Classical Free Fall Calculation

This Mathematica Notebook performs the calculation of the time of free fall from the orbit of the Earth to the Sun, using simple Newtonian functions.

Glen Langston,  NRAO GB, 2004 October 1

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The Newtonian Acceleration of a particle in free fall is the derivative of the potential

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During Free Fall, energy is conserved, so falling at rest from A to r yields

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Now work with the special case of falling from rest at 1 AU towards the Sun

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Calculate the function, T[r], the time it takes a particle to fall from rest at 1 AU, to a ratius r. The Integral function of a freely falling particle is surprisingly complicated:

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Now calculate the duration of the fall from the limits at one AU and zero AU.

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Calculation Units: Conversion to Days

The Distance Units for this calculation are AU.  The time units for this calculation are days.  In these units, GM is a small number, indicating that the system is not relativistic.  GC = gravitational constant = 6.67300 × 10^-11 m3 kg^-1 s^-1; Astronomical Unit = 149 598 000 000;  One Day = 86400 s

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Calculate the Speed of Light in AU per day (Its a fairly small number)

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The Distance Units for this calculation are AU.  The time units for this calculation are days.  In these units, GM (gm) is a small number, indicating that our solar system is not general-relativistic.

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Formally, the velocity of the particle goes to infinity as r goes to zero.  To avoid this, the calculation will stop at the Solar Radius (RS).   The Sun can not be treated as a point source inside a Solar Radius anyway.

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In Astronomical Units (AU) the Solar Radius is 0.47 % of an AU.

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Calculation of the Speed: v[r]

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t[r] is the time to fall a distance r(AU) in units of days

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Calculate the speed of the falling object in units of the speed of light.

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The maximum velocity is the velocity when reaching the surface of the Sun (in units of c).

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Converting  back to km/s

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Created by Mathematica  (October 1, 2004)