World Science Scholars
2.2 Modified Newtonian Dynamics
summary
Galaxies Are Surprisingly Simple Objects

• Galaxies are huge structures composed of a wide variety of objects – dark matter, stars, planets, gases, dust, black holes, supernovae, etc. Despite their complex compositions, galaxies are remarkably simple.
• The rotation curves of observed galaxies (that relate the velocity of galactic objects to distance from the galactic center) are very flat, approaching an asymptotic angular velocity.
• A strange relationship exists between that asymptotic angular velocity and the observable, visible mass of a galaxy. This Tully-Fisher relationship says that visible mass of a galaxy is proportional to the asymptotic velocity of the galaxy to the fourth power. This relationship is remarkably consistent and accurate compared to actual observations of galaxies.
• This is strange because the asymptotic rotational velocity of a galaxy should depend on how much dark matter it contains, not visible matter, since dark matter makes up most of a galaxy. Therefore this relationship suggests that the amount of visible matter in a galaxy is precisely related to the amount of dark matter it contains. It is not understood why this happens.

The Missing Satellite Problem

• Simulations that only include dark matter find that, in addition to the large clumps of dark matter that form the basis of galaxies like our own, there should be small satellite clumps of dark matter that precipitate small galaxies.
• Based on these simulations the Milky Way should be surrounded by hundreds of these satellite dwarf galaxies, but observations have found only about 30 such satellites.
• Another discrepancy exists with regard to the distribution of these satellite galaxies. Simulations suggest that they should be distributed isotropically around the Milky Way; observation actually shows that these dwarf galaxies lie along specific planes in space, and not evenly distributed in all directions.

Modified Newtonian Dynamics

• Modified Newtonian dynamics (MOND) is a theory that was proposed by Mordehai Milgrom over thirty years ago.
• His theory relates to the problem of galaxy rotation curves and the subsequent inference of massive dark matter halos. In Milgrom’s theory there is no dark matter – instead, Newtonian gravity must be altered in the regime of very low acceleration. This is captured in the equation:$$F_N = m\mu(\frac{a}{a_0})a$$
• $\mu()$ is the interpolating function between standard Newtonian mechanics and “deep-MOND” mechanics. The threshold acceleration ($a_0$) marks the transition between Newtonian and deep-MOND regimes. This transition acceleration is about $10^{-10} \frac{m}{s^2}$ .
• The interpolating function µ(a/a0) takes the following form in the deep-MOND regime:$$\mu (\frac{a}{a_0}) = \sqrt{1+(\frac{a}{a_0})^2}$$
• This theory perfectly matches the rotation curves of galaxies. Furthermore, the Tully-Fisher relation is an exact prediction of MOND.
• Unfortunately, while MOND works exceptionally well for galaxies, it fails on larger scales. In particular, predictions of MOND do not mesh well with observed properties of galaxy clusters and the cosmic web.

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