Question

For a moon orbiting its planet, rp is the shortest distance between the moon and its planet and ra is the longest distance between the moon and its planet. What is a moon's orbital eccentricity if rp is equal to 0.27ra ?(1 point)
Responses

0.57
0.57

0.65
0.65

0.48
0.48

0.27

Answer 1

The orbital eccentricity \( e \) of an orbit can be determined from the relationship between the closest approach (periapsis, \( r_p \)) and the furthest distance (apoapsis, \( r_a \)). The formula to calculate eccentricity is:

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

Given that \( r_p = 0.27 r_a \), we can express \( r_a \) in terms of \( r_p \):

\[
r_a = \frac{r_p}{0.27}
\]

Substituting \( r_a \) into the eccentricity formula:

1. First express \( r_a \):
\[
r_a = \frac{r_p}{0.27}
\]

2. Substitute this and \( r_p \) into the eccentricity formula:
\[
e = \frac{r_a - r_p}{r_a + r_p}
= \frac{\left( \frac{r_p}{0.27} \right) - r_p}{\left( \frac{r_p}{0.27} \right) + r_p}
\]

3. Factor out \( r_p \):
\[
e = \frac{r_p \left( \frac{1}{0.27} - 1 \right)}{r_p \left( \frac{1}{0.27} + 1 \right)}
\]

4. This simplifies to:
\[
e = \frac{\frac{1}{0.27} - 1}{\frac{1}{0.27} + 1}
= \frac{\frac{1 - 0.27}{0.27}}{\frac{1 + 0.27}{0.27}}
= \frac{0.73}{1.27}
\]

5. Calculating \( \frac{0.73}{1.27} \):
\[
e \approx 0.574
\]

Therefore, the closest answer to this calculation from the provided options is:

**0.57**