REFERENCE: Six Orbital Elements

A typical asteroidal orbit can best be described by six orbital elements which are briefly described in following content.

NOTE: Kepler’s Third Law:  After many years of research, Johann Kepler discovered a relation between orbit’s semi-major axis, a, and orbit’s period, P, travel time for object to complete entire orbit. ORBITS ARE ELLIPTICAL. All orbits have a semi-major axis (a), a semi-minor axis (b) and a focus (c) with a Pythagorean relationship as shown. CARTESIAN COORDINATES. With origin (0,0) at center. P2 = a3        (For P in years and a in AUs) Example: Let P = 2 years, then a = 22/3 AU = 1.587 AU

Q, aphelion, is orbit’s most distant point from Sol, our Sun. Perihelion, q, is closest point. Readily measured, Q and q, can help compute several orbital elements. EXAMPLES: Semimajor axis (a): a = (Q + q)/2 = 1.587 AU Focus Distance (c): c = (Q - q)/2 = 0.965 AU Eccentricity (e): e = (Q - q)/(Q + q) = 0.608 Semi-latus rectum, p (sometimes designated as “ℓ”, script L), is perpendicular to major axis (from Q to q) at focus;  p’s distance measures from focus to orbit. p = b2/a = 1 AU = 2Qq/(Q + q) = 1.0 AU

	REFERENCE PLANE: ECLIPTIC For the Solar System, the reference plane is usually the Ecliptic, the plane in which the Earth orbits the Sun.  Above view presumes observation from North of Earth; thus, direction of revolution is Counter Clock-Wise (CCW) around Sol.   First point of Aries is determined by position of Earth during Vernal Equinox (about March 20).  Over the centuries, this position moves; thus, it used to point to the Aries constellation, but it now points toward Sagittarius.
OTHER SOLAR ORBITS INCLINE AXIOM:  In the Solar System, most asteroidal objects must orbit Sol, our sun.  However, most of these orbits are not co-located with Earth's orbit in the Ecliptic; thus, virtually all other Solar orbits are tilted with respect to Earth's orbit, and they must pass through the Ecliptic.   At one point, the orbit pierces the Ecliptic as object ascends from South to North (Ascending Point,  ☊).  At another point, the orbit again pierces the Ecliptic as it descends from North to South (Descending Point,  ☋).  "Line of nodes" connects  ☊  with  ☋, the line of intersection between two planes.  TE assumes Sol normally to be on this line between the two nodes.
FIRST TWO ORBITAL ELEMENTS First two elements define the size and shape of the asteroid’s elliptical orbit: 1.     Semimajor axis (a)—longest distance from orbit’s center to any orbital point; it averages perihelion (q) and aphelion (Q) distances, [a = (q+Q)/2]. 2.     Eccentricity (e)—measures orbit’s elongation compared to a circle; the quotient of the difference of Q and q by their sum. [e = (Q-q)/(Q+q)].
	NEXT THREE ELEMENTS:  ORBIT ORIENTATION Argument of Perihelion (ω): angle from the ascending node (☊) to the perihelion (the closest point of orbit to Sol). It is measured along orbit's motion (most Solar objects orbit Counter ClockWise (CCW) as observed north of the ecliptic). Inclination (i): angle corresponding to tilt of the asteroid’s orbital plane with respect to the ecliptic. At the ascending node (☊), where the orbit passes upward through the reference plane, inclination (value is from 0° to 90°) measures from  ecliptic to orbital plane. Longitude of Ascending Node (Ω): angle, measured CCW from 0° to 360° on the Solar Ecliptic.  It starts from the First Point of Aries ♈︎ and proceeds to a notional ray from Sol to the ascending node (☊).
Aircraft maneuvers do differ from orbital planes; HOWEVER, celestial mechanics does use analogous terms to describe orbital plane’s spatial relationship with Ecliptic plane; these 3 orbital elements are described in following content. Yaw, pitch and roll could be “conceptual aids” to help students visualize the three orbital elements of ---Longitude of Ascending Node, LAN (Ω)  ---Inclination (i)  ---Argument of Perihelion (ω)
	Longitude of the ascending node (Ω)  ...enables us to precisely place the orbit’s ascending node (☊).  Ω can help determine the date when an orbiting object pierces the ecliptic from below to above.  EXAMPLE:  If Ω = 90⁰; then, ascension happens on Summer Solstice (about June 20) as in diagram.
Inclination (i) The asteroid path passes upward (North of Earth) through the Ascending Node (☊).  It then travels 180⁰ to travel downward through the Descending Node (☋). Between the Ascending and Descending Nodes is a notional “Line of Nodes” which passes through Sol.  This line shows where the asteroidal plane intersects the Ecliptic plane.  Inclination (0° to 90°) measures from the ecliptic to the orbital plane.
	Argument of perihelion (ω) ...defines orientation of the elliptical orbit in the orbital plane. ω is angular distance from the ascending node to q (closest point to Sol). Unlike Ω (measured around Ecliptic), ω is measured around the orbit. KEY DIFFERENCE between aeronautical terms (Yaw, Pitch, Roll) and celestial terms ( Ω, i, ω), the aero terms are highly dynamic which operators use to guide their craft.  HOWEVER, celestial terms are relatively static and used by observers to describe paths of orbiting objects.

FINAL ORBITAL ELEMENT: True Anomaly Final element enables us to find the asteroid’s precise location on its orbit: True anomaly (ν or θ) is an angle from 0° to 360°.  It defines the position of the orbiting body along the elliptical orbit at a specific time.  While the other five elements remain virtually constant over many millennia, True Anomaly (θ) is a highly dynamic variable.  θ continually changes value as object travels throughout its orbit.
	SIDEBAR: Four Cardinal Directions: Perihelion, q, is orbit’s nearest distance to Sol, a reference ray from Sol to q has value of θ = 0°. Aphelion, Q, farthest distance from Sol is always at θ = 180°.  AXIOMATIC: Q and q are seldom equal; thus, ---Line of nodes (notional line between ☊ and ☋) is seldom at the center of an orbit. ---Line of Apses (notional line between q and Q) is seldom perpendicular to Line of Nodes. ---Sol is seldom equidistant to both ☊ and ☋. However, First Semilatus Rectum (p) is always at θ = 90°. 2nd p is always at θ = 270°.
SUMMARY Six Classic Orbital Elements. Symbol Name Description a Semi-major axis Orbit size e Eccentricity Orbit shape I Inclination Intersection angle of 2 planes: asteroid and Ecliptic. Ω Right Ascension of ascending node.  Swivel angle from vernal equinox to ascending node.  ω Argument of perihelion Angle from ascending node to perihelion ν True anomaly Angle from perihelion to object's position Source: UNDERSTANDING SPACE, An Introduction to Astronautics; by Jerry Jon Sellers. See page 161, Table 5-3.
For much more on orbital basics. SLIDESHOW VOLUME 0: ELEVATIONAL VOLUME I: ASTEROIDAL VOLUME II: INTERPLANETARY VOLUME III: INTERSTELLAR	CONCLUSION SIX ORBITAL ELEMENTS... ...now determine orbital paths of many thousands of asteroids throughout our Solar System. HOWEVER, many of these asteroids can be refashioned as habitats for millions of human residents.