Solution
= Solution
Average the phase probability over the <thermal eccentricity distribution>:
$$
f_{\rm unbound}
=\int_0^1 2e\left(\frac12-\frac e\pi\right)de.
$$
Therefore
$$
\boxed{
f_{\rm unbound}=\frac12-\frac{2}{3\pi}
\simeq0.288}.
$$
= Solution
Average the phase probability over the <thermal eccentricity distribution>:
$$
f_{\rm unbound}
=\int_0^1 2e\left(\frac12-\frac e\pi\right)de.
$$
Therefore
$$
\boxed{
f_{\rm unbound}=\frac12-\frac{2}{3\pi}
\simeq0.288}.
$$