An Approach to Mathematic Functions Basics

by Frank C. Fung ( 1st published in December, 2014. )

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Section XXXIX - A Proposal for Further Study of Orbital Shapes

Summary for the Section :

• In this Section , we shall outline our proposal for further study of the Orbital Shape of a Satellite .

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The Circular Orbit for any value of [ N ] :

• Let us again bring in a Circular Orbit for any { finite value of [ N ] greater than [ zero ] } .

  

  Equating the Centripetal Force and the Central Force the yields us this relation :

  • 

  Consequently ,

  • 

    yielding :

  • 

    and finally :

  • 

  Therefore ,

  • [ G-sub-N ] may be expressed as a function of [ N ] , [ R-sub-zero ] , [ V-sub-zero ] and [ M ] ;

    i.e. :

    • 

      once the Circular Orbit is established .

Standardizing the Circular Orbits :

• Let us now standardize the Circular Orbit for any given value of [ N ] :

  STEP ONE :

  • We can adjust the [ length unit ] so that the [ numeric value ] of [ R-Sub-zero ] is [ 1 ] .

  STEP TWO :

  • If after STEP ONE , the [ numeric value ] of [ V-sub-zero ] is not [ 1 ] ,
    • we can adjust the [ time unit ] so that [ V-sub-zero ] is [ 1 ] .

  STEP THREE :

  • We can then adjust the [ mass unit ] so that the [ numeric value ] of [ M ] is [ 1 ] .

  STEP FOUR :

  • We bring in this equation for the Circular Orbit for any { finite value of [ N ] greater than[ zero ] } :
    • 

    Since [ R-sub-zero ] , [ V-sub-zero ] and [ M ] now all spot the numeric value [ 1 ] ,

    • this necessarily dictates that the [ numeric value ] of [ G-sub-N ] is also [ 1 ] .

• Isn't that wonderful !

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The Proposal for Further Study of Oribtal Shapes:

• Our proposal here is then as follows :
  • For any value of [ N ] greater that [ 0 ] :
    • We use the standardized Circular Orbit with :
      • 

      • 

      • 

      • 

      as our starting point .

  • We then let [ M ] drop to [ zero ] all-of-a-sudden so that :
    • the satellite fly-off in a [ straight-line ] orbit .

    The Circular Orbit and the Straight-Line Orbit are then the 2 anchoring [ orbital shapes ] for any given value of [ N ] .

  • We then let the value of [ M ] vary :
    • from [ + infinity ] to [ 1 ] ,
    • from [ 1 ] to [ 0 ] , and
    • from [ 0 ] to [ - infinity ] ;

      to come up with other [ orbital shapes ] for the same value of [ N ] .

  • We can then do the same for other values of [ N ] and then :
    • group the curves for the same value of [ M ] to form 3-Dimensional Surfaces ,
      • with the { axis-for-[ N ] } being orthogonal to the 2-Dimesnional { orbital-shape-plane } .

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Splitting the range of [ N ] :

• Based on our discussion in the last Section ,
  • we do propose to split the range of [ N ] into 5 regions :
    • Region 1 : 0 < N < 1 ,
    • Region 2 : N = 1 ,
    • Region 3 : 1 < N < 3 ,
    • Region 4 : N = 3 ,
    • Region 5 : 3 < N < infinity .

• Lots of work ahead ,
  • and we do hope that someone will take up the challenge .

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go to the next section : Section XL - A Statement of Purpose for Part VII

go to the last section : Section XXXVII - A Graphic Look at the Solutions for [ N = 3 ]

return to the MATH Functions Basics HomePage

Original dated 2014-12-07