Sunday, October 12, 2008

Reference Material: Orbital Elements

Please note that following books differ slightly in their presentation. Further note that authors are not responsible for errors that I may have unintentionly introduced into following extracts from their excellent material. Before we start computing above elements, let's go back to basics.


http://www.keplersdiscovery.com/Asteroid.html


T=2 π SQRT(a3/μ)

μ = 132,712,440,018

Please note that following books differ slightly in their presentation. Further note that authors are not responsible for errors that I may have unintentionly introduced into following extracts from their excellent material. Before we start computing above elements, let's go back to basics.

Six Classic Orbital Elements.
SymbolNameDescription

a

Semimajor axisOrbit size

e

EccentricityOrbit shape

I

Inclination Angle of intersection between asteroid's orbit and Ecliptic.

Ω

Right Ascension of ascending node. Swivel angel from vernal equinox to ascending node.

ω

Argument of perigee Angle from ascending node to perigee

ν

True anomaly Angle from perigee to object's position

Source: UNDERSTANDING SPACE, An Introduction to Astronautics
by Jerry Jon Sellers et al. See page 161, Table 5-3.



13 Steps to Compute Classic Orbital Elements
Seq.NameSymbolEquation
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 perigee

13. True anomaly.

ν

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.



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

Periapsis is the point at which an orbiting object is closest to the body it is orbiting. This point is sometimes given a name that is specific to the body being orbited. For example, the periapsis of an object orbiting the Sun is its perihelion (from helios, the Greek word for Sun). If an orbit can be pictured as an ellipse and the Sun is pictured as one of the foci, then the periapsis is the point on the ellipse which coincides with the major axis endpoint closest to the Sun. Distance from focus to periapsis (or perihelion in case of the Sun) is always, semi-major axis minus focal length, c (a - c).

Plane of asteroid's orbit is usually inclined to plane of Earth's orbit, the Ecliptic. The line of intersection is the "line of nodes" where the line's defining points are the ascending node and the descending node.

xeccentricxx
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

True Anomaly: ν = 360 - arccos((x-c)/r) after E>180 deg.

  • 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, let's see if we can determine areas of certain figures.

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

  • C is center of Auxilliary Circle which coincides with orbital center.
  • P is perihelion of orbit.
  • ACIR is Apollo's position superimposed upon Auxilliary Circle as described in preceding text.
  • This area is a circular sector and easily computed.

TRUE ANAMOLY. 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.

  • S, Sun, is a focus of the orbit and the apex of the angle ASP.
  • P is perihelion of orbit.
  • AORB, Apollo's position in orbit.
  • 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 ANAMOLY. 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 noncircular sector. Once we determine this circular sector, then computing its area would be a simple task.

If fact, Kepler discovered this circular sector and named it, Mean Anamoly. 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. This will be discussed shortly.



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