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

?(1 point)
Responses

0.27
0.27

0.65
0.65

0.57
0.57

0.48

Answer 1

The eccentricity $e$ of an orbit can be related to the closest and farthest distances (periapsis $R_p$ and apoapsis $R_a$, respectively) using the formula:

$$e=\frac{R_a-R_p}{R_a+R_p}$$

Given that $R_p=0.27R_a$, we can express $R_a$ in terms of $R_p$:

$$R_a=\frac{R_p}{0.27}$$

Now we can substitute $R_a$ into the eccentricity formula:

1. Substituting $R_p=0.27R_a$:

$$e=\frac{R_a-R_p}{R_a+R_p}$$

2. Substitute $R_p=0.27R_a$:

$$e=\frac{R_a-0.27R_a}{R_a+0.27R_a}$$

3. Simplifying this gives us:

$$e=\frac{R_a(1-0.27)}{R_a(1+0.27)}=\frac{1-0.27}{1+0.27}=\frac{0.73}{1.27}$$

4. Calculating this:

$$e\approx 0.5748$$

Thus, rounding gives us approximately $0.57$.

Therefore, the correct response is:

0.57