Monday, September 15, 2008

Compare Orbits: Apollo vs. Earth




Polar Equation

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 (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 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 = (Q-q)/2 2c = Q - q ; finally, this leads to following equation for eccentricity:

e =
c
a
=
2 c
2 a
=
Q - q
Q + q

Constant, l, can be computed due to the following initial conditions:

  • Zero degrees (oo) 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(0o) = q, distance from Sun to perihelion.

R(0o) = q =
l
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 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.

First 90o of Earth's orbit.

True AnomalyDist fm Sun
νR-value
(deg)(AU)
01.000
151.000
301.000
451.000
601.000
751.000
901.000
Given
l
1+e Cos(ν)
e =
Q - q
Q + q
=
1.0-1.0
1.0+1.0
=
0
2
=
0
l =
2Qq
Q + q
=
2*1*1
1.0+1.0
=
2
2
=
1

First 90o of Apollo's orbit.

True AnomalyDist fm Sun
νR-value
(deg)(AU)
00.647
150.655
300.680
450.724
600.789
750.882
901.010
Given
l
1+e Cos(ν)

e =
Q - q
Q + q
=
2.295-0.647
2.295+0.647
=
1.648
2.942
=
0.56
l =
2Qq
Q + q
=
2*2.295*0.647
2.295+0.647
=
2.969
2.942
=
1.01

...




Apollo Semi-orbit: Cartesian Coordinates

Convert polar coordinates (angle, radius) to the x, y coordinates commonly used for the abscissa and ordinate of the Cartesian coordinate system.

Easily done by recalling some very common trigonometric identities.

x = R Cos(ν); abscissa value, x, obtained by multiplying radius distance times cosine of true anomaly.

y = R Sin(ν); ordinate value, y, obtained by multiplying radius distance times sine of true anomaly.

Earth's Semi-orbit.

True AnomalyDist fm SunCartesian Coord.
νR-valueX-valueY-value
(deg)(AU)(AU)(AU)
01.0001.000.00
151.0000.970.26
301.0000.870.50
451.0000.710.71
601.0000.500.87
751.0000.260.97
901.0000.001.00
1051.000-0.260.97
1201.000-0.500.87
1351.000-0.710.71
1501.000-0.870.50
1651.000-0.970.26
1801.000-1.000.00
Given
l 
1+e Cos(ν)
1.0AU * Cos(ν)1.0AU * Sin(ν)

Since Earth's orbit is almost exactly circular, and the radius is by definition a static 1.0 AU; converting polar coordinates is simply determining components of a unit vector, a simple high school exercise. (Values shown in above table but not in diagram.)

Apollo's Semi-orbit.

True AnomalyDist fm SunCartesian Coord.
νR-valueX-valueY-value
(deg)(AU)(AU)(AU)
00.6471.470.00
150.6551.460.17
300.6801.410.34
450.7241.330.51
600.7891.220.68
750.8821.050.85
901.0100.821.01
1051.1810.521.14
1201.4030.121.21
1351.672-0.361.18
1501.961-0.880.98
1652.200-1.300.57
1802.295-1.470.00
Given
l 
1+e Cos(ν)
c+R*Cos(ν)R * Sin(ν)

Apollo's orbit is definitely eccentric; thus, conversion must consider following factors.

  1. Since focus is not at center, X value must be translated by c, distance of focus from center.
  2. R value varies from perihelion, q, to aphelion, Q.
...




Orbital Velocities

Kepler's Laws point out that orbital velocity varies per some relationship with distance from the Sun. Newton built on this work to derive the now famous equation: v = √(G * MSol/R) which can determine orbital speeds for circular orbits. For Earth's Solar orbit, use following values:

  • Universal gravitational constant: G = 6.667 x 10-11 N-m2 /kg2
  • Mass of Sun: MSol = 2.0 x 1030 kilograms

NOTE: Since values G and MSol can be considered constant, the product, G*MSol , is often expressed as the standard gravitational parameter:
μSol = 132,712,440,018 km3 / sec2 = 13.27 x 1019 m3/sec2
Radius of Earth's orbit: R = 1.5 x 1011 meters Thus, we determine Earth's orbital velocity,

v =G * MSol
R
=μSol
R
v =13.27 x 1019 m3/sec2
1.5 x 1011 m
v =8.847 x108m3/sec2
1.0 m
=29.74x103 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
r
-
1
a
))

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 1019 m3/sec2 * (2/151,093,849,398m - 1/219,908,869,916m)) √(μ (2/r - 1/a)) = √(13.27 x 1019 m3/sec2 * (1.32368 x 10-11m - 0.45473 x 10-11m)) √(μ (2/r - 1/a)) = √(13.27 x 1019 m3/sec2 * 0.86895 x 10-11m) √(μ (2/r - 1/a)) = √(11.531 x 108 m2/sec2)

c. Accomplish square root calculation: √(μ (2/r - 1/a)) = 3.396 x 104 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 10-11N-m2 
kg2
2 x 1030 kg 
R
)

Recall N, Newton, is Unit of force, Newton. N = 1.0 kg m/sec2.
√(
kg * m 
sec2
6.667 x 10-11 m2 
kg2
2 x 1030 kg 
R
)
Introduce conversion expression so R value can be in AUs.

√(
m2 
sec2
6.667 x 10-11 m 
kg2
1.0 AU 
1.5 * 1011 m
2 x 1030 kg2 
R
)

Rearrange powers of ten.
√(
(103m)(103m) 
sec2
6.667 x 1011 m 
1.5 * 1011 m
1.0 AU * 2 x 102
R
)

Substitute: 1 km = 1,000 m

√(
km2 
sec2
4.4444 AU
1.0
200
R
)

Final Rearrangement.
(Assume R = 1 AU.)

√(
888.888 AU-km2 
sec2 R
) = 
29.81 km/sec

Use μ = 888.888 AU-km2/sec2

RULE OF THUMB: Infer μ = 888.888 AU-km2/sec2 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 AU-km2/sec2 (2/1.01AU - 1/1.47AU)) √(μ (2/r - 1/a)) = √(888.888 AU-km2/sec2 (1.98/1.0AU - 0.68/1.0AU)) √(μ (2/r - 1/a)) = √(888.888 km2/sec2 (1.30)) √(μ (2/r - 1/a)) = √(1155.5544 km2/sec2) = 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.

More Orbital Speeds

√[μ (2/R - 1/a)]

Four simple steps are shown again for additional Apollo ranges, distances from Sun.

Orbital
Parameter

Rmk

R

X1

X2

X3

V

q

Min R

0.65 AU

3.08/1AU

2.48/1AU

2,204 km2/s2

47 km/s

a

Ave R

1.47 AU

1.36/1AU

0.68/1AU

604 km2/s2

24.6 km/s

Q

Max R

2.30 AU

0.87/1AU

0.19/1AU

169 km2/s2

13 km/s

Observed

2/R

X1-0.68

μ*X2

√(X3)

To compute V, elliptical orbital speed, we suggest breaking down equation as shown.

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, X1 = 2/R.√[μ (2/R - 1/a)]

√[μ (X1 - 1/a)]

2. Subtract 0.68 from X1 for 2nd term, X2 = X1-.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 (X2)]

3. X3= 888.888 AU-km2/s2 * X2.
{Above heuristic trades minor accuracy loss for major convenience gain.}
√[μ (2/R - 1/a)]

√[X3]

4. For 4th and last step, determine V, orbital speed for specific range from Sun by determing square root of X3.√[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).





















































































































































































































































































































































































































































































































































































































Asteroid Apollo Table of Elements

Computed for every 15°
TrueDist fm SunCart. Coord.DistanceVelocityTimeAdjusted
Anomaly (ν)r-valueX-valueY-valuex,y to x',y'Vt/SectorCumm
Time
Forecast
(deg)(AU)(AU)(AU)(AU)(km/sec)(days)(days)Ordinal-Date
0

0.647

1.470.00Start SectorStart SectorStart Sector Start Sector

101-2009
(11Apr09)

15

0.655

1.460.170.17046.26.4

6.5

107-2009

30

0.680

1.410.340.17645.46.7

13.2

114-2009

45

0.724

1.330.510.18844.07.4

20.7

121-2009

60

0.789

1.220.680.20841.88.6

29.4

130-2009

75

0.882

1.050.850.23739.110.5

40.0

141-2009

90

1.010

0.821.010.27835.713.4

53.6

154-2009

105

1.181

0.521.140.33331.918.0

71.8

172-2009

120

1.403

0.121.210.40327.825.1

97.1

198-2009

135

1.672

-0.361.180.48223.535.5

133.0

234-2009

150

1.961

-0.880.980.55419.449.5

183.1

284-2009

165

2.200

-1.300.570.59315.864.8

248.6

349-2009

180

2.295

-1.470.000.59413.775.3

324.7

50-2010

195

2.200

-1.30-0.570.59413.775.3

400.8

136-2010

210

1.961

-0.88-0.980.59315.864.8

466.3

192-2010

225

1.672

-0.36-1.180.55419.449.5

516.3

242-2010

240

1.403

0.12-1.210.48223.535.5

552.2

288-2010

255

1.181

0.52-1.140.40327.825.1

577.5

313-2010

270

1.010

0.82-1.010.33331.918.0

595.7

331-2010

285

0.882

1.05-0.850.27835.713.4

609.3

345-2010

300

0.789

1.22-0.680.23739.110.5

619.9

355-2010

315

0.724

1.33-0.510.20841.88.6

628.6

363-2010

330

0.680

1.41-0.340.18844.07.4

636.1

007-2011

345

0.655

1.46-0.170.17645.46.7

642.9

015-2011

360

0.647

1.470.000.17046.26.4

649.3

020-2011
(20Jan11)

Given
l
1+eCos(ν)
c+rCos(ν)
rSin(ν)
√(Δx2y2)
√[μ (
2
RAve
-
1
a
)]
d
V
t'=1.0105t
 
Center 0.82 0.000.00Δx = x' - xt'i=t'i-1+t'i
Sun 0.0 0.820.00Δy = y' - y
Perihelion 0.65 AU 1.470.00Σ = Approx. Circum. =Σ = App. Per. =
Aphelion 2.29 AU -1.470.008.4 (AUs)643 Days = 1.8 Yrs
  • ν=True Anomaly Angle originates from Sun and starts with line to perihelion.
  • a=semi-major axis. For Apollo, a = 1.47 AU = 2.205 x 1011m
  • b=semi-minor 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 * m2/kg2
  • MSol=Sun's Mass = 2.0 x 1030 kg
  • RAve=Average radius for sector
  • AU=Astronomical Unit 1.5 x 1011 m=1.5 x 108 km
  • π/12 * Rave / (VAve * 86,400)

Cross Check: Compute Apollo's period, T:
T = 2 π √[(a3/(G*MSol)]
T = 2 π √[(2.205 x 1011m)3)/(6.667 x 10-11N * m2/kg2*2.0 x 1030 kg)
]
T = 2 π √[(10.72 x 1033m3))/(13.33 x 1019N * m3/sec2)]
T = 2 π √[(0.804 x 1014sec2)]
T = 2 π 0.8966 x 107sec = 5.61 x 107sec
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).

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