Monday, October 28, 2013

MIX ORBITS

Marsonance
T= 1.68 years, Cycler shares same period as Mars, rendezvous with Mars at same orbital position once per orbit. (for more, see "Marsonance"). guaranteed rendezvous with Mars once per Martian orbit.
Terrasonance
T=2.00, Earth Resonant cycler.  Same date, every two years, returns to Earth, guaranteed rendezvous with Earth every 2 years.  NOTE:  could add another habitat exactly one year after current one.  Thus, Earth would rendezvous with a habitat once per year.
Synodic
Every 2.135 years, Mars and Earth achieve same spatial relationship; thus, Earth dwellers observe Mars in a synodic orbit with period of 2.135 years (about 778 days).

NOTE:  Orbit is about 2 1/7 years; thus, after 7 orbits (15 years), habitat would be very close to another quick transport opportunity.  Futhermore, if we added 6 more habitats and evenly spaced all 7 habitats throughout the obit's period; then every synodic orbit a habitat could accomplish quick transport from Earth to Mars.①②③④⑤⑥⑦
①In an ideal situation, habitat will first rendezvous with Earth with Mars leading by about 5°.

As habitat continues its orbit past Earth, Mars continues its orbit to its interception point as shown.

②About 65 days later, habitat and Mars both arrive at interception point for another rendezvous.

 

 
Orbital Elements
 MarsonanceTerrasonanceSynodic
Sidereal PeriodTH =1.878 Yr GivenTH =  2.000 Yr GivenTH = 2.135 Yr Given
Semi-latus RectumH = 1.00 AU GivenH = 1.00 AUGivenH = 1.00 AU Given
Semi-major axisaH =1.522 AU  =  (T2)aH =1.587 AU  =  (T2)a = 1.658 AU  =  (T2)
Semi-minor axisbH = 1.234 AU  = ( × a) bH =1.260 AU  = ( × a) b = 1.288 AU  = ( × a)
FocuscH =  0.892 AU  = (a2 - b2)cH = 0.966 AU  = (a2 - b2)c =  1.045 AU  =  (a2 - b2)
EccentricityeH = 0.586 = c / aeH =  0.608  = c / ae = 0.630  = c / a
PerihelionRMin = qH = 0.631 AUsee belowRMin = qH =  0.622 AUsee belowRMin = qH =  0.614 AU see below
AphelionRMax = QH =2.414 AUsee belowRMax = QH = 2.553 AUsee belowRMax = QH =   2.703 AUsee below
Max Velocity VMax = 47.4 kps see belowVMax = 48.0 kpssee below VMax = 48.7 kps see below
Min Velocity VMin  = 12.4 kps see belowVMin =11.7 kps see below VMin = 11.1 kps see below
CONSIDER FOLLOWING:
to be added

Earth Resonance
Period (T)
Semi-Major Axis (a)
2 Years
(T2) = 1.587 AU
Semilatus
Rectum 
Semiminor
Axis
Focus
Eccen-
tricity
Perihelion
Aphelion
Max
Velocity
Min
Velocity
 
b
c
e
RMin = q

θ=0°
RMax = Q

θ=180°
VMax

θ=0°
VMin

θ=180°
0.5 AU
0.891 AU
1.314 AU
0.828
0.274 AU
2.901 AU
77.19 kps
7.28 kps
1 AU
1.260 AU
0.966 AU
0.608
0.622 AU
2.553 AU
48.03 kps
11.70 kps
1.5 AU
1.543 AU
0.372 AU
0.235
1.215 AU
1.960 AU
30.10 kps
18.66 kps
Given
( × a) 
(a2 - b2)
 c ÷ a


(1 + e × Cos(θ))
(μSol(2

R
-1

a
))
1 AU
1.260 AU
0.966 AU
0.608
0.622 AU
2.553 AU
48.03 kps
11.70 kps
1.37 AU
1.475 AU
0.587 AU
0.370
1.00 AU
2.175AU
34.96 kps
16.07 kps
b2 ÷ a
(a2 - c2)
a - q
 c ÷ a
Given
a + c
(μSol(2

R
-1

a
))
 
b
c
e
q
Q
VMax

θ=0°
VMin

θ=180
xaxaxx

μSol = 891.906 AU-km2 / sec2
μSol = 29.865 AU-km / sec


Rθ=H

1 + eH × Cos(θ)
Position


qH = RMin=H

1 + eH × Cos(0°)




QH = RMax=H

1 + eH × Cos(180°)



Vθ=√(μSol(2

Rθ
-

aH

) ) 
Velocity

VMax=√(Sol

q
- × μSol

aH
)



VMin=√(Sol

Q
- × μSol

aH
)



Cycler Relevant Positions
Pos.TimeRangeCycler
Velocity
ν
Days
AU
km/sec
00.614 AU 48.6
90°501.000 AU 35.2
127°1121.611 AU23.8
180° 3922.703 AU11.02
Given
Deg
AU
km/sec
xxxxxxxxxxxxxx
Semimajor Axis(a)
Given: a = 1.58 AU
Perihelion (q)

q = a - c
    Focus (c)

c = a - q
    c2 = a2 -2aq + q2  
Semiminor Axis (b)

b2 = a2 - c2  
b2 = 2aq - q2  = q (2a - q)
b = (q (2a - q))
Semi Latis Rectum ()
  = b2 / a
q
b
(AU)(AU)(AU)
0.10.550.19
0.20.770.37
0.30.930.54
0.41.050.70
0.51.150.84
0.61.240.97
0.71.311.09
0.81.371.19
0.91.431.29
11.471.37
xxxxxx


(μSol(2

R
-

a
))

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