Orbital elements: Difference between revisions

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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:
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 = l/(1 + e cos(angle)) </big>
<big> r = L/(1 + e cos(angle)) </big>


The &nbsp; <big> '''<tt>l</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.
<big> '''<tt>L</tt>''' </big>, <big> '''e''' </big> and <big> '''angle''' </big> are respectively called ''semi-latus rectum'', ''eccentricity'' and ''true anomaly''. The eccentricity and the true anomaly are two of the six so-called [[wp:orbital elements|orbital elements]] 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 four other parameters are the ''semi-major axis'', the ''longitude of the ascending node'', the ''inclination'' and the ''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:
The semi-major axis is half the distance between [[wp:perihelion and aphelion|perihelion and aphelion]]. It is often noted <big> '''a'''</big>, and it's not too hard to see how it's related to the semi-latus-rectum:


<big> a = l/(1 - e<sup>2</sup>) </big>
<big> a = L/(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.
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 implying notably the &nbsp; [[wp:vis-viva equation|vis-viva equation]], &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]].
Those six parameters, along with dynamical considerations explained below, allow for the determination of both the position and the speed of the orbiting object in [[wp:cartesian coordinates|cartesian coordinates]], those two vectors constituting the so-called [[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>
<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.
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.)
TODO: 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.)
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