# Modified Newtonian dynamics

In physics, Modified Newtonian dynamics (MOND) is a theory that proposes a modification of Newton's Second Law of Dynamics (F = ma) to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects that are farther out will have lower velocities. For example, planets in the Solar System orbit with velocities that decrease as their distance from the Sun increases. MOND theory posits that acceleration is not linearly proportional to force at low values. The galaxy rotation problem may be understood without MOND if a halo of dark matter provides an overall mass distribution different from the observed distribution of normal matter.

MOND was proposed by Mordehai Milgrom in 1981 to model the observed uniform velocity data without the dark matter assumption. He noted that Newton's Second Law for gravitational force has only been verified when gravitational acceleration is large.

## Overview: Galaxy dynamics

Observations of the rotation rates of spiral galaxies began in 1978. By the early 1980s it was clear that galaxies did not exhibit the same pattern of decreasing orbital velocity with increasing distance from the center of mass observed in the Solar System. A spiral galaxy consists of a bulge of stars at the centre with a vast disc of stars orbiting around the central group. If the orbits of the stars were governed solely by gravitational force and the observed distribution of normal matter, it was expected that stars at the outer edge of the disc would have a much lower orbital velocity than those near the middle. In the observed galaxies this pattern is not apparent. Stars near the outer edge orbit at the same speed as stars closer to the middle.

Figure 1 - Expected (A) and observed (B) star velocities as a function of distance from the galactic center.
Figure 2 - Postulated dark-matter halo around a spiral galaxy

The dotted curve A in Figure 1 at left shows the predicted orbital velocity as a function of distance from the galactic center assuming neither MOND nor dark matter. The solid curve B shows the observed distribution. Instead of decreasing asymptotically to zero as the effect of gravity wanes, this curve remains flat, showing the same velocity at increasing distances from the bulge. Astronomers call this phenomenon the "flattening of galaxies' rotation curves".

Scientists hypothesized that the flatness of the rotation of galaxies is caused by matter outside the galaxy's visible disc. Since all large galaxies show the same characteristic, large galaxies must, according to this line of reasoning, be embedded in a halo of invisible "dark" matter as shown in Figure 2.

## The MOND Theory

In 1983, Mordehai Milgrom, a physicist at the Weizmann Institute in Israel, published two papers in Astrophysical Journal to propose a modification of Newton's second law of motion. 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 acceleration 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 ( 0 <x≪1 ). The term a0 is a proposed 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 rotation 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 could not 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.

### 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:

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

Assuming 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 does not 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)1/4 allows one to calculate a0 from the observed v and M. Milgrom found a0=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."[1]

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.

## Consistency with the observations

According to the Modified Newtonian Dynamics theory, every physical process that involves small accelerations will have an outcome different from that predicted by the simple law F=ma. Therefore, astronomers need to look for all such processes and verify that MOND remains compatible with observations, that is, within the limit of the uncertainties on the data. There is, however, a complication overlooked up to this point but that strongly affects the compatibility between MOND and the observed world: 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 astronomers must consider. Also critical is the process's 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.

This affects MOND's application to experimental observation and empirical data because all experiments done on Earth or its neighborhood are subject to the Sun's gravitational field, and this field is so strong that all objects in the Solar system undergo an acceleration greater than a0. This explains why the flattening of galaxies' rotation curve, or the MOND effect, had not been detected until the early 1980s, when astronomers first gathered empirical data on the rotation of galaxies.

Therefore, only galaxies and other large systems are expected to exhibit the dynamics that will allow astronomers to verify that MOND agrees 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 the 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. Indeed, conditions for conducting an experiment that could confirm or disprove MOND can only be performed outside the Solar system — farther even than the positions that the Pioneer and Voyager space probes have reached.

In search of 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 an 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 which is 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 some astronomers have claimed their data invalidated MOND. There is evidence that a contradiction exists.[2]

An exception to MOND other than LSB is prediction of the speeds of galaxies that gyrate around the center of a galaxy cluster. Our galaxy is part of the Virgo supercluster. MOND predicts a rate of rotation of these galaxies about their center, and temperature distributions, that are contrary to observation.[3][4]

One experiment that might test MOND would be to observe the particles proposed to contribute to the majority of the Universe’s mass; several experiments are endeavoring to do this under the assumption that the particles have weak interactions. Another approach to test MOND is to apply it to the evolution of cosmic structure or to the dynamics and evolution of observed galaxies..

Lee Smolin and co-workers have tried unsuccessfully to obtain a theoretical basis for MOND from quantum gravity. His conclusion is "MOND is a tantalizing mystery, but not one that can be resolved now."[5] Another attempt to provide a basis for MOND is Allen Rothwarf's aether model.[6]

## 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 $\mu(\xi) \sim 1$ for ξ > > 1 (Newtonian regime), $\mu(\xi) \sim \xi$ for ξ < < 1 (Deep-MOND regime). In the deep-MOND regime, the 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

An empirical criticism of MOND, released in August 2006, involves the Bullet cluster (Milgrom's comments[1]) , a system of two colliding galaxy clusters. In most instances where phenomena associated with either MOND or dark matter are present, they appear to flow from physical locations with similar centers of gravity. But, the dark matter-like effects in this colliding galactic cluster system appears to emanate from different points in space than the center of mass of the visible matter in the system, which is unusually easy to discern due to the high energy collisions of the gas in the vicinity of the colliding galactic clusters.[2]. MOND proponents admit that a purely baryonic MOND is not able to explain this observation. Therefore a “marriage” of MOND with ordinary hot neutrinos of 2eV has been proposed to save the hypothesis [3]. Beside MOND, three other notable theories try to explain the mystery of the rotational curves and/or the apparent missing dark matter, these are Nonsymmetric Gravitational Theory proposed by John Moffat, Weyl's conformal gravity by Philip Mannheim, and the more recently published Dynamic Newtonian Advanced gravitation (DNAg).[7]

## Tensor-vector-scalar gravity

Tensor-Vector-Scalar gravity (TeVeS) is a proposed relativistic theory that is equivalent to Modified Newtonian dynamics (MOND) in the non-relativistic limit, which purports to explain the galaxy rotation problem without invoking dark matter. Originated by Jacob Bekenstein in 2004, it incorporates various dynamical and non-dynamical tensor fields, vector fields and scalar fields.[8]

The break-through of TeVeS over MOND is that it can explain the phenomenon of gravitational lensing, a cosmic phenomenon in which nearby matter bends light, which has been confirmed many times.

A recent preliminary finding is that it can explain structure formation without cold dark matter (CDM), but requiring ~2eV massive neutrinos. [4] and [5]. However, other authors (see Slosar, Melchiorri and Silk [6]) claim that TeVeS can't explain cosmic microwave background anisotropies and structure formation at the same time, i.e. ruling out those models at high significance.

## In-line references

1. ^ The actual result is within an order of magnitude of the lifetime of the universe. It would take 79.2 billion years, about 5.8 times the current age of the universe, to reach the speed of light with an acceleration of a0. Conversely, starting from zero velocity with an acceleration of a0, one would reach about 17.3% of the speed of light at the current age of the universe.
2. ^ RH Sanders (2001). "Modified Newtonian dynamics and its implications". in Mario Livio. The Dark Universe: Matter, Energy and Gravity, Proceedings of the Space Telescope Science Institute Symposium. Cambridge University Press. p. 62. ISBN 0521822270.
3. ^ Charles Seife (2004). Alpha and Omega. Penguin Books. pp. 100-101. ISBN 0142004464.
4. ^ Anthony Aguirre, Joop Schaye & Eliot Quataert (2001). "Problems for Modified Newtonian Dynamics in Clusters and the Lyα Forest?". The Astrophysical Journal 561: 550–558. doi:10.1086/323376.
5. ^ Lee Smolin (2007). The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. Mariner Books. p. 215. ISBN 061891868X.
6. ^ F Rothwarf, S Roy (2007). "Quantum Vacuum and a Matter - Antimatter Cosmology". Arxiv preprint.
7. ^ A.Worsley (2008). An advanced dynamic adaptation of Newtonian equations of gravity. Physics Essays 21: 3, 222-228 (2008).
8. ^ Jacob D. Bekenstein (2004). "Relativistic gravitation theory for the MOND paradigm". Phys. Rev. D70.