Orbital elements: Difference between revisions
→{{header|REXX}}: added REXX version 2. |
m →version 2: simplified the REXX code by eliding the need to use a slash separator for vectors, other improvements. |
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<lang rexx>/*REXX pgm converts orbital elements ──► orbital state vectors (angles are in radians).*/ |
<lang rexx>/*REXX pgm converts orbital elements ──► orbital state vectors (angles are in radians).*/ |
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numeric digits length(pi() ) - 1 /*limited to pi len, but show 1/3 digs.*/ |
numeric digits length(pi() ) - 1 /*limited to pi len, but show 1/3 digs.*/ |
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parse value orbV(1, 0.1, 0, 355 / (113*6), 0, 0) |
parse value orbV(1, 0.1, 0, 355 / (113*6), 0, 0) with position '~' speed |
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say ' position:' show(position) |
say ' position:' show(position) |
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say ' speed:' show(speed) |
say ' speed:' show(speed) |
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exit /*stick a fork in it, we're all done. */ |
exit /*stick a fork in it, we're all done. */ |
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/*──────────────────────────────────────────────────────────────────────────────────────*/ |
/*──────────────────────────────────────────────────────────────────────────────────────*/ |
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absV: procedure; parse arg x |
absV: procedure; parse arg x y z; return sqrt(x**2 + y**2 + z**2) |
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divV: procedure; parse arg x |
divV: procedure; parse arg x y z,d; return (x / d) (y / d) (z / d) |
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mulV: procedure; parse arg x |
mulV: procedure; parse arg x y z,d; return (x * d) (y * d) (z * d) |
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show: procedure; parse arg a |
show: procedure; parse arg a b c; return '('fmt(a)"," fmt(b)',' fmt(c)")" |
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fmt: procedure; parse arg #; |
fmt: procedure; parse arg #; return left('', #>=0)format(#, , digits() % 3) /1 |
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pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923; return pi |
pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923; return pi |
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rot: procedure; parse arg i,j,$; return MA(i, |
rot: procedure; parse arg i,j,$; return MA(i,cos($),j,sin($))'~'MA(i, -sin($), j, cos($)) |
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r2r: return arg(1) // (pi() *2) |
r2r: return arg(1) // (pi() * 2) /*normalize radians ──► a unit circle*/ |
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/*──────────────────────────────────────────────────────────────────────────────────────*/ |
/*──────────────────────────────────────────────────────────────────────────────────────*/ |
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orbV: procedure; parse arg semiMaj, eccentricity, inclination, node, periapsis, anomaly |
orbV: procedure; parse arg semiMaj, eccentricity, inclination, node, periapsis, anomaly |
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i= 1 0 0; j= 0 1 0; k= 0 0 1 |
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parse value rot(i, |
parse value rot(i, j, node) with i '~' j /*rotate ascending node longitude.*/ |
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parse value rot(j, k, inclination) with j '~' /*rotate the inclination. */ |
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parse value rot(i, |
parse value rot(i, j, periapsis) with i '~' j /*rotate the argument of periapsis*/ |
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if eccentricity=1 then L= 2 |
if eccentricity=1 then L= 2 |
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else L= 1 - eccentricity**2 |
else L= 1 - eccentricity**2 |
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r= L / (1 + eccentricity * c) |
r= L / (1 + eccentricity * c) |
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@= s*r**2 / L; speed= MA(i, @*c - r*s, j, @*s + r*c) |
@= s*r**2 / L; speed= MA(i, @*c - r*s, j, @*s + r*c) |
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return mulV(MA(i, |
return mulV( MA(i,c,j,s), r)'~'mulV( divV( speed, absV(speed)),sqrt(2/r -1/semiMaj)) |
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/*──────────────────────────────────────────────────────────────────────────────────────*/ |
/*──────────────────────────────────────────────────────────────────────────────────────*/ |
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MA: procedure; parse arg x |
MA: procedure; parse arg x y z,a,xx yy zz,b; return (x*a+xx*b) (y*a+yy*b) (z*a+zz*b) |
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return (x*a + xx*b)slash || (y*a + yy*b)slash || (z*a + zz*b) |
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/*──────────────────────────────────────────────────────────────────────────────────────*/ |
/*──────────────────────────────────────────────────────────────────────────────────────*/ |
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cos: procedure; parse arg x; x= r2r(x); a=abs(x); hpi= pi * .5 |
cos: procedure; parse arg x; x= r2r(x); a=abs(x); hpi= pi * .5 |
Revision as of 02:07, 29 October 2018
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
<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
<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
<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
<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
<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
<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
- 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 Say 'Vector:'||x'/'y'/'z
- method tostring
expose x y z Return '('||x','y','z')'
- method abs
expose x y z Return rxCalcsqrt(x**2+y**2+z**2)
- method '*'
expose x y z Parse Arg f Return .vector~new(x*f,y*f,z*f)
- method '/'
expose x y z Parse Arg f Return .vector~new(x/f,y/f,z/f)
- method '+'
expose x y z Use Arg v2 Return .vector~new(x+v2~x,y+v2~y,z+v2~z)
- routine orbitalStateVectors
Use Arg semimajorAxis,,
eccentricity,, inclination,, longitudeOfAscendingNode,, argumentOfPeriapsis,, trueAnomaly
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) s=sin(trueAnomaly) 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)) /* position~print speed~print
- /
Return .v2~new(position,speed)
- routine muladd
Use Arg v1,x1,v2,x2 w1=v1~'*'(x1) w2=v2~'*'(x2) Return w1~'+'(w2)
- routine rotate
Use Arg i,j,alpha 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 : (0.779422844,0.450000035,0) Speed : (-0.552770842,0.957427085,0)
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:
$VAR1 = { 'position' => bless( [ '0.77942284339868', '0.450000034653684', '0' ], 'Math::Vector::Real' ), 'speed' => bless( [ '-0.552770840960444', '0.957427083179762', '0' ], 'Math::Vector::Real' ) };
Perl 6
We'll use the Clifford geometric algebra library but only for the vector operations. <lang perl6>sub orbital-state-vectors(
Real :$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
<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
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) s=my_sin(trueAnomaly) 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)) 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)
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,'R')) yy=mulAdd(i,-my_sin(alpha,16,'R'),j,my_cos(alpha,16,'R')) 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=pi(2*prec) 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 : (-0.5527708425678772,0.9574270859639177,0)
version 2
Translation of REXX version 1. <lang rexx>/*REXX pgm converts orbital elements ──► orbital state vectors (angles are in radians).*/ numeric digits length(pi() ) - 1 /*limited to pi len, but show 1/3 digs.*/ parse value orbV(1, 0.1, 0, 355 / (113*6), 0, 0) with position '~' speed say ' position:' show(position) say ' speed:' show(speed) exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ absV: procedure; parse arg x y z; return sqrt(x**2 + y**2 + z**2) divV: procedure; parse arg x y z,d; return (x / d) (y / d) (z / d) mulV: procedure; parse arg x y z,d; return (x * d) (y * d) (z * d) show: procedure; parse arg a b c; return '('fmt(a)"," fmt(b)',' fmt(c)")" fmt: procedure; parse arg #; return left(, #>=0)format(#, , digits() % 3) /1 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*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ orbV: procedure; parse arg semiMaj, eccentricity, inclination, node, periapsis, anomaly
i= 1 0 0; j= 0 1 0; k= 0 0 1 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 anomoly*/ r= L / (1 + eccentricity * c) @= s*r**2 / L; speed= MA(i, @*c - r*s, j, @*s + r*c) return mulV( MA(i,c,j,s), r)'~'mulV( divV( speed, absV(speed)),sqrt(2/r -1/semiMaj))
/*──────────────────────────────────────────────────────────────────────────────────────*/ MA: procedure; parse arg x y z,a,xx yy zz,b; return (x*a+xx*b) (y*a+yy*b) (z*a+zz*b) /*──────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; parse arg x; x= r2r(x); 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; parse arg x; x=r2r(x); numeric fuzz min(5, max(1, digits() -3))
if x=pi * .5 then return 1; if x==pi * 1.5 then return -1 if abs(x)=pi | x=0 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; parse 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:
position: ( 0.779422843398679832042, 0.450000034653684237432, 0) speed: (-0.552770840960443759673, 0.957427083179761535246, 0)
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
) {
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
<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))