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

0.57
0.57

0.65
0.65

0.27

Answer 1

To find the orbital eccentricity (e) of a moon given the relationship \( r_p = 0.27 r_a \), we can use the formula that relates the periapsis (rp), apoapsis (ra), and eccentricity:

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

First, let's express \( r_p \) in terms of \( r_a \): \[ r_p = 0.27 r_a \]

Now we can substitute this into the equation for eccentricity:

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

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

3. Now substitute these values back into the formula for eccentricity: \[ e = \frac{r_a - r_p}{r_a + r_p} = \frac{0.73 r_a}{1.27 r_a} = \frac{0.73}{1.27} \]

Now, calculate \( \frac{0.73}{1.27} \): \[ e \approx 0.575 \]

This is approximately \( 0.57 \) when rounded to two decimal places.

Therefore, the orbital eccentricity of the moon is 0.57.