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MathGroup Archive 2006

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Re: Area of ellipse between major axis and ray through focus, given angle

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71939] Re: Area of ellipse between major axis and ray through focus, given angle
  • From: rip pelletier <bitbucket at comcast.net>
  • Date: Tue, 5 Dec 2006 06:04:37 -0500 (EST)
  • References: <ek96ij$fd5$1@smc.vnet.net>

In article <ek96ij$fd5$1 at smc.vnet.net>,
 "Kelly Jones" <kelly.terry.jones at gmail.com> wrote:

> Given:
> 
> 1) an ellipse with eccentricity "ec", one focus on the origin, and
> the major axis along the x-axis
> 
> 2) a ray through the origin at angle theta to the x-axis
> 
> Question:
> 
> What Mathematica function gives the relation/inverse relation between
> the angle theta and the area of the ellipse between the x-axis and the ray?
> 

two books which i find very useful for doing orbital mechanics are:

prussing & conway, "orbital mechanics", 1993, oxford.    

bate, mueller & white, "fundamentals of astrodynamics", 1971, dover.

kepler's equation for position-on-orbit is pretty and simple; and a 
geometric proof of it leads to a a pretty and simple solution for the 
area. the key is to define an auxiliary angle called the eccentric 
anomaly, for which see mathworld.

http://mathworld.wolfram.com/EccentricAnomaly.html

especially, note the last sentence, which tells you how to get the area: 
"M can also be interpreted as the area of the shaded region in the above 
figure"

vale,
   rip

-- 
NB eddress is r i p 1 AT c o m c a s t DOT n e t


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