One alternative to dark matter, particularly as an explanation for the non-Keplerian motions of rotating bodies, is called MOND (MOdified Newtonian Dynamics). In 1983 M. Milgrom proposed that the flat rotation curves observed in many galaxies may be explained without postulating any sort of missing mass in the universe. [86] He instead introduced an acceleration constant to modify Newton's second law, which would at small accelerations account for the radius independent nature of stellar motion.

Rather than the usual
=
*m*, the
equation at the heart of MOND is

(30) |

where is the
force acting on an object of mass *m* and acceleration *a* =
||, and
*a*_{0}
2 × 10^{-8} cm s^{-2} is the acceleration constant
determined by Milgrom (many other MOND theories have emerged
with differing values for *a*_{0}). For accelerations greater
than or equal to *a*_{0} (most accelerations we see in
everyday life, including the motions of planets within our solar system),
*x* 1, and
Newtonian dynamics can be used as usual.
However, for very small accelerations such as for the orbits of
objects far away from the galactic center, *a*_{0} becomes
significant; this is how MOND predicts and explains the flat
rotation curves.

To demonstrate how MOND can explain flat rotation curves, we first consider the expression for the force of gravity on a star and Milgrom's modification of Newton's second law:

(31) |

where *G* is the gravitational constant, *m* and
*M* are the masses of the star and galaxy respectively, and
*r* is the radius of the star's orbit. If we cancel *m* from
both sides and assume that *µ* (*a* /
*a*_{0}) = (*a*/*a*_{0}) at a very
large *r*, we are left with

(32) |

Solving for *a* and using the relationship of
acceleration with velocity and radius (*a* = *v*^{2} /
*r*), we find

(33) |

(34) |

where *v* has no dependence on *r*. This relation
has allowed various studies to use MOND to fit flat rotation
curves quite successfully for several low and high surface
brightness galaxies (LSB and HSB galaxies, respectively) based
on luminous mass alone.
[87,
88,
89]
As MOND predicts, LSB galaxies show a larger departure from
Newtonian dynamics where HSB galaxies show discrepancies only
in their outer regions where gravitational attraction is
considerably smaller. MOND and TeVeS (the MONDian version of
General Relativity)
[90]
have had success in predicting and describing other observed galactic
dynamics as well. For a recent review, see R. H. Sanders.
[91]

Despite these successes, MOND faces several major and critical
challenges it has not been able to overcome. For example, when
considering galaxy clusters, MOND cannot account for density and
temperature profiles and requires unseen matter.
[92]
Evidence for dark matter exists on
many distance scales and MOND essentially only works on galactic
scales. Also, extremely low acceleration experiments (below
*a*_{0}) have been conducted, finding no departure from
Newton's second law and thus constraining MOND to reduce to Newton's
second law in laboratory conditions.
[93,
94]
And finally, gravitational lensing
evidence such as in the Bullet Cluster show that, in effect, the
gravitational force points back not towards regular, observed baryonic
matter but rather some form of dark matter which is not observed
optically. MOND theories in their current forms cannot account for such
a discrepancy easily. For these reasons and others, we feel that dark
matter is a more promising solution to the puzzle of missing mass in the
universe.