## Friday, May 1, 2015

### HP Prime and Casio Prizm: Orbital Elements

HP Prime and Casio Prizm:  Orbital Elements

 Satellite:  Longitude Angle, Inclination Angle, Node Vector, Radius Vector

Input:

Three types of unit systems that can be used:

MI-LB-S: Miles-Pounds-Seconds
KM-KG-S:  Kilograms-Kilometers-Seconds
MU=1:  Sets the gravitational parameter (μ) to 1.  Used for problems involving canonical units.

Radius vector:  position where the satellite is located relative to the celestial object the satellite is orbiting around.  Celestial objects include planets and stars.

Velocity vector:  velocity of the satellite.

Canonical Units:  The distance unit is set between the average distance between the satellite and its reference celestial object.  The time unit is selected such that the velocity of the satellite is 1 distance unit per time unit.  This ensures that μ = 1.

Output:

Angular Momentum Vector

Node Vector

Inclination Angle of the satellite:  angle between the z-axis (J vector [0,0,1]) and the angular momentum vector.

Longitude Angle:  the angle between the I vector ([1,0,0]) and when the satellite crosses the fundamental plane.  For orbits around Earth, the measure statements from the vector pointing towards the Vernal Equinox (known as the First Point of Aries*) and going eastward.

(Even though today, in 2015, the Vernal Equinox is in the constellation Pisces, slowly moving towards Aquarius.)

Eccentricity:  The eccentricity of the orbit.  The vector and the norm is given.  The eccentricity determines the path of the orbit:
If e = 0, the orbit is a circle.
If e < 1, the orbit is an ellipse.
If e = 1, the orbit is a parabola.
If e > 1, the orbit is a hyperbola.

HP Prime:  ORBELEM

** The HP Prime version sets the calculator to Fixed 5 decimal mode.  All results shown are rounded to five digits.

EXPORT ORBELEM(M,vr,vv)
BEGIN
// 2015-05-01 EWS, Orbital Elements
// mass, radius vector, velocity vector

LOCAL mu,vh,vn,ve;
LOCAL R,H,N,E,I,L,C;
LOCAL str,s;

// Degrees Mode
HAngle:=1;

// Set to Fixed 5 Mode
HFormat:=1;
HDigits:=5;

// Main
CHOOSE(C,"Unit System","MI-LB-S",
"KM-KG-S","mu=1");
IF C==1 THEN
mu:=95629.523435*M; END;
IF C==2 THEN
mu:=398600.4418*M; END;
IF C==3 THEN
mu:=1; END;
vh:=CROSS(vr,vv);
MSGBOX("Angular Momentum Vector: "+vh);
vn:=CROSS([0,0,1],vh);
MSGBOX("Node Vector: "+vn);
H:=ABS(vh);
IF H≠0 THEN
I:=ACOS(vh(3)/H);
MSGBOX("Inclination:" +I+"°");
ELSE
MSGBOX("No Inclination Angle");
END;
N:=ABS(vn);
IF N≠0 THEN
L:=ACOS(vn(1)/N);
MSGBOX("Longitude: "+L+"°");
ELSE
MSGBOX("No Longitude Angle");
END;
V:=ABS(vv);
R:=ABS(vr);
ve:=1/mu*((V^2-mu/R)*vr-DOT(vr,vv)*vv);
E:=ABS(ve);

str:={"Circle","Ellipse","Parabola",
"Hyperbola"};
IF E=0 THEN s:=1; END;
IF E<1 THEN s:=2; END;
IF E=1 THEN s:=3; END;
IF E>1 THEN s:=4; END;
MSGBOX("Eccentricity: "+E+"; "+str(s));

END;

Casio Prizm:  ORBELEM

Deg
Menu “UNIT SYSTEM”, “MI-LB-S”, 1, “KM-KG-S”, 2, “MU=1”, 3
Lbl 1
“MASS”?→M
95629.5234325*M→U
Goto 0
Lbl 2
“MASS”?→M
398600.4418*M→U
Goto 0
Lbl 3
1→U
Goto 0
Lbl 0
“VELOCITY VECTOR”?→Mat V
CrossP(Mat R, Mat V)→Mat H
“ANGULAR MOMENTUM VECTOR:”  (right triangle symbol)
Mat H
CrossP([[0,0,1]],Mat H) →Mat N
“NODE VECTOR:”
Mat N
“INCLINATION:”
If H≠0
Then
cosˉ¹(Mat H[1,3]÷H)→ I
I
Else
“UNDEF”
IfEnd
Norm(Mat N)→N
“LONGITUDE ANGLE:”
If N≠0
Then
cosˉ¹(Mat N[1,1]÷N)→ L
L
Else
“UNDEF”
IfEnd
Norm(Mat V)→V
Norm(Mat R)→R
1÷U*((V^2-U÷R)*Mat R-DotP(Mat R,Mat V)*Mat V)→Mat E
Norm(Mat E)→Mat E
“ECCENTRICITY:”
E
E=0 “CIRCLE”
E<1 “ELLIPSE”
E=1 “PARABOLA”
E>1 “HYPERBOLA”

Example:
A satellite orbiting Earth (mass of 1.3170 * 10^25 pounds) has a radius vector of [[ 14700, 18268, 11500 ]] (miles) with the velocity of [[ 3.5, 4.5, 3.2 ]] (miles/sec).  Recall that an hour is 3600 seconds.

Then:
Angular Momentum Vector: [[ 6707.6, -6790, 2212]]
Node Vector:  [[6790, 6707.6, 0]]
Incidence Angle:  76.95158555°
Longitude Angle: 44.6502257°
Eccentricity:  1, Parabola

Source:

Roger R. Bate, Donald D. Mueller, Jerry E. White.  “Fundamental of Astrodynamics” Dover Publications, Inc.  New York:  1971

Eddie

This blog is property of Edward Shore.  2015

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