Relationship between orbital radius, mass, and orbital velocity

Question

I have a question regarding the relationship between a body's orbital radius, mass, and orbital velocity.

I understand that there is the equation $V = \sqrt{\frac{GM}{R}}$, where $V$ is the orbital velocity, $G$ is the gravitational constant, and $R$ is the orbital radius. Does the equation imply that the orbital velocity is inversely proportional to the square root of the orbital radius? So, can it be accurately stated that the orbital velocity decreases as the orbital radius increases?

However, this article states that "In general, the speed with which stars orbit the centre of their galaxy is independent of their separation from the centre; indeed, orbital velocity is either constant or increases slightly with distance rather than dropping off as expected."

These observations seem contradictory. Can someone please help clear up the misinterpretation?

Answer 1

Your understanding that the orbital velocity decreases as the radius increases is correct. Yet, as the article states, we see that orbiting stars seem to have a uniform speed. The resolution comes from the fact that the $M$ in the formula refers to the mass enclosed by the orbit (really the $M$ refers to a point mass, but an object a distance $R$ from the center of a spherically symmetric mass distribution feels the gravitational force that would result if all the mass inside the radius $R$ had been concentrated at the origin and the outside mass was removed completely).

For bigger orbits, $R$ is bigger as you noted, but also the orbits enclose more mass (mostly dark matter, as the article says), and so the orbital speed stays more or less constant. In fact it is precisely from this reasoning that the existence of dark matter was deduced.

Answer 2

The article states:

To account for this, the mass of the galaxy within the orbit of the stars must increase linearly with the distance of the stars from the galaxy’s centre.

So the mass inside the orbit of radius $R$ is $M = M_oR$ where $M_o$ is a constant.

$\Rightarrow V = \sqrt{\dfrac{GM}{R}}= \sqrt{\dfrac{GM_oR}{R}} = \sqrt{GM_o} = \rm constant$