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

Question

?(1 point)
Responses

0.27
0.27

0.65
0.65

0.57
0.57

0.48

Answer 1

The eccentricity \( e \) of an orbit can be related to the closest and farthest distances (periapsis \( R_p \) and apoapsis \( R_a \), respectively) using the formula:

\[ e = \frac{R_a - R_p}{R_a + R_p} \]

Given that \( R_p = 0.27 R_a \), we can express \( R_a \) in terms of \( R_p \):

\[ R_a = \frac{R_p}{0.27} \]

Now we can substitute \( R_a \) into the eccentricity formula:

1. Substituting \( R_p = 0.27 R_a \):

\[ e = \frac{R_a - R_p}{R_a + R_p} \]

2. Substitute \( R_p = 0.27 R_a \):

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

3. Simplifying this gives us:

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

4. Calculating this:

\[ e \approx 0.5748 \]

Thus, rounding gives us approximately \( 0.57 \).

Therefore, the correct response is:

0.57