Monday, September 15, 2008

REFERENCE: Compare Apollo Orbit vs. Earth Orbit








1862 Apollo is a near Earth asteroid, approx 1.5 kms in diameter. Discovered by German astronomer Karl Reinmuth at Heidelberg Observatory on 24 April 1932; then, not observed again until 1973. It is the namesake and the first recognized member of the Apollo asteroids, a group of NEAs which seem to cross the orbit of Earth when viewed perpendicular to the ecliptic plane (NOTE: object’s inclination might prevent actual intersection). (See NASA's description of the NEO groups.)








Free Orbit Radius Equation

R =

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 (semi-latis rectum) must be determined prior to calculating R values.
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 semi-major axis. Thus,
  • Orbit's average distance to Sun is semi-major axis,
    a = (Q + q)/2
RECALL: Semi-major 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 = (Q-q)/2 
c = Q - q
Finally, this leads to following equation for eccentricity:
e =  c

a
 =  2 c

2 a
 =  Q - q

Q + q
Constant, , is computed with following initial conditions:
  • Zero degrees (oo) is the "reference ray" to measure other angles. 
  • R(0o) = q, distance from Sun to perihelion.
R(0o) = q =

1 + e × Cos(0o)
ℓ = q (1 + e × Cos(0o)) = q (1 + e )
ℓ = q (1 + Q - q

Q + q
) = q  Q + q + Q - q

Q + q
Leads to following equation
for semi-latus 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.
Compare Solar Distances ...
... for first 90° of orbit. This helps determine semi-latus rectum, ℓ, of each orbit. 
(NOTE:  R(90o) = , distance of semi-latus rectum.)
R(90o) = ℓ =

1 + e × Cos(90°)
First 90° of Earth's Orbit.
True Anomaly (θ)Dist fm Sun (R)
1 AU
15°1 AU
30°1 AU
45°1 AU
60°1 AU
75°1 AU
90°1 AU
Given
q=.983 AU;      Q=1.017 AU
R =

1 + e × Cos(θ)
e = Q - q

Q + q
= .017
ℓ =2×Q×q

Q + q
= 1AU
Average Radius of Earth's orbit: R 149,597,870.7 km = 1 AU. 
With a very small eccentricty (e=.017), TE assumes constant radius (R = 1.0 AU).
 Thus, assume Earth's semi-latus rectum () = 1 AU.
First 90° of Apollo's Orbit.
True Anomaly (θ)Dist fm Sun (R)
0.647 AU
15°0.655 AU
30°0.680 AU
45°0.724 AU
60°0.789 AU
75°0.882 AU
90°1.010 AU
Given
q=.65 AU;     Q=2.295 AU
R =

1 + e × Cos(θ)
e = Q-q

Q+q
= 0.56
ℓ =2×Q×q

Q + q
= 1.01 AU
From Sol, Apollo's orbit ranges from .65 AU to 2.23 AU.
Reference ray starts at θ =0° and .65 AU.
At 90°, the ray perpendicular to Major Axis is ℓ, at 1.01 AU.

Compare Orientations

Convert polar coordinates (angle, radius) to the x, y coordinates commonly used for the abscissa and ordinate of the Cartesian coordinate system. Apollo's orbit is definitely eccentric; thus, conversion must consider that R value varies from perihelion, q, to aphelion, Q.
Apollo's Semi-Orbit.
True
Anomaly
Distance
from Sun
Cartesian
Coordinates
νR-valueX-valueY-value
0.647 AU0.65 AU0.00 AU
15°0.655 AU0.64 AU0.17 AU
30°0.680 AU0.59 AU0.34 AU
45°0.724 AU0.51 AU0.51 AU
60°0.789 AU0.40 AU0.68 AU
75°0.882 AU0.23 AU0.85 AU
90°1.010 AU0.00 AU1.01 AU
105°1.181 AU-0.30 AU1.14 AU
120°1.403 AU-0.70 AU1.21 AU
135°1.672 AU-1.18 AU1.18 AU
150°1.961 AU-1.70 AU0.98 AU
165°2.200 AU-2.12 AU0.57 AU
180°2.295 AU-2.29 AU0.00 AU
Given

1+e Cos(θ)
R×Cos(θ)R×Sin(θ)
Recall very basic trig identities: x = R Cos(θ); abscissa value, x, obtained by multiplying radius distance times cosine of true anomaly. y = R Sin(θ); ordinate value, y, obtained by multiplying radius distance times sine of true anomaly.
Earth's Semi-Orbit
True
Anomaly
Distance
 from Sun
Cartesian
Coordinates
νR-valueX-valueY-value
1 AU1.00 AU0.00 AU
15°1 AU0.97 AU0.26 AU
30°1 AU0.87 AU0.50 AU
45°1 AU0.71 AU0.71 AU
60°1 AU0.50 AU0.87 AU
75°1 AU0.26 AU0.97 AU
90°1 AU0.00 AU1.00 AU
105°1 AU-0.26 AU0.97 AU
120°1 AU-0.50 AU0.87 AU
135°1 AU-0.71 AU0.71 AU
150°1 AU-0.87 AU0.50 AU
165°1 AU-0.97 AU0.26 AU
180°1 AU-1.00 AU0.00 AU
Given

1+eCos(θ)
1.0×Cos(θ)1.0×Sin(θ)
With eccentricty of e = 0.0167,  Earth's orbit is nearly circular; thus, TE assumes radius as a static 1.0 AU.  Therefore,  converting polar to Cartesian coordinates is simply determining components of a unit vector, a simple high school exercise. 

Kepler shows how orbital velocity varies per with distance from the Sun. From Kepler and Galileo,  Newton derived the now famous equation:
v = (G × MSol/R
to determine orbital speeds for circular orbits. For Solar orbits, use following values:
●  Universal gravitational constant: 
       G = 6.67408×10-11 N-m2/kg2
●  Mass of Sun: MSol = 1.989×1030 kilograms
NOTE: Since values G and MSol are constant, their product, G×MSol , is also constant, and it is often expressed as μSol, the Solar System's standard gravitational parameter:
μSol = 132,712,440,018 km3 / sec2 = 13.27 x 1019 m3/sec2

Compare Orbital Velocities

Compare orbital velocities for similar  points on both orbits, Apollo and Earth.  For Apollo, 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.

Apollo's Orbital Velocities
Angular
Dist. (θ)
 Linear Distance
from Sun (R)
Orbital
Velocity(V)
degAstro UnitsKilometerkm/sec
0.647 AU96,855,032 km46.23 kps
45°0.724 AU108,234,983 km43.00 kps
90°1.010 AU151,093,849 km33.96 kps
135°1.672 AU250,147,013 km21.39 kps
180°2.295 AU343,395,112 km13.02 kps
Given

1+e Cos(θ)
v = (μ(2

R
-1

a
))
Terra's Orbital Velocities
Angular
Dist. (θ)
 Linear Distance
from Sun (R)
Orbital
Velocity(V)
degAstro UnitsKilometerkm/sec
0.983 AU147,098,074 km30.29 kps
45°0.988 AU147,809,610 km30.14 kps
90°1.000 AU149,556,115 km29.79 kps
135°1.012 AU151,344,387 km29.44 kps
180°1.017 AU152,097,701 km29.29 kps
Given

1+e Cos(θ)
v = (μ(2

R
-1

a
))
Since orbits are usually elliptical, use following equation to determine most orbital velocities.  (NOTE: Even though Earth's orbit is nearly circular with tiny eccentricity (e = .017), it is still elliptical with distinct difference between q (nearest point) and Q (farthest point)):
v = (μ(2

R
- 1

a
) )
TE Proposes Heuristic Express μSol, the Solar System's standard gravitational parameter in a slightly different way. Instead of meters (or kilometers), input values as Astronomical Units (AUs) for a, semi-major axis, and R, distance from Sol.  To do this, we must adjust value for μSolas shown below.
μSol = 132,712,440,018 km3 / sec2 = 13.27 x 1019 m3/sec2
μSol=  AU × 132,712,440,018 km3

149,597,870.7 km × sec2
=887.123 AU-km²

sec2
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.
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, semi-major axis, a, remains a constant 1.47 AU (=219,908,869,916 m) throughout entire Apollo orbit.)
TRADITIONAL CALCULATIONS: 
=(μ (2/r - 1/a))
(13.27×1019 m3/sec2 × (2/151,093,849,398 m - 1/219,908,869,916 m)
= (13.27×1019 m3/sec2 × (1.32368×10-11/m - 0.45473×10-11/m)
v= (13.27×1019 m3/sec2 × 0.86895×10-11/m)
v= (11.531×108 m2/sec2)
Determine square root: 
= 3.396×104 m/sec = 33,960 m/sec
Convert to km/sec. 
v = 33.96 kps
TE PROPOSED CALCULATIONS: 
v = (887.123 AU-km²

sec2
(2

1.01 AU
-1

1.47 AU
))
v = (887.123 AU-km²

sec2
×1.980-0.680

AU
)
v = (887.123 km² × 1.3

sec2
)
v = (1,153.23 km²

sec2
)
v = 33.96 km/sec
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.

More Orbital Speeds

V = ( 887.123 AU-km2

sec2
×(2

R
-1

1.47 AU
))
Four simple steps show straight forward calculations 
for Apollo's entire range of orbital velocities.
Orbit
Parm
Step 0:
R
Step 1:
X1
Step 2:
X2
Step 3:
X3
Step 4:
V
q
0.65 AU
3.08

1 AU
2.48

1 AU
2,204 km2

sec2
47 km

sec
a
1.47 AU
1.36

1 AU
0.68

1 AU
604 km2

sec2
24.6 km

sec
Q
2.30 AU
0.87

1 AU
0.19

1 AU
169 km2

sec2
13 km

sec
Observed
2

R 
X1-0.68
μ×X2
(X3)
To more easily compute V, elliptical orbital speed, 
we suggest breaking down equation as shown above.

TRAVEL TIMES: Compare Apollo versus Earth

Ang. Dist.Sol. Dist.Cart. Coord.Incr. Dist.Ave. Vel.Incr. timeCum. timeForecast
θRXYΔ dVΔ tΣ tDate
DegreesAstronomical Units (km/sec)Earth DaysGregorian
0.650 AU0.650 AU0.00 AUn/a AU n/a kpsn/a dyn/a dy4/11/2009
15°0.658 AU0.636 AU0.170 AU0.1709 AU 46.94 kps13.19 dy13.19 dy4/24/2009
30°0.683 AU0.591 AU0.170 AU0.1767 AU 45.21 kps13.63 dy26.82 dy5/7/2009
45°0.726 AU0.513 AU0.513 AU0.1889 AU 43.77 kps14.57 dy41.39 dy5/22/2009
60°0.792 AU0.396 AU0.686 AU0.3333 AU 41.65 kps16.09 dy57.48 dy6/7/2009
75°0.885 AU0.229 AU0.855 AU0.2376 AU 38.90 kps18.33 dy75.81 dy6/25/2009
90°1.013 AU0.000 AU1.013 AU0.2783 AU35.60 kps21.47 dy97.29 dy7/17/2009
105°1.184 AU-0.306 AU1.144 AU0.3333 AU31.82 kps25.71 dy123.00 dy8/11/2009
120°1.406 AU-0.703 AU1.217 AU0.4030 AU27.71 kps31.09 dy154.09 dy9/12/2009
135°1.674 AU-1.184 AU1.184 AU0.4823 AU23.44 kps37.21 dy191.30 dy10/19/2009
150°1.963 AU-1.999 AU0.981 AU0.5539 AU19.32 kps42.73 dy234.04 dy12/1/2009
165°2.200 AU-2.125 AU0.569 AU0.5923 AU15.81 kps45.69 dy279.73 dy1/15/2010
180°2.295 AU-2.295 AU0.000AU0.5942 AU13.67 kps45.84 dy325.57 dy3/2/2010
Given

1+e×Cos(θ)
R×Cos(θ)R×Sin(θ)(ΔX2 +ΔY2)
[μ (2

RAve
-1

a
)]
Δ d

VAve
Σti=Σti-1+Δti
**Adjusted
By
Inspection.
180°1.017 AU-1.017 AU  0.000 AU0.2653 AU29.30 kps15.47 dy182.625 dy7/1/2016
195°1.016 AU-0.981 AU  -0.263 AU0.2653 AU29.30 kps15.47 dy198.09 dy7/17/2016
210°1.014 AU-0.878 AU  -0.507 AU0.2650 AU29.33 kps15.45 dy213.54 dy8/1/2016
225°1.012 AU-0.715 AU  -0.715 AU0.2645 AU29.40 kps15.42 dy228.96 dy8/16/2016
240°1.008 AU-0.504 AU  -0.873 AU0.2637 AU29.49 kps15.37 dy244.33 dy9/1/2016
255°1.004 AU-0.260 AU  -0.970 AU0.2627 AU29.60 kps15.31 dy259.64 dy9/16/2016
270°1.000 AU0.000 AU  -1.000 AU0.2616 AU29.73 kps15.25 dy274.89 dy10/1/2016
285°0.995 AU0.258 AU  -0.961 AU0.2605 AU29.86 kps15.18 dy290.08 dy10/17/2016
300°0.991 AU0.496 AU  -0.859 AU0.2594 AU29.98 kps15.12 dy305.20 dy11/1/2016
315°0.988 AU0.699 AU  -0.699 AU0.2584 AU30.09 kps15.06 dy310.26 dy11/16/2016
330°0.985 AU0.853 AU  -0.493 AU0.2576 AU30.18 kps15.02 dy335.28 dy12/1/2016
345°0.984 AU0.950 AU  -0.255 AU0.2571 AU30.24 kps14.98 dy350.26 dy12/16/2016
360°0.983 AU0.983 AU  -0.000 AU0.2568 AU30.28 kps14.97 dy365.23 dy12/31/2016
DegreesAstronomical Units(km/sec)Earth DaysGregorian
θRXYΔ dVΔ tΣ tDate
Ang. Dist.Sol. Dist.Cart. Coord.Incr. Dist.Ave. Vel.Incr. timeCum. timeForecast

TRAVEL TIMES: Compare Earth versus Apollo

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).  

SLIDESHOW




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




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