Solution To Application Of Kepler's 2nd Law Problem

  The problem:

An asteroid moves in an elliptic orbit around the Sun. The lengths of the major and minor axes are 2a and 2b, respectively. If the asteroid’s velocity at the point of closest approach (where it crosses the major axis) is   v _o   then how much time is needed for the object to make one complete orbit?  

Solution:

We note:  dA/dt =  h/ 2,     

Then:

  d A/ dt     =   ½  r ²   dq /  dt   

ò  d A    =   ½  r ²  ò    dq  (dt) / dt

ò  d A       = ½  r ²  ò  ^2p  _o    dq

A  =   p ab   =   ½  r ²  ò  ^2p  _o    dq

But:    ò  ^2p  _o    dq   =   2 p   radians  so that:

p ab   =   ½   (2 p )   r <sup>2

  Subst.</sup>  r =  v _o / w     so:

ab   =     ( v _o / w  ) ²

But:  w    = 2 p  /  T

So:  ab =  v _o 2  T ²   /  (2 p) ²

  T ²    =   (2 p) ²  ab /   v _o 2


T     =     2 p  Ö (ab)  /   v _o