Solution
= Solution
At <periapsis>, $r=a(1-e)=a\delta$. The grain's energy from part (ii) is nonnegative when
$$
-\frac1{2a}+\frac\beta r\geq0,
$$
so
$$
\boxed{\beta\geq\frac\delta2}.
$$
The 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 $\beta>\delta/2$.