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Clarification about Earth's Precession

  1. Jan 30, 2016 #1
    I haven't a textbook where I can study Earth Precession and I'm not sure to have correctly understood... Could you tell me if I do mistakes?

    I consider two points P and Q on the equatorial bulge. The torques of the gravitation forces between P-sun and Q-sun (calculated wrt the Earth center) are not equal and opposite, so I have a ##\tau_{tot} \neq 0## that generates ##\Delta L## (##\tau=dL/dt##).

    ##\vec{\tau_{tot}}=\vec{\tau_p}-\vec{\tau_q}=\vec{R}\times\vec{F_p}-\vec{R}\times\vec{F_q}=\vec{R}\times\vec{\Delta F}##

    ##\Delta F=\displaystyle \frac{4G mMR^2 cos \theta}{a^3}##

    so ##\tau_{tot}=\displaystyle \frac{4G mMR^2 cos \theta}{a^3} Rsin \theta##

    but ##\tau_{tot}=\displaystyle\vec{\Omega_p}\times{L}=\Omega_p L sin \theta## . From this last relations, I obtain:

    ##\displaystyle \frac{4G mMR^2 cos \theta}{a^3} Rsin \theta=\Omega_p L sin \theta## -> ##\Omega_p=\displaystyle\frac{4GmMR^2 cos \theta}{a^3 I \omega_{rot} }##

    Is it correct? My collegue wrote on his blocknotes ##|\Omega_p||L|=|\tau_{tot}|## so ##\Omega_p=\frac{2GmMR^2 sin2 \theta}{a^3 I \omega_{rot} }##
     
  2. jcsd
  3. Jan 30, 2016 #2

    ORF

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    Hello

    I'm not sure, but maybe it's that the angle between L and Ω_p is 90deg, and not θ. Drawing the vectors may help.

    Greetings
     
  4. Jan 31, 2016 #3
    Mmmmh. ##\theta## is the angle between the Earth rotation axis and the Earth direction axis that we would have if the Earth wasn't "crooked"......
     
  5. Jan 31, 2016 #4

    ORF

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    Hello

    I'm not completely sure, but if nobody answers... I think that the Ω_p is always perpendicular to the ω_{rot}; the tricky point here is to see that the direction of Ω_p is not constant, but its module is (ie, the frequency of precession is constant).

    So the angle between Ω_p and L would be 90deg, and not theta, and you will keep the sin(theta) in the cross product, in order to get 2sin(theta)cos(theta) = sin(2theta), as your classmate. I still think that a drawing is useful (I'm not native English speaker, and it would be easier also for me ;) ).

    Another point: I think there is another mistake in the second equation; maybe it's just R not R squared (magically it desappears in the next steps ;) ).

    Greetings! :D
     
  6. Jan 31, 2016 #5
    The setting should be this:
    XkJeU7U.jpg

    I can't understand the reason why ##\Omega_p## should be perpendicular to ##w_r##.. see also

    http://oceanworld.tamu.edu/students/iceage/images/precession_1.jpg
     
  7. Feb 2, 2016 #6

    ORF

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