Interferometric baseline ($\mathbf{b},{\mathbf{b}}_{\perp }/\lambda$)

An interferometric baseline is the displacement $\mathbf{b}={\mathbf{r}}_2-{\mathbf{r}}_1$ between two antennas. For a unit line of sight ${\mathbf{s}}_0$ toward the phase centre, its projection onto the plane of the sky isHere $c$ is the speed of light. The signed delay depends on which receiver is delayed; the geometric path difference's magnitude does not. If ${\mathbf{e}}_{\theta }$ is a unit tangent in the direction being imaged, the dimensionless coordinate $u=\mathbf{b}\cdot {\mathbf{e}}_{\theta }/\lambda$ is a spatial frequency, measured in cycles per radian of small sky offset. A single pair gives fringes of angular period $1/|u|$ along that tangent. This period specifies one measured oscillation; it does not specify a whole image's angular resolution or eliminate ambiguous source arrangements.

The left panel separates the component producing the reference delay from the component measuring angular structure. The right panel shows a possible Earth-rotation aperture synthesis track, with its opposite complex conjugate track. Those opposite samples provide the symmetry required by a real brightness distribution; they are not independent extra observations.

Earth-rotation aperture synthesis uses Earth's changing orientation to vary a fixed ground baseline's projection onto the plane of the sky. In two dimensions, $u$ and $v$ are the components of ${\mathbf{b}}_{\perp }/\lambda$ along two perpendicular sky tangents. Tracking one field traces a curve in this $(u,v)$ plane; changing the physical baseline supplies other curves. A track samples Fourier transform coordinates, rather than the brightness along a track across the sky.

For a horizontal east–west baseline of length $B$, choose the sky-axis signs so thatwhere $H$ is hour angle and $\delta$ is declination. The ellipse follows by projecting the rotating baseline onto sky tangents. At a celestial pole it is a circle; near the celestial equator its minor axis collapses. Thus the same ground layout constrains different sky directions unequally. Rotation improves coverage, but does not fill every spatial frequency, and synthesis across many hours assumes that the source changes negligibly during that interval.