Source: The United States Naval Observatory (USNO)
Link: https://web.archive.org/web/20190611151431/https://aa.usno.navy.mil/faq/docs/crescent.php
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Crescent Moon Visibility
Although the date and time of each New Moon can be computed exactly, 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. During the first two days after New Moon, the young crescent Moon appears very low in the western sky after sunset, must be viewed through bright twilight, and 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. Naked-eye sightings as early as 15.5 hours after New Moon have been reliably reported while observers with telescopes have made reliable reports as early as 12.1 hours after New Moon. Because these observations are exceptional, crescent sightings this early in the lunar month should not be expected as the norm.
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 during these months. The steep angle means the Moon's altitude will be greater just after sunset.
Ignoring local conditions for the moment and visualizing 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 the value of the elongation at any instant is known, the width of the crescent can be computed.
The elongation as a function of the Moon's age depends on several factors:
1. The Moon's elongation at New Moon. The elongation of the Moon at New Moon is not necessarily 0. The Moon's center may pass directly in front of the Sun at New Moon (when a solar eclipse will occur) or it may be as much as five degrees to the north or south of the Sun. 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.
2. The speed of the Moon in its orbit. The Moon's orbit is elliptical, and its speed is greatest when it is near perigee (closest to the Earth), least near apogee (furthest from the Earth). The change in speed is caused by conservation of angular momentum; the same principle causes a spinning ice skater to speed up when she pulls her arms inward. 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.
3. The distance of the Moon: Because of its elliptical orbit, the distance of the Moon varies. 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. Similarly, a nearby automobile appears to be moving quicker than a more distant one, even if they are actually moving at the same speed.
4. The observer's location (parallax). 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 position used for most almanac calculations. Similarly, if you look at a foreground object with one eye closed and then close that eye and open the other, the object makes an apparent jump against the background. The change in the observed elongation is less for observers at middle or high latitudes; however, 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%. The combined effect of the first three factors gives geocentric elongation of the Moon from the Sun at an age of one day can vary between about 10 and 15 degrees. The last factor can subtract about a degree for an observer at the equator.
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