Umbra Shapes
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- Visualizations by:
- Ernie Wright
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For centuries, eclipse maps have depicted the shape of the Moon's umbra on the ground as a smooth ellipse. But as this visualization shows — in a way never seen before — the shape is dramatically altered by both the rugged lunar terrain and the elevations of observers on the Earth.
The lunar umbra is the part of the Moon's shadow where the entire Sun is blocked by the Moon. In space, it's a cone extending some 400,000 kilometers behind the Moon. When the small end of this cone hits the Earth, we experience a total solar eclipse. The umbra shape discussed here is the intersection of the umbra cone with the surface of the Earth. On an eclipse map, this tells you where to stand in order to experience totality.
The true shape of the umbra is more like an irregular polygon with slightly curved edges. Each edge corresponds to a single valley on the lunar limb, the last (or first) spot on the limb that lets sunlight through. This is the location of the diamond part of the diamond ring effect visible in the seconds just before or just after totality. An observer standing at the cusp where two edges meet will be treated to a double diamond ring.
As these edges pass over mountain ranges (for the 2017 eclipse, the Cascades, Rockies, and Appalachians), they are scalloped by the peaks and valleys of the landscape. The higher elevations in the western states in 2017 also shift the umbra toward the southeast (in the direction of the Sun's azimuth) by as much as 3 kilometers.
In the animation, the red ellipse is the shape that results from assuming that the Moon and the Earth are both smooth. This is the shape most commonly seen on eclipse maps. The white shape shows the effect of the mountains and valleys along the silhouette edge of the Moon (the lunar limb). The dark gray shape adds the effect of elevations on the Earth's surface.
Details
The math used currently to predict and map eclipses was first described by Friedrich Wilhelm Bessel in 1829 and was expressed in its modern form by William Chauvenet in 1863. Bessel's method uses a coordinate system based on a plane, called the fundamental plane, passing through the center of the Earth and perpendicular to the Sun-Moon line. This greatly simplifies the calculations. The intersection of the Moon's shadow with the plane is always a circle, for example, and its size depends only on the Moon's z-coordinate.Using this coordinate system, it's possible to calculate just a handful of numbers, called Besselian elements, that can be plugged into various equations to predict almost anything you'd want to know about an eclipse. This was especially important in the 19th and early 20th centuries, when the math had to be done by hand. Even now, the simplicity of this approach allows us to compare hundreds or even thousands of eclipses far into the past and the future, using a reasonable amount of computer time.
Mountains and valleys near the south pole of the Moon are visible in this image of a partial solar eclipse taken from space by the Solar Dynamics Observatory spacecraft on October 7, 2010.
Bessel's method for predicting eclipses pretends that the Moon is a smooth sphere, when in fact its terrain is more rugged and extreme than the Earth's. The valleys along the silhouette edge, or limb, of the Moon affect the timing and duration of an eclipse by allowing sunlight to sneak through in places where a smooth Moon would block it. Eclipse calculations can correct for this by using a limb profile, a description of the surface elevations around the disk of the Moon.
Until quite recently, everyone used the limb profiles published in 1963 by Chester Burleigh Watts. To produce his profiles, Watts designed a machine that traced some 700 photographs of the Moon covering the full range of angles, or librations, visible from Earth, an effort that spanned 17 years. Eclipse calculations are now moving to much more accurate limb profiles based on data from NASA's Lunar Reconnaissance Orbiter (LRO) and the Japan Space Agency's Kaguya spacecraft.
The lunar limb in the present work is based on LRO laser altimetry and on a hybrid LRO/Kaguya dataset called SLDEM2015. To create a limb profile, each point in an elevation map is transformed into 3D cartesian coordinates in a Moon body-fixed frame. At each time step in the eclipse calculation, the point cloud is rotated into fundamental plane coordinates. The limb profile then comprises the set of points lying farthest from the shadow axis. The animation above shows how this profile changes as the Moon librates.
Observer elevations are taken from SRTM, a digital elevation map of the Earth based on radar data collected during the February, 2000 flight of the Space Shuttle Endeavor. As illustrated by the following cartoon, higher elevations can lift the observer either into or out of the umbra cone. The overall effect is to shift the umbra toward the Sun.
For More Information
Credits
Please give credit for this item to:
NASA's Scientific Visualization Studio
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Visualizer
- Ernie Wright (USRA) [Lead]
Missions
This visualization is related to the following missions:Series
This visualization can be found in the following series:Datasets used in this visualization
SRTM DEM (Collected with the SIR-C sensor)
LRO DEM (A.K.A. Digital Elevation Map) (Collected with the LOLA sensor)
LRO/SELENE SLDEM2015 (A.K.A. DIgital Elevation Model) (Collected with the LOLA/TC sensor)
A digital elevation model of the Moon derived from the Lunar Orbiter Laser Altimeter and the SELENE Terrain Camera. See the description in Icarus. The data is here.
See more visualizations using this data setDE421 (A.K.A. JPL DE421)
Planetary ephemerides
Dataset can be found at: http://ssd.jpl.nasa.gov/?ephemerides#planets
See more visualizations using this data setNote: While we identify the data sets used in these visualizations, we do not store any further details, nor the data sets themselves on our site.
Related pages
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ID: 4515 Visualization
2017 Path of Totality
Read moreThis visualization closely follows the Moon's umbra shadow as it passes over the United States during the August 21, 2017 total solar eclipse. It covers the one hour and 40 minutes between 10:12 am PDT and 2:52 pm EDT. Through the use of a number of NASA datasets, notably the global elevation maps from Lunar Reconnaissance Orbiter, the shape and location of the shadow is depicted with unprecedented accuracy. || During the August 21, 2017 total solar eclipse, the Moon's umbral shadow will fly across the United States, from Oregon to South Carolina, in a little over 90 minutes. The path of this shadow, the path of totality, is where observers will see the Moon completely cover the Sun for about two and a half minutes.People traveling to see totality, likely numbering in the millions for this eclipse, will rely on maps that show the predicted location of this path. The math used to make eclipse maps was worked out by Friedrich Wilhelm Bessel and William Chauvenet in the 19th century, long before computers and the precise astronomical data gathered during the Space Age.In keeping with their paper and pencil origins, traditional eclipse calculations pretend that all observers are at sea level and that the Moon is a smooth sphere centered on its center of mass. Reasonably accurate maps, including this one, are drawn based on those simplifying assumptions. Those who want greater accuracy are usually referred to elevation tables and plots of the lunar limb.This animation shows the umbra and its path in a new way. Elevations on the Earth's surface and the irregular lunar limb (the silhouette edge of the Moon's disk) are both fully accounted for, and they both have dramatic and surprising effects on the shape of the umbra and the location of the path. To read more about these effects, go here.The animation provides an overhead view of the umbra and runs at a rate of 30× real time — every minute of the eclipse takes two seconds in the animation. For an oblique view that emphasizes the terrain of the path, go here.Earth radius6378.137 kmEllipsoidWGS84GeoidEGM96Moon radius1737.4 kmSun radius696,000 km (959.645 arcsec at 1 AU)EphemerisDE 421Earth orientationearth_070425_370426_predict.bpc (ΔT corrected)Delta UTC69.184 seconds (TT – TAI + 37 leap seconds)ΔT68.917 seconds ||
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ID: 4516 Visualization
2017 Path of Totality: Oblique View
Read moreThis animation closely follows the Moon's umbra shadow as it passes over the United States during the August 21, 2017 total solar eclipse. Through the use of a number of NASA datasets, notably the global elevation maps from Lunar Reconnaissance Orbiter, the shape and location of the shadow is depicted with unprecedented accuracy. || During the August 21, 2017 total solar eclipse, the Moon's umbral shadow will fly across the United States, from Oregon to South Carolina, in a little over 90 minutes. The path of this shadow, the path of totality, is where observers will see the Moon completely cover the Sun for about two and a half minutes.People traveling to see totality, likely numbering in the millions for this eclipse, will rely on maps that show the predicted location of this path. The math used to make eclipse maps was worked out by Friedrich Wilhelm Bessel and William Chauvenet in the 19th century, long before computers and the precise astronomical data gathered during the Space Age.In keeping with their paper and pencil origins, traditional eclipse calculations pretend that all observers are at sea level and that the Moon is a smooth sphere centered on its center of mass. Reasonably accurate maps, including this one, are drawn based on those simplifying assumptions. Those who want greater accuracy are usually referred to elevation tables and plots of the lunar limb.This animation shows the umbra and its path in a new way. Elevations on the Earth's surface and the irregular lunar limb (the silhouette edge of the Moon's disk) are both fully accounted for, and they both have dramatic and surprising effects on the shape of the umbra and the location of the path. To read more about these effects, go here.The animation runs at a rate of 30× real time — every minute of the eclipse takes two seconds in the animation. The oblique view emphasizes the terrain of the umbral path. For an overhead view, go here.Earth radius6378.137 kmEllipsoidWGS84GeoidEGM96Moon radius1737.4 kmSun radius696,000 km (959.645 arcsec at 1 AU)EphemerisDE 421Earth orientationearth_070425_370426_predict.bpc (ΔT corrected)Delta UTC69.184 seconds (TT – TAI + 37 leap seconds)ΔT68.917 seconds ||
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ID: 4518 Visualization
2017 Total Solar Eclipse Map and Shapefiles
Read moreA map of the United States showing the path of totality for the August 21, 2017 total solar eclipse. This is version 2 of the map, available at both 5400 × 2700 and 10,800 × 5400. || This map of the United States shows the path of the Moon's umbral shadow — the path of totality — during the total solar eclipse on August 21, 2017, as well as the obscuration (the fraction of the Sun's area covered by the Moon) in places outside the umbral path. Features include state boundaries, major highways, and 833 place names. At 18" × 9" (45 × 22.5 cm), the scale of the map is approximately 1:10,000,000.The umbra is shown at 10-minute intervals. Umbra shapes within U.S. time zones are labeled in local time. To read about the reason the shapes aren't smooth ovals, go here.The map uses a number of NASA data products. The land color is based on Blue Marble Next Generation, a global mosaic of MODIS images assembled by NASA's Earth Observatory. Elevations are from SRTM, a radar instrument flown on Space Shuttle Endeavour during the STS-99 mission. Lunar topography, used for precise shadow calculations, is from NASA LRO laser altimetry and JAXA Kaguya stereo imaging. Planetary positions are from the JPL DE421 ephemeris. The lunar limb profile and eclipse calculations are by the visualizer. ||
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ID: 4314 Visualization
2017 Total Solar Eclipse in the U.S.
Read moreA view of the United States during the total solar eclipse of August 21, 2017, showing the umbra (black oval), penumbra (concentric shaded ovals), and path of totality (red) through or very near several major cities. || On Monday, August 21, 2017, the Moon will pass in front of the Sun, casting its shadow across all of North America. This will be the first total solar eclipse visible in the contiguous United States in 38 years.The Moon's shadow can be divided into areas called the umbra and the penumbra. Within the penumbra, the Sun is only partially blocked, and observers experience a partial eclipse. The much smaller umbra lies at the very center of the shadow cone, and anyone there sees the Moon entirely cover the Sun in a total solar eclipse.In the animation, the umbra is the small black oval. The red streak behind this oval is the path of totality. Anyone within this path will see a total eclipse when the umbra passes over them. The much larger shaded bullseye pattern represents the penumbra. Steps in the shading denote different percentages of Sun coverage (eclipse magnitude), at levels of 90%, 75%, 50% and 25%. The yellow and orange contours map the path of the penumbra. The outermost yellow contour is the edge of the penumbra path. Outside this limit, no part of the Sun is covered by the Moon.The numbers in the lower left corner give the latitude and longitude of the center of the umbra as it moves eastward, along with the altitude of the Sun above the horizon at that point. Also shown is the duration of totality: for anyone standing at the center point, this is how long the total solar eclipse will last. Note that the duration varies from just 2 minutes on the West Coast to 2 minutes 40 seconds east of the Mississippi River.About AccuracyYou might think that calculating the circumstances of an eclipse would be, if not easy, then at least precise. If you do the math correctly, you’d expect to get exactly the same answers as everyone else. But the universe is more subtle than that. The Earth is neither smooth nor perfectly spherical, nor does it rotate at a perfectly constant, predictable speed. The Moon isn’t smooth, either, which means that the shadow it casts isn’t a simple circle. And our knowledge of the size of the Sun is uncertain by a factor of about 0.2%, enough to affect the duration of totality by several seconds.Everyone who performs these calculations will make certain choices to simplify the math or to precisely define an imperfectly known number. The choices often depend on the goals and the computing resources of the calculator, and as you'd expect, the results will differ slightly. You can get quite good results with a relatively simple approach, but it sometimes takes an enormous effort to get only slightly better answers.The following table lists some of the constants and data used for this animation.Earth radius6378.137 kmEarth flattening1 / 298.257 (the WGS 84 ellipsoid)Moon radius1737.4 km (k = 0.2723993)Sun radius696,000 km (959.634 arcsec at 1 AU)EphemerisDE 421Earth orientationearth_070425_370426_predict.bpc (ΔT corrected)Delta UTC68.184 seconds (TT – TAI + 36 leap seconds)A number of sources explain Bessel’s method of solar eclipse calculation, including chapter 9 of Astronomy on the Personal Computer by Oliver Montenbruck and Thomas Pflager and the eclipses chapter of The Explanatory Supplement to the Astronomical Almanac. The method was adapted to the routines available in NAIF's SPICE software library.The value for the radius of the Moon is slightly larger than the one used by Fred Espenak and slightly smaller than the one used by the Astronomical Almanac. The Sun radius is the one used most often, but see figure 1 in M. Emilio et al., Measuring the Solar Radius from Space during the 2003 and 2006 Mercury Transits for a sense of the uncertainty in this number.Both the elevations of locations on the Earth and the irregular limb of the Moon were ignored. The resulting small errors mostly affect the totality duration calculation, but they tend to cancel out—elevations above sea level slightly lengthen totality, while valleys along the lunar limb slightly shorten it. The effect on the rendered images is negligible (smaller than a pixel).Another minor complication that's ignored here is the difference between the Moon's center of mass (the position reported in the ephemeris) and its center of figure (the center of the disk as seen from Earth). These two centers don't exactly coincide because the Moon's mass isn't distributed evenly, but the difference is quite small, about 0.5 kilometers. ||
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ID: 12412 Produced Video
Tracing the 2017 Solar Eclipse
Read moreWhen depicting an eclipse path, data visualizers have usually chosen to represent the moon's shadow as an oval. By bringing in a variety of NASA data sets, visualizer Ernie Wright has created a new and more accurate representation of the eclipse. For the first time, we are able to see that the moon's shadow is better represented as a polygon. This more complicated shape is based NASA's Lunar Reconnaissance Orbiter's view of the mountains and valleys that form the moon's jagged edge. By combining moon's terrain, heights of land forms on Earth, and the angle of the sun, Wright is able to show the eclipse path with the greatest accuracy to date. ||
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