How small can you see? — diffraction & interferometry at interplanetary distances

A diffraction-limited telescope of diameter D viewing at wavelength λ from a distance L can separate features no smaller than θ·L, where θ = k·λ/D. An array of N telescopes spaced s apart acts — for resolution only — like a single mirror of diameter equal to the longest baseline, B = (N−1)·s. Everything below is reported in metres, not radians.

Diffraction-limited system (Wikipedia) Astronomical interferometer (Wikipedia) Rayleigh criterion Angular resolution

Target

1.22: circular aperture, sinθ = 1.22 λ/D 1.00: λ/D, uniform/simple aperture

Single telescope

Interferometer

Single telescope — smallest resolvable object—
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Working

Resolution vs distance

single telescope
Log–log: resolvable size grows linearly with distance, so doubling the range doubles the detail needed. Reference objects are drawn as horizontal bars to show where a given object becomes resolvable.

The physics

θ = k · λ / D   [radians]

The diffraction-limited angular resolution of a perfect circular aperture is proportional to wavelength and inversely proportional to the entrance-aperture diameter. Under the Rayleigh criterion (k = 1.22) two point sources are just resolved when the maximum of one Airy pattern falls on the first minimum of the other, which occurs at sin θ = 1.22 λ / D. The corresponding image detail is the Airy disk, whose full width to the first null is 2.44 λ/D. The Abbe form d = λ / (2·n·sinθ) = λ / (2·NA) is the microscope convention and is why "diffraction limit" factors of 0.5–1.22 all appear in the literature.

s_min = θ · L = k · λ · L / D   [metres]

To convert an angle into a length at distance L, use the small-angle relation s = L·tanθ ≈ L·θ. This is what gives the object size in metres on this page. Note the inverse-square-feeling intuition break: because θ is fixed by the optics, the resolvable size at Jupiter is simply the Earth–Jupiter distance times the same angle, i.e. it scales linearly with distance, not quadratically.

B = (N − 1) · s   →   θ_if = k · λ / B,   s_min,if = θ_if · L

For N telescopes on an equidistant line with spacing s, the longest (maximum) projected baseline is the end-to-end span B = (N−1)s. Fringe spacing of a two-element interferometer is λ/B, so the array resolves the same angles as one ideal mirror of diameter B — the classic statement that an interferometer has the resolution of a telescope the size of the separation between its apertures. Adding telescopes therefore buys resolution only in proportion to how far apart the outermost elements are placed; a compact array of many small dishes is limited by B, not by N.

θ_max ≈ λ / s   [largest source angular size measurable]

The complement of the resolution limit: the sparsely-sampled pupil function of a discrete array has strong grating lobes, so the array cannot cleanly measure an extended source much larger than about λ/s. This is the "maximum angular size of a detectable emission source is limited by the minimum gap between detectors" caveat in the interferometer literature.

  • Thinned-array curse: collecting area scales with the real aperture area, not with B². The array resolves beautifully but shows almost nothing too faint.
  • Coherence & baselines: at visible wavelengths the optical paths from each telescope to the beam combiner must be matched to a fraction of a wavelength across the whole span, which is why real optical interferometers such as VLTI span only ~130 m equivalent while radio VLBI spans thousands of kilometres.
  • Seeing: a ground telescope is usually seeing-limited (~1″), far above its diffraction limit; only space telescopes or adaptive optics approach the numbers computed here.
  • Orientation: a baseline perpendicular to the source's position angle resolves finest. The (N−1)s figure is the best case (array laid broadside).
L: 1 AU = 1.495978707×1011 m  ·  1 light-minute = c×60 s = 1.79875×1010 m  ·  1 light-hour = 1.07925×1012 m