Wednesday, November 07, 2012

Longitude of Ascending Node

Cardinal longitudes
  • Reference longitude (0°) always points to q, perihelion, orbit's closest point to Sol. This axis is commonly used as the positive abscissa (+X).
  • 90° longitude points to L, semi-Latus rectum. This is the numerator "l" used to compute R, distance from focus, Sol, to orbit positions for any angle position, ν. (positive ordinate, +Y).
  • 180° longitude points to Q, aphelion, orbit's most distant point from Sol. (negative abscissa, -X).
  • 270° longitude points to other L. (negative ordinate, -Y).
  • 360° longitude takes us back to reference.

Define ascending node:  passes from beneath Ecliptic (south) to above Ecliptic (south to north).  Where value of Z coord is precisely zero.

Define subject: Longitude of ascending node is measured from reference longitiude (ray from Sol to q, perihelion) around the orbit til
Alternatively: measure longitude directly on Ecliptic. (pg 23) The next element to yield is the longitude of the ascending node, for the plane intersects the ecliptic at Z = 0 in the line aX + bY = 0, from which we get:
sin(Ω) = a/(a2 + b2)
cos(Ω) = -b/(a2 + b2)
Eq. 10.10.9a,b  with no quadrant ambiguity.

Calculate an ephemeris from the orbital elements

The six elements of an elliptic orbit, then, are as follows.
asemi major axis, usually expressed in astronomical units (AU).
longitude of the ascending node
argument of perihelion
time of perihelion passage
Ω, measured eastward from Aries to ascending node, is the ecliptic longitude of the ascending node.
(The word “ecliptic” is usually omitted as understood.) It goes from 0° to 360°.
Angle ω, measured in the direction of the planet’s motion from Aries to P, is the argument of perihelion.
It goes from 0° to 360°.


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