 Solution

At periapsis, $r=a(1-e)=a\delta$. The grain's energy from part (ii) is nonnegative whensoThe strict inequality gives a hyperbolic Kepler orbit, while equality gives a parabolic Kepler orbit. The printed eccentricity formula yields $\beta >\delta /(2+\delta )$, which has the same requested lowest-order limit $\beta >\delta /2$.