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.

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**Monte Carlo**. Show all postsShowing posts with label

**Monte Carlo**. Show all posts## Tuesday, February 4, 2020

### Simulating Uncertainty

Labels:moon Apollo Lunar Module orbit
Apollo 10,
Descent Stage,
Lunar Module,
Lunar Orbit,
Monte Carlo,
Simulation,
Snoopy

## Monday, January 20, 2020

### Introduction

The descent stage of the Apollo 10 Lunar Module ("Snoopy") may still be in lunar orbit today. This defies conventional wisdom. It goes against all expectation about how things behave in lunar orbit. It is the last thing I expected to find when I set out to look for an impact crater that I assumed would be the final resting place of the stage. Nonetheless, this is what simulations of the stage orbit show. In this blog I will show how I arrived at this surprising conclusion.

When I started looking for the stage, I was expecting a quick orbital decay. Everything I read said that's what happened. I thought this would mean less uncertainty about the impact point...a smaller search area. So I was disappointed when I started running simulations, showing that the stage stayed in orbit for months. That was not helpful for finding a crater. My remaining hope was that some high piece of lunar terrain might have snatched the stage if it slowly drifted down to lower and lower altitudes. Perhaps I could focus the search on lunar mountaintops. So I kept looking.

The stage orbit was unusual, in terms of Apollo orbits. In order to demonstrate undocking, firing the LM descent engine to approach the moon, and then firing the ascent engine for the rendezvous, NASA had a problem. Without any landing, they needed a way to arrange for the right timing of the maneuvers. The descent would put the LM in a lower orbit, moving it ahead of the Command Service Module. (The "CSM".) Demonstrating the ascent and rendezvous required that the CSM be leading the LM. The solution was the "Phasing" maneuver. This special burn, never performed by any other Apollo mission, would raise the high side of the LM orbit to 190 nautical miles above the far side of the moon, slowing down the LM's orbital period enough to allow the CSM to overtake it.

While the LM was in the Phasing orbit, 12 n.m. at its low point, and 190 at the high point, the descent stage was jettisoned, with an initial velocity relative to the ascent stage of around 2 feet per second. (Both parts were zipping along at a mile a second at this point.) The goal was to kick the stage forward, but unexpected problems with the attitude controls during staging altered this, and the stage was pushed "upward" relative to the local horizontal at the time of staging. (Notice that the moon is "upside down" in the picture above taken during staging...it wasn't supposed to be this way.) Regardless of the extra drama, ten minutes later, the stage was at a safe distance, and the crew fired the ascent engine, slowing their velocity and lowering the high side of their orbit, putting them on track for a successful rendezvous and docking. The stage was left behind in the phasing orbit. It was assumed that this orbit would quickly decay, impacting the moon within days or weeks.

As I starting running simulations of the stage orbit, the hope for a quick demise did not pan out. I ran the simulations out longer and longer, out 10 years, and still the stage kept going. Finally I decided to run the simulation out to the present. This took about 40 hours on my laptop. At the end, the stage remained in orbit all the way to the present, with no sign of decay or orbital instability. As I build out this blog I will share more details, and show you how to try it to see for yourself.

_{}^{}This picture of the descent stage ladder and footpad comes from the 16mm "DAC" film taken on May 22, 1969, during the dramatic moments when the stage was jettisoned. |

Snoopy's tail was jettisoned into lunar orbit on May 22, 1969, during a daring mission that paved the way for the first moon landing less than two months later. Apollo 10 was the first mission to take a Lunar Module to the moon; the first test of all the hardware and procedures. All except landing. It was the first demonstration of Lunar Orbit Rendezvous, the risky, radical, "sine qua non" of Apollo.

The stage orbit was unusual, in terms of Apollo orbits. In order to demonstrate undocking, firing the LM descent engine to approach the moon, and then firing the ascent engine for the rendezvous, NASA had a problem. Without any landing, they needed a way to arrange for the right timing of the maneuvers. The descent would put the LM in a lower orbit, moving it ahead of the Command Service Module. (The "CSM".) Demonstrating the ascent and rendezvous required that the CSM be leading the LM. The solution was the "Phasing" maneuver. This special burn, never performed by any other Apollo mission, would raise the high side of the LM orbit to 190 nautical miles above the far side of the moon, slowing down the LM's orbital period enough to allow the CSM to overtake it.

While the LM was in the Phasing orbit, 12 n.m. at its low point, and 190 at the high point, the descent stage was jettisoned, with an initial velocity relative to the ascent stage of around 2 feet per second. (Both parts were zipping along at a mile a second at this point.) The goal was to kick the stage forward, but unexpected problems with the attitude controls during staging altered this, and the stage was pushed "upward" relative to the local horizontal at the time of staging. (Notice that the moon is "upside down" in the picture above taken during staging...it wasn't supposed to be this way.) Regardless of the extra drama, ten minutes later, the stage was at a safe distance, and the crew fired the ascent engine, slowing their velocity and lowering the high side of their orbit, putting them on track for a successful rendezvous and docking. The stage was left behind in the phasing orbit. It was assumed that this orbit would quickly decay, impacting the moon within days or weeks.

As I starting running simulations of the stage orbit, the hope for a quick demise did not pan out. I ran the simulations out longer and longer, out 10 years, and still the stage kept going. Finally I decided to run the simulation out to the present. This took about 40 hours on my laptop. At the end, the stage remained in orbit all the way to the present, with no sign of decay or orbital instability. As I build out this blog I will share more details, and show you how to try it to see for yourself.

Labels:moon Apollo Lunar Module orbit
Apollo 10,
Descent Stage,
GMAT,
Hypergolic propellants,
JPL,
Lunar Exosphere,
Lunar Module,
Lunar Orbit,
Monte Carlo,
NASA,
Near Miss,
Precession,
Radar,
Simulation,
Snoopy,
supercritical helium,
Tom Stafford

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