10230

# Dark Fraction of the Moon

The fraction of the Moon that is dark is a function of the phase angle . This function is and can be found by taking an area integral over the circle. A sphere with one hemisphere black, the other hemisphere white, and rotated by is shown in the top-left corner. The area integral being performed is shown in the top-right corner.

### DETAILS

For a unit radius sphere, the dividing equatorial line between the light and dark hemispheres projects to when , and to for . Therefore when , the dark area is given by the area integral from the bottom of the sphere, , to the dividing line . Similarly, when the dark area is given by the area integral from the dividing line to the top of the sphere, , and is . Because the total area of the sphere is , dividing these area integrals by gives us the fraction that is dark, . The fraction of the moon that is illuminated is (1 – the fraction that is dark), and is given by .
This quantity is derived in a different way in the following reference. Note that by the double-angle formula, .
Reference
[1] D. B. Taylor, S. A. Bell, J. L. Hilton, and A. T. Sinclair, Computation of the Quantities Describing the Lunar Librations in the Astronomical Almanac, Ft. Belvoir: Defense Technical Information Center, 2010.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.