Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the $\operatorname{Sun}$ is $\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\beta<1$, show that the time the comet spends within the orbit of Earth is given by
$$
\sqrt{2(1-\beta)} \cdot(1+2 \beta) / 3 \pi \times 1 \text { year }
$$
If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?
76