Off hand, I don't know the answer to this. It would depend on where the star is located.
If it is in a dense star cluster where the mean stellar separations are a light year or less, then this distance would have to be, perhaps, 10 billion miles ( 100 AU) or less otherwise the stars would be so loosely bound together that a passing star could disrupt the binary system. In the vicinity of the Sun, I think the largest separations between stars in identifiable binary systems runs to something like 100 billion miles or 1000 Astronomical Units; some 25 times the distance to Pluto. In terms of observing binary stars with separations more than a few billion miles, they move so little in their orbits in a few decades that they can be easily overlooked. I believe many of the most common binary stars have orbits less than 50 years. There are about 70,000 binary stars known with several such as Castor with periods of 420 years, and 61 Cygni with a period of 720 years. The ones with the longest periods are also the closest so that their motion can be detectable. Here is a short list of some of the widest binaries I have come across:
Name....................Separation.........Period........Distance +43 1326 44 156 AU 3000 11.5 Delta Geminorum 6.8 111 AU 1200 53 +50 1725 19 86 AU 1000 14.8 61 Cygni 25 85 AU 700 11.0 Eta Cass 11.9" 70 AU 478 yrs 19 LY Castor 5.8" 81 AU 349 45 40 Eridani BC 6.8 33 AU 247 15.8 Gamma Virginis 3.7 42 AU 171 36 Xi Bootes 4.9 32 AU 149 21 .................................................................
A recent search for wide binaries in the Gleise Catalog of nearby stars by Arcadio Poveda at the Institute of Astronomy in Mexico revealed that of the 118 nearby binary stars, 33 had separations greater than 500 AU, and 14 had separations greater than 2000 AU.
Copyright 1997 Dr. Sten OdenwaldReturn to Ask the Astronomer.