Solve H_{O}=4E/pia_r^2text{se}a_{r}>d_{p}

$$H _ { O } = \frac { 4 E } { \pi a _ { r } ^ { 2 } } \text { se } a _ { r } > d _ { p }$$

Solve for d_p

$$d_{p}<a_{r}$$

$$H_{O}=\frac{4E}{\pi a_{r}^{2}}\text{ and }a_{r}\neq 0$$

Solve for a_r

$$\left\{\begin{matrix}a_{r}>d_{p}\text{, }&d_{p}\geq 0\text{ and }H_{O}=0\text{ and }E=0\\a_{r}\in \left(d_{p},0\right)\cup \left(0,\infty\right)\text{, }&d_{p}<0\text{ and }H_{O}=0\text{ and }E=0\\a_{r}=-2\sqrt{\frac{E}{\pi H_{O}}}\text{, }&H_{O}>0\text{ and }E>0\text{ and }H_{O}>\frac{4E}{\pi d_{p}^{2}}\text{ and }d_{p}<0\\a_{r}=2\sqrt{\frac{E}{\pi H_{O}}}\text{, }&H_{O}>0\text{ and }E>0\text{ and }\left(d_{p}\leq 0\text{ or }H_{O}<\frac{4E}{\pi d_{p}^{2}}\right)\\a_{r}=-2\sqrt{\frac{E}{\pi H_{O}}}\text{, }&H_{O}<0\text{ and }E<0\text{ and }H_{O}<\frac{4E}{\pi d_{p}^{2}}\text{ and }d_{p}<0\\a_{r}=2\sqrt{\frac{E}{\pi H_{O}}}\text{, }&H_{O}<0\text{ and }E<0\text{ and }\left(H_{O}>\frac{4E}{\pi d_{p}^{2}}\text{ or }d_{p}\leq 0\right)\end{matrix}\right.$$