In the beginning of the 1980s, the first observational evidence was reported that galaxies do not spin as expected according to then current theories. A galaxy is a collection of stars orbiting the *bulge* (the center of the galaxy). Since the orbit of stars is driven solely by the gravitational force, it was expected that stars at the edge would have an orbital period much larger than those near the bulge. For example, the Earth which is 150 million kilometers away from the Sun completes an orbit in one year, while it takes Saturn 30 years to do the same at a distance of 1.4 billion kilometers.

A similar behavior was expected from galaxies, even if the distribution of stars is more cloud-like. However, it became more and more apparent that stars at the edge of a galaxy move faster than expected.

Astronomers call this phenomenon the "flattening of galaxies' rotation curve". Basically, if one draws a curve describing the velocity of stars as a function of the distance from the center, he or she should obtain curve A in fig. 2 (dashed line). Data from telescopes give curve B (plain line). This curve, instead of decreasing asymptotically to zero, remains flat at large distance from the bulge. For comparison purpose, the same curve for the Solar system -- (properly scaled) -- is provided (curve C in fig. 2).

Reluctant to change Newton's law as well as Einstein's theory of relativity for galaxies only, scientists simply assumed that the rotation curve was flat because of the presence of a large amount of matter outside the galaxies. The new theory was that galaxies are embedded in a spherical halo of invisible, "dark" matter (see fig. 3). Since then, the search for dark matter has kept many astronomers busy, with mitigated success.

As time has passed, the hypothesis of dark matter halos encountered many problems, casting doubt on the validity of this model (although it is still the most widely accepted model). Alternate approaches have therefore been considered, one of them called the Modified Newtonian Dynamics (MOND) theory.

In 1989, Mordehai Milgrom, a physicist at the Weizmann Institute in Israel, proposed a modification of Newton's second law of motion. Basically, this law states that an object of mass *m*, subject to a force *F* undergoes an acceleration *a* satisfying the simple equation *F=ma*. This law is well known to students, and has been verified in a variety of situations. However, it has never been verified in the case where the acceleration *a* is extremely small. And that is exactly what's happening at the scale of galaxies, where the distances between stars are so large that the gravitational force is extremely small.

The modification proposed by Milgrom is the following: instead of *F=ma*, the equation should be *F=mµ(a/a _{0})a*, where

Here is the simple set of equations for the Modified Newtonian Dynamics:

*F=mµ(a/a*_{0})a

*µ(x)=1*if*x>>1*

*µ(x)=x*if*x<<1*

In the every day world, *a* is greater than *a _{0}* for all physical effects, therefore

Since MOND was inspired by the desire to solve the flat rotation curve problem, it is not a surprise that using the MOND theory with observations reconciled this problem. This can be shown by a calculation of the new rotation curve.

Far away from the center of a galaxy, the gravitational force a star undergoes is, with good approximation:

*F=(GMm)/r*^{2}

*F=(GMm)/r*^{2}=mµ(a/a_{0})a

We assume that, at this large distance *r*, *a* is smaller than *a _{0}* and thus

*(GM)/r*^{2}=a^{2}/a_{0}

Since the equation that relates the velocity to the acceleration for a circular orbit is *a=V ^{2}/r* one has

*a=V*^{2}/r=(GMa_{0})^{½}/r

Consequently, the velocity of stars on a circular orbit far from the center is a constant, and doesn't depend on the distance *r*: the rotation curve is flat.

At the same time, there is a clear relationship between the velocity and the constant *a _{0}*. The equation

Retrospectively, the impact of assumed value of *a>>a _{0}* for physical effects on Earth remains valid. Had

In the life of physical theories in general, there are some steps a newborn theory usually follows:

- It is designed to explain an observation or the result of an experiment for which other theories fails. Therefore the new theory must describe how is explains the observation, with new equations that fit the data, and at the same time remain consistent (that is, assumptions made in the design of the theory must not be proved wrong after the new equations are obtained). As we have seen, MOND passed this test.
- After it explains one particular observation, it must not be in clear contraction with all other observations and experimental results.
- If competing with other new theories, it should describe how an experiment can distinguish it from these other theories. Unfortunately, experiments are not easily done in astrophysics, so it should tell what new data from telescopes should look like in a particular case. Again, in the purpose of distinguish it from other, competing theories.
- Ultimately, a theory becomes accepted by the scientific community only if it makes predictions that are verified later on.

According to the Modified Newtonian Dynamics theory, every physical process that involves small accelerations will have an outcome different than predicted by the simple law *F=ma*. Therefore, one needs to look for all such processes and verify that MOND remains compatible with observations, i.e. within the limit of the uncertainties on the data. There is, however, a complication overlooked until now but that strongly affect our discussion on the compatibility between MOND and the observed world.

Here is the problem: in a system considered as isolated, for example a single satellite orbiting a planet, the effect of MOND results in an increased velocity beyond a given range (actually, below a given acceleration, but for circular orbits it's the same thing), that depends on the mass of both the planet and the satellite. However, if the same system is actually orbiting a star, the planet and the satellite will be accelerated in the star's gravitational field. For the satellite, the sum of the two fields could yield an acceleration greater than *a _{0}*, and the orbit would not be the same than if the system was isolated.

For this reason, the typical acceleration of any physical process is not the only parameter one must consider. Also critical is the process' environment, that is all external forces that are usually neglected. In his paper, Milgrom arranged the typical acceleration of various physical process in a two-dimensional diagram (see fig. 4). One parameter is the acceleration of the process itself, the other parameter is the acceleration induced by the environment.

How does this affect our discussion about the adequation of MOND to the real world? Very simply: all experiments done on Earth or its neighborhood are subject to the Sun's gravitational field. This field is so strong that all objects in the Solar system undergo an acceleration greater than *a _{0}*. That's why MOND effects have escaped detection.

Therefore, only the dynamics of galaxies and larger systems need be examined to check that MOND is compatible with observation. Since 1989 and the outcome of Milgrom's theory, the most accurate data has come from observation of distant galaxies and neighbors of the Milky Way. Within the uncertainties of the data, MOND has remained valid. The Milky way itself is spawned with clouds of gas and interstellar dust, and until now it has not been possible to draw a rotation curve for our Galaxy. Finally, the uncertainties on the velocity of galaxies within clusters and larger systems has been too large to conclude in favor of or against MOND.

Is it possible to design an experiment that would confirm MOND predictions, or rule it out? Unfortunately, conditions for conducting this experiment can be found only outside the Solar system. However, the Pioneer and Voyager probes are currently traveling beyond Pluto and perhaps they have already reached this zone. To check that, let's calculate the radius of the gravitational sphere of influence of the Sun, inside which a probe undergoes an acceleration greater than *a _{0}*.

We have seen above that the equation relating the acceleration *a* to the distance *r* from the Sun is

*(GM)/r*^{2}=µ(a/a_{0})a

In search for observations that would validate his theory, Milgrom noticed that a special class of objects, the low surface brightness galaxies (LSB) are of particular interest: the radius of a LSB is large compared to its mass, and thus almost all stars are within the flat part of the rotation curve. Also, other theories predict that the velocity at the edge depends on the average surface brightness in addition to the LSB mass. Finally, no data on the rotation curve of these galaxies was available at the time. Milgrom thus could make the prediction that LSBs would have a rotation curve essentially flat, and with a relation between the flat velocity and the mass of the LSB identical to that of brighter galaxies.

Since then, many such LSBs have been observed, and while some astronomers have claimed their data invalidated MOND, others said it confirmed the prediction. At the time of this writing, the debate is still hot, and scientists are waiting for more accurate observations.

One reason why some astronomers find MOND difficult to accept is that it's an effective theory, not a physical theory. As an effective theory, it describe the dynamics of accelerated object with an equation, without any physical justification. This approach is completely different than Einstein's, who assumed that some fundamental physical principles were true (continuity, smoothness and isotropy of space-time, conservation of energy, principle of equivalence) and derived new equations from these principles, including the famous *E=mc ^{2}* and the less famous but extremely powerful

Attempts in this direction have essentially been modifications of Einstein's theory of gravitation. When one looks at the equation *F=mµ(a/a _{0})a*, the value of

In the eyes of astronomers, MOND is just an alternative to the more widely accepted theory of dark matter. As new data is coming from telescopes, MOND as well as dark matter is sometimes invalidated and sometimes supported, and no clear-cut observation has yet helped decide which theory is the one. Toward this goal, supporters of MOND have concentrated their effort on specific areas:

- To look for new predictions of MOND that could be tested. For example, the dynamics of satellites of our Galaxy could be distorted by MOND effects, in a way difficult to explain with a dark matter halo.
- To obtain the relativistic extension of MOND, that would incidentally help understand how light is bent by galaxies' gravitational field, one feature MOND cannot explain.
- Alternatively, to establish MOND as a theory of inertia and find its fundamental principles, although progress in this direction has been pretty small.

**External links:**

- Preprints related to MOND
- Mordehai Milgrom:
*Does Dark Matter Really Exist?*, Scientific American, August 2002