How the Milky Way Rotates
"Everything must be made as simple as possible. But not simpler."
~ Albert Einstein[7]
In order to show that dark matter must exist in our galaxy, the Milky Way Galaxy, we are going to need to find a relation for how the mass and orbits are related. The shape of the galaxy is essentially a disk where the height of the disk is much much smaller than the width of the disk, meaning we can consider all of the stars to be in the same plane (we can ignore height when calculating the positions of stars). Our other main assumption is that the stars orbit in circular orbits around the galactic center as this makes the math much simpler and we have no practical way of guessing the eccentricity of galactic orbits.
In order for an star to have circular motion, the forces acting on the star must be equal to the centripetal acceleration required to keep it in that orbit, which we know is
where F is a force acting on the star, m is the mass of the star, V is the linear velocity of the star, R is the radius of the star's orbit, and ê is the radial direction.
In a galactic system, the dominant force acting is gravity. From Newton's Law of Universal Gravitation or Gauss' Law for Gravity, we know that the gravitational force is
where g is the gravitational acceleration, G is the gravitational constant, and M is the mass interior to the orbit. When we combine these two equations together, we are left with
Now we can see that the mass interior to a star's orbit is only dependent on its radius and velocity. Te are going to need to find these two things from our data.