Theory MOND

The theory MOND (for the English Modified newtonian dynamics , “dynamic Newtonian modified”) is an adaptation of the traditional Mécanique proposed to explain the problem of the curve of rotation punt of the spiral galaxies. It constitutes an alternative to the concept of black Matière, whose existence still could not be highlighted.

It was calculated that if the black Matière existed, approximately 22% to 27% of the Univers of it would be made up.

Theory MOND rests on a modification of the second law of Newton to the very weak Accélération S. It is generalized within the framework of a relativistic theory, the Théorie tensor-scalar.

The problem of the gravitation

See also: black Matter

In 1932, the American astrophysicist Fritz Zwicky notes that in large the Galaxy cluster the speed of these last compared with the ideal models reveals a very important variation. Indeed, for these speeds, the mass of the galaxy (which one deduces from the luminosity of this one) would not suffice to maintain them: they should move away. He proposed whereas this variation was related to the presence of a nonvisible source of Gravitation, i.e. other that stellar.

Nevertheless, its calculations to determine the proportion of this “black Matière” showed that it was represented much than the visible mass.

Starting from 1978, one starts to observe the phenomenon on a smaller scale. It is noticed that in the galaxies, plus stars are far away from the galactic Noyau, plus their angular Velocity is high. The initial observation of this uniformity speed was unexpected because the theory of the gravity of Newton predicted that the more distant objects have a less speed. For example, the planets of our Solar system orbit with a respective speed which decrease whereas their respective distance grows compared to the Sun. One finds oneself with the same problem: how to explain that at a point given measurement is higher than the theoretical value?

Relativity since the theory of Newton in is a simplified form) says to us that the only force which maintains a system dynamic is the Gravité. If speed increases, gravity also increases. -->

In the same way, speed that the objects can maintain subjected to the gravitation must correspond, according to the theory of Newton, with the force exerted by it, i.e. mass present. It is however observed that the Galaxie S are more luminous in the center than in edge: the theory of Newton is not checked that if there exists an additional mass: the black Matière again returned to make speak about it.

One can however “observe” the black Matière: the effect of gravitational Lens — not explained by theory MOND — allows to deduce the Masse according to the equations from the General relativity: one thus realizes that the mass observed does not correspond to the predicted mass. An equivalent interpretation, used by the generalization of theory MOND, called Théorie tensor-scalar (TeVeS), is that actually it is the field of Gravitation which is modified.

These two approaches cannot be decided between by the current observations, if one admits the existence of energy sinks and of a certain form of black Matière in the MOND/TeVeS theory.

Birth of theory MOND

In 1983, the physicist and Israeli professor Mordehai Milgrom proposes a small modification of the theory of Newton. It shows that makes it possible to solve the problem of the too fast rotation of stars and the galaxies: he baptizes his theory “MOND” ( Modified Newton Dynamics , or “dynamics of Newton modified”).

However, this approach was with the entirely empirical departure. Moreover it appeared recently that it was in contradiction with certain astronomical observations. In particular, contrary to the theory of the black Matter, it did not explain the aspect of the Amas of the ball. That was solved by admitting in theory MOND the existence of a certain quantity of black matter in the form of neutrinos.

In 2004, the Physical Review of October publishes work of another Israeli researcher, Jacob Bekenstein. This one showed that theory MOND was in agreement with the Principe of relativity, just like the theory of Newton: they are thus two possible solutions with energies and the weak fields of gravity.

In May 2005, a team of the University of Oxford, directed by Constantinos Skordis, calculated the effects of gravity on the small condensations produced 300.000 years after the Big Bang by using the description of theory MOND. Their simulation evokes the cosmological diffuse Fond, observed by the satellite WMAP.

The theory MOND and black matter

WMAP could measure with a high degree of accuracy the Anisotropie S of the cosmological diffuse Fond. These measurements are in agreement with the existence of the black matter. But of the work published in 2007 by Zlosnik and his/her collaborators showed that it was possible to produce models within the theory of Bekenstein which are in also good agreement. If only one black matter particle is directly detected, theory MOND is invalidated. The physicists currently activate themselves to trace the effects predicted by this theory in the solar system. The abnormal deceleration of the probe Pioneer, whose one is unaware of still the causes, could be one about it.

The verdict between MOND and black matter could come from a perfectly reliable measurement of this effect, which would undoubtedly require a space mission dedicated to this measurement.

Theory MOND

Principles of the modification

The keystone of this theory is that the second law of Newton on the force of gravitation was checked only with high accelerations.

The second law is stated as follows:

$\backslash \; mathbf\; F\; =\; m\; \backslash \; frac\; \{\backslash \; mathrm\; d^2\; \backslash \; mathbf\; R\}\; \{\backslash \; mathrm\; dt^2\}\; =\; m\; \backslash \; mathbf\; a$

where F is the force, m the Masse and R the position (or has the Accélération).

If the force in question is the force of Gravitation, then the acceleration of an object subjected to the attraction of a body is given by the formula:

$\backslash \; mathbf\; has\; =\; G\; \backslash \; frac\; \{M\}\; \{r^2\}$

with G the constant of gravitation, M the Mass of the body which attracts and R the distance between this body and the object which one considers.

In the theory of Newton, the attraction force between two bodies decrease like the square of their distance. In the theory MOND, that is true only up to one certain threshold: beyond that, it decrease like the reverse of their distance. That makes it possible to follow the curve of rotation of stars (or of the galaxies).

Mathematical description

In 1983, Mordehai Milgrom, physicist with the Institute Weizmann in Israel, published two papers in Astrophysical Journal proposing a modification of the basic principle of dynamics of Newton. Initially, this law states that for any object of inertial mass m , subjected to a force $\backslash \; vec\; F$, has an acceleration $\backslash \; vec\; has$ checking $\backslash \; vec\; F\; =\; m\; \backslash \; vec\; has$.

This law is well-known, and was always confirmed in all the experiments of traditional physics. However, it tested forever in situations where acceleration is extremely weak, which is the case on a galactic scale: the distances are so large there that gravitational attraction is tiny.

Modification of the law of Newton

The modification suggested by Milgrom is the following one: instead of $\backslash \; vec\; F\; =\; m\; \backslash \; vec\; has$, it postulates that one a:

$\backslash \; vec\; F\; =\; Mr.\; \backslash \; driven\; \backslash !\; \backslash \; left\; (\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; right).\; \backslash \; vec\; has$, with $a\; =\; |\; \backslash \; vec\; has\; |~~$ and

$\backslash \; driven\; (X)\; =\; 1\; \backslash \; mbox\; \{if\}\; X\; \backslash \; gg\; 1$

$\backslash \; driven\; (X)\; =\; X\; \backslash \; mbox\; \{if\}\; |X|\backslash \; L\; 1$

The term a₀ being supposed to be a new constant of physics having the dimension of an acceleration.

The exact definition of µ is not specified, only is specified its behavior for the extreme values of X . Moreover, Milgrom showed that the formula of µ does not change the principal consequences of its theory, such flatness of the curve number of revolutions of the edges of the galaxies.

Traditional limit

Under the terms of the principle of correspondence, one must find the physics of Newton which one usually observes, under the conditions where it seems true.

In usual physics, has is much larger than a₀ , thus $\backslash \; driven\; \backslash \; left\; (\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; right)\; =\; 1$ and thus $\backslash \; \backslash \; vec\; F\; =\; Mr.\; \backslash \; vec\; has$. Consequently, the modification of the basic principle of dynamics is negligible and Newton could not have realized some.

Force with weak field

• In the case of an object on the edge of a galactic disc, acceleration has is much smaller than the constant a₀ because the gravitational force is very weak, therefore $\backslash \; driven\; \backslash \; left\; (\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; right)\; =\; \backslash \; frac\; \{has\}\; \{a\_0\}$ and $F\; =\; Mr.\; \backslash \; frac\; \{a^2\}\; \{a\_0\}$: the gravitational force is always the same one as in the Newtonian theory, but acceleration $\backslash \; has$ is clearly modified.

Far from the center of the galaxy, the gravitational force undergone by a star is, with a good approximation:

$F\; =\; \backslash \; frac\; \{GMm\}\; \{r^2\}$

where G is the gravitational Constante, M the mass of the galaxy, m the mass of star and R the distance between the center of the galaxy and star.

With the new law of dynamics, we have:

$F\; =\; \backslash \; frac\; \{GMm\}\; \{r^2\}\; =\; m\; \backslash \; driven\; \{\backslash \; left\; (\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; right)\}$ has from where: $\backslash \; frac\; \{GM\}\; \{r^2\}\; =\; \backslash \; driven\; \{\backslash \; left\; (\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; right)\}$

has

As the distance R is very large, has is much smaller than a₀ and thus $\backslash \; driven\; \{\backslash \; left\; (\backslash \; frac\; \{has\}\; \{a\_0\}\; \backslash \; right)\}\; =\; \backslash \; frac\; \{has\}\; \{a\_0\}$, which gives:

$\backslash \; frac\; \{GM\}\; \{r^2\}\; =\; \backslash \; frac\; \{a^2\}\; \{a\_0\}$, and as follows: $has\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{G\; M\; a\_0\}\}\; \{R\}$

As the equation giving the Accélération centrifuges on a circular orbit is $has\; =\; \backslash \; frac\; \{v^2\}\; \{R\}$ one a:

$has\; =\; \backslash \; frac\; \{v^2\}\; \{R\}\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{G\; M\; a\_0\}\}\; \{R\}$ from where the tangential speed of rotation: $v\; =\; \backslash \; sqrt\; \{G\; M\; a\_0\}$

Thus, the number of revolutions of stars at the edge of a galaxy is constant, and does not depend on the distance R : the curve number of revolutions is punt. As theory MOND was created to solve the problem of the flatness of the curve number of revolutions, there is not to be surprised to note only it agrees with the observations of this phenomenon.

Starting from the astrophysical observations, Milgrom deduced a value from its constant:

a₀ =1,2×10^{− 10}  ms^{− 2}

He noticed that this value is also: “… the acceleration which one would need, on the basis of null speed at time zero of the universe, to arrive at speed of light at time present. ”

See too

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