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A perfectly reflecting solar sail is deployed and controlled so that its normal vector is always aligned with the radius vector from the sun.

The force exerted on the spacecraft due to solar radiation pressure is

$$

\mathbf{F} = \frac{2 S A}{c}\frac{\mathbf{r}}{r^3}

$$

where $S$ is the solar intensity, $A$ is the sail area and $c$ is the speed of light.

Show that the equation of motion for the spacecraft is

$$

\ddot{\mathbf{r}} + \left(\mu - \frac{2 S A}{cm}\right)\frac{\mathbf{r}}{r^3} = 0.

$$

The equation of motion for a spacecraft in the absence of a solar sail is

$$

\ddot{\mathbf{r}} = -\frac{\mu}{r^3}\mathbf{r}.

$$

Summing the forces, we have

\begin{alignat*}{3}

\ddot{\mathbf{r}} & = & \frac{2 S A}{c}\frac{\mathbf{r}}{r^3} - \frac{\mu}{r^3}\mathbf{r}\\

\ddot{\mathbf{r}} - \frac{2 S A}{c}\frac{\mathbf{r}}{r^3} + \frac{\mu}{r^3}\mathbf{r} & = & 0\\

\ddot{\mathbf{r}} + \left(\mu - \frac{2 S A}{c}\right)\frac{\mathbf{r}}{r^3} & = & 0

\end{alignat*}