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@SF.'s question What are the parameters of the new Iodine electrical rocket engine developed by RSC Energia? links to the short RT article 'Ten times cheaper': Russian space company testing iodine rocket engine which contains a sentence which was likely altered during translation as well as in paraphrasing for a non-space general news publication:

Furthermore, xenon-run engines are incapable of long-distance flights, like going to the Moon.

But my question is only about the orbital mechanics of going to lunar orbit from LEO using a low-thrust form of propulsion such as ions.

From LEO you can take your time slowly spiraling outwards in Earth's gravity well as shown here and is done by some of the newer "all electric propulsion" communications satellites that use ion propulsion to get to GEO.

And if you enter a high lunar orbit, you may be able to lower it using ion propulsion as well, provided it can handle the lumpy gravity and perturbations from the Earth and Sun.

But my question is about the transfer between Earth-bound to Moon-bound orbits. I'm wondering if you can make the transfer with very low propulsion at all times, or if there is some point where a high impulse would be needed in this four-body problem to change from one to the other without risk of getting lost or thrown into a heliocentric orbit.

For "how low is low thrust?" you could choose an existing spacecraft like an all-electric-to-GEO communications satellite, or a deep-space explorer like DAWN.

Sound mathematical arguments would be great, or a reference to a published study with conclusive results would be good as well. Even if possible, I'd still like to understand if it was tricky to make it work, or if it is actually not as hard as I might imagine.

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    $\begingroup$ "without risk of getting lost or thrown into a heliocentric orbit." At the point where your orbit transitions from Earth to Moon, you are still about 100m/s slower than escape velocity from Earth. You could use a moon gravity sling to eject your craft into heliocentric orbit, but it would require good planning. You could also use the same gravity from a very slightly different angle to facilitate a capture to the Moon. This as a comment not an answer, because the math is ... owch. $\endgroup$
    – CuteKItty_pleaseStopBArking
    Jul 19, 2021 at 8:57
  • $\begingroup$ @PcMan have a look at MarkAdler's insightful answer to Was the Apollo spacecraft always gravitationally bound to the Earth-Moon system? $\endgroup$
    – uhoh
    Jul 19, 2021 at 9:03

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The piece that you are asking about not only can be done, but it has been done. SMART-1 was launched in to GTO in 2003 and entered orbit around the Moon in 2004, using only an Ion engine to do so. It gradually reduced its orbit around the Moon, eventually colliding in to it. The trick is to use the Lagrange points to give one more time to do a proper orbit.

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    $\begingroup$ Oh, use a Lagrange point. My, that's a "smart-one"! $\endgroup$
    – uhoh
    Jun 22, 2018 at 16:27
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    $\begingroup$ I'm not sure I understand why using a Lagrange point would help. Couldn't one simply raise the orbit slowly until it passes the sphere of influence of the Earth (which is a virtual boundary not a real one of course), and then slow down such that the vehicle is in a highly elliptical orbit around the Moon, and then circularize it? Going to a Lagrange point itself intuitively seems more complicated than needed. $\endgroup$
    – ChrisR
    Jun 23, 2018 at 2:44
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    $\begingroup$ I'm not sure if it does to me either, but I do know they used it for the SMART-1 mission. I think it said it was the L-1 point, which is more or less what you are indicating. $\endgroup$
    – PearsonArtPhoto ♦
    Jun 23, 2018 at 3:08
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    $\begingroup$ @ChrisR I've used your comment here, thank you for bringing it up! It doesn't really answer your question though. Why not ask it as a new question? Something like "Could SMART-1 have been smarter?" or maybe "Challenges SMART-1 would have encountered getting to lunar orbit but avoiding L1, or all Lagrange points?" $\endgroup$
    – uhoh
    Jun 24, 2018 at 10:33
  • $\begingroup$ @uhoh , the Flight section of the WP article is especially interesting. It does in fact make sense to go through the Earth-Moon L1, cf. this Lagrange point illustration. The goal was to reach the Moon, and passing the SOI, hence, let's target specifically the closest border region to Earth, which is L1. And then "fall" into the Moon system. Good job on the mission designers! $\endgroup$
    – ChrisR
    Jun 24, 2018 at 18:05
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@ChrisR's comment

I'm not sure I understand why using a Lagrange point would help. Couldn't one simply raise the orbit slowly until it passes the sphere of influence of the Earth (which is a virtual boundary not a real one of course), and then slow down such that the vehicle is in a highly elliptical orbit around the Moon, and then circularize it? Going to a Lagrange point itself intuitively seems more complicated than needed.

really cuts to the chase of the matter!

The concept of "sphere of influence" is one of the lies we tell to children which is an expression meaning that it's not true but it makes simple explanations easer. Of course, it's also a lie upon which KSP is based as well. The reason I've mentioned "four body problem" in the question is that gravity from everything is always pulling on you. Sphere of influence is a lie, or an approximation, depending on context.

Of course if you are making videos like Scott Manley then it's an approximation. If you want to understand orbital mechanics rather than explain it, it's best treated as a lie and ignored, or as Monty Python would say:

Of the sparkling wines, the most famous is 'Perth Pink'. This is a bottle with a message in, and the message is Beware! This is not a wine for drinking -- this is a wine for laying down and avoiding.

Before ion propulsion, it didn't matter quite as much because the impulse from chemical rockets are so fast that you can fairly quickly make changes to your trajectory.

The motivation behind my current question is that with ion propulsion, changes are extremely slow, and so in a multi-body scenario you are in a bit more trouble.

The solar arrays made 1,190 W available for powering the thruster, giving a nominal thrust of 68 mN, hence an acceleration of 0.2 mm/s² or 0.7 m/s per hour (i.e., just under 0.00002 g of acceleration).

The Lagrange point solves this problem by using the three body problem effects constructively instead of fighting them.

Looking at the Wikipedia page linked in @PearsonArtPhoto's answer you can see that it took over a year of thrust for SMART-1 to slowly spiral farther and farther out in Earth orbit to get to the region of the Earth-Moon L1 point. It took some very careful vectoring of the tiny thrust to "thread the needle" and pass it towards the Moon with a low-enough relative velocity to remain in orbit. SMART-1 would probably have had a really serious navigation challenge had it tried to do this transfer any other way.

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