For a moon orbiting its planet, rp

To find the orbital eccentricity \( e \) of a moon given the shortest distance (periapsis, \( r_p \)) and the longest distance (apoapsis, \( r_a \)), you can use the following formula for eccentricity:

\[
e = \frac{r_a - r_p}{r_a + r_p}
\]

From the problem, you are given that \( r_p = 0.27 r_a \).

First, let's express \( r_p \) in terms of \( r_a \):

\[
r_p = 0.27 r_a
\]

Now substitute \( r_p \) into the eccentricity formula:

\[
e = \frac{r_a - 0.27 r_a}{r_a + 0.27 r_a}
\]

This simplifies to:

\[
e = \frac{r_a(1 - 0.27)}{r_a(1 + 0.27)} = \frac{1 - 0.27}{1 + 0.27} = \frac{0.73}{1.27}
\]

Now, calculate the value:

\[
e = \frac{0.73}{1.27} \approx 0.5748016
\]

This rounds to approximately \( 0.57 \).

Thus, the moon's orbital eccentricity is approximately **0.57**.