Orbits and Eccentricity

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Presentation transcript:

Orbits and Eccentricity More with Ellipses in Sec. 8.2b

Orbits and Eccentricity Many moons ago, it was discovered that many celestial bodies (for example, those orbiting the sun) followed elliptical paths… Perihelion – point closest to the sun in such an orbit Aphelion – point farthest from the sun in such an orbit The shape of an ellipse (including these orbital paths) is related to its eccentricity…

Orbits and Eccentricity a – c a + c Center Sun at focus Semimajor Axis Orbiting Object

Definition: Eccentricity of an Ellipse The eccentricity of an ellipse is where a is the semimajor axis, b is the semiminor axis, and c is the distance from the center of the ellipse to either focus. What is the range of possible “e” values for an ellipse? What happens when “e” is zero?  A CIRCLE!!!

Practice Problems The semiminor axis is only 0.014% shorter The Earth’s orbit has a semimajor axis and an eccentricity of . Calculate and interpret b and c. Semiminor Axis The semiminor axis is only 0.014% shorter than the semimajor axis…

Practice Problems The Earth’s orbit is nearly a perfect circle, but The Earth’s orbit has a semimajor axis and an eccentricity of . Calculate and interpret b and c. Aphelion of Earth: Perihelion of Earth: The Earth’s orbit is nearly a perfect circle, but the eccentricity as a percentage is 1.67%; this measures how far off-center the Sun is…

Other Types of Practice Problems Prove that the graph of the equation is an ellipse, and find its vertices, foci, and eccentricity. Put into standard form:

Other Types of Practice Problems Prove that the graph of the equation is an ellipse, and find its vertices, foci, and eccentricity. Standard Form: Vertices: Eccentricity: Foci:

Other Types of Practice Problems Write an equation for the given ellipse. (–4, 5) Center: C(–4, 2) (0, 2) Semimajor, semiminor axes: Standard Form: