# Re: OT: Orbital Radius / Semi-major Axis

From: Greg Roberts (grr@telkomsa.net)
Date: Mon Mar 16 2009 - 21:50:07 UTC

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```Good evening John

I have several formulae that Ive used over the years - just hope I can type them
in okay:

(1) Zhongolovich and Amelin in "A COLLECTION OF TABLES AND NOMOGRAMS
FOR THE PROCESSING OF OBSERVATIONS MADE ON ARTIFICIAL EARTH
SATELLITES"  -published in 1961 gives the following:

Basically n (squared)  times  a (cubed)  equals a constant where the constant is
equal to  fM = 398600 km sec (raised to power -2)
and n = 2* Phi/T where n is the argument of the mean diurnal motion
T= period of revolution  and n = 518400/T where T is in minutes
n is expressed in degrees per mean solar day, a is expressed in kilometres

example1:   if n = 2656.36 degrees then T = 195.16 minutes and a = 11145.0 km

example2:   if T = 158.317 minutes then n = 518400/T = 3274.44 degrees and
a = 9694.2 km

(2) Macko in the book SATELLITE TRACKING ( 1962) gives a very simple formula
a = 205.82 (T) raised to power 2/3
where a= semi major axis in statute miles, T = period in minutes
so a satellite 100 miles up has a semi-major axis of (3963 + 100 ) = 4063
statute miles
so T then becomes (4063/205.82) raised to power 3/2 = (19.741)raised to power
3/2
giving a period of 87.7 minutes

rationalized orbital elements ( Independent Tracking Coordination Program -1964)
gives
n = 17.0435726/a (raised to power 1.5)
where n = mean motion and a = semi major axis.

(4) From a booklet published by the Volunteer Satellite Tracking Program -1962
called "A LETTER TO GREGORY ROBERTS -Part II" by Wilcox P.Overbeck
we find (page 91-equation 6.8.1)

n (squared ) x  a (cubed) = 1.80838 x 10 (raised to power 13)

and further refines this by taking into account the effects of perturbing forces
that include eccentricity and inclination but I think its beyond my ability to
type this correctly into an email message and is quite small anyway and only
necessary in analytical work.

(5) Desmond King Hele in  "SATELLITE ORBITS IN AN ATMOSPHERE-
Theory and applications " (1987) gives

n (squared) x a (cubed ) = 398600 km(cubed)  s (raised to power-2) and
n in radians per second and goes into more detail on page 35 but again Im
not going to try and type this - do I hear cheering ?

This is all actually an application of Keplers third law that tells us that
a(cubed) is proportional to T(squared) where T is the orbital period for
one revolution.

I am sure there are many variations of this basic rule.

Hope this helps -- bet my mailer will mangle all this! ( if I havent already!)

Cheers
Greg

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