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.
a  e  i  w  Node  M  q  Q  T  H  MOID  ref  class  

Object  (AU)  (deg)  (deg)  (deg)  (deg)  (AU)  (AU)  (yr)  (mag)  (AU)  
1  3753 Cruithne  1.00  0.515  19.8  43.7  126.3  332.6  0.484  1.51  1  15.10  0.070941  81  ATE 
2  85770 (1998 UP1)  1.00  0.345  33.2  234.3  18.4  144.9  0.654  1.34  1  20.37  0.08372  63  ATE 
3  164207 (2004 GU9)  1.00  0.136  13.6  281.0  38.8  115.8  0.864  1.14  1  21.15  0.000047  24  APO* 
4  138852 (2000 WN10)  1.00  0.299  21.5  225.1  61.0  112.8  0.702  1.30  1  20.06  0.126007  73  APO 
5  (2008 FH)  1.59  0.505  3.5  264.1  5.2  331.3  0.785  2.39  2  24.39  0.0154  4  APO* 
6  (2006 UP)  1.59  0.302  2.3  334.7  48.1  24.2  1.108  2.07  2  23.11  0.112261  18  AMO 
7  169352 (2001 UY16)  2.08  0.518  32.3  48.4  32.3  113.3  1.003  3.16  3  19.10  0.122497  30  APO 
8  26166 (1995 QN3)  3.30  0.645  14.8  62.9  185.8  98.0  1.173  5.43  6  17.30  0.270169  31  AMO 
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 xaxis CCW to asteroid.
Finally, equation of orbital shape (ellipse) can be described:
(xc)^{2}/a^{2} + y^{2}/b^{2} = 1
y = √[a^{2}  (xc)^{2}] b/a
y = √[a^{2}  (xc)^{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.
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 * M_{Sol}/R) which can determine orbital speeds for circular orbits.For Earth's Solar orbit, use following values:
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 * M_{Sol}/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} Nm^{2} /kg^{2}
Mass of Sun: M_{Sol} = 2.0 x 10^{20}kilograms
Radius of Earth's orbit: R = 1.5 x 10^{11} meters
Mass of Sun: M_{Sol} = 2.0 x 10^{20}kilograms
Radius of Earth's orbit: R = 1.5 x 10^{11} meters
Earth's orbital velocity,v = 29.81 km/sec, a well known value.
NOTE: Since values G and M_{Sol} are constant, the product, G*M_{Sol} , is often expressed as the standard gravitational parameter, μ = 132,712,440,018 km^{3} / sec^{2} = 13.27 x 10^{19} m^{3}/sec^{2}.
NOTE: Since values G and M_{Sol} are constant, the product, G*M_{Sol} , is often expressed as the standard gravitational parameter, μ = 132,712,440,018 km^{3} / sec^{2} = 13.27 x 10^{19} m^{3}/sec^{2}.
Convenience: use approx equivalent value, μ = 39 AU^{3}/yr^{2} since we prefer to use AUs for a and years for T (versus meters and seconds).
Cross Check: Compute period, T = 2 π √(a^{3}/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.
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)?
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
 Appropriate Conversions
 New value

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