For a moon orbiting its planet, rp

To find the orbital eccentricity (e) of a moon given the relationship \( r_p = 0.27 r_a \), we can use the formula that relates the periapsis (rp), apoapsis (ra), and eccentricity:

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

First, let's express \( r_p \) in terms of \( r_a \):
\[
r_p = 0.27 r_a
\]

Now we can substitute this into the equation for eccentricity:

1. Calculate \( r_a + r_p \):
\[
r_a + r_p = r_a + 0.27 r_a = 1.27 r_a
\]

2. Calculate \( r_a - r_p \):
\[
r_a - r_p = r_a - 0.27 r_a = 0.73 r_a
\]

3. Now substitute these values back into the formula for eccentricity:
\[
e = \frac{r_a - r_p}{r_a + r_p} = \frac{0.73 r_a}{1.27 r_a} = \frac{0.73}{1.27}
\]

Now, calculate \( \frac{0.73}{1.27} \):
\[
e \approx 0.575
\]

This is approximately \( 0.57 \) when rounded to two decimal places.

Therefore, the orbital eccentricity of the moon is **0.57**.