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bznm
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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 colleague wrote on his blocknotes ##|\Omega_p||L|=|\tau_{tot}|## so ##\Omega_p=\frac{2GmMR^2 sin2 \theta}{a^3 I \omega_{rot} }##
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 colleague wrote on his blocknotes ##|\Omega_p||L|=|\tau_{tot}|## so ##\Omega_p=\frac{2GmMR^2 sin2 \theta}{a^3 I \omega_{rot} }##