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Home/Astronomy & The Sky/Lunar Eclipse Geometry

Lunar Eclipse Geometry

Lunar eclipses occur when the Moon passes through the shadow cast by Earth. This simulator visualizes the geometric alignment of the Sun, Earth, and Moon required for these events. It models the two distinct parts of Earth's shadow: the central, dark umbra and the surrounding, partially illuminated penumbra. The core physics involves the geometry of conical shadows, governed by the relative sizes of the celestial bodies and their distances. The simulator uses the principle of similar triangles to calculate the extent and taper of the umbral and penumbral cones. Key parameters include the radii of the Sun (R_s), Earth (R_e), and Moon (R_m), and the distances from Earth to Sun (d_s) and Earth to Moon (d_m). For instance, the length of Earth's umbra (L_u) can be approximated by L_u ≈ (R_e * d_s) / (R_s - R_e). The model simplifies the solar system to a 2D plane, treating orbits as circular and ignoring orbital inclinations and the Moon's libration to focus on the fundamental shadow geometry. By interacting with the controls, students learn how varying alignments produce total, partial, and penumbral lunar eclipses, observe the curvature of Earth's shadow on the lunar surface, and understand why eclipses do not occur every full moon.

Who it's for: High school and introductory undergraduate astronomy students learning celestial mechanics and eclipse geometry, as well as educators seeking a dynamic tool to illustrate shadow cones and orbital alignments.

Key terms

  • Umbra
  • Penumbra
  • Shadow Cone
  • Orbital Plane
  • Syzygy
  • Total Lunar Eclipse
  • Partial Eclipse
  • Penumbral Eclipse

Geometry

1
0°

Sun–Earth–Moon alignment: Earth’s shadow sweeps the Moon. The Moon is drawn redder in the umbra (schematic “blood moon”); partial and penumbral shading are stylized. Real colors depend on Earth’s atmosphere.

Measured values

θ_Earth from Moon3419 arcsec
θ_Sun from Moon960 arcsec
Umbra vs Sun (size)total possible
Moon vs shadow (center)total (umbral)

How it works

A lunar eclipse happens at full Moon when the Moon enters Earth’s shadow. The umbra is where the Sun is completely hidden by Earth; the penumbra receives only part of the sunlight. The Moon’s orbit is tilted, so not every full Moon is eclipsed.

Key equations

Same tangent construction as solar eclipses, but Earth is the occluder and the Moon is in the shadow cone.

Frequently asked questions

Why is the Moon sometimes red during a total lunar eclipse?
This red color, often called a 'Blood Moon,' is caused by Rayleigh scattering of sunlight in Earth's atmosphere. As sunlight passes through our atmosphere, shorter blue wavelengths are scattered away, while longer red wavelengths are bent (refracted) into Earth's umbral shadow, softly illuminating the Moon. This simulator uses schematic colors for clarity but does not model this atmospheric scattering effect.
If the Sun is so much bigger than Earth, how can Earth cast such a large, dark shadow?
Earth's shadow appears large because it is a long, tapering cone. While the Sun is enormous, it is also very far away. This geometry creates a shadow cone that is narrow at Earth but extends over a great distance. At the Moon's distance, the umbra's cross-section is still about three times the Moon's width, allowing it to be fully enveloped.
Why don't we see a lunar eclipse every full moon?
The Moon's orbit is tilted by about 5 degrees relative to Earth's orbital plane (the ecliptic). Most full moons occur when the Moon is above or below this plane, so it passes north or south of Earth's shadow. Eclipses only happen during a 'syzygy'—when the Sun, Earth, and Moon align nearly perfectly in a straight line, which typically occurs 2-4 times per year.
What is the difference between a penumbral eclipse and a partial eclipse?
In a partial eclipse, part of the Moon enters Earth's dark central umbra, creating a distinct, sharp 'bite' out of the lunar disk. In a penumbral eclipse, the Moon only passes through Earth's faint outer penumbra. The dimming is very subtle and often difficult to notice with the naked eye, as the Moon is still illuminated by part of the Sun's disk.