The Eagle May Still be Flying

Exploded View of the Apollo Lunar Module showing the descent and ascent stages, source:

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.
Exploded view of the Apollo Lunar Module showing the descent and ascent stages, source:

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.

Six month simulation results of the Eagle orbit showing the lowest pericentre altitudes as function of the selenographic longitude
Six month simulation results of the Eagle orbit showing the lowest pericentre altitudes as function of the selenographic longitude, source: Michael Khan

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.

Topographic map of the lunar surface between longitudes 15 and 45 deg E and latitudes 15 deg N and S, showing colour coded surface elevation with respect to the reference radius.
Topographic map of the lunar surface between longitudes 15 and 45 deg E and latitudes 15 deg N and S, showing colour coded surface elevation with respect to the reference radius. Source: Michael Khan

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 campaign 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.

James Meador: Long-term Orbit Stability of the Apollo 11 Eagle Lunar Module Ascent Stage, arXiv:2105:10088 []


Ich bin Luft- und Raumfahrtingenieur und arbeite bei einer Raumfahrtagentur als Missionsanalytiker. Alle in meinen Artikeln geäußerten Meinungen sind aber meine eigenen und geben nicht notwendigerweise die Sichtweise meines Arbeitgebers wieder.

13 Kommentare

  1. Great,
    “This is not just any spaceship. It is the Eagle. ”
    Schreiben Sie einen Brief an Elon Musk, er wird die Suche und Bergung finanzieren.

  2. NSSDCA Master Catalog
    The fate of the LM is not known, but it is assumed that it crashed into the lunar surface sometime within the following 1 to 4 months.

    In der Ascent Stage befanden sich Reste von Inhaltsstoffe wie Treibstoffe, Sauerstoff, Helium, Wasser und Batterien. Das Lunar Module wurde nicht für die Ewigkeit gebaut, dadurch gehe ich von aus, dass irgendwann die Tanks oder die Behälter oder die Leitungen undicht geworden sind und dadurch die Ascent Stage einen Antrieb herhalten hat. Aber wohin?
    Beim Anflug an den Landplatz war der Eagle zu schnell gewesen, war noch in den erlaubten Werten gewesen. Der Grund dafür war das der Eagle beim Abdocken von der Columbia eine zusätzliche Geschwindigkeit erhalten hatte. Und wenn dies beim zweiten Abdocken auch passiert ist, wie will man da was berechnen?
    Was damit Aussagen will die Ascent Stage war kein toter Körper gewesen, sondern ein Objekt mit Energie, wo keiner sagen wie die sich ausgewirkt hatte.

    • Sehr geehrte Frau Schäfer, vielen Dank für Ihren Kommentar. Ich werde mich bei der Beantwortung kurz fassen, muss aber zunächst die Frage voranstellen, ob Sie meinen Blog-Artikel und auch die darin verlinkte Veröffentlichung auf arXiv gelesen haben. An mehreren Stellen dort wird nämlich bereits genau auf die Punkte Bezug genommen, die Sie ansprechen.

      Zunächst zur Aussage dim NSSDCA Master Catalog (und anderswo), die Aufstiegsstufe des Mondmoduls “Eagle” sei binnen weniger Monate abgestürzt. Das ist die Aussage der NASA zur damaligen Zeit und ist wohlgemerkt nur eine Vermutung. Ich habe bereits ausführlich dargelegt, wie es zu dieser Aussage gekommen sein muss und warum es damals unmöglich war, eine genaue Vorausberechnung vorzunehmen. Später hat niemand das nachgerechnet. Zumindest kann ich keine Veröffentlichung dazu finden. Bis jetzt.

      Was die Möglichkeit eines unvorhersehbaren, vielleicht katastrophalen Ereignisses angeht: Jim Meador nimmt damit in seinem Paper am Ende von Kapitel 4 (Seite 11) ausdrücklich Stellung und ich greife seine Aussage in meinem Blog-Artikel (Abschnitt “Jim Meador’s Paper” auf. Die Sache ist ganz klar: Wenn der Eagle explodiert ist, ist er weg. Wenn es durch Treibstoff-Austritt ein Manöver gegeben hat, kann die Bahn sehr wohl aus dem engen Bereich gestoßen worden sein, in dem Langzeitstabilität erreicht wird. Niemand kann was wissen. Was Jim Meador sagt, ist, dass bahnmechanisch gesehen ein Überleben im Mondorbit möglich gewesen wäre. (Das ist keineswegs selbstverständlich, sondern ein Ausnahmefall im niedrigen Mondorbit) Er schlägt vor, das mit einer relativ einfach durchzuführenden Radarkampagne zu klären, wie sie bereits bei der Suche nach einem indischen Mondorbiter erfolgreich war. Mehr nicht.

      Woher stammt die Aussage mit der “zusätzlichen Geschwindigkeit beim Abdocken”? Ich habe dazu anderes gelesen, und zwar in technischen Veröffentlichungen der NASA aus der Zeit kurz nach Apollo 11, als noch alle Informationen vorlagen.

      Erstens ist hier die Timeline der Ereignisse der Apollo-11-Mission zu nennen. “CSM/LM undocked” passiert bei 100 Stunden und 12 Minuten nach Liftoff. Zur Zeit 100:39:52.9, also fast 28 Minuten später, erfolgte das kleine “CSM/LM separation maneuver”. Bis dahin hätten die also eine Diskrepanz aufgrund eines nennenswerten zusätzlichen delta-v bei der Abtrennung nicht gemerkt? Nochmals fast eine Stunde später, um 101:36:14, also mittlerweile fast drei Viertel eines Umlaufs nach der Abtrennung begann das “LM descent orbit insertion”, also das Manöver, das den eigentlichen Abstieg einleitete. Dieses Manöver war mit 75 fps (feet per second, ich zitiere das so, weil es in der relevanten Literatur so steht) erheblich.

      Der Grund für den Overshoot des eigentlich anvisierten Landepunkts um “mehr als 3 Meilen” ist laut diesen technischen Berichten von Floyd Bennett, verantwortlich für die flugdynamische Analyse der Abstiegs- und Aufstiegsphase des Lunar Module ein Initialisierungsfehler von etwa 18 fps (aber in radialer Richtung!) des PGNCS (Primary Guidance, Navigation and Control System). Siehe hier und hier. Gibt es eine plausible Alternativerklärung für den Navigationsfehler und wenn ja, wo ist diese dokumentiert?

      Hätte Jim Meador in seinem Paper die mögliche Unsicherheit in den Ausgangsbahnelementen nicht berücksichtigt, dann hätte ich ihm das sofort angekreidet (und die Peer Reviewer auch), weil das ein schwerwiegendes Versäumnis dargestellt hätte. Er hat aber die Unsicherheiten im Rahmnen einer Monte Carlo-Analyse einbezogen, in der die Anfangsparameter innerhalb vorgegebener Bandbreiten mit einem Zufallsgenerator variiert und somit 100 jeweils unterschiedliche Einzelfälle jeweils über eine Simulationsperiode von 51 Jahren numerisch propagiert werden. Trotz Unterschieden im individuellen Verlauf sind die Ergebnisse in allen Fällen vergleichbar und in keinem Fall kommt die Simulation zu einem Absturz.

      Offenbar ist die kleine Stabilitätstasche, in der sich diese Bahn zufällig befand, groß genug, um auch bei realistischen Unsicherheiten in den Bahnparametern nicht verlassen zu werden. Natürlich kann niemand sagen, welches die genauen Bahnparameter sind, aber man kann sehr wohl die zu erwartenden Variationsbandbreiten mithilfe des gewählten Analyseverfahrens abschätzen. Das wird in der Navigation von Satelliten und Raumsonden immer so gemacht.

      • Die Abweichung ist erst später aufgefallen. Für die Orientierung wurden Bezugspunkte auf der Mondoberfläche ausgewählt und Neil Armstong stellte fest das diese um 2 bis 3 Sekunden zu früh überflogen werden. Bei Flugzeit 102:36:18 meldelt Armstong: Our position checks down range show us to be a little long.“)

        While monitoring the Lunar Module’s position and velocity he came close to calling an abort when it became clear a navigational error had occurred. The spacecraft was moving 20 feet per second (6 m/s) faster than it should have been and was halfway to its abort limits. However, Bales continued to watch the data and the situation remained stable.

        Ron Wells macht darauf aufmerksam, wie Gene Kranz in Failure is Not an Option diese Positionsabweichung erklärt. Der Verbindungstunnel zwischen LM und CSM war nicht vollständig entlüftet, sodass ein gewisser, wenn auch geringer, Überdruck beim Abdocken (bei 100:12:00) einen Impuls verursachte.

        • Ich denke, man sollte vorsichtig mit irgendwelchen anekdotischen Begründungen umgehen. Floyd Bennett zumindest hat seine Ausführungen technisch begründet und nachvollziehbar dokumentiert (Link in meiner vorherigen Antwort)

          Wie auch immer – die Berechnungen in dem Paper von Meador enthalten eine Unsicherheitsanalyse unter Berücksichtigung recht erheblicher Abweichungen vom abgegebenen Anfangsorbit, die nahelegen, dass der Stabilitätsbereich, in dem sich die Bahn des Eagle befindet, nicht ganz schmal ist. Das Paper von Meador sagt aus, dass gravitative Störungen allein nicht zwangsläufig die Bahn des Eagle zum Absturz gebracht haben müssen. Das Paper listet aber auch diverse Gründe auf, weshalb das Raumschiff dennoch zerstört worden oder abgestürzt sein könnte. Es gibt aber zumindest gute Gründe für den Versuch, mit einer Radar-Messkampagne nachzuschauen.

      • Natürlich kann niemand sagen, welches die genauen Bahnparameter sind

        Die Bahnparameter des LM sind im Apollo Flight Journal an zwei Stellen explizit genannt (siehe bevor 135:47:24 und bevor 137:30:12 ). Dies war etwa 5 und 7 Stunden nach der Abtrennung des LM und die Exzentrizität der Bahn war demnach 0,0048 und 0,0050. Dies war also beträchtlich mehr als von dem Autor angenommen (0.0037, 0,0038 und 0,0035 für seine Nominal, Maximum und Minimum Fälle)

  3. Ich würde mir wirklich wünschen, dass nicht nur Behauptungen und Zitate geliefert werden, sondern auch mal etwas nachgerechnet wird. Gerade durch eigenes Nachrechnen kann man viele Sachen klären und ist nicht immer auf Aussagen anderer Leute angewiesen.

    Beispielsweise die Sache mit dem angeblich durch den Luftaustritt verursachten Impuls. Für die Nachrechnung braucht man nicht mal mehr als die Grundrechenarten.

    Ich mache mich an die Extremwertbetrachtung. Angenommen, die komplette Atmosphäre aus dem Aufstiegsmodul entweicht. Was kann das maximal für ein delta-v bewirken? Die Zahlen, die ich brauche, hole ich mir aus der Wikipedia.

    Das Innenvolumen des Lunar Module is 6.7 m^3. Die Atmosphäre ist Sauerstoff unter 330 hPa Druck, also 1/3 des Luftdrucks bei Normal 0. Wieviele kg Gas werden das sein? Wohl weniger als 3 kg.

    Wie schnell kann das ausgestoßen werden? Wir kennen Kaltgastriebwerke. Da wird ein Gas unter Druck durch eine Düse ausgestoßen. Der typische spezifische Impuls ist so rund 60 Sekunden. Hier aber haben wir es nicht mit hohem Druck zu tun, sondern nur 330 hPa, und es gibt auch keine Düse. Also wird der spezifische Impuls höchstens so rund 50 s, sein, d.h., die Ausströmgeschwindigkeit 490.5 m/s.

    Wie lange das Ausströmen dauert, ist für das delta-v egal, aber nehmen wir mal eine Minute an. Dann erzeugt das Ausströmen einen “Schub” von 24.5 N. In Wirklichkeit ist das alles etwas komplizierter und sogar weniger, weil der Druck ja nachlässt.

    Die leere Aufstiegsstufe hat eine Masse von mehr als 2500 kg, aber ich bin großzügig und sage 2500. Die Beschleunigung wäre also weniger als 0.01 m/s^2 und das gesamte delta-v knapp 0.6 m/s, oder wenn man fps will, dann knapp 2 davon.

    Das wäre also das maximale delta-v, wenn alles Atemgas aus dem Habitat in eine Richtung ausgestoßen wird und die Treibstofftanks leer sind. Schlappe 0.6 m/s. Weit innerhalb der Unsicherheit, die Meador in seiner Monte Carlo-Rechnung vorgibt.

    Und man ganz nebenbei:

    Was nun das angebliche Ausgasen bei der CM/LM-Trennung vor der Landung angeht, da war das ganze LM noch 15 Tonnen schwer und es ging nur um das Gas im Verbindungstunnel, also viel weniger Gas als oben angenommen. Es dürfte also sehr schwierig sein, mit dem verbleibenden, winzigen Impuls den Overshoot von 3 Meilen auch nur ansatzweise zu erklären.

    Das ist zwar im Rahmen des hier behandelten Themas ganz egal. Aber trotzdem kam die Behauptung hier hoch. Ich denke, auf Basis meiner Berechnungen hier, dass das Ganze hinten und vorne nicht stimmen kann. Ich lasse mich natürlich gern überzeugen. Aber nicht, indem man mich mit Zitaten zuschüttet. Da muss mir bitteschön schon jemand was vorrechnen.

    Zurück zum Thema des Artikels und dem zitierten Paper: Der Autor hat Unsicherheiten von bis zu 10 fps angenommen, die maximal 2 fps beim vollständigen Ausstoß des gesamten Gasinhalts würden also nicht ausreichen, um den Absturz herbeizuführen.

    Da müsste schon der Resttreibstoff in den Tanks beteiligt sein. Das aber stellt niemand in Abrede. Jim Meador hat es sogar ausdrücklich angeführt. Das Paper muss man aber schon lesen.

  4. The Apollo flight journal gives the orbital parameters of the LM at around 135:47:24 and 137:30:12 mission time (see , comments by Public Affairs Officer just before these times). These imply eccentricities of 0.0049 and 0.0050 respectively, which is substantially higher than those the numerical simulations were based on (0.0037, 0.0038 and 0.0035 for the nominal, maximum and minimum case). The latter figures were apparently based on the parameters for the command module at the moment when the LM was jettisoned

    • The numerical propagation of the orbital state following the epoch of separation, starting out from the orbital state stated in James Meador’s paper for the first hours does show an increase of eccentricity (see here for eccentricity and here for peri-/aposelene). After around 5 hours, the osculating eccentricity peaks at about 0.0041 and after around 7 hours at about 0.0043. Though this is still less than the values you state, there is a distinct increase with respect to the starting value. It would now be important to have some more information on the accuracy of the orbit determination that led to the “last fix” on the Eagle’s orbit as stated in the Apollo Flight Journal. How long were the observation arcs, what is the standard deviation on the elements etc.

  5. OK, thanks, this is useful information (which I can’t see being mentioned in James Meador’s paper though). The question is whether adjusting the starting parameters to be consistent with the values mentioned in the Apollo flight journal would make any difference to the conclusions.

    Also, I wonder why the orbit of PFS-2 (Apollo 16 sub-satellite) has not been considered as a test comparison here. It would make the results of any simulation of an event with an unknown outcome much more conclusive if the same algorithm gives the correct result for an event where we know the outcome exactly.

    • For clarification: the short term results I presented in my previous reply followed from a numerical simulation run that I performed, not James Meador (though he will also have done such calculations). I think that Mr. Meador’s procedure was correct: He based his calculations on a presumably consolidated, complete set of orbital data, which was published in an official NASA document, the Apollo 11 mission report from November 1969, with some cross-checking and verification from a technical paper by J.P. Murphy, issued in 1970 (see the list of references in Meador’s paper). I would agree with the author that this is the most reliable source of data, much more so than a brief mention, while the mission was still under way, of just a subset of orbital elements which is not complete and does not have an attached time tag.

      How consolidated are those mentioned orbit fixes for the Eagle ascent stage in the flight journal? What was the duration of the observation arc? The number, type and quality of measurements used? As someone who has spent much of his professional career in spacecraft control centres, I learnt (the hard way) that these factors matter at lot. The orbit of the CM/SM was continuously being determined. Precise knowledge of the orbital data of this mission element was mission-critical. It was vital both for the deployment of the LM and for the timing and execution of the TEI. Therefore, the spacecraft would have been designed to permit precise OD and the frequency and quality of the measurements would be commensurate with the precision required.

      Conversely, the mission profile of the LM was such that critical parts of the navigation during its short flight phases were relative, not absolute. This is a rather different situation from the CM/SM. Also, after the final jettison, any orbital data obtained for the LM, while interesting, would no longer have been considered as mission-critical and therefore would have been accorded a lesser priority. I assume that this would have been reflected in the achieved accuracy of the OD.

      You suggest that the Apollo 16 Particles and Fields Subsatellites should be used as a comparison. That certainly would be an interesting numerical experiment, but I am not sure I understand what you expect to gain from it. The accuracy of a numerical simulation depends on two things: 1.) The accuracy of the physical model of the environment that affects the dynamics of the system studied and 2.) the accuracy of the parameters defining the initial state of the system. James Meador used a professionally developed, widely available and tested tool for his calculations. Verification of the model was achieved by having someone else perform the same runs on a completely independent piece of software that itself has been multiply verified and also used operationally. I’d say that this adequately addresses point 1.) above. This leaves point 2.), the question of the parameters of the starting point. We’ve already discussed those. At any rate, I don’t see how simulating PFS-2 would shed any further light on the matter of the post-separation state of the Eagle.

      I am not saying that a simulation of PFS-2 would not be of interest and should not be done; I just don’t see how that would significantly contribute to the results in James Meador’s paper. Basically, anyone could do the simulation of the PFS-2 orbit, so if you do, please publish your results; I would read them with interest.

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