A binary star to the SUN ?!

A less likely possibility.

An upper limit of distance between two binary stars is around o.25 light years. Considering that and the fact that we are able to observe stars and celestial objects with apparent magnitudes up to 32, we get using the expression…

(where m = apparent magnitude i.e., 32 in this case, M = absolute magnitude, r is distance in parsecs )

M ≈ 42.57

This value is so high ( meaning that the star is too dim ). In fact the dimmest red dwarfs ever observed, are only about 15 - 20 in absolute magnitude. If we still want to theoretically prove it wrong, we can go ahead and use the Stefan- Boltzmann Law to make an attempt to find its surface temperature.

Assuming a Sun sized binary star, with an area of 6.09 * 10 ^18 m^2, plugging in some values we get T, the surface temperature of the binary star to be a mere 1.02 Kelvin.

If we try to make the size of the binary star even smaller, ( in an attempt to increase the surface temperature) by pushing it to the smallest size of a star ever, which has a radius around 0.12 times the radius of the Sun, we still get an upper limit of 1.66 Kelvin.

So this case can dismissed by the trivial argument that a star can never be so cold as close to 1 kelvin. This temperature that we just calculated, is in fact, lesser than the average background temperature of the observable Universe around us which is about 2.73 Kelvin.

Hence, by my calculations, it is almost impossible that the Sun has a binary star of any size orbiting anywhere close to it to feel its gravity significantly, of we were to believe that we know the extremities of star sizes, temperatures, and luminosity.