the helium plumbing and into the depressurized fuel tanks. Kaboom!

It bothers me to say it, but I just can't make any solid prediction about what happened to these propellants. Ultimately there are two possibilities. Either the stage reached an inert state, and remained in orbit, or it didn't. The only way to find out is to look, and I hope someday the looking will occur.

The Stage Returns

On May 23, 1969, Tom Stafford was concerned. The day before, he and Gene Cernan had performed the first lunar rendezvous in history. Now back in the Command Module, fully rested, preparing for the Trans Earth Injection (TEI) maneuver that would bring them home, Tom saw something out the window that gave him pause. Snoopy's tail was back.

(This transcript comes from the wonderful Apollo Flight Journal, but it also happened to occur during a TV broadcast, which you can find on YouTube.)

The stage had been cut loose in a looping orbit, low on the near side above the Sea of Tranquility, and high on the lunar far side. This "phasing" or "dwell" orbit was designed to be slower than that of the CSM, allowing "Charlie Brown" to overtake the LM and set up the right timing for the rendezvous. When Tom and Gene in the ascent stage ran through the rendezvous they caught up to the CSM and docked. The descent stage was left in the slower orbit.

The difference in orbital periods meant that one day later, the CSM was lapping the stage. Stafford wondered if the stage would be safely out of the way in time for TEI. One fortunate consequence of Stafford's concern was a "hack" on the position of the stage that he called out to Houston. Again, you can hear this on the TV broadcast, about 8 minutes into the YouTube video.

He calls out the stage position "between Taruntius P and K...I'm looking down now at 257 He's right down below us." This is a very valuable clue about the stage orbit. The craters Tom mentions are close to the lunar equator, at about 51.55 degrees East longitude. (You can see them with the LRO Quickmap tool here.) The "257" refers to the angle towards the stage, relative to the local horizontal. It means the stage was about 13 degrees below and behind the CSM. Here is a diagram to illustrate the situation. From simulations, the stage was at an altitude of about 93 km, and the CSM was about 99 km high. From the geometry, this means the CSM was at around 50.82 East, and Snoopy's tail was at 50.86 East at that moment.

From the mission transcript this occurs at the Mission Elapsed Time of 132:16:10. This is really helpful to nail down the exact timing of the stage orbit. Think about checking the accuracy of a watch...you set it to a known time and then check it a day later...you can see how many seconds it has drifted ahead or behind. Same idea here. We simulate the stage orbit out to this time, and then look to see if the stage is coming up ahead or behind 50.86 East.

The process is easy. Start from data published in the Apollo 10 Mission Report, and run a simulation of the stage orbit. Stop this simulation at the time of the sighting, and check the longitude. The sighting time translates to UTC of 5/24/1969 05:05:10.00. When I run the stage simulation out to that time, using the nominal values from the Mission Report, the longitude from the simulation comes out at 49.85 East. Not bad! It's off by about a degree, which translates to about 30 km or about 18 miles at the lunar equator. Remember that these things were moving about a mile per second in their orbits, so that means the simulator ran the stage past the target longitude about 18 seconds early.

For this adjustment, I keep almost everything as it is in the mission report, adjusting only the initial velocity. This is where orbital mechanics gets fun! Since the stage is going too far, we need to...speed up? Yes indeed...we need to go faster to slow down!  It's about as counter-intuitive as it can be, but that's how it works. By adjusting the initial stage velocity upward, the stage is driven into a (slightly) higher orbit. Because it's in a higher orbit, it takes longer to go around. Since it takes longer to go around, when we stop the simulation at the right moment, the stage is farther back. It only takes a few tries to dial it in, and these simulations take only 20 seconds or so to run, so very quickly I have a simulation that puts the stage right at 50.88 East at 132:16:10. The tweak to the stage velocity to get this to line up is just 0.27 feet per second. Beauty!

After playing the same game with the CSM orbit, now I have two simulations that put the two spacecraft in the right place at the right time. Here comes some real fun! GMAT, the simulator, can co-simulate two different spacecraft. Now we can see the dance that was making Tom Stafford so nervous. Here is a link to the GMAT co-simulation script that I posted on GitHub.

First off, let's look at what happened while the crew was catching up on their sleep. The CSM, in the faster orbit, pulled away from the stage. Initially, around the time they jettisoned the LM upper stage, they were about 100 degrees of longitude ahead of the lower stage. Over the course of the night (it was night time in Houston) and next day, this lead increased, until they were 360 degrees ahead...they had caught up to Snoopy.

The "wiggles" in this line are due to Snoopy's eccentric orbit...down to within 20 or 25 km of the surface, then up to 340 km. When Snoopy dropped down lower, he would move faster than the CSM, and gain ground. When he went around the back side and went higher, he would move slower than the CSM, and fall behind. Overall he was losing ground. (Remember, that was the purpose of the "dwell" orbit.)

The plot below shows the position of Snoopy, relative to the CSM, and it makes clear how the stage would gain and lose ground during each orbit. The horizontal scale is compressed by 5x to fit several orbits onto the plot. Notice that Snoopy would mover farther ahead as he dropped down below the CSM, then fall farther behind as he climbed up higher. When the TEI burn started, Snoopy was safely away, about 1200 km behind, so it turns out that Tom's fears were unfounded.

Notice how close Snoopy came to the CSM, in the red circle marked "Close Encounter". (On this plot, by definition the CSM is at 0,0 in the center of that circle.) By zooming in we can see just how close they came:

As Snoopy dropped down behind the CSM, the stage seems to have passed within 3-4 kilometers before moving away. Nowhere near an actual collision, but too close for comfort!

Fortunately for all concerned, the stage drifted past harmlessly, and was quickly forgotten...until now. Perhaps we might hear some news about this long lost artifact someday if, indeed, it has remained in a stable orbit for all these years.