5. Let $A$ and $B$ be two moving points on the ellipse $x^{2}+3 y^{2}=1$, and $O A \perp O B$ ($O$ is the origin). Then the product of the maximum and minimum values of $|A B|$ is $\qquad$

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5. $\frac{2 \sqrt{3}}{3}$.

Establish a polar coordinate system with $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. Then the equation of the ellipse is
$$
\rho^{2}\left(\cos ^{2} \theta+3 \sin ^{2} \theta\right)=1(\theta \in[0,2 \pi)) .
$$

Let $A\left(\rho_{1}, \alpha\right), B\left(\rho_{2}, \alpha+\frac{\pi}{2}\right)$. Then
$$
\begin{array}{l}
|A B|^{2}=\rho_{1}^{2}+\rho_{2}^{2} \\
=\frac{1}{\cos ^{2} \alpha+3 \sin ^{2} \alpha}+\frac{1}{\sin ^{2} \alpha+3 \cos ^{2} \alpha} \\
=\frac{4}{3+\sin ^{2} 2 \alpha} .
\end{array}
$$

Therefore, the product of the maximum and minimum values of $|A B|$ is
$$
\frac{2 \sqrt{3}}{3} \times 1=\frac{2 \sqrt{3}}{3} \text {. }
$$