How do we know the Sun was not at one time a member of a binary system?

---

A very good question!

If we can assume that the secondary companion is a star, its mass must be at least 0.04 times the sun---or it wouldn't be a star in the first place. The next issue is, how close can this body be to the sun...or could it have been to the sun, without having disturbed the orbits of the planets? Most binary stars have separations within 100 AU, or two and a half times the distance to Pluto. Any companion star of the sun that close would have gravitationally ejected the outer planets, leaving perhaps only Jupiter ( 0.5 billion miles or 5.4 AU) and the inner planets remaining. The orbital planes of the planets are so similar, that the hypothetical companion star must have also orbited the sun very near the orbital plane of the solar system, or it would have produced tilted orbit planets among the outer planets. The orbital eccentricities of the planets are also so small that the companion star must have had a negligeable tidal effect, which again means that the hypothetical companion can not have been closer than the average 100 AU scale of typical binaries.

Without a detailed calculation, it is hard to make specific estimates, but given the shape and scale of our solar system today, it is unlikely there ever was a companion star of any significance, within the boundaries of the gaseous disk out of which the primitive solar system formed. We were just lucky I guess!