Circular motion

Since many planetary orbits are nearly circular, it is useful to obtain the condition for a circular orbit. In this case, the magnitude of the acceleration ${\mathbf a}$ is related to the radius by

$$a=\frac{v^2}{r}$$

where $v$ is the speed of the object. The acceleration is always directed toward the center. Hence

$$\frac{mv^2}{r}=\frac{mMG}{r^2}$$

or

$$v=(\frac{MG}{r})^{1/2}.$$ (32)

This is a general condition for the circular orbit. We can also find the dependence of the period $T$ on the radius of a circular orbit. Using the relation

$$T=\frac{2\pi r}{v},$$

we obtain

$$T^2=\frac{4\pi^2r^3}{GM}$$