Tuesday, August 28, 2012

Ref. Table: Synodic Cycler Orbit Details

Numerical Methods
Determine Angle Values Between Two Successive Degrees
Given
Time
True
Anom.
Radius Sun Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
tνRXYXACYACEtΔt
days Deg AU AU AU AU AU Degdays days
149°2.174 -1.86 1.12 -0.82 1.44 119.6°191.08 na
192149.19°2.179 -1.87 1.12 -0.83 1.44 119.9°192.0 0.92
193149.40°2.185 -1.88 1.11 -0.84 1.43 120.3°193.0 1.92
194149.60°2.190 -1.89 1.11 -0.84 1.43 120.6°194.0 2.92
195149.81°2.196 -1.90 1.10 -0.85 1.42 121.0°195.0 3.92
150°2.201 -1.911.10-0.861.42121.3°195.934.85
ν. . .tνn/a
tGivenνt. . .tCompΔtν

ν+1. . .tν+1Δtν+1
Approximate νt
Arbitrarily choose tGiven between tν and tν+1. Example:191.08 < t < 195.93
where t could be: 192, 193, 194 or 195. Note that True Anonmaly, ν= 149°, results in duration of 191.08 days, and ν+1 (150°) results in 195.93 days.
Best guess νt such that resulting tComp = tGiven
Δtν = tGiven - tν
Δtν+1 = tν+1 - tν
Understood but not shown is Δν = 1° = 150° - 149°.
νt = ν + Δtν

Δtν+1
(Δν)
For convenience, following table designed with some arbitrary considerations:
--Cycler's perihelion conveniently chosen to be colinear with Sol and perihelion of Mars.

--Orbit inclinations disregarded.

Given.
Time
True
Anom.
Radius Sun () Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
Earth (⊕) Based Mars (♂) Based
tν RXYXACYACE t Δt ν XYνRX Y
days Deg AU AU AU AU AU Deg days days Deg AU AU Deg AU AU AU
00.6140.610.001.660.000.0°0.0n/a40.96°0.760.6669.35°1.4620.521.37
1025.52°0.6380.580.271.620.3512.3°10.010.0050.82°0.630.7874.59°1.4740.391.42
2047.71°0.7020.470.521.520.6723.8°20.010.0060.68°0.490.8779.83°1.4860.261.46
3065.49°0.7930.330.721.370.9334.1°30.010.0070.53°0.330.9485.07°1.4980.131.49
4079.38°0.8960.170.881.211.1343.2°40.010.0080.39°0.170.9990.31°1.511-0.011.51
5090.34°1.004-0.011.001.041.2951.2°50.010.0090.24°0.001.0095.55°1.524-0.151.52
6099.18°1.112-0.181.100.871.4158.5°60.010.00100.10°-0.180.98100.79°1.537-0.291.51
Given.
Time
True
Anom.
Radius Sun () Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
Earth (⊕) Based Mars (♂) Based
70106.45°1.217-0.341.170.701.5065.0°70.010.00109.95°-0.340.94106.03°1.550-0.431.49
80112.59°1.319-0.511.220.541.5771.1°80.010.00119.81°-0.500.87111.27°1.563-0.571.46
90117.86°1.417-0.661.250.381.6176.7°90.010.00129.67°-0.640.77116.51°1.576-0.701.41
100122.46°1.511-0.811.270.231.6481.9°100.010.00139.52°-0.760.65121.75°1.588-0.841.35
110126.53°1.600-0.951.290.091.6686.8°110.010.00149.37°-0.860.51126.99°1.600-0.961.28
120130.19°1.685-1.091.29-0.041.6691.5°120.010.00159.23°-0.940.35132.23°1.611-1.081.19
tν RXYXACYACE t Δt ν XYνRX Y
130133.50°1.766-1.221.28-0.171.6595.9°130.010.00169.08°-0.980.19137.47°1.622-1.201.10
140136.53°1.842-1.341.27-0.291.63100.2°140.010.00178.94°-1.000.02142.71°1.632-1.300.99
150139.32°1.915-1.451.25-0.411.61104.2°150.010.00188.80°-0.99-0.15147.95v1.640-1.390.87
160141.92°1.984-1.561.22-0.521.58108.2°160.010.01198.66°-0.95-0.32153.19°1.648-1.470.74
170144.34°2.049-1.661.19-0.621.54112.0°170.09.99208.51°-0.88-0.48158.43°1.654-1.540.61
180146.62°2.110-1.761.16-0.721.49115.6°180.010.00218.36°-0.78-0.62163.67°1.659-1.590.47
days Deg AU AU AU AU AU Deg days days Deg AU AU Deg AU AU AU
190148.77°2.168-1.851.12-0.811.45119.2°190.010.00228.22°-0.67-0.75168.91°1.663-1.630.32
200150.82°2.222-1.941.08-0.901.40122.7°200.010.00238.08°-0.53-0.85174.15°1.665-1.660.17
210152.77°2.274-2.021.04-0.981.34126.1°210.010.00247.93°-0.38-0.93179.39°1.666-1.670.02
220154.63°2.322-2.100.99-1.051.28129.4°220.010.00257.79°-0.21-0.98184.63°1.665-1.66-0.13
230156.43°2.366-2.170.95-1.121.22132.7°230.010.00267.64°-0.04-1.00189.87°1.663-1.64-0.29
240158.16°2.408-2.240.90-1.191.15135.9°240.010.01277.51°0.13-0.99195.11°1.660-1.60-0.43
Given.
Time
True
Anom.
Radius Sun () Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
Earth (⊕) Based Mars (♂) Based
250159.83°2.447-2.300.84-1.251.09139.1°250.010.00287.36°0.30-0.95200.35°1.655-1.55-0.58
260161.45°2.483-2.350.79-1.311.02142.2°260.09.99297.21°0.46-0.89205.59°1.649-1.49-0.71
270163.03°2.516-2.410.73-1.360.95145.2°270.010.01307.08°0.60-0.80210.84°1.642-1.41-0.84
280164.57°2.546-2.450.68-1.410.87148.2°280.09.99316.93°0.73-0.68216.07°1.634-1.32-0.96
290166.07°2.574-2.500.62-1.450.80151.2°290.010.01326.79°0.84-0.55221.31°1.624-1.22-1.07
300167.54°2.598-2.540.56-1.490.72154.2°300.09.99336.64°0.92-0.40226.55°1.614-1.11-1.17
tν RXYXACYACE t Δt ν XYνRX Y
310168.99°2.621-2.570.50-1.530.64157.1°310.010.01346.51°0.97-0.23231.80°1.603-0.99-1.26
320170.42°2.640-2.600.44-1.560.57160.0°320.09.99356.36°1.00-0.06237.03°1.591-0.87-1.33
330171.82°2.657-2.630.38-1.590.49162.9°330.010.006.22°0.990.11242.27°1.579-0.73-1.40
340173.21°2.671-2.650.32-1.610.41165.8°340.010.0016.07°0.960.28247.52°1.566-0.60-1.45
350174.59°2.682-2.670.25-1.630.33168.7°350.010.0025.93°0.900.44252.75°1.553-0.46-1.48
360175.95°2.691-2.680.19-1.640.24171.5°360.010.0035.79°0.810.58258.00°1.540-0.32-1.51
days Deg AU AU AU AU AU Deg days days Deg AU AU Deg AU AU AU
370177.31°2.698-2.690.13-1.650.16174.4°370.010.0045.64°0.700.71263.24°1.527-0.18-1.52
380178.66°2.701-2.700.06-1.660.08177.2°380.010.0055.50°0.570.82268.48°1.514-0.04-1.51
390180.01°2.703-2.700.00-1.660.00180.0°390.010.0065.36°0.420.91273.72°1.5010.10-1.50
400181.36°2.701-2.70-0.06-1.66-0.08182.9°400.010.0075.22°0.260.97278.96°1.4890.23-1.47
410182.71°2.697-2.69-0.13-1.65-0.16185.7°410.010.0085.07°0.091.00284.20°1.4770.36-1.43
420184.07°2.691-2.68-0.19-1.64-0.25188.5°420.010.0094.93°-0.091.00289.44°1.4650.49-1.38
Given.
Time
True
Anom.
Radius Sun () Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
Earth (⊕) Based Mars (♂) Based
430185.44°2.682-2.67-0.25-1.63-0.33191.4°430.010.00104.79°-0.260.97294.68°1.4540.61-1.32
440186.81°2.670-2.65-0.32-1.61-0.41194.2°440.010.00114.64°-0.420.91299.92°1.4430.72-1.25
450188.20°2.656-2.63-0.38-1.58-0.49197.1°450.010.00124.50°-0.570.82305.16°1.4330.83-1.17
460189.61°2.640-2.60-0.44-1.56-0.57200.0°460.010.00134.36°-0.700.71310.40°1.4240.92-1.08
470191.03°2.620-2.57-0.50-1.53-0.65202.9°470.09.99144.21°-0.810.58315.64°1.4161.01-0.99
480192.48°2.598-2.54-0.56-1.49-0.72205.9°480.010.01154.07°-0.900.44320.89°1.4081.09-0.89
tν RXYXACYACE t Δt ν XYνRX Y
490193.95°2.573-2.50-0.62-1.45-0.80208.8°490.09.99163.92°-0.960.28326.12°1.4021.16-0.78
500195.46°2.546-2.45-0.68-1.41-0.87211.8°500.010.01173.79°-0.990.11331.37°1.3961.23-0.67
510197.00°2.516-2.41-0.74-1.36-0.95214.8°510.09.99183.63°-1.00-0.06336.60°1.3911.28-0.55
520198.58°2.482-2.35-0.79-1.31-1.02217.9°520.010.01193.50°-0.97-0.23341.85°1.3871.32-0.43
530200.20°2.446-2.30-0.84-1.25-1.09221.0°530.010.00203.35°-0.92-0.40347.09°1.3841.35-0.31
540201.87°2.408-2.23-0.90-1.19-1.15224.1°540.010.00213.21°-0.84-0.55352.32°1.3831.37-0.18
days Deg AU AU AU AU AU Deg days days Deg AU AU Deg AU AU AU
550203.60°2.366-2.17-0.95-1.12-1.22227.4°550.010.01223.07°-0.73-0.68357.57°1.3821.38-0.06
560205.40°2.321-2.10-1.00-1.05-1.28230.6°560.01.00232.93°-0.60-0.802.81°1.3821.380.07
570207.27°2.273-2.02-1.04-0.98-1.34234.0°570.010.00242.78°-0.46-0.898.05°1.3831.370.19
580209.22°2.221-1.94-1.08-0.89-1.40237.4°580.010.00252.64°-0.30-0.9513.29°1.3851.350.32
590211.26°2.167-1.85-1.12-0.81-1.45240.9°590.010.00262.49°-0.13-0.9918.53°1.3881.320.44
600213.42°2.109-1.76-1.16-0.72-1.50244.4°600.010.00272.35°0.04-1.0023.77°1.3921.270.56
Given.
Time
True
Anom.
Radius Sun () Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
Earth (⊕) Based Mars (♂) Based
610215.70°2.047-1.66-1.19-0.62-1.54248.1°610.010.00282.21°0.21-0.9829.01°1.3961.220.68
620218.13°1.982-1.56-1.22-0.51-1.58251.9°620.09.99292.06°0.38-0.9334.25°1.4021.160.79
630220.73°1.914-1.45-1.25-0.41-1.61255.8°630.010.00301.92°0.53-0.8539.49°1.4091.090.90
640223.53°1.841-1.33-1.27-0.29-1.63259.9°640.010.00311.78°0.67-0.7544.73°1.4161.011.00
650226.56°1.764-1.21-1.28-0.17-1.65264.2°650.010.00321.63°0.78-0.6249.97°1.4250.921.09
660229.87°1.684-1.09-1.29-0.04-1.66268.6°660.09.99331.48°0.88-0.4855.21°1.4340.821.18
tν RXYXACYACE t Δt ν XYνRX Y
670233.54°1.599-0.95-1.290.09-1.66273.3°670.010.00341.34°0.95-0.3260.45°1.4440.711.26
680237.62°1.509-0.81-1.270.24-1.64278.2°680.010.00351.20°0.99-0.1565.69°1.4550.601.33
690242.23°1.416-0.66-1.250.39-1.61283.4°690.010.001.05°1.000.0270.93°1.4660.481.39
700247.51°1.317-0.50-1.220.54-1.57289.0°700.010.0010.91°0.980.1976.17°1.4770.351.43
710253.66°1.215-0.34-1.170.70-1.50295.1°710.010.0020.76°0.940.3581.41°1.4900.221.47
720260.96°1.110-0.17-1.100.87-1.41301.7°720.010.0030.61°0.860.5186.64°1.5020.091.50
days Deg AU AU AU AU AU Deg days days Deg AU AU Deg AU AU AU
730269.83°1.0020.00-1.001.04-1.29308.9°730.010.0040.47°0.760.6591.89°1.515-0.051.51
740280.83°0.8940.17-0.881.21-1.13317.0°740.010.0050.33°0.640.7797.12°1.528-0.191.52
750294.79°0.7910.33-0.721.38-0.92326.1°750.010.0060.18°0.500.87102.36°1.541-0.331.51
760312.64°0.7010.47-0.521.52-0.66336.4°760.010.0070.04°0.340.94107.61°1.554-0.471.48
770334.91°0.6370.58-0.271.62-0.35347.9°770.010.0079.90°0.180.98112.84°1.567-0.611.44
779.8360.00°0.6140.610.001.660.00360.0°779.89.8289.58°0.011.00117.99°1.580-0.741.39
GivenNum.
Meth.
See R
below.
R×
cos(ν)
R×
sin(ν)
XO + c
YOa

b
See E belowSee tν
below.
tν-tν-1ω=
.9856°/dy
1AU ×
cos(ν)
1AU ×
sin(ν)
ω=
.5240°/dy
See R
below
R ×cos(ν)R ×sin(ν)
tν RXYXACYACE t Δt ν XYνRX Y
Given.
Time
True
Anom.
Radius Sun () Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
Earth (⊕) Based Mars (♂) Based
.*****************************
Independent variable; time (t): incremented 10 days throughout cycler period.
Estimate cycler angle, ν (Greek letter: "nu"), via Numerical Methods (see top of page).
****************************
Semi-latus Rectum, l, is chosen as intersection point of Earth's orbit with cyler's orbit.l =1.0000 AU

Period, T, chosen to coincide with Earth-Mars synodic period.T =2.135 Yr

Semi-major axis, a, must comply with Kepler's Third Law.a=1.6581 AU =(T2)1/3
R = l

1 + e × cos(ν)
X = R × cos(ν)Y= R × sin(ν)
Cycler Eccentricitye  =0.6300 =c / a
*****************************
Semi-minor axisb=1.2877 AU =( l × a)1/2
Focusc=1.0446 AU =(a2 - b2)1/2
XAC = X- cYAC =Y× a / b
E =tan-1YAC

XAC
t = (E - e × Sin(E)) × a × (a/μ)
*****************************
Martian Orbital Elements  (Slightly elliptical Solar Orbit)
Average Angular Velocity of Mars, ω = 0.524°/day = 360°/T
When cycler angle is 0°, Terran angle, ν , is 69.4° degs (leading Earth by 28.4°).
Thus, ν 69.4° + t × ω
R = l

1 + e × cos(ν)
X= Rcos(ν)
Y= Rsin(ν)
 Semi-major axis a = 1.5237 AU Given
R = l

1 + e × cos(ν)
Sidereal Period T = 1.88 Yr = (a3)1/2
Semi-minor axis b = 1.5170 AU = (a2 - c2)1/2
Focus c = 0.1422 AU = a × e
Eccentricity e = 0.0933 Given
Semi-latus Rectum l = 1.5104 AU = b2 /a
Sidereal Period T = 1.88 Yr = (a3)1/2
*****************************

Terran Orbital Elements (Assume Circular Solar Orbit)
Earth's Angular Velocity ω= 0.986°/day = 360°/T
When cycler angle is 0°, Terran angle, ν , is 41° degs.
Thus, ν = 41° + t × ω 
Radius:R=1.0 AU
 
X-coordinate:X=
R × cos(ν)
 
Y-coordinate:Y=
R × sin( ν)
 
Aphelion:Q=
1.0 AU   = a + c
 
Perihelion:q=
1.0 AU   = a - c
 
Semi-major axis:a =
1.0 AU   = R
 
Semi-minor axis:b =
1.0 AU   = R
 
Focus:c =
0.0 AU  = (a2-b2)
 
Semi-latus Rectum: l=
1.0 AU  = b2 / a
 
Sidereal Period:T= 1.0 Yr   = a3
 
 *****************************


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