Saros cycle - from Greiner's book

[babylonia]

saros cycle -- from Greiner's book

Hi,

I'm having some trouble about the calculation of saros cycle in example 13.9 of Greiner's book "classical mechanics: system of particles and hamiltonian dynamics". I am confused about the factor "3" of eqn. (13.35). This equation is derived from (13.27). However as I did it, I only got a factor "2". Obviously the factor "3" is necessary for the final answer to be close to the true value. However I can't figure out how this factor comes. (I've no idea whether this could be posted here. This is neither a homework nor a problem in the course I'm taking.)

It will really be very much appreciated if you can give me some help.

Thanks.

 

[Valerie Park]

The factor of 3 is the number of times the moon's orbit around the Earth must overlap itself in order to complete a saros cycle. This is because the moon's orbit is slightly tilted relative to its orbit around the Sun, and it takes three full orbits of the moon around the Earth for it to return to the same orientation relative to the Sun.