## Saturday, September 22, 2012

### Ref. Mat'l: Orbit Inclination (Z-coord)

From Wikipedia
http://en.wikipedia.org/wiki/Orbital_inclination
Inclination is one of the six orbital parameters describing a celestial orbit
Of these six parameters, two of them orient the plane of the object's orbit:

• Longitude of the ascending node (Ω) horizontally orients the ascending node of the ellipse (where the orbit passes upward through the reference plane) with respect to the reference frame's vernal point.
• Inclination (i) is the vertical tilt of the ellipse with respect to the reference plane. i is best displayed at the ascending node (where the orbit passes upward through the reference plane). Inclination is best measured by the orbital plane's angle (in degrees) to the plane of reference (usually the ecliptic).The ecliptic is the Earth's orbital path; thus, the plane containing the ecliptic is a very practical reference for Earth-bound observers

• Most planetary orbits in the Solar System have relatively small inclinations, both in relation to the ecliptic as well as to each other. However, inclinations of smaller bodies vary widely.EXAMPLES: Dwarf planets Pluto and Eris,have inclinations of 17° and 44° respectively. The large asteroid Pallas, is inclined at 34°.
• Inclination of 0° means the orbiting body orbits Sol in same plane as Earth (ecliptic) in the same direction as Earth, counterclock wise (CCW) as observed from north of the ecliptic.
• Inclination greater than -90° and less than 90° is a prograde orbit (CCW). Most solar objects orbit Sol in a prograde manner.
• Inclination greater than 90° and less than 270° is a retrograde orbit (clockwise as observed from due north). Very few solar objects have such orbits.
• Inclination of exactly 90° is a solar polar orbit, in which the object passes over the north and south poles of the Sun.
• Inclination of exactly 180° is a retrograde ecliptic orbit (direction is exact opposite of Earth's orbit).

• Calculation
In astrodynamics, inclination $i$ can be computed from the orbital momentum vector $\mathbf{h}\,$ (or any vector perpendicular to the orbital plane) as $i=\arccos{h_\mathrm{z}\over\left|\mathbf{h}\right|}$, where $h_\mathrm{z}$ is the z-component of $\mathbf{h}$.

Cross product is defined by the formula

$\mathbf{a} \times \mathbf{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin \theta \ \mathbf{n}$
where θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), ‖a‖ and ‖b‖ are the magnitudes of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule as illustrated. If the vectors a and b are parallel (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

Orbital Momentum
In an elliptical orbit, a specific relative angular momentum is twice the area per unit time swept out by a chord from the primary to the secondary: this is the area referred to by Kepler's second law of planetary motion. Since the entire orbital area is swept out in one orbital period, orbital momentum, h, equals  twice the area of the ellipse divided by the orbital period, giving the equation:
h =2πab

(a3/G(M+m)
$a\,$ is the semi-major axis.
$b\,$ is the semi-minor axis.

G is the gravitational constant, 6.67 × 10 -20 km3kg -1s -2.
For Solar orbiting objects MSol = mass of the Sun, 1.989 × 1030 kg
m is mass of solar orbiting object. Mass of m is relatively insignificant and can be disregarded.

h =2πab

(a3Sol)
Thus, it's more convenient to use, μ the standard gravitational parameter:
μSol = G×MSol
For solar orbits, use μSol, the heliocentric gravitational constant.
μSol =1.327×1011 km3s−2
h = b × Sol/a)
h = ( μSol ×b2/a)
h = ( μSol × )
is the semi-latus rectum

$h = \frac{ 2\pi ab }{2\pi \sqrt{ \frac{a^3}{ G(M\!+\!m) }}} = b \sqrt{\frac{ G(M\!+\!m) }{a} } = \sqrt{a(1-e^2) G(M\!+\!m) } = \sqrt{ p G(M\!+\!m) }$

Mutual inclination of two orbits may be also be calculated from their inclinations to another plane using cosine rule for angles.
Inclination = 60.89°
Mars (♂) Based
ν Time R X Y Z
Deg Days AU AU AU AU
TBD 0.787 0.79 0.00 0.00
15°TBD 0.796 0.77 0.21 0.18
30°TBD 0.823 0.71 0.41 0.36
45°TBD 0.870 0.62 0.62 0.54
60°TBD 0.940 0.47 0.81 0.71
75°TBD 1.038 0.27 1.00 0.88
90°TBD 1.168 0.00 1.17 1.02
105°TBD 1.335 -0.35 1.29 1.13
120°TBD 1.541 -0.77 1.33 1.17
135°TBD 1.776 -1.26 1.26 1.10
150°TBD 2.011 -1.74 1.01 0.88
165°TBD 2.194 -2.12 0.57 0.50
180°TBD 2.264 -2.26 0.00 0.00
195°TBD 2.194 -2.12 -0.57 -0.50
210°TBD 2.011 -1.74 -1.01 -0.88
225°TBD 1.776 -1.26 -1.26 -1.10
240°TBD 1.541 -0.77 -1.33 -1.17
255°TBD 1.335 -0.35 -1.29 -1.13
270°TBD 1.168 0.00 -1.17 -1.02
285°TBD 1.038 0.27 -1.00 -0.88
300°TBD 0.940 0.47 -0.81 -0.71
315°TBD 0.870 0.62 -0.62 -0.54
330°TBD 0.823 0.71 -0.41 -0.36
345°TBD 0.796 0.77 -0.21 -0.18
360°TBD 0.787 0.79 0.00 0.00