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

0.57
0.57

0.27
0.27

0.48
0.48

0.65

1 answer

To find the orbital eccentricity \(e\) of a moon given that \(rp\) (the periapsis distance) is equal to \(0.27ra\) (the apoapsis distance), we can use the formula for eccentricity in terms of periapsis and apoapsis distances:

\[ e = \frac{ra - rp}{ra + rp} \]

From the problem, we know that:

\[ rp = 0.27ra \]

Now, we can express \(ra\) in terms of \(rp\):

Let \(rp = x\), then \(ra = \frac{rp}{0.27} = \frac{x}{0.27}\).

Substituting these into the equation for eccentricity:

\[ e = \frac{ \frac{x}{0.27} - x }{ \frac{x}{0.27} + x } \]

Next, we can factor \(x\) out:

\[ e = \frac{ \frac{x - 0.27x}{0.27} }{ \frac{x + 0.27x}{0.27} } \]

Simplifying further:

\[ e = \frac{ 1 - 0.27 }{ 1 + 0.27 } \]

Calculating the numerator and denominator gives:

\[ e = \frac{0.73}{1.27} \]

Now, calculating the numerical value:

\[ e \approx \frac{0.73}{1.27} \approx 0.574 \]

Since we can round this value, the closest option given is:

0.57.