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

Responses

0.48
0.48 - no response given

0.27
0.27 - no response given

0.65
0.65 - no response given

0.57

Answer 1

The orbital eccentricity \( e \) of an orbit can be determined using the closest (perigee or periapsis) and farthest (apogee or apoapsis) distances from the central body. The formula for eccentricity in terms of \( r_p \) (the closest distance) and \( r_a \) (the farthest distance) is:

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

In this case, we know that \( r_p = 0.27 r_a \). Let's denote \( r_a \) as \( r_a \) and \( r_p \) can be expressed in terms of \( r_a \):

\[ r_p = 0.27 r_a \]

Now we can plug this into the formula for eccentricity:

1. Calculate \( r_a - r_p \): \[ r_a - r_p = r_a - 0.27 r_a = 0.73 r_a \]

2. Calculate \( r_a + r_p \): \[ r_a + r_p = r_a + 0.27 r_a = 1.27 r_a \]

3. Using these values in the eccentricity formula: \[ e = \frac{0.73 r_a}{1.27 r_a} \] The \( r_a \) terms will cancel out: \[ e = \frac{0.73}{1.27} \]

4. Now, calculate that value: \[ e \approx 0.5748 \]

This approximates to 0.57 when rounding.

So, the answer is 0.57.