A satellite orbits Earth in an elliptical path where the Earth is at one focus. The semi-major axis is 15,000 km and the eccentricity is 0.2. What is the distance from the satellite to the Earth’s center at the closest approach (perigee)?
["Understanding Satellite Orbits: Calculating the Perigee Distance for an Elliptical Path", "Satellites play a vital role in modern communication, navigation, weather forecasting, and scientific research—most orbit Earth in elliptical paths rather than perfect circles. One key feature of these orbits is that Earth occupies one of the two foci of the ellipse. For practical orbit calculations, understanding the satellite’s closest approach—known as the perigee—is essential.", "### What Is an Elliptical Orbit?", "An elliptical orbit is defined by two main parameters:\n- Semi-major axis (a): Half the longest diameter of the ellipse, determining the orbit’s size.\n- Eccentricity (e): A measure of how elongated the ellipse is, ranging from 0 (circular) to values approaching 1 (highly elongated).", "When Earth is located at one focus of this ellipse, the satellite’s distance from Earth varies between two extremes: the perigee (closest point) and the apogee (farthest point).", "### Given Orbital Parameters", "- Semi-major axis ( a = 15,000 , \ ext{km} )\n- Eccentricity ( e = 0.2 )", "### Perigee Distance Formula", "The perigee is the closest distance between the satellite and Earth’s center and is calculated using the formula:", "[\n\ ext{Perigee distance} = a \ imes (1 - e)\n]", "Substituting the known values:", "[\n\ ext{Perigee distance} = 15,000 \ imes (1 - 0.2) = 15,000 \ imes 0.8 = 12,000 , \ ext{km}\n]", "### Meaning and Implications", "At perigee, the satellite approaches Earth to within 12,000 kilometers of its center. This makes it crucial for missions requiring detailed observation or high-resolution imaging. At apogee, the satellite reaches up to:", "[\n\ ext{Apogee distance} = a \ imes (1 + e) = 15,000 \ imes 1.2 = 18,000 , \ ext{km}\n]", "Understanding these distances helps engineers optimize mission design, fuel usage, and communication timing.", "### Conclusion", "In elliptical satellite orbits, precise knowledge of perigee distance—calculated simply from the semi-major axis and eccentricity—supports accurate navigation, rendezvous planning, and scientific measurements. For a satellite with a 15,000 km semi-major axis and eccentricity 0.2, the closest approach to Earth’s center is 12,000 km—a critical parameter for satellite operations worldwide.", "---", "Keywords: elliptical orbit, perigee distance, satellite orbit, Earth satellite, semi-major axis, eccentricity 0.2, orbital mechanics, perigee calculation"]
