Modified Newtonian dynamics
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The modified Newtonian dynamics (MOND) is a physical theory which attempts to explain the galaxy rotation problem by changing Newton's law of motion. (The more mainstream approach to explain this problem postulates the existence of dark matter.) MOND was proposed in 1983 by Mordehai Milgrom.
Contents 
Overview: Galaxies dynamics
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 cloudlike. 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 should obtain curve A in figure 1 (dashed line). Data from telescopes give curve B (plain line). This curve, instead of decreasing asymptotically to zero, remains flat at large distances from the bulge. For comparison purpose, the same curve for the Solar system  (properly scaled)  is provided (curve C in figure 1).
Newtonianfig.png
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 figure 2). Since then, the search for dark matter has kept many astronomers busy, with mitigated success.
Newtonianfig3.png
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.
The MOND Theory
In 1983, Mordehai Milgrom, a physicist at the Weizmann Institute in Israel, published two papers on the Astrophysical Journal to propose 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 change
The modification proposed by Milgrom is the following: instead of F=ma, the equation should be F=mµ(a/a_{0})a, where µ(x) is a function that for a given variable x gives 1 if x is much larger than 1 ( x≫1 ) and gives x if x is much smaller than 1 ( x≪1 ). The term a_{0} is some new constant, in the same sense that c (the speed of light) is a constant, except that a_{0} is an acceleration whereas c is a speed.
Here is the simple set of equations for the Modified Newtonian Dynamics:
 <math> \vec{F} = m \cdot \mu\left( { a \over a_0 } \right) \ \vec{a} <math>
 <math> \mu (x) = 1 \quad if \ \ x >> 1 <math>
 <math> \mu (x) = x \quad if \ \ x << 1 <math>
The exact form of µ is unspecified, only its behavior when the argument x is small or large. As Milgrom proved in his original paper, the form of µ doesn't change the consequences of the theory.
In the every day world, a is greater than a_{0} for all physical effects, therefore µ(a/a_{0})=1 and F=ma as usual. Consequently, the change in Newton's second law is negligible and Newton couldn't have seen it.
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.
MOND predicted rotation curve
Far away from the center of a galaxy, the gravitational force a star undergoes is, with good approximation:
 <math> F = \frac{GMm}{r^2} <math>
with G the gravitation constant, M the mass of the galaxy, m the mass of the star and r the distance between the center and the star. Using the new law of dynamics gives:
 <math> F = \frac{GMm}{r^2} = m \mu{ \left( \frac{a}{a_0}\right)} a <math>
Eliminating m gives (GM)/r^{2}=µ(a/a_{0})a
We assume that, at this large distance r, a is smaller than a_{0} and thus µ(a/a_{0})=a/a_{0}, which gives:
 <math> \frac{GM}{r^2} = \frac{a^2}{a_0} <math>
Therefore a=((GMa_{0})/r^{2})^{½}=(GMa_{0})^{½}/r
 <math> a = \left( \frac{ G M a_0 }{ r^2 } \right) ^{1/2} = \frac{\sqrt{ G M a_0 }}{r} <math>
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
Eliminating r gives V^{2}=(GMa_{0})^{½} and therefore V=(GMa_{0})^{ĵ}
 <math> v = \left( { G M a_0 } \right)^{\frac{1}{4}} <math>
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.
The proportion between the "flat" rotation velocity to the observed mass derived here is matching the observed relation between "flat" velocity to luminosity known as the TullyFisher relation.
At the same time, there is a clear relationship between the velocity and the constant a_{0}. The equation V=(GMa_{0})^{ĵ} allows one to calculate a_{0} from the observed V and M. Milgrom found a_{0}=1.2 10^{10} ms^{2}. Milgrom has noted that this value is also "... the acceleration you get by dividing the speed of light by the lifetime of the universe. If you start from zero velocity, with this acceleration you will reach the speed of light roughly in the lifetime of the universe."
Retrospectively, the impact of assumed value of a>>a_{0} for physical effects on Earth remains valid. Had a_{0} been larger, its consequences would have been visible on Earth and, since it is not the case, the new theory would have been inconsistent.
Consistence with the observations
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 fail. Therefore the new theory must describe how it 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 contradiction 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, to serve the purpose of distinguishing it from other, competing theories.
 Ultimately, a theory becomes accepted by the scientific community only if it makes predictions that are verified later on.
MOND satisfies criterion 1, but what about criteria 2 to 4?
According to the Modified Newtonian Dynamics theory, every physical process that involves small accelerations will have an outcome different from 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 affects our discussion of 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 is 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 as that in an isolated system.
Newtonianfig4.png
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 twodimensional diagram (see figure 3). 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 of MOND's application 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 to be examined to check that MOND is compatible with observation. Since Milgrom's theory first appeared in 1983, the most accurate data has come from observations of distant galaxies and neighbors of the Milky Way. Within the uncertainties of the data, MOND has remained valid. The Milky Way itself is scattered 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
 <math>\frac{GM}{r^2}=\mu\left(\frac{a}{a_0}\right)a<math>
So, for a=a_{0}, assuming µ(a/a_{0})=µ(1)=1, with G=6.67 10^{8} in cgs units and M (the mass of the Sun)=2 10^{33} g, we get r=1.05 10^{17} cm. This is roughly 0.034 parsecs or 0.1 light years, over 100 times the distance between Voyager 1, the most remote probe, and the Sun. It is therefore doubtful that an experiment could be accurate enough to test the departure from Newton's second law. Perhaps µ(1) is less than 1, but it is very likely greater than 0.2. Consequently, experiments on MOND will have to wait for the next age of space exploration.
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 have been able to confirm their predictions. At the time of this writing, the debate is still hot, and scientists are waiting for more accurate observations.
The mathematics of MOND
In nonrelativistic Modified Newtonian Dynamics, Poisson's equation,
 <math>\nabla^2 \Phi_N = 4 \pi G \rho<math>
(where <math>\Phi_N<math> is the gravitational potential and <math>\rho<math> is the density distribution) is modified as
 <math>\nabla\cdot\left[ \mu \left( \frac{\left\ \nabla\Phi \right\}{a_0} \right) \nabla\Phi\right] = 4\pi G \rho<math>
where <math>\Phi<math> is the MOND potential. The equation is to be solved with boundary condition <math>\left\ \nabla\Phi \right\ \rightarrow 0<math> for <math>\left\ \mathbf{r} \right\ \rightarrow \infty<math>. The exact form of <math>\mu(\xi)<math> is not constrained by observations, but must have the behaviour <math>\mu(\xi) \sim 1<math> for <math>\xi >> 1<math> (Newtonian regime), <math>\mu(\xi) \sim \xi<math> for <math>\xi << 1<math> (DeepMOND regime). Under deepMOND regime, modified Poisson equation may be rewritten as
 <math>
\nabla \cdot \left[ \frac{\left\ \nabla\Phi \right\}{a_0} \nabla\Phi  \nabla\Phi_N \right] = 0 <math>
and that simplifies to
<math> \frac{\left\ \nabla\Phi \right\}{a_0} \nabla\Phi  \nabla\Phi_N = \nabla \times \mathbf{h} <math>
The vector field <math>\mathbf{h}<math> is unknown, but is null whenever the density distribution is spherical, cylindrical or planar. In that case, MOND acceleration field is given by the simple formula
 <math>
\mathbf{g}_M = \mathbf{g}_N \sqrt{\frac{a_0}{\left\ \mathbf{g}_N \right \}} <math>
where <math>\mathbf{g}_N<math> is the normal Newtonian field.
Discussion and Criticisms
One reason why some astronomers find MOND difficult to accept is that it is 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 from Einstein's, who assumed that some fundamental physical principles were true (continuity, smoothness and isotropy of spacetime, conservation of energy, principle of equivalence) and derived new equations from these principles, including the famous E=mc² and the less famous but extremely powerful G=8πT. For many, MOND lacks a physical ground, some new fundamental principle about matter, vacuum, or spacetime that would lead to the modified equation F=mµ(a/a_{0})a.
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 a, and the parameter of µ seem to depend on m as well as F. However, for the gravitational force, F also depends on m. Therefore, a change in Newton's second law can be a change of the gravitational force or a change of inertia. The two are indistinguishable. Note that this is not true, for example, for the electromagnetic force: moving in the same weak electromagnetic field, two particles with the same charge but with different masses would follow fundamentally different trajectories. With the same charge, the F term in the MOND equation is the same for the two particles. However, with a different value for m, one could have a MONDian trajectory and not the other, even though they are subject to the same force. However, in interstellar space, gravity is the main acting force, and since no experiment could be performed on Earth to check whether MOND is a new theory of inertia or a new theory of gravity, physicists have concentrated their effort on the later. Until now, they have achieved only partial success, coming up with some more complicated version of Einstein's theory of gravitation. Although it doesn't look like a big trouble (after all, relativity is much more complicated than Newton's law of gravity) one must remember that each and every relativistic theory of gravitation proposed since 1915, and the first appearance of Einstein's theory, has been ruled out or abandoned since. One way or another, only the simplest form of Einstein's theory has passed the many tests physicists put it on. Future will tell if this new one stands or falls. The most successful relativistic version of MOND, which was proposed in 2004, is known as "TeVeS" for TensorVectorScalar and was proposed by black hole physicist and Milgrom associate Bekenstein. It is currently receiving scholarly review in the field.
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 clearcut 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.
Another problem with MOND is that it violates the principle of least astonishment (also known as the principle of maximum boredom). This principle is that the explanation which is the least astonishing and which is the most boring is usually (but not always) the right one. Any modifications in Newton's laws can also be explained in terms of distributions of dark matter, and the second explanation is more boring in that it requires fewer changes to what we think we know. However, proponents of MOND point out that some of the candidates for dark matter such as WIMPs are as astonishing as asserting that gravity behaves differently over long distances than we normally think.
Beside MOND, another theory that tries to explain the mystery of the rotational curves is Nonsymmetric Gravitational Theory proposed by John Moffat.
External links
 Preprints related to MOND (http://arXiv.org/find/physics,astroph/1/abs:+mond/0/1/0/all/0/1)
 Mordehai Milgrom: Does Dark Matter Really Exist?, Scientific American, August 2002
 MOND  A Pedagogical Review (http://nedwww.ipac.caltech.edu/level5/Sept01/Milgrom2/Milgrom_contents.html)de:Modifizierte Newtonsche Dynamik