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Orbital elements (view source)

Revision as of 10:17, 4 August 2016

161 bytes removed ,  7 years ago
various rephrasing + changing 'l' to 'L' in order to avoid confusion with 1
m (added whitespace and highlighting to the task's preamble, used a shaded window for the formulae.)
(various rephrasing + changing 'l' to 'L' in order to avoid confusion with 1)
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When neglecting the influence of other objects, two celestial bodies orbit one another along a [[wp:conic section|conic]] trajectory. In the orbital plane, the radial equation is thus:
 
<big> r = lL/(1 + e cos(angle)) </big>
 
The &nbsp; <big> '''<tt>lL</tt>''' </big> &nbsp; (unity), &nbsp; <big> '''e''' </big> &nbsp; and &nbsp; <big> '''angle''' </big> &nbsp; are respectively called the &nbsp; ''semi-latus rectum'', &nbsp; the ''eccentricity'' &nbsp; and the &nbsp; ''true anomaly''. &nbsp; The eccentricity and the true anomaly are two of the six so-called &nbsp; [[wp:orbital elements|orbital elements]] &nbsp; often used to specify an orbit and the position of a point on this orbit.
 
The four other parameters are the &nbsp; ''semi-major axis'', &nbsp; the &nbsp; ''longitude of the ascending node'', &nbsp; the &nbsp; ''inclination'' &nbsp; and the &nbsp; ''argument of periapsis''.
 
The semi-major axis is half the distance between &nbsp; [[wp:perihelion and aphelion|perihelion and aphelion]]. &nbsp; It is often noted &nbsp; <big> '''a'''</big>, &nbsp; and it's not too hard to see how it's related to the semi-latus-rectum:
 
<big> a = lL/(1 - e<sup>2</sup>) </big>
 
The longitude of the ascending node, the inclination and the argument of the periapsis specify the orientation of the orbiting plane with respect to a reference plane defined with three arbitrarily chosen reference distant stars.
 
Those six parameters, along with dynamical considerations implyingexplained notably the &nbsp; [[wp:vis-viva equation|vis-viva equation]]below, &nbsp; allow for the determination of both the position and the speed of the orbiting object in &nbsp; [[wp:cartesian coordinates|cartesian coordinates]], &nbsp; those two vectors constituting the so-called &nbsp; [[wp:orbital state vectors|orbital state vectors]].
 
The aforementioned dynamical considerations imply the so-called [[wp:vis-viva equation|vis-viva equation]]:
The determination of the speed also requires the knowledge of the so-called &nbsp; ''gravitational parameter'' &nbsp; which we shall consider equal to one here:
 
<big>v<sup>2</sup> = GM(2/r - 1/a)</big>
 
<tt>GM</tt> is the gravitational parameter sometimes noted <tt>µ</tt>. It will be chosen as one here for the sake of simplicity:
 
<big> µ = GM = 1 </big>
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The purpose of this task is to show how to perform this conversion from orbital elements to orbital state vectors in your programming language.
 
TODO: &nbsp; pick an example from a reputable source, and bring the algorithm description onto this site. (Restating those pages in concise a fashion comprehensible to the coders and readers of this site will be a good exercise.)
<br><br>
 
Grondilu
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