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

0.27
0.27

0.65
0.65

0.48
0.48

0.57

Answer 1

The orbital eccentricity \( e \) of an orbit can be calculated using the relationship between the closest distance (periapsis, \( r_p \)) and the farthest distance (apoapsis, \( r_a \)):

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

In your case, \( r_p = 0.27r_a \). We can express \( r_a \) in terms of \( r_p \):

Let \( r_p = r \) (the periapsis). Then, \[ r_a = \frac{r_p}{0.27} = \frac{r}{0.27} \approx 3.70r \]

Now we can substitute \( r_p \) and \( r_a \) into the formula for eccentricity:

\[ e = \frac{r_a - r_p}{r_a + r_p} = \frac{\left(\frac{r}{0.27}\right) - r}{\left(\frac{r}{0.27}\right) + r} \]

Calculating the numerator and denominator separately:

1. Numerator: \[ \frac{r}{0.27} - r = \frac{r - 0.27r}{0.27} = \frac{0.73r}{0.27} \]

2. Denominator: \[ \frac{r}{0.27} + r = \frac{r + 0.27r}{0.27} = \frac{1.27r}{0.27} \]

Now placing these into the eccentricity formula:

\[ e = \frac{\frac{0.73r}{0.27}}{\frac{1.27r}{0.27}} = \frac{0.73r}{1.27r} = \frac{0.73}{1.27} \approx 0.57 \]

Therefore, the eccentricity \( e \) of the moon's orbit is approximately 0.57.

So the correct response is:

0.57