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Kepler’s Laws and Newton’s Synthesis

Table of Contents
Introduction
Problems
Answers

    

Introduction

Kepler’s Laws of Planetary Motion

Kepler’s First Law:  The path of each planet about the Sun is an ellipse with the Sun at one focus.
    
Kepler’s Second Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.

The time to travel from point 1 to point 2 is the same amount of time to travel from point 3 to point 4.

The blue area is the same as the red area.

 

Kepler’s Third law: the ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of cubes of their mean distance from the Sun. That is if T1 and T2 represent the periods for any two planets and r1 and r2 represent their average distances from the Sun, then
    

T12         r13
------ =  --------   This can be derived from Newton's Law of Universal Gravitation.
T22        r23  

     

    
For a circular Orbit      F = ma

       m1ms          m1v12                          2r1 
G -----------  =  ----------        and  v1 = ----------
         r12              r1                                 T1 
   

Substituting v1 and canceling m1.
   

      ms         42 r1 2                                   ms         42 r1       
G ------  =  ------------        Simplifying    G ------  =  ------------
      r12         r1T12                                       r12           T12    
   

Rearranging to get r and T on one side and all other terms on the other side.   Repeat the equation for 2nd period.
   

T12       42                 T22       42                                                                       T12        T22  
----- = ----------    and   ----- = ----------     Since G, ms, and are constants        ----- =   -------
r13        G ms               r23        G ms                                                                     r13          r23 
   

This can be rearranged to get

   
T12         r13 
-----  =  -------
T22        r23 
   

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Problems

1. Mars' period is about 684 days. Calculate the distance from the Sun to Mars if the Earth is 1.496 x 108 km from the Sun.
   
2.     Calculate the mass of the sun if the Earth has a period of 365.25 days.
   
3. Calculate the height of a geosynchronous satellite above the Earth given in relationship of the Moon’s distance From Earth.

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Answers

1. Mars' period is about 684 days. Calculate the distance from the Sun to Mars if the Earth is 1.496 x 108 km from the Sun.
   
     
Tm2         rm3 
-----  =  -------          Solve for rm   
TE2        rE3 
   

              Tm2  rE3                     (654 d)2 x (1.496 x 10 8 km)3 
rm  =  ( -------------- ) 1/3   = ( ----------------------------------------- ) 1/3  = 2.21 x 10 8 km  
                TE2                                     (365.25 d)2  
    
   

   
2.     Calculate the mass of the sun if the Earth has a period of 365.25 days.
   
   
      ms         42 r      
G ------  =  ------------  Simplifying and solving for ms.       365.25 d = 3.156 x 10 7 s
     r2              T2    
   

           42 r3                   42   (1.496 x 10 11 m)3
ms  =  ------------ = ---------------------------------------------------------- =  1.99 x 10 30 kg
            G T2            6.67 z 10 -11 N m2 kg -2 x (3.156 x 10 7 s)2 
   
   

   
3. Calculate the height of a geosynchronous satellite above the Earth given in relationship of the Moon’s distance From Earth.
   
     
Ts2         rs3 
-----  =  -------          Solve for rs   
Tm2        rm3 
   

              Ts2  rm3                  (1 d)2 x rm3    
rs  =  ( -------------- ) 1/3   = ( ----------------- ) 1/3  =  0.329 rm   
                Tm2                        (28 d)2  
    

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