[Keysh]

It's kind of amusing that you present "keplerian rotation curves" as a "prediction" of MOND, given that the whole point of MOND is a kludge to produce non-Keplerian rotation curves. That is, by definition MOND cannot produce Keplerian rotation curves. This is why the (small) number of dwarf galaxies (not "elliptical and lenticular galaxies"!) which apparently lack dark matter -- and which do not follow the Tully-Fisher relation-- are serious problems for MOND.

[throwawaymaths]

> by definition MOND cannot produce Keplerian rotation curves.

thats a common misunderstanding of MOND and means you did not stop to understand the definition. look at the formula carefully; in high gravitational regimes it looks newtonian (obviously, e.g. solar system). MOND, by definition predicts keplerian curves when the galaxies are rotating quickly (high gravitation). You will find that ellpitical and lenticular galaxies are almost always rotating fast. it also can through EFE but i dont fully understand the math enough to explain that.

[Keysh]

"Keplerian" is a term of art based on the rotation curve in Newtonian gravity when all the mass of a system is concentrated at its center. For the Solar System, the rotation curve outside the radius of the Sun is pretty much pure Keplerian: the velocity decreases proportional to the square root of the radius.

For galaxies, which are extended objects, the rotation curve is not Keplerian when you are well inside the galaxy: it first rises to larger radii, then levels off. But since the baryonic matter (stars, gas, dust) in galaxy is rather centrally concentrated, the rotation curve should start looking more and more Keplerian as you get further and further into the outskirts and outside the galaxy.

But that is not what we see. Instead, we see the rotation curves staying roughly constant with radius ("flat"); we call this "non-Keplerian". This is true for almost all galaxies, including ellipticals and lenticulars (this is a recent study of three lenticular galaxies: https://www.aanda.org/articles/aa/full_html/2020/09/aa38184-.... Note the rotation curves in the bottom panels of Figure 3: they do not at any point start decaying, let alone decaying as fast as R^(-1/2).)

Figure 5 of that paper (https://www.aanda.org/articles/aa/full_html/2020/09/aa38184-...) shows the observed rotation curves; it also shows the predicted curves if just the stars and standard Newtonian gravity were operating, with the dotted red lines. Note how these lines first rise to a local peak at small radii, and then decline to larger radii: this is a (quasi-)Keplerian decline. It fails to match the actual rotation curve at large radii.

The conventional response is to postulate some additional form of matter, distributed in a much more extended fashion than the baryonic matter (this produces the dashed black lines in Figure 5 of that paper). The MOND response is to modify gravity (or: to modify the acceleration due to gravity) such that it doesn't show anything like a Keplerian falloff at large radii, even at radii where the gravitating matter (assumed to be just the visible baryonic matter) is well inside.

In the case of the Solar System, the Keplerian decline starts right outside the Sun, where the acceleration is strong enough to be above the MOND threshold. But if you went far enough out and could measure the circular orbital speed, MOND would start to deviate from Keplerian. In the case of galaxies, the outer radii where the rotation curves appear "flat" are where the acceleration due to gravity is low enough for MOND to matter, and so the predicted MOND curves will not be Keplerian.

(I should perhaps point out that I'm a professional astronomer whose been studying galaxies, including lenticulars and ellipticals, for almost 30 years, so attempts to mansplain my field to me won't really impress me.)