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. 

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!!!!