Saturday, May 3, 2014

Spin too much

Earth has too much spin.  If there was angular momentum before Earth was captured by the Sun, ie. Earth spinning very fast,  after capture, this momentum would have been divided between rotation and spin in linear proportion to the angular moments of inertia,

moment of inertia about axis =  0.4*\({r}^2_{e}\) = 0.4*6371^2 = 1.63*10^7 m2

moment of inertia about sun  =  0.4*\({r}^2_{e}\) +  \({O}^2_{e}\) = 1.63*10^7 + 149598261^2 =2.237*10^16 m2

Proportion of angular momentum with spin \(\cfrac{0.4*{r}^{2}_{e}}{2*0.4*{r}_{e}^{2} + {O}^{2}_{e}}\) = 0.00000000072

Proportion of angular momentum with rotation   \(\cfrac{0.4*{r}^{2}_{e} +  {O}^{2}_{e}}{2*0.4*{r}^{2}_{e} +  {O}^{2}_{e}}\) = 0.9999999992

Since moment of inertia for rotation is large compared to the moment of inertia for spin, almost none of the initial angular momentum is allocated to spin, but yet,

angular momentum of spin = 0.4*\({r}_{e}\)v =0.4*6371*0.4651 = 1.1850*10^3 km2s-1

angular momentum of rotation = \({O}_{e}\)V = 149598261*29.78 = 4.4550*10^9  km2s-1

actual angular momentum \(\cfrac {spin}{rotation}\) = 0.00000026

where Earth spin speed, v = 0.4651 kms-1,\({r}_{e}\) = Earth radius = 6371 km and \({O}_{e}\) is orbital distance and V is Earth's Rotational speed.

Earth must have acquired more spins through collisions after being captured by the Sun.  (Every thing is in per unit mass, so the unit may not add up.)