The Eagle May Still be Flying
On 20 July 1969, two human beings landed on the Moon in a spaceship called Eagle. Less than 24 hours later, on 21 July 1969 the Eagle and its two occupants took back off and rendezvoused with a spaceship called Columbia waiting in a low orbit with another human being on board. The Eagle docked to Columbia and its two occupants changed ship. The Eagle was then undocked and abandoned. As its orbit was unstable, it crashed on the lunar surface some time later. Or so everybody assumed.
The Eagle’s two occupants were Neil Armstrong and Buzz Aldrin. The third man was Michael Collins. The mission was Apollo 11. Its journey is one of humankind’s crowning achievements. The first time human beings set foot on a different celestial body.
At the time, nobody cared much about the fate of the Eagle. After having transported its crew safely to the waiting Columbia, it was no longer relevant. The major thing that mattered was whether it would pose a danger to Columbia after the undocking. Everybody working for the Apollo project was concentrating on getting Columbia safely back to the Earth, which it did three days later, or on preparing the upcoming Apollo missions. Nobody would have had time to worry about a piece of abandoned spacecraft hardware.
Why did everyone think the Eagle crashed?
At the time of the Apollo missions, not very much was known about the Moon, though much progress had been made in that decade – most of it in the context of the Apollo project and its preparatory missions. It was known that the Moon was, to put it simply, a rather lumpy body. Its gravitational field was not that of a perfect sphere with uniformly distributed mass, but showed a significant degree of inhomogeneity, some of it in the form of “mascons”, local mass concentrations.
The inhomogeneity of the lunar gravity potential, together with the gravitational pull of the sun and the nearby Earth, work together to perturb the trajectory of a body orbiting the Moon. The main effect is that the orbit will become eccentric. The eccentricity, or ellipticity, describes how much the shape of an orbit deviates from that of a pure circle.
As eccentricity builds up and the orbit becomes an ellipse rather than a circle, there will be one point on the orbit that is lower than anywhere else on that orbit. This point is called the pericentre (or periapsis, or for lunar orbits, periselene or perilune). If the mean orbital altitude (which is equal to the orbital altitude of the initial, circular orbit) is low, then you don’t need much eccentricity to have a pericentre that is so low that the orbiting body will hit some high mountain or even the lowland areas.
This is what makes lunar orbits unstable. With lunar orbits, things are a lot different than with Earth orbits. On a low Earth orbit, the dominating perturbation source is air drag. Air drag is a dissipative force. The resulting friction converts orbital energy to heat. If you plot the trajectory over time, the result will look like a spiral. You’ll see a mean altitude that decreases, first slowly, then faster and faster. Conversely, on a low lunar orbit, the main perturbative forces are gravity forces. These are conservative; they do not change the orbital energy but build up the eccentricity.
If there is a crash in the end, such niceties may may seem like a moot point. If you get hit by a truck at speed, does it matter whether the truck is blue or red? But here, it does matter! Air drag will always lead to a crash, eventually. But eccentricity variations are cyclical; the eccentricity alternatingly increases and decreases. If conditions happen to be just right, it may happen that the eccentricity peaks out at a value that is too small to lead to a crash.
The fact that gravitational perturbations lead to an eccentricity buildup was known to astrodynamics experts in the Apollo era. They also knew that the lunar gravity potential had a strong degree of inhomogeneity, and that the effect of the sun and Earth gravity on a lunar orbiter would be significant because the gravitional attraction of these bodies is strong, compared to that of the fairly small Moon.
However, the exact parameters of the lunar gravity potential were not known. To obtain these, you need spacecraft in low lunar orbit. These have to be operated for long times, during which their orbits are continuously determined, using radiometric measurements derived from the signals exchanged between the spacecraft and ground stations on the Earth. The orbits must also be corrected regularly via a propulsion system on the spacecraft.
From the measured changes in the orbital elements, an increasingly voluminous set of parameters can be derived that describes the gravity potential. This takes a.) a long time and b.) enormous computational power. Two things that NASA did not have back then, but that became available in the following half-century.
In the 1960s, NASA had operated only a handful of spacecraft in lunar orbit for extended periods of time. The five Lunar Orbiters had been used to provide high resolution surface imaging data in preparation of the planned human missions. Although it was known that low lunar orbits would be subject to significant perturbations, and existing data also gave some indication of the expected size of those perturbations, the orbit experts did not have the data and they did not have the computing power to perform meaningful long-term analysis.
Based on the limited knowledge and resources available, conventional wisdom at the time was that low lunar orbits are generally unstable. A spacecraft placed on a low lunar orbit was expected to crash within an unspecified but limited period of time. The prediction that this applied to the ascent stage of the Eagle, the Apollo 11 lunar module, appeared like a pretty safe assumption.
The Eagle, the Apollo 11 Lunar Module
For the Apollo lunar missions, the Lunar Module was the mission element that
- brought two people and everything they would need to survive and work from low lunar orbit down to the surface,
- provided a habitat with a a life support system and
- carried the two people and things they wanted to bring back from the lunar surface to low lunar orbit.
Each lunar module had its own name that was chosen by its crew. Apollo 11’s was called Eagle. The Lunar Module had a “wet mass” of around 15 metric tons, three quarters of which were propellant. It was a two-stage vehicle.
The descent stage slowed the module down during descent and cushioned the impact at touchdown. It also carried most of the gear that was needed on the surface. The ascent stage contained the pressurised crew habitat and the engine for the ascent from the lunar surface. When the time for return had come, the descent stage remained on the surface and the ascent stage with its two occupants separated and launched itself into low lunar orbit.
The Apollo 11 Eagle Lunar Module descent stage still stands where Armstrong and Aldrin landed it in Mare Tranquillitatis. This ascent stage was believed to have have crashed on the lunar surface.
Until now, that is.
Enter Jim Meador
A few months ago, I received an e-mail from an engineer called Jim Meador from Mountain View, California. Jim was not a professional in the space industry, but a space buff who had attempted to find out where the Eagle might have crashed. He thought that constraining the impact region might be of help when searching for the impact crater.
Jim understood that he would need two things:
- Knowledge of the final set of orbital parameters. As these can not be known exactly, then at least the approximate values and the range of the uncertainty should be known
- An orbit propagation tool that allows inclusion of all relevant perturbation sources and contains the most up-to-date model of the inhomogeneity of the lunar gravity potential
For the first of these two, he had to rely on Apollo-era documentation. Nobody had seen the ascent stage since, and nobody had performed an orbit determination. What was known was the orbit of Columbia and (albeit approximately) the delta-v imparted at the final separation of the Eagle from Columbia.
For the second, Jim used NASA’s GMAT (General Mission Analysis Tool), an open source numerical trajectory software. Unlike the 1960s, computing power no longer is an issue nowadays. Any high end PC now has sufficient number-crunching capability to handle even high-precision numerical orbit propagation runs that cover the period of over 50 years since the Eagle was abandoned in space.
To his surprise, GMAT did not tell Jim where the Eagle may have crashed. In fact, GMAT told him that the orbit appeared to show long-term stability. Though the eccentricity did undergo cyclical variations, as it should, the peak eccentricity was not large enough to lead to an impact on the surface. The pericentre did decrease to about 15 km of altitude periodically, but it always went up again and did not show any secular trend, as a truly decaying orbit would.
Jim Meador’s Paper
Jim did what everybody who has made a scientific discovery should. He wrote a paper summarising his findings and submitted it to a journal. But then he encountered some problems getting it published. The editor of the journal suggested that he should contact me for advice, which led to Jim’s e-mail. I was immediately interested, but also cautious. People contact me all the time, sometimes with the most outlandish claims. But this looked different right from the start. I could see from the original paper Jim had submitted that he had been doing his homework.
The first thing I did was to check whether the parameters of the initial orbit assumed by Jim made sense. They did. Then I looked at his assumptions for the perturbation model set in GMAT. That looked good as well. I then proceeded to propagate the trajectory with my software, which is completely independent of GMAT. My results completely confirmed his.
By now I had become extremely interested. I made some suggestions and listed some things that in my opinion should be changed. As a result, Jim’s paper was considerably rearranged and extended. In its re-worked form, it qualified for resubmission. Jim then resubmitted it to the Journal of Planetary and Space Science, where it is currently undergoing peer review. He also uploaded it to arXiv (see link at the end of this article) so anyone can download and read it.
In a nutshell, the paper contains the following:
- Presentation and justification of the assumptions for the starting orbit
- Explanation of the analysis method used
- Results of the initial orbital propagation
- Analysis of the possible effects of dispersion in the initial orbital state
- Analysis of the added perturbative effect through solar radiation pressure
- Discussion of the resulting probability of the Eagle having survived to this day
- Suggestion of a radar detection campaign to re-locate the Eagle.
Jim isn’t claiming that it is a certainty that the Eagle is still in orbit. In fact he explicitly states that this spacecraft and was designed for a limited duration mission and so were all its systems. It is entirely possible e.g., that fuel has leaked and caused it to blow up or deorbit. But this is just a possibility, not a foregone conclusion. We simply don’t know. But if we look at the celestial mechanics, the orbit appears to be sufficiently stable to ensure survival to this day.
Conclusions of the Paper
The results of the extensive numerical analysis presented in the paper are entirely consistent with the assumption that there is a strong possibility that the ascent stage of the Apollo 11 lunar module, the Eagle, may still be orbiting the Moon and will continue to do so in the foreseeable future.
Jim Meador recommends a radar tracking campaign to relocated the Eagle. The orbital period is just under 2 hours. Due to the near-equatorial inclination, the Eagle, if it is still in orbit, should regularly appear above the Eastern or Western limb of the Moon. Therefore, a sequence of radar tracking passes with durations of a few hours each should suffice to find it. A precedent for this is the campaign that led to detection of the Indian Chandrayaan orbiter in 2017.
A few more findings from my side
Out of the extensive sets of results of my verification runs that completely confirm Jim Meador’s calculations, I will show only only one diagram of mine that, for a limited time frame of around 6 months, shows only those parts of the trajectory where the altitude is below 30 km, as function of the selenographic latitude. This confirms Jim Meador’s finding that low pericentre altitudes coincide with a very limited longitude range.
As Jim Meador goes on to demonstrate, with some variation of the initial orbital parameters, the minimum altitude may in some cases go below 10 km, but the location will then still be in the same longitude range, i.e., above Mare Tranquillitatis. Keeping in mind that the latitude range is constrained to within around 3 degrees straddling the lunar equator, there does not appear to be any risk of impact with surface features.
And Finally: My Opinion
As stated, I have had the chance to discuss all aspects of the work in detail with the author. I have verified the salient results independently. Unless some catastrophic, destructive event overtook the Eagle, there is a good chance that the Eagle is still flying.
I wholeheartedly agree with Jim’s recommendation to conduct a dedicated radar tracking compain with the aim of relocating the Eagle after more than five decades. This would be more than just a technical exercise. The societal value of any recoverable artefact associated with the historic Apollo 11 mission cannot be overstated.
This is not just any spaceship. It is the Eagle. The ship that landed the first human crew on another celestial body.
It is the single most important ship in human history. We should try to find it.
Link to Jim’s Paper on arXiv
James Meador: Long-term Orbit Stability of the Apollo 11 Eagle Lunar Module Ascent Stage, arXiv:2105:10088 [physics.space-ph]