Reference: 13 Steps to Compute Classic Orbital Elements

13 Steps to Compute Classic Orbital Elements

To compute the six classic orbital elements, some experts suggest 13 steps as described in textbook published by Embry Riddle University, Space Mechanics by Howard D. Curtis, pages 6.10-12, Sep 2003.

Seq.	Name	Symbol	Equation
1.	Distance	r
2.	Speed	v
3.	Radial velocity	vr
4.	Specific angular momentum	h
5.	Magnitude of specific angular momentum
6.	Inclination	i
7.	Node line	N
8.	Magnitude of N
9.	Right Ascension (RA) of ascending node	Ω
10.	Eccentricity vector
11.	Eccentricity
12.	Argument of perihelion
13.	True anomaly.	ν

---

Auxilliary Circle is the circumcircle of an orbit. The circle's centercoincides with the orbit's center, and the circle's radius equals the orbit's semimajor axis.

---

TRUE ANOMALY: Θ = 360 - cos^-1((x-c)/r) after E>180°

Eccentric	Position in Orbit		Dist fm Sun	True	Mean
Anomaly	X-value	Y-value	R-value	Anomaly	Anomaly
(Deg)	(AU)	(AU)	(AU)	(Deg)	(Deg)
0	1.47	0.00	0.650	0.0	0.0
15	1.42	0.32	0.678	27.8	6.7
30	1.27	0.61	0.760	53.4	14.0
45	1.04	0.86	0.890	75.7	22.3
60	0.74	1.06	1.060	94.6	32.2
75	0.38	1.18	1.258	110.5	44.0
90	0.00	1.22	1.470	123.9	57.9
105	-0.38	1.18	1.682	135.5	74.0
120	-0.74	1.06	1.880	145.8	92.2
135	-1.04	0.86	2.050	155.1	112.3
150	-1.27	0.61	2.180	163.8	134.0
165	-1.42	0.32	2.262	172.0	156.7
180	-1.47	0.00	2.290	180.0	180.0
195	-1.42	-0.32	2.262	188.0	203.3
210	-1.27	-0.61	2.180	196.2	226.0
225	-1.04	-0.86	2.050	204.9	247.7
240	-0.74	-1.06	1.880	214.2	267.8
255	-0.38	-1.18	1.682	224.5	286.0
270	0.00	-1.22	1.470	236.1	302.1
285	0.38	-1.18	1.258	249.5	316.0
300	0.74	-1.06	1.060	265.4	327.8
315	1.04	-0.86	0.890	284.3	337.7
330	1.27	-0.61	0.760	306.6	346.0
345	1.42	-0.32	0.678	332.2	353.3
360	1.47	0.00	0.650	360.0	360.0
Given	a×cos(E)	b×sin(E)	SQRT((x - c)2 + y2)	cos-1((x-c)/r)	E - ν

• E=Eccentric Anomaly: Angle originates from elliptical center and starts with line to perihelion.
• a=semi-major axis. For Apollo, a = 1.47 AU
• b=semi-minor axis. For Apollo, b = 1.22 AU
• c=focal length. For Apollo, c = 0.82 AU, distance of Sun from orbit center.
• ν=True Anomaly: Angle originates from Sun and starts with line to perihelion.

Source material: Fundamentals of Astrodynamics, by Roger R. Bate, Donald D. Mueller, Jerry E. White.  Observing enclosed figures described above, determine areas of certain figures.

---

ECCENTRIC ANOMALY. If we assume this figure to be a circular sector defined by points, C, P, and A_CIR. Points are shown in figure in above table and defined below.

1. C is center of Auxiliary Circle which coincides with orbital center.
2. P is perihelion of orbit.
3. A_CIR is Apollo's position superimposed upon Auxiliary Circle as described in preceding text.
4. This area is a circular sector and easily computed.

---

TRUE ANOMALY. The outline of this figure vaguely resembles previous figure but with some significant differences. The three points which define the border of this area are shown in above figure and described below.

1. S, Sun, is a focus of the orbit and the apex of the angle ASP.
2. P is perihelion of orbit.
3. A_ORB, Apollo's position in orbit.
4. This area is not so easy to compute.
• The enclosing arc is a portion of an ellipse, thus not circular.
• The angle apex is at a focus, not at the center.

Thus, computing the enclosed area seems far from simple. However, Kepler discovered a straightforward way to do it.

---

MEAN ANOMALY. Observing the above figure which shows both of above, it seems the area of eccentric anomaly must always contain area bounded by true anomaly. Intuitively, it seems there must be a circular sector with area less then that of eccentric anomaly which equals area of the true anomaly figure, a non-circular sector. Once we determine this circular sector, then computing its area would be a simple task.
In fact, Kepler discovered this circular sector and named it, Mean Anomaly. It's computed as follows.

• M = E - e sin E
• M = mean anomaly (radians)
• E = eccentric anomaly (radians)
• e = eccentricity

Purpose of determining enclosed areas: Recall Kepler's 2nd Law of orbital motion.

---

MEAN MOTION. As the positional vector, r, sweeps out areas during Apollo's orbit, areas per time can be related to motion of Apollo throughout it's elliptical orbit.

---

For much more on orbital basics.