Asteroidal: Initial speed descriptions
Celestial Body  Dimensions  Inclination  Orbital  

Type  Name  Semimajor Axis  Eccentricity  Perihelion  Aphelion  Angle  Speed (Ave)  Period  
Asteroid

1.471 AU

0.560

0.647 AU

2.294 AU

6.36^{o}

22.5 km/sec

1.81 yr

^{}
 
Planet

Earth

1.0000001 AU

0.017

0.983 AU

1.017 AU

n/a

29.8 km/sec

1.0 yr

Given three orbital parameters: circumference, average speed, period, it seems intuitive that any two of these three items will enable us to determine the third.Computing Circumference
Circumference for elliptical orbits: C = 2 π √((a^{2}+b^{2})/2) For Apollo's orbit: C = 2 π √((1.47^{2} + 1.22^{2})/2) AU = 8.49 AU = 12.735 x 10^{8} km (Formula for circumference of ellipse from Handbook of Chemistry and Physics, 10th ed. (1954), Mathematical Tables, p. 315. )
Using CircumferenceOf course, Earth's period is one year; therefore, we readily use circumference to determine average speed.
v_{ave} = 6.28 AU / 1 yr = (6.28AU *150,000,000 km/AU) / (365.25 days * 86,400 sec/day) v_{ave} = 29.85 km/sec On the other hand, one can also use the circumference to determine the period, T, from the average speed. We can find the average orbital speed of Apollo from the Internet (see Table 2) v_{ave} = 22.5 km/sec = 22.5 km/sec * 86,400 sec/day = 1,944,000 km/day = 0.01296 AU/day T_{Apollo} = C / v_{ave} = 8.49AU / 0.01296 AU/day = 655 days = 1.79 years Computing Speed: Circular Orbits(With patience, one can always measure an object's parameters by taking precise observations over an extended duration. These observations will readily confirm repetitions of orbital positions to establish speeds and periodicity. Since many astronomers already do just that, we'll seek a more convenient method. Thus, instead of taking measurements for several years, let's use some formulas to quickly compute estimates.)Knowing neither the circumference nor the period, we can use textbook formulas to compute the speed of an orbiting body; for example, Earth in its orbit around the Sun. A well known formula for circular orbits around the Sun: v = √(G*M_{Sol}/R_{Sol}) v = √(μ /r) ; where μ = G*M_{Sol} Newton's Universal gravitational constant, G = 6.667 x 10^{11} N * m^{2}/kg^{2} Sun's mass, M_{Sol}, = 2 x 10^{30} kgs. R_{Sol} is radius of orbit from Sun. For further simplicity, also assume Earth's orbit to be circular. Radius of Earth's orbit = 1 AU = 1.5 x 10^{11} m Linear velocity of Earth in Sol's orbit V_{E} = √(6.667 x 10^{11} N * m^{2}/kg^{2} * 2 x 10^{30} kgs/ [1.5 x 10^{11} m]) = 29.81 km/sec Computing T: Period of Elliptical OrbitsComputing Apollo's T is straightforward; it comes directly from Kepler's Third Law. T = 2 π √(a^{3}/μ)μ = G * M_{Sol} = (6.667 x 10^{11} N * m^{2}/kg^{2}) (2 x 10^{30} kg) = 13.2 x 10^{19} m^{3}/sec^{2 } μ = G * M_{Sol} = 132,712,440,018 km^{3}/sec^{2} = 13.2 x 10^{10} km^{3}/sec^{2} a = 1.47 AU (1.50 x 10^{11} m/AU) = 2.205 x 10^{11} m T = 2 π √(10.72 x10^{33} m^{3}/13.2 x 10^{19} m^{3}/sec^{2}) T = 2 π √(0.812 x 10^{14} sec^{2}) = 5.66 x 10^{7} secs T= 5.66 x 10^{7} secs * (1 day/86,400 sec) = 655 days (1 yr/365.25 days) = 1.79 yrs Computing Apollo's average orbital speed is straightforward; divide C, circumference, by T, period. v_{ave }= C/T v_{ave }= 8.49 AU / 1.79 yrs v_{ave }= 12.735 x 10^{8} km / 5.66 x 10^{7} secs v_{ave} = 22.5 km/sec However, determining specific orbital speeds for any particular orbital position is more difficult for elliptical orbit. For one thing, orbital speed formula is more complicated. elliptic orbital velocity: v = √(μ (2/r  1/a))

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