Monday, October 08, 2007

Asteroidal: Initial speed descriptions

Orbits: Apollo vs. Earth

Celestial Body DimensionsInclinationOrbital
TypeName Semimajor
Axis
EccentricityPerihelionAphelionAngle Speed (Ave) Period
Asteroid
1.471 AU
0.560
0.647 AU
2.294 AU
6.36o
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

 

T: Speed and Period

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
Circular orbits could use the well known C = 2 π rA well known example is Earth's orbit: C = 2 π 1 AU = 6.28 AU = 9.42 x 108 km
Circumference for elliptical orbits: C = 2 π ((a2+b2)/2)
For Apollo's orbit: C = 2 π ((1.472 + 1.222)/2) AU = 8.49 AU = 12.735 x 108 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.
vave = 6.28 AU / 1 yr = (6.28AU *150,000,000 km/AU) / (365.25 days * 86,400 sec/day)
vave = 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)
vave = 22.5 km/sec = 22.5 km/sec * 86,400 sec/day = 1,944,000 km/day = 0.01296 AU/day
TApollo = C / vave = 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*MSol/RSol)
v = (μ /r) ; where μ = G*MSol
Newton's Universal gravitational constant, G = 6.667 x 10-11 N * m2/kg2

Sun's mass, MSol, = 2 x 1030 kgs.

RSol 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 1011 m

Linear velocity of Earth in Sol's orbit
VE = (6.667 x 10-11 N * m2/kg2 * 2 x 1030 kgs/ [1.5 x 1011 m]) = 29.81 km/sec




Computing T: Period of Elliptical Orbits

Computing Apollo's T is straightforward; it comes directly from Kepler's Third Law. T = 2 π (a3/μ)
μ = G * MSol = (6.667 x 10-11 N * m2/kg2) (2 x 1030 kg) = 13.2 x 1019 m3/sec2
μ = G * MSol = 132,712,440,018 km3/sec2 = 13.2 x 1010 km3/sec2
a = 1.47 AU (1.50 x 1011 m/AU) = 2.205 x 1011 m
T = 2 π (10.72 x1033 m3/13.2 x 1019 m3/sec2)
T = 2 π (0.812 x 1014 sec2) = 5.66 x 107 secs
T= 5.66 x 107 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.
vave = C/T
vave = 8.49 AU / 1.79 yrs
vave = 12.735 x 108 km / 5.66 x 107 secs
vave = 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)) Thus, we can readily determine an asteroid's period and average speed. However, an elliptical orbits differ from circular orbits in at least one respect, velocity range. While circular orbits have a very narrow range of velocities (we often think of Earth's orbit as having only one speed, 29.81 kms/sec). However, elliptical have a wide range of speeds, thus, next chapter covers this.







 

 


 

 
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