Segmented Radio Telescope Dish (2D)

What the model computes

A circular aperture of diameter D collects a plane radio wave of wavelength λ = c / f. The paraboloid z = r² / (4F) sends every ray to the focus with the same path length, so a perfect dish adds no phase error (check 3). The segments are 90 panels in 5 rings; the drawing is the 20 m reference design.

Equations

• Far field. With illumination E(ρ) = c + (1 − c)(1 − ρ²)² (c = 10−T/20 for a feed edge taper of T dB) the pattern is the Hankel transform F(u) = ∫ E(ρ) J0(uρ) ρ dρ / ∫ E ρ dρ, u = π D sinθ / λ, power F². Uniform illumination gives the Airy pattern: first null at 1.22 λ/D, FWHM 1.03 λ/D. A taper widens the beam and lowers the sidelobes.
• Gain. G = η (π D / λ)², effective area Aeff = η π D² / 4, with η = ηtaper ηspill ηRuze ηblock ηmisc. ηtaper = (∫E dA)² / (A ∫E² dA); ηspill is the fraction of a Gaussian feed pattern (edge taper T at the rim angle for F/D = 0.4) that lands on the dish. Blockage 0.95 and ohmic/phase losses 0.97 are fixed typical values.
• Surface errors (Ruze 1966). An rms surface error ε normal to the surface changes the path by about twice that, so ηRuze = exp[−(4πε/λ)²]. The rule of thumb ε = λ/16 costs about 2.7 dB. The lost power goes into a broad error pattern whose width is set by the panel correlation length (taken as 2 m here); the dashed floor in the first graph is its first ten terms.
• Gravity. The reflector is rigged at 50° elevation. Away from it the sag grows as εg = gs |cos(el) − cos 50°| and adds in quadrature: ε² = εpanels² + εg². This is a one-parameter caricature of homologous deformation, not a finite-element result.
• Pointing. An offset of p beamwidths costs the tabulated pattern value at that angle (about 3 dB at p = 0.5).
• Sensitivity. SEFD = 2 k Tsys / Aeff; total-power noise for two polarisations σS = SEFD / √(2 Δν τ) (1 Jy = 10⁻²⁶ W m⁻² Hz⁻¹). Time to a 5σ detection of a 100 mJy source follows from the same formula.

Checks (run in this page)

1. Parseval: the far-field power integral equals the aperture power integral.
2. Uniform aperture reproduces the Airy numbers 1.2197 and 1.0290 λ/D.
3. Equal path length wavefront → parabola → focus.
4. A brute-force coherent sum over 90 panels with random errors reproduces the Ruze factor plus its finite-panel correction.

Honest simplifications

Prime-focus feed with an ideal Gaussian pattern; no subreflector, no quadripod scattering, no atmosphere, no gain compression or baseline ripple, and no sky noise (Tsys is a number you set). Panel errors are independent pistons (no tip/tilt, no thermal gradients). The 3D picture exaggerates the panel offsets (slider) and the gravity term is a caricature. This is an illustrative generic design and not a copy of any real observatory. A real large dish is also limited by wind, temperature and pointing models, which are not simulated here.

What to try

• Choose the preset "Perfect surface" and read the FWHM (about 1.17 λ/D with a 10 dB taper); then set the taper to 0 dB and look at the sidelobes in the first graph.
• Raise the frequency at 0.8 mm panel error: the gain curve bends away from the ideal line near λ = 16 ε (about 23 GHz).
• Set gravity sag to 6 mm and sweep the elevation: the gain peaks at the 50° rigging angle.
• Re-phase the segments and watch the wave crests line up at the feed.