Engineering >> Computer Science & EngineeringComputer Modeling/Simulation: Satellite Orbitby Seth Kenney
Submitted : Spring 2011 A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system. Computer simulations have become a useful part of mathematical modeling of many natural systems in physics (computational physics), astrophysics, chemistry and biology, human systems in economics, psychology, and social science and in the process of engineering new technology, to gain insight into the operation of those. Computer simulations vary from computer programs that run a few minutes, to networkbased groups of computers running for hours, to ongoing simulations that run for days. The scale of events being simulated by computer simulations has far exceeded anything possible (or perhaps even imaginable) using the traditional paperandpencil mathematical modeling. Computer modeling is also used heavily in architecture. Computer modeling/simulations are used to find techniques to make structures more durable and stable from natural calamities like earthquakes and artificial ones like a bomb explosion. Satellites in orbit around the earth are commonplace. They are used to transmit television signals, telephone calls, weather data, airplane navigational information, and radio programs. And, many of today’s cars are equipped with GPS (global positioning satellite) locators that help them get around in traffic. Manned satellites, such as the Space Shuttle, allow for all types of valuable experimentation and space repair work. The motion of objects is governed by Newton's laws. The same simple laws that govern the motion of objects on earth also extend to the heavens to govern the motion of planets, moons, and other satellites. Consider a satellite with mass M_{sat} orbiting a central body with a mass of mass M_{Central}. The central body could be a planet, the sun or other large mass capable of causing sufficient acceleration on a less massive nearby object. Part A: Using the gravitational constant (G = 6.673 x 10^{11} Nm^{2}/kg^{2}), M_{Central} and radius of orbit for the satellite (R), derive the equation for the velocity of a satellite (v) moving about a central body in circular motion. Part B: Derive the equation for the acceleration of a satellite (a) in circular motion about a central body. Part C: Using Newton's form of Kepler's third law, derive the period of the satellite (T) in circular motion about a central body. Part D: A satellite is going to orbit the earth at a height of 150 km above the surface of the earth. Determine the speed, acceleration and orbital period of the satellite. (Given: M_{earth} = 5.98 x 10^{24} kg, R_{earth} = 6.37 x 10^{6} m) A geosynchronous satellite is a satellite that orbits the earth with an orbital period of 24 hours, thus matching the period of the earth's rotational motion. A special class of geosynchronous satellites is a geostationary satellite. A geostationary satellite orbits the earth in 24 hours along an orbital path that is parallel to an imaginary plane drawn through the Earth's equator. Such a satellite appears permanently fixed above the same location on the Earth. Part E: If a geostationary satellite wishes to orbit the earth in 23h 56m, then how high above the earth's surface must it be located? Part F: Consider a satellite which is in a low orbit about the Earth at an altitude of 200 km above Earth's surface. Determine the speed, acceleration and orbital period of the satellite. Part G: The period of the moon is approximately 27.2 days (2.35 x 10^{6} s). Determine the radius of the moon's orbit and the orbital speed of the moon.
