# Modified Newtonian dynamics

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In astrophysics, the modified Newtonian dynamics (MOND) is a burgeoning theory that attempts to explain the galaxy rotation problem by modifying Newton's second law of motion. (The most widely accepted approach to explaining this problem postulates the existence of dark matter.) MOND was proposed in 1983 by Mordehai Milgrom. The central pillar of MOND is the assumption that Newton's Second Law (F = ma) is a high-acceleration approximation of a more accurate law that describes all accelerations. The proposed modification would only become relevant when the total acceleration of a body falls significantly below the constant a0. Consequently, observations of this behavior could never be made on Earth.

Although Milgrom and others have catalogued a respectable sum of persuasive evidence in favor of MOND, there has not been enough compelling research to conclusively substantiate or disprove the theory. Calculations suggest that the parameters required to gather direct experimental evidence can be found only far outside our solar system (beyond the overwhelming influence of the Sun's gravitational field). Therefore, it will be some time before the validity of MOND can be directly tested. Phenomena such as low surface brightness galaxies have yielded an abundance of indirect evidence on the matter, but in the absence of more varied data, such successes have been viewed as isolated. Many researchers have criticized MOND for 'adapting' its mathematics to the observed mass discrepancy instead of hypothesizing a true physical explanation. The theory is the minority view in the astrophysicist community.

## Overview: Galaxy dynamics

At the start of the 1980s, the first observational evidence was reported that galaxies do not spin as current theories predict. A galaxy is a vast group of stars orbiting a bulge at the center of a galaxy. Since the orbit of stars is driven solely by gravitational force, it was expected that stars at the edge would have a much larger orbital period than those near the bulge. For example, the Earth, which is 150 million km from the sun, completes an orbit in one year, while it takes Saturn, at a distance of 1.4 billion kilometers from the sun, 30 years to do the same. A similar behavior was expected from galaxies, even if the distribution of stars is more cloud-like . However, it became increasingly apparent that stars at the edge of a galaxy move faster than predicted by conventional theory.

File:GalacticRotation.png
Expected (A) and observed (B) galactic rotational curves, along with solar rotational curve (C)
File:DarkMatterHalo.png
Postulated dark-matter halo around a spiral galaxy

Astronomers call this phenomenon the "flattening of galaxies' rotation curve". In simple terms, drawing a curve to describe the velocity of stars as a function of their distance from the center of a galaxy should yield curve A (the dashed line in Figure 1); however, data from telescopes yield curve B (the plain line). Instead of decreasing asymptotically to zero as the effect of gravity wanes, this curve remains flat at large distances from the bulge. By comparison, the same curve for our solar system—properly scaled—is provided (curve C in Figure 1).

Reluctant to change Newton's law and Einstein's theory of relativity for galaxies alone, scientists have simply assumed that the rotation curve was flat because of the presence of a large amount of matter outside galaxies. Their new theory was that galaxies are embedded in a spherical halo of invisible, "dark" matter (Figure 2); however, the search for dark matter has since been met with limited success. The hypothesis of dark matter halos has encountered many problems that have cast doubt on the validity of this model. While it is still the most widely accepted model, alternative approaches have been considered, of which MOND is one.

## 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/a0)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 a0 is some new constant, in the same sense that c (the speed of light) is a constant, except that a0 is acceleration whereas c is speed.

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

$\vec{F} = m \cdot \mu\!\left( { a \over a_0 } \right) \ \vec{a}$
$\mu (x) = 1 \mbox{ if } x\gg 1$
$\mu (x) = x \mbox{ if } x\ll 1$

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 does not change most of the consequences of the theory, such as the flattening of the rotational curve.

In the everyday world, a is much greater than a0 for all physical effects, therefore µ(a/a0)=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:

$F = \frac{GMm}{r^2}$

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:

$F = \frac{GMm}{r^2} = m \mu{ \left( \frac{a}{a_0}\right)} a$

Eliminating m gives:

$\frac{GM}{r^2} = \mu{ \left( \frac{a}{a_0}\right)} a$

We assume that, at this large distance r, a is smaller than a0 and thus $\mu{ \left( \frac{a}{a_0}\right)} = \frac{a}{a_0}$, which gives:

$\frac{GM}{r^2} = \frac{a^2}{a_0}$

Therefore:

$a = \frac{\sqrt{ G M a_0 }}{r}$

Since the equation that relates the velocity to the acceleration for a circular orbit is $a = \frac{v^2}{r}$ one has:

$a = \frac{v^2}{r} = \frac{\sqrt{ G M a_0 }}{r}$

and therefore:

$v = \sqrt[4]{ G M a_0 }$

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 Tully-Fisher relation.

At the same time, there is a clear relationship between the velocity and the constant a0. The equation v=(GMa0)¼ allows one to calculate a0 from the observed v and M. Milgrom found a0=1.2 10-10ms-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>>a0 for physical effects on Earth remains valid. Had a0 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

See scientific method for background information

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 acceleration greater than a0, and the orbit would not be the same as that in an isolated system.

For this reason, the typical acceleration of any physical process is not the only parameter one must consider. Also critical is the process' environment, which is all external forces that are usually neglected. In his paper, Milgrom arranged the typical acceleration of various physical processes in a two-dimensional diagram. 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 a0. 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 have 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. (See Pioneer anomaly.) To check that, let's calculate the radius of the gravitational sphere of influence of the Sun, inside which a probe undergoes acceleration greater than a0.

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

$\frac{GM}{r^2}=\mu\left(\frac{a}{a_0}\right)a$

So, for a=a0, assuming µ(a/a0)=µ(1)=1, with G=6.67 10-8 in cgs units and M (the mass of the Sun)=2 1033 g, we get r=1.05 1017 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) is 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 non-relativistic Modified Newtonian Dynamics, Poisson's equation,

$\nabla^2 \Phi_N = 4 \pi G \rho$

(where ΦN is the gravitational potential and ρ is the density distribution) is modified as

$\nabla\cdot\left[ \mu \left( \frac{\left\| \nabla\Phi \right\|}{a_0} \right) \nabla\Phi\right] = 4\pi G \rho$

where Φ is the MOND potential. The equation is to be solved with boundary condition $\left\| \nabla\Phi \right\| \rightarrow 0$ for $\left\| \mathbf{r} \right\| \rightarrow \infty$. The exact form of μ(ξ) is not constrained by observations, but must have the behaviour μ(ξ)˜1 for ξ > > 1 (Newtonian regime), μ(ξ)˜ξ for ξ < < 1 (Deep-MOND regime). Under deep-MOND regime, modified Poisson equation may be rewritten as

$\nabla \cdot \left[ \frac{\left\| \nabla\Phi \right\|}{a_0} \nabla\Phi - \nabla\Phi_N \right] = 0$

and that simplifies to

$\frac{\left\| \nabla\Phi \right\|}{a_0} \nabla\Phi - \nabla\Phi_N = \nabla \times \mathbf{h}$

The vector field $\mathbf{h}$ 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

$\mathbf{g}_M = \mathbf{g}_N \sqrt{\frac{a_0}{\left\| \mathbf{g}_N \right \|}}$

where $\mathbf{g}_N$ 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 describes 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 space-time, 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 seems to lack a physical ground, some new fundamental principle about matter, vacuum, or space-time that would lead to the modified equation F=mµ(a/a0)a. Supporters of MOND, on the other hand, have compared theories of dark matter to the now obscure aether hypothesis, which was discarded in favor of a fundamental change in our understanding of light. Instead of postulating the existence of an invisible influence to explain the observed mass discrepancy, supporters of MOND believe we should re-examine our understanding of acceleration.

Attempts in this direction have essentially been modifications of Einstein's theory of gravitation. When one looks at the equation F=mµ(a/a0)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 determine whether MOND is a new theory of inertia or a new theory of gravity, physicists have concentrated their effort on the latter. Presently, they have achieved only partial success, devising a more complicated version of Einstein's theory of gravitation. The most successful relativistic version of MOND, which was proposed in 2004, is known as "TeVeS" for Tensor-Vector-Scalar. Proposed by black hole physicist and Milgrom associate Jacob D. Bekenstein, it is currently undergoing scholarly review. Although this most recent incarnation looks promising and even successfully predicts gravitational lensing, one must remember that every relativistic theory of gravitation proposed since the emergence of Einstein's theory in 1915 has been ruled out or abandoned since. One way or another, only the simplest form of Einstein's theory has survived the scientific rigors of the past century.

In the eyes of most cosmologists and astrophysicists, MOND is considered a possible but unlikely alternative to the more widely accepted theory of dark matter. As new data is collected, both MOND and dark matter are occaisionally invalidated and occasionally supported, and no observation has yet conclusively settled the debate. Toward this goal, supporters of MOND have concentrated their effort on specific areas:

• To determine phenomena predicted by MOND and to search for them. For example, the dynamics of satellites of our Galaxy could be distorted by the effects of MOND, in such a way that would be difficult to explain with a dark matter halo.
• To obtain the relativistic extension of MOND that would incidentally help scientists understand how light is bent by galaxies' gravitational field. Although MOND cannot currently cope with this, TeVeS has shown promise.
• To otherwise establish MOND as a theory of inertia and determine its fundamental principles. Progress in this direction has been minimal.

Another criticism of MOND is that it violates Occam's Razor, which states that the simplest explanation is usually correct. Any modifications to Newton's laws can also be explained in terms of distributions of dark matter, and the second explanation is simpler in that it requires fewer changes to rigorously-established scientific theory. However, proponents of MOND are quick to point out that extremely hypothetical candidates for dark matter such as WIMPs, six-dimensional dark matter theories, and the general requirement that dark matter be evenly distributed through space are no simpler than the idea that gravity acts differently at low accelerations.

Beside MOND, another theory that tries to explain the mystery of the rotational curves is Nonsymmetric Gravitational Theory proposed by John Moffat.

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