Question

For a moon orbiting its planet RP is the shortest distance between the Moon and its planet are a is the longest distance between the Moon and the planet what is the Moon's orbital eccentricle if RP is equal to 0.27ra

Answer 1

To find the orbital eccentricity (\(e\)) of a moon orbiting its planet, we can use the relationship between the shortest distance (periapsis) and the longest distance (apoapsis) in an elliptical orbit.

Let:
- \(r_p\) = periapsis (shortest distance)
- \(r_a\) = apoapsis (longest distance)
- \(P\) = distance between the perapsis and apoapsis (\(P = r_a - r_p\))
- \(a\) = semi-major axis

Given that \(RP = 0.27r_a\):
- \(r_p = r_a - 0.27r_a = 0.73r_a\)

The semi-major axis \(a\) is the average of \(r_p\) and \(r_a\):
\[
a = \frac{r_p + r_a}{2}
\]

Substituting for \(r_p\):
\[
a = \frac{0.73r_a + r_a}{2} = \frac{1.73r_a}{2} = 0.865r_a
\]

The eccentricity \(e\) can be calculated using the formula:
\[
e = \frac{r_a - r_p}{r_a + r_p}
\]

Substituting the known values:
\[
e = \frac{r_a - 0.73r_a}{r_a + 0.73r_a} = \frac{0.27r_a}{1.73r_a} = \frac{0.27}{1.73}
\]

Now, calculate \(e\):
\[
e \approx 0.156
\]

Thus, the orbital eccentricity of the moon is approximately **0.156**.