Hour angle 0 h Elevation 0 ° Path diff. w 0 m = 0 λ uv samples 0 FPS 0
Drag to orbit, wheel or pinch to zoom. Yellow bar: extra path the wavefront travels to reach the later dish (w = b·ŝ); the dish separation is drawn compressed (not to scale). Pale planes: incoming wavefronts from the radio source.
Radio Interferometer: Two-Dish Fringes, uv Coverage and Earth-Rotation Aperture Synthesis
Two dishes make fringes cos(2πw/λ). As the Earth turns, each baseline sweeps an ellipse in the uv plane; more dishes and more hours fill it in and the dirty image sharpens toward the true sky.
Pause Restart observation Observe all 12 h now Quality: High
View: array View: dish close-up View: from above
Array Layout
Sky Source model
Readouts
Correlator output, first baseline (±1 min of sky time) Delay is not tracked here, so the raw fringe r = cos(2πw/λ) rotates through many cycles per minute as w changes.
Fringe pattern on the sky (first baseline, now) Projected-baseline fringes cos(2π q θ) under the dish primary beam; dotted verticals mark sources. Fringe spacing = λ/|(u,v)|.
uv coverage (kλ) Show true sources
Experiment: resolution vs baseline length
Four full 12 h syntheses of the current layout with the longest baseline set to 250 m … 2 km. Measured = area-equivalent FWHM of the dirty beam computed by summing the (u,v) samples; predicted = λ/Bmax . The prediction is a scale, not an exact width: the ratio depends on the uv fill (an elliptical track is narrower than λ/Bmax along its long axis, a one-dimensional east-west line leaves a long ridge), but the ratio must stay constant when B changes, because the beam scales as 1/B.
Run resolution sweep
Independent numerical checks
Controls Layout and longest-baseline sliders set the antenna positions; frequency sets λ = c/ν; declination sets which arc of the sky the source follows. The clock runs the Earth's rotation at the chosen speed (sky hours per real second). Every 6 sky minutes each baseline adds one uv sample (and its conjugate) to the dirty image. Below the horizon nothing is recorded. “Observe all 12 h now” fills the whole −6 h … +6 h range at once; changing any physical parameter restarts the observation. The science
Baseline and delay. For antennas separated by a vector b and a source in direction ŝ, the wavefront reaches them with a geometric path difference w = b·ŝ. A correlator multiplying the two voltages outputs r = S cos(2πw/λ) (with a point source at the phase centre and no delay tracking). In the equatorial frame (X,Y,Z) = (−bN sinφ + bU cosφ, bE , bN cosφ + bU sinφ):
u = X sinH + Y cosH, v = −X sinδ cosH + Y sinδ sinH + Z cosδ, w = X cosδ cosH − Y cosδ sinH + Z sinδ.
Visibility. With delay and phase tracking each baseline measures one Fourier component of the sky: V(u,v) = ∫ I(l,m) e−2πi(ul+vm) dl dm (van Cittert–Zernike). A point source at (l₀,m₀) gives a pure phase ramp; a uniform disk of angular diameter θ gives |V| = |2J₁(x)/x| with x = πθ|uv|/λ, so a resolved source loses amplitude on long baselines.
Dirty image. ID (l,m) = Σk Re[Vk e+2πi(uk l+vk m) ] over the sampled points and their conjugates (−u,−v). It equals the true sky convolved with the dirty beam (the same sum with V = 1). The beam width is about λ/Bmax ; gaps in uv coverage produce sidelobes.
Simplifications. Flat-sky small-field approximation, no w-term (non-coplanar) correction, no primary-beam attenuation in the image, no noise, no atmosphere or ionosphere, no CLEAN deconvolution (the dirty image is deliberately shown), identical dishes of 25 m diameter. The site latitude 34.08° and the layouts are illustrative, not a real array's coordinates. The sky models are invented examples.