# Modified Newtonian dynamics

In physics, Modified Newtonian Dynamics (MoND) is a theory that proposes a modification of Newton's law of gravity to explain the galaxy rotation problem.[1] 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 was proposed by Mordehai Milgrom in 1983 as a way to model this observed uniform velocity data.[2][3] Milgrom noted that Newton's law for gravitational force has been verified only where gravitational acceleration is large, and suggested that for extremely small accelerations the theory may not hold. MoND theory posits that acceleration is not linearly proportional to gravitational force at small values.

MoND is one alternative to the more widely accepted theory of dark matter. Dark matter theory suggests that each galaxy contains a halo of an as yet unidentified type of matter that provides an overall mass distribution different from the observed distribution of normal matter. This dark matter accounts for the uniform rotation velocity data without modifying Newton's law of gravity.

## Overview: Galaxy dynamics

The original purpose of MoND was to explain the galactic rotation curves for spiral galaxies. 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 (stars, gas clouds, dust, etc.), 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 the centre of the galaxy 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.

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.

### Publication of the MoND theory

In 1983, Mordehai Milgrom, a physicist at the Weizmann Institute in Israel, published three papers in Astrophysical Journal to propose a modification of Newton's law of gravity.[2] A pedagogical introduction to MoND can be found in Bekenstein, who characterizes MoND as follows:[4] "Relativistic MoND as here described has developed from the ground up, rather than coming down from the sky: phenomenology, rather than pure theoretical ideas."

Actually, Milgrom provided several interpretations of his proposal, one being a modification of Newton's second law of motion. However, this proposed interpretation is inconsistent with conservation of momentum, requiring some unconventional physical assumptions to regain plausibility.[4] A second interpretation, as a modification of the law of gravity, requires that the acceleration due to gravitational force not depend simply upon the mass m, but upon the form m/μ(a/a0), where μ is some function approaching the value one for large arguments and a/a0 for small arguments, and a is the acceleration caused by gravity and a0 is a natural constant, a0 ≈ 10−10 m/s2.[4] The centripetal accelerations of stars and gas clouds at the outskirts of spiral galaxies tend to be below a0.[4] This interpretation is also inconsistent with momentum conservation; but this can be repaired by substituting a Lagrangian-based theory known as AQUAL; it reproduces MoND for spherical or disc-like galaxies, and is very similar even for general elliptical galaxies.[4]

A third interpretation views MoND simply as a description of the behavior of the dark matter in a galaxy; this leads to the conclusion that the dark matter must be tightly correlated with the visible matter in accordance with a fixed law, at least for stable galaxies (not engaged in strong interactions with other galaxies). But one would expect instead that the ratio of dark-to-visible matter would depend on the history of the galaxy.[4]

The exact form of µ is unspecified, only its asymptotic behavior when the argument a/a0 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. Empirically, a good description of galaxy rotation curves is given by [5]

$\mu{\left( \frac{a}{a_0} \right)} = \left(1 + \frac{a_0}{a}\right)^{-1}$ ;

a possible physical motivation of this functional form is provided by the assumption that gravity is mediated by gravitons with non-zero mass.[6]

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 law of gravity is negligible and Newton could not have seen it.

### Predicted rotation curve

Far away from the center of a galaxy, the gravitational acceleration, a, that a star undergoes is predicted by MoND to be roughly:

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

with G the gravitation constant, M the mass of the galaxy, and r the distance between the center and the star.

Assuming that, at this large distance r, a is smaller than a0, $\mu{ \left( \frac{a}{a_0}\right)} = \frac{a}{a_0}$. This 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. As expected, this quantity is far smaller than any acceleration typically found in solar system-scale interactions.

To explain the meaning of this constant, Milgrom said : "... It is roughly the acceleration that will take an object from rest to the speed of light in the lifetime of the universe. It is also of the order of the recently discovered acceleration of the universe."[7][8]

## Consistency with the observations

According to the Modified Newtonian Dynamics theory, every physical process that involves small accelerations due to gravity 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 of 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 of the velocities 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 may only be possible outside the Solar system. A couple of near-to-Earth tests of MoND have been proposed though: one involves flying the LISA Pathfinder spacecraft through the Earth-Sun saddlepoint;[9] another involves using a precisely controlled spinning disk to cancel out the acceleration effects of Earth's orbit around the Sun, and Sun's orbit around the galaxy;[10] if either of these tests are carried out, and if MoND holds true, then an accelerometer on either body should detect a slight kick as they approach the very low acceleration levels required by MoND.

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, the majority of LSBs observed has been consistent with the rotational curve predicted by MoND.[11]

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.[12][13]

Computer simulations show that MoND is generally very precise at predicting individual galaxy rotation curves, of all kinds of galaxies: spirals, ellipticals,[14] dwarfs,[15] etc. However, MoND and MoND-like theories are not so good at predicting galactic cluster-scale, or cosmological scale structures.[citation needed]

A test that might disprove MoND would be to discover any of the theorized Dark Matter particles, such as the WIMPs.

A recent proposal is that MoND successfully predicts the local galactic escape speed of the Milky Way, a measure of the mass beyond the galactocentric radius of the Sun.[16]

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."[17]

In 2011 University of Maryland Astronomy Professor, Stacy McGaugh, examined the rotation of gas rich galaxies, which have relatively fewer stars and a prevalence of mass in the form of interstellar gas. This allowed the mass of the galaxy to be more accurately determined since matter in the form of gas is easier to see and measure than matter in the form of stars or planets. McGaugh studied a sample of 47 galaxies and compared each one's mass and speed of rotation with the ratio expected from MoND predictions. All 47 galaxies fell on or very close to the MoND prediction. No dark matter model performed as well.[18] On the other hand, another 2011 study observing the gravity-induced redshift of galactic clusters found results that strongly supported general relativity, but were inconsistent with MoND.[19][20] A recent work has found mistakes in the work by Wojtak, Hansen, and Hjorth, and confirmed that MoND can fit the determined redshifts only slightly worse than does general relativity with dark-matter halos.[21]

## The mathematics of MoND

In non-relativistic Modified Newtonian Dynamics, Poisson's equation,

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

(where $\Phi_N$ is the gravitational potential and $\rho$ 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 $\Phi$ is the MOND potential. The $a_0$ is a natural constant, approximately equal to $10^{-10}\,m/s^2$. It is a fact that the centripetal accelerations of stars and gas clouds in the outskirts of spiral galaxies tend to be below $a_0$.[4]

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 $\mu(\xi)$ is not constrained by observations, but must have the behaviour $\mu(\xi) \sim 1$ for $\xi \gg 1$ (Newtonian regime), $\mu(\xi) \sim \xi$ for $\xi \ll 1$ (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.

## The External Field Effect (EFE)

In MoND it turns out that if a weakly gravitationally bound system s, whose inner accelerations are expected to be of the order of 10−10 m s−2 from a Newtonian calculation, is embedded in an external gravitational field $E_g$ generated by a larger array of masses S, then, even if $E_g$ is uniform throughout the spatial extension of s, the internal dynamics of the latter is influenced by $E_g$ in such a way that the total acceleration within s is, actually, larger than 10−10 m s−2. In other words, the Strong Equivalence Principle is violated. Milgrom originally introduced[2] such a concept to explain the fact that the expected phenomenology of dark matter—-to be explained in terms of MoND—-was absent just in some systems (open clusters) in which it should have, instead, been present. It was shown later by R. Scarpa and collaborators[22] that also a number of globular clusters in the neighborhood of the Milky Way behave in the same way, that is MoND effects are seen even though the total (internal+external) field is above MoND acceleration limit.

It has recently been shown that simulations show a significant increase in the velocity dispersion of general globular clusters in MOND,especially in the outer parts of the cluster, and a uniform external gravitational field tends to weaken this effect and pushes the profile towards that of the Newtonian regime. However, it is not sufficiently efficient to cause the velocity dispersion profile, that is predicted by the MONDian EFE for a real globular cluster (NGC 2419) to match the observational data, and the Newtonian prediction is better than that of the MONDian to an order of magnitude, even with taking the external Galactic field into account.[23]

## Discussion and criticisms

An empirical criticism of MoND, released in August 2006, involves the Bullet cluster,[24] 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.[25] 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.[26]

Beside MoND, three other notable theories that try to explain the mystery of the rotational curves are Nonsymmetric Gravitational Theory proposed by John Moffat, Conformal gravity by Philip Mannheim, and Dynamic Newtonian Advanced gravity (DNAg)[27]

## 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.[4]

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.[citation needed]

A recent preliminary finding is that it can explain structure formation without cold dark matter (CDM), but requiring ~2eV massive neutrinos.[28][29] However, other authors[30] 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.

## References

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2. ^ a b c Milgrom, M. (1983). "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis". Astrophysical Journal 270: 365–370. Bibcode:1983ApJ...270..365M. doi:10.1086/161130.. Milgrom, M. (1983). "A modification of the Newtonian dynamics - Implications for galaxies". Astrophysical Journal 270: 371–389. Bibcode:1983ApJ...270..371M. doi:10.1086/161131..
3. ^ Milgrom, M. (2010). "MD or DM? Modified dynamics at low accelerations vs dark matter". Proceedings of Science 1101: 5122. arXiv:1101.5122. Bibcode:2011arXiv1101.5122M.
4. Jacob D. Bekenstein (2006). "The modified Newtonian dynamics-MOND-and its implications for new physics". Contemporary Physics 47 (6): 387. arXiv:astro-ph/0701848. Bibcode:2006ConPh..47..387B. doi:10.1080/00107510701244055.
5. ^ G. Gentile, B. Famaey, W.J.G. de Blok (2011). "THINGS about MOND", Astron. Astrophys. 527, A76. arXiv:1011.4148
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7. ^ Dark-matter heretic, interview of Physicist Mordehai Milgrom, American Scientist, January–February 2003, Volume 91, Number 1, Page: 1.
8. ^ 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.
9. ^ Christian Trenkel, Steve Kemble, Neil Bevis, Joao Magueijo (2010). "Testing MOND/TEVES with LISA Pathfinder" arXiv:1001.1303
10. ^ V. A. De Lorenci, M. Faundez-Abans, J. P. Pereira (2010). "Testing the Newton second law in the regime of small accelerations" arXiv:1002.2766
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12. ^ Charles Seife (2004). Alpha and Omega. Penguin Books. pp. 100–101. ISBN 0-14-200446-4.
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14. ^ Riccardo Scarpa (2003). "MOND and the fundamental plane of elliptical galaxies" arXiv:astro-ph/0302445
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16. ^ Benoit Famaey, Jean-Philippe Bruneton, HongSheng Zhao; Bruneton; Zhao (2007). "Escaping from MOND". Mon.Not.Roy.Astron.Soc. 377: L79. arXiv:astro-ph/0702275. Bibcode:2007MNRAS.377L..79F. doi:10.1111/j.1745-3933.2007.00308.x.
17. ^ 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 0-618-91868-X.
18. ^ "Gas rich galaxies confirm prediction of modified gravity theory". Physorg.com. February 23, 2011. Retrieved 2011-02-27.
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23. ^ K.Derakhshani (2014). "The MOND External Field Effect on the Dynamics of the Globular Clusters: General Considerations and Application to NGC 2419". The Astrophysical Journal 783: 48. arXiv:astro-ph/1404.2202. Bibcode:2014ApJ...783.48D. doi:10.1088/0004-637X/783/1/48.