jmc

First, we solve for $b=g(a)$.

$$\begin{array}{rl}f(g(a))& =8\\ f(b)& =8\\ b^2+8& =8\\ b^2& =0\end{array}$$

Therefore, $b=g(a)=0.$ Now, we solve for $a$ using $g(a)=a^2-4=0.$ From this, $a=\pm 2.$ Since we are given that $a>0$, our answer is $a=\boxed{2}.$