Y'all are better at this than I am, but it is much easier than I thought. [And I have been limited to an iphone lately.]
Originally Posted by grapes
F = mA
A is acceleration; we will use “a” for the semi-major axis of the system later
1) s1 =1/2 * A1 * t2
2) A1 = 2*s1/ t2 = 2*r1/ t2…. Because r1 = s1
F1 = F2….. from the inverse square law (i.e. gravity)
From, F = mA
m1*A1 = m2*A2
Substituting from eq. 2…
m1*(2*r1/ t2) = m2*(2*r2/ t2)
3) m1*r1 = m2 * r2……. This is the seesaw equation; summation of moments = 0 stuff.
To get it to the common barycenter equation….
4) a = r1 + r2………… where “a” is the semi-major axis.
From eq. 3…
r2 = m1*r1/m2
a = r1 + r2 = r1+ (m1*r1/m2)
a = r1*(1 + m1/m2)
r1 = a/(1+m1/m2)
As Hornblower has stated, no orbital dynamics are needed to give us a c.g. or barycenter, though as a layperson I suggest the use of "barycenter" be associated with orbital masses. [When I was very little, I still recall seeing a picture (perhaps my first) of an airplane in the air and it was just.... resting there in the air. I don't see two masses in space just sitting there worth determining their barycenter; if they aren't going to fly, let 'em fall.]
The seesaw analogy is helpful but it assigns the force, F, perpendicular to the board (moment arm), whereas here we use F (from gravity) aligned radially with the "board". [The perpendicular F for seesaws is really the same because the F used there is radially with the c.g. of the Earth (gravity). We will ignore the case for a vertical seesaw if for safety only. ]
I had thought that something more than F=ma, presented by Grant, would be needed because I couldn't get the orbital motions out of my head. But they aren't needed to derive the barycenter, after all.
Last edited by George; 2017-Mar-26 at 06:43 PM.
We know time flies, we just can't see its wings.