CYLERS TO MARS: MARSONANCE
Most Solar objects orbit in a counter clockwise (CCW) direction as observed from north of the Earth.
Most solar objects use a reference ray at orbit's perihelion, q. Thus, Habitat's angular distance (Θ) is 0° at q, and increases in a CCW direction.
Arbitrarily design Habitat's orbit so q happens at 40° behind Earth and 69° behind Mars. This arrangement enables Habitat to rendezvous with Earth 51 days after q; 67 days after Earth, it could rendezvous with Mars.
IT TURNS OUT that such a rendezvous would be rare for Earth, because this same configuration would seldom repeat.
However, orbital resonance enables Habitat to rendezvous with Mars at same position for every orbital cycle.

Orbital Period  Semimajor Axis  Semilatus Rectum  Semiminor Axis  Focus  Eccen tricity  Angular Positions  
Object  T  a  ℓ  b  c  e  Θ_{i }  _{}Θ_{i+1} 
Earth (ⴲ)  1.00 Yrs  1.000 AU  1.00 AU  1.000 AU  0.017 AU  0.0167  TBD  TBD 
Habitat (H)  1.88 Yrs  1.523 AU  1.00 AU  1.234 AU  0.896 AU  0.586  0°  1° 
Mars (♂)  1.88 Yrs  1.523 AU  1.51AU  1.516 AU  0.142 AU  0.0933  TBD  TBD 
Given  Observed  ∛(T^{2})  b^{2} ÷ a  √(ℓ×a)  √(a^{2}b^{2})  c ÷ a  Given 
Habitat velocities range from over 47 kilometers per second (kps) to a low of 12.33 kps.
① At Θ = 0°, diagram shows Habitat's semiorbit at nearest point to Sol, perihelion (q). This particular semiorbit has Earth leading habitat by 40.16° and Mars leading by 68.60°. ② At Θ = 45°, all three objects (Habitat, Earth, Mars) have advanced, but the fast moving Habitat has gained on the other two. ③ At Θ = 90°, Habitat will cross orbit of Earth, because Habitat's orbit is designed that way. Diagrams shows a rendezvous with Earth for this particular orbit; unfortunately, Habitat orbit does not resonate with orbit of Earth, and this fortuitous event will be relatively rare. Fortunately, travelers will use "parking orbits (described later) to complete most transits to/from Earth. ④ At Θ = 129°, Habitat completes rendezvous with Mars. Since Habitat's orbit is designed with same period (T) as orbit of Mars, Habitat will rendezvous with Mars in same position for subsequent orbits; thus, travelers can count on this Habitat for consistent transport to/fm Mars. ⑤ At Θ = 180°, Habitat's semiorbit is at farthest point from Sol, perihelion (Q). Mars is far away, and Earth has almost completed a full orbit (almost a year) since the first position on the diagram. 
H A B I T A T  Earth  Mars  

Ang. Pos.  Radius  XCoord  YCoord  Incr. Distance  Ave. Velocity  Incr. Time  Ang. Pos.  Ang. Pos.  
Θ_{i }  Θ_{i+1}  R_{i }  X_{i }  Y_{i }  Δd  V_{Ave }  Δt  Θ_{i}  Θ_{i+1}  Θ_{i}  Θ_{i+1}  
0°  1°  0.630 AU  0.63 AU  0.00 AU  0.0110 AU = 1,646,197 km  47.24 km/sec  0.40 day  40.16°  40.56°  68.60°  68.81°  
45°  46°  0.707 AU  0.50 AU  0.50 AU  0.0128 AU = 1,916,977 km  43.75 km/sec  0.51 day  59.53°  60.03°  78.90°  79.17°  
90°  91°  1.000 AU  0.00 AU  1.00 AU  0.0204 AU = 3,045,924 km  34.26 km/sec  1.01 day  90.86°  91.87°  95.56°  96.09°  
129°  130°  1.584 AU  1.00 AU  1.23 AU  0.0339 AU = 5,070,645 km  23.183 km/sec  2.53 day  154.20°  156.76°  129.23°  130.59°  
180°  181°  2.416 AU  2.416 AU  0.00 AU  0.0422 AU = 6,307,907 km  12.33 km/sec  5.92 day  20.31°  26.14°  249.44°  252.55°  
Given 
ℓ
1+e×Cosθ_{ }  R × Cos(θ)  R × Sin(θ)  ΔX = X_{i+1}  X_{i} ΔY = Y_{i+1}  Y_{i} Δd = √[(Δx)^{2}+(Δy)^{2}] 

 Given  Θ_{i} + ω_{ⴲ}×Δt  Given  Θ_{i} + ω_{♂}×Δt  
Radius readily computed from Sol to Habitat; then, translate into 2 Dimensional (X.Y) coordinates.
Pythagorean Theorem helps determine incremental distances between selected pairs of positions.
Determine average velocity for each positional pair; divide distance by V_{Ave} for approximate travel times.
 Assume Earth's angular velocity ω_{ⴲ}=360°/365.25 day ω_{ⴲ }= 0.986°/day  Assume Mars's angular velocity ω_{♂ } =360°/686.67day ω_{♂ } = 0.524°/day 
Design Habitat orbit to have same semimajor axis as orbit of Mars; thus, both share same orbital period (1.88 year) which exceeds Earth's orbital period (1 year). Thus, Earth will repeat much of its orbit while Mars and Habitat complete their respective orbits. Thus, using Earth dates to track progress of Habitat can result in ambiguities. Clarify by labeling Earth positions with both date and year. However, positions of Mars/Habitat are not ambiguous; thus, they can be clearly marked with Earth date and days of travel (time since Habitat's perihelion, q). 
HABITAT  EARTH  MARS  
Ang. Pos.  Incr. Time  Cum. Time  Date  Ang. Pos.  Radius  Coordinates  Ang. Pos.  Radius  Coordinates  
Θ_{H}  Δt  T  MonDy  Θ_{ⴲ}  R_{ⴲ}  X_{ⴲ}  Y_{ⴲ}  Θ_{♂}  R_{♂}  X_{♂}  Y_{♂} 
0°  0.40 day  0.40 day  Y1: Aug 1  40.16°  1.00 AU  0.76 AU  0.64 AU  68.60°  1.461 AU  0.53 AU  1.36 AU 
1°  0.40 day  0.81 day  Y1: Aug 1  40.56°  1.00 AU  0.76 AU  0.65 AU  68.81°  1.461 AU  0.53 AU  1.36 AU 
2°  0.40 day  1.21 day  Y1: Aug 2  40.96°  1.00 AU  0.76 AU  0.66 AU  69.02°  1.461 AU  0.52 AU  1.36 AU 
...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ... 
129°  2.53 day  115.70 day  Y1: Nov 25  154.20°  1.00 AU  0.90 AU  0.44 AU  129.23°  1.605 AU  1.02 AU  1.24 AU 
...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ... 
180°  5.92 day  345.11 day  Y1: Jul 12  20.31°  1.00 AU  0.94 AU  0.35 AU  249.44°  1.562 AU  0.55 AU  1.46 AU 
...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ... 
231°  2.56 day  571.84 day  Y2: Feb 24  243.78°  1.00 AU  0.44 AU  0.90 AU  8.26°  1.383 AU  1.37 AU  0.20 AU 
...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ...  ... 
358°  0.40 day  685.85 day  Y2: Jun 18  356.15°  1.00 AU  1.00 AU  0.07 AU  68.00°  1.459 AU  0.55 AU  1.35 AU 
359°  0.40 day  686.25 day  Y2: Jun 18  356.55°  1.00 AU  1.00 AU  0.06 AU  68.21°  1.460 AU  0.54 AU  1.36 AU 
360°  0.40 day  686.65 day  Y2: Jun 19  356.94°  1.00 AU  1.00 AU  0.05 AU  68.42°  1.460 AU  0.54 AU  1.36 AU 
Given  Prev Table  ΣΔt_{i}  Convert T  Θ_{i1} + ω_{ⴲ}×Δt  ℓ_{ⴲ} 1+e×Cos(θ)  R_{ⴲ}Cos(θ_{ⴲ})  R_{ⴲ}Sin(θ_{ⴲ})  Θ_{i1} + ω_{♂}×Δt 
ℓ_{♂}
1+e×Cos(θ_{♂})  R_{♂}Cos(θ_{♂})  R_{♂}Sin(θ_{♂}) 
Habitat Orbit Starts
 
Habitat Intercepts
② 1st Intercept: 116 days after q (Θ=0°), Habitat will rendezvous with Mars.Orbit Twice; Mars Once ③ Habitat's position at 231 days is Θ=159° after q. ④ Aphelion. 345 days (Θ=180°) into orbit, Habitat reaches farthest point from Sol, also the orbit's slowest point. ⑤ Habitat's position at459 days ; Θ=201° past q. ⑥ 2nd Intercept: 574 days (Θ=231°), Habitat again intercepts orbit of Mars; however, Mars, the planet, is far away and out of sight. 
Future Intercepts
Since Habitat orbit has same period as Mars orbit, Habitat will continue Mars intercepts for future orbits.
These will occur at same angle (129°) and same travel time (116 days since q).
 
Consider Earth Intercepts During First Orbit.  

Traditionally, q, orbit's perihelion, is an orbit's starting point. For the first orbit, we arbitrarily start this orbit at August 1, which puts Earth leading Habitat by 40°. Furthermore, we arbitrarily place Mars leading by 69°. Earth's initial position enables rapid rendezvous with Earth after 51 days of orbit from q.  From Earth, Habitat needs 65 days to travel on to Mars. ALTERNATE SCENARIO: Habitat could launch from Earth when MarsEarth are in Sep 22 positions.) Habitat orbital period (686 days) greatly exceeds period of Earth orbit. Thus, clarity requires Earth positions to label with both date and year.
At Mar 22 (Yr2), Earth reaches 2nd possible intercept point. This is over a year (476 days) since the Mars rendezvous.
.  Habitat completes an entire orbit in 686 days (Jun 19, Yr2); then, it starts another orbit. Fortunately, Mars once again leads by 69°. Unfortunately, Earth no longer leads by 40°; instead it now lags q by 3°. Thus, habitat's 2nd orbit will likely not intercept Earth in same way.
Fortyone days later (Apr 28), Habitat reaches same position. Thus, intercept not accomplished.

Consider Habitat's Subsequent Orbits.  
① Start 2nd Orbit 686 days after start of first orbit. Earth slips from a lead of 40° to a lag of 3°, but Mars maintains lead of 69°. ② 51 days later, Habitat intercepts orbit of Earth but not Earth. Earth arrives at this point 44 days later. ③ 65 days later, Habitat again transits Mars. ④ Aphelion, orbit's farthest point from Sol is also orbits slowest point. Positions of Earth/Mars not shown, but be assured they are both far away. ⑤ Habitat approaches Earth, coincidental. ⑥ 2nd Orbit Ends with Earth slipping another 43° behind. We observe a trend of Earth increasing its lag of 43° per orbit; thus, we conclude that this orbit will not facilitate constant rendezvous with Earth. 
VOLUME I: ASTEROIDAL 

VOLUME II: INTERPLANETARY 
VOLUME III: INTERSTELLAR 
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