Telescope Shootout Lab (2D): Aberrations, Focal Ratio, Resolution Limit and Eyepiece Magnification

Presets

Live readouts

Graphs

Intensity profile through the double star: ideal optics against this telescope and sky

Resolution limits against aperture (log scale), with the seeing floor

Strehl ratio against focal ratio for each aberration (0.8 = diffraction limited)

Numerical checks

running...

How to use

Pick a preset, or change the optical design, aperture and focal ratio. Try: (1) 100 mm f/9 achromat then raise N: the colour fringe ratio falls; (2) switch to ED apochromat; (3) Newtonian with a spherical mirror: the Airy disc dissolves, bring N above 8 to recover; (4) drag the off-axis angle on an f/4 Newtonian to turn stars into comets; (5) set the separation near the Dawes limit and watch the dip in the profile graph and the verdict; (6) add seeing until a 200 mm telescope splits no better than a 100 mm one; (7) change the eyepiece and see the field of view, exit pupil and limiting magnitude change. On the 3D page use the camera buttons, the guided tour and the three telescope models on the mount.

The science

What this lab shows. What limits a telescope: diffraction at the aperture, aberrations of the optics, turbulence in the air and the eyepiece that feeds the eye. Four designs share one model: an achromatic refractor, an extra-low-dispersion (ED) apochromat, a Newtonian reflector and a Schmidt-Cassegrain. The target is a double star.

1. Focal ratio, magnification, field, exit pupil.

N = F / D, M = F / f_e, true field = AFOV / M, exit pupil = D / M = f_e / N

A telescope is an afocal pair of lenses (or mirror and lens). Ray-transfer matrices give an angular magnification F/f_e and an exit-pupil diameter D f_e/F; both are verified in the page. If the exit pupil is larger than the eye pupil the outer part of the aperture is wasted; below about 0.5 mm the image dims and floaters show. Useful power runs from D/7 (exit pupil 7 mm) to about 2D (D in mm). The visual limiting magnitude is modelled as 2.7 + 5 log₁₀(D_eff) with a throughput correction, a rough empirical rule, not a measurement.

2. Diffraction: the Airy disc. A circular aperture of diameter D forms the pattern I(x) = [2 J₁(x)/x]², with x = π D θ / λ. The first dark ring lies at 1.2197 λ/D (138/D arcsec at 550 nm, D in mm) and holds 83.8 % of the light.

Rayleigh: θ = 1.22 λ/D, Sparrow: 0.947 λ/D, Dawes (empirical): 116/D arcsec

Two equal stars at the Rayleigh distance leave a 26.5 % dip between them (middle/peak = 0.735); at Sparrow's distance the dip just vanishes. Dawes' value comes from visual observations and corresponds to a dip of a few percent. All three are computed numerically in the checks.

3. Central obstruction. A secondary mirror of diameter εD gives the amplitude [2J₁(x)/x - ε² 2J₁(εx)/(εx)] / (1 - ε²). The core gets slightly narrower, the first ring brighter, and the light grasp falls with 1 - ε². The check compares this closed form with a direct numerical integral over the annular pupil.

4. Aberrations (Seidel theory). A wavefront error with rms σ (in waves) lowers the peak of the image to the Strehl ratio S ≈ exp[-(2πσ)²] (Maréchal; S = 0.8 is the diffraction-limited threshold). This page uses a diffraction core weighted by S plus a Gaussian halo that holds the remaining light, a deliberate simplification of the true point-spread function.

Chromatic aberration. A single glass lens focuses blue and red at distances that differ by F/V (V is the Abbe number; the check recomputes it for BK7 from the Sellmeier dispersion formula). An achromatic doublet brings two colours to one focus and leaves a secondary spectrum of about F/2200 (typical), an ED apochromat about F/15000 (typical, depends on the glass). Whether the colour fringes show depends on the focal ratio, because the tolerable focus error is 4 λ N², so long focal ratios hide chromatic aberration. This is why f/15 achromats were long.

Spherical aberration. A spherical mirror focuses the edge light closer than the centre: longitudinal error D/(32 N), checked against an exact ray trace. The edge wavefront error is D/(512 N³) (about D/(2048 N³) peak-to-valley after refocusing), so a sphere is acceptable only for long focal ratios (roughly N > 8 for 150 mm). A paraboloid removes it exactly for an object at infinity.

Coma. Off axis, a paraboloid gives comet-shaped images with tangential length 3θ/(16 N²) (angular); the wave coefficient is θD/(32 N²), rms = coefficient / √72. The check traces a tilted bundle through a paraboloid exactly and compares. The Schmidt-Cassegrain is modelled with the coma of a paraboloid of the same effective focal ratio, the refractors with 10 % of it: simplifications.

Defocus. The focus error is entered in units of the quarter-wave tolerance 2 λ N²; its wavefront rms is |Δz| / (8 N² λ) / (2√3).

5. Seeing. Turbulence smears every star with a Gaussian of full width s (seeing). The Fried parameter r₀ = 0.98 λ/s is the aperture over which the wavefront stays coherent; when D is larger than r₀, extra aperture gathers more light but does not sharpen the image. The pair profile and the 5 % split limit are computed with the seeing-convolved core.

Checks (run in the page). Bessel J₁ (series and asymptotic) against the defining integral; first Airy zero; energy inside the first ring; Rayleigh and Sparrow separations; obstructed aperture against pupil integration; BK7 chromatic spread against F/V; spherical-mirror ray trace; parabola coma ray trace; ray-transfer matrices for magnification and exit pupil.

Simplifications. No thermal effects, no diagonal-mirror diffraction spikes, no atmospheric dispersion, no eye aberrations; the SCT corrector is treated as ideal; star colours are fixed (warm and cool white); limiting magnitude and Dawes limit are empirical rules.