Final answer:
To find the required altitude for a satellite to have 3000 km spaced ground tracks at the equator, calculate the time for the satellite to complete an orbit, use Kepler's Third Law to find the orbit radius, and subtract Earth's radius to determine the altitude.
Step-by-step explanation:
To determine the required altitude for a satellite to achieve 3000 km spaced ground tracks at the equator, we can use the fact that the Earth is approximately 40,075 km in circumference at the equator. This means that for a satellite to have ground tracks 3000 km apart, it would need to complete one orbit in the time it takes the Earth to rotate 3000 km. As Earth rotates 360 degrees per day, this is equivalent to Earth rotating 3000 km/40075 km * 360 degrees = 26.9 degrees. Therefore, the satellite needs to complete an orbit in the time it takes for the Earth to rotate 26.9 degrees.
To calculate the actual satellite orbit period, we need to use the sidereal day, which is approximately 23 hours, 56 minutes, or 1436 minutes total. Since the Earth rotates 360 degrees during one sidereal day, the satellite must complete an orbit in (26.9/360) * 1436 minutes = 107.12 minutes.
To find the altitude, we can use Kepler's Third Law, which relates the period of an orbit to its radius. For circular orbits, the period T in minutes is related to the orbit radius r in kilometres by the formula T = 2π √(r³/GM), where GM is Earth's gravitational parameter (approximately 398,600 km³/s²). By rearranging this formula to solve for r and then subtracting the Earth's radius (approximately 6371 km) from r, we can find the satellite's altitude above the Earth's surface.
After calculating the orbit radius, we will arrive at the satellite's altitude needed to have ground tracks 3000 km apart on the equator.