2. Orbits
Topics Orbit types Kepler and Newton laws Coverage area Influence of Earth 1
Orbit types According to inclination angle Equatorial Polar Inclinational orbit According to shape Circular orbit Ellipsoid orbit 2
Orbit types Geosynchronous orbit Satellites in low orbits around Earth at height of 1000 km have period of 2 hours. Moon circles around Earth at height of 380000 km and has period of 27 days. Satellite at height of 35786,6 km has period of app. 24 hours or 1 day. This orbit is called geosynchronous orbit. Satellite in geosynchronous orbit circles the Earth with the same angular speed of Earth rotation. 3
Orbit types Geosynchronous orbit The circumference (2rπ) of the Earth at the equator is app. 40000 km. The Earth rotates in about 24 hours. If a person were to hang above the surface of the Earth at the equator without moving, he would see 40000 km pass by in 24 hours, at a speed of 40000/24 or app. 1670 km per hour. To see how fast is Earth is moving at Zagreb, the above value must be multiplied with cosine of Zagreb latitude (45 degrees). 1600 x cos (45) = 1130 km/h is the speed of Earth rotation at Zagreb 4
Orbit types Geosynchronous orbit Earth is also moving around the Sun. Earth's average distance to the Sun is 150 mil. km. The distance the Earth travels in one year is 2rπ =9,42 x10 5 km or 107 000 km/hours (30 km/s). 5
Orbit types Circular orbits LEO - Low Earth Orbit MEO Medium Earth Orbit GEO Geostationary Earth Orbit Geostationary orbit is a geosinchronous orbit with inclination angle equal to 0 degrees. 6
Orbit types Geostationary orbit (GEO) GEO satellite appears motionless to a observer from Earth fixed antennas Watching from a certain point on Earth, all GEO satellites have same elevation angle and different azimuth angle. 7
Orbit types LEO MEO GEO Height (km) 1000 5000 5000-20000 35786,6 Period (hr) 2-4 4-20 24 Advantage Small delay (20 ms) Low transmitter power Smaller loss than GEO Smaller Doppler than LEO Moderate launch expenses Drawback Large Doppler Delay 150 ms Many satellites required Larger losses than LEO Small coverage Higher transmitter power Application Remote sensing Navigation Satellite photos Voice and data communication Fixed satellite Large coverage No Doppler Large Delay High losses Poles not covered Broadcasting VSAT systems High data speed 8
Kepler and Newton laws Mathemathical description of satellite orbits Kepler laws of planetary motion Newton laws of motion and gravity German astronomer Johannes Kepler (1571.-1630.) English physicist and astronomer Isaac Newton (1643.-1728.) 9
Kepler and Newton laws 1. Kepler law The orbit of every planet is an ellipse with the Sun at one of the two focuses. 10
Kepler and Newton laws Solar system planet trajectories 11
Kepler and Newton laws 2. Kepler law The vector from the Sun to the planet sweeps equal areas in equal times. 12
Kepler and Newton laws 3. Kepler law T 2 3 The ratio of the square of the period T of revolution of a planet around the Sun to the cube of the semi-major axis of the ellipse is the same for all planets. Example 1 : Planet Mars travels around the Sun 687 days. = R What is its average distance from the Sun? Distance from Earth to Sun is app. 150 mil. km (1 astronomical unit) Mars year is 687/365,25 = 1,88 Earth years. Since T 2 = R 3, it follows that (1,88) 2 = R 3 or further 3,5344=R 3 It follows that the distance from Mars to Sun is app. 1,52 astronomical units or 228 mil km. 13
Kepler and Newton laws 1. Newton law Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to. 2. Newton law Force is equal to the change in momentum (mv) per change in time. For a constant mass, force equals mass times acceleration. dmv ( ) F = = m a dt 3. Newton law The mutual forces of action and reaction between two bodies are equal. 14
Kepler and Newton laws Newton law of gravitation Two bodies of mass m 1 and m 2 attract each other with a force proportional to their masses and inversely proportional to the square of the distance r between them: m m F = F = F = G r G = m [kg] F [N] r [m] 1 2 1 2 2-11 2-2 6,673 10 Nm kg. G gravitational constant 15
Kepler and Newton laws Example 2: V-2 Rocket V-2 rocket weighs around 12 tons with fuel and 3 tons empty. Rocket thrust is 240 000 N. If g = 10 m/s 2, what is the rocket acceleration at launch and when there is no more fuel in rocket? Force up must be above 12 000*10 = 120 000 N; otherwise the rocket would not leave Earth. Total force up is: F = +240 000 N 120 000 N = 120 000 N Acceleration at the time of launching is a = F/m = 120 000 N/12 000 kg = 10 m/s 2 = 1 g As the fuel is used, the mass gets smaller, but the force remains the same, resulting in a higher acceleration. When the fuel is spent F = +240 000 N 30 000 N = 210 000 N Acceleration is at that time a = F/m = 210 000 N/3 000 kg = 70 m/s 2 = 7 g Increased acceleration affects the people traveling into space. Multiple rocket stage lowers the acceleration. 16
Coverage area Satellites at larger distance from the surface of Earth cover larger areas, but also require higher transmitter power. GEO satellites have much larger coverage area than LEO satellites. 3 GEO satellites cover the entire Earth (except poles). 17
Coverage area S 2 = 2π RE 1 h E R + E R E 18
Coverage area Global coverage of a GEO satellite located at 105 degrees W according to the minimum elevation angle value 19
Coverage area Receive antenna size necessary depending on the geographical position 20
Influence of Earth Inhomogeneity of gravitational field Earth is not a perfect sphere (flattened at poles). Results in sliding of a satellite to east and west P and P position of observer on Earth Z zenith distance measured by using elipsoid model of Earth Z - zenith distance measured by using sphere model of Earth 21
Influence of Earth Atmosphere drag Below height of 1000 km, atmosphere slows the satellite Earth magnetic fields 22
Influence of Earth Van Allen belts are radiation belts with charged particles (protons and electrons) attracted by Earth magnetic fields. Inner belt has high energy protons (50 MeV) LEO satellites Outer belt has both electrons and protons having energy from 1 to 100 kev MEO satellites. Electrons in inner belt create polar lights. Van Allen belts demand satellite and space station shielding. 23