Wednesday, February 04, 2009

Designing Orbits: Paths of Cycler Habitats


Gather list of NEAs to validate computed a, as well as poss source of raw materials.
Recall Lewis, author of Mining the Asteroids, and his cycler concept.



Resonant NEAs

Looking for resonance from a list of 5961 Near-Earth Asteroids.

to 36 then to the following 8.
aeiwNodeMqQTHMOIDrefclass
Object(AU)(deg)(deg)(deg)(deg)(AU)(AU)(yr)(mag)(AU)
1 3753 Cruithne1.000.51519.843.7126.3332.60.4841.51115.100.07094181ATE
2 85770 (1998 UP1)1.000.34533.2234.318.4144.90.6541.34120.370.0837263ATE
3 164207 (2004 GU9)1.000.13613.6281.038.8115.80.8641.14121.150.00004724APO*
4 138852 (2000 WN10)1.000.29921.5225.161.0112.80.7021.30120.060.12600773APO
5 (2008 FH)1.590.5053.5264.15.2331.30.7852.39224.390.01544APO*
6 (2006 UP)1.590.3022.3334.748.124.21.1082.07223.110.11226118AMO
7 169352 (2001 UY16)2.080.51832.348.432.3113.31.0033.16319.100.12249730APO
8 26166 (1995 QN3)3.300.64514.862.9185.898.01.1735.43617.300.27016931AMO
Define "resonance".


Define coord system such that x axis (abscissa) has following values:
Q (aphelion) = -a -c
q (perihelion) = a - c
Orbit center is at (-c, 0)
Origin is offset to right focus (location of Sun): (0,0).
Then ray from Sun to asteriod position can be described
y = mx
where m (slope) is tan of angle from x-axis CCW to asteroid.
Finally, equation of orbital shape (ellipse) can be described:
(x-c)2/a2 + y2/b2 = 1

y = √[a2 - (x-c)2] b/a
Let T = 2 years to determine resonance
then a = 1.59 AU
let e = 1/3, an arbitrary value picked for our convenience.
Then a = 1.5 AU and c = .53 AU

Lewis, author Mining the Asteroids, suggested orbit of 2 years for each of 6 cyclers; thus, one cycler roundevous with near Earth location every 4 months.

Suggest following items:

Review Alpha/Omega, near Earth orbiting habitats. Suggest these be the rondevous .

Review Kepler's 3rd law; thus, 2 year orbits indicates specific a, starting orbital element. Semimajor axis, a, would be common to all cycles, but other orbital elements would probably vary.

Orbital Velocities
Kepler's Laws point out that orbital velocity varies per some relationship with distance from the Sun. Newton built on this work to derive the now famous equation: v = √(G * MSol/R) which can determine orbital speeds for circular orbits.For Earth's Solar orbit, use following values:
Universal gravitational constant: G = 6.667 x 10-11 N-m2 /kg2
Mass of Sun: MSol = 2.0 x 1020kilograms
Radius of Earth's orbit: R = 1.5 x 1011 meters

Earth's orbital velocity,v = 29.81 km/sec, a well known value.
NOTE: Since values G and MSol are constant, the product, G*MSol , is often expressed as the standard gravitational parameter, μ = 132,712,440,018 km3 / sec2 = 13.27 x 1019 m3/sec2.
Convenience: use approx equivalent value, μ = 39 AU3/yr2 since we prefer to use AUs for a and years for T (versus meters and seconds).


Cross Check: Compute period, T = 2 π √(a3/μ)
For elliptical orbits, use following equation to determine orbital velocities at various positions throughout the orbit: v = √(μ (2/r - 1/a)) Poss sidebar, determine conversion constants such that we could input a in AUs and output T in years. Traditional formula takes a in meters and outputs T in seconds.

Poss sidebar, velocity analysis of proposed orbits.
Rondevous analysis,
should cycler perihelion be short or long of Α/Ω?
If Α/Ω are ± 60° of Earth's Solar orbit, should cycler rondevous midway between Earth and orbiter (30° away)?

Traditional Value

μ =13.27 x 1019 m3
sec2

Appropriate Conversions

*
AU3

(1.5 * 1011 m)3
*(365.25 * 86,400 sec)2
yr2

New value

=39 AU3
yr2
=μ

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