# Prove acceleration in orbit with newtons second law

I want to prove that the acceleration in a orbit at a given point r=(x,y) is $$a=-\frac{GM}{R^3}r$$ (My professor said this can be proven by newtons second law, but he never explained in detail how). I know that N2 gives $$F=ma$$ substituted into $$F=\frac{GMm}{R^2}r$$ gives $$a=\frac{GM}{R^2}r$$ (Where r is the positional vector of the orbiting point). How can you prove $$a=-\frac{GM}{R^3}r$$ ?

Gravitational force is attractive, so its direction is opposite to the radial vector, which points outward. And its magnitude is $$\frac{GMm}{r^2}$$.
So $$\vec{F}=-\frac{GMm}{r^2}\hat{r}= -\frac{GMm}{r^3}\vec{r}$$, where $$\hat{r}=\frac{\vec{r}}{r}$$ is the unit vector joining the two point masses $$M$$ and $$m$$.