Re: Eccentricity of orbit
Would someone kindly help me solve this problem correctly?
Problem: The moon travels an elliptical path with Earth as one focus. The maximum distance from the moon to Earth is 405,500 km and the minimum distance is 363,300 km.
What is the eccentricity of the orbit?
An ellipse is the locus of a point, P, moving in such a way that the sum of its distances from two fixed points, F and F', called foci, is a constant. Draw yourself two perpendicular axes, X and Y with origin O. Locate two points, the foci, on the X axis, equidistant on either side of the origin O, say 1 inch, if you want to draw your own picture. Label the points F on the left and F' on the right. Each represents a focus of the ellipse. Now locate two points, on the X axis, 2 inches on either side of the origin. Label these A on the left and B on the right. AB is the major axis of our ellipse. With a compass, using AO as a radius and F as the center, swing an arc that intersects the +Y axis at C and the -Y axis at D. CD represents the minor axis of our ellipse. Let AO = a, CO = b, and FO = c, the typical terms used to represent the major and minor axes of an ellipse and the distance to the focus from the center O. Note that CF also equals "a" since we used radius AO to locate point B. (As you can see, from the pythagorean theorem, a^2 = b^2 + c^2.) Therefore if you are given the major and minor axes of an ellipse, you can determine the location of the focus points from c = sqrt(a^2 - b^2) where a = semi-major axis and b = semi-minor axis. The eccentricity of the ellipse is e = c/a = <1. The terms a, b, and c are also related by b^2 = a^2(1 -e^2) or e^2 = 1 - b^2/a^2.
That said, an elliptical orbit is the elliptical shaped path of a body, or satellite, around another body having a gravitational field. As Kepler's First Law states, the orbits of all the planets in our solar system, as well as the orbits of all Earth satellites, are ellipses. In the case of the planets, the Sun is the focus of the planetary orbits. In the case of Earth satellites, the Earth is the focus. Geometrically, an ellipse is the locus of a point moving in such a way that the sum of its distances from two fixed points, f and f', called foci, is a constant. Pictorially, draw yourself two perpendicular axes, X and Y, with origin O. Locate the two foci points, on the X axis, equidistant on either side of the origin O, say 1 inch for example, labeling the points f on the right and f' on the left. Each represents a focus of the ellipse. Now locate two points, on the X axis, 2 inches on either side of the origin O. Label these A on the left and B on the right. AB is the major axis of our ellipse. With a compass, using AO as a radius and f as the center, swing an arc that intersects the +Y axis at C and the -Y axis at D. CD represents the minor axis of our ellipse. Note that fB equals AO since we used radius AO to locate point B. (As you can see, from the pythagorean theorem, (fB)^2 = (fO)^2 + (OB)^2.) Therefore if you are given the major and minor axes of an ellipse, you can determine the location of the focus points from fO = sqrt[(fB)a^2 - (OB)^2] where AO = semi-major axis and OB = semi-minor axis. The eccentricity of the ellipse is e = fO/AO = <1. The terms a, b, and c are also related by b^2 = a^2(1 -e^2) or e^2 = 1 - b^2/a^2.
