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
“RADIUS VECTOR”?→Mat
R
“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
