In the lecture it was shown that the speed of a body on a circular orbit decreased as a function of distance from the object it orbits (e.g. the Sun) as v(r)∝1/r√. This followed from Newton's gravitational law
F=GmMr2
and that for circular motion ma=mv2/r. It was discussed how it was discovered that galactic rotation curves are flat, i.e. v(r)∝constant instead of v(r)∝1/r√, and this is evidence for dark matter. What is the distribution of dark matter that would lead to such an effect? Let the density of a dark matter halo decrease with r as a power law:
ρ(r)=arn,a is a constant
What is the power n that leads to a flat rotation curve? To do this, you can replace the mass that the body orbits M with the total mass within the orbit M(r):
F=GmM(r)r2
To find the total mass M(r) within a given radius r, you simply add up all the mass within the volume of a sphere of radius r. This "adding up'' is done by taking an integral of the density within the volume M(r)=∫ρ(r)dV.
Answer for Question 9