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A coronagraph is an optical instrument that suppresses a bright source's light before detection while transmitting some light from nearby directions. Its purpose is to reveal faint structure buried in the bright source's point spread function, as in exoplanet direct imaging. A Lyot coronagraph combines a coronagraphic focal plane mask with a Lyot stop in a reimaged optical pupil; blocking the central image alone does not remove its diffraction rings. ^[ref].

The original motivation was the solar corona: hiding the bright photosphere would allow longer, repeated observations of faint surrounding light. In his 1930 paper, Lyot explained that short total eclipses restricted observing time and prevented some long exposures for spectral lines. The attraction is an instrument that makes the faint neighborhood accessible on demand. ^[ref]. The Sun is an extended source; the point-source calculation below illustrates imaging of an unresolved star and is not a quantitative model of the solar corona.

The coronagraph inner working angle describes how close to the blocked source a specified coronagraph can usefully transmit or detect a faint neighbor. It requires a criterion, such as a chosen fraction of the far off-axis coronagraph throughput, and is often expressed in units of $\lambda /D$. It is not automatically the coronagraphic focal plane mask radius or a guarantee of detection. Small separations make useful source transmission and accurate control of wavefront errors particularly demanding. ^[ref].

Consider an ideal coherent monochromatic wave from an unresolved source on the optical axis, a uniform circular entrance pupil of diameter $D$, and ideal relay optics without wavefront errors. In the scalar Fraunhofer diffraction approximation, a Fourier transform carries the pupil's complex amplitude into the image plane. Choose dimensionless pupil coordinates $\mathbf{x}$ in units of $D$ and angular coordinates $\mathbf{u}=D\mathbf{\theta }/\lambda$, where $\lambda$ is the wavelength. DefineThe coronagraphic focal plane mask has transmission $M=1-W$, with $W=1$ inside the occulting disk and $W=0$ outside. Thus the field immediately after the mask is $MA$. Outside the disk $M=1$, so the local wave intensity is still $|A{|}^2$: hiding the core has left those rings intact. Reimaging an unlimited pupil giveswhere $\ast$ is convolution, $L$ is the Lyot stop transmission, and the proportionality constant in the irradiance is omitted. Coordinate reversals from physical relay optics are absorbed into the coordinate choice. These equations use wave amplitudes throughout propagation; applying a Fourier transform to wave intensity would give a different and incorrect model. This is the pupil/image calculation described by Sivaramakrishnan et al.; the circular disk and numerical choices below are our illustrative specialization. ^[ref].

The subtraction explains the beauty of the arrangement: in much of the interior, the original pupil field and the convolution term nearly cancel. Near a boundary, that cancellation is incomplete. The Lyot stop can therefore block a region containing disproportionately much unwanted light. A hard-edged coronagraphic focal plane mask has a transform with extended tails, so the interior need not be perfectly dark. With an unlimited relay, $L=1$ gives $A_{\mathrm{final}}=MA$ exactly: no further ring suppression occurs. A stop covering the full original optical pupil instead uses $L=P$ and already clips field outside the optical pupil; an undersized Lyot stop clips more of the remaining rim.

The toy uses a coronagraphic focal plane mask of angular radius $3\lambda /D$ and a Lyot stop of diameter $0.8D$. The discrete Fourier transform grid has $1024\times 1024$ samples, with 128 across $D$ and angular sampling $0.125\lambda /D$; it approximates the continuous equations on a finite periodic domain. The plot averages wave intensity in annuli of width $0.25\lambda /D$. Over the annulus $4\le |\mathbf{u}|\le 10$, the mean wave intensity divided by the direct stellar peak is approximately $1.40\times 10^{-4}$ without a coronagraph, the same with only the focal mask and an unlimited relay, $4.44\times 10^{-5}$ with the mask and full pupil, and $1.66\times 10^{-6}$ with the mask and $0.8D$ Lyot stop. Thus the undersized opening reduces this particular annular mean about 27-fold relative to the full-pupil comparison. It does not suppress every direction by that factor, and these values do not predict a real instrument's detection limit.

The ideal toy excludes a central obstruction, support vanes, scattering of light, wavefront errors, atmospheric effects, a spread of wavelengths, pointing drift and detector noise. It also contains only the star: no simulated planet or solar corona. For a far off-axis source whose image misses the coronagraphic focal plane mask, the continuous geometric coronagraph throughput of the $0.8D$ opening tends to $0.8^2=0.64$ relative to the full disk. Rejecting stellar light therefore costs some desired light as well. Enlarging the coronagraphic focal plane mask narrows the characteristic scale of its pupil-plane transform by the scaling property of the Fourier transform, but also hides a larger neighborhood of the source.

Three detection methods are exoplanet transit photometry, the radial-velocity method, and exoplanet direct imaging. Transits detect obscuration of the star; radial velocities detect stellar reflex motion; imaging separates the planet's own reflected or thermal light from the star.

Exoplanet transit photometry supports wavelength-dependent exoplanet transmission spectra, while the same orbital geometry supports exoplanet secondary eclipses and phase-resolved planetary spectra. Exoplanet direct imaging provides resolved light for atmospheric spectroscopy. Stellar radial-velocity discovery alone does not measure an atmosphere, although high-resolution follow-up can separate a moving planetary molecular spectrum through its changing Doppler effect.

Two favorable conditions for detecting wide-orbit planets by exoplanet direct imaging are:

A wide orbit improves separation but weakens reflected-light illumination and lengthens the orbital period. These conditions therefore describe imaging sensitivity, not an assertion that every detection method becomes easier at large separation.

Assume the planet and present-day Jupiter have the same mass, comparable composition and initial cooling normalization, and both can be described by the supplied power-law planetary cooling over the relevant ages. Take the age of present-day Jupiter to be $t_J\simeq 4.5\mathrm{Gyr}$ and the young planet's age to be $0.050\mathrm{Gyr}$. The normalization cancels:The 5-AU orbit is used to regard stellar irradiation as modest compared with a hot-Jupiter orbit; the ratio is specifically intrinsic cooling, not the sum of intrinsic and reradiated luminosity. Initial entropy and irradiation can change the assumed normalization, so this estimate is not independent of formation conditions.

Such a young self-luminous giant is a favourable target for near-infrared exoplanet direct imaging. The increased intrinsic luminosity improves its contrast with the host star, and a wide physical orbit is more readily separated on the sky than a hot-Jupiter orbit. Two essential observing considerations are:

Thus direct imaging of a nearby young system, with sufficient angular resolution and contrast, is the natural discovery method, subject to the actual stellar brightness and instrumental limits.

At maximum projected separation, an Earth analogue at $10\mathrm{pc}$ is separated from its star by $\vartheta =a/D_{\mathrm{sys}}=0.1\mathrm{arcsec}=4.85\times 10^{-7}\mathrm{rad}$. The Rayleigh criterion for a circular pupil givesThe relevant quantity is the star–planet separation, not the tiny angular diameter of the planet itself. Simple angular optical resolution is insufficient: the planet lies in the star's diffraction pattern at reflected contrast of order $10^{-10}$, or thermal contrast of order $10^{-6}$. Residual speckles, wavefront errors, scattered light, detector/background noise and, in the infrared, strong telescope and sky thermal emission obstruct detection. A coronagraph or other stellar suppression method generally has an inner working angle of several $\lambda /D$, making practical aperture requirements larger than the bare Rayleigh criterion. The question's instrumentation claim is interpreted in its 2019 context; the physical requirements remain the basis of the estimate. Maximum reflected brightness near full phase also coincides with small projected separation, so maximum brightness and maximum resolvable separation cannot generally be obtained simultaneously.