Question

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 moons orbital eccentricity if rp is equal to 0.27 ra






























































































dont get this wrong

Answer 1

To calculate the orbital eccentricity \( e \) of a moon, you can use the relationship between the shortest distance \( r_p \) (perigee) and the longest distance \( r_a \) (apogee) from the planet. The formulas to find the eccentricity in terms of the perigee and apogee are as follows:

1. The semi-major axis \( a \) is given by:
\[
a = \frac{r_p + r_a}{2}
\]

2. The eccentricity \( e \) can be calculated using:
\[
e = \frac{r_a - r_p}{r_a + r_p}
\]

Given that \( r_p = 0.27 r_a \), we can express \( r_a \) in terms of \( r_p \):
\[
r_a = \frac{r_p}{0.27}
\]

Now, plug this into the formulas:

### Step 1: Calculate \( r_a \)
Let \( r_p = x \), then:
\[
r_a = \frac{x}{0.27} \approx 3.7037x
\]

### Step 2: Calculate \( a \)
Now calculate the semi-major axis \( a \):
\[
a = \frac{r_p + r_a}{2} = \frac{x + 3.7037x}{2} = \frac{4.7037x}{2} = 2.35185x
\]

### Step 3: Calculate \( e \)
Now plug \( r_a \) and \( r_p \) into the eccentricity formula:
\[
e = \frac{r_a - r_p}{r_a + r_p}
\]
Substituting for \( r_a \):
\[
e = \frac{\frac{x}{0.27} - x}{\frac{x}{0.27} + x}
\]

Now simplify:
\[
e = \frac{\frac{x - 0.27x}{0.27}}{\frac{x + 0.27x}{0.27}} = \frac{(1 - 0.27)x}{(1 + 0.27)x} = \frac{0.73}{1.27}
\]

### Step 4: Calculate the final value
Now calculate the value:
\[
e = \frac{0.73}{1.27} \approx 0.5748
\]

Thus, the eccentricity \( e \) of the moon's orbit is approximately:
\[
\boxed{0.5748}
\]