3C 66B · AN INTERACTIVE GEOMETRY STUDY
The orbit behind the ellipses.
Two black holes. A moving jet. Two radio cores.
Follow the three-dimensional motion all the way to the sky.
Based on Sudou et al. (2003)
Figure 3 → Figures 1 & 2
Candidate binary · illustrative reconstruction
Figure 3 → Figures 1 & 2
Candidate binary · illustrative reconstruction
Observed mean period1.05 ± 0.03 yr
8.4 GHz ellipse scale45 ± 4 μas
2.3 GHz ellipse scale243 ± 30 μas
Paper’s adopted distance85 Mpc
01 / THE SOURCE
ANALYTICAL MODELBinary + jet, in three dimensions
Drag to rotate · scroll to zoom · arrow keys also rotate
● 8.4 GHz core◆ 2.3 GHz core● Jet-emitting BH, m● Companion, M
02 / WHAT THE TELESCOPE SEES
Project the motion. Draw the ellipse.
Each colored point is the same radio core in the 3D scene and the charts. Centers are removed to compare the motions; the two sky plots use independent scales.
2.3 GHz farther along the jet
SKY PLANERight ascension Δα · μas
Declination Δδ · μas
Solid lines: this analytical model. Dashed ellipses: Table 1 geometry, centered independently. Hollow dots: six evenly spaced model phases, not the six measured epochs. Time traces use an arbitrary zero phase; no VLBI data points are fabricated.
03 / FROM GEOMETRY TO ASTROMETRY
What is actually moving?
Orbit the central engine
The less massive black hole follows a circular orbit about the barycenter. The companion moves in the opposite direction, with radius q times smaller. Their sizes are symbolic.
Sample the radio jet
The radio core is an approximate optical-depth-one (τ ≈ 1) emission region. The 8.4 GHz core lies nearer the engine; the 2.3 GHz core lies farther out. These are emission centroids, not individual plasma particles.
Look along the sightline
Remove the line-of-sight coordinate. A sinusoidal 3D trajectory projects to an ellipse on the sky. Plotting its east and north components against time gives the two sinusoidal traces.
The analytical model and its limits
Orbital motion
rₘ(φ) = R [u cos φ + v sin φ]
rᴹ(φ) = −q rₘ(φ)
φ = 2π tobs / Pobs
u and v span the orbit; i is the angle between its angular momentum and the line of sight. The node angle Ω is measured east of north. We use Psrc = Pobs / (1 + z) for the orbital speed and Newtonian Kepler mass.
A first-order jet response
δj(φ) = [vₘ − (vₘ · j₀)j₀] / (βc)
rν(φ) = rₘ(φ) + sν[j₀ + δj(φ)]
The mean jet direction j₀ may be misaligned with the orbital angular momentum. Adding the orbital velocity to the jet produces a transverse wobble, amplified with core distance sν. This is a small-angle, instantaneous kinematic illustration of the mechanism in Figure 3, not an equation supplied or fitted by Sudou et al.
Projection onto the sky
Δα = (rν − ⟨rν⟩) · E / D
Δδ = (rν − ⟨rν⟩) · N / D
E points east, N points north, and n points toward the observer. Angles are converted from radians to μas. Astronomy convention is used: east is left in the sky plots. Here Δα is the eastward tangent-plane angular offset (often written Δα cos δ). Rotating the inspection camera never changes the physical viewing angle.
The model is a linear combination A cos φ + B sin φ, so its projected paths are ellipses. Their semiaxes are the singular values of the 2 × 2 sky coefficient matrix.
Interpretation matters
The model omits propagation delays, relativistic velocity addition, radiative transfer, lensing, inspiral, and differential light-travel time. It does not uniquely reproduce both fitted ellipses. Changing the sliders explores the proposed mechanism; it does not infer the binary’s true parameters. The motion selector removes terms from the radio-core trajectory; the black-hole orbits remain visible as context.
Published values · Table 1 and Figure 3
| Quantity | 2.3 GHz | 8.4 GHz |
|---|---|---|
| “Major axis” / plotted semiamplitude | 243 ± 30 μas | 45 ± 4 μas |
| Axial ratio, b/a | 0.31 ± 0.17 | 0.24 ± 0.14 |
| Period | 1.10 ± 0.06 yr | 1.02 ± 0.04 yr |
| Position angle, east of north | 113 ± 9° | 89 ± 6° |
| Offset angle (paper’s convention) | 60 ± 7° | 101 ± 5° |
| RA center | 1.441 ± 0.020 mas | 0.970 ± 0.002 mas |
| Dec center | −0.888 ± 0.033 mas | −1.861 ± 0.004 mas |
| Angle from mean jet PA ≈ 50° | 63 ± 9° | 39 ± 6° |
We use 243 and 45 μas as semiamplitudes, matching the coordinate extents in Figures 1–2. The paper labels these “major axis”; the full displayed diameters are twice these values. Centers refer to the paper’s relative astrometry; they are not used to claim an absolute black-hole location or a measured distance along the jet.
Other paper values: z = 0.0215; D ≈ 85 Mpc for H₀ = 75 km s⁻¹ Mpc⁻¹; mean P = 1.05 ± 0.03 yr. Its interpretation gives R ≲ 5.4 × 10¹⁶ cm and rmax ≈ 5.4(1 + q) × 10¹⁶ cm. These are historical model-dependent upper bounds, not measured separations. Masses displayed above are recalculated from the illustrative orbit, not set to the paper’s upper-bound masses.