Explanation:
In an elliptical orbit, the distances from the center of the ellipse to the foci are denoted by a (semi-major axis) and c (distance from the center to the focus). The value of b (semi-minor axis) is related to a and c by the equation:
[tex]b = \sqrt{ {a}^{2} - {c}^{2} } [/tex]
. The eccentricity e of the ellipse is defined as:
[tex]e = \frac{c}{a}[/tex]
Given that the perigee is the closest distance to Earth and the apogee is the farthest distance, we can deduce that a is half the sum of the perigee and apogee distances:
[tex]\[a = \frac{221463 + 252710}{2} = \frac{474173}{2} = 237086.5\text{ miles}\][/tex]
The distance from the center of the ellipse to one focus c is half the difference between the apogee and perigee distances:
[tex]\[c = \frac{252710 - 221463}{2} = \frac{31397}{2} = 15698.5\text{ miles}\][/tex]
Using the formula for b, we get:
[tex]\[b = \sqrt{237086.5^2 - 15698.5^2} \approx \sqrt{56103928503.25 - 246393052.25} \approx \sqrt{55857535451} \approx 236316.69\text{ miles}\][/tex]
The eccentricity e can be calculated as:
[tex]\[e = \frac{15698.5}{237086.5} \approx \frac{15698.5}{237086.5} \approx 0.066\][/tex]
