HABITAT PATH TIMES and KEPLER'S LAWS
Consider a possible path of a "cycler",
an asteroid transformed into a space habitat
with a convenient orbit.
As an orbiting object, cycler must obey
Laws of Kepler
I. Solar orbit is an ellipse with the Sun at a focus.II. Line from Sol sweeps equal areas in equal times.
III. Square of period relates to cube of semimajor axis.
T^{2} = a^{3}
BACKGROUND: In his excellent book, Mining the Sky, Dr. John S. Lewis describes concept of cycler orbits with 2 year period.(pg. 115, "formerly a Near Earth Asteroid (NEA) whose orbit has been slightly modified to assure a metastable resonant relationship to Earth. Small amounts of propulsion will be expended from time to time to keep the orbit at resonance....Each cycler passes by Earth every two years...though transport to the Belt is the main function, it frequently passes near Mars...")
RESONANCE
If the asteroid's orbital period is an exact multiple of the Earth's orbit (i.e., integer number of years); then, asteroid will arrive at nearest position to Earth at same date every cycle. Consider such an asteroid with a period of exactly two years.As an example, let asteroid's closest approach to Earth happen on September 25, year X. For every two years thereafter (X+2, X+4, .....), that resonant asteroid would approach Earth at same position on September 25. Humanity could leverage that resonance by transforming asteroid into a habitable habitat and hitching a ride for the two year orbit from Earth to beyond orbit of Mars and return back to Earth. 

Major/Minor Axii
All features of an ellipse apply to an orbit,.
Thus, major axis and minor axis intersect at the center.
Orbit positions can be described by the Cartesian elliptical equation

Focus and Eccentricity
Since Kepler's first law places Sol at a focus,
define focus as c, distance from center to Sun.
Eccentricity, e, is the "flatness" of an ellipse. It is defined
e = c ÷ a. Common usage often substitutes term, ae, for c.
EXAMPLE: Arbitrarily pick e=1/3 which produces
c = ae = 1.59 AU × .3333 = 0.53 AU
c = 0.53 AU
 

Pythagorean Properties
Semimajor axis (a), semiminor axis (b) and focus (c)
have a Pythagorean relationship:
a^{2} = b^{2} + c^{2}
a and c define range of distance from Sol.
Aphelion, Q, is orbit's furthest point.
Perihelion, q, is closest.

Auxiliary Circle Auxiliary Circle (AC) circumscribes the elliptical orbit.
Semimajor axis, a, is also the AC radius.
AC radius remains the same.While orbit radius varies with angle from major axis. Orbit and AC share two endpoints of the major axis, a and +a.
Next diagram shows further relationship:
For every point, x, between a and +a, there are two corresponding y points directly above. Y_{E} is on the elliptical orbit. Y_{C} is on the auxiliary circle. 
Angle  Radius  ν  R 
0°
1.06 AU
45°
1.14 AU
90°
1.41 AU
135°
1.85 AU
180°
2.12 AU
ℓ=1.41 AU 

For time, t_{1}, let orbiting object proceed counter clock wise (CCW) from point q to next position as shown in diagram. The associated positional vector would then sweep an area shown by the green sector, A_{1}.
Similarly, for time, t_{2}, positional vector would sweep out another sector,A_{2}, of different shape but same size area. Thus, for same duration, asteroid travels greater orbital distances when closer to the Sun and a lesser distance when further away. Per Isaac Newton's work in gravity, Sun's closer satellites have greater speeds then those further away. Thus, if we can readily determine sector areas, then we can readily determine satellite flight times throughout the orbit. 
True Anomaly Area
True Anomaly, ν (Greek "nu"), is angle from Xaxis to object's current position.
It uses Sol as the apex; recall that Sol is an orbit focus and is not the orbit center.




Eccentric Anomaly (E)
is similar to the True Anomaly (ν). It is an angle (∠_{E} ) from Xaxis (reference ray), and it depends on object's current position in elliptical orbit. Unlike ν, E uses the orbit center as the apex. Also, ∠_{E} measures from ref. ray to a ray from orbit center to AC point directly above object (see Y_{E} in figure).


EA triangle area (A_{E∠}) can be easily calculated:
A_{E∠} = 0.5 (x + c) Y_{E} = 0.573 AU^{2}
Fortunately, EA edge segment area (A_{E∇}) can also be easily calculated:
A_{E∇ }= A_{E } A_{E∠}
A_{E∇ }= (0.719  0.573) AU^{2} = 0.146 AU^{2}  Subdivide EA Sector


^{} 

Kepler's Third Law
The square of the Solar orbit's period (time^{2}) is proportional to the cube of its semimajor axis (distance^{3}).
Also known as the Harmonic Law, this is one of the most powerful statements of physical law in astronomy.
NOTE: Above has been rephrased for simplicity. For example, Kepler said "average distance from Sun" vs. "semimajor axis". There's no intent to change Kepler's content.

0 Comments:
Post a Comment
Links to this post:
Create a Link
<< Home