**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