Orbital elements
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.)
11l
<lang 11l>F mulAdd(v1, x1, v2, x2)
R v1 * x1 + v2 * x2
F rotate(i, j, alpha)
R [mulAdd(i, cos(alpha), j, sin(alpha)), mulAdd(i, -sin(alpha), j, cos(alpha))]
F orbitalStateVectors(semimajorAxis, eccentricity, inclination, longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly)
V i = (1.0, 0.0, 0.0) V j = (0.0, 1.0, 0.0) V k = (0.0, 0.0, 1.0)
V 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]
V l = I (eccentricity == 1.0) {2.0} E 1.0 - eccentricity * eccentricity
l *= semimajorAxis
V c = cos(trueAnomaly)
V s = sin(trueAnomaly)
V r = 1 / (1.0 + eccentricity * c)
V rprime = s * r * r / l
V position = mulAdd(i, c, j, s) * r
V speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c)
speed = normalize(speed) * sqrt(2.0 / r - 1.0 / semimajorAxis)
R [position, speed]
V 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 : (0.787295801, 0.45454549, 0) Speed : (-0.5477226, 0.948683274, 0)
Ada
<lang Ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Generic_Real_Arrays; with Ada.Numerics.Generic_Elementary_Functions;
procedure Orbit is
type Real is new Long_Float;
package Real_Arrays is
new Ada.Numerics.Generic_Real_Arrays (Real => Real);
use Real_Arrays;
package Math is
new Ada.Numerics.Generic_Elementary_Functions (Float_Type => Real);
subtype Vector_3D is Real_Vector (1 .. 3);
procedure Put (V : Vector_3D) is
package Real_IO is
new Ada.Text_Io.Float_IO (Num => Real);
begin
Put ("(");
Real_IO.Put (V (1), Exp => 0, Aft => 6); Put (",");
Real_IO.Put (V (2), Exp => 0, Aft => 6); Put (",");
Real_IO.Put (V (3), Exp => 0, Aft => 6); Put (")");
end Put;
function Mul_Add (V1 : Vector_3D;
X1 : Real;
V2 : Vector_3D;
X2 : Real)
return Vector_3D is
begin
return V1 * X1 + V2 * X2;
end Mul_Add;
procedure Rotate (R1 : out Vector_3D;
R2 : out Vector_3D;
I : Vector_3D;
J : Vector_3D;
Alpha : Real) is
begin
R1 := Mul_Add (I, +Math.Cos (Alpha), J, Math.Sin (Alpha));
R2 := Mul_Add (I, -Math.Sin (Alpha), J, Math.Cos (Alpha));
end Rotate;
type Orbital_State_Vectors is record
Position : Vector_3D;
Speed : Vector_3D;
end record;
function Calculate_Orbital_State
(Semimajor_Axis : Real;
Eccentricity : Real;
Inclination : Real;
Longitude_Of_Ascending_Node : Real;
Argument_Of_Periapsis : Real;
True_Anomaly : Real)
return Orbital_State_Vectors
is
I : Vector_3D := (1.0, 0.0, 0.0);
J : Vector_3D := (0.0, 1.0, 0.0);
K : constant Vector_3D := (0.0, 0.0, 1.0);
P_R1, P_R2 : Vector_3D;
State : Orbital_State_Vectors;
Position : Vector_3D renames State.Position;
Speed : Vector_3D renames State.Speed;
begin
Rotate (P_R1, P_R2, I, J, Longitude_Of_Ascending_Node);
I := P_R1; J := P_R2;
Rotate (P_R1, P_R2, J, K, Inclination);
J := P_R1;
Rotate (P_R1, P_R2, I, J, Argument_Of_Periapsis);
I := P_R1; J := P_R2;
declare
L : constant Real :=
Semimajor_Axis * (if (Eccentricity = 1.0) then 2.0
else (1.0 - Eccentricity * Eccentricity));
C : constant Real := Math.Cos (True_Anomaly);
S : constant Real := Math.Sin (True_Anomaly);
R : constant Real := L / (1.0 + Eccentricity * C);
Rprime : constant Real := S * R * R / L;
begin
Position := Mul_Add (I, C, J, S) * R;
Speed := Mul_Add (I, Rprime * C - R * S,
J, Rprime * S + R * C);
Speed := Speed / abs (Speed);
Speed := Speed * Math.Sqrt (2.0 / R - 1.0 / Semimajor_Axis);
end;
return State;
end Calculate_Orbital_State;
Longitude : constant Real := 355.000 / (113.000 * 6.000); State : constant Orbital_State_Vectors :=
Calculate_Orbital_State
(Semimajor_Axis => 1.000,
Eccentricity => 0.100,
Inclination => 0.000,
Longitude_Of_Ascending_Node => Longitude,
Argument_Of_Periapsis => 0.000,
True_Anomaly => 0.000);
begin
Put ("Position : "); Put (State.Position); New_Line;
Put ("Speed : "); Put (State.Speed); New_Line;
end Orbit;</lang>
- Output:
Position : ( 0.779423, 0.450000, 0.000000) Speed : (-0.552771, 0.957427, 0.000000)
ALGOL W
(which is a translation of Kotlin which is a translation of ...).
<lang algolw>begin
% compute orbital elements %
% 3-element vector %
record Vector( real x, y, z );
% prints the components of the vector v %
procedure writeOnVector( reference(Vector) value v ) ;
writeon( r_format := "A", r_w := 10, r_d := 6, s_w := 0, "( ", x(v), ", ", y(v), ", ", z(v), " )" );
% returns a vector whose elements are the sum of the elements of v & w %
reference(Vector) procedure add( reference(Vector) value v, w ) ;
Vector( x(v) + x(w), y(v) + y(w), z(v) + z(w) );
% returns a vector whose elements are those of v multiplied by m %
reference(Vector) procedure mul( reference(Vector) value v
; real value m
) ;
Vector( x(v) * m, y(v) * m, z(v) * m );
% returns a vector whose elements are those of v divided by d %
reference(Vector) procedure divVR( reference(Vector) value v
; real value d
) ;
mul( v, 1 / d );
% returns the norm of the vector v %
real procedure vabs( reference(Vector) value v ) ;
sqrt( ( x(v) * x(v) ) + y(v) * y(v) + z(v) * z(v) );
% returns the sum of v1 * x1 and v2 * x2 %
reference(Vector) procedure mulAdd( reference(Vector) value v1, v2
; real value x1, x2
) ;
add( mul( v1, x1 ), mul( v2, x2 ) );
% sets ps to rotations of i and j by alpha %
procedure rotate( reference(Vector) value i, j
; real value alpha
; reference(Vector) array ps ( * )
) ;
begin
ps( 0 ) := mulAdd( i, j, cos( alpha ), sin( alpha ) );
ps( 1 ) := mulAdd( i, j, -sin( alpha ), cos( alpha ) )
end rotate ;
% sets position and speed vectors from the supplied elements %
procedure orbitalStateVectors( real value semimajorAxis
, eccentricity
, inclination
, longitudeOfAscendingNode
, argumentOfPeriapsis
, trueAnomaly
; reference(Vector) result position
, speed
) ;
begin
reference(Vector) i, j, k;
reference(Vector) array qs ( 0 :: 1 );
real L, c, s, r, rprime;
i := Vector( 1.0, 0.0, 0.0 );
j := Vector( 0.0, 1.0, 0.0 );
k := Vector( 0.0, 0.0, 1.0 );
L := 2.0;
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 not = 1 then L := 1 - eccentricity * eccentricity;
l := L * semimajorAxis;
c := cos( trueAnomaly );
s := sin( trueAnomaly );
r := L / ( 1.0 + eccentricity * c );
rprime := s * r * r / L;
position := mulAdd( i, j, c, s );
position := mul( position, r ) ;
speed := mulAdd( i, j, rprime * c - r * s, rprime * s + r * c );
speed := divVR( speed, vabs( speed ) );
speed := mul( speed, sqrt( 2 / r - 1 / semimajorAxis ) );
end orbitalStateVectors ;
% test the orbitalStateVectors routine %
begin
real longitude;
reference(Vector) position, speed;
longitude := 355.0 / ( 113.0 * 6.0 );
orbitalStateVectors( 1.0, 0.1, 0.0, longitude, 0.0, 0.0, position, speed );
write( "Position : " ); writeOnVector( position );
write( "Speed : " ); writeOnVector( speed )
end
end.</lang>
- Output:
Position : ( 0.779422, 0.450000, 0.000000 ) Speed : ( -0.552770, 0.957427, 0.000000 )
C
<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)
C#
<lang csharp>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);
}
}
}</lang>
- Output:
Position : (0.77942284339868, 0.450000034653684, 0) Speed : (-0.552770840960444, 0.957427083179762, 0)
C++
<lang cpp>#include <iostream>
- include <tuple>
class Vector { private:
double _x, _y, _z;
public:
Vector(double x, double y, double z) : _x(x), _y(y), _z(z) {
// empty
}
double getX() {
return _x;
}
double getY() {
return _y;
}
double getZ() {
return _z;
}
double abs() {
return sqrt(_x * _x + _y * _y + _z * _z);
}
Vector operator+(const Vector& rhs) const {
return Vector(_x + rhs._x, _y + rhs._y, _z + rhs._z);
}
Vector operator*(double m) const {
return Vector(_x * m, _y * m, _z * m);
}
Vector operator/(double m) const {
return Vector(_x / m, _y / m, _z / m);
}
friend std::ostream& operator<<(std::ostream& os, const Vector& v);
};
std::ostream& operator<<(std::ostream& os, const Vector& v) {
return os << '(' << v._x << ", " << v._y << ", " << v._z << ')';
}
std::pair<Vector, Vector> orbitalStateVectors(
double semiMajorAxis, double eccentricity, double inclination, double longitudeOfAscendingNode, double argumentOfPeriapsis, double trueAnomaly
) {
auto mulAdd = [](const Vector& v1, double x1, const Vector& v2, double x2) {
return v1 * x1 + v2 * x2;
};
auto rotate = [mulAdd](const Vector& iv, const Vector& jv, double alpha) {
return std::make_pair(
mulAdd(iv, +cos(alpha), jv, sin(alpha)),
mulAdd(iv, -sin(alpha), jv, cos(alpha))
);
};
Vector i(1, 0, 0); Vector j(0, 1, 0); Vector k(0, 0, 1);
auto 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;
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 / speed.abs(); speed = speed * sqrt(2.0 / r - 1.0 / semiMajorAxis);
return std::make_pair(position, speed);
}
int main() {
auto res = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0); std::cout << "Position : " << res.first << '\n'; std::cout << "Speed : " << res.second << '\n';
return 0;
}</lang>
- Output:
Position : (0.779423, 0.45, 0) Speed : (-0.552771, 0.957427, 0)
D
<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)
Go
<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)
J
<lang J>NB. euler rotation matrix, left hand rule
NB. x: axis (0, 1 or 2), y: angle in radians R=: Template:((2 1,:1 2) o.(,-)y* 1^2 X=: +/ .* NB. inner product norm=: % %:@X~
orbitalStateVectors=: {{ 'a e i Om w f'=. y
NB. a: semi-major axis
NB. e: eccentricity
NB. i: inclination
NB. Om: Longitude of the ascending node
NB. w: argument of Periapsis (the other "omega")
NB. f: true anomaly
L=. a*2:`]@.*1-*:e
'c s'=. 2{.,F=. 2 R f
ra=. L % 1+ e*c
rp=. s*ra*ra%L
ijk=. F X (2 R w)X(0 R i)X(2 R Om)
position=. ra*{.ijk
speed=. (%:(2%ra)-%a)*norm(rp,ra,0) X ijk
position,:speed
}}</lang>
Task example:<lang J> orbitalStateVectors 1 0.1 0 355r678 0 0
0.779423 0.45 0
_0.552771 0.957427 0</lang>
Java
<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)
jq
Works with gojq, the Go implementation of jq <lang jq># Array/vector operations def addVectors: transpose | map(add);
def multiply($m): map(. * $m);
def divide($d): map(1/$d);
def abs: map(.*.) | add | sqrt;
def orbitalStateVectors(semimajorAxis; eccentricity; inclination;
longitudeOfAscendingNode; argumentOfPeriapsis; trueAnomaly):
def mulAdd($v1; $x1; $v2; $x2):
[($v1|multiply($x1)), ($v2|multiply($x2))] | addVectors;
def rotate($i; $j; $alpha):
[mulAdd($i; $alpha|cos; $j; $alpha|sin),
mulAdd($i; -$alpha|sin; $j; $alpha|cos)];
[1, 0, 0] as $i | [0, 1, 0] as $j | [0, 0, 1] as $k | rotate($i; $j; longitudeOfAscendingNode) as [$i, $j] | rotate($j; $k; inclination) as [$j, $_] | rotate($i; $j; argumentOfPeriapsis) as [$i, $j] | (semimajorAxis * (if (eccentricity == 1) then 2 else (1 - eccentricity * eccentricity) end)) as $l | (trueAnomaly|cos) as $c | (trueAnomaly|sin) as $s | ($l / (1 + eccentricity * $c)) as $r | ($s * $r * $r / $l) as $rprime
| mulAdd($i; $c; $j; $s) | multiply($r) as $position
| mulAdd($i; $rprime * $c - $r * $s; $j; $rprime * $s + $r * $c) | divide(abs) | multiply( ((2 / $r) - (1 / semimajorAxis))|sqrt) as $speed | [$position, $speed] ;</lang>
The Task <lang jq>orbitalStateVectors(1; 0.1; 0; 355 / (113 * 6); 0; 0) | "Position : \(.[0])",
"Speed : \(.[1])"</lang>
- Output:
Position : [0.7794228433986797,0.45000003465368416,0] Speed : [1.228379551983482,1.228379551983482,1.228379551983482]
Julia
<lang julia>using GeometryTypes import Base.abs, Base.print
Vect = Point3 Base.abs(p::Vect) = sqrt(sum(x -> x*x, p)) Base.print(io::IO, p::Vect) = print(io, "(", p[1], ", ", p[2], ", ", p[3], ")") muladd(v1, x1, v2, x2) = v1 * x1 + v2 * x2 rotate(i, j, a) = Pair(muladd(i, cos(a), j, sin(a)), muladd(i, -sin(a), j, cos(a)))
function orbitalStateVectors(semimajorAxis, eccentricity, inclination,
longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly) i, j, k = Vect(1.0, 0.0, 0.0), Vect(0.0, 1.0, 0.0), Vect(0.0, 0.0, 1.0) p = rotate(i, j, longitudeOfAscendingNode) i, j = p p = rotate(j, k, inclination) p = rotate(i, p[1], argumentOfPeriapsis) i, j = p l = semimajorAxis * (eccentricity == 1.0 ? 2.0 : (1.0 - eccentricity * eccentricity)) c, s = cos(trueAnomaly), sin(trueAnomaly) r = l / (1.0 + eccentricity * c) rprime, position = s * r * r / l, muladd(i, c, j, s) * r speed = muladd(i, rprime * c - r * s, j, rprime * s + r * c) speed /= abs(speed) speed *= sqrt(2.0 / r - 1.0 / semimajorAxis) return Pair(position, speed)
end
function testorbitalmath()
(position, speed) = orbitalStateVectors(1.0, 0.1, 0.0, 355.0 / (113.0 * 6.0), 0.0, 0.0)
println("Position : $position\nSpeed : $speed")
end
testorbitalmath()
</lang>
- Output:
Position : (0.7794228433986797, 0.45000003465368416, 0.0) Speed : (-0.5527708409604438, 0.9574270831797618, 0.0)
Kotlin
<lang scala>// version 1.1.4-3
class Vector(val x: Double, val y: Double, val z: Double) {
operator fun plus(other: Vector) = Vector(x + other.x, y + other.y, z + other.z) operator fun times(m: Double) = Vector(x * m, y * m, z * m)
operator fun div(d: Double) = this * (1.0 / d)
fun abs() = Math.sqrt(x * x + y * y + z * z)
override fun toString() = "($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 : $speed")
}</lang>
- Output:
Position : (0.7794228433986797, 0.45000003465368416, 0.0) Speed : (-0.5527708409604438, 0.9574270831797618, 0.0)
Nim
<lang Nim>import math, strformat
type Vector = tuple[x, y, z: float]
func `+`(v1, v2: Vector): Vector = (v1.x + v2.x, v1.y + v2.y, v1.z + v2.z) func `*`(v: Vector; m: float): Vector = (v.x * m, v.y * m, v.z * m) func `*=`(v: var Vector; m: float) = v.x *= m; v.y *= m; v.z *= m func `/=`(v: var Vector; d: float) = v.x /= d; v.y /= d; v.z /= d func abs(v: Vector): float = sqrt(v.x * v.x + v.y * v.y + v.z * v.z) func `$`(v: Vector): string = &"({v.x}, {v.y}, {v.z})"
func orbitalStateVectors(semimajorAxis: float,
eccentricity: float,
inclination: float,
longitudeOfAscendingNode: float,
argumentOfPeriapsis: float,
trueAnomaly: float): tuple[position, speed: Vector] =
var i: Vector = (1.0, 0.0, 0.0) j: Vector = (0.0, 1.0, 0.0) k: Vector = (0.0, 0.0, 1.0)
func mulAdd(v1: Vector; x1: float; v2: Vector; x2: float): Vector = v1 * x1 + v2 * x2
func rotate(a, b: Vector; alpha: float): (Vector, Vector) = (mulAdd(a, cos(alpha), b, sin(alpha)), mulAdd(a, -sin(alpha), b, cos(alpha)))
var p = rotate(i, j, longitudeOfAscendingNode) (i, j) = p p = rotate(j, k, inclination) j = p[0] p = rotate(i, j, argumentOfPeriapsis) (i, j) = p
let l = semimajorAxis * (if eccentricity == 1: 2.0 else: 1.0 - eccentricity * eccentricity) c = cos(trueAnomaly) s = sin(trueAnomaly) r = l / (1.0 + eccentricity * c) rprime = s * r * r / l
result.position = mulAdd(i, c, j, s) * r result.speed = mulAdd(i, rprime * c - r * s, j, rprime * s + r * c) result.speed /= abs(result.speed) result.speed *= sqrt(2 / r - 1 / semimajorAxis)
let (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)
echo "Position: ", position echo "Speed: ", speed</lang>
- Output:
Position: (0.7794228433986797, 0.4500000346536842, 0.0) Speed: (-0.5527708409604438, 0.9574270831797618, 0.0)
ooRexx
<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 'Raku:' pi=rxCalcpi(16) ps=orbitalStateVectors(1,.1,pi/18,pi/6,pi/4,0) /*Raku*/ 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) Raku: Position : (0.2377712839822067,0.8609602616977158,0.1105090235720755) Speed : (-1.061933017480060,0.2758500205692495,0.1357470248655981)
Perl
<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' )
};
Phix
with javascript_semantics 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}
Prolog
This implementation uses the CLP/R library of swi-prolog, but doesn't have to. This removes the need for a vector divide and has limited capability to reverse the functionality (eg: given the position/speed find some orbital elements).
<lang Prolog>:- use_module(library(clpr)).
v3_add(v(X1,Y1,Z1),v(X2,Y2,Z2),v(X,Y,Z)) :- { X = X1 + X2 }, { Y = Y1 + Y2 }, { Z = Z1 + Z2 }.
v3_mul(v(X1,Y1,Z1),M,v(X,Y,Z)) :- { X = X1 * M }, { Y = Y1 * M }, { Z = Z1 * M }.
v3_muladd(V1,X1,V2,X2,R) :- v3_mul(V1,X1,V1X1), v3_mul(V2,X2,V2X2), v3_add(V1X1,V2X2,R).
v3_rotate(IV, JV, Alpha, R1, R2) :- { SinA = sin(Alpha) }, { CosA = cos(Alpha) }, { NegSinA = 0 - SinA }, v3_muladd(IV, CosA, JV, SinA, R1), v3_muladd(IV, NegSinA, JV, CosA, R2).
v3_abs(v(X,Y,Z), Abs) :- { Abs = (X * X + Y * Y + Z * Z) ^ 0.5 }.
orbital_state_vectors( o(SemiMajor,Ecc,Inc,LongAscNode,ArgPer,TrueAnon), Position, Speed) :-
v3_rotate(v(1,0,0),v(0,1,0),LongAscNode,I1,J1), v3_rotate(J1,v(0,0,1),Inc,J2,_), v3_rotate(I1,J2,ArgPer,I,J),
find_l(Ecc, SemiMajor, L),
{ C = cos(TrueAnon) }, { S = sin(TrueAnon) }, { R = L / (1.0 + Ecc * C ) }, { RPrime = S * R * R / L },
v3_muladd(I, C, J, S, P1), v3_mul(P1, R, Position),
{ SpeedIr = RPrime * C - R * S }, { SpeedJr = RPrime * S + R * C }, v3_muladd(I, SpeedIr, J, SpeedJr, SpeedA), v3_abs(SpeedA, SpeedAbs), v3_mul(SpeedDiv, SpeedAbs, SpeedA), { Sf = (2.0 / R - 1.0 / SemiMajor ) ^ 0.5 }, v3_mul(SpeedDiv, Sf, Speed).
find_l(1.0, SemiMajor, L) :-
{ L = SemiMajor * 2.0 }.
find_l(Ecc, SemiMajor, L) :-
dif(Ecc,1.0),
{ L = SemiMajor * (1.0 - Ecc * Ecc) }.</lang>
- Output:
?- { T = 355 / (113 * 6) }, orbital_state_vectors(o(1.0,0.1,0.0,T,0,0), P, S).
T = 0.523598820058997,
P = v(0.7794228433986797, 0.45000003465368416, 0.0),
S = v(-0.5527708409604438, 0.9574270831797618, 0.0) .
?-
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)
Raku
(formerly 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}
REXX
version 1
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, but with greater decimal digits precision. <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($)) r2r: return arg(1) // (pi() * 2) /*normalize radians ──► a unit circle*/ .sinCos: arg z 1 _,i; do k=2 by 2 until p=z; p=z; _= -_*$ /(k*(k+i)); z=z+_; end; return z /*──────────────────────────────────────────────────────────────────────────────────────*/ 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'; $= x * x; 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; $= x * x; return .sinCos(x, 1)
/*──────────────────────────────────────────────────────────────────────────────────────*/ 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
<lang ruby>func orbital_state_vectors(
semimajor_axis, eccentricity, inclination, longitude_of_ascending_node, argument_of_periapsis, true_anomaly
) {
var (i, j, k) = (
Vector(1, 0, 0),
Vector(0, 1, 0),
Vector(0, 0, 1),
)
func muladd(v1, x1, v2, x2) {
(v1 * x1) + (v2 * 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)*r
var speed = muladd(i, rprime*c - r*s, j, rprime*s + r*c) speed /= speed.abs speed *= sqrt(2/r - 1/semimajor_axis)
struct Result { position, speed }
Result(position, speed)
}
for args in ([
[1, 0.1, 0, 355/(113*6), 0, 0], [1, 0.1, Num.pi/18, Num.pi/6, Num.pi/4, 0]
]) {
var r = orbital_state_vectors(args...)
say "Arguments: #{args}:"
say "Position : #{r.position}"
say "Speed : #{r.speed}\n"
}</lang>
- Output:
Arguments: [1, 1/10, 0, 355/678, 0, 0]: Position : Vector(0.779422843398679832042176328223663037464703527986, 0.450000034653684237432302249506712706822033851071, 0) Speed : Vector(-0.552770840960443759673279062314259546277084494097, 0.957427083179761535246200368614952095349966503287, 0) Arguments: [1, 1/10, 0.174532925199432957692369076848861271344287188854, 0.523598775598298873077107230546583814032861566563, 0.785398163397448309615660845819875721049292349844, 0]: Position : Vector(0.23777128398220654779107184959165027147748809404, 0.860960261697715834668966272382699039216399966872, 0.110509023572075562109405412890808505271310143909) Speed : Vector(-1.06193301748006004757467368094494935655538772696, 0.275850020569249507846452830330085489348356659642, 0.135747024865598167166145512759280712986072818844)
Swift
<lang swift>import Foundation
public struct Vector {
public var x = 0.0 public var y = 0.0 public var z = 0.0
public init(x: Double, y: Double, z: Double) {
(self.x, self.y, self.z) = (x, y, z)
}
public func mod() -> Double {
(x * x + y * y + z * z).squareRoot()
}
public static func + (lhs: Vector, rhs: Vector) -> Vector {
return Vector(
x: lhs.x + rhs.x,
y: lhs.y + rhs.y,
z: lhs.z + rhs.z
)
}
public static func * (lhs: Vector, rhs: Double) -> Vector {
return Vector(
x: lhs.x * rhs,
y: lhs.y * rhs,
z: lhs.z * rhs
)
}
public static func *= (lhs: inout Vector, rhs: Double) {
lhs.x *= rhs
lhs.y *= rhs
lhs.z *= rhs
}
public static func / (lhs: Vector, rhs: Double) -> Vector {
return lhs * (1 / rhs)
}
public static func /= (lhs: inout Vector, rhs: Double) {
lhs = lhs * (1 / rhs)
}
}
extension Vector: CustomStringConvertible {
public var description: String {
return String(format: "%.6f\t%.6f\t%.6f", x, y, z)
}
}
private func mulAdd(v1: Vector, x1: Double, v2: Vector, x2: Double) -> Vector {
return v1 * x1 + v2 * x2
}
private func rotate(_ i: Vector, _ j: Vector, alpha: Double) -> (Vector, Vector) {
return ( mulAdd(v1: i, x1: +cos(alpha), v2: j, x2: sin(alpha)), mulAdd(v1: i, x1: -sin(alpha), v2: j, x2: cos(alpha)) )
}
public func orbitalStateVectors(
semimajorAxis: Double, eccentricity: Double, inclination: Double, longitudeOfAscendingNode: Double, argumentOfPeriapsis: Double, trueAnomaly: Double
) -> (Vector, Vector) {
var i = Vector(x: 1.0, y: 0.0, z: 0.0) var j = Vector(x: 0.0, y: 1.0, z: 0.0) let k = Vector(x: 0.0, y: 0.0, z: 1.0)
(i, j) = rotate(i, j, alpha: longitudeOfAscendingNode) (j, _) = rotate(j, k, alpha: inclination) (i, j) = rotate(i, j, alpha: argumentOfPeriapsis)
let l = eccentricity == 1.0 ? 2.0 : 1.0 - eccentricity * eccentricity let c = cos(trueAnomaly) let s = sin(trueAnomaly) let r = l / (1.0 + eccentricity * c) let rPrime = s * r * r / l let position = mulAdd(v1: i, x1: c, v2: j, x2: s) * r var speed = mulAdd(v1: i, x1: rPrime * c - r * s, v2: j, x2: rPrime * s + r * c)
speed /= speed.mod() speed *= (2.0 / r - 1.0 / semimajorAxis).squareRoot()
return (position, speed)
}
let (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
)
print("Position: \(position); Speed: \(speed)")</lang>
- Output:
Position: 0.779423 0.450000 0.000000; Speed: -0.552771 0.957427 0.000000
Wren
<lang ecmascript>class Vector {
construct new(x, y, z) {
_x = x
_y = y
_z = z
}
x { _x }
y { _y }
z { _z }
+(other) { Vector.new(_x + other.x, _y + other.y, _z + other.z) }
*(m) { Vector.new(_x * m, _y * m, _z * m) }
/(d) { this * (1/d) }
abs { (_x *_x + _y *_y + _z * _z).sqrt }
toString { "(%(_x), %(_y), %(_z))" }
}
var orbitalStateVectors = Fn.new { |semimajorAxis, eccentricity, inclination,
longitudeOfAscendingNode, argumentOfPeriapsis, trueAnomaly| var i = Vector.new(1, 0, 0) var j = Vector.new(0, 1, 0) var k = Vector.new(0, 0, 1)
var mulAdd = Fn.new { |v1, x1, v2, x2| v1 * x1 + v2 * x2 }
var rotate = Fn.new { |i, j, alpha|
return [mulAdd.call(i, alpha.cos, j, alpha.sin),
mulAdd.call(i, -alpha.sin, j, alpha.cos)]
}
var p = rotate.call(i, j, longitudeOfAscendingNode) i = p[0] j = p[1] p = rotate.call(j, k, inclination) j = p[0] p = rotate.call(i, j, argumentOfPeriapsis) i = p[0] j = p[1]
var l = semimajorAxis * ((eccentricity == 1) ? 2 : (1 - eccentricity * eccentricity)) var c = trueAnomaly.cos var s = trueAnomaly.sin var r = l / (1 + eccentricity * c) var rprime = s * r * r / l var position = mulAdd.call(i, c, j, s) * r var speed = mulAdd.call(i, rprime * c - r * s, j, rprime * s + r * c) speed = speed / speed.abs speed = speed * (2 / r - 1 / semimajorAxis).sqrt return [position, speed]
}
var ps = orbitalStateVectors.call(1, 0.1, 0, 355 / (113 * 6), 0, 0) System.print("Position : %(ps[0])") System.print("Speed : %(ps[1])")</lang>
- Output:
Position : (0.77942284339868, 0.45000003465368, 0) Speed : (-0.55277084096044, 0.95742708317976, 0)
zkl
<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))