For a moon orbiting its planet, rp

Question

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

0.48
0.48

0.27
0.27

0.65
0.65

0.57

Answer 1

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.