On Saturday, February 20, 2016 at 6:30:02 PM UTC+8, Poutnik wrote:
I performed analysis of circular orbit case of twin binary stars.

I have realized the semi major axis in context of the 3rd Kepler law
must be surprisingly considered wrt the other star
and not wrt their barycentre, as I expected.

Otherwise formula of the 3rd Kepler law
does not match circular case requirement
of equality of gravitational and centripetal acceleration.

3rd Kepler law formula is usually presented as

G.(M1+M2) = 4.pi^2 . a^3/T^2

G is gravitational constant
M1,M2 object masses
a is semi major axis of orbit
T is orbit period.

For units of mass in Solar masses Ms,
distance in AU and time in years,
G/(4.pi^2 ) is approximately equal to 1.

1 + m/Ms = (AUs)^3/(years)^2

Let imagine our Solar system contains
twin binary stars of same masses 1 Ms,
on circular orbit with distance 2 AU,
with the radius of orbits 1 AU.

The gravitational acceleration
caused by the other star is 1/4 of that acting on the Earth,
as Sun stars have doubled distance 2 AU.

But the orbit radius is the same as for Earth 1AU,
so does centripetal acceleration for the same period 1 year.

Therefore period must be doubled,
as centripetal acceleration is proportional to 1/T^2.

It leads to Kepler 3rd law equation in Ms, AU, year units

2 = 1^3/2^2 = 1/4
for case we take as semi major axis the circle radius as usually.
This is obviously wrong.

But if we take as coordination centre not the barycentre,
but the centre of the other star,
then a = 2R = 2AU.

Then 2 = 2^3/2^2 = 2 and equality is reached.

So for comparable objects it looks like
G.(M1+M2) = 4.pi^2 . D^3/T^2
where D is maximum object distance.

--
Poutnik ( the Czech word for a wanderer )

Knowledge makes great men humble, but small men arrogant.

I posted on 17/2/16 in sci.physics:
'Science has not grasped the significance of Kepler's third'.

Sam Wormley replied:
"Kepler's third law (from the early 1600s) gave us the relative sizes of the orbits, T^2 ~ a^3.

Isaac Newton's version of Kepler's third law T^2 = (2π)^2 a^3 / G(M+m)
where (2π)^2/G is just a constant of proportionality.

Orbits of earth and solar system satellites today are described by their Keplerian Elements.

https://en.wikipedia.org/wiki/Orbital_elements

Science not only grasped the significance of Kepler's third law,
but makes use of it 24/7".

Both Sam and Herr Spitzohr make sense, you do not.