Pull Here, Push There

We know that ${F}_{ob}$ acts on the system of total mass ${M}_{e} + {M}_{s}$, the force per unit mass is

${Fp}_{ob} = \frac{1}{{M}_{e} + {M}_{s}}{F}_{ob}$,

Let the forces on the masses be,

${F}_{e}$,  ${F}_{m}$

These forces act on both ends of the system keeping Orbs constant, the force per unit mass must be equal for the acceleration to be equal (and opposite).

$\frac{{F}_{e}} {{M}_{e}} = \frac{{F}_{m}}{ {M}_{m}}$

and are equal to ${Fp}_{ob}$; because that is the force per unit mass restraining the system as a whole. Therefore,

$\frac{{F}_{e}} {{M}_{e}} = \frac{1}{{M}_{e} + {M}_{s}}{F}_{ob}$

And so,

${F}_{e}= \frac{ {M}_{e}}{{M}_{e} + {M}_{s}}{F}_{ob}$

${F}_{m}= \frac{ {M}_{m}}{{M}_{e} + {M}_{s}}{F}_{ob}$

The direction of these forces are such that they oppose the increase in Orbs, the orbital distance, acting on the masses at both ends of the system.