Question

numbers, the other with the equations. Put a box around the answer to each part. 1. An object's circular orbital velocity vc around the earth can be calculated using the equation ", where ve is the circular orbital velocity, μ-3.98601 E5 km3/s, is the earth's gravitational parameter, and r is the distance of the object from the earth's center. Some commercial satellites, like DirecTV, are in an orbit where they take one sidereal day (23h 56m 4.09s) to orbit the earth. This makes it so that the satellite appears to be stationary to an observer on the earth's surface, which also takes one sidereal day to rotate about its axis That is a good situation because each receiving antenna only needs to be aimed once, which is a lot easier than tracking a satellite. This type of orbit is called "geosynchronous" (since it is synchronized with geo-earth) and is abbreviated GEO, for Geosynchronous Earth Orbit. (a) Using ONLY the circular orbital velocity equation given above AND knowledge about the circumference of a circle, calculate the required GEO orbital radius rceo, in km.(b) For this to work, what angle must the orbit have with respect to the equator? (c) Calculate the GEO satellite's circular orbital velocity y^-ceo around the earth, in km/s (d) Explain why a sidereal day is not exactly 24h 0m 0s

Answer 1

(a) given mu = 3.98601 times 10^5 km^3/s^2 [gravitational parameter] Time period T = 23 hrs 56 m 4.09 s = (23 times 60 times 60) + (56 times 60 + 4.09) 8 = 86,164.9 s mu = squareroot mu/2 (given) for GEO - orbital radius mu = squareroot mu/r_GEO - (1) similarly, from time period of covering in conference 2 pi r_GEO, velocity is mu = 2 pi r_GEO /T - (2) equating (1) and (2) squareroot mu/r_GEO = 2 pi r_GEO /T equating both sides, mu/r_GEO = 4 pi^2 r^2_GEO/T^2 implies r^3_GEO = mu T^2/4 pi^2 = 3.98601 times 10^5 times (86,164.9)/4 times (3.14)^2 r^3_GEO = 75.037 times 10^+ 12 km^3 r_GEO = 4.2178 times 10^+ 4 km \r\n geo synchronous orbit always lies in the plane of equate or i.e. Angle that the orbot must have speed, respect to equate is 0 degree Angle = 0 degree with respect to equates mu_GEO = squareroot 3.98601 times 10^5/4.2178 times 10^4 km/s = squareroot 39.8601 times 10^4/4.2178 times 10^4 km/s