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.

Orion's Impact Area

In a recent post I showed that one event in the seismic catalog of the Moon seems to have recorded the impact of the Apollo 16 Lunar Module “Orion”. This event occurred late on May 29th, 1972, about five weeks after Orion was jettisoned. Then in my last post I described a way to “nudge” the initial conditions of a simulation in order to move the impact date/time towards the time of this event. Using this nudging technique, I have been able to generate several hundred simulations, all random variants of the nominal orbit of Orion, all of which impact the Moon within an hour of the target event at around 21:14 UTC. I have posted csv and Excel versions of the combined result files on GitHub. The files include the initial orbital state used for the simulations plus other initial state data, along with the impact location and time for each case.

We can’t have perfect knowledge about the initial orbital state of Orion. These simulations represent a set of initial conditions that vary randomly around my best guess at the nominal state, allowing us to get a reasonable picture of the possible outcomes for Orion given the uncertainties. What is exciting about the results is that the simulated impacts are concentrated in four “high terrain” areas of the Moon. These are the same four impact areas I found earlier with a smaller set of simulations. That’s good! The search area didn’t expand even though we have a larger database.

Figure 1: Impacts from the new database superimposed on a map of the Moon. There are over 350 simulated impacts, all striking the surface within an hour of the target event on May 29th, 1972.

Figure 1 shows the impacts superimposed on a map of the Moon. You can see that each impact cluster is in a place where the terrain is higher…mostly along the ridges surrounding craters. Again, this makes sense: as the orbit destabilizes, the spacecraft is on a flat trajectory at its low point, and it will strike the first piece of high ground it encounters. Overall the possible locations for Orion’s final impact seem pretty well constrained.

Could we tighten things up even more? In looking deeper at the data, it appears that we can. In the result files mentioned above, one extra parameter included for each parameter set is Orion’s initial inclination. Using this data, we can look for any correspondence between inclination and impact point, as plotted in Figure 2. Lo and behold, there is a pattern! The impact longitudes cluster into bands depending on the initial inclination. If we could determine the inclination more precisely, we could focus in on one or two of the clusters.

Figure 2: Orion's Impact Longitude versus Initial Inclination. All the simulations close to the nominal inclination value result in impacts near 104.3° East longitude. This leads to a very small area to search for Orion's impact crater.

As it happens, we can get a very good guess at Orion’s initial inclination, thanks to the Metric Camera database. Prior to casting off Orion, the Apollo 16 crew ran a camera pass, exposing a 70 mm film picture of the Moon’s surface every 10 seconds. Meanwhile another camera took simultaneous pictures of reference stars, so as to know exactly which way the mapping camera was pointing for each shot. This allowed NASA to determine the latitude and longitude of each picture with great precision, which works back to the latitude/longitude of the spacecraft. 

Inclination means how much the orbit is tilted away from the equator, so if we look at all the pictures and find the one that is farthest north or south of the Moon’s equator, that tells us the inclination. It turns out that during revolution 60, a few hours prior to when Orion was jettisoned, there was a mapping camera pass, and we can see from the image database that image AS16-M-2828 was the south-most picture in the run, taken from a point above 10.55 °S. Therefore, the orbit was tilted 10.55 degrees away from the Moon’s equator. Since the orbit was “retrograde”, or against the Moon’s rotation, we reference the inclination to 180°, so it is expressed as 180-10.55 = 169.45°.

Take a look at Figure 2 again. If we limit the inclination values from 169.4° to 169.5° All the impact longitudes are in a narrow band around 104.3°. Wow! That gives us a very small area to look for Orion. Figure 3 is a plot of the impacts from this narrow inclination range. They are clustered within +/- 0.1 degrees in both latitude and longitude. That translates to a square-ish area about 6 km on each side.

Figure 3: A plot of impact locations after applying the inclination constraint. This is an area roughly 6 km on a side. Based on all the evidence, this seems to be the most likely area where Orion struck the Moon in 1972.

To give a sense of scale, Figure 4 compares this impact area to a part of Pasadena, California that is similar in size. The California Institute of Technology is at the lower right corner and the Jet Propulsion Laboratory is at the upper left. The Rose Bowl stadium, along the left about 1/3 of the way from the bottom, gives a sense for the scale of the craters.

Figure 4. Comparison of the impact area to a section of Pasadena, California.

I'm really surprised at how far this analysis has come. When I started, I was hoping that perhaps one of those Mapping Camera pictures should show the impact area BEFORE impact, making it possible to compare with modern images, and perhaps identify any "new" crater. It turns out that isn't feasible. None of the pictures from the mission provide the needed coverage. Given the relatively small size of the area I have identified, perhaps an exhaustive search may turn up some craters or features of interest? 

I have been very impressed by the work of Dr. Phil Stooke, who has been able to identify Lunar Module impact locations for Apollo 12 and Apollo 15, among other notable finds. Perhaps, with the above analysis as a starting point, Dr. Stooke or others might be able to locate the final resting place of Orion someday. 

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!

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!!!!

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.

Supercritical Helium

In previous posts I showed that the Apollo 10 descent stage orbit was stable over decades and that the lunar atmosphere could not have slowed the stage significantly. Is the stage still in lunar orbit today?

It seems clear that an inert object would still be in orbit today, but the stage was hardly inert. It was much closer to a flying bomb, with over 8 tons of highly reactive propellants, plus a tank with 40-odd pounds of liquid helium, which was slowly warming up, slowly increasing its pressure, slowly approaching the breaking point. Let's look at what would have happened next.

Film taken as it was jettisoned shows the ladder and footpad of the stage. Note the "upside down" lunar horizon above the pad.

To understand what might have happened, we need to understand the design of the descent engine. To keep things simple, reliable, and light, the Descent Propulsion System (DPS) employed a pressure-fed system. Helium gas was used to pressurize the propellant tanks, so that when valves were opened the fuel and oxidizer would flow into the combustion chamber. The propellants were hypergolic, so they would burn as soon as they came into contact. The image at right, excerpted from this presentation, shows a simplified view of the system.

To save weight, the helium was not stored as a gas. That would have required a very heavy tank, able to withstand very high pressure. Instead it was stored as a "supercritical" liquid, at very low temperature and modest pressure, inside an insulated tank along the lines of a big thermos bottle. This worked as a lightweight way to store the helium, but it was not designed to work for a long time. Heat would leak into the helium tank during the mission, raising its temperature and pressure. Eventually this rising pressure could cause a "burst disc" safety valve to open, and the helium would vent out to space.

On a normal mission the engine would be fired long before the pressure reached the breaking point. Once on the lunar surface the extra helium would be vented. For Apollo 10 most of the original helium was still in the tank when it was cut loose. (The Mission Report states that 44 pounds were loaded at launch, and the DOI and phasing burns consumed only 4% of the fuel.) Sometime after staging, the pressure would have climbed to the breaking point. How long did that take?

The pressure inside the helium tank was monitored by mission control, and the Mission Report states that the pressure was rising at 5.9 psi per hour after launch. A report on the DPS showed that the tank pressure at the end of the phasing burn was at 1160 psi (and still rising, due to the way fuel was piped through the helium during the burn) and that the burst disc was designed to open at 1881-1967 psi. From this it seems very likely that Snoopy's descent stage vented the helium between 4 and 6 days after staging, sometime between May 26-28, 1969.

How did the venting affect the orbit? Could it bring the stage down? This document can help answer the question. Figure 9.1-3 in the document, copied below, shows the time and force for a full tank to vent, and the resulting impulse. The total impulse to vent a full tank is ~1700 lb-secs, over the course of 120 seconds. Let's ignore the fact that the stage is tumbling, and assume this impulse all contributes to a change in velocity. With the stage weighing a total of 21,000 pounds, the venting could change the velocity by 2.6 feet per second at most.

This graph shows the thrust generated by helium as it vented to space after blowing the burst valves

Keep in mind, the stage was moving about 5600 feet per second in it's orbit. So even if this 2.6 fps change was exactly aligned to the direction of motion, it's much less than the variation I simulated in my Monte Carlo run. I simulated velocity variations of plus or minus 65 fps, and all the runs stayed in orbit for at least 10 years with no sign of decay. Furthermore, since the stage was tumbling in orbit, the overall impact of the venting would certainly have been much less than 2.6 fps. As it spun around, forward thrust would cancel out backward thrust. It seems very likely that the overall change in velocity was less than 1 fps.

There is another data point to consider. During Apollo 13, as they were returning to Earth using the LM as a lifeboat, the descent helium tank vented. (In addition to the serious problem Apollo 13 faced, this tank apparently had an insulation problem, and warmed up faster than it should have. It vented about 109 hours into the mission.) Although the venting did affect the roll rate of the combined LM/CSM stack, it did not affect the trajectory in any substantial way, and no final course correction was needed. This is good evidence that venting the tank would not strongly affect Snoopy's orbit.

My conclusion is that the venting of the helium didn't bring down the stage. Now we are left with a mere 8 tons of hypergolic propellants, held back by aging valves and decaying seals, and exposed to pure hot sunlight for one hour, then to the freezing black of space the next, orbit after orbit after orbit. In an upcoming post I'll examine what might have happened to all this fuel.

A deeper understanding

There is a strong pattern in the behavior of the stage, which shows itself as soon as one simulates a month or two of its orbit. The orbit oscillates, becoming more circular for about 12 days, and then moving back towards being more eccentric...more egg shaped. What is driving this pattern? In this post we will dig in to understand it in more detail.

This is a plot I shared in an earlier post, simulating the first two months after the stage was placed in orbit, and the oscillation is visible as the waviness of the minimum and maximum altitudes. The peaks occur about once every 25 days.

A plot of stage altitude versus days in orbit shows a pattern of oscillating eccentricity. This pattern, repeating every 25 days or so, remains a fixture of the orbit even when simulated to the present day.

Here is another plot, simulating the stage orbit in the year leading up to the 50th anniversary of the mission. In this plot only the perilune points of each orbit are shown, but the same 25-ish day oscillation is still a major feature of the orbit.

The 25 day period is intriguing, since it is very close to the 27.32 days required for the moon to complete one rotation around its axis. But the difference between 25 and 27 is critical, since it means that the stage orbit oscillation is not directly tied of the moon's rotation. At least not in any obvious way.

When I first started running simulations of the stage, I was hoping to find an impact crater, and this required that the stage decay out of orbit quickly. Once I got going with the simulations, and the stage seemed to be long lived, there was one thing I noticed which kept me looking. I was interested in the lowest points of the orbit, the ones where the stage got below 20 km. Interestingly, these points always occurred at a similar location, around 30 degrees east longitude.

This plot shows low perilune points from a simulation of 3000 days. All the lowest points were centered around 30 degrees East longitude.

This result gave me some sliver of hope that I might yet find a crater...perhaps there was a lone mountain on the moon at 30 East, near the lunar equator, that acted as a giant catchers mitt, snagging the stage as it came whizzing by on a particularly low pass. So as I began to run longer simulations, I didn't just record the altitude of the perilune points. I also recorded their latitude and longitude.

One day, by accident, as I was plotting the results from a simulation run, I happened to plot both perilune altitude and longitude on the same graph, and something like this appeared:

The blue lines are longitude, while the orange line is altitude, from a simulation of a few months of 2019. What is important to notice is that there is a direct correlation between longitude and altitude. The peaks in perilune always occur near longitude 180 (or minus 180...same place) which is on the far side of the moon.

OK. Wow! That was a surprise to me. But something was not adding up. The plot is saying that the stage orbit perilune point moves all the way around the moon in 25 days. More correctly, what is happening is that the moon is rotating beneath the orbit, and it should take about 28 days to complete a full rotation. So how is the perilune point getting all the way back to the same longitude in less than 25 days? The answer is "precession".

Perhaps you have seen a gyroscope precessing as it balances on a pedestal. There is a nice video example here. Orbits also precess, and you can see a nice illustration (greatly exaggerated) here. In the case of the stage orbit, it is also slowly precessing, in a direction opposite to the rotation of the moon. The illustration below is not to scale, but shows how the major axis of the stage orbit changes over one month. The net effect is that the moon does not have to complete a full rotation for a given longitude line to come back under the low point of the stage orbit. It gets there a few days early. Just under 25 days.

So now the 25-ish day period of the perilune altitude oscillation makes more sense. It is a combination of the 27.32 day rotation of the moon under the stage orbit, combined with a slower precession of the orbit. This precession may also tie in to the slower cycles that are apparent in longer runs. The rate of the precession suggests that the stage orbit would precess all the way around the moon in about 10 months, which is about the period of the longer cycle.

I don't know what exactly is driving the change to the eccentricity of the stage orbit, but the fact that it is tied to longitude certainly points to a couple of possible explanations. It could be mascons along the lunar equator, tugging the stage to a greater or lesser extent. Or perhaps, since the same side of the moon is always facing Earth, and therefore the moon's rotation is closely related to its position along its own orbit, the relative positions of the Earth and Moon are somehow conspiring to tug the stage orbit in different directions. It remains a mystery to me. Nonetheless, I am happy to have a better understanding of the most prominent feature of the orbit of Snoopy's tail end.