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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 $a=-\frac{GM}{R^3}r$ ?

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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 $m$.

And Newton's second law says force is mass times acceleration.

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