page_title: Modified Newtonian dynamics
section_title: Overview
hierarchical_section_title: Modified Newtonian dynamics / Overview
caption_attribution_description: English: Mordehai Milgrom is an Israeli physicist and professor in the department of Condensed Matter Physics at the Weizmann Institute in Rehovot, Israel.
context_page_description: Modified Newtonian dynamics is a hypothesis that proposes a modification of Newton's laws to account for observed properties of galaxies. It is an alternative to the hypothesis of dark matter in terms of explaining why galaxies do not appear to obey the currently understood laws of physics.
  Created in 1982 and first published in 1983 by Israeli physicist Mordehai Milgrom, the hypothesis' original motivation was to explain why the velocities of stars in galaxies were observed to be larger than expected based on Newtonian mechanics. Milgrom noted that this discrepancy could be resolved if the gravitational force experienced by a star in the outer regions of a galaxy was proportional to the square of its centripetal acceleration, or alternatively if gravitational force came to vary inversely with radius. In MOND, violation of Newton's laws occurs at extremely small accelerations, characteristic of galaxies yet far below anything typically encountered in the Solar System or on Earth.
  MOND is an example of a class of theories known as modified gravity, and is an alternative to the hypothesis that the dynamics of galaxies are determined by massive, invisible dark matter halos.
context_section_description: Several independent observations point to the fact that the visible mass in galaxies and galaxy clusters is insufficient to account for their dynamics, when analysed using Newton's laws. This discrepancy – known as the "missing mass problem" – was first identified for clusters by Swiss astronomer Fritz Zwicky in 1933 (who studied the Coma cluster), and subsequently extended to include spiral galaxies by the 1939 work of Horace Babcock on Andromeda. These early studies were augmented and brought to the attention of the astronomical community in the 1960s and 1970s by the work of Vera Rubin at the Carnegie Institute in Washington, who mapped in detail the rotation velocities of stars in a large sample of spirals. While Newton's Laws predict that stellar rotation velocities should decrease with distance from the galactic centre, Rubin and collaborators found instead that they remain almost constant – the rotation curves are said to be "flat". This observation necessitates at least one of the following: 1) There exists in galaxies large quantities of unseen matter which boosts the stars' velocities beyond what would be expected on the basis of the visible mass alone, or 2) Newton's Laws do not apply to galaxies. The former leads to the dark matter hypothesis; the latter leads to MOND.
  The basic premise of MOND is that while Newton's laws have been extensively tested in high-acceleration environments (in the Solar System and on Earth), they have not been verified for objects with extremely low acceleration, such as stars in the outer parts of galaxies. This led Milgrom to postulate a new effective gravitational force law (sometimes referred to as "Milgrom's law") that relates the true acceleration of an object to the acceleration that would be predicted for it on the basis of Newtonian mechanics. This law, the keystone of MOND, is chosen to reduce to the Newtonian result at high acceleration but leads to different ("deep-MOND") behaviour at low acceleration:
  Here FN is the Newtonian force, m is the object's (gravitational) mass, a is its acceleration, μ(x) is an as-yet unspecified function (known as the "interpolating function"), and a₀ is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires 
  and consistency with astronomical observations requires
  Beyond these limits, the interpolating function is not specified by the hypothesis, although it is possible to weakly constrain it empirically. Two common choices are the "simple interpolating function":
  and the "standard interpolating function":
  Thus, in the deep-MOND regime (a ≪ a₀):
  Applying this to an object of mass m in circular orbit around a point mass M (a crude approximation for a star in the outer regions of a galaxy), we find:
  that is, the star's rotation velocity is independent of r, its distance from the centre of the galaxy – the rotation curve is flat, as required. By fitting his law to rotation curve data, Milgrom found  to be optimal. This simple law is sufficient to make predictions for a broad range of galactic phenomena.
  Milgrom's law can be interpreted in two different ways. One possibility is to treat it as a modification to the classical law of inertia (Newton's second law), so that the force on an object is not proportional to the particle's acceleration a but rather to . In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other forces, for example electromagnetism. Alternatively, Milgrom's law can be viewed as leaving Newton's Second Law intact and instead modifying the inverse-square law of gravity, so that the true gravitational force on an object of mass m due to another of mass M is roughly of the form
  In this interpretation, Milgrom's modification would apply exclusively to gravitational phenomena.
  By itself, Milgrom's law is not a complete and self-contained physical theory, but rather an ad-hoc empirically motivated variant of one of the several equations that constitute classical mechanics. Its status 
caption: MOND was proposed by Mordehai Milgrom in 1983
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