Explanation of the velocity of the stars in the galaxies in the dynamic medium of reference (DMR) theory
Authors: Pignard Olivier 
Published: March 2022

The theory of the dynamic medium of reference (DMR) has already been presented in several articles, in particular: “Dynamic medium of reference: A new theory of gravitation” [O. Pignard, Phys. Essays 32, 422 (2019)]. The article “Theory of the dynamic medium
of reference: Exterior case and interior case” [O. Pignard, Phys. Essays 34, 280 (2021)] gives an explanation and mathematical developments of the gravitational acceleration from atomic nuclei of a massive body. The objective of this article is to explain the velocity of the stars
in galaxies within the framework of the DMR theory. The DMR theory proposes to modify the law of gravitation at long distance. The demonstration allowing to obtain the gravitational acceleration makes it possible to establish that: the gravitational acceleration generated by a massive body
of mass M one of whose dimensions is much smaller than the other two becomes <mml:math display="inline"> <mml:msub> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo>
<mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> for distances greater than a certain limit distance RL from the
massive body, and the gravitational acceleration generated by a massive body of mass M of spherical shape (a star, for example) becomes <mml:math display="inline"> <mml:msub> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi>
</mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow>
<mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:math> for distances greater than a certain limit distance RL
from the massive body. The first law of gravitation at long-distance <mml:math display="inline"> <mml:msub> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo>
<mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> makes it possible to explain a constant star rotation curve from a certain distance
from the center of the galaxy. Among 126 galaxies analyzed, this corresponds to the profile of 76 galaxies. For this, it is assumed the existence of dark matter located in the center of the galaxy in the form of a flat disk of thickness much less than its diameter. For rotating stars in this
type of galaxy, this causes that beyond the distance RL from the center of the galaxy, the velocity of the stars becomes constant and equals to <mml:math display="inline"> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mi>G</mml:mi>
<mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> </mml:msqrt> </mml:math> . The dark matter required
by the DMR theory has a mass that is only about 30% that of ordinary matter in a galaxy (stars and interstellar gas) instead of the immense quantities required by current theories. The second law of gravitation at long-distance <mml:math display="inline"> <mml:msub> <mml:mrow>
<mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:msubsup> <mml:mrow>
<mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> makes it possible to explain an increasing star rotation curve. Among
126 galaxies analyzed, this corresponds to the profile of 50 galaxies. For this type of galaxy, it is not necessary to assume the existence of dark matter, and all the stars contained in the galaxy are sufficient to explain the star rotation curve. For this type of galaxy, the velocity of
the stars increases approximately in proportion to <mml:math display="inline"> <mml:msqrt> <mml:mi>r</mml:mi> </mml:msqrt> </mml:math> . Finally, the modifications of the law of gravitation proposed by the DMR theory would also explain the observed values
of the deflection of light rays by galaxies (Einstein lenses and rings), which the modified Newtonian dynamics (MOND) theory cannot do.