Final answer:
To estimate dark matter density from rotation curves, one compares the actual rotational velocities of galaxies to the expected velocities from visible matter. Discrepancies indicate dark matter presence, which can then be estimated by deducing the excess mass required to produce the observed rotation speeds and considering the volume over which this mass would be distributed.
Step-by-step explanation:
To estimate the dark matter density in specific galaxies using rotation curve data (radius vs rotational velocity), one would analyze how these velocities differ from what is expected if only visible matter were present. Normally, in a system like our solar system, the rotation velocity decreases with increasing distance from the central mass. This is in line with Kepler's laws of planetary motion, which tell us that the velocity should decrease with the square root of the distance from the center if most of the mass is located there.
However, the rotation curves of galaxies, such as the Milky Way and Andromeda, show that stars at the outer edges rotate at similar speeds to those near the core, which implies the presence of dark matter. To quantify this, one would use the observed rotation speeds and apply Newton's law of gravitation to deduce the mass distribution that would be necessary to produce the observed curve. By comparing this with the mass of visible matter from luminous stars and dust, the difference can be attributed to dark matter. One can then estimate the density of dark matter by considering the volume over which this mass is distributed, which often extends well beyond the luminous edge of the galaxy into an extensive halo.
Other techniques, such as gravitational lensing and X-ray emissions, also support the findings from rotation curves, reinforcing the hypothesis of dark matter. The mass-to-light ratio of spiral galaxies, deduced from rotational velocities, can be used to estimate the total mass of galaxies, including dark matter.