Welcome to another edition of the Fans Week Podcast, also on mediumwaves 1575 KHz. From Paris I’m Ami Carter-Wilson.
Your Eclipse Percentage Is a Lie Until You Solve a Differential Equation
By Ami Carter-Wilson | Pressbeat.org | Paris, August 11, 2026
Today’s news: Solar eclipse obscuration percentage varies by observer location, and can be computed from local coordinates relative to the eclipse’s central line.
Somewhere right now, a man in a folding chair is telling his neighbor he is “getting 94 percent,” and he is wrong, and he will never know he is wrong, because nobody in the history of backyard astronomy has ever actually done the math. They looked at an app. The app rounded. The rounding was a lie. The lie is now load-bearing for an entire afternoon of smugness.
The Innocent Beginning
It starts simply enough. Your percentage of solar occultation depends on where you are standing relative to the Moon’s shadow cone, expressed — at the level normal people are willing to accept — as a function of your distance from the path of totality:
where d is your perpendicular distance from the centerline and D_max is the shadow’s outer radius. Charming. Rustic, even. A formula you could explain to a golden retriever.
Nobody stops here. Nobody has ever stopped here.
Where Rustic Goes to Die
Because the Moon is not a coin held up against a flashlight, it is an oblate, librating, non-uniformly cratered mass orbiting an oblate, wobbling planet, and the “shadow” is not a shadow but a time-dependent umbral ellipse whose boundary must be solved via the full lunar limb profile integrated against solar limb darkening. Which means the honest observer is obligated — obligated, I am told, by people who use the word “obligated” recreationally — to work with the local obscuration rate:
where ω_m is the Moon’s angular shadow velocity, R_s and R_m are the apparent solar and lunar radii, ρ(t) is the time-varying angular separation of the two disk centers, and ε(φ,λ) is a correction term for your latitude and longitude that three separate observatories define three separate ways, none of which agree, and all of which are published as though this were normal.
“There is no shame in the correction term,” said Dr. Aurélie Fontanel, a researcher at an institute she declined to name, “there is only shame in pretending you understand it.”
The Part Where We Stop Pretending This Is About the Sun
To get your actual, defensible, publishable obscuration percentage, you integrate the rate equation over the full contact window, apply a second-order correction for atmospheric refraction bending the solar limb by a few arcseconds you will never notice with your naked eye but which the formula insists matters enormously:
where κ is an empirical refractive-coupling constant nobody can cite a primary source for, and ∇²n(h) is the Laplacian of the atmospheric refractive index profile as a function of altitude, which changes if a cloud passes over Nebraska.
Solve this properly — actually solve it, with real boundary conditions, not the “eh, close enough” boundary conditions your cousin used for his physics minor — and you arrive at a number. That number, when checked against six independent solar-geometry models run by six people who have never spoken to each other and would prefer to keep it that way, agrees with itself to within a margin of error of plus or minus 28 percent.
Twenty-eight percent. On a percentage. A margin of error nearly a third the size of the thing it is measuring, arrived at through three integrals, a Laplacian, and a correction term that requires knowing your own latitude to a precision your phone’s GPS chip was never asked to provide.
The man in the folding chair was closer to the truth doing nothing at all.
That’s all from today: from Paris, I’m Ami Carter-Wilson.
