About Umbra Atlas
This project generates high-precision shadow paths for over 11,000 solar eclipses spanning 5,000 years (from 2000 BCE to 3000 CE). While the resulting maps look simple, creating them requires navigating complex orbital mechanics and carefully matching math to what observers actually see on Earth.
How It Works (The Methodology)
We use Besselian Elements as our starting point. These are polynomial equations provided by NASA (Fred Espenak and Jean Meeus) that describe where the Moon’s shadow is relative to the center of the Earth at any given second.
From that mathematical foundation, we apply several layers of corrections to align the geometric data with the real world:
1. The Shape of the Earth (Ellipsoidal Geometry)
Most simple calculations assume the Earth is a perfect sphere. It isn’t; it is flattened at the poles and bulges at the equator.
What we did: We use the WGS84 ellipsoid (the standard used by GPS) to calculate exactly where the shadow cone intersects the Earth’s surface. This corrects errors of up to 10-20 km that can occur at high latitudes compared to spherical calculations.
2. The Size of the Moon (Visual vs. Strict)
This is the most common reason eclipse maps disagree.
Strict Totality: NASA’s raw data uses a conservative “Valley Radius” (k ≈ 0.27228). This assumes the Moon is slightly smaller to guarantee that total darkness occurs even if sunlight peeks through the deepest lunar valley, resulting in a narrower path.
Visual Reality: Observers and interactive maps (like Google Maps or Xavier Jubier’s) use the “Mean Radius” (k ≈ 0.27250), representing the average edge of the Moon.
Our choice: We apply a correction to use the Mean Radius. This widens the path by approximately 2-4 km, matching what observers actually see and aligning with modern visualization standards.
3. Center of Figure vs. Center of Mass
The Moon’s center of gravity (mass) is not exactly at the center of its visible shape (figure); the visible center is offset by about 0.5 km.
What we did: We apply a coordinate shift to align the shadow with the Moon’s Center of Figure, slightly shifting the entire path (usually south/east) to improve accuracy.
4. Atmospheric Refraction
At sunrise and sunset, the Sun appears above the horizon even when it has geometrically set, because the Earth’s atmosphere bends light.
What we did: We calculate the path slightly beyond the geometric horizon (up to a refraction limit of ~1.002 Earth radii). This ensures the eclipse path extends all the way to the visual sunrise/sunset line, preventing the path from looking “cut off” at the ends.
5. Hybrid Eclipses
Hybrid eclipses are rare events where the path shifts between total and annular.
What we did: We calculate the radius of the shadow cone at the Earth’s surface for every second:
- If the shadow vertex is below the ground, we output a total polygon (labeled HT).
- If the shadow vertex is above the ground, we output an annular polygon (labeled HA).
These segments are stitched together to form the complete track for each hybrid event.
6. Delta T (Rotation Correction)
The Earth’s rotation speed is slowing down over millennia. To map an eclipse from 1000 BCE correctly, you must account for the discrepancy between atomic time and Earth rotation time.
What we did: We apply the Delta T correction provided in the dataset to rotate the Earth to the correct longitude for each historical or future eclipse.