Help deriving this elliptical orbit equation?

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Discussion Overview

The discussion revolves around deriving the equation for an elliptical orbit, specifically the equation \(\frac{(x + ae)^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where \(e\) is the eccentricity of the ellipse. Participants explore both geometric and mechanical approaches to this derivation, reflecting on their experiences in physics courses.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant seeks help with deriving the elliptical orbit equation, expressing difficulty in finding resources and understanding where to start.
  • Another participant mentions that David Morin's book on mechanics provides a clear explanation for the derivation.
  • A different participant suggests using the geometry of an ellipse as a potential approach to the derivation.
  • It is noted that the book by Kleppner & Kolenkow begins with conservation principles to derive the polar form of the equation, which is considered a good method.
  • One participant questions whether the problem is a mechanics question related to forces in an electric or gravitational field or a geometry question focused on the ellipse itself.
  • Another participant emphasizes the importance of Newton's second law in classical mechanics and suggests focusing on the "transverse" direction for insights.
  • A later post requests assistance specifically with finding the equation for the eccentricity of an object in an elliptical orbit.

Areas of Agreement / Disagreement

Participants express varying levels of familiarity with resources and approaches, indicating that there is no consensus on the best method for deriving the equation. Multiple perspectives on how to approach the problem remain present.

Contextual Notes

Some participants reference specific textbooks and methods, but there is no agreement on a singular approach or resolution of the derivation process. The discussion reflects a range of experiences and understandings related to classical mechanics.

skiz
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help deriving this elliptical orbit equation??

Hi guys, this is my first post on these boards. just found out about this forum and I am really happy because i often find i need a place like this to ask questions and my prayers are answered!

im a physics/computer science major in my second year at the University of the Witwatersrand in Johannesburg, South Africa.

any way.. enough history..

i need help deriving this equation for an elliptical orbit :

[tex]\frac{(x + ae)^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1[/tex]

where e is the eccentricity and for an ellipse e < 1

i can't find anything useful in my books and don't even know where to start.

any help would be awesome! thanks!

-skiz
 
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David Morin's book on mechanics explains very clearly on how to derive this equation
 
hmm i don't have david morin's book on mechanics..

any tips on how to go about deriving this? am i supposed to use the geometry of an ellipse?

bah
 
I assume that you are doing a course in Introductory Mechanics. The book by Kleppner & Kolenkow is really good for such a course. However in that book, the authors have begun from Law of conservation of energy and angular momentum and have derived the polar form of the equation. I believe that this is a good way to do this as it begins from conservation principles.
 
Welcome to PF!

Hi skiz! Welcome to PF! :smile:

Is this a mechanics question (find the orbit of a particle in an electric or gravitational field), or a geometry question (find the equation for an ellipse)?

Assuming it's mechanics, start with Newton's second law, and remember that the force perpendicular to the "radius" vector is zero. :smile:
 
hey tim, thanks for the welcome!

yeah its a mechanics course. Our first 6 months are split between "Classical Mechanics" and "Modern Physics/Relativity"

i find modern physics and relativity really interesting and easy to grasp but classical mechanics is kicking my ass...
 
Hi skiz! :smile:

Just remember that nearly all classical mechanics boils down to good ol' Newton's second law …

force in a particular direction = (rate of) change of momentum in that direction :smile:​

… and in each case you just have to work out which direction gives you the best information (in this case, it's the "transverse" direction).
 
please anyone help me?

please anyone help me to find the equation of eccentricity of object moves in elliptical orbit
 

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