2. The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, then the eccentricity of the hyperbola is $\qquad$
2. $\sqrt{3}$.

Let the equation of the hyperbola be
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0) \text {, }
$$

with the semi-focal distance being $c$.
Then $c^{2}=a^{2}+b^{2}$.
By $\left|F_{1} A\right|-\left|F_{2} A\right|=\left|F_{1} B\right|-\left|F_{2} B\right|=2 a$,
$$
\left|F_{1} A\right|=\left|F_{2} A\right|+\left|F_{2} B\right|=\left|F_{1} B\right|,
$$

we solve to get $\left|F_{2} A\right|=\left|F_{2} B\right|=2 a$.
This indicates that $A B \perp x$ axis.
Thus, $\left|F_{2} A\right|=\left|F_{2} B\right|=\frac{b^{2}}{a}$.
Combining $c^{2}=a^{2}+b^{2}$, we solve to get the eccentricity of the hyperbola as $e=\frac{c}{a}=\sqrt{3}$.