Umbra
- Radius at the Moon
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Lunar eclipse
Geometry of the shadow cone. The Earth radius is the GRS80 mean radius, …. The Sun is at 1 AU, and the opening distance of the Moon is 384,400 km. The umbra’s radius at the Moon is set beside the Moon’s angular size, so it is plain whether a total eclipse fits in the cone. An eclipse also needs the Moon near a node. This page does not look up the next eclipse and has no lunar ephemeris.
The cone on the left is a schematic, not to scale. The two circles on the right are the angular radius of the umbra and the angular radius of the Moon, to scale with each other. The brown circle is the Moon.
- Earth and the umbra
- Moon
Moon
- Angular radius
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Umbra
- Angular radius
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What this page shows
The Sun and the Earth fix a cone of shadow. With Re the Earth radius, Rs the solar radius, and Ds the Sun’s distance, the apex lies beyond the Earth at L = Re × Ds / (Rs − Re). In the plane at distance Dm from the Earth’s centre, the umbra’s radius is Re × (L − Dm) / L.
The angular radius is the arctangent of that radius over Dm, and the same for the Moon’s own radius over Dm. When the umbra is the larger of the two, the Moon can sit inside the shadow. That is the geometric room for a total eclipse. It is not a prediction of one.
An eclipse also needs the Moon near a node, so that it passes through the cone rather than above or below it. This page does not look up the next eclipse and has no lunar ephemeris.
Assumptions
- The Earth radius is the GRS80 mean radius from the shared geometry module. The cone uses that sphere.
- The solar radius is the IAU 2015 nominal solar radius, 695,700 km. It is a conversion constant, not the measured radius on a given day.
- The Sun’s distance is 1 astronomical unit, 149,597,870,700 m, the IAU 2012 definition.
- The Moon’s radius is 1,737.4 km, the volumetric mean radius on the NASA Moon Fact Sheet. The opening distance is 384,400 km, that sheet’s semimajor axis, 0.3844 × 10⁶ km. Perigee 363,300 km and apogee 405,500 km are the same sheet. They are round figures at the precision the sheet prints, not a distance for a named date.
- There is no penumbra on the page, no refraction, and no flattening of the cone. Past the apex the umbra radius is not given, because the cone has ended.
Sources
- NASA Goddard Space Flight Center, Moon Fact Sheet. Semimajor axis 0.3844 × 10⁶ km, perigee 0.3633 × 10⁶ km, apogee 0.4055 × 10⁶ km, volumetric mean radius 1,737.4 km. nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html.
- IAU 2015 Resolution B3. Nominal solar radius 6.957 × 10⁸ m.
- IAU 2012 Resolution B2. The astronomical unit is 149,597,870,700 m.
- Moritz, H. (1980). Geodetic Reference System 1980. The Earth radius is the mean radius from the shared geometry module.
- The need for a node is the geometry of the Moon’s orbit against the ecliptic. This page stops at the size of the cone.