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This video: Is Apollo 11's Lunar Module Still In Orbit Around The Moon 52 Years Later? claims, based on orbital simulations, that there is a chanche that the lunar module of Apollo 11 might be still orbiting the Moon.

Is this claim reasonable from an astronomical point of view? Ie are there telescopes on Earth which would've spotted this probably shiny object on Lunar orbit? Afterall, there are space rocks we know about smaller than this spacecraft.

A related but different question in Space SE: Could the ascent stage of Apollo 11's Eagle still orbit the Moon?

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Background/Video

Starting around minute 6 of the video frozen orbits are discussed, and some example GMAT simulations are run to explain how lunar orbits can be simulated including up to order 160 in the Moon's gravitational spherical harmonics, which have been very accurately mapped by the GRAIL mission.

The video then links to the study itself:

The Apollo 11 “Eagle” Lunar Module ascent stage was abandoned in lunar orbit after the historic landing in 1969. Its fate is unknown. Numerical analysis described here provides evidence that this object might have remained in lunar orbit to the present day. The simulations show a periodic variation in eccentricity of the orbit, correlated to the selenographic longitude of the apsidal line. The rate of apsidal precession is correlated to eccentricity. These two factors appear to interact to stabilize the orbit over the long term.

"Would we have spotted the ascent stage of Apollo 11's Eagle if it was still in orbit around the Moon?"

It's unclear; maybe.

Here's how it might have been spotted.

See also Chasing Snoopy's Tail; My search for the descent stage of the Apollo 10 Lunar Module.

The Space SE question Why was the 100m Green Bank dish needed together with DSN's 70m Goldstone dish to detect Chandrayaan-1 in lunar orbit? and its answers explain that delay-doppler radar from the very large diameter (= narrow beam) radar transmitter at Arecibo and the very large diameter (= narrow beam + sensitive) receiver at Green Bank were able to pick up the tiny reflection of Chandrayaan-1 in it's orbit 200 km away from the surface of the Moon.

This was done by separating the spacecraft's reflection from that of the 1012 times larger Moon both spatially (using these large $\lambda/D$ dishes) and in frequency space since the doppler shift of a 200 km circular lunar orbit with a velocity of about 1600 m/s at 2380 MHz will be about 25 kHz.

note: I don't know yet if they did in fact use the highest frequency available at Arecibo; there are lower ERP's at 430 and 47 MHz as well, Green bank can receive down to 290 MHz.

They found Chandrayaan-1 this way, and it's unclear if that data would have picked up the much larger Eagle as well or not. 1.22 $\lambda/D$ for Aredibo's radar is only 1.7 arc minutes, so unless they really looked for the Eagle they could have easily missed it.

From the linked question in Space SE:

"Radar imagery acquired of the Chandrayaan-1 spacecraft as it flew over the moon's south pole on July 3, 2016. The imagery was acquired using NASA's 70-meter (230-foot) antenna at the Goldstone Deep Space Communications Complex in California. This is one of four detections of Chandrayaan-1 from that day."

above: "Radar imagery acquired of the Chandrayaan-1 spacecraft as it flew over the moon's south pole on July 3, 2016. The imagery was acquired using NASA's 70-meter (230-foot) antenna at the Goldstone Deep Space Communications Complex in California. This is one of four detections of Chandrayaan-1 from that day." Credit: NASA/JPL-Caltech. From here

Cropped section of the previous figure to draw attention to "The white box in the upper-right corner of the animation depicts the strength of echo."

above: Cropped section of the previous figure to draw attention to "The white box in the upper-right corner of the animation depicts the strength of echo." Credit: NASA/JPL-Caltech. From here

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There are a couple of inconsistencies with theory presented in the video and the paper it is based on. Apparently, the author(s) are suggesting that Eagle did not crash as it was/is in a 'frozen' orbit. However, frozen orbits are thought to occur only for orbits with higher inclinations (in fact only for 4 specific inclinations: 27,50,76 and 86 deg) (see https://science.nasa.gov/science-news/science-at-nasa/2006/06nov_loworbit/ ). But the Apollo 11 module had a nearly equatorial orbit, and a small satellite called PFS-2 released by a later Apollo mission, having a small inclination as well, crashed after only about 1 month. So the argument doesn't quite add up, with both objects having a very similar orbit. The author(s) should run their calculations as well for PFS-2, which we know crashed, and see whether they can replicate this in their calculations. If not, it would show that their calculations are erroneous and their conclusion therefore unfounded.

The brightness of the Eagle ascent module should make it in principle detectable in parts of the orbit where it is not in front of the moon. From its distance and size one can estimate that it would be about one million times less bright than the Hubble Space Telescope (which appears as about a star of magnitude 1), so it should be like a star of magnitude 16. With a telescope of 20 inch diameter or more, this should theoretically be spottable if a sufficient magnification is used (the surface brightness of the background will be diminished with the inverse of the magnification squared, whilst the object, being essentially a point source, will stay at the same brightness). I have heard from some amateur astronomers that they can see magnitude 3 stars during broad daylight with their telescope if they know where to look. Now the moon is 14 magnitudes less bright than the sun, so one should be able to see objects with magnitude 17 during broad (full)moonlight, maybe even fainter if increased magnification is used (provided of course the telescope is powerful enough for this in the first place, i.e. if it has a diameter of 20 inches or more).

And with radar, other satellites orbiting the moon have indeed been detected (see https://moon.nasa.gov/news/12/new-radar-technique-finds-lost-lunar-spacecraft/ ) but not the Eagle module.

(note: I have substantially edited my answer, so some of the comments below may now not apply anymore)

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    $\begingroup$ @zabop The landing modules were crashed deliberately as a seismic experiment, but in case of Apollo 11 the seismometer failed after 3 weeks, which is why it may have missed the crash (see references in my edited answer). The estimate regarding the optical visibility is my own calculation. In any case, satellites of this size (or even smaller) orbiting the moon can and have been detected by radar (see edited answer) $\endgroup$
    – Thomas
    Jul 17, 2021 at 22:27
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    $\begingroup$ @JohnHoltz It may be borderline for viewing trough the atmosphere, but 60 miles above the moon would be about an arcminute, so it is fair bit from the moon's disc if you have a sufficient magnification, and with maybe some post-processing you could make it better visible (I am just making an educated guess here; I am not in a position to confirm this from practical experience (my last telescope observation sessions date many years back)). It definitely should not be a problem from outside the earth;s atmosphere. And with radar you should be able to detect it as well (see my edited answer) $\endgroup$
    – Thomas
    Jul 17, 2021 at 22:51
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    $\begingroup$ According to Wikipedia, at least, the ascent stages of the LMs for Apollo 12 and 14–17 were crashed into the lunar surface, whereas the ascent stage of Apollo 11's LM was left in lunar orbit, so this means that even if we restrict ourselves to those Apollo missions that actually landed on the moon, it is still not correct that all LMs were crashed into the moon. Eagle's ascent stage may have crashed into the moon as well during the last 50 years, but it was not deliberate, and there are in fact papers showing that it may not have crashed at all. $\endgroup$
    – Jörg W Mittag
    Jul 18, 2021 at 9:20
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    $\begingroup$ @Thomas, you should have watched the OP's video before posting this. It is well-reasoned and informative, especially about lunar orbit degradation. $\endgroup$
    – TonyK
    Jul 18, 2021 at 9:35
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    $\begingroup$ So, what you are saying is that when you say in your answer that "all the lunar ascent modules were crashed deliberately back onto the moon", what you really mean is "all the lunar ascent modules were crashed deliberately back onto the moon, except the ones from Apollo 4, 5, 6, 8, 10, 11, and 13"? In that case, it might be a good idea to clarify your answer, since the question is explicitly about Apollo 11's LM, which was left in lunar orbit and not "crashed deliberately back onto the moon", and your answer is explicitly only about the LMs that were crashed deliberately into the moon. $\endgroup$
    – Jörg W Mittag
    Jul 18, 2021 at 9:57
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I first wanted to edit this into my other answer here, but since this, even though being highly relevant for the issue, does not directly address the OP's question(s), I decided to add this as a separate answer :

The orbital elements used in the cited work for simulating the orbit of the Eagle ascent stage after being jettisoned from Apollo 11 are actually apparently those of the command module as calculated from the orbit data in the Apollo 11 mission report (Table 7-II) (the lunar module was not really of any interest anymore at this point, so it probably was not systematically tracked after that). This resulted in the following orbital elements they tried for their simulations for the lunar module

enter image description here

The eccentricity of the orbit for these three cases is practically the same: 0.0037, 0.0038, 0.0035 for the nominal, maximum and minimum case respectively.

However, in the Apollo 11 Flight Journal they mention these figures explicitly for the lunar module shortly after 'Ignition of Trans-Earth Injection burn' (about 5 and 7 hours after jettison of the LM)

just before 135:47:24 mission time: Perilune 100.7 km, Apolune 118.7 km

just before 137:30:12 mission time: Perilune 100.7 km Apolune 119.3 km

(they are saying '-cynthion' instead of '-lune' there)

This results in eccentricities 0.0049 and 0.0050 respectively, so substantially higher than assumed for the simulations based on the command module orbit at the time of separation.

So the author may want to revise the orbital parameters in this sense, and also apply the simulation to the PFS-2 satellite in order to remove any ambiguities here and make his results more conclusive.

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