The physics
Two stars pull each other with equal and opposite forces. So both stars move. No star sits still at the centre.
They circle their centre of mass. The heavy star sits nearer it, on a smaller circle.
m₁ r₁ = m₂ r₂ r₁ + r₂ = d
The pull acts across the full distance d between them. Both stars go round in the same time, so they share one ω.
G m₁ m₂ / d² = m₁ ω² r₁ ω² = G (m₁ + m₂) / d³
In years, AU and Suns, the lap time of a pair is simple. The Earth and the Sun set the units.
T² = d³ / (M₁ + M₂)
Equal stars on a ring can turn as one. Each star feels a pull from every other star. Add only the parts toward the centre. Set the total equal to mv²/R.
The Earth circles the Sun at 30 km/s, 1 AU away. So the Sun has GM = 900 (km/s)²·AU.