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.

Atmospheric Drag

Suppose it is true that Apollo 10 Lunar Module descent stage, aka "Snoopy", has not decayed out of orbit due to lumpy lunar gravity. What other factors might have brought the stage down in the intervening 50 years? That's a long time, so even subtle effects could come into play, including the nearly non-existent lunar atmosphere.

The space around the moon is nearly a perfect vacuum, by Earthly standards. But "nearly perfect" means that there are still thousands of molecules in each cubic centimeter of space around the moon. In fact scientists refer to the lunar atmosphere as an "exosphere" because there are so few molecules that they rarely collide with each other. They behave more like an army of tiny satellites, moving in response to gravity or electrostatic forces.

This instrument was placed on the moon's surface during Apollo 17 to study the moon's exosphere.

Source

According to this source, there are around 155,000 molecules in each cubic centimeter of the lunar exosphere. Yikes! That seems like a very big number. When you think that the descent stage is 12 feet on a side, and moving at almost a mile a second, that is a LOT of collisions with lunar molecules. Then take that out 50 years...now it's REALLY a lot. If we calculate the mass of all those molecules, and compare that to the mass of the stage, we can get an idea about whether the exospheric drag might have brought down the stage. If the mass of the gas is even close to the mass of the stage, that would be a real drag.

First of all lets estimate how many cubic centimeters the stage passes through every second. Just to get an upper bound on it, let's assume the stage is always moving with it's largest side facing forwards, sweeping through the largest area. We can approximate it as a square that is 12 feet on a side, which works out to an area of 140,000 square centimeters. (To get to this square I add the area of the foot pads into the missing corners of the stage, which is actually octagonal. Remember we are just estimating here.)

Now, the stage is moving about one mile every second. It moves faster when it drops down closer to the moon, and slower when it reaches it's "apolune" high point, but we can average that out and just assume a fixed velocity, as if it were in a circular orbit. One mile is 1.6 kilometers or 160,000 centimeters. Every second. Day after day. Month after month. Perhaps even decade after decade?

Well, now it seems like we might have a problem. If the stage is sweeping through an area of 140,000 square centimeters, and covering 160,000 centimeters per second, that works out to 22 billion cubic centimeters every second. And remember, if every cubic centimeter has 155,000 molecules, it means the stage is colliding with around 3.5 quadrillion molecules. Every second. Day after day. OK, then, perhaps all these collisions slowed down the stage, and it hit the moon after all?

Now we have to figure out how much all those gajillions of molecules weigh. Fortunately, remembering high school chemistry, this is easy. With a periodic table of elements, we take the atomic weight of each molecule, and then add them up. So for instance each Helium4 molecule has an atomic weight of 4 (or close enough for this estimate) so 40,000 of them total to 160,000 atomic units. Total up all the other molecules in the same way and it comes out to about 2.5 million atomic mass units.

OK, so now how much is that in units I can comprehend? It seems like a lot! Well, it turns out to be very little, because each molecule weighs very, very, very little. For instance one Helium atom, in grams, weighs about 0.00000000000000000000000166 grams. We better start using scientific notation, so that would be written as 1.66E-24. Now multiply that by the 2.5 million atomic mass units, and it come out to 4.13E-18 grams per cubic centimeter. Which is not much. (Yes, I am assuming the lunar exosphere is uniformly dense, which I know is not true. At this point I just want to get a feel for the problem.)

So now we have a war between big numbers and small numbers, that is, between the very large number of cubic centimeters (or cc's) that the stage sweeps through every second, 22 billion, versus the very small weight of the gas in each cc. So 22 billion times 4.13e-18 comes out to 9.2E-8 grams. Let's write that out as 0.000000092 grams. Every second. So far the small numbers are winning. That is the weight of the stuff the stage is running into every second, and it isn't much. And if we imagine each molecule as stationary when the stage strikes it, and moving away at the velocity of the stage after the collision, each collision will rob the stage of that tiny amount of momentum. (Think of a billiard ball striking another, stopping the first ball and sending the second one off with the momentum of the first.) So again, 9.2E-8 grams of collisions every second.

So now we just multiply by the number of seconds in 50 years. One year is 31.5 million seconds, and 50 years works out to 1.58E+9 seconds. Now the seconds help the big numbers to win the war, but not by much. The 9.2E-8 grams times 1.58E+9 seconds comes out to 146 grams. About one-third of a pound. Compare that to the weight of the stage. It's dry weight was 4700 pounds. So colliding with one-third pound of lunar exosphere over the course of 50 years could have slowed down the stage by 0.007%...i.e. nada!

So far, it seems that lumpy gravity didn't bring the stage down, and neither did atmospheric (or exospheric) drag. Perhaps Snoopy is still out there!

Simulating Uncertainty

Previously I posted about a simulation of the Apollo 10 LM descent stage which shows that the stage remains in lunar orbit to the present day. How robust is this result? The data for the initial stage orbit comes from the Mission Report...no doubt it was the best information they had. But fifty years in lunar orbit is a very long time, and lunar gravity is notoriously "lumpy". What if a slight change in the initial stage orbit state meant the difference between stability and decay?

To answer this question, I ran a set of 50 simulations, each with the initial conditions randomly varied to cover any possible miss in the initial state of the stage. I tried to keep the variation wide, to insure I covered the real conditions, but I also stuck to reasonable limits. In fact the variations I applied were so wide that many of the orbits were not viable. To cover this, I ran each parameter set through one orbit, recording the apolune and perilune...the low and high points of the orbit. I cut any set that was lower or higher by more than 20% from the values NASA reported. Only about one third of the random sets passed this test. To get 50 sets for the final test I passed more than 150 sets through the initial 1-orbit screen.

Here is a plot of the perilune points for all 50 random parameter sets, showing their minimum orbit altitude after 10 years in orbit. It's a bit messy, as these orbits show quite a bit of variation.

But the important thing to note is that in all 50 cases, the stage was still in orbit after 10 years. Each one of these plots is very similar to the "nominal" orbit I simulated initially. What if we just find the one of these, out of the 50, that got lower than any of the others, and plot it out by itself? Here it is.

You can see that this one did indeed make a rather low pass, in December of 1979, to about 12 km above the mean radius. (Still well above any lunar mountains.) And if you saw my earlier post about the stage orbit behavior, you see the same patterns here. The oscillation over a period of 25 days, and a longer oscillation with a period of around 5 months. Why did this one get lower than the others? It was one of the lowest initially, so it is hardly surprising. The real question is whether this one is any less stable over decades  than the "nominal" orbit that I showed before. What happens if we simulate this orbit out to the present? Here is the answer:

It is every bit as stable in it's orbit as the "nominal" case. There is no long term decay in evidence, and the simulated stage remains in orbit to the present day.

To me, this represents rather convincing proof. The result I got the first time I ran a simulation out to 50 years was no fluke. The nominal stage orbit is just one of a family of similar orbits that all exhibit long term stability. If something knocked the stage out of orbit during those 50 years, it wasn't the moon's lumpy gravity.

How could Snoopy be found?

If you've read my earlier posts you have seen how simulations of the Apollo 10 Lunar Module descent stage...aka "Snoopy"...show that the stage might still be in lunar orbit today, despite widespread expectations to the contrary.

If, indeed, by some miracle the stage orbit never did decay, and is still in orbit, how might we locate it? Experts have told me that optical methods won't work. However, there is another method that has been proven to work, and was used to locate a "lost" Indian lunar satellite in 2017. This method uses radar. Specifically, a dish in Goldstone is used as a transmitter, and another dish in Green Bank is used as a receiver.

In 2017, they knew that the lost satellite was in a polar orbit, so they could aim the radar just off of the moon's pole and wait for it to appear. Objects orbiting the moon come around about every two hours, so even if they were unlucky and the thing had just passed by when they turned on the machine, they would have to watch for two hours at the most.

How about Snoopy? If you read my earlier post about the stage orbit, you know that the stage is in a low inclination orbit...i.e. it follows the lunar equator. So to look for the stage they would aim the beam just above the rim of the moon, right at the lunar equator. And wait. And hope. For two hours at most? Not quite.

The same face of the moon is always facing the Earth. (Mostly.) Using the simulator, we can record the latitude and altitude of the stage as it crosses 90 degrees East longitude, that is, as it comes around from the back side of the moon. Here is a plot showing how that might have looked over 6 months of 2018. I have also drawn in a circle representing the size of the radar beam at lunar distance, which I was told is about 200 km wide.

Each orange dot represents one crossing of the stage above the lunar horizon, so there are two hours between each dot. Some days the stage is coming "over the hill" at a low altitude, below 100 km, and then other days it is coming over at a higher altitude. At this point, after 50 years, although the simulations give an idea that the stage is still in orbit, there is too much uncertainty to know exactly where it might be in its orbit, so you can take the above plot as a kind of "probability map" of how high the stage might be as it comes around on any given orbit. But the key takeaway here is that with a single two-hour radar observation, the stage might pass under the beam.

Aha, I have a brilliant idea! Aim the beam closer to the moon! Unfortunately, an expert told me that they need to aim the beam "several hundred kilometers" away from the surface, so that the receiver doesn't get overwhelmed with reflections of the moon itself. So how long would they need to observe with the radar to either find the stage or conclusively prove it was not there? To answer this question, you need to look at how the altitude at the limb changes with time.

As the moon rotates once each month, always keeping one face towards the Earth, it also rotates under the eccentricity of the stage orbit. So the altitude of the stage as it comes into view slowly varies, over the course of a month, from the lowest part of the orbit to the highest, and back again. So if the first attempted radar observation happened to be on a "low horizon-crossing altitude" day, and found nothing, they could wait half a month and try again. And oh, by the way, when the orbit is crossing the eastern limb of the moon at low altitude, it is crossing the western limb at high altitude, so another strategy would be to observe each equatorial horizon of the moon for two hours on a single night.

Wouldn't it be awesome to locate this amazing artifact after 50 years? Please tweet a link to this post to @NASAJPL if you agree.

Simulating Decades

Using a simulation environment developed by NASA, and a highly accurate lunar gravity model, we have seen how the orbit of the Apollo 10 Lunar Module ("Snoopy") descent stage behaved its first year in lunar orbit. Surprisingly, the simulation gives no sign of the orbital decay and lunar impact that was widely assumed. Now lets see what happens if we run longer simulations, out to 10 years, or 25, or even 50 years...to the present. Surely the stage will decay out of orbit over decades.

OK, so here are the results for a simulation of the first 10 years of the stage orbit. On my laptop, using a GRAIL gravity model with degree/order set to 200/200, it takes about 9 hours to run this simulation. As described in an earlier post I am only plotting the low "perilune" points of each orbit.

We can still see the 25-day cycles of oscillation in the orbit eccentricity that we saw in the first year, but now they start to blur together. The longer 5-month cycles are still easy to discern. But the key point of this plot is that there is no downward trend. Here and there you see the perilune dropping below 20 km, but overall there is no sign that the stage will be impacting the surface any time soon.

If we zoom in to look at the data for year 10, the pattern closely resembles the pattern of the first year in orbit.

So it seems possible that as John Young guided the first Space Shuttle flight to a successful landing in 1981, the stage he had last seen in lunar orbit ten years earlier might still have been orbiting the moon.

All right then, let's run this simulation out to the present day. The 50th anniversary of the mission was in May of 2019, and that is when I first got the hair-brained idea to go looking for this stage.

At the resolution of this image, the individual 25-day oscillations are no longer visible, and the 5-month oscillations are nearly lost. We see the orbit dipping below 20 km four or five times in any given decade. But the key takeaway from this plot is that the simulation gives no hint of any decay of the stage orbit.

And once more, zooming in on the results of the simulation for the year leading up to the 50th anniversary of Apollo 10, there is nothing to distinguish the behavior of the orbit in that year from the first year or the tenth. The same pattern of oscillating eccentricity is present.

It seems possible from this result that on the 50th anniversary of the mission, in May of 2019, the stage was still orbiting the moon, screaming along at over a mile a second, just as easily, as silently, as majestically as it was in May of 1969.

Could this really be true? If the stage were still in orbit, isn't there some way to detect it? That will be the subject of my next post.

Running Longer Simulations

Last time I posted an XY plot generated by GMAT, showing the stage altitude versus time. If you run the simulation yourself you will notice that GMAT spends a lot of time drawing and redrawing this plot while the simulation is running, and it is not very efficient.

For longer simulations, simulating months or years, I have found that it is better to shut down the XY plot, and instead use the GMAT "report" function to capture the data into a csv data file, which can then be processed after the simulation finishes to generate plots, etc. Here is an example of the GUI to set it up:

The next thing I do for longer simulations is to cut down on the number of data points that go into the report file. I am mostly interested in the minimum altitude, i.e. the perilune, for each orbit, so in the setup of the GMAT mission I build a loop that propagates the orbit around to the perilune and records that point. In that way I get a single line in the report file for every two hours of the stage simulation. That's 12 points per simulated day, or about 4300 per year. That keeps the report files manageable for longer simulations.

The other thing about saving the data to a csv file is that I can import it into a spreadsheet to generate more meaningful plots. Compare the XY plot from GMAT with a plot of the same simulation generated with a spreadsheet program. I overlaid the blue line on the GMAT plot so that you can see how it relates to my plot. 

In my plot I can control the labeling and scaling of the of the axes to make things more clear.

This chart is a bit more abstract, in that it is only showing how the low point of the orbit evolves over time, and ignoring the higher parts of the orbit. On the other hand, the labeling makes it easier to relate to what it means, and its easier to see some important features. The GMAT XY plot appears to show the altitude as "0" in the lower left corner, but that is really "0 elapsed days", and the initial altitude of 20+ km is not clear. In my plot the altitude is much more clearly labelled. The other thing that is much easier to notice is that the perilune is showing an upward trend! Unexpectedly, not only was the stage still in orbit on July 20 when the Eagle landed, its perilune altitude was apparently HIGHER than at any time since staging, two months earlier.

One misleading thing about plotting only the perilune points is that it appears the stage is defying gravity by rising in its orbit, but look back up at the GMAT XY plot. As the perilune rises, the apolune is dropping. Another way of saying this is that the energy of the stage orbit is conserved. A bump in the perilune altitude is always accompanied by the dip in the apolune, and vice versa. The eccentricity of the orbit is oscillating, but the overall average height of the orbit remains the same. Keep that in mind when looking at the perilune plots.

OK, lets run the simulation out a full year to see if the trend of the first two months continues.

Now we see a new aspect of the oscillation of the orbit. The initial upward trend in the perilune doesn't hold up. Over about 5 months, the oscillations drop back down, and the minimum altitude of the stage is actually lower than it was in May, for the first time. But then it starts trending up again, so that by November, while Pete Conrad and Al Bean were down on the surface standing next to Surveyor, with Dick Gordon orbiting overhead, the stage was still whizzing around.

If you were hoping to find a crater marking the final resting place of the stage, as I was when I began this exercise, at this point you are realizing that these simulations won't be very helpful. (Actually there is a pattern in this data, which I will explain in a later post, that did give me some slim hope. What can I say...I am an optimist!) 

Next time I will show what happens when you run the simulations out 10 years and more.

Stage Orbit Basics

In this post I will show the behavior of the stage in the first days, weeks, and months after it was jettisoned on May 22, 1969.

The stage was jettisoned into the "Phasing" orbit, which rose to around 352 kilometers high on the far side of the moon, and dropped down to around 22 km on the near side. The low point in the orbit, called the perilune, was near the Sea of Tranquility where Apollo 11 would be landing in two months.

Using the GMAT simulator I described earlier, you can enter the orbital parameters from the Mission Report, Table 6-II, and run the simulation for a single orbit. First off, here is the GroundTrack view that GMAT generates:

You see the stage in a low inclination orbit, very closely following the lunar equator. I added a small red dot showing where Apollo 11 landed, and you see that they passed just south of that site, just right of the center of this plot.

Here is GMAT's "Default Orbit View" of the same orbit, with a view from above the moon's north pole.

You see that the orbit drops down closer to the surface on the lower left, and then rises higher around on the far side. The red axis points towards Earth, and the blue axis is the north polar axis. In this view the orbit looks very nearly circular, which is indeed the case. The eccentricity of the orbit is actually not so high, relatively speaking.

Finally, here is a GMAT X-Y plot of the stage Altitude versus elapsed seconds.

Now the low and high points of the orbit are much more exaggerated. The "Altitude" is relative to the "mean lunar radius", which is sort of like "sea level"...the size the moon would be if it melted into a perfect sphere. (This altitude does not take into account lunar terrain.) The orbit drops down to within 22 km, and then rises up to around 350 km. (BTW, that "0" down in the lower left corner is zero seconds, not zero km.) This is one complete orbit, and the total time, i.e. the period, is around 7611 seconds...just over 2 hours.

So what happens when you run the simulation out longer? Here is a similar plot showing the altitude over the first 5 days.

You see the same highs and lows, but a new pattern begins to emerge. The lows are slowly getting higher, while the highs are getting lower. The orbit is becoming more circular, and less eccentric. This seems to be one of the first reasons that the stage does not quickly decay out of orbit. Later I will post the results of a study that shows more about this pattern.

OK, now let's run even longer, simulating the first 60 days in orbit. This will put the simulated time into late July, 1969, when Apollo 11 returned to the moon.

The first thing to notice is that the stage is still in orbit at the time of the Apollo 11 landing. Something might have happened since then to bring the stage down, but I feel quite certain that it was still orbiting the moon during Apollo 11. Then notice the pattern of oscillation in the eccentricity. The orbit becomes more circular for the first 12 days or so, but then starts to become more eccentric again. After about 25 days the orbit is nearly back to its original apolune/perilune. (But not quite, you notice.) This 25 day period of oscillation of the eccentricity is a pattern that will continue, and when I run simulations out to the present, this same 25 day oscillation remains a key feature of the stage orbit.