Monday, November 11, 2013

Synodic Cyclers

The Earth-Mars Synodic orbit is 780 days (2.135 years).

If a cycler period is exactly 2.135 Earth years, a given Earth-Mars angular relationship would repeat once per cycler orbit. We're particularly interested in the "correct" angular relationship where Earth's angular position leads position of Mars such that a cycler could pass near Earth then intersect orbit of Mars precisely when Mars gets there. This fortuitous happenstance would enable a payload of passengers and cargo to board cycler near Earth; then, depart cycler near Mars.

For this fortuitous double rendezvous to occur, the cycler's perihelion must itself lead this Earth-Mars angular relationship at certain angular interval. Thus, an optimum "shuttle" from Earth to Mars requires a simultaneous angular relation amongst all three celestial objects: 1) cycler perihelion must lead Earth by a certain angle; 2) Earth position must lead Mars by a certain angle.  This circumstance would be a rare occurence.

With human help, this three part angular relationship could recur more frequently.


DIAGRAM DETAILS
Two green dots inside Earth's orbit represent Mercury and Venus.
Large arrowheads near Earth and Mars indicate that planets orbit in a
counter clockwise direction  as observed from a Heavenly Observer
north of Earth.
This example arbitrarily picked a start date of Dec 21.
In olden days, ancient star gazers thought that Mother Terra was the center of all Creation; thus, "Mars orbits Earth" seemed to reflect common sense.

Shepherds were perhaps the first stargazers.  They had to watch their flocks at night to protect them against wolves and lions and other nocturnal predators; thus, they had lots of time to observe the night sky. 

It's likely that most shepherds noticed the red dot which didn't twinkle; perhaps, some of them noticed that every couple of years, this red dot (Mars) rose exactly at dusk, reached its peak exactly at midnight; then, set exactly at dawn. This rare occurrence would happen when the Sun and Mars would be on opposite sides of the Earth or "in opposition" as shown in the diagram. Most Earthly observers assumed this event marked the beginning of a new orbit of Mars around the Earth.
Orbital Elements
Earth
Nearly Circular
Mars
Slightly Elliptical
1.0 AU
Aphelion (Q)
1.6659 AU

1.0 AU
Perihelion (q)
1.3815 AU

1.0 AU
Semi-major axis (a)
1.5237 AU

1.0 AU
Semi-minor axis (b)
1.5170 AU

0.0 AU
Focus (c)
0.1422 AU

0.0
Eccentricity (e)
 0.0933

1.0 AU
Semi-latus Rectum (l) 
1.5104 AU

1.0Yr=365.25dys
Sidereal Period (T)
1.88Yr=780dys

0.986 °/day
Ave Ang Velocity (ω)
0.524 °/day


We further surmise that some patient star gazers eventually measured the interval between this recurring phenomena to be  2 years plus about 49 days.

This duration eventually became known as the Earth-Mars Synodic Period. We now consider this to be the duration between repetitions of same relative alignment.  An obvious example is the Earth-Mars Opposition as shown; we can expect Opposition to repeat every 2.135 years.
During First Year
Earth completes its first orbit and returns to starting position of Dec 21. 
Diagrams accentuate the fact the two planetary orbits are not precisely concentric.


Earth's 2nd orbit starts with Mars halfway through its orbit.
Mars completes its first orbit with Earth not quite finished with its second.Earth completes orbit 2; however, Mars has moved on.Earth once again lines up with Sol and Mars about 45° past previous such alignment.


During Second Year and Beyond

In this example, Earth goes 48.6° past previous Opposition
to once again line up precisely between Mars and the Sun.

This starts another Synodic Orbit.

Earth-Mars Synodic Orbit
Synodic Velocity ωΔ=
0.462°/day
=ωE- ωM
Synodic Period P=779.2 dy =2.135 yr=360°/ωΔ
Compute synodic period:
1) Determine average angular velocities (Earth & Mars) as shown in diagram..
2) Determine ωΔ, difference between two angular velocities.
3) Divide 360° by ωΔ.
Describe Synodic Cycle

Orbital Elements
Earth-Mars Synodic Cycler
Semi-latus Rectum =1.0000 AU
Given
PeriodT =2.135 Yr
Given
Semi-major axisa=1.6581 AU =(T2)1/3
Semi-minor axisb=1.2877 AU =l × a)1/2
Focusc=1.0446 AU =(a2 - b2)1/2
Eccentricity=0.6300 =c / a
AphelionQ=2.7026 AU =a + c
Perihelionq=0.6135 AU =a - c
Typical reference ray (ν = 0°) extends from Sol to the perehelion, orbit's closest point to Sol. An asteroid is fastest at its perihelion; for this orbit,  object takes only 50 days to travel the first 90° as shown.
Asteroid is slowest at its farthest point (aphelion).

It takes 340 days from 90° point (Semi-latus rectum) to 180° (aphelion).
This orbit is engineered for the asteriod ("cycler") to intersect Earth's orbit at ν=90°. For cycler to rendezvous with Earth, Terra's position 50 days prior to the rendezvous must lead the cycler by 41° as shown above.After leaving Earth, cycler will subsequently intersect orbit of Mars at 112 days past perihelion postion. For cycler to rendezvous with Mars, the red planet must lead the cycler by 69° when cycler is at perihelion.
First Year
Arbitrarily start E-M synodic orbit such that Mars leads Earth by 28°.

  This same E-M relation should reoccur at the completion of the Synodic Oribit.
After 50 days, cycler will rendezvous with Earth; pax and payload can board.

Lead of Mars reduces to only 5°.
After only 62 more days, cycler can perform rendezvous with Mars.

Pax and payload can disembark.

 At the end of first year, cycler approaches the aphelion.

Second Year and Beyond
After considerable progress around its orbit, cycler once again intercepts orbit of Mars, but Mars is literally nowhere to be seen.

It hides behind Sol!
After 2 years, cycler once again intercepts orbit of Earth. 

However, Earth is far away, and Mars still hides.
Cycler completes its orbit, and the Earth-Mars  angular relation repeats.

Mars again leads Earth by 28°.
However, cycler can not repeat rendezvous with either Earth or Mars.

 They both lead cycler perihelion by 48.6° too far.
For a Synodic Cycler, perhaps one orbit of seven would bring the cycler close to Earth. 


Cycler Table

Rν=lC

1 + eC × cos(ν)
Vν=√(Sol

Rν
+μSol

aC
)
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

x Remarks
At Perihelion
Nearest Earth
Nearest Mars
At Aphelion
x

Future content:
Multiple cyclers
inorbit fuel burns to adjust orbit
retrograde rendezvous, payloads from Mars to Earth
Synodic Cyclers.  For cyclers with periods exactly 2.135 Earth years, their period would coincide with the Mars's synodic orbit viewed from Earth. Thus, when synodic cycler intersects Earth's orbit, the Earth-Mars angular relationship would be the same as the previous such intersection.
PROBLEM: This intersection point is not resonant and strays all over Earth's orbit.  Thus, it would seldom be near Earth's position.
Orbital Elements
Earth (ⴲ) OrbitHabitat (H) OrbitMars (♂) Orbit
AphelionQ = 1.0 AUObservedQH = 2.5530 AU = a + cQ =  1.6659 AUObserved
Perihelionq = 1.0 AUObservedqH = 0.6218 AU = a - cq =  1.3815 AUObserved
Semi-major axisa = 1.0 AU = (Q + q) ÷ 2aH = 1.5874 AU =  (T2)a = 1.5237 AU = (Q + q) ÷ 2
Semi-minor axisb = 1.0 AU = (Q - q) ÷ 2bH = 1.2599 AU = ( × a)b = 1.5170 AU = (Q - q) ÷ 2
Focusc = 0.0 AU (a2 - b2)cH = 0.9656 AU (a2 - b2)c =  0.1422 AU =  (a2 - b2)
Eccentricitye = 0.0 AU = c ÷ aeH = 0.6053 = c ÷ ae =  0.0933 = c ÷ a
Semi-latus Rectum = 1.0 AU = b÷ aH = 1.00 AUGiven 1.5104 AU = b2 ÷ a
Sidereal PeriodT  =1.00  Yr =  (a3)TH = 2.00 YrGivenT  =  1.88  Yr = (a3)
Angular Velocityω 0.986°/dy = 360° ÷ TωH = VariableTBDω  =  0.524°/dy = 360° ÷ T
CONSIDER FOLLOWING ORBITS:
  1. 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.
  2. T=2.00, Earth Resonant cycler.  Same date, every two years, returns to Earth, guaranteed rendezvous with Earth every 2 years.
  3. Synodic orbit, T=2.135 years. Will seldom rendezvous with either Mars or Earth, but if it ever does, it'll rendezvous with Earth first, then quickly move on to Mars for 2nd rendezvous; IT'LL BE AWESOME!!!

Rθ=H

1 + eH × Cos(θ)
Position


qH = RMin=H

1 + eH × Cos(0°)


QH = RMax=H

1 + eH × Cos(180°)


Vθ=√(Sol

Rθ
+μSol

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 AU48.6
90°501.000 AU35.2
127°1121.611 AU23.8
180° 3922.703 AU11.02
Given
Deg
AU
km/sec


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