Although the date and time of each New Moon can be computed exactly (see, for example, Phases of the Moon in Data Services), the visibility of the lunar crescent as a function of the Moon's "age" - the time counted from New Moon - depends upon many factors and cannot be predicted with certainty. In the first two days after New Moon, the young crescent Moon appears very low in the western sky after sunset, and must be viewed through bright twilight. It sets shortly after sunset. The sighting of the lunar crescent within one day of New Moon is usually difficult. The crescent at this time is quite thin, has a low surface brightness, and can easily be lost in the twilight. Generally, the lunar crescent will become visible to suitably-located, experienced observers with good sky conditions about one day after New Moon. However, the time that the crescent actually becomes visible varies quite a bit from one month to another. The record for an early sighting of a lunar crescent, with a telescope, is 12.1 hours after New Moon; for naked-eye sightings, the record is 15.5 hours from New Moon. These are exceptional observations and crescent sightings this early in the lunar month should not be expected as the norm.
Obviously, the visibility of the young lunar crescent depends on sky conditions and the location, experience, and preparation of the observer. Generally, low latitude and high altitude observers who know exactly where and when to look will be favored. For observers at mid-northern latitudes, months near the spring equinox are also favored, because the ecliptic makes a relatively steep angle to the western horizon at sunset during these months (tending to make the Moon's altitude greater).
If we ignore local conditions for the moment, and visualize the problem from outside the Earth's atmosphere, the size and brightness of the lunar crescent depend on only one astronomical quantity - the elongation of the Moon from the Sun, which is the apparent angular distance between their centers. For this reason the elongation has also been called the arc of light. If we know the value of the elongation at any instant, we can immediately compute the width of the crescent.
What is the value of the elongation when the Moon's age is one day? It varies, depending on several factors:
- The elongation at New Moon. The Moon can pass directly in front of the Sun at New Moon (when a solar eclipse will occur) or can pass as far as five degrees away. That is, the Moon can start the month with an elongation ranging from zero to five degrees. A minor complicating factor involves the definition of New Moon in the almanacs. Astronomical New Moon is defined to occur when the Sun and Moon have the same geocentric ecliptic longitude, which may not occur precisely when the Sun and Moon are closest together in the sky.
- The speed of the Moon in its orbit. The Moon's orbit is elliptical, and its speed is greatest when it is near perigee, least near apogee. If perigee occurs near New Moon, the Moon will appear to be moving away from the Sun in the sky at a greater than average rate.
- The distance of the Moon. Again, because of its elliptical orbit, the distance of the Moon varies, so even if the Moon moved with a constant speed, its angular motion as viewed from the Earth would be greater when the Moon is near perigee.
- The location of the observer. If the observer is located in the tropics such that the one-day-old-Moon is observed just before it sets, its elongation as seen by the observer will be about a degree less than that seen by a fictitious observer at the center of the Earth, which is the basis for most almanac calculations. This decrease in observed elongation is less for observers at middle or high latitudes (although other geometric factors are less favorable for these observers).
Factors (2) and (3) are linked by Kepler's second law, which predicts that the angular speed of the Moon as seen from the Earth will vary by about 22%. If we combine all these factors we find that geocentric elongation of the Moon from the Sun at an age of one day can vary between about 10 and 15 degrees.
This large range of possible elongations in the one-day-old Moon is critical, because at this time the width of the crescent is increasing with the square of the elongation, and the surface brightness of the crescent is also rapidly increasing. Some of the earliest reliable sightings of the crescent occur near elongations of around 10 degrees. Obviously, simply specifying the age of the Moon cannot tell the whole story. Of course, the elongation of the Moon does not tell the full story, either. But, of the two parameters, the elongation is a much more reliable parameter to use as a starting point in assessing the lunar crescent visibility at any given date and time.
The prediction of the first sighting of the early crescent Moon is an interesting problem because it simultaneously involves a number of highly non-linear effects. Stated in less technical language, a lot of things are changing very rapidly. Effects to be considered are the geometry of the Sun, Moon, and horizon; the width and surface brightness of the crescent; the absorption of the Moon's light and the scattering of the Sun's light in the Earth's atmosphere; and the physiology of human vision. The problem has a rich literature. Some modern astronomical references are:
- Schaefer, B. E., 1988: "Visibility of the Lunar Crescent", Quarterly Journal of the Royal Astronomical Society, Vol. 29, pp. 511-523.
Schaefer, B. E., Ahmad, I. A., Doggett, L. E., 1993: "Records for Young Moon Sightings", Quarterly Journal of the Royal Astronomical Society, Vol. 34, pp. 53-56.
- Ilyas, M., 1994: "Lunar Crescent Visibility Criterion and Islamic Calendar", Quarterly Journal of the Royal Astronomical Society, Vol. 35, pp. 425-461.
- Doggett, L. E., Schaefer, B. E., 1994: "Lunar Crescent Visibility", Icarus, Vol. 107, pp. 388-403.
- M. B. Pepin, 1996: "In Quest of the Youngest Moon", Sky & Telescope, December 1996, pp. 104-106.
Her Majesty's Nautical Almanac Office computes and distributes predictions of lunar crescent visibility. The Astronomical Calendar by Guy Ottewell includes good diagrams of the positions of young and old Moons during the year (drawn for the eastern U.S.) and an explanation of the factors affecting their visibility.
Related information on these web pages includes:
Phases of the Moon and Percent of the Moon Illuminated (definitions) in FAQ
Dates of Primary Phases of the Moon in Data Services
Fraction of the Moon Illuminated in Data Services
What the Moon Looks Like Today in Data Services
Complete Sun and Moon Data for One Day in Data Services