Observational Evidence of the Dark matter Existence
Problem 1
Analyze the problem of a small satellite galaxy moving in the gravitational field of a large galaxy (the problem of non-decreasing behavior of the rotational curves.)
Many galaxies have several small satellite galaxies. For the latter the second Newton law reads: \[G\frac{mM(r)}{r^{2} } =\frac{mv^2}{r} \] For the rotation velocity one then obtains \[v(r)=\sqrt{\frac{GM(r)}{r} } \] where $M(r)$ is the total mass of the matter inside the sphere of radius $r$. In the spherically symmetric case the effect of the outside mass vanishes, and it follows that \[M(r)=\bar{\rho }\frac{4}{3} \pi r^{3} \Rightarrow v(r)=\sqrt{\frac{4}{3} \pi G\bar{\rho }r^{2} } \sim r,\] where $\bar{\rho }$ is the distribution of the mass $M.$ Thus linear growth of the rotation velocity with the galactocentric radius is expected in the inner part of the galaxy. In the outer region of the galaxy the dependence of the rotation velocity on the radius corresponds to the case of point mass in the center of the galaxy: \[v(r)\sim 1/\sqrt{r} \] The rotation velocity $v(r)$ can be in particular determined by the measured Doppler shift in the emission spectrum of the corresponding satellite galaxies. The behavior of the observed rotation curves measured for some thousands galaxies does not show the expected decrease $v(r)$ with radius growing. A pronounced approximate constance of $v(r)$ is observed instead for large radii (See Figure). This fact implies that the mass $M(r)$ must increase with the radius growing: $M(r)\sim r$. Then one can conclude that mass of the central is far from being exhausted by the one contained in the visible volume: it is distributed instead in much larger region. Unlike the stars, the latter additional mass does not emit light (it is invisible on pictures), but creates considerable gravity force, additional to that of the visible stars. That is how the concept of the hidden mass originates.
