2. The graph of the quadratic function $y=x^{2}+b x+c$ intersects the positive half-axis of the $x$-axis at points $A$ and $B$, and intersects the positive half-axis of the $y$-axis at point $C$. It is known that $A B=\sqrt{3} A C, \angle C A O=30^{\circ}$. Then $c=$ $\qquad$.
2. $\frac{1}{9}$.

From the problem, we know $C(0, c), O C=c$.
Let $A\left(x_{1}, 0\right)$ and $B\left(x_{2}, 0\right)$.
Then $x_{1}$ and $x_{2}$ are the roots of the equation $x^{2}+b x+c=0$.
By the relationship between roots and coefficients, we have $x_{1} x_{2}=c$.
Also, $\angle C A O=30^{\circ}$, so
$A C=2 c, A B=\sqrt{3} A C=2 \sqrt{3} c$.
Thus, $x_{1}=O A=A C \cos 30^{\circ}=\sqrt{3} c$, $x_{2}=O B=O A+A B=3 \sqrt{3} c$.
From $x_{1} x_{2}=9 c^{2}=c$, we get $c=\frac{1}{9}$.