Compare Orbits: Apollo vs. Earth
R =  ℓ 1 + e × Cos(ν) 

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,
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 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 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 

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.
True Anomaly (ν)  Dist fm Sun (R)  

0°  0.647  
15°  0.655  
30°  0.680  
45°  0.724  
60°  0.789  
75°  0.882  
90°  1.010  
Given q=.647 AU; Q=2.295 
ℓ
1 + e Cos(ν)
 





Orbital Velocities
Kepler points out that orbital velocity varies per with distance from the Sun. Furthermore, Newton derived the now famous equation:
v = √(G × M_{Sol}/R)
to determine orbital speeds for circular orbits.
For Earth's Solar orbit, use following values:  Universal gravitational constant:
 G = 6.667 × 10^{11} Nm^{2}/kg^{2}
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}

>
 μ_{Sol} R  

v =  √  13.27 x 10^{19} m^{3}/sec^{2} 1.5 x 10^{11} m 

v =  √  8.847 x10^{8}m^{3}/sec^{2} 1.0 m  =  29.74x10^{3} m sec  =  29.74km sec 

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.Traditional Values. All distances in meters.
√(  6.667 x 10^{11}Nm^{2}  2 x 10^{30} kg  ) 

Recall N, Newton, is Unit of force, Newton. N = 1.0 kg m/sec^{2}.
√(  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}  ) 

Rearrange powers of ten.
√(  (10^{3}m)(10^{3}m)  6.667 x 10^{11} m  1.0 AU * 2 x 10^{2}  ) 

√(  km^{2}  4.4444 AU  200  ) 

(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
Parameter 
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
1.470.00
Start Sector
Start Sector
Start Sector
Start Sector
1012009
(11Apr09)
(11Apr09)
15
0.655
1.460.17
0.170
46.2
6.4
6.5
107200930
0.680
1.410.34
0.176
45.4
6.7
13.2
114200945
0.724
1.330.51
0.188
44.0
7.4
20.7
121200960
0.789
1.220.68
0.208
41.8
8.6
29.4
130200975
0.882
1.050.85
0.237
39.1
10.5
40.0
141200990
1.010
0.821.01
0.278
35.7
13.4
53.6
1542009105
1.181
0.521.14
0.333
31.9
18.0
71.8
1722009120
1.403
0.121.21
0.403
27.8
25.1
97.1
1982009135
1.672
0.361.18
0.482
23.5
35.5
133.0
2342009150
1.961
0.880.98
0.554
19.4
49.5
183.1
2842009165
2.200
1.300.57
0.593
15.8
64.8
248.6
3492009180
2.295
1.470.00
0.594
13.7
75.3
324.7
502010195
2.200
1.300.57
0.594
13.7
75.3
400.8
1362010210
1.961
0.880.98
0.593
15.8
64.8
466.3
1922010225
1.672
0.361.18
0.554
19.4
49.5
516.3
2422010240
1.403
0.121.21
0.482
23.5
35.5
552.2
2882010255
1.181
0.521.14
0.403
27.8
25.1
577.5
3132010270
1.010
0.821.01
0.333
31.9
18.0
595.7
3312010285
0.882
1.050.85
0.278
35.7
13.4
609.3
3452010300
0.789
1.220.68
0.237
39.1
10.5
619.9
3552010315
0.724
1.330.51
0.208
41.8
8.6
628.6
3632010330
0.680
1.410.34
0.188
44.0
7.4
636.1
0072011345
0.655
1.460.17
0.176
45.4
6.7
642.9
0152011360
0.647
1.470.00
0.170
46.2
6.4
649.3
0202011
(20Jan11)
(20Jan11)
Given
l
1+eCos(ν)
c+rCos(ν)
rSin(ν)
√(Δ_{x}^{2} +Δ_{y}^{2})
√[μ (
 2
 
 1
 )]


d
V
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
 ν=True Anomaly Angle originates from Sun and starts with line to perihelion.
 a=semimajor axis. For Apollo, a = 1.47 AU = 2.205 x 10^{11}m
 b=semiminor axis. For Apollo, b = 1.22 AU
 c=focal length. For Apollo, c = 0.82 AU, distance of Sun from orbit center.
 e=eccentricity. For Apollo, e = 0.56 AU, c/a.
 G = Universal Gravitation Constant = 6.667 x 10^{11} N * m^{2}/kg^{2}
 M_{Sol}=Sun's Mass = 2.0 x 10^{30} kg
 R_{Ave}=Average radius for sector
 AU=Astronomical Unit 1.5 x 10^{11} m=1.5 x 10^{8} km
 π/12 * R_{ave} / (V_{Ave} * 86,400)
Cross Check: Compute Apollo's period, T:
T = 2 π √[(a^{3}/(G*M_{Sol})]
T = 2 π √[(2.205 x 10^{11}m)^{3})/(6.667 x 10^{11}N * m^{2}/kg^{2}*2.0 x 10^{30} kg)]
T = 2 π √[(10.72 x 10^{33}m^{3}))/(13.33 x 10^{19}N * m^{3}/sec^{2})]
T = 2 π √[(0.804 x 10^{14}sec^{2})]
T = 2 π 0.8966 x 10^{7}sec = 5.61 x 10^{7}sec
T =649.3 days = 1.777 yrs
NOTE: This time is slightly greater then table's total sector times; thus, CUMM TIME column reflects adjusted times.
T = 2 π √[(a^{3}/(G*M_{Sol})]
T = 2 π √[(2.205 x 10^{11}m)^{3})/(6.667 x 10^{11}N * m^{2}/kg^{2}*2.0 x 10^{30} kg)]
T = 2 π √[(10.72 x 10^{33}m^{3}))/(13.33 x 10^{19}N * m^{3}/sec^{2})]
T = 2 π √[(0.804 x 10^{14}sec^{2})]
T = 2 π 0.8966 x 10^{7}sec = 5.61 x 10^{7}sec
T =649.3 days = 1.777 yrs
NOTE: This time is slightly greater then table's total sector times; thus, CUMM TIME column reflects adjusted times.
SECTOR TIMES NEED ADJUSTMENT when accummulated.
There is an error due to straight line calculation of sector distances; however, all sectors of an orbit MUST have some curvature. Unfortunately, this curvature is elliptical (not circular) and not so easily calculated. Thus, this table makes an arbitrary adjustment (1.0105) which brings summed sector times closer to actual period of Apollo.
NOTE: Kepler did NOT arbitrarily adjust his figures. Instead, he invented his 2nd law (equal areas result in equal times). He came up with an involved method to compute areas of elliptical sectors with apexes at the relevant focus (location of Sun).
There is an error due to straight line calculation of sector distances; however, all sectors of an orbit MUST have some curvature. Unfortunately, this curvature is elliptical (not circular) and not so easily calculated. Thus, this table makes an arbitrary adjustment (1.0105) which brings summed sector times closer to actual period of Apollo.
NOTE: Kepler did NOT arbitrarily adjust his figures. Instead, he invented his 2nd law (equal areas result in equal times). He came up with an involved method to compute areas of elliptical sectors with apexes at the relevant focus (location of Sun).
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