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

0.27
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**