Saturday, April 13, 2013

LINE OF NODES

CONTENTS

The paths of habitats will essentially be those of asteroids;
thus, following material describes orbits of asteroids,
material objects orbiting the Sun.



Asteroid orbits intersect Ecliptic (orbital plane of Earth about Sol) in two points (i.e. "nodes");
one "ascending"; other "descending".
connect these for "line of nodes".



Kepler proved that all orbits are elliptical; thus, we must leverage
basic definition of an ellipse.
 Show all points (x,y) such that sum of the distances (d1 + d2) to
two fixed points (f1 and f2) is always the same value.

Compute Distance
 
Distance to one fixed point can be computed via Pythagorean method if we plot points via Cartesian Coordinate system.Let an asteroid represent an arbitrary point (0,4), and let the Sun (aka "Sol) represent a fixed point (3, 0).
d = √(Δx2+Δy2) 
d = √(32+42) = 5
2 Fixed Points
 
Distance to second fixed point (d2) can be computed with same, Pythagorean method as for first distance (d1) . Let the second fixed point be represented by a notional object, "Not Sol", arbitrarily placed at coordinates (-3, 0).
dTtl = d1 + d2
Consistent Distance
 Construct an ellipse
such that any randomly selected point
always gives same total distance to two fixed points
(dTtl = d1 + d2)
as shown on the diagram.

Of  the points shown,
only the one above the center has two equal distances such that
d1 = d2
We arbitrarily designate this distance as "a"; thus,
dTtl / 2 = a
ORBITAL SHAPE
As object orbits, distance to center varies. 
Minimum distance to center is "b", semi-minor axis.
 
Maximum distance from the center is "a", semi-major axis.
"a" happens to be hypotenuse of right triangle formed by
∠1: Center     ∠2: Sol     ∠3: Orbit Top

MATHEMATICAL DEFINITION of ellipse
is often given as following equation:
x2

a2
+y2

b2
=1
FIXED FOCI
Fixed points, focus-1 (f1) and focus-2 (f2) anchor ellipse.

In the Solar System, Sol occupies one of the focus positions.

Objects which orbit Sol are "anchored" by Sol's massive gravity.

"c" is distance from center to either focus.
It can be determined from observed values of "a" and "b".
c  =(a2 - b2)
ECCENTRICITY
Eccentricity is a measure of an ellipse's "flatness" or departure from a circle's symmetry.
It can be easily computedcompute = c/a
To obtain e directly from a and b, substitute for c: e = (a2 - b2)/a ; and rearrange:
e =  ( 1 - b2

a2
 ) = (1 - 42

52
)= 0.6 
SOLITARY SOL
Positions of orbit centers and non-Solar foci can vary considerably.
However, Sol contains sole point which is a focus of every Solar orbit.
Thus, Sol is the best reference for comparing different orbits;
they'll always share Sol as a common focus.
 Q (=a+c) is distance to aphelion, orbit's farthest point from Sol.
q (=a-c) is distance to perihelion, orbit's closest point to Sol.
From readily observed values for Q and q, one can determine following:
a = (Q + q) ÷ 2
c = (Q - q) ÷ 2
e = (Q-q) ÷ (Q+q)

b = (a2 - c2)  = √[(a-c)(a+c)] = (Q×q)
Named after the Greek god, 1862 Apollo was initially discovered by Karl Reinmuth in 1932.
As the first discovered of the Apollo asteroids, the group has taken its name.
Ironically, 1862 got lost and rediscovered in 1973; thus, its asteroid number (1862) is higher than some other Apollo asteroids (i.e., 1566 Icarus).
It was the first asteroid recognized to cross Earth's orbit. It also crosses orbits of Venus and Mars (see online diagram).
Celestial Body DimensionsInclination
TypeName Semimajor
Axis
EccentricityPerihelionAphelionAngle
Asteroid
1.471 AU
0.560
0.647 AU
2.294 AU
6.36o
Planet
Earth
1.0000001 AU
0.017
0.983 AU
1.017 AU
n/a
From values in above table, compute and draw some basic orbital views.
SEMI-MAJOR AXIS (a): Compare Earth's with Apollo's.
For ease of comparison, TE temporarily assumes a common center for the orbits of Earth and Apollo as shown above.
In actual fact, Apollo is distinguished as first asteroid discovered to periodically cross orbit of Earth.
Thus, orbit of Earth can NOT be completely contained inside Apollo's orbit.
Kepler's first Law puts our sun, Sol, at a focus of all Solar orbits.
For Apollo's elliptical orbit, each focus is .822 AU from center.
We've arbitrarily chosen the left focus to show Sun's position.
All Solar orbits share Sol as a common focus.
Since Earth's orbit is nearly circular, both foci lie within Sol at the center.
With Sol being well off center in Apollo's elliptical orbit,
the two orbits intersect as shown below.
Given archived eccentricity, e, compute focus: c (= a × e)
For Apollo, semimajor axis, a = 1.47 AU; and e= 0.56; thus,
FOCUS: c = .56 × 1.47 AU = .822 AU
Given a and c, compute distances to perihelion and aphelion.
PERIHELION: q = a - c = 1.47 AU - .82 AU = .65 AU
APHELION: Q = a + c = 1.47 AU + .82 AU = 2.29 AU
Rotate Orbit Views
 To best observe the angle of inclination, we must rotate a full face on view of each orbit until we achieve the edge on views.

Still observing edge on view, join orbits at Sun to show angle of inclination. Earth's orbit is often used a reference for other Solar orbits; thus, it's often called the "Ecliptic".
The orbits of Earth and Apollo share a common focus (occupied by the Sun), where the two orbital planes intersect at the "angle of inclination".
Rotate view to show "line of nodes",
intersection between Apollo's orbit and the plane of the ecliptic (mean plane of Earth's orbit around the Sun).

Line of nodes is defined by two points:
Objects of opportunity must intersect Ecliptic near "nodes".

Sidebar: Johannes Kepler

A Short Biography. Johannes Kepler was born at 2:30 PM on December 27, 1571, in Weil der Stadt, Württemburg, (in now Germany). Though sickly and from a poor family, his obvious intelligence earned him a scholarship to the University of Tübingen to study for the Lutheran ministry. There, he learned and delighted in the ideas of Copernicus.

During his life, Kepler worked as a teacher and mathematician, along with duties as court astrologer and astronomer. However, he is best known for his work with the last great astronomical observer of the pre-telescope age, Tycho Brahe. Brahe spent years carefully measuring the position of astronomical objects, including Mars. His goal was to understand the motions of the planets.

Tyco spent over twenty years meticulously collecting observational data on all the visible planets. Tyco's data were the best available before the invention of the telescope. While his celestial measurement skills were superb, Brahe lacked math skills to analyze his collected data. This gave Kepler an opportunity to use his considerable mathematical expertise.

Kepler Works on Motion of Mars. Brahe gave Kepler the data for Mars, including measurements of Mars at successive oppositions over a period of many years. Of all the planets, the predicted position of Mars had the largest errors and therefore posed the greatest problem. Mars's motion never quite fit that of a circular orbit, nor an orbit of epicycles (multiple circles upon circles, a model favored by Brahe and other contemporary astronomers). After ten years of work, Kepler determined the orbit of Mars to be an ellipse.

Kepler's calculations were based on observational data and added mathematical credibility to Copernicus's Solar centric concept. This marks the beginning of modern, scientific astronomy. He based his explanations upon observations, rather than making the observations fit an assumed model of the universe. Today, we call this the scientific method.

In 1601, Kepler inherited Tycho's post as Imperial Mathematician. He spent the rest of his life working with Brahe’s data and publishing many works. Johannes Kepler died in Regensburg in 1630, while on a journey from his home in Sagan. His grave was demolished within two years because of the Thirty Years War. Though Kepler lived and died in turbulent times, his scientific works had a far greater effect on humanity then all the historical events during his time.
Kepler's Three Laws of Planetary Motion. Kepler discovered three relationships, now called "Kepler's laws" that describe the orbital motion of the planets. Prior to Kepler, planets were believed to orbit Earth in circular paths. The many variations of their motions were accommodated by adding more and more complex sets of circles upon circles (epicycles) to rationalize the observations. Although this worked moderately well, the future position of planets could only be roughly predicted.

1. Law of Ellipses (1609). The orbit of each planet is an ellipse, with the Sun at one focus. Earth is closest to the Sun in January and farthest from the Sun in July as it travels along its elliptical orbit.

2. Law of Equal Areas (1609). A line from the planet to the Sun sweeps out equal areas in equal times. This geometric description describes the fact that a planet's orbital velocity varies in a regular way -- the farther the planet is from the Sun, the more slowly it moves along its orbit. For example, Earth moves fastest in January (when it’s further from the Sun) and slowest in July (when it’s closest).

3. Harmonic Law (1618). The square of the sidereal period of a planet is directly proportional to the cube of its orbit’s semi-major axis. This is described by following equation:
T2= a3 * k
T2= a3 * 4 π2 / μ
T = 2 π (a3/μ)
For Solar orbits:
μ = G * MSol = (6.667 x 10-11 N m2/kg2) * (2 x 1030 kgs)
μ = 132,712,440,018 km3/sec2 = 13.2 x 1010 km3/sec2
Remarks: Period (T) doesn't care about shape, e, or circumference, C. However, it does care about a, semimajor axis. For example, any orbit with a= 1.0 AU will take exactly one year to complete. Thus, the first item in Table 2, a (semimajor axis), relates directly to the last item, T, period.

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