Optics 014 · Ray worlds, boundaries, and natural optics

Eclipse & Transit Shadow Observatory

A deep-space shadow observatory derives umbra, penumbra, and antumbra from exact common tangents, then closes the geometry with analytic apparent-disk overlap. Its first scene contrasts a sharp point-source cone with a finite-source shadow; its solar scene maps IAU and NASA scales into total, partial, and annular views; its transit scene turns the same overlap into a live normalized light curve.

Apparent-radius ratio or inferred scale

q

$0.50$

Obscuration or visible flux

\mathcal O

$50\%$

Contact and shadow-region audit

\mathcal C

$0.00\pi$

Model regime $\text{valid model regime}$

Circular-disk geometric occultation in the observer tangent plane. Stage distances and body sizes are display mappings; readouts retain the stated angular and astronomical scales. Atmospheric refraction, Earth topography, diffraction, corona radiance, stellar limb darkening, spots, and transmission spectroscopy are explicit omissions.

Physics tutorial

Audit the physics of Eclipse & Transit Shadow Observatory

BackgroundAn eclipse is determined by what the observer measures on the sky, not by comparing two unscaled stage spheres. Let the source and occluder have apparent angular radii and angular center separation summarized by $q=\frac{\alpha_{\mathrm o}}{\alpha_{\mathrm s}},\qquad z=\frac{\delta}{\alpha_{\mathrm s}}$ . These dimensionless quantities drive the apparent disks, the physical foreground-body displacement, the contact labels, and every flux readout together.

Why it mattersHow do two measured apparent disks generate sharp point-source shadow, umbra, penumbra, antumbra, eclipse contacts, and a transit light curve from one geometry?

Start with the essentials

Focus question: How do two measured apparent disks generate sharp point-source shadow, umbra, penumbra, antumbra, eclipse contacts, and a transit light curve from one geometry?
One-sentence intuition: The two contact thresholds are exact for circular apparent disks: first and fourth contact occur at the sum of the radii, while second and third contact occur at their absolute difference, so the normalized boundaries are $z=1+q,\qquad z=\lvert1-q\rvert$ . Inside the inner boundary, a larger occluder produces totality and a smaller one produces annularity. Between the boundaries the event is partial.

Core mathematical model

Umbra, antumbra, and penumbra radii

r_{\mathrm u}(x)=R_{\mathrm o}-\frac{R_{\mathrm s}-R_{\mathrm o}}{D}x,\qquad r_{\mathrm p}(x)=R_{\mathrm o}+\frac{R_{\mathrm s}+R_{\mathrm o}}{D}x,\qquad x_{\mathrm apex}=\frac{D R_{\mathrm o}}{R_{\mathrm s}-R_{\mathrm o}}

Similar triangles give the coaxial cross-section. The signed inner radius vanishes at the umbral apex; beyond it, its negative magnitude is the antumbral radius. The rendered three-dimensional surfaces are generated from the corresponding exact common tangents rather than this diagrammatic shortcut.

Exact apparent-disk overlap and flux

A_{\mathrm{ov}}=\begin{cases}0,&z\ge 1+q,\\[2pt]\pi\min(1,q)^2,&z\le\lvert1-q\rvert,\\[2pt]\cos^{-1}\!\left(\frac{z^2+1-q^2}{2z}\right)+q^2\cos^{-1}\!\left(\frac{z^2+q^2-1}{2zq}\right)-\frac12\sqrt{(-z+1+q)(z+1-q)(z-1+q)(z+1+q)},&\text{otherwise},\end{cases}\qquad \frac{F}{F_0}=1-\frac{A_{\mathrm{ov}}}{\pi}

The overlap lens is evaluated analytically with a unit source radius. Dividing by the source-disk area gives obscuration, and subtracting it from unity gives the visible flux of a uniformly bright source. The live photometer curve is sampled from this same function.

Angular calibration and the solar preset

\alpha=\sin^{-1}\!\left(\frac{R}{d}\right),\qquad d_{\mathrm M}=\frac{R_{\mathrm M}}{\sin(q\alpha_{\odot})},\qquad \delta_0=\left(\frac{R_{\mathrm p}}{R_\star}\right)^2\ \ (R_{\mathrm p}<R_\star)

The solar scene infers the lunar distance from the requested apparent-radius ratio using the IAU nominal solar radius, NASA lunar radius, and mean Earth–Sun distance. The transit scene recovers the familiar squared-radius-ratio depth only as the centered, uniform-star limit.

Common difficulties

Treating every shadow edge as sharp

Typical misconceptionAn opaque body blocks each ray completely, so a finite luminous source must still cast one hard-edged dark cone just like a point source.

Better mental modelOpacity decides what happens to an intercepted ray; source extent decides how many source points remain visible. External and internal tangent families partition full, partial, and zero source visibility, creating umbra, penumbra, and antumbra. Only the point-source limit has one sharp shadow boundary.

Run the experiment

01
Scene 1: Point versus finite-source shadow anatomy
Begin with shadow anatomy. Orbit around the translucent volumes, drag the opaque sphere across the optical axis, and distinguish the gold point-source cone from the two finite-source tangent families. Locate the inner-cone apex and follow the signed-radius readout as umbra becomes antumbra.
What to observe: The point-source limit has one tangent cone and no penumbra. Giving the source a finite angular size splits the boundary into an inner and outer family; a location between them sees only part of the luminous disk.
02
Scene 2: Sun–Moon–Earth total and annular eclipse
Switch to the Sun–Moon–Earth scene. Set the apparent-radius ratio just above and just below unity, center the disks, and compare totality with annularity. Then increase center separation through inner and outer contact while watching the inferred lunar distance and signed terrestrial umbra.
What to observe: Near equal apparent radii, a small distance change can switch the central event between total and annular even though the bodies retain their physical radii. The decisive comparison is angular, and the stage compression does not enter that calculation.
03
Scene 3: Exoplanet transit photometry
Enter transit photometry. Choose a small planet, drag it from before ingress through mid-transit to after egress, and match the apparent-sky overlap to the moving flux marker. At exact center, compare the measured depth with the squared radius ratio.
What to observe: For the uniform-star model, ingress and egress are curved rather than linear because circle-overlap area is nonlinear in separation. Limb darkening would reshape the curve, so it is named as an omitted effect rather than silently approximated.