Tuesday, July 31, 2012

Plot Orbits and Tell Time

1. Plot Dots
Plot orbital positions with Sol based Cartesian Coordinates.
Determine as follows:
X =Rν cos(ν) Y = Rν sin(ν)
Examples for ν values of 160° and 180°:
X = R160 cos(160°) = -2.3 AU Y= R160 sin(160°) = 0.84 AU  
X= R180 cos(180°) = -2.7 AUY= R180 sin(180°)  = 0.00 AU
To determine length of radius vector, R, from Sol to satellite, use following polar equation:
Rν= l

1 + e × cos(ν)
TRUE ANOMALY: ν, is angle of the R vector. Ref. Angle (ν= 0°) is the ray connecting Sol to perihelion.
(In table, ν independently varies from 0° to 180° in 20° increments).
SEMI-LATIS RECTUM: l, is key to radius formula; distance from Sol to cycler at ν = 90°.
l = 1.0 AU =149,597,870.7 km (arbitrarily chosen to be same distance as from Sol to Terra).
SEMI-MAJOR AXIS: a, is longest distance from center to orbit; cycler at ν = 0°.
let a = 1.66 AU (arbitrarily chosen to give orbit a syndodic period, 2.135 years).
SEMI-MINOR AXIS: b, shortest distance from center to orbit. (a×l)
b=(1.587AU×1.0AU)=1.288 AU
FOCUS: c, distance from orbit center to Sol; defined: (a2-b2)
c= (a2-b2) = 1.045 AU
ECCENTRICITY: e, measure of orbit's "flatness".
e = c/a = .623
ν Rν X  Y
Deg AU AU AU
0.614 0.61 0.00
20°0.628 0.59 0.21
40°0.674 0.52 0.43
60°0.760 0.38 0.66
80°0.901 0.16 0.89
100°1.123 -0.19 1.11
120°1.460 -0.73 1.26
140°1.933 -1.48 1.24
160°2.451 -2.30 0.84
180°2.703 -2.70 0.00
2. Connect Dots
Determine straight line distance increments with Pythagorean formula.
Δd=√((ΔX)2+(ΔY)2)=((X2-X1)2+(Y2-Y1)2)
μSol, Solar standard gravitational parameter, = 132,712,440,018 km3s−2
Use following formulas to approximate incremental times.
Vν=(Sol

Rν
+μSol

a
)
Δt=Δd

Vν
×day

86,400 sec
νRν XYΔdVΔt
Deg AU AU AU AU km km/sec secs days
160°2.451 -2.30 0.84
180°2.703 -2.70 0.00 0.9286
138,914,962
11.021 12,604,896
145.89
Given See above See aboveSee aboveSee aboveΔdAU * 149,597,871See above.See above.ΔtSee/86,400
 To approximate half of orbital period,
summ increments from ν = 0° to 180°.

νRν__X__YDistance (Δd)V ΔtΣt
degAUAUAUAUkmkm/secdaysdays
0.614
0.61
0.00
n/a
48.549
n/a
20°0.628
0.59
0.21 0.2161 32,326,752 47.849 7.82 7.82
40°0.674
0.52
0.43 0.2308 34,521,127 45.776 8.73 16.55
60°0.760
0.38
0.66 0.2632 39,369,196 42.404 10.75 27.29
80°0.901
0.16
0.89 0.3202 47,903,956 37.859 14.64 41.94
100°1.123
-0.19
1.11 0.4137 61,882,247 32.328 22.15 64.09
120°1.460
-0.73
1.26 0.5579 83,464,320 26.083 37.04 101.13
140°1.933
-1.48
1.24 0.7510 112,341,873 19.569 66.44 167.57
160°2.451
-2.30
0.84 0.9165 137,100,294 13.743 115.46 283.04
180°2.703
-2.70
0.00 0.9286 138,914,962 11.021 145.89 428.93

Validate this approximation with
Kepler's 3rd Law
a3 = k * T2
3. Use Kepler's Anamolies to compute precise times for any angle.
Use previous table's approximate orbit times to validate this model.
Recall Kepler's Second Law: For undisturbed Solar orbits, the line joining the object to the Sun sweeps out equal areas in equal intervals of time.
Recall simple geometry gives the total area inside an orbit (AEllipse = π a b) and Kepler's Third Law gives us the total orbit( T= √(a3/μ).  Thus, finding times for parts of orbits should be a simple proportionality problem.
PROBLEM: Finding the "True Anomaly Area" is not a straightforward task. The vector, "r", from the Sun to the orbiting object does not sweep easy circular sectors; instead, it sweeps focus centered elliptical sectors.
Four hundred years ago, Kepler solved this problem by deriving his famous "Kepler's Equation". Since then, his equation has been derived by many different methods. Some pertinent concepts are briefly discussed below:
Aux. circle
Translate X-Y coordinates
---from Sun based where Sol is origin. (X,Y) = (0,0)
---to Auxilliary Circle (AC) based, where origin is AC center.   (XAC,YAC) = (0,0)
XAC = X xxxx
YAC = Y xxxxxx
The AC based coordinates enable us to determine Ecentric Anomaly (E).
Thru Kepler's Equation, E enables us to determine transition duration for any ν.
True
Anom.
Radius Sun Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Total
Time
Incr.
Time
ν Rν X Y XAC YAC E tνΔt
0.614 0.61 0.00 1.66 0.00 0.0°0.00 n/a
20°0.628 0.59 0.21 1.63 0.28 9.6°7.76 7.76
40°0.674 0.52 0.43 1.56 0.56 19.7°16.30 8.54
60°0.760 0.38 0.66 1.42 0.85 30.8°26.64 10.35
80°0.901 0.16 0.89 1.20 1.14 43.6° 40.50 13.86
100°1.123 -0.19 1.11 0.85 1.42 59.2°61.04 20.54
120°1.460 -0.73 1.26 0.31 1.63 79.1°94.49 33.45
140°1.933 -1.48 1.24 -0.44 1.60 105.2°152.54 58.05
160°2.451 -2.30 0.84 -1.26 1.08 139.4°251.03 98.48
180°2.703 -2.70 0.00 -1.66 0.00 180.0°389.91 138.89
200°2.451 -2.30 -0.84 -1.26 -1.08 220.6°528.80 138.89
220°1.933 -1.48 -1.24 -0.44 -1.60 254.8°627.28 98.48
240°1.460 -0.73 -1.26 0.31 -1.63 280.9°685.33 58.05
260°1.123 -0.19 -1.11 0.85 -1.42 300.8°718.78 33.45
280°0.901 0.16 -0.89 1.20 -1.14 316.4°739.32 20.54
300°0.760 0.38 -0.66 1.42 -0.85 329.2°753.18 13.86
320°0.674 0.52 -0.43 1.56 -0.56 340.3°763.53 10.35
340°0.628 0.59 -0.21 1.63 -0.28 350.4°772.06 8.54
360°0.614 0.61 0.00 1.66 0.00 360.0°779.82 7.76
See above "Plot Dots"See belowtν-prev.
XAC=X-  c
YAC=Y× a ÷ b
E=tan-1(YAC / XAC)
t=(E-e×Sin(E)) × a × (a/μ)

4. Use numerical methods to plot orbit times on this orbit.

Numerical Methods
Given True
Anom.
Radius Sun Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Total
Time
Incr.
Time
tνR?X?Y?XACYAC Et Δt
DaysDegAUAUAUAUAUDegDays Days
00.614 0.61 0.00 1.66 0.000.0°0.0 30.0
3065.49°0.7930.330.72 1.37 0.93 34.1°30.0 30.0
6099.18°1.112-0.181.10 0.87 1.41 58.5°60.0 30.0
90117.86°1.417-0.661.25 0.38 1.61 76.7°90.0 30.0
120130.19°1.685-1.091.29 -0.04 1.66 91.5°120.0 30.0
150139.32°1.915-1.451.25 -0.41 1.61 104.2°150.030.0
180146.62°2.110-1.761.16 -0.72 1.49 115.6°180.030.0
210152.77°2.274-2.021.04 -0.98 1.34 126.1°210.0 30.0
240158.16°2.408-2.240.90 -1.19 1.15 135.9°240.0 30.0
270163.03°2.516-2.410.73 -1.36 0.95 145.2°270.030.0
300167.54°2.598-2.540.56 -1.49 0.72 154.2°300.0 30.0
330171.82°2.657-2.630.38 -1.59 0.49 162.9°330.0 30.0
360175.95°2.691-2.680.19 -1.64 0.24 171.5°360.0 30.0
390180.01°2.703-2.700.00 -1.66 0.00 180.0°390.0 30.0
420184.07°2.691-2.68-0.19 -1.64 -0.25 188.5°420.030.0
450188.20° 2.656-2.63-0.38 -1.58 -0.49 197.1°450.0 30.0
480192.48°2.598-2.54-0.56 -1.49 -0.72 205.9°480.0 30.0
510197.00°2.516-2.41-0.74 -1.36 -0.95 214.8°510.0 30.0
540201.87°2.408-2.23-0.90 -1.19 -1.15 224.1°540.030.0
570207.27°2.273-2.02-1.04 -0.98 -1.34 234.0°570.030.0
600213.42°2.109-1.76-1.16 -0.72 -1.50 244.4°600.30.0
630220.73°1.914-1.45-1.25 -0.41 -1.61 255.8°630.0 30.0
660229.87°1.684-1.09-1.29 -0.04 -1.66 268.6°660.030.0
690242.23°1.416-0.66-1.25 0.39 -1.61 283.4°690.0 30.0
720260.96°1.110-0.17-1.10 0.87 -1.41 301.7°720.030.0
750294.79°0.791 0.33 -0.72 1.38 -0.92 326.1°750.0 30.0
779.8360.00°0.614 0.61 0.00 1.66 0.00 360.0°779.8 29.8
Numerical
Methods
See R?
below.
R?cos(ν) R?sin(ν) XO +c (a/b)*YO
tan-1YAC

XAC
See tEarth
below.
ti-ti-01
Methods to Improve Precision
Increase Granularity: Instead of computing a row for every 30 days, compute one for every day.  "Brute Force" method.
Increase Elegance: Instead of linear interpretation of durations between degrees, try weighted interpretation.  Weigh values according to relevant velocities which is ever changine on an eccentric orbit.
Previous work enables us to readily compute positions and durations for any orbiting body given the body's angular position. Thus, one can create a table with values for all integer degree values from 0° to 360°. One can also create a much smaller table with angular positions of 65° and 66°.
True
Anom.
Radius Sun Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
νRXYXACYACEtΔt
Deg AU AU AU AU AU Deg days days
65 0°0.790 0.33 0.72 1.38 0.9233.77°29.69n/a
66 0°0.796 0.32 0.73 1.37 0.9434.39°30.33 0.64
One can see respective durations of 29.69 days and 30.33 days. However, one might want to conduct reverse calculations to determine what values would result at a duration of exactly 30.0 days duration.
Unfortunately, considerable work by many scholars on Kepler's Equation still show that a straight forward solution is not obvious; thus, we must use numerical methods to approximate angles given an arbitrary duration. Therefore, following tables use linear interpolation to approximate true anomaly values for given durations.
Value Between Two Successive Degrees
Given
Time
True
Anom.
Radius Sun Based
Coordinates
Aux. Circle
Coordinates
Eccen.
Anom.
Comp.
Time
Incr.
Time
tνRXYXACYACEtΔt
DaysDeg AU AU AU AU AU Deg days days
65.0°0.790 0.33 0.72 1.38 0.9233.77°29.69n/a
3065.49°0.7930.330.72 1.37 0.93 34.1°30.0 0.31
66 0°0.796 0.32 0.73 1.37 0.9434.39°30.330.64
ν. . .tνn/a
tGivenνt. . .tCompΔtν

ν+1. . .tν+1Δtν+1
Approximate νt 
Arbitrarily choose tGiven between tν and tν+1. Example:  29.69 < 30.00 < 30.33
Note that True Anonmaly, ν= 65°, results in duration of 29.69 days, and ν+1 (66°) results in 30.33 days.
Best guess νt such that resulting tComp = tGiven
 Δtν    = tGiven - tν
Δtν+1 = tν+1      - tν

Understood but not shown is Δν = 1° = 66° - 65°.
 

νt = ν +
Δtν

Δtν+1
(Δν)
 


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