For any given orbit, average velocity is simply computed;
circumference distance divided by period duration:
V_{Ave} = 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 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 
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)

C_{A} = 2 π √((a^{2}+b^{2})/2)

C_{♁}=6.28AU  ×  149,597,871 km
1 AU 

1.495×10^{8} km
1 AU  ×  2π√[(1.47 AU)^{2}+(1.22 AU)^{2}]
√2 

C_{♁ }= 9.42 × 10^{8} km

C_{A }= 12.735 × 10^{8} km

Period
To compute period of any orbit, use Kepler's Third Law:
Square of orbit's period (T) is proportional to square of semimajor axis(a): T^{2} ∝ a^{3}
Often expressed: T = 2 π √(a^{3}/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} m^{3}/(kg sec^{2})
Sun's mass: M_{☉ }= 2 × 10^{30} kgs
μ= (6.667 × 10^{11} m^{3}/(kg sec^{2})) (2 x 10^{30} kg) = 13.2 × 10^{19} m^{3}/^{ }sec^{2}
Convert to kilometers: μ = μ × km ^{3}/(1,000m) ^{3 }= μ × km^{3}/(10^{9}m^{3})^{ }
μ = 132,712,440,018 km ^{3}/sec ^{2} = 13.2 × 10 ^{10} km ^{3}/sec ^{2}
Earth Orbit  Apollo Orbit 
T_{♁} = 2 π √(r^{3}/μ)= 2 π √(1AU^{3}/μ)

T_{A} = 2 π √(a^{3}/μ) = 2 π √(1.47AU)^{3}/μ)

T_{♁}= 6.28  ×  √(1.495 × 10^{8} km)^{3}
√(13.2 × 10^{10} km^{3}/sec^{2}) 

T_{A}=6.28  ×  √(2.197 × 10^{8} km)^{3}
√(13.2 × 10^{10 }km^{3}/sec^{2}) 

T_{♁} = 3.15 × 10^{7} sec ≈ 1 year

T_{A} = 5.62 × 10^{7} 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.
V_{ave} = C ÷ T
Since orbital velocities are more useful in kilometers per second, select units accordingly from above tables.
Earth Orbit  Apollo Orbit 
V_{Ave} = C_{♁} ÷ T_{♁}

V_{ave} = C_{A} ÷ T_{A}

9.42 × 10^{8} km
3.15 × 10^{7} sec  12.735 × 10^{8} km
5.62 × 10^{7} sec 
V_{♁ave} = 29.9 km/sec

V_{Aave} = 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


1.471 AU

1.218 AU

8.49 AU

12.74 ×10^{8} km

1.81 yr

57,119,256sec

22.30 km/sec

Planet

Earth

1.0000001 AU

0.9998555 AU

6.28 AU

9.42 ×10^{8} km

1.0 yr

31,557,600sec

29.85 km/sec



Observed

Observed
 2 π √((a^{2}+b^{2})/2)  2 π √(a^{3}/μ) 
V_{Ave} = C_{km} ÷ T_{sec}

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:
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 = b^{2}/a = .99971 AU  ℓ = b^{2}/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


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_{♁}  R_{A} 
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.
μ is the standard gravitational parameter
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 }m^{3}
kgsec^{2}


×


2 × 10^{30} kg

)

= μ =

(

6.667 m^{2}
sec^{2}


×


20 × 10^{18}

)


Add CONVERSION CONSTANT and rearrange. 
(

6.667 m^{2}
sec^{2}

1.0 AU
1.495 × 10^{11} m


20×10^{18} m

)

= μ =

(

10^{6}m^{2}
sec^{2}

10^{11} m
10^{11} m


200 × 6.667 AU
1.495 
)


Substitute:
1 km=1,000m=10^{3}m

(

(10^{3}m)(10^{3}m)
sec^{2}

×

6.667 × 200 AU
1.495


) = μ =

(

891.906 AUkm^{2}
sec^{2}


)



√(  μ 
)

=

29.865 km √(AU)
sec


V = 29.865 km/sec × √  (  AU  (  2
R    ℓ
a  )) 
a is the semimajor axis, which is constant for any given orbit; as is b, the semiminor axis.
b^{2}/a = ℓ, orbit's semilatis rectum, perpendicular distance from semimajor axis to orbit at Sol, the focus.
ℓ is also parallel to semiminor 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 = b^{2}/a = .99971 AU⊥ ℓ/a = .99971

ℓ = b^{2}/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 (A_{Vel}) varies as shown below.

θ  A_{Rad}  A_{Vel} 
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 

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