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:

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

<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.

<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 ''semi-major axis'', the ''longitude of the ascending node'', the ''inclination'' and the ''argument of periapsis''. An other parameter, called the ''gravitational parameter'', along with dynamical considerations described further, also allows for the determination of the speed of the orbiting object.

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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:

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.

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The gravitational parameter is the coefficent GM in Newton's gravitational force. It is sometimes noted <tt>µ</tt> and will be chosen as one here for the sake of simplicity:

As mentioned, dynamical considerations allow for the determination of the speed. They result in the so-called [[wp:vis-viva equation|vis-viva equation]]:

This only gives the magnitude of the speed. The direction is easily determined since it's tangent to the conic.

Those parameters 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]].

;Task:

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=={{header|Java}}==

<lang Java>public class OrbitalElements {

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println("Speed : $speed")

}</lang>

<pre>Position : (0.7794228433986797, 0.45000003465368416, 0.0)

<pre>

~~Position~~ : (0.~~7794228433986797~~, 0.~~45000003465368416~~, 0.0)

Speed : (-0.5527708409604438, 0.9574270831797618, 0.0)</pre>

Speed : (-0.5527708409604438, 0.9574270831797618, 0.0)

</pre>

=={{header|Perl}}==

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0, # argument of periapsis

0 # true-anomaly

;

;</lang>

⚫

</lang>

<pre>$VAR1 = {

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=={{header|Perl 6}}==

We'll use the [https://github.com/grondilu/clifford Clifford geometric algebra library] but only for the vector operations.

<lang perl6>sub orbital-state-vectors(

Real :$semimajor-axis where * >= 0,

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longitude-of-ascending-node => pi/6,

argument-of-periapsis => pi/4,

true-anomaly => 0;

true-anomaly => 0;</lang>

</lang>

<pre>{position => 0.237771283982207*e0+0.860960261697716*e1+0.110509023572076*e2, speed => -1.06193301748006*e0+0.27585002056925*e1+0.135747024865598*e2}</pre>

=={{header|Scala}}==

<lang Scala>import scala.language.existentials

object OrbitalElements extends App {

private val ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)

println(f"Position : ${ps(0)}%s%nSpeed : ${ps(1)}%s")

private def orbitalStateVectors(semimajorAxis: Double,

eccentricity: Double,

inclination: Double,

longitudeOfAscendingNode: Double,

argumentOfPeriapsis: Double,

trueAnomaly: Double) = {

def mulAdd(v1: Vector, x1: Double, v2: Vector, x2: Double) = v1 * x1 + v2 * x2

case class Vector(x: Double, y: Double, z: Double) {

def +(term: Vector) =

Vector(x + term.x, y + term.y, z + term.z)

def *(factor: Double) = Vector(factor * x, factor * y, factor * z)

def /(divisor: Double) = Vector(x / divisor, y / divisor, z / divisor)

def abs: Double = math.sqrt(x * x + y * y + z * z)

override def toString: String = f"($x%.16f, $y%.16f, $z%.16f)"

}

def rotate(i: Vector, j: Vector, alpha: Double) =

Array[Vector](mulAdd(i, math.cos(alpha), j, math.sin(alpha)),

mulAdd(i, -math.sin(alpha), j, math.cos(alpha)))

val p = rotate(Vector(1, 0, 0), Vector(0, 1, 0), longitudeOfAscendingNode)

val p2 = rotate(p(0),

rotate(p(1), Vector(0, 0, 1), inclination)(0),

argumentOfPeriapsis)

val l = semimajorAxis *

(if (eccentricity == 1.0) 2.0 else 1.0 - eccentricity * eccentricity)

val (c, s) = (math.cos(trueAnomaly), math.sin(trueAnomaly))

val r = l / (1.0 + eccentricity * c)

val rprime = s * r * r / l

val speed = mulAdd(p2(0), rprime * c - r * s, p2(1), rprime * s + r * c)

Array[Vector](mulAdd(p(0), c, p2(1), s) * r,

speed / speed.abs * math.sqrt(2.0 / r - 1.0 / semimajorAxis))

}

⚫

}</lang>

{{Out}}Best seen running in your browser either by [https://scalafiddle.io/sf/ac17jh2/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/2NQNgj4OQkazxZNvSzcexQ Scastie (remote JVM)].

=={{header|Sidef}}==

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say '['+r.speed.join(', ')+']'</lang>

<pre>[0.77942284339868, 0.450000034653684, 0]

<pre>

[0.~~77942284339868~~, 0.~~450000034653684~~, 0]

[-0.552770840960444, 0.957427083179761, 0]</pre>

[-0.552770840960444, 0.957427083179761, 0]

</pre>

=={{header|zkl}}==

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).println();</lang>

⚫

<pre>

⚫

L(L(0.779423,0.45,0),L(-0.552771,0.957427,0))

</pre>