If the companion acted like a planet orbiting our sun, and the orbit periodicity was close to the precession periodicity, then standard calculations would put our binary counterpart somewhere between 848.5 AU and 1515 AU depending on its mass and eccentricity. For detailed calculations under this scenario, please download the PDF Document below.
Size/Distance Calculations PDF
However, there are other possibilities, including the possibility that the solar system itself is moving much faster than any of the planets and therefore our companion star may be a nearby visible star.
While the solar system speed is difficult to measure (the question is always “compared to what?”), astrophysicist Reg Cahill of Australia has suggested that the solar system is moving in excess of 430km/s, relative to the cosmic microwave background (CMB), and this opens the possibility that our solar system might be orbiting anyone of a number of local stars, but more than likely one of the larger masses that lies not too far inclined to the plane of the solar system. Such a scenario might seem improbable given our current understanding of gravity and visible star distances, however there are compelling theories, such as MOND theory, and there is unusual evidentiary information, such as the data from Voyager 1 and 2 or the anomalous acceleration of the Pioneer 10 and 11 spacecraft that make this scenario attractive to investigation.
In addition to the odd behavior of distant satellites the most distant dwarf planets also display the signature of a potential companion star in their strange orbits. Mike Brown, Professor of Planetary Astronomy at Caltech, and the discoverer of Sedna, said simply:
“Sedna shouldn't be there. ...There's no way to put Sedna where it is. It never comes close enough to be affected by the Sun, but it never goes far enough away from the Sun to be affected by other stars.” (Space.com, “Sun’s Nemesis…” 3-11-10)
Other astronomers, including John Matese, Emeritus Professor of Physics at the University of Louisiana at Lafayette and physicist Daniel P. Whitmire, of the same institution, and Richard Muller of the University of California Berkeley, have come to similar conclusions (based mostly on comet data), albeit without correlating orbit periodicity to the precession observable. Given such research we urge the scientific community to keep an open mind on possible forces that might be affecting the sun’s motion, and in turn the earth’s orientation.