Monday, September 15, 2008

Compare Orbits: Apollo vs. Earth



























Polar Equation

R =

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,
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 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 =

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.

First 90° of Earth's Orbit.

True Anomaly (ν)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

1 + e Cos(ν)
e = Q - q

Q + q
=1.0-1.0

1.0+1.0
=0

2
= 0
ℓ =2×Q×q

Q + q
=2×1×1

1.0+1.0
=2

2
= 1


First 90° of Apollo's Orbit.

True Anomaly (ν)Dist fm Sun (R)
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

1 + e Cos(ν)
e = Q-q

Q+q
= 0.56
ℓ =2×Q×q

Q + q
= 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. 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(ν)
R×Cos(ν)R×Sin(ν)
Recall very basic trig 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
Anomaly
Distance
 from Sun
Cartesian
Coordinates
νR-valueX-valueY-value
1.000 AU1.00 AU0.00 AU
15°1.000 AU0.97 AU0.26 AU
30°1.000 AU0.87 AU0.50 AU
45°1.000 AU0.71 AU0.71 AU
60°1.000 AU0.50 AU0.87 AU
75°1.000 AU0.26 AU0.97 AU
90°1.000 AU0.00 AU1.00 AU
105°1.000 AU-0.26 AU0.97 AU
120°1.000 AU-0.50 AU0.87 AU
135°1.000 AU-0.71 AU0.71 AU
150°1.000 AU-0.87 AU0.50 AU
165°1.000 AU-0.97 AU0.26 AU
180°1.000 AU-1.00 AU0.00 AU
Given

1+eCos(ν)
1.0×Cos(ν)1.0×Sin(ν)
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. 




Orbital Velocities

Kepler points out that orbital velocity varies per with distance from the Sun. Furthermore, Newton derived the now famous equation:
v = (G × MSol/R
to determine orbital speeds for circular orbits. For Earth's Solar orbit, use following values:
  • Universal gravitational constant: 
  • G = 6.667 × 10-11 N-m2/kg2


  •  Mass of Sun: MSol = 2.0 × 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
    0.65 AU
    3.08/1AU
    2.48/1AU
    2,204 km2/s2
    47 km/s
    a
    1.47 AU
    1.36/1AU
    0.68/1AU
    604 km2/s2
    24.6 km/s
    Q
    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
    1.47
    0.00
    Start Sector
    Start Sector
    Start Sector
    Start Sector
    101-2009
    (11Apr09)


    15
    0.655
    1.46
    0.17
    0.170
    46.2
    6.4
    6.5
    107-2009


    30
    0.680
    1.41
    0.34
    0.176
    45.4
    6.7
    13.2
    114-2009


    45
    0.724
    1.33
    0.51
    0.188
    44.0
    7.4
    20.7
    121-2009


    60
    0.789
    1.22
    0.68
    0.208
    41.8
    8.6
    29.4
    130-2009


    75
    0.882
    1.05
    0.85
    0.237
    39.1
    10.5
    40.0
    141-2009


    90
    1.010
    0.82
    1.01
    0.278
    35.7
    13.4
    53.6
    154-2009


    105
    1.181
    0.52
    1.14
    0.333
    31.9
    18.0
    71.8
    172-2009


    120
    1.403
    0.12
    1.21
    0.403
    27.8
    25.1
    97.1
    198-2009


    135
    1.672
    -0.36
    1.18
    0.482
    23.5
    35.5
    133.0
    234-2009


    150
    1.961
    -0.88
    0.98
    0.554
    19.4
    49.5
    183.1
    284-2009


    165
    2.200
    -1.30
    0.57
    0.593
    15.8
    64.8
    248.6
    349-2009


    180
    2.295
    -1.47
    0.00
    0.594
    13.7
    75.3
    324.7
    50-2010


    195
    2.200
    -1.30
    -0.57
    0.594
    13.7
    75.3
    400.8
    136-2010


    210
    1.961
    -0.88
    -0.98
    0.593
    15.8
    64.8
    466.3
    192-2010


    225
    1.672
    -0.36
    -1.18
    0.554
    19.4
    49.5
    516.3
    242-2010


    240
    1.403
    0.12
    -1.21
    0.482
    23.5
    35.5
    552.2
    288-2010


    255
    1.181
    0.52
    -1.14
    0.403
    27.8
    25.1
    577.5
    313-2010


    270
    1.010
    0.82
    -1.01
    0.333
    31.9
    18.0
    595.7
    331-2010


    285
    0.882
    1.05
    -0.85
    0.278
    35.7
    13.4
    609.3
    345-2010


    300
    0.789
    1.22
    -0.68
    0.237
    39.1
    10.5
    619.9
    355-2010


    315
    0.724
    1.33
    -0.51
    0.208
    41.8
    8.6
    628.6
    363-2010


    330
    0.680
    1.41
    -0.34
    0.188
    44.0
    7.4
    636.1
    007-2011


    345
    0.655
    1.46
    -0.17
    0.176
    45.4
    6.7
    642.9
    015-2011


    360
    0.647
    1.47
    0.00
    0.170
    46.2
    6.4
    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
    0.00
    0.00
    Δx = x' - x




    t'i=t'i-1+t'i




    Sun
    0.0
    0.82
    0.00
    Δy = 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=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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