Orbital elements: Difference between revisions

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When neglecting the influence of other objects, two celestial bodies orbit one another along a conic trajectory. In the orbital plane, the radial equation is thus:

r = L/(1 + e cos(angle))

L , e and angle are respectively called semi-latus rectum, eccentricity and true anomaly. The eccentricity and the true anomaly are two of the six so-called 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.

The semi-major axis is half the distance between perihelion and aphelion. It is often noted a, and it's not too hard to see how it's related to the semi-latus-rectum:

a = L/(1 - e²)

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

µ = GM = 1

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

v² = GM(2/r - 1/a)

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 cartesian coordinates, those two vectors constituting the so-called orbital state vectors.

Task

Show how to perform this conversion from orbital elements to orbital state vectors in your programming language.

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.)

C

Translation of: Kotlin

<lang c>#include <stdio.h>

include <math.h>

typedef struct {

   double x, y, z;

} vector;

vector add(vector v, vector w) {

   return (vector){v.x + w.x, v.y + w.y, v.z + w.z};

}

vector mul(vector v, double m) {

   return (vector){v.x * m, v.y * m, v.z * m};

}

vector div(vector v, double d) {

   return mul(v, 1.0 / d);

}

double vabs(vector v) {

   return sqrt(v.x * v.x + v.y * v.y + v.z * v.z);

}

vector mulAdd(vector v1, vector v2, double x1, double x2) {

   return add(mul(v1, x1), mul(v2, x2));

}

void vecAsStr(char buffer[], vector v) {

   sprintf(buffer, "(%.17g, %.17g, %.17g)", v.x, v.y, v.z);

}

void rotate(vector i, vector j, double alpha, vector ps[]) {

   ps[0] = mulAdd(i, j, cos(alpha), sin(alpha));
   ps[1] = mulAdd(i, j, -sin(alpha), cos(alpha));

}

void orbitalStateVectors(

   double semimajorAxis, double eccentricity, double inclination,
   double longitudeOfAscendingNode, double argumentOfPeriapsis,
   double trueAnomaly, vector ps[]) {

   vector i = {1.0, 0.0, 0.0};
   vector j = {0.0, 1.0, 0.0};
   vector k = {0.0, 0.0, 1.0};
   double l = 2.0, c, s, r, rprime;
   vector qs[2];
   rotate(i, j, longitudeOfAscendingNode, qs);
   i = qs[0]; j = qs[1];
   rotate(j, k, inclination, qs);
   j = qs[0];
   rotate(i, j, argumentOfPeriapsis, qs);
   i = qs[0]; j = qs[1];
   if (eccentricity != 1.0)  l = 1.0 - eccentricity * eccentricity;
   l *= semimajorAxis;
   c = cos(trueAnomaly);
   s = sin(trueAnomaly);
   r = l / (1.0 + eccentricity * c);
   rprime = s * r * r / l;
   ps[0] = mulAdd(i, j, c, s);
   ps[0] = mul(ps[0], r);
   ps[1] = mulAdd(i, j, rprime * c - r * s, rprime * s + r * c);
   ps[1] = div(ps[1], vabs(ps[1]));
   ps[1] = mul(ps[1], sqrt(2.0 / r - 1.0 / semimajorAxis));

}

int main() {

   double longitude = 355.0 / (113.0 * 6.0);
   vector ps[2];
   char buffer[80];
   orbitalStateVectors(1.0, 0.1, 0.0, longitude, 0.0, 0.0, ps);
   vecAsStr(buffer, ps[0]);
   printf("Position : %s\n", buffer);
   vecAsStr(buffer, ps[1]);
   printf("Speed    : %s\n", buffer);
   return 0;

}</lang>

Output:

Position : (0.77942284339867973, 0.45000003465368416, 0)
Speed    : (-0.55277084096044382, 0.95742708317976177, 0)

D

Translation of: Kotlin

<lang D>import std.math; import std.stdio; import std.typecons;

struct Vector {

   double x, y, z;

   auto opBinary(string op : "+")(Vector rhs) {
       return Vector(x+rhs.x, y+rhs.y, z+rhs.z);
   }

   auto opBinary(string op : "*")(double m) {
       return Vector(x*m, y*m, z*m);
   }
   auto opOpAssign(string op : "*")(double m) {
       this.x *= m;
       this.y *= m;
       this.z *= m;
       return this;
   }

   auto opBinary(string op : "/")(double d) {
       return Vector(x/d, y/d, z/d);
   }
   auto opOpAssign(string op : "/")(double m) {
       this.x /= m;
       this.y /= m;
       this.z /= m;
       return this;
   }

   auto abs() {
       return sqrt(x * x + y * y + z * z);
   }

   void toString(scope void delegate(const(char)[]) sink) const {
       import std.format;
       sink("(");
       formattedWrite(sink, "%.16f", x);
       sink(", ");
       formattedWrite(sink, "%.16f", y);
       sink(", ");
       formattedWrite(sink, "%.16f", z);
       sink(")");
   }

}

auto orbitalStateVectors(

   double semiMajorAxis,
   double eccentricity,
   double inclination,
   double longitudeOfAscendingNode,
   double argumentOfPeriapsis,
   double trueAnomaly

) {

   auto i = Vector(1.0, 0.0, 0.0);
   auto j = Vector(0.0, 1.0, 0.0);
   auto k = Vector(0.0, 0.0, 1.0);

   auto mulAdd = (Vector v1, double x1, Vector v2, double x2) => v1 * x1 + v2 * x2;

   auto rotate = (Vector i, Vector j, double alpha) =>
       tuple(mulAdd(i, +cos(alpha), j, sin(alpha)),
             mulAdd(i, -sin(alpha), j, cos(alpha)));

   auto p = rotate(i, j, longitudeOfAscendingNode);
   i = p[0]; j = p[1];
   p = rotate(j, k, inclination);
   j = p[0];
   p = rotate(i, j, argumentOfPeriapsis);
   i = p[0]; j = p[1];

   auto l = semiMajorAxis * ((eccentricity == 1.0) ? 2.0 : (1.0 - eccentricity * eccentricity));
   auto c = cos(trueAnomaly);
   auto s = sin(trueAnomaly);
   auto r = l / (1.0 + eccentricity * c);
   auto rprime = s * r * r / l;
   auto position = mulAdd(i, c, j, s) * r;
   auto speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c);
   speed /= speed.abs();
   speed *= sqrt(2.0 / r - 1.0 / semiMajorAxis);
   return tuple(position, speed);

}

void main() {

   auto res = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0);
   writeln("Position : ", res[0]);
   writeln("Speed    : ", res[1]);

}</lang>

Output:

Position : (0.7794228433986798, 0.4500000346536842, 0.0000000000000000)
Speed    : (-0.5527708409604437, 0.9574270831797614, 0.0000000000000000)
Edit source/app.d to start your project.

Go

Translation of: Kotlin

<lang go>package main

import (

   "fmt"
   "math"

)

type vector struct{ x, y, z float64 }

func (v vector) add(w vector) vector {

   return vector{v.x + w.x, v.y + w.y, v.z + w.z}

}

func (v vector) mul(m float64) vector {

   return vector{v.x * m, v.y * m, v.z * m}

}

func (v vector) div(d float64) vector {

   return v.mul(1.0 / d)

}

func (v vector) abs() float64 {

   return math.Sqrt(v.x*v.x + v.y*v.y + v.z*v.z)

}

func (v vector) String() string {

   return fmt.Sprintf("(%g, %g, %g)", v.x, v.y, v.z)

}

func orbitalStateVectors(

   semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode,
   argumentOfPeriapsis, trueAnomaly float64) (position vector, speed vector) {

   i := vector{1, 0, 0}
   j := vector{0, 1, 0}
   k := vector{0, 0, 1}

   mulAdd := func(v1, v2 vector, x1, x2 float64) vector {
       return v1.mul(x1).add(v2.mul(x2))
   }

   rotate := func(i, j vector, alpha float64) (vector, vector) {
       return mulAdd(i, j, math.Cos(alpha), math.Sin(alpha)),
           mulAdd(i, j, -math.Sin(alpha), math.Cos(alpha))
   }

   i, j = rotate(i, j, longitudeOfAscendingNode)
   j, _ = rotate(j, k, inclination)
   i, j = rotate(i, j, argumentOfPeriapsis)

   l := 2.0
   if eccentricity != 1.0 {
       l = 1.0 - eccentricity*eccentricity
   }
   l *= semimajorAxis
   c := math.Cos(trueAnomaly)
   s := math.Sin(trueAnomaly)
   r := l / (1.0 + eccentricity*c)
   rprime := s * r * r / l
   position = mulAdd(i, j, c, s).mul(r)
   speed = mulAdd(i, j, rprime*c-r*s, rprime*s+r*c)
   speed = speed.div(speed.abs())
   speed = speed.mul(math.Sqrt(2.0/r - 1.0/semimajorAxis))
   return

}

func main() {

   long := 355.0 / (113.0 * 6.0)
   position, speed := orbitalStateVectors(1.0, 0.1, 0.0, long, 0.0, 0.0)
   fmt.Println("Position :", position)
   fmt.Println("Speed    :", speed)

}</lang>

Output:

Position : (0.7794228433986797, 0.45000003465368416, 0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0)

Java

Translation of: Kotlin

<lang Java>public class OrbitalElements {

   private static class Vector {
       private double x, y, z;

       public Vector(double x, double y, double z) {
           this.x = x;
           this.y = y;
           this.z = z;
       }

       public Vector plus(Vector rhs) {
           return new Vector(x + rhs.x, y + rhs.y, z + rhs.z);
       }

       public Vector times(double s) {
           return new Vector(s * x, s * y, s * z);
       }

       public Vector div(double d) {
           return new Vector(x / d, y / d, z / d);
       }

       public double abs() {
           return Math.sqrt(x * x + y * y + z * z);
       }

       @Override
       public String toString() {
           return String.format("(%.16f, %.16f, %.16f)", x, y, z);
       }
   }

   private static Vector mulAdd(Vector v1, Double x1, Vector v2, Double x2) {
       return v1.times(x1).plus(v2.times(x2));
   }

   private static Vector[] rotate(Vector i, Vector j, double alpha) {
       return new Vector[]{
           mulAdd(i, Math.cos(alpha), j, Math.sin(alpha)),
           mulAdd(i, -Math.sin(alpha), j, Math.cos(alpha))
       };
   }

   private static Vector[] orbitalStateVectors(
       double semimajorAxis, double eccentricity, 
       double inclination, double longitudeOfAscendingNode, 
       double argumentOfPeriapsis, double trueAnomaly
   ) {
       Vector i = new Vector(1, 0, 0);
       Vector j = new Vector(0, 1, 0);
       Vector k = new Vector(0, 0, 1);

       Vector[] p = rotate(i, j, longitudeOfAscendingNode);
       i = p[0];
       j = p[1];
       p = rotate(j, k, inclination);
       j = p[0];
       p = rotate(i, j, argumentOfPeriapsis);
       i = p[0];
       j = p[1];

       double l = (eccentricity == 1.0) ? 2.0 : 1.0 - eccentricity * eccentricity;
       l *= semimajorAxis;
       double c = Math.cos(trueAnomaly);
       double s = Math.sin(trueAnomaly);
       double r = l / (1.0 + eccentricity * c);
       double rprime = s * r * r / l;
       Vector position = mulAdd(i, c, j, s).times(r);
       Vector speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c);
       speed = speed.div(speed.abs());
       speed = speed.times(Math.sqrt(2.0 / r - 1.0 / semimajorAxis));

       return new Vector[]{position, speed};
   }

   public static void main(String[] args) {
       Vector[] ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0);
       System.out.printf("Position : %s\n", ps[0]);
       System.out.printf("Speed : %s\n", ps[1]);
   }

}</lang>

Output:

Position : (0.7794228433986797, 0.4500000346536842, 0.0000000000000000)
Speed : (-0.5527708409604438, 0.9574270831797618, 0.0000000000000000)

Kotlin

Translation of: Sidef

<lang scala>// version 1.1.4-3

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

   operator fun plus(other: Vector) = Vector(x + other.x, y + other.y, z + other.z)
   
   operator fun times(m: Double) = Vector(x * m, y * m, z * m)

   operator fun div(d: Double) = this * (1.0 / d)

   fun abs() = Math.sqrt(x * x + y * y + z * z)

   override fun toString() = "($x, $y, $z)"

}

fun orbitalStateVectors(

   semimajorAxis: Double,
   eccentricity: Double,
   inclination: Double,
   longitudeOfAscendingNode: Double,
   argumentOfPeriapsis: Double,
   trueAnomaly: Double

): Pair<Vector, Vector> {

   var i = Vector(1.0, 0.0, 0.0)
   var j = Vector(0.0, 1.0, 0.0)
   var k = Vector(0.0, 0.0, 1.0)

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

   fun rotate(i: Vector, j: Vector, alpha: Double) = 
       Pair(mulAdd(i, +Math.cos(alpha), j, Math.sin(alpha)),
            mulAdd(i, -Math.sin(alpha), j, Math.cos(alpha)))

   var p = rotate(i, j, longitudeOfAscendingNode)
   i = p.first; j = p.second
   p = rotate(j, k, inclination)
   j = p.first
   p = rotate(i, j, argumentOfPeriapsis)
   i = p.first; j = p.second

   val l = semimajorAxis * (if (eccentricity == 1.0) 2.0 else (1.0 - eccentricity * eccentricity))
   val c = Math.cos(trueAnomaly)
   val s = Math.sin(trueAnomaly)
   val r = l / (1.0 + eccentricity * c)
   val rprime = s * r * r / l
   val position = mulAdd(i, c, j, s) * r
   var speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c)
   speed /= speed.abs()
   speed *= Math.sqrt(2.0 / r - 1.0 / semimajorAxis)
   return Pair(position, speed)

}

fun main(args: Array<String>) {

   val (position, speed) = orbitalStateVectors(
       semimajorAxis = 1.0,
       eccentricity = 0.1,
       inclination = 0.0,
       longitudeOfAscendingNode = 355.0 / (113.0 * 6.0),
       argumentOfPeriapsis = 0.0,
       trueAnomaly = 0.0
   ) 
   println("Position : $position")
   println("Speed    : $speed")

}</lang>

Output:

Position : (0.7794228433986797, 0.45000003465368416, 0.0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0.0)

ooRexx

Translation of: Java

<lang oorexx>/* REXX */ Numeric Digits 16 ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0) Say "Position :" ps~x~tostring Say "Speed :" ps~y~tostring Say 'Perl6:' pi=rxCalcpi(16) ps=orbitalStateVectors(1,.1,pi/18,pi/6,pi/4,0) /*Perl6*/ Say "Position :" ps~x~tostring Say "Speed :" ps~y~tostring

class v2

method init

 expose x y
 Use Arg x,y

attribute x

attribute y

class vector

method init

 expose x y z
 use strict arg x = 0, y = 0, z = 0  -- defaults to 0 for any non-specified coordinates

attribute x

attribute y

attribute z

method print

 expose x y z
 Numeric Digits 16
 Say 'Vector:'||x'/'y'/'z

method tostring

 expose x y z
 Return '('||x','y','z')'

method abs

 expose x y z
 Numeric Digits 16
 Return rxCalcsqrt(x**2+y**2+z**2,16)

method '*'

 expose x y z
 Parse Arg f
 Numeric Digits 16
 Return .vector~new(x*f,y*f,z*f)

method '/'

 expose x y z
 Parse Arg f
 Numeric Digits 16
 Return .vector~new(x/f,y/f,z/f)

method '+'

 expose x y z
 Use Arg v2
 Numeric Digits 16
 Return .vector~new(x+v2~x,y+v2~y,z+v2~z)

routine orbitalStateVectors

Use Arg semimajorAxis,,

        eccentricity,,
        inclination,,
        longitudeOfAscendingNode,,
        argumentOfPeriapsis,,
        trueAnomaly

Numeric Digits 16 i = .vector~new(1, 0, 0) j = .vector~new(0, 1, 0) k = .vector~new(0, 0, 1) p = rotate(i, j, longitudeOfAscendingNode) i = p~x j = p~y p = rotate(j, k, inclination) j = p~x p = rotate(i, j, argumentOfPeriapsis) i = p~x j = p~y If eccentricity=1 Then l=2 Else l=1-eccentricity*eccentricity l*=semimajorAxis c=rxCalccos(trueAnomaly,16,'R') s=rxCalcsin(trueAnomaly,16,'R') r=l/(1+eccentricity*c) rprime=s*r*r/l position=mulAdd(i,c,j,s)~'*'(r) speed=mulAdd(i,rprime*c-r*s,j,rprime*s+r*c) speed=speed~'/'(speed~abs) speed=speed~'*'(rxCalcsqrt(2.0/r-1.0/semimajorAxis,16)) Return .v2~new(position,speed)

routine muladd

 Use Arg v1,x1,v2,x2
 Numeric Digits 16
 w1=v1~'*'(x1)
 w2=v2~'*'(x2)
 Return w1~'+'(w2)

routine rotate

 Use Arg i,j,alpha
 Numeric Digits 16
 xx=mulAdd(i,rxCalccos(alpha,16,'R'),j,rxCalcsin(alpha,16,'R'))
 yy=mulAdd(i,-rxCalcsin(alpha,16,'R'),j,rxCalccos(alpha,16,'R'))
 res=.v2~new(xx,yy)
 Return res

requires 'rxmath' LIBRARY</lang>

Output:

Position : (0.7794228433986798,0.4500000346536842,0)
Speed    : (-0.5527708409604436,0.9574270831797613,0)
Perl6:
Position : (0.2377712839822067,0.8609602616977158,0.1105090235720755)
Speed    : (-1.061933017480060,0.2758500205692495,0.1357470248655981)

Perl

Translation of: Perl 6

<lang perl>use strict; use warnings; use Math::Vector::Real;

sub orbital_state_vectors {

   my (
       $semimajor_axis,
       $eccentricity,
       $inclination,
       $longitude_of_ascending_node,
       $argument_of_periapsis,
       $true_anomaly
   ) = @_[0..5];

   my ($i, $j, $k) = (V(1,0,0), V(0,1,0), V(0,0,1));
   
   sub rotate {
       my $alpha = shift;
       @_[0,1] = (
           +cos($alpha)*$_[0] + sin($alpha)*$_[1],
           -sin($alpha)*$_[0] + cos($alpha)*$_[1]
       );
   }

   rotate $longitude_of_ascending_node, $i, $j;
   rotate $inclination,                 $j, $k;
   rotate $argument_of_periapsis,       $i, $j;

   my $l = $eccentricity == 1 ? # PARABOLIC CASE
       2*$semimajor_axis :
       $semimajor_axis*(1 - $eccentricity**2);

   my ($c, $s) = (cos($true_anomaly), sin($true_anomaly));

   my $r = $l/(1 + $eccentricity*$c);
   my $rprime = $s*$r**2/$l;

   my $position = $r*($c*$i + $s*$j);

   my $speed = 
   ($rprime*$c - $r*$s)*$i + ($rprime*$s + $r*$c)*$j;
   $speed /= abs($speed);
   $speed *= sqrt(2/$r - 1/$semimajor_axis);

   {
       position => $position,
       speed    => $speed
   }

}

use Data::Dumper;

print Dumper orbital_state_vectors

   1,                             # semimajor axis
   0.1,                           # eccentricity
   0,                             # inclination
   355/113/6,                     # longitude of ascending node
   0,                             # argument of periapsis
   0                              # true-anomaly
   ;</lang>

Output:

$VAR1 = {
          'position' => bless( [
                                 '0.77942284339868',
                                 '0.450000034653684',
                                 '0'
                               ], 'Math::Vector::Real' ),
          'speed' => bless( [
                              '-0.552770840960444',
                              '0.957427083179762',
                              '0'
                            ], 'Math::Vector::Real' )
        };

Perl 6

We'll use the Clifford geometric algebra library but only for the vector operations. <lang perl6>sub orbital-state-vectors(

   Real :$semimajor-axis where * >= 0,
   Real :$eccentricity   where * >= 0,
   Real :$inclination,
   Real :$longitude-of-ascending-node,
   Real :$argument-of-periapsis,
   Real :$true-anomaly

) {

   use Clifford;
   my ($i, $j, $k) = @e[^3];

   sub rotate($a is rw, $b is rw, Real \α) {
       ($a, $b) = cos(α)*$a + sin(α)*$b, -sin(α)*$a + cos(α)*$b;
   }
   rotate($i, $j, $longitude-of-ascending-node);
   rotate($j, $k, $inclination);
   rotate($i, $j, $argument-of-periapsis);

   my \l = $eccentricity == 1 ?? # PARABOLIC CASE
       2*$semimajor-axis !!
       $semimajor-axis*(1 - $eccentricity**2);

   my ($c, $s) = .cos, .sin given $true-anomaly;

   my \r = l/(1 + $eccentricity*$c);
   my \rprime = $s*r**2/l;

   my $position = r*($c*$i + $s*$j);

   my $speed = 
   (rprime*$c - r*$s)*$i + (rprime*$s + r*$c)*$j;
   $speed /= sqrt($speed**2);
   $speed *= sqrt(2/r - 1/$semimajor-axis);

   { :$position, :$speed }

}

say orbital-state-vectors

   semimajor-axis => 1,
   eccentricity => 0.1,
   inclination => pi/18,
   longitude-of-ascending-node => pi/6,
   argument-of-periapsis => pi/4,
   true-anomaly => 0;</lang>

Output:

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

Phix

Translation of: Python

<lang Phix>function vabs(sequence v)

   return sqrt(sum(sq_power(v,2)))

end function

function mulAdd(sequence v1, atom x1, sequence v2, atom x2)

   return sq_add(sq_mul(v1,x1),sq_mul(v2,x2))

end function

function rotate(sequence i, j, atom alpha)

   atom ca = cos(alpha),
        sa = sin(alpha)
   return {mulAdd(i,ca,j,sa),mulAdd(i,-sa,j,ca)}

end function

procedure orbitalStateVectors(atom semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly)

   sequence i = {1, 0, 0},
            j = {0, 1, 0},
            k = {0, 0, 1}

   {i,j} = rotate(i, j, longitudeOfAscendingNode)
   {j} = rotate(j, k, inclination)
   {i,j} = rotate(i, j, argumentOfPeriapsis)

   atom l = iff(eccentricity=1?2:1-eccentricity*eccentricity)*semimajorAxis,
        c = cos(trueAnomaly),
        s = sin(trueAnomaly),
        r = 1 / (1+eccentricity*c),
        rprime = s * r * r / l
   sequence posn = sq_mul(mulAdd(i, c, j, s),r),
            speed = mulAdd(i, rprime*c-r*s, j, rprime*s+r*c)
   speed = sq_div(speed,vabs(speed))
   speed = sq_mul(speed,sqrt(2/r - 1/semimajorAxis))

   puts(1,"Position :") ?posn
   puts(1,"Speed    :") ?speed

end procedure

orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)</lang>

Output:

Position :{0.7872958014,0.4545454895,0}
Speed    :{-0.5477225997,0.9486832737,0}

Python

<lang python>import math

class Vector:

   def __init__(self, x, y, z):
       self.x = x
       self.y = y
       self.z = z

   def __add__(self, other):
       return Vector(self.x + other.x, self.y + other.y, self.z + other.z)

   def __mul__(self, other):
       return Vector(self.x * other, self.y * other, self.z * other)

   def __div__(self, other):
       return Vector(self.x / other, self.y / other, self.z / other)

   def __str__(self):
       return '({x}, {y}, {z})'.format(x=self.x, y=self.y, z=self.z)

   def abs(self):
       return math.sqrt(self.x*self.x + self.y*self.y + self.z*self.z)

def mulAdd(v1, x1, v2, x2):

   return v1 * x1 + v2 * x2

def rotate(i, j, alpha):

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

def orbitalStateVectors(semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly):

   i = Vector(1, 0, 0)
   j = Vector(0, 1, 0)
   k = Vector(0, 0, 1)

   p = rotate(i, j, longitudeOfAscendingNode)
   i = p[0]
   j = p[1]
   p = rotate(j, k, inclination)
   j = p[0]
   p  =rotate(i, j, argumentOfPeriapsis)
   i = p[0]
   j = p[1]

   l = 2.0 if (eccentricity == 1.0) else 1.0 - eccentricity * eccentricity
   l *= semimajorAxis
   c = math.cos(trueAnomaly)
   s = math.sin(trueAnomaly)
   r = 1 / (1.0 + eccentricity * c)
   rprime = s * r * r / l
   position = mulAdd(i, c, j, s) * r
   speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c)
   speed = speed / speed.abs()
   speed = speed * math.sqrt(2.0 / r - 1.0 / semimajorAxis)

   return [position, speed]

ps = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0) print "Position :", ps[0] print "Speed :", ps[1]</lang>

Output:

Position : (0.787295801413, 0.454545489549, 0.0)
Speed    : (-0.547722599684, 0.948683273698, 0.0)

REXX

version 1

Translation of: Java

Vectors are represented by strings: 'x/y/z' <lang rexx>/* REXX */ Numeric Digits 16

 Parse Value orbitalStateVectors(1.0,0.1,0.0,355.0/(113.0*6.0),0.0,0.0),
     With position speed
 Say "Position :" tostring(position)
 Say "Speed    :" tostring(speed)
 Exit

orbitalStateVectors: Procedure

 Parse Arg semimajorAxis,,
           eccentricity,,
           inclination,,
           longitudeOfAscendingNode,,
           argumentOfPeriapsis,,
           trueAnomaly
 i='1/0/0'
 j='0/1/0'
 k='0/0/1'
 Parse Value rotate(i, j, longitudeOfAscendingNode) With i j
 Parse Value rotate(j, k, inclination) With j p
 Parse Value rotate(i, j, argumentOfPeriapsis) With i j
 If eccentricity=1 Then l=2
 Else l=1-eccentricity*eccentricity
 l=l*semimajorAxis
 c=my_cos(trueAnomaly,16)
 s=my_sin(trueAnomaly,16)
 r=l/(1+eccentricity*c)
 rprime=s*r*r/l
 position=vmultiply(mulAdd(i,c,j,s),r)
 speed=mulAdd(i,rprime*c-r*s,j,rprime*s+r*c)
 speed=vdivide(speed,abs(speed))
 speed=vmultiply(speed,my_sqrt(2.0/r-1.0/semimajorAxis,16))
 Return position speed

abs: Procedure

 Parse Arg v.x '/' v.y '/' v.z
 Return my_sqrt(v.x**2+v.y**2+v.z**2,16)

muladd: Procedure

 Parse Arg v1,x1,v2,x2
 Parse Var v1 v1.x '/' v1.y '/' v1.z
 Parse Var v2 v2.x '/' v2.y '/' v2.z
 z=(v1.x*x1+v2.x*x2)||'/'||(v1.y*x1+v2.y*x2)||'/'||(v1.z*x1+v2.z*x2)
 Return z

rotate: Procedure Parse Arg i,j,alpha

 xx=mulAdd(i,my_cos(alpha,16,'R'),j,my_sin(alpha,16))
 yy=mulAdd(i,-my_sin(alpha,16,'R'),j,my_cos(alpha,16))
 Return xx yy

vmultiply: Procedure

 Parse Arg v,d
 Parse Var v v.x '/' v.y '/' v.z
 Return (v.x*d)||'/'||(v.y*d)||'/'||(v.z*d)

vdivide: Procedure

 Parse Arg v,d
 Parse Var v v.x '/' v.y '/' v.z
 Return (v.x/d)||'/'||(v.y/d)||'/'||(v.z/d)

tostring:

 Parse Arg v.x '/' v.y '/' v.z
 Return '('v.x','v.y','v.z')'

my_sqrt: Procedure /* REXX ***************************************************************

EXEC to calculate the square root of a = 2 with high precision
- - - - /

 Parse Arg x,prec
 If prec<9 Then prec=9
 prec1=2*prec
 eps=10**(-prec1)
 k = 1
 Numeric Digits 3
 r0= x
 r = 1
 Do i=1 By 1 Until r=r0 | ('ABS'(r*r-x)<eps)
   r0 = r
   r  = (r + x/r) / 2
   k  = min(prec1,2*k)
   Numeric Digits (k + 5)
   End
 Numeric Digits prec
 Return r+0

my_sin: Procedure /* REXX ****************************************************************

Return my_sin(x<,p>) -- with the specified precision
my_sin(x) = x-(x**3/3!)+(x**5/5!)-(x**7/7!)+-...
- - - - /

 Parse Arg x,prec
 If prec= Then prec=9
 Numeric Digits (2*prec)
 Numeric Fuzz   3
 pi=left('3.1415926535897932384626433832795028841971693993751058209749445923',2*prec+1)
 Do While x>pi
   x=x-pi
   End
 Do While x<-pi
   x=x+pi
   End
 o=x
 u=1
 r=x
 Do i=3 By 2
   ra=r
   o=-o*x*x
   u=u*i*(i-1)
   r=r+(o/u)
   If r=ra Then Leave
   End
 Numeric Digits prec
 Return r+0

my_cos: Procedure /* REXX ****************************************************************

Return my_cos(x) -- with specified precision
my_cos(x) = 1-(x**2/2!)+(x**4/4!)-(x**6/6!)+-...
- - - - /

 Parse Arg x,prec
 If prec= Then prec=9
 Numeric Digits (2*prec)
 Numeric Fuzz 3
 o=1
 u=1
 r=1
 Do i=1 By 2
   ra=r
   o=-o*x*x
   u=u*i*(i+1)
   r=r+(o/u)
   If r=ra Then Leave
   End
 Numeric Digits prec
 Return r+0</lang>

Output:

Position : (0.7794228433986798,0.4500000346536842,0)
Speed    : (-0.5527708409604436,0.9574270831797613,0)

version 2

Re-coding of REXX version 1. <lang rexx>/*REXX pgm converts orbital elements ──► orbital state vectors (angles are in radians).*/ numeric digits length( pi() ) - length(.) /*limited to pi len, but show 1/3 digs.*/ call orbV 1, .1, 0, 355/113/6, 0, 0 /*orbital elements taken from: Java */ call orbV 1, .1, pi/18, pi/6, pi/4, 0 /* " " " " Perl 6 */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ orbV: procedure; parse arg semiMaj, eccentricity, inclination, node, periapsis, anomaly

     say;     say center(' orbital elements ', 99, "═")
     say '            semi-major axis:'  fmt(semiMaj)
     say '               eccentricity:'  fmt(eccentricity)
     say '                inclination:'  fmt(inclination)
     say '   ascending node longitude:'  fmt(node)
     say '      argument of periapsis:'  fmt(periapsis)
     say '               true anomaly:'  fmt(anomaly)
     i= 1 0 0;          j= 0 1 0;        k= 0 0 1    /*define the  I,  J,  K   vectors.*/
     parse value rot(i, j, node)   with  i '~' j     /*rotate ascending node longitude.*/
     parse value rot(j, k, inclination) with j '~'   /*rotate the inclination.         */
     parse value rot(i, j, periapsis)   with i '~' j /*rotate the argument of periapsis*/
     if eccentricity=1  then L= 2
                        else L= 1 - eccentricity**2
     L= L * semiMaj                                  /*calculate the semi─latus rectum.*/
     c= cos(anomaly);               s= sin(anomaly)  /*calculate COS and SIN of anomaly*/
     r= L / (1 + eccentricity * c)
     @= s*r**2 / L;        speed= MA(i,  @*c - r*s,  j,   @*s + r*c)
     speed=    mulV( divV( speed, absV(speed) ), sqrt(2 / r  - 1 / semiMaj) )
     say '                   position:'  show( mulV( MA(i, c, j, s),  r) )
     say '                      speed:'  show( speed)
     return

/*──────────────────────────────────────────────────────────────────────────────────────*/ absV: procedure; parse arg x y z; return sqrt(x**2 + y**2 + z**2) divV: procedure; parse arg x y z, div; return (x / div) (y / div) (z / div) mulV: procedure; parse arg x y z, mul; return (x * mul) (y * mul) (z * mul) show: procedure; parse arg a b c; return '('fmt(a)"," fmt(b)',' fmt(c)")" fmt: procedure; parse arg #; return strip( left( left(, #>=0)# / 1, digits() %3), 'T') MA: procedure; parse arg x y z,@,a b c,$; return (x*@ + a*$) (y*@ + b*$) (z*@ + c*$) pi: pi= 3.1415926535897932384626433832795028841971693993751058209749445923; return pi rot: procedure; parse arg i,j,$; return MA(i,cos($),j,sin($))'~'MA(i, -sin($), j, cos($)) r2r: return arg(1) // (pi() * 2) /*normalize radians ──► a unit circle*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; arg x; x= r2r(x); if x=0 then return 1; a= abs(x); Hpi= pi * .5

     numeric fuzz min(6, digits() - 3);        if a=pi       then return '-1'
     if a=Hpi | a=Hpi*3  then return   0;      if a=pi / 3   then return .5
     if a=pi * 2 / 3     then return '-.5';                       return .sinCos(1, '-1')

/*──────────────────────────────────────────────────────────────────────────────────────*/ sin: procedure; arg x; x= r2r(x); numeric fuzz min(5, max(1, digits() - 3) )

     if x=0  then return 0;   if x=pi*.5  then return 1;   if x==pi*1.5  then return '-1'
     if abs(x)=pi  then return 0;                          return .sinCos(x, 1)

/*──────────────────────────────────────────────────────────────────────────────────────*/ .sinCos: parse arg z 1 _,i; xx= x*x

          do k=2  by 2  until p=z; p=z;  _= -_ * xx / (k*(k+i));  z= z+_;  end;  return z

/*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; arg x; if x=0 then return 0; d= digits(); numeric form; m.= 9; h= d+6

     numeric digits;  parse value format(x,2,1,,0) 'E0' with g 'E' _ .;  g= g *.5'e'_ % 2
       do j=0  while h>9;        m.j= h;              h= h % 2  +  1;    end
       do k=j+5  to 0  by '-1';  numeric digits m.k;  g= (g+x/g) * .5;   end;    return g</lang>

output when using the default internal inputs:

════════════════════════════════════════ orbital elements ═════════════════════════════════════════
            semi-major axis:  1
               eccentricity:  0.1
                inclination:  0
   ascending node longitude:  0.523598820058997050
      argument of periapsis:  0
               true anomaly:  0
                   position: ( 0.779422843398679832,  0.450000034653684237,  0)
                      speed: (-0.552770840960443759,  0.957427083179761535,  0)

════════════════════════════════════════ orbital elements ═════════════════════════════════════════
            semi-major axis:  1
               eccentricity:  0.1
                inclination:  0.174532925199432957
   ascending node longitude:  0.523598775598298873
      argument of periapsis:  0.785398163397448309
               true anomaly:  0
                   position: ( 0.237771283982206547,  0.860960261697715834,  0.110509023572075562)
                      speed: (-1.061933017480060047,  0.275850020569249507,  0.135747024865598167)

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>

Output:

Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

Sidef

Translation of: Perl

<lang ruby>func orbital_state_vectors(

   semimajor_axis,
   eccentricity,
   inclination,
   longitude_of_ascending_node,
   argument_of_periapsis,
   true_anomaly

) {

   static vec = frequire('Math::Vector::Real')
   var (i, j, k) = (vec.V(1,0,0), vec.V(0,1,0), vec.V(0,0,1))

   func muladd(v1, x1, v2, x2) {
       v1.mul(x1).add(v2.mul(x2))
   }

   func rotate(Ref i, Ref j, α) {
       (*i, *j) = (
           muladd(*i, +cos(α), *j, sin(α)),
           muladd(*i, -sin(α), *j, cos(α)),
       )
   }

   rotate(\i, \j, longitude_of_ascending_node)
   rotate(\j, \k, inclination)
   rotate(\i, \j, argument_of_periapsis)

   var l = (eccentricity == 1 ? 2*semimajor_axis
                              : semimajor_axis*(1 - eccentricity**2))

   var (c, s) = with(true_anomaly) { (.cos, .sin) }

   var r = l/(1 + eccentricity*c)
   var rprime = (s * r**2 / l)
   var position = muladd(i, c, j, s).mul(r)

   var speed = muladd(i, rprime*c - r*s, j, rprime*s + r*c)
   speed.div!(abs(speed))
   speed.mul!(sqrt(2/r - 1/semimajor_axis))

   struct Result { position, speed }
   Result([position,speed].map {|v| [v{:module}[]].map{Num(_)} }...)

}

var r = orbital_state_vectors(

   semimajor_axis: 1,
   eccentricity: 0.1,
   inclination: 0,
   longitude_of_ascending_node: 355/(113*6),
   argument_of_periapsis: 0,
   true_anomaly: 0,

)

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

Output:

[0.77942284339868, 0.450000034653684, 0]
[-0.552770840960444, 0.957427083179761, 0]

zkl

Translation of: Perl

<lang zkl>fcn orbital_state_vectors(semimajor_axis, eccentricity, inclination,

       longitude_of_ascending_node, argument_of_periapsis, true_anomaly){
  i,j,k:=T(1.0, 0.0, 0.0), T(0.0, 1.0, 0.0), T(0.0, 0.0, 1.0);

  vdot:=fcn(c,vector){ vector.apply('*,c) };
  vsum:=fcn(v1,v2)   { v1.zipWith('+,v2)  };
  rotate:='wrap(alpha, a,b){  // a&b are vectors: (x,y,z)
     return(vsum(vdot( alpha.cos(),a), vdot(alpha.sin(),b)), #cos(alpha)*a + sin(alpha)*b
            vsum(vdot(-alpha.sin(),a), vdot(alpha.cos(),b)));
  };
  i,j=rotate(longitude_of_ascending_node,i,j);
  j,k=rotate(inclination,		  j,k);
  i,j=rotate(argument_of_periapsis,      i,j);

  l:=if(eccentricity==1)   # PARABOLIC CASE
       semimajor_axis*2  else
       semimajor_axis*(1.0 - eccentricity.pow(2));;
  c,s,r:=true_anomaly.cos(), true_anomaly.sin(), l/(eccentricity*c + 1);
  rprime:=s*r.pow(2)/l;

  position:=vdot(r,vsum(vdot(c,i), vdot(s,j)));  #r*(c*i + s*j)

  speed:=vsum(vdot(rprime*c - r*s,i), vdot(rprime*s + r*c,j)); #(rprime*c - r*s)*i + (rprime*s + r*c)*j
  z:=speed.zipWith('*,speed).sum(0.0).sqrt();  #sqrt(speed**2)
  speed=vdot(1.0/z,speed);			#speed/z

  speed=vdot((2.0/r - 1.0/semimajor_axis).sqrt(),speed); #speed*sqrt(2/r - 1/semimajor_axis)

  return(position,speed);

}</lang> <lang zkl>orbital_state_vectors(

   1.0,                           # semimajor axis
   0.1,                           # eccentricity
   0.0,                           # inclination
   (0.0).pi/6,                    # longitude of ascending node
   0.0,                           # argument of periapsis
   0.0                            # true-anomaly

).println();</lang>

Output:

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