Designing a Satellite Orbit?

📌 CONCEPT: The problem of designing a satellite orbit involves finding the maximum and minimum distance from the satellite to the center of the Earth, given the altitude of the periapsis above the Earth's surface.

📐 RULE / FORMULA: The key principle involved is the definition of an elliptical orbit, with the sum of the distances from any point on the ellipse to the two foci being constant.

💡 WORKED EXAMPLE: To find the maximum and minimum distance from the satellite to the center of the Earth, let's consider the given altitude of the periapsis at 200 km above the Earth's surface. The Earth's radius is 6371 km, so the distance from the center of the Earth to the periapsis is 6371 + 200 = 6571 km. Using the property of an ellipse, the sum of the distances from the satellite to the two foci is constant, and the maximum distance will be at the apoapsis, which is 2 * (6571 - 6371) = 400 km more than the Earth's radius, i.e., 8001 km.

⚠️ COMMON MISTAKE: Students often get confused between the closest and farthest distance from the Earth's surface and the distance from the satellite to the center of the Earth. They should remember that the distance from the satellite to the center of the Earth is always greater than the distance from the satellite to the Earth's surface.