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