Compare Orbits: Apollo vs. Earth
R =  l 

There is a simple math tool available to compute elliptical orbits. Above equation can readily determine polar coordinates (ν, R) of orbital path.
 Radius, R, is the straight line distance from the focus of interest.
 ν (Greek letter, "nu") is sometimes called "true anomaly". This is the angular distance from a reference ray, which originates from relevant focus to orbit's periapse (point closest to relevant focus).
 R's value will depend on the independent variable, ν.
 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,
Further recall that semimajor axis, a, includes length to focus, c, and length to perihelion, q. a=c + q = (Q + q)/2 = Q/2 + q/2 c = Q/2 + q/2  q = (Qq)/2 2c = Q  q ; finally, this leads to following equation for eccentricity:
e =  c  =  2 c  =  Q  q 

Constant, l, can be computed due to the following initial conditions:
 Zero degrees (o^{o}) is defined as ray from focus to periapse, nearest point of orbit. In case of Solar orbits, the Sun's position is the relevant focus, and periapse is commonly called perihelion.
 R(0^{o}) = q, distance from Sun to perihelion.
R(0^{o}) = q =  l 

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

l  =  q  (1 +  Q  q  ) =  q  Q + q + Q  q  
Leads to following equation  
l  =  q  2 * Q  =  q * 2 * Q 
Fortunately, values Q and q have been determined for most observed asteroids and are readily available in many online sources.
True Anomaly  Dist fm Sun  

ν  Rvalue  
(deg)  (AU)  
0  1.000  
15  1.000  
30  1.000  
45  1.000  
60  1.000  
75  1.000  
90  1.000  
Given  l 
e =  Q  q  =  1.01.0  =  0  =  0 

l =  2Qq  =  2*1*1  =  2  =  1 



NOTE: Since values G and M_{Sol }can be considered constant, the product, G*M_{Sol} , is often expressed as the standard gravitational parameter: μ_{Sol} = 132,712,440,018 km^{3} / sec^{2} = 13.27 x 10^{19} m^{3}/sec^{2} 
 

 
 
For elliptical orbits, use following equation to determine orbital velocities at various positions throughout the orbit:
v = √(μ (2/r  1/a))
Note that for circular orbits, semimajor axis, a, would equal r, radius. Above equation would then reduce to the simpler equation for a circular orbit: v = √(μ/r). 
Traditional Method
Computing v:
√(μ(  2    1  )) 

For an example of determining velocity for one point on an elliptical orbit, compute velocity for Apollo's orbit at the latis rectum. Thus, use r = l = 1.01 AU which is very close to Earth's radius; consequently, we'll be able compare Apollo's velocity vs. Earth's velocity at same distance from Sun.
a. For consistent distance units, convert AU to meters.
 Radius: R = 1.01 AU = 151,093,849,398 meters. This value varies throughout the orbit. This value of R, semilatis rectum, happens to be near radius of Earth's orbit, but we think it's likely that Apollo's velocity at this point will significantly differ from Earth's velocity at similar R value.
 Semimajor axis: a = 1.47 AU = 219,908,869,916 meters. Since an ellipse has only one major axis, this value is constant for each orbit.
b. Accomplish calculations inside radical: √(μ (2/r  1/a)) = √(13.27 x 10^{19} m^{3}/sec^{2} * (2/151,093,849,398m  1/219,908,869,916m)) √(μ (2/r  1/a)) = √(13.27 x 10^{19} m^{3}/sec^{2} * (1.32368 x 10^{11}m  0.45473 x 10^{11}m)) √(μ (2/r  1/a)) = √(13.27 x 10^{19} m^{3}/sec^{2} * 0.86895 x 10^{11}m) √(μ (2/r  1/a)) = √(11.531 x 10^{8} m^{2}/sec^{2})
c. Accomplish square root calculation: √(μ (2/r  1/a)) = 3.396 x 10^{4} m/sec = 33,960 m/sec
d. Convert to km/sec. √(μ (2/r  1/a)) = 33.96 km/sec
Proposed Heuristic
How can we determine orbital velocities in km/sec when given orbital radius in AUs? Perhaps we can use a more convenient value for μ.√(  6.667 x 10^{11}Nm^{2}  2 x 10^{30} kg  ) 

√(  kg * m  6.667 x 10^{11} m^{2}  2 x 10^{30} kg  ) 

√(  m^{2}  6.667 x 10^{11} m  1.0 AU  2 x 10^{30} kg^{2}  ) 

√(  (10^{3}m)(10^{3}m)  6.667 x 10^{11} m  1.0 AU * 2 x 10^{2}  ) 

Substitute: 1 km = 1,000 m
√(  km^{2}  4.4444 AU  200  ) 

Final Rearrangement.
(Assume R = 1 AU.)
√(  888.888 AUkm^{2}  ) =  29.81 km/sec 

Use μ = 888.888 AUkm^{2}/sec^{2}
RULE OF THUMB: Infer μ = 888.888 AUkm^{2}/sec^{2 }to enable radius in Astronomical Units with resulting v in km/sec.
To confirm above rule of thumb, quickly reaccomplish elliptical orbit velocity with R = 1.01 AU. (Of course, semimajor axis, a, remains 1.47 AU for entire Apollo orbit.)
√(μ (2/r  1/a)) = √(888.888 AUkm^{2}/sec^{2} (2/1.01AU  1/1.47AU)) √(μ (2/r  1/a)) = √(888.888 AUkm^{2}/sec^{2} (1.98/1.0AU  0.68/1.0AU)) √(μ (2/r  1/a)) = √(888.888 km^{2}/sec^{2} (1.30)) √(μ (2/r  1/a)) = √(1155.5544 km^{2}/sec^{2}) = 33.99 km/sec
The lengthy, traditional method reduces to four simple steps. A little bit of accuracy is sacrificed for a whole lot of convenience.
Orbital  Rmk  R  X_{1}  X_{2}  X_{3}  V 

0.65 AU  3.08/1AU  2.48/1AU  2,204 km^{2}/s^{2}  47 km/s  
1.47 AU  1.36/1AU  0.68/1AU  604 km^{2}/s^{2}  24.6 km/s  
2.30 AU  0.87/1AU  0.19/1AU  169 km^{2}/s^{2}  13 km/s  
Observed  2/R  X_{1}0.68  μ*X_{2}  √(X_{3}) 
0. Prior to starting process, determine R, range of Apollo from Sun.  √[μ (2/R  1/a)]  √[μ (2/R  1/a)] 
1. Divide R into 2 for first term, X_{1} = 2/R.  √[μ (2/R  1/a)]  √[μ (X_{1}  1/a)] 
2. Subtract 0.68 from X_{1} for 2nd term, X_{2} = X_{1}.68. {Recall Apollo's a = 1.47 AU; thus, 1/a remains constant 0.68 for Apollo's entire orbit.}  √[μ (2/R  1/a)]  √[μ (X_{2})] 
3. X_{3}= 888.888 AUkm^{2}/s^{2} * X_{2}. {Above heuristic trades minor accuracy loss for major convenience gain.}  √[μ (2/R  1/a)]  √[X_{3}] 
4. For 4th and last step, determine V, orbital speed for specific range from Sun by determing square root of X_{3}.  √[μ (2/R  1/a)]  V 
Discuss mid vs. mean velocities (note mid range R doesn't necc result in mean v for elliptical orbit).
True  Dist fm Sun  Cart. Coord.  Distance  Velocity  Time  Adjusted  

Anomaly (ν)  rvalue  Xvalue  Yvalue  x,y to x',y'  V  t/Sector  Cumm Time  Forecast  
(deg)  (AU)  (AU)  (AU)  (AU)  (km/sec)  (days)  (days)  OrdinalDate  
0  0.647  1.47  0.00  Start Sector  Start Sector  Start Sector  Start Sector  1012009  
15  0.655  1.46  0.17  0.170  46.2  6.4  6.5  1072009  
30  0.680  1.41  0.34  0.176  45.4  6.7  13.2  1142009  
45  0.724  1.33  0.51  0.188  44.0  7.4  20.7  1212009  
60  0.789  1.22  0.68  0.208  41.8  8.6  29.4  1302009  
75  0.882  1.05  0.85  0.237  39.1  10.5  40.0  1412009  
90  1.010  0.82  1.01  0.278  35.7  13.4  53.6  1542009  
105  1.181  0.52  1.14  0.333  31.9  18.0  71.8  1722009  
120  1.403  0.12  1.21  0.403  27.8  25.1  97.1  1982009  
135  1.672  0.36  1.18  0.482  23.5  35.5  133.0  2342009  
150  1.961  0.88  0.98  0.554  19.4  49.5  183.1  2842009  
165  2.200  1.30  0.57  0.593  15.8  64.8  248.6  3492009  
180  2.295  1.47  0.00  0.594  13.7  75.3  324.7  502010  
195  2.200  1.30  0.57  0.594  13.7  75.3  400.8  1362010  
210  1.961  0.88  0.98  0.593  15.8  64.8  466.3  1922010  
225  1.672  0.36  1.18  0.554  19.4  49.5  516.3  2422010  
240  1.403  0.12  1.21  0.482  23.5  35.5  552.2  2882010  
255  1.181  0.52  1.14  0.403  27.8  25.1  577.5  3132010  
270  1.010  0.82  1.01  0.333  31.9  18.0  595.7  3312010  
285  0.882  1.05  0.85  0.278  35.7  13.4  609.3  3452010  
300  0.789  1.22  0.68  0.237  39.1  10.5  619.9  3552010  
315  0.724  1.33  0.51  0.208  41.8  8.6  628.6  3632010  
330  0.680  1.41  0.34  0.188  44.0  7.4  636.1  0072011  
345  0.655  1.46  0.17  0.176  45.4  6.7  642.9  0152011  
360  0.647  1.47  0.00  0.170  46.2  6.4  649.3  0202011  
Given  l  c+rCos(ν)  rSin(ν)  √(Δ_{x}^{2} +Δ_{y}^{2}) 
 d  t'=1.0105t   
Center  0.82  0.00  0.00  Δ_{x} = x'  x  t'_{i}=t'_{i1}+t'_{i}  
Sun  0.0  0.82  0.00  Δ_{y} = y'  y  
Perihelion  0.65 AU  1.47  0.00  Σ = Approx. Circum. =  Σ = App. Per. =  
Aphelion  2.29 AU  1.47  0.00  8.4 (AUs)  643 Days = 1.8 Yrs 
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