Explanation Given: The equation of the ellipse is r = p ( 1 + e cos θ ) Where r is the radial distance e is the eccentricity θ is the angle a is the semimajor axis r a p + r p e r = 2 a Where r a p is the aphelion r p e r is the Perihelion Calculation: The Perihelion and aphelion are the extremum radial distances. To find the extremum distances one must take the derivative and equating it to zero. r = p ( 1 + e cos θ ) Here the only variable is θ so the derivative is taken with respect to θ d r d θ = d d θ p ( 1 + e cos θ ) d r d θ = p e sin θ ( 1 + e cos θ ) 2 Equating to zero to find the extremum d r d θ = p e sin θ e x t ( 1 + e cos θ e x t ) 2 = 0 p e sin θ e x t = 0 θ e x t = 0 , π The radial distance is extremum at two values of θ which are θ e x t = 0 , π . To find the maximum and erihel find the double derivative. d 2 r d θ 2 = d d θ p e sin θ ( 1 + e cos θ ) 2 d 2 r d θ 2 = p e cos θ ( 1 + e cos θ ) 2 − 2 p e 2 sin 2 θ ( 1 + e cos θ ) ( 1 + e cos θ ) 4 For maximum d 2 r d θ 2 < 0 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 14.6, Problem 13E

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Perihelion and aphelion of an ellipse

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Chapter 14 Solutions

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Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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Perihelion and aphelion of an ellipse

Solution Summary: The author explains the equation of the ellipse: r = p ( 1 + e cos ) and aphelion.

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Perihelion and aphelion of an ellipse

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