Thursday, November 01, 2012

REFERENCE: Inclination, the Z factor

Consider the Cartesian X-Y-Z axis where X-Y is the traditional 2 dimensional (2 D) plane, and Z is a factor which adds elevation for a third dimension.

Ecliptic contains Earth orbit.
Two dimensional Ecliptic is the plane of Earth's orbit around our sun, Sol.  It could be described with Sol at origin (0,0) with X,Y coordinates given in Astronomical Units (AUs); an AU is the average distance from Sol to Terran orbit. As the reference plan, all Z coordinates are zero; thus, we might consider the Ecliptic as having a zero Z-factor.
For convenience, Thought Experiment (TE) will assume Earth’s orbit to be exactly circular with constant Radius (R) of 1.0 AU.  NOTE:  Terran orbit is actually elliptic with small eccentricity of .0167.

Unlikely Asteroid Orbit in Ecliptic.
In the highly unlikely event that an asteroid orbit resided completely in same plane as Earth’s Solar orbit; then, it would also have a zero Z-factor.

Typical Asteroid Orbit Pierces Ecliptic.
If desired, one could assign a Z-value to every point on the typical asteroid's path; this Z-value would indicate its elevation above/below the Ecliptic plane. At two points, the Z-value would be zero.
1) The ascending node is one of two places where an orbiting object passes through the ecliptic, an imaginary plane of Earth's orbit about Sol. From below the Ecliptic, the asteroid path passes upward (North of Earth) through the Ascending Node (☊), first point where Z = 0.
2) It then travels 180⁰ to travel downward through the Descending Node (☊), 2nd point where Z = 0.
3) On the Ecliptic, between these two Nodes, is a notional “Line of Nodes” which passes through Sol. This line shows where the asteroidal plane intersects the Ecliptic plane.
4) The inclination (i) is the angle between these two planes.

Consider the asteroidal orbit of 1862 Apollo.
FACE-ON VIEW of Asteroidal Orbit would be same as FACE-ON View of Ecliptic if asteroid orbital inclination = 0°..
EDGE-ON VIEW of Ecliptic: One observes asteroid orbit from level of Ecliptic plane. From this vantage point, asteroid orbit looks like a line which usually passes through the Ecliptic; however, in the unusual case of zero inclination, the line would parallel the Ecliptic plane.
NOTE: TE chooses X-axis to coincide with the θ = 0° ray from Sol to the orbit's perihelion (q).
Furthermore, TE chooses Y-axis to coincide with the θ = 90° ray from Sol to the orbit's semi-latus rectum ( ).

ω = 90⁰
Node of ascension coincides with a semi-latus rectum.  Line of nodes coincides with line connecting Sol with orbit's latus rectum. The argument of perihelion  (ω) is an angle from Sol's ray to ascending node (☊) to Sol's ray to orbiting body's  perihelion (q, closest point to Sol). This angle is measured in the asteroid's orbital plane and in the direction of motion.

Let Apollo inclination (i) = 6° @ ω = 270⁰.
ω = 270⁰
The argument of perihelion (ω) of 270⁰ causes inclination to reverse orientation from ω = 90⁰. With i = 6°, highest z value is now 0.2397 AU at orbit's aphelion (Q), and the most negative z value (-0.0676) AU) is now at orbit's perihelion (q).

NOTE:  Spot velocity is determined by following:
v={\sqrt  {\mu \left({2 \over r}-{1 \over a}\right)}}
Variable r  (or R) is radius, asteroid's distance from Sol in AUs.

Constant a is orbit's semi-major axis in AUs, distance from orbit's center to orbit's aphelion, Q.

Constant μ is Solar Standard Gravitational Parameter.  TE proposes μ = 887.123 AU km²/sec². to quickly compute velocity as kilometers per second (kps).

The orbital plane intersects a reference plane; for solar orbits, it is the ecliptic plane. The intersection is called the line of nodes, as it connects our sun, Solar System's center of mass, with the ascending and descending nodes.

Orbital inclination measures the tilt of an object's orbitIt is the angle between a reference plane and the object’s orbital plane or axis.

Precise observations of an asteroid’s orbit will determine:

  • Perihelion, q, the closest distance to Sol, our sun.
  • Aphelion, Q, the farthest distance from Sol.
For the Solar System, the reference plane is usually the ecliptic, the plane in which the Earth orbits the Sun.
This proves very practical for Earth-based observers; thus, Earth's orbital inclination is by definition zero (0°).

Inclination is one of the orbit’s six orbital elements which describe its shape and orientation compared to the ecliptic.

An inclination of 30° could also be described using an angle of 150°.


  • the normal orbit is prograde, an orbit in the same direction as the planet rotates.
  • Inclinations greater than 90° describe retrograde orbits.
Components of the calculation of the orbital inclination from the momentum vector
In astro-dynamics, the inclination i can be computed from the orbital momentum vector h (or any vector perpendicular to the orbital plane) as
While most planetary orbits in the Solar System have relatively small inclinations, both in relation to each other and to the Sun's equator.
However, there are notable exceptions.  See following table:


Other meaning

For planets and other rotating celestial bodies, the angle of the equatorial plane relative to the orbital plane — such as the tilt of the Earth's poles toward or away from the Sun — is sometimes also called inclination, but less ambiguous terms are axial tilt or obliquity.
The angle of intersection of a reference plane
Used in sentence:
The astronomer calculated the inclination of the orbital planes of each visible heavenly body.
^ Chobotov, Vladimir A. (2002). Orbital Mechanics (3rd ed.). AIAA. pp. 28–30, . ISBN 1-56347-537-5.
^ McBride, Neil; Bland, Philip A.; Gilmour, Iain (2004). An Introduction to the Solar System. Cambridge University Press. p. 248. ISBN 0-521-54620-6.
From Wikipedia
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

    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

    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.
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