REFERENCE: Compare Apollo Orbit vs. Earth Orbit
R =  ℓ 1 + e × Cos(θ) 

A simple math tool can help determine orbital paths. Above equation can readily compute polar coordinates (ν, R) of orbital path.
 Radius, R, is the straight line distance from the focus of interest (often our sun, Sol).
 θ (Greek letter, "theta") is sometimes called "true anomaly". This is the angular distance from a reference ray, which originates from Sol to orbit's perihelion (point closest to Sol).(NOTE: Older literature often uses Greek letter, nu (ν), for true anomaly.
 R's value will depend on the independent variable, θ.
 Constants e (eccentricity) and ℓ (semilatis rectum) must be determined prior to calculating R values.
 Eccentricity (e) is measure of "flatness" of ellipse.
 SemiLatis Rectum (ℓ) distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. (See "First 90° of Apollo's Orbit" below.)
Eccentricity Given following distances:
we can determine orbit's eccentricity, e, for above equation. Recall that Q and q are endpoints of the orbit's major axis; thus, Q+q=2a where a = length of semimajor axis. Thus,
 Orbit's average distance to Sun is semimajor axis,
a = (Q + q)/2
RECALL: Semimajor axis, a, includes length to focus, c, plus length to perihelion, q.
a= c + q = (Q + q)/2 = Q/2 + q/2
c = Q/2 + q/2  q = (Qq)/2
2 c = Q  q
Finally, this leads to following equation for eccentricity:e =  c a  =  2 c
2 a  =  Q  q
Q + q 

Constant, ℓ, is computed with following initial conditions:
 Zero degrees (o^{o}) is the "reference ray" to measure other angles.
 R(0^{o}) = q, distance from Sun to perihelion.
R(0^{o}) = q =  ℓ 1 + e × Cos(0^{o}) 

ℓ = q (1 + e × Cos(0^{o})) = q (1 + e  ) 

ℓ = q (1 +  Q  q Q + q 
) = q  Q + q + Q  q
Q + q 

Leads to following equation
for semilatus rectum.  

ℓ = q  2 × Q Q + q 
=  q × 2 × Q
Q + q 
Fortunately, values Q and q have been determined for most observed asteroids and are readily available in many online sources.



Kepler shows how orbital velocity varies per with distance from the Sun. From Kepler and Galileo, Newton derived the now famous equation:
v = √(G × M_{Sol}/R)
to determine orbital speeds for circular orbits.
For Solar orbits, use following values:G = 6.67408×10^{11} Nm^{2}/kg^{2}
● Mass of Sun: M_{Sol} = 1.989×10^{30} kilograms
NOTE: Since values G and M_{Sol }are constant, their product, G×M_{Sol} , is also constant, and it is often expressed as μ_{Sol}, the Solar System's standard gravitational parameter:
μ_{Sol} = 132,712,440,018 km^{3} / sec^{2} = 13.27 x 10^{19} m^{3}/sec^{2}



v =  √  (  μ  (  2 R    1 a 
)  ) 

μ_{Sol} = 132,712,440,018 km^{3} / sec^{2} = 13.27 x 10^{19} m^{3}/sec^{2}
Resultant values for orbital velocity at R would be in kilometers per second (kps). Thus, we can more easily determine orbital velocities in km/sec when given orbital values in AUs.μ_{Sol}  =  AU × 132,712,440,018 km^{3} 149,597,870.7 km × sec^{2}  =  887.123 AUkm² sec^{2} 

To demonstrate this heuristic, compare traditional method versus TE proposed method to compute Apollo's orbit velocity at R = 1.01 AU (=151,093,849,398 m). (Of course, semimajor axis, a, remains a constant 1.47 AU (=219,908,869,916 m) throughout entire Apollo orbit.)
TRADITIONAL CALCULATIONS:
v =√(μ (2/r  1/a))
v = √(13.27×10^{19} m^{3}/sec^{2} × (2/151,093,849,398 m  1/219,908,869,916 m))
v = √(13.27×10^{19} m^{3}/sec^{2} × (1.32368×10^{11}/m  0.45473×10^{11}/m))
v= √(13.27×10^{19} m^{3}/sec^{2} × 0.86895×10^{11}/m)
v= √(11.531×10^{8} m^{2}/sec^{2})
Determine square root:
v = 3.396×10^{4} m/sec = 33,960 m/sec
Convert to km/sec.
v = 33.96 kps

TE PROPOSED CALCULATIONS:

COMPARISON: The traditional method uses large unwieldy numbers. On the other hand, the TE method trades a tiny loss of precision for a whole lot lot of convenience.
V =  √  (  887.123 AUkm^{2} sec^{2} 
×  (  2 R    1 1.47 AU  )  ) 

Parm
R
X_{1}
X_{2}
X_{3}
V
0.65 AU
3.08
1 AU
1 AU
2.48
1 AU
1 AU
2,204 km^{2}
sec^{2}
sec^{2}
47 km
sec
sec
1.47 AU
1.36
1 AU
1 AU
0.68
1 AU
1 AU
604 km^{2}
sec^{2}
sec^{2}
24.6 km
sec
sec
2.30 AU
0.87
1 AU
1 AU
0.19
1 AU
1 AU
169 km^{2}
sec^{2}
sec^{2}
13 km
sec
sec
Observed
2
R
X_{1}0.68
μ×X_{2}
√(X_{3})
To more easily compute V, elliptical orbital speed,
we suggest breaking down equation as shown above.
we suggest breaking down equation as shown above.
Ang. Dist.  Sol. Dist.  Cart. Coord.  Incr. Dist.  Ave. Vel.  Incr. time  Cum. time  Forecast  

θ  R  X  Y  Δ d  V  Δ t  Σ t  Date  
Degrees  Astronomical Units  (km/sec)  Earth Days  Gregorian  
0°  0.650 AU  0.650 AU  0.00 AU  n/a AU  n/a kps  n/a dy  n/a dy  4/11/2009  
15°  0.658 AU  0.636 AU  0.170 AU  0.1709 AU  46.94 kps  13.19 dy  13.19 dy  4/24/2009  
30°  0.683 AU  0.591 AU  0.170 AU  0.1767 AU  45.21 kps  13.63 dy  26.82 dy  5/7/2009  
45°  0.726 AU  0.513 AU  0.513 AU  0.1889 AU  43.77 kps  14.57 dy  41.39 dy  5/22/2009  
60°  0.792 AU  0.396 AU  0.686 AU  0.3333 AU  41.65 kps  16.09 dy  57.48 dy  6/7/2009  
75°  0.885 AU  0.229 AU  0.855 AU  0.2376 AU  38.90 kps  18.33 dy  75.81 dy  6/25/2009  
90°  1.013 AU  0.000 AU  1.013 AU  0.2783 AU  35.60 kps  21.47 dy  97.29 dy  7/17/2009  
105°  1.184 AU  0.306 AU  1.144 AU  0.3333 AU  31.82 kps  25.71 dy  123.00 dy  8/11/2009  
120°  1.406 AU  0.703 AU  1.217 AU  0.4030 AU  27.71 kps  31.09 dy  154.09 dy  9/12/2009  
135°  1.674 AU  1.184 AU  1.184 AU  0.4823 AU  23.44 kps  37.21 dy  191.30 dy  10/19/2009  
150°  1.963 AU  1.999 AU  0.981 AU  0.5539 AU  19.32 kps  42.73 dy  234.04 dy  12/1/2009  
165°  2.200 AU  2.125 AU  0.569 AU  0.5923 AU  15.81 kps  45.69 dy  279.73 dy  1/15/2010  
180°  2.295 AU  2.295 AU  0.000AU  0.5942 AU  13.67 kps  45.84 dy  325.57 dy  3/2/2010  
Given  ℓ 1+e×Cos(θ)  R×Cos(θ)  R×Sin(θ)  √(ΔX^{2} +ΔY^{2}) 
 Δ d V_{Ave}  Σt_{i}=Σt_{i1}+Δt_{i} **Adjusted 
By Inspection.  
180°  1.017 AU  1.017 AU  0.000 AU  0.2653 AU  29.30 kps  15.47 dy  182.625 dy  7/1/2016  
195°  1.016 AU  0.981 AU  0.263 AU  0.2653 AU  29.30 kps  15.47 dy  198.09 dy  7/17/2016  
210°  1.014 AU  0.878 AU  0.507 AU  0.2650 AU  29.33 kps  15.45 dy  213.54 dy  8/1/2016  
225°  1.012 AU  0.715 AU  0.715 AU  0.2645 AU  29.40 kps  15.42 dy  228.96 dy  8/16/2016  
240°  1.008 AU  0.504 AU  0.873 AU  0.2637 AU  29.49 kps  15.37 dy  244.33 dy  9/1/2016  
255°  1.004 AU  0.260 AU  0.970 AU  0.2627 AU  29.60 kps  15.31 dy  259.64 dy  9/16/2016  
270°  1.000 AU  0.000 AU  1.000 AU  0.2616 AU  29.73 kps  15.25 dy  274.89 dy  10/1/2016  
285°  0.995 AU  0.258 AU  0.961 AU  0.2605 AU  29.86 kps  15.18 dy  290.08 dy  10/17/2016  
300°  0.991 AU  0.496 AU  0.859 AU  0.2594 AU  29.98 kps  15.12 dy  305.20 dy  11/1/2016  
315°  0.988 AU  0.699 AU  0.699 AU  0.2584 AU  30.09 kps  15.06 dy  310.26 dy  11/16/2016  
330°  0.985 AU  0.853 AU  0.493 AU  0.2576 AU  30.18 kps  15.02 dy  335.28 dy  12/1/2016  
345°  0.984 AU  0.950 AU  0.255 AU  0.2571 AU  30.24 kps  14.98 dy  350.26 dy  12/16/2016  
360°  0.983 AU  0.983 AU  0.000 AU  0.2568 AU  30.28 kps  14.97 dy  365.23 dy  12/31/2016  
Degrees  Astronomical Units  (km/sec)  Earth Days  Gregorian  
θ  R  X  Y  Δ d  V  Δ t  Σ t  Date  
Ang. Dist.  Sol. Dist.  Cart. Coord.  Incr. Dist.  Ave. Vel.  Incr. time  Cum. time  Forecast 
SUMMARY
Contrast orbits of Apollo vs. Earth:
Eccentricity  Earth orbit is nearly circular (e=.017) while Apollo orbit is highly elliptical (e=.56).
Distances  Earth distances range from closest point, q=.983 AU, to farthest point, Q=1.017 AU. Apollo distances range much more widely from q=.65 AU to Q=2.295 AU.
Orientation  Earth's orbit centered on Sol which is very near Apollo's perihelion.
Velocities  Apollo's velocities range from 46.94 kilometers per second (kps) to 13.67 kps; by contrast, Earth's range more tightly from 29.30 kps to 30.28 kps. NOTE: Heuristic enables a quick, straight forward calculation of these times.
Sector travel times  Time to traverse 15° sectors vary considerably for Apollo's orbit; sector times range widely from 13.19 days to 45.34 days. By contrast, Earth orbit's sector times closely cluster slightly above 15 days (15.47 days to 14.97 days).
VOLUME O: ELEVATIONAL 

VOLUME I: ASTEROIDAL 
VOLUME II: INTERPLANETARY 
VOLUME III: INTERSTELLAR 
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