Orbital elements

From Rosetta Code
Orbital elements is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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[edit]

Translation of: Kotlin
#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;
}
Output:
Position : (0.77942284339867973, 0.45000003465368416, 0)
Speed    : (-0.55277084096044382, 0.95742708317976177, 0)

C#[edit]

Translation of: D
using System;
 
namespace OrbitalElements {
class Vector {
public Vector(double x, double y, double z) {
X = x;
Y = y;
Z = z;
}
 
public double X { get; set; }
public double Y { get; set; }
public double Z { get; set; }
 
public double Abs() {
return Math.Sqrt(X * X + Y * Y + Z * Z);
}
 
public static Vector operator +(Vector lhs, Vector rhs) {
return new Vector(lhs.X + rhs.X, lhs.Y + rhs.Y, lhs.Z + rhs.Z);
}
 
public static Vector operator *(Vector self, double m) {
return new Vector(self.X * m, self.Y * m, self.Z * m);
}
 
public static Vector operator /(Vector self, double m) {
return new Vector(self.X / m, self.Y / m, self.Z / m);
}
 
public override string ToString() {
return string.Format("({0}, {1}, {2})", X, Y, Z);
}
}
 
class Program {
static Tuple<Vector, Vector> OrbitalStateVectors(
double semiMajorAxis,
double eccentricity,
double inclination,
double longitudeOfAscendingNode,
double argumentOfPeriapsis,
double trueAnomaly
) {
Vector mulAdd(Vector v1, double x1, Vector v2, double x2) {
return v1 * x1 + v2 * x2;
}
 
Tuple<Vector, Vector> rotate(Vector iv, Vector jv, double alpha) {
return new Tuple<Vector, Vector>(
mulAdd(iv, +Math.Cos(alpha), jv, Math.Sin(alpha)),
mulAdd(iv, -Math.Sin(alpha), jv, Math.Cos(alpha))
);
}
 
var i = new Vector(1, 0, 0);
var j = new Vector(0, 1, 0);
var k = new Vector(0, 0, 1);
 
var p = rotate(i, j, longitudeOfAscendingNode);
i = p.Item1; j = p.Item2;
p = rotate(j, k, inclination);
j = p.Item1;
p = rotate(i, j, argumentOfPeriapsis);
i = p.Item1; j = p.Item2;
 
var l = semiMajorAxis * ((eccentricity == 1.0) ? 2.0 : (1.0 - eccentricity * eccentricity));
var c = Math.Cos(trueAnomaly);
var s = Math.Sin(trueAnomaly);
var r = l / (1.0 + eccentricity * c);
var rprime = s * r * r / l;
var 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 new Tuple<Vector, Vector>(position, speed);
}
 
static void Main(string[] args) {
var res = OrbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0);
Console.WriteLine("Position : {0}", res.Item1);
Console.WriteLine("Speed  : {0}", res.Item2);
}
}
}
Output:
Position : (0.77942284339868, 0.450000034653684, 0)
Speed    : (-0.552770840960444, 0.957427083179762, 0)

D[edit]

Translation of: Kotlin
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]);
}
Output:
Position : (0.7794228433986798, 0.4500000346536842, 0.0000000000000000)
Speed    : (-0.5527708409604437, 0.9574270831797614, 0.0000000000000000)

Go[edit]

Translation of: Kotlin
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)
}
Output:
Position : (0.7794228433986797, 0.45000003465368416, 0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0)

Java[edit]

Translation of: Kotlin
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]);
}
}
Output:
Position : (0.7794228433986797, 0.4500000346536842, 0.0000000000000000)
Speed : (-0.5527708409604438, 0.9574270831797618, 0.0000000000000000)

Kotlin[edit]

Translation of: Sidef
// 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")
}
Output:
Position : (0.7794228433986797, 0.45000003465368416, 0.0)
Speed    : (-0.5527708409604438, 0.9574270831797618, 0.0)

ooRexx[edit]

Translation of: Java
/* 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
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[edit]

Translation of: Perl 6
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
;
Output:
$VAR1 = {
          'position' => bless( [
                                 '0.77942284339868',
                                 '0.450000034653684',
                                 '0'
                               ], 'Math::Vector::Real' ),
          'speed' => bless( [
                              '-0.552770840960444',
                              '0.957427083179762',
                              '0'
                            ], 'Math::Vector::Real' )
        };

Perl 6[edit]

We'll use the Clifford geometric algebra library but only for the vector operations.

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

Phix[edit]

Translation of: Python
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)
Output:
Position :{0.7872958014,0.4545454895,0}
Speed    :{-0.5477225997,0.9486832737,0}

Python[edit]

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

REXX[edit]

version 1[edit]

Translation of: Java

Vectors are represented by strings: 'x/y/z'

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

version 2[edit]

Re-coding of REXX version 1.

/*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
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[edit]

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

Sidef[edit]

Translation of: Perl
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(', ')+']'
Output:
[0.77942284339868, 0.450000034653684, 0]
[-0.552770840960444, 0.957427083179761, 0]

zkl[edit]

Translation of: Perl
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);
}
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();
Output:
L(L(0.779423,0.45,0),L(-0.552771,0.957427,0))