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

0.48
0.48

0.65
0.65

0.57
0.57

0.27

Answer 1

The orbital eccentricity \( e \) of an orbiting body can be calculated using the relationship between the closest approach distance (periapsis, \( r_p \)) and the farthest distance (apoapsis, \( r_a \)). The formulas for \( r_p \) and \( r_a \) in terms of the semi-major axis \( a \) and eccentricity \( e \) are:

\[
r_p = a(1 - e)
\]
\[
r_a = a(1 + e)
\]

Given that \( r_p = 0.27 r_a \), we can substitute the expressions for \( r_p \) and \( r_a \):

\[
a(1 - e) = 0.27 \cdot a(1 + e)
\]

Since \( a \) is on both sides, it can be canceled out (assuming \( a \neq 0 \)):

\[
1 - e = 0.27(1 + e)
\]

Now, distribute \( 0.27 \):

\[
1 - e = 0.27 + 0.27e
\]

Next, rearrange the equation to isolate \( e \):

\[
1 - 0.27 = e + 0.27e
\]
\[
0.73 = e(1 + 0.27)
\]
\[
0.73 = e(1.27)
\]

Now, solve for \( e \):

\[
e = \frac{0.73}{1.27}
\]

Calculating this gives:

\[
e \approx 0.5755
\]

Rounding this to two decimal places, we find:

\[
e \approx 0.57
\]

Thus, the correct answer is:

**0.57**