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Here's the latest "dot theory". These three dots are Barbarossa:
A2. POSS-I (Red) plate date 1954.154, geocentric position RA 11 02 25.16 Decl -5 56 11.3
C3. SERC (Red) plate 1987.08215, RA 11 18 37.6 Decl -7 54 09.5
D. SERC-I (Optical Infrared) plate 1997.16711, RA 11 22 16.77 Decl -8 29 30.9.
These three dots are Frey:
A. POSS-I (Red) plate date 1954.154, geocentric position RA 11 03 12.4 Decl -5 58 09
C. SERC (Red) plate 1987.08215, RA 11 18 03.18 Decl -7 58 46.1
D2. SERC-I (Optical Infrared) plate 1997.16711, RA 11 22 32.9 Decl -8 26 56.
(Some of the other dots are Freya. There are some disappearing dots on the 1986 and 1995 plates which could be these bodies too.)
I assumed that A2 & D are Barbarossa, then making my most accurate correction for Earth parallax, interpolated the expected position for Barbarossa on the 1987 plate. Both C and C3 are a small distance away from that position.
Then I drew lines between C3 & C, A2 & A, D & D2. If these are Barbarossa & Frey in mutual orbit, the center of mass should be displaced at a constant rate. This is best seen by graphing all six bodies on the same sheet, each body relative to the presumed Barbarossa of the pair for its epoch. Generally there will be one mass ratio which makes the centers of mass collinear.
However, when the centers of mass became collinear, they also assumed the correct distance ratio, i.e., constant speed, to within 2% accuracy. (I refer to the residual small speed remaining after the speed from A2 to D is deducted.) This is a very precise and unlikely result. The implied period for circular orbit around the sun was 2847 yr (vs 2688 yr for the J:S resonance progression). Furthermore the mass ratio which gave this precisely constant-velocity center of mass, was 1:1. The conditioning of the graphical solution was such that a 1.2 :1 ratio either way might occur, but certainly not 1.5 :1. Alpha Centauri A & B are said to have a 1.2 :1 ratio, as do Earth & Venus.
The mutual orbit cannot be perfectly circular, because no ellipse centered on the center of mass, fit the points. Slight displacement of the ellipse center (if a noncircular orbital ellipse is tilted, the center of mass generally is not even a focus) allows an infinitude of ellipses. I chose one such that was especially easy to calculate, and found constant angular speed between A, C & D, within 10%; distance between Barbarossa & Frey, 0.7 AU; inclination 18 deg; tilt to Barbarossa's solar orbit, 25 deg; tilt to orbital plane, 30.5 deg; combined mass of Barbarossa & Frey, 0.0036 solar masses.
The trajectory of the presumed center of mass of Barbarossa & Frey, is so constant that Freya likely would have to be of much smaller mass than Frey, or much more distant from Barbarossa. Alternatively, let 1954 be t=0, 1997 be t=1. The midpoint of the interval A-C then is t=3/8 and the midpoint of C-D is t=7/8. If Freya were at conjunction (near our line of sight to the center of mass of Barbarossa & Frey) at t=5/8, then to a first-order approximation the acceleration due to Freya would be zero in the plane of the celestial sphere.
In this model, Barbarossa & Frey are always within 15 arcminutes of the more recent of my various predicted positions. So, the best I have to offer now, is to keep looking within 15' of those coordinates.
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