In 1707, Philippe de La Hire made a curious observation about Earth's umbra. The predicted radius of the shadow needed to be enlarged by about 1/41 in order to fit timings made during a lunar eclipse. The enlargement is attributed to Earth's atmosphere, which becomes increasingly less transparent at lower levels. Additional observations over the next two centuries revealed that the shadow enlargement was somewhat variable from one eclipse to the next.

William Chauvenet (1891) formulated one method to account for the shadow enlargement while André-Louis Danjon (1951) devised another. Both methods assumed a circular cross section for the umbral shadow. However, Earth is flattened at the poles and bulges at the Equator, so an oblate spheroid more closely represents its shape. The projection of each of the planet's shadows is an ellipse rather than a circle. Furthermore, Earth's axial tilt towards or away from the Sun throughout the year means the elliptical shape of the penumbral and umbral shadows varies as well.

In an analysis of 22,539 observations made at 94 lunar eclipses from 1842 to 2011, Herald and Sinnott (2014) found that the size and shape of the umbra are consistent with an oblate spheroid at the time of each eclipse, enlarged by the empirically determined occulting layer that uniformly surrounds Earth. The effective height of this layer was found to be 87 kilometers. Based on this work, the authors developed a new method to calculate the shadow enlargement including its elliptical shape.

The 21st Century Canon of Lunar Eclipses uses the Herald and Sinnott shadow enlargement method in calculating lunar eclipse circumstances and magnitudes. It serves as the basis of lunar eclipse predictions published at EclipseWise.com for eclipses during the years 2001 through 2100.

Read a detailed description the Herald and Sinnott elliptical shadow model used both here and in the 21st Century Canon of Lunar Eclipses.

References

Herald, D., and Sinnott, R. W., “Analysis of Lunar Crater Timings, 1842–2011,” J. Br. Astron. Assoc., 124, 5, 2014.