Dark matter is the only explanation that astronomers can conjure up for the various mass discrepancies. But if we accept a departure from these standard laws, we might do away with dark matter.
The diverse appearances of the mass discrepancy, as reflected in the motions inside galactic systems, stem from the use of a single formula in Newtonian physics. This formula combines two basic laws: Newton’s law of gravity (which relates the force of gravity between bodies to the bodies’ masses and separation) and Newton’s second law (which relates force to acceleration). The acceleration of a body in orbit depends on the velocity and size of the orbit.
These laws accurately explain the flight of a ballistic missile and the motions of the planets. But their extrapolation to galaxies has never been directly tested. If the laws break down, then modifying them might obviate dark matter.
Such a modification would not be without precedent. Two drastic changes to Newtonian physics have already proved necessary. The first upgraded Newtonian dynamics to the theory of relativity—both the special theory (which changed Newton’s second law) and the general theory (which altered the law of gravity). The second led to the quantum theory, which accounts for the behavior of microscopic systems under certain circumstances. The two were proven extensions of Newtonian dynamics come into play under extreme conditions such as extreme speeds (special relativity) or extremely strong gravity (general relativity).
What attributes of galactic systems are so extreme that they might require yet another modification? The first possibility is size. Perhaps gravity departs from the Newtonian law at large distances. As early as 1923, English astronomer James Jeans proposed modifying the distance dependence of the law of gravity on galactic scales. But the observations he sought to explain were unrelated to dark matter. Another modified distance dependence was proposed in 1963 by Arrigo Finzi, as a possible solution to the dark matter problem in clusters. But in the early 1980s I showed that such modifications of distance dependence of gravity fail to reproduce these observations.
What then can work? The acceleration in galactic systems is many orders of magnitude smaller than in everyday experience. The acceleration of the solar system towards the center of our galaxy is one hundred billionth the acceleration of the space shuttle toward the center of the earth. 20 years ago I proposed a modification to Newton’s second law that changed the relation between force and acceleration when the acceleration is low. This is the beginning of the idea called MOND for Modified Newtonian Dynamics.
Building up Speed
MOND introduces a new constant nature with the dimensions of acceleration called . When the acceleration is lager than , Newton’s second law applies as usual. But when the acceleration is small compared with , Newton’s second law is altered: force becomes proportional to the square of the acceleration. The force needed to impart a given acceleration is always smaller than Newtonian dynamics requires. To account for the observed accelerations in galaxies, MOND predicts a smaller force- hence, less gravity-producing mass- than Newtonian dynamics does. In this way, it can eliminate the need for dark matter.
In the outskirts of galaxies, the acceleration produced by gravity decreases with distance and eventually goes below . Exactly where this happens depends on the value of and on the mass. The higher the mass, the father out the effects of MOND set in. For the value of that is required by the data, and for a galaxy of typical mass, the transition takes place several tens of thousands of light-years from the center. For the mass of a typical cluster of galaxies, it happens at a few million light-years from the center.
Suppose the bulk of a galaxy is contained within a certain radius. Then, by the Newtonian dynamics, beyond this radius the speed of objects in circular orbits (such as gas or stars) should decrease with increasing radius. This is what happens in the solar system. The bulk of the solar system’s mass is contained in the sun, and the orbital velocity of the planets decreases with distance. Mercury trundles around the sun much faster than the earth does, for example. Where MOND applies, however, the situation is radically different. At sufficiently large distances from the center of a galaxy, the orbital velocity should stop decreasing and reach a constant value. This constant velocity should be proportional to the fourth root of the galaxy’s mass.
How does MOND fare when confronted with the data? Orbital velocities in spiral galaxies, instead of declining with increasing distance from the galactic center, flatten out to a constant value, as predicted by MOND. Moreover, according to an observed correlation known as the Tully-Fisher relation, this constant velocity is proportional to the fourth root of the galaxy’s luminosity. This, too, emerges naturally from MOND. The key assumption is that the luminosity, in turn, is proportional to mass or nearly so. Recent observations vindicate the assumption: a direct velocity-mass correlation is even tighter than the velocity-luminosity correlation.
Glimpses of these regularities were already in were already in sight when I proposed MOND; indeed, they furnished clues for its construction. What makes MOND particularly intriguing is that it predicted many effects that could not even be tested when I formulated it. One example is the nature of low-surface-brightness galaxies- stellar agglomerations so wispy that they can barely be seen at all. Whereas the acceleration in ordinary galaxies exceeds
