An Approach to Mathematic Functions Basics

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

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Section XXIX - Setting up the Standardized Set of Initial Conditions

Summary for the Section :

• In this Section , we shall set up our { Standardized Set of [ Initial Conditions ] }
  • for the Satellite moving under a Central Force .

  This is done for a specific purpose :

  • It is our intention to start-off our analysis on [ orbit shapes ]
    • with the Circular Orbit as the [ base scenario ] ,

    and thereafter :

    • move further onto other [ orbital shapes ] as the [ initial conditions ] varies .

• We shall therefore set up the [ Standardized Set of Initial Conditions ] here as such , such that :
  • the [ circular orbit ] is always a viable possibility under the said [ Standardized Set of Initial Conditions ] .

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The Initial Conditions for the Satellite :

• In general , the [ initial conditions ] for a Satellite moving under a Central Force are defined via :
  • its Position Vector at time [ t = 0 ] , and

  • its Velocity Vector at time [ t = 0 ] .

  And the Satellite's subsequent Position at [ t = T ] is always determined by :

  • the Differential Equation involving the derivatives with respect to time .

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The Standardized Set of Initial Conditions :

• Let us now place two (2) sets of restrictions on the [ initial conditions ] .

  FIRST RESTRICTION - restrictions on the Position Vector .

  In general , the Position Vector is defined by the distance [ R ] and the angle [ Angle Theta ] ,

  • as per the diagram on-the-right below .

  

  For our [ Standardized Set of Initial Condidtions ] :

  • we shall now start-off at [ t = 0 ] with the Satellite being at a { position on the [ U-Axis ] } ,
    • as per the diagram on-the-left above ,

    so that :

    • the { distance [ O-P ] at [ t = 0 ] } may now be designated as the quantity [ R-sub-Zero ] .

  We then have :

  • 

  • 

  SECOND RESTRICTION - restrictions on the Velocity Vector .

  In general , the Velocity Vector is defined by the quantities [ V-sub-R ] and [ V-sub-Theta ] ,

  • as per the diagram on-the-right below .

  

  For our [ Standardized Set of Initial Condidtions ] :

  • we shall now set up :
    • 

    • 

      where the quantity [ V-sub-Zero ] may take-on any non-zero value .

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Summarizing on the [ Standardized Set of Initial Conditions ] :

• Let us quickly highlight the key features of our [ Standardized Set of Initial Conditions ] here :
  • FIRST :
    • the Velocity Vector and the Position Vector are orthogonal to one-another at [ t = 0 ] .

  • SECOND :
    • 

  We then have this diagram below on-the-left ,

  • showing this [ Standardized Set of Initial Conditions ] .

  

• We also take note here that , in general :
  • the [ Velocity in the Radial-Direction ] can be expressed in this mannner :
    • 

  Since we have set up :

  • 

  • 

    we do have this relation below at [ t = 0 ] :

    • 

  But :

  • 

    because :

    • 

• We can therefore safely conclude that :
  • 

    at [ t = 0 ] .

  And this is an important [ initial condition ] that needs to be satisfied ,

  • when we attempt to solve the Differential Equation involving [ R ] and [ Theta ] ,
    • to come up with an expression expressing [ R ] in terms of [ Theta ] ;

    in later Sections .

• Let us now move on :
  • to look at the [ circular orbit ] , next .

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go to the next section : Section XXX - Setting up the Circular Orbit as the Base Scenario

go to the last section : Section XXVIII - The Differential Equation involving [ R ] and [ Theta ]

return to the MATH Functions Basics HomePage

Original dated 2014-12-07