For a moon orbiting its planet, rp (radius of perihelion) is the shortest distance between the moon and its planet, and ra (radius of aphelion) is the longest distance between the moon and its planet. What is a moon's orbital eccentricity if rp is equal to 0.27ra? (Hint: e=ra−rpra+rp

Question

, or e=ra(1−0.27)ra(1+0.27)
)
(1 point)
Responses

0.48
0.48

0.57
0.57

0.65
0.65

0.27
0.27

Answer 1

To find the orbital eccentricity $e$ of the moon, we can use the given relation and the hint provided.

From the hint, the eccentricity can be calculated using the formula: $$e=\frac{ra-rp}{ra+rp}$$

We're given that $rp=0.27ra$.

Substituting this into the formula:

1. $ra=ra$
2. $rp=0.27ra$

Now, substituting $rp$ in the equation for $e$:

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

Now, calculating this value: $$e=0.57$$

Thus, the orbital eccentricity $e$ is $0.57$.

So the correct response is: 0.57