7. If the ellipse $W: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has its two foci and one endpoint of the minor axis forming the three vertices of an equilateral triangle, and the shortest distance from the foci of the ellipse $W$ to the ellipse is $\sqrt{3}$, then the equation of $W$ is $\qquad$ .
According to the problem, $\left\{\begin{array}{l}b=\sqrt{3} c, \\ a-c=\sqrt{3}\end{array} \Rightarrow\left\{\begin{array}{l}a=2 \sqrt{3}, \\ b=3 .\end{array}\right.\right.$ Therefore, the equation of $W$ is $\frac{x^{2}}{12}+\frac{y^{2}}{9}=1$.