Average Velocity vs. Specific Velocity

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For any given orbit, average velocity is simply computed;

circumference distance divided by period duration: 

V_Ave = C ÷ T

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However, utility of average velocity is limited.

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Radial distance (R) is measured from Sol to orbital object.

Like R, specific velocity varies from q to Q.

Thus, utility is greater.

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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 Orbit	Apollo Orbit
Semimajor axis: a = 1.0000001 AU	Semimajor axis: a= 1.471 AU
Semiminor axis: b = 0.9998555 AU	Semiminor 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 × 2π√[(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 Orbit Apollo Orbit T♁ = 2 π √(r3/μ)= 2 π √(1AU3/μ) TA = 2 π √(a3/μ) = 2 π √(1.47AU)3/μ) T♁= 6.28 × √(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 Orbit Apollo Orbit VAve = C♁ ÷ T♁  Vave = CA ÷ TA  9.42 × 108 km  3.15 × 107 sec 12.735 × 108 km 5.62 × 107 sec V♁ave = 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 s Period Ave Orbital Type Name Semimajor Axis (a) Semiminor Axis (b) Circumference (C) Year (T) Seconds Speed (v) Asteroid Apollo 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 = ℓ 1+ e × Cos(θ) Earth Orbit Apollo Orbit e = c/a= 0.17 AU ÷ 1.0000001 AU = 0.017 e = c/a = 0.82376 AU ÷ 1.471 AU = 0.560 ℓ = b2/a = .99971 AU ℓ = 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. θ R♁ RA Deg.  AU AU  0° 0.98 0.65 30° 0.99 0.68 60° 0.99 0.79 90° 1.00 1.01 120° 1.01 1.40 150° 1.01 1.96 180° 1.02 2.29 210° 1.01 1.96 240° 1.01 1.40 270° 1.00 1.01 300° 0.99 0.79 330° 0.99 0.68 360° 0.98 0.65

Specific Orbital Velocities  Determining specific orbital speeds for any particular orbital position requires following formula. V =√ ( μ ( 2 R - ℓ 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 - ℓ a )) a is the semi-major axis, which is constant for any given orbit; as is b, the semi-minor axis. b2/a = ℓ, 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 "ℓ/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 ℓ = b2/a = .99971 AU⊥ ℓ/a =  .99971 ℓ = b2/a =1.0085 AU⊥ ℓ/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.   θ ARad AVel Deg. AU kps 0° 0.65 46.34 30° 0.68 44.89 60° 0.79 40.65 90° 1.01 34.02 120° 1.40 25.73 150° 1.96 17.31 180° 2.29 12.91 210° 1.96 17.31 240° 1.40 25.73 270° 1.01 34.02 300° 0.79 40.65 330° 0.68 44.89 360° 0.65 46.34