How Big Is The MOND Acceleration Constant?

In Modified Newtonian Dynamics (MOND), a phenomenological tweak to Newtonian gravity proposed in 1983 by Mordehai Milgrom to replicate the galactic rotation curves attributed with dark matter without using dark matter, there is one experimentally determined physical constant:

 a₀ = 1.2 x 10^-10 ms^-2 

In MOND, this constant is the characteristic acceleration due to a gravitational field, above which ordinary Newtonian gravity applies and below which an enhanced gravitational field strength leading to flat rotation curves in spiral galaxies is present.

What does this mean in reference to something understandable?

The gravitational field of the Sun alone falls to that strength at a distance of about 1,052 billion km (about 7000 astronomical units (AU). An AU, which is 149.6 million km, is the average distance of the Earth from the Sun.

This is about 1/9th of the light year from the Sun, which is about 175 times the average distance of Pluto from the Sun. Pluto's average distance from the Sun is about 6 billion km). 

This is about 58 times more distant from the Sun that the heliosphere, which is a functional definition of where the solar system ends and deep interstellar space begins, that is about 18 billion km (120 astronomical units) from the Sun. 

As of February 2018, Voyager 1, the most distant man made object from Earth, was about 21 billion km from Earth, and Voyager 2, the second most distant man made object from Earth is about 17 billion km from Earth. Both were launched 43 years ago in 1977. These probes (which will run about of power around the year 2025), will reach this distance from the Sun about 2000 years from now in around 4000 CE.

This doesn't necessarily mean that MOND effects are actually predicted to occur at that distance from the Sun, because the theory has "external field effects" which prevent this effect from occurring if gravitational effects from other object bring the overall strength of the local gravitational field from all sources combined are above the acceleration constant threshold (in which case the behavior is Newtonian), or the field from external sources is below that constant but above the strength of the gravitational field from the source you are trying to observe (in which case there is only a partial MOND effect).

Also, this effect is very subtle at first and it is only discernible when Newtonian gravity, which is a function of one divided by the distance from the source squared, is significantly different from the MOND value which is a function of one divided distance from source when beyond the acceleration constant threshold.

For example, in MOND, at twice the critical distance from a mass the size of the Sun with no external field effect (i.e. at a distance of 2/9th of a light year which is about 14000 AU), the gravitational force from the Sun is twice its predicted Newtonian gravity value.

Since acceleration due to gravity at this distances is very small, and the observable is distance and not acceleration, which is the second derivative of distance (i.e. the rate at which velocity changes which is in turn the rate at which distance changes), even a fairly substantial change in gravitational acceleration is not easy to observe in objects that are, by definition, very far away from any realistic terrestrial observer.

The external field effect in MOND violates the strong equivalence principle of general relativity that "The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime," but not the weak equivalence principle that "All test particles at the alike spacetime point, in a given gravitational field, will undergo the same acceleration, independent of their properties, including their rest mass."

To be clear, toy model MOND is not an accurate description of reality. It is a non-relativistic theory, it underestimates the magnitude of dark matter phenomena at the scale of galactic clusters, and it doesn't predict the dynamics of stars outside the galactic plane in a spiral galaxy as well as inferred dark matter halos fitted to a spiral galaxies rotation curve speed.

But MOND does accurately describe rotation curves of objects up to the scale of the largest galaxies in a way that can make concrete predictions (which dark matter particle theories don't do very well at all), and can inform your intuition of how dark matter phenomena behave and in which circumstances they arise, with just one simple physical constant and one simple equation.