Coronagraph Contrast: Imaging an Exoplanet Next to Its Star

Coronagraph Mask

Raw contrast floor 1e-6

Inner working angle 3.0 λ/D

Companion Planet

Angular separation 4.0 λ/D

Planet/star contrast 1e-8

Readouts

Detection SNR

0.0σ

Separation

0.00″

Status (5σ threshold)

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How it works

A perfect circular telescope aperture of diameter D does not form a point image of a star — diffraction spreads it into the Airy pattern:

I(θ) = [2 J₁(x) / x]²,  x = π·θ·D/λ
first dark ring at θ ≈ 1.22 λ/D

The grid below plots this pattern (in units of λ/D) as a log-scaled height field: the tall central peak is starlight, the surrounding ripples are diffraction rings. A coronagraph mask removes on-axis starlight so a companion millions of times fainter can be seen. Two numbers set how well it does that:

Raw contrast floor — the residual starlight left everywhere outside the mask after all suppression and post-processing, expressed as a fraction of the star's peak brightness. Real coronagraphs (Lyot, vortex, PIAA) reach 10⁻⁵ to 10⁻⁹.
Inner working angle (IWA) — the radius inside which the mask itself, not the residual floor, dominates: leakage there is modeled as scaling like (θ/IWA)⁴, so anything closer than the IWA is essentially unrecoverable.

The residual field also carries slowly-drifting quasi-static speckles — the bumps you see shifting over time — a real effect of uncorrected wavefront errors in the optics that limits ground- and space-based direct imaging even after the coronagraph does its job.

Detection significance follows the standard high-contrast-imaging criterion:

SNR = planet peak intensity / local residual+speckle noise
detected when SNR ≥ 5σ

Move the planet closer than the IWA, or raise the contrast floor, and the signal drowns in the mask's own leakage — exactly the trade every real coronagraph design has to make.

Star + residual halo Planet signal

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