World Builders™                                                                       Session One   --  Astronomy             
          Calculating Orbits

                                   for Your Moons 

             
More "we can do it" math from our good friend Gerald Nordley.  Go for it, World Builders!

You will need to know the mass of your planet for this
.
 
Just as we are able to calculate how long it will take our worlds to circle their suns, so we can also calculate how long it will take our moons to go around our worlds. Will you have two moons in the sky at once? Three? Let's find out!

   Here is the formula that will help us:

 

Translation:

T = the Time in hours that it takes the moon to go around the planet once:

This is one complete orbit.

r = the average distance of the moon from its world when we use the radius of the earth as 1.

M = the mass of the planet that the moon is orbiting if we count the mass of the earth as 1.

   Mr Nordley was kind enough to provide the following examples:


  Example 1:  

  Calculate the time that it takes for the earth's moon to orbit the earth once.

What we know:

                The moon is 60 earth radii away from the earth's center
                The mass of the earth is 1 earth mass  =1

Here is our formula:

Let's put in the numbers:

        T = 1.4  *   

 

T = 1.4  *     = 

T = 1.4  *             =

                     

                   The square root of 216,000 = 464.7

         T = 1.4  * 464.7 = 650.6 hours

These are a lot of hours. 
Divide by 24 hours to turn this into days.

T = 650.6 divided by 24 = 27.1 days

The moon goes around the earth in about 27 days. 

            It works!


  Example 2:

  Calculate the time that it takes for the moon Phobos to orbit Mars once.

What we know:

Phobos is 1.5 earth radii away from the center of Mars.

The mass of Mars is 0.11 earth mass  = 0.11

Here is our formula:

   Let's put in the numbers

                      T = 1.4   * 

                      T = 1.4   *    = 

                      T = 1.4   *                      

 

            
                      
(3.375 divided by .11) = 30.7    (Round off the decimals)

                         T = 1.4  * 

             The square root of 30.7 = 5.5

                       T = 1.4 times 5.5 = 7.7 hours

                       Phobos zips around Mars every 7.7 hours!


   Example 3: 

  Calculate the time that it takes for the moon Io to orbit Jupiter once.

 
     What we know:

Io is about 66 earth radii away from the center of Jupiter.

The mass of Jupiter is about 318 earth masses. 
Remember, Jupiter is a huge planet!
                     

Here is our formula:

Let's put in the numbers:

            T = 1.4 

               T = 1.4    = 

            T = 1.4        = 

                               (287,496 divided by 318) = 904   (Round off the decimals)

    T = 1.4  

     The square root of 904 = 30

               T = 1.4  * 30 = 42 hours

               Io zips around Jupiter every 42 hours!

If a moon is close to your planet, it will go around quickly. A distant moon will take longer to complete its orbit.

Do not put your moon too close to your planet!  The moon will disintegrate!

Check The Roche Limit.

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© 1998, 2003. Elizabeth Anne Viau. All rights reserved. This material may be used by individuals for instructional purposes but not sold. Please inform the author if you use it at eviau@earthlink.net.