We know that a change in Orbs changes \(GPE\).
\({F}_{ob}=\cfrac{d{GPE}_{e}}{d Orbs}=-{ g }_{ e }e^{ -\cfrac { Orbs }{ { r }_{ e } } }+{ g }_{ e }e^{ (\cfrac { x-Orbs }{ { r }_{ e } }) }, {F}_{ob}( x = Orbs ) = {g}_{e}\)
A force develops that opposes an increase in Orbs. It acts on the system and is shared by both the Sun and Earth. It is proposed that \({F}_{ob}\) is divided among the two bodies in proportion to their masses.
\({F}_{e} + {F}_{s} = {F}_{ob}\)
\({F}_{e}=\cfrac{{M}_{e}}{{M}_{e}+{M}_{s}}{F}_{ob}\), and
\({F}_{s}=\cfrac{{M}_{s}}{{M}_{e}+{M}_{s}}{F}_{ob}\)
and acts as a centripetal force keeping Orbs constant.
\({F}_{e}\)=5.97219*10^24/(1.98855*10^30+5.97219*10^24)*0.00980665 = 0.0000000295 kms-2
It is this force that provides the centripetal force, ie
\(\cfrac{{V}^{2}_{o}}{{O}_{e}}={F}_{e}\) and so Earth's Orbital speed is
\({V}_{e}=\sqrt{{O}_{e}{F}_{e}}\) = (149597870*0.0000000295)^(0.5) = 2.100 kms-1
This compares very badly with the measured average speed of 29.78 kms-1
The previous theoretical answer based on \(\triangle GPE\) was \({V}_{e}\) = 13.69 kms-1 which was proven consistent once. By logic, bravery and an inflated ego... if the Sun mass is estimated wrongly, by calculating backwards assumming that 13.69 is the consistent answer we have a new estimated Sun Mass
\({M}_{s}\) = 9.942779860*10^(27) kg
And that's how the Sun slim downed.
