Monday, May 27, 2013

Average Velocity vs. Specific Velocity




For any given orbit, average velocity is simply computed;
circumference distance divided by period duration: 
VAve = C ÷ T

However, utility of average velocity is limited.



Radial distance (R) is measured from Sol to orbital object.
Like R, specific velocity varies from q to Q.
Thus, utility is greater.



Recall previous work which describes orbit of Apollo,
an asteroid discovered in 1932.
Apollo is a Near Earth Object (NEO) which penetrates the Ecliptic
near Earth's orbit at the ascending and descending nodes.
Earth OrbitApollo Orbit
Semimajor axis: a = 1.0000001 AU Semimajor axis: a= 1.471 AU
Semiminor axis: b = 0.9998555 AUSemiminor axis:  b = 1.218 AU
 

Average Velocity

For any given orbit, use circumference and period to determine average speed,

Circumference
1 AU = 149,597,871 km 
Earth Orbit
(Circular)
r = radius from Sun to orbit.
Apollo Orbit
(Elliptical)
a=Semimajor axis ⊥ b=Semiminor axis
C = 2 π r = 2 π × (1 AU)
 CA = 2 π ((a2+b2)/2)
C=6.28AU×149,597,871 km 

1 AU
1.495×108 km

1 AU
×[(1.47 AU)2+(1.22 AU)2]

2
C= 9.42 × 108 km
CA = 12.735 × 108 km
(Formula for circumference of ellipse from Handbook of Chemistry and Physics, 10th ed. (1954), Mathematical Tables, p. 315. )

Period
To compute period of any orbit, use Kepler's Third Law:
Square of orbit's period (T) is proportional to square of semi-major axis(a): T2 ∝ a3 
Often expressed: T = 2 π (a3/μ)
Standard Gravitation Parameter (μ) is product of Universal Gravitation Constant (G) times mass of Sun (M)
 μ = G × M
Newton's Universal gravitational constant: G = 6.667 × 10-11 m3/(kg sec2)
Sun's mass: M= 2 × 1030 kgs
μ= (6.667 × 10-11  m3/(kg sec2)) (2 x 1030 kg) = 13.2 × 1019 m3/ sec2
Convert to kilometers: μ = μ × km3/(1,000m)3 = μ × km3/(109m3)
μ = 132,712,440,018 km3/sec2 = 13.2 × 1010 km3/sec2
Earth OrbitApollo Orbit
T = 2 π (r3/μ)= 2 π (1AU3/μ)
TA = 2 π (a3/μ) = 2 π (1.47AU)3/μ)
T6.28×(1.495 × 108 km)3

√(13.2 × 1010 km3/sec2)
TA=6.28×(2.197 × 108 km)3

√(13.2 × 1010 km3/sec2)
T = 3.15 × 107 sec ≈ 1 year
TA = 5.62 × 107 sec ≈ 1.79 year
1 year = 365.25 days × 86,400 sec/day = 31,557,600 sec

Average Speed 
Divide circumference by period to determine average speed.
Vave = C ÷ T 
Since orbital velocities are more useful in kilometers per second, select units accordingly from above tables.
Earth OrbitApollo Orbit
VAve = C ÷ T 
Vave = CA ÷ TA 
9.42 × 108 km

 3.15 × 107 sec
12.735 × 108 km

5.62 × 107 sec
Vave = 29.9 km/sec
VA-ave = 22.66 km/sec

Average Orbital Speeds SUMMARY: Apollo vs. Earth

Celestial Body D i m e n s i o n sPeriodAve Orbital
TypeName Semimajor
Axis (a)
Semiminor
Axis (b)
Circumference
(C)
Year
(T)
SecondsSpeed (v)
Asteroid
1.471 AU
1.218 AU
8.49 AU
12.74 ×108 km
1.81 yr
57,119,256sec
22.30 km/sec
Planet
Earth
1.0000001 AU
0.9998555 AU
6.28 AU
9.42 ×108 km
1.0 yr
31,557,600sec
29.85 km/sec 
 
 
Observed
Observed
2 π ((a2+b2)/2)2 π (a3/μ)
 VAve = Ckm ÷ Tsec 
Using average velocity to determine position has limited utility.
Recall that Sol "anchors" all Solar orbits at a focus of each orbit;
furthermore, gravity compels an orbiting object to move faster when near Sol and slower when further away.
Thus, object's max velocity will be at q, perihelion, orbit's closest point to Sol; slowest at Q, aphelion, furthest point. 
At each point in the orbit, object's specific velocity will vary accordingly.
Next, determine specific radial distance (R) for any given orbital position; then, determine associated velocity.
Orbital Ranges


R is the distance between the orbiting body and the central body.

Value for R varies between minimum, q, and max, Q, per following formula:
R =

1+ e × Cos(θ)
Earth OrbitApollo Orbit
e = c/a= 0.17 AU ÷ 1.0000001 AU = 0.017e = c/a = 0.82376 AU ÷ 1.471 AU = 0.560
= b2/a = .99971 AU = b2/a =1.0085 AU
R = .99971 AU

1+ 0.017 × Cos(θ)
R = 1.0085 AU

1+  0.560  × Cos(θ)

Orbital radial distances have a range of radial distances (r) from Sol.

 Earth
Orbit
(Near Circular)
Apollo
Orbit
(Elliptical)
Perihelion, q:
Minimum distance from Sol
0.98329 AU 
0.647 AU
Aphelion, Q:
Maximum distance from Sol
1.017 AU
2.294 AU
Between q, min distance from Sol, and Q, max distance,
there is a range of distances; examples follow. 
Near circular orbits (like Earth's) have narrow range.
Elliptical orbits (like Apollo's) have much wider range of distances from Sol.
θRRA
Deg.
 AU
AU 
0.980.65
30°0.990.68
60°0.990.79
90°1.001.01
120°1.011.40
150°1.011.96
180°1.022.29
210°1.011.96
240°1.011.40
270°1.001.01
300°0.990.79
330°0.990.68
360°0.980.65
Specific Orbital Velocities 
Determining specific orbital speeds for any particular orbital position requires following formula.
V =(μ(2

R
-

a
))
μ is the standard gravitational parameter

Proposed Heuristic

Determine orbital velocities in km/sec when given orbital radius in AUs?
Perhaps we can use a more convenient value for μ.
TRADITIONAL
μ = G × M
Distances in meters.
(
6.667 × 10-11 m3

kg-sec2

×

2 × 1030 kg
)
= μ =
(
6.667 m2

sec2

×

20 × 1018
)
Add CONVERSION CONSTANT and rearrange.
(
6.667 m2

sec2
1.0 AU

1.495 × 1011 m
× 
20×1018 m
)
= μ =
(
106m2

sec2
1011 m

1011 m
× 
200 × 6.667 AU

1.495
)
Substitute:
1 km=1,000m=103m
(
(103m)(103m)

sec2
×
6.667 × 200 AU

1.495

) = μ =
(
891.906 AU-km2

sec2

)
Square root both sides.
√(μ
)
=
29.865 km √(AU)

sec

V = 29.865 km/sec  ×   (AU(2

R
-

a
))
  • a is the semi-major axis, which is constant for any given orbit; as is b, the semi-minor axis.
  • b2/a = , orbit's semi-latis rectum, perpendicular distance from semi-major axis to orbit at Sol, the focus.
    ℓ is also parallel to semi-minor axis and constant for any given orbit.
  • Thus, term "/a" will be constant for any given orbit.
    Earth Orbital Velocity
    Apollo Orbital Velocity
    a = 1.0000001 AU ⊥ b = 0.9998555 AU
    a= 1.471 AU ⊥ b = 1.218 AU
    = b2/a = .99971 AU⊥ /a =  .99971
    = b2/a =1.0085 AU⊥ /a = 0.6856
    V = 29.865 km/sec ×
    (
    AU
    (
    2

    R
    -
     .99971
    ))
    V = 29.865 km/sec ×
    (
    AU
    (
    2

    R
    -
     
    0.6856))
    Input Radial distance (R) as AUs; output orbital velocity (V) as kilometers per second (kps)
    Assume Earth has constant orbital Radius (♁Rad) = 1.0 AU;
    thus, assume constant orbital Velocity (♁Vel)  =  29.86 kps.
    As shown above, Apollo's orbital Radius (ARad) ranges from q to Q,
    thus, orbital Velocity (AVel) varies as shown below.
 
θARadAVel
Deg.
AU
kps
0.6546.34
30°0.6844.89
60°0.7940.65
90°1.0134.02
120°1.4025.73
150°1.9617.31
180°2.2912.91
210°1.9617.31
240°1.4025.73
270°1.0134.02
300°0.7940.65
330°0.6844.89
360°0.6546.34

 

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