The Science Behind the Crescent: Yallop vs. Odeh Criteria Explained

For centuries, looking for the new crescent (the hilāl) has marked the beginning of Islamic lunar months. It is a tradition rooted in observation, linking faith to the cosmos. But determining whether a thin sliver of light will actually be seen on a given evening is one of the harder problems in positional astronomy. It depends not on a single number but on a cluster of geometric and photometric quantities, all working together at the precise moment when the Sun has just set and the sky is still bright.

At Hilal Vision we merge this timeless tradition with modern computational astronomy. To predict where and when the crescent will be visible, we rely on two world-renowned mathematical models: the Yallop criterion and the Odeh criterion. This article unpacks both, defines the quantities they depend on, walks through a worked example, and is honest about where the models stop being reliable.

Conjunction Is Not the Crescent

A common misconception is that the "new moon" you can see in the sky is the astronomical new moon. It is not. The astronomical new moon is the instant of conjunction, when the Moon passes directly between the Earth and the Sun in ecliptic longitude. At that instant the illuminated hemisphere faces entirely away from us and the Moon is lost in solar glare, so it is invisible from everywhere on Earth.

The hilāl is something different: it is the first visible crescent, the thin arc of reflected sunlight that appears typically 15 to 40 or more hours after conjunction. Calling the visible crescent the "new moon" blurs an important distinction. If you want the full terminology, our companion piece on what the hilal actually is sets it out carefully. The job of a visibility criterion is to predict the gap between these two events: how long after conjunction, and under what geometry, the sliver becomes thick enough and far enough from the Sun's glare to be perceived.

The Quantities That Actually Matter

Visibility is decided by geometry and photometry, not by the calendar. The crucial parameters are the following.

• ARCL (arc of light): the angular separation between the centres of the Moon and Sun, also called the elongation. ARCL is the master quantity. It drives how much of the disc is lit and therefore the width of the crescent, and it underpins the Danjon limit discussed below.
• ARCV (arc of vision): the difference in altitude between the centre of the Sun and the centre of the Moon at sunset. ARCV measures how high the Moon sits above the Sun, and therefore how far it has climbed out of the bright twilight glow near the horizon.
• DAZ (difference in azimuth): the horizontal angle between the Sun and the Moon measured along the horizon. When the Moon is offset sideways from the Sun, part of ARCL is "spent" on azimuth rather than altitude, which changes the visibility geometry.
• W (crescent width): the topocentric width of the crescent at its thickest point, measured in arcminutes. W is a direct function of ARCL.
• Lag time: the interval between sunset and moonset. A longer lag means more time to search in a darkening sky.
• Topocentric parallax: the Moon's horizontal parallax is about 57 arcminutes, so the Moon seen from your location can sit up to roughly one degree lower than its geocentric position when it is near the horizon. All serious criteria use topocentric, not geocentric, coordinates.

These three angles, ARCL, ARCV and DAZ, form a spherical triangle on the sky. Knowing any two essentially fixes the third, which is why the criteria can be expressed compactly.

From ARCL to Crescent Width W

The width of the crescent is set by the elongation. For a thin crescent the topocentric width in arcminutes is well approximated by

W = (s/2)·(1 - cos ARCL)

scaled to the Moon's apparent semi-diameter s in arcminutes (s is roughly 15 to 16 arcminutes). Because (1 - cos ARCL) grows steeply from zero as the elongation opens up, the crescent thickens rapidly in the first few degrees past the Danjon limit. A crescent at 7 degrees of elongation is a barely perceptible thread perhaps 0.1 arcminutes wide; by 12 degrees it has roughly tripled. This is why ARCL, not moon age, is the quantity that governs how much light there is to see.

The Age Myth

It is worth stating plainly: moon age, the number of hours since conjunction, is a poor predictor of visibility and is not a parameter in either the Yallop or the Odeh model. Two crescents of identical age can have very different elongations because the Moon's orbital speed varies with its position (faster near perigee, slower near apogee), and because the tilt of the ecliptic relative to the horizon varies with latitude and season. What actually decides visibility is ARCV, ARCL, W and DAZ. Age is a rough proxy at best, and it routinely fails for marginal cases.

The Yallop Criterion (1997)

Dr Bernard Yallop, a former astronomer at HM Nautical Almanac Office, published his method in 1997 as NAO Technical Note No. 69. His criterion distils the geometry into a single number, the q-value, computed not at sunset but at Yallop's "best time", an instant roughly four ninths of the lag time after sunset, when the sky has darkened enough yet the Moon has not sunk too low.

The q-value is defined as

q = (ARCV - (11.8371 - 6.3226·W + 0.7319·W² - 0.1018·W³)) / 10

where W is the topocentric crescent width in arcminutes and ARCV is in degrees. The polynomial in W is Yallop's empirical fit to the minimum ARCV required for visibility at a given crescent width: the wider the crescent, the lower it can be and still be seen. The q-value is essentially the margin by which the actual ARCV exceeds (or falls short of) that empirical threshold.

Crucially, Yallop's zones are mutually exclusive bands, not a cumulative "greater-than" ladder. This is the single most common error people make when reading the criterion. The zones are non-overlapping ranges:

Note carefully what Zone F means: the crescent is below the Danjon limit and cannot be seen by any means. It does not mean the Moon is below the horizon. A Moon can be comfortably above the horizon and still fall in Zone F because its elongation is simply too small.

When you toggle the Yallop layer on Hilal Vision's global visibility map, you are seeing a live computation of this q-value across thousands of grid points worldwide, each cell coloured by its zone.

A Worked Example

Suppose that, at Yallop's best time for a particular location, the topocentric ARCV is 10.5 degrees, the elongation ARCL is 12 degrees, and the Moon's apparent semi-diameter SD is 16 arcminutes (a typical mid-range value). We first derive the crescent width W from the elongation and semi-diameter:

W = SD × (1 - cos ARCL) = 16 × (1 - cos 12°) = 16 × (1 - 0.97815) ≈ 16 × 0.02185 ≈ 0.350 arcminutes

Now we evaluate the Yallop threshold polynomial with W = 0.350:

11.8371 - 6.3226 × 0.350 + 0.7319 × 0.350² - 0.1018 × 0.350³ = 11.8371 - 2.2129 + 0.0897 - 0.00436 ≈ 9.719 degrees

Then:

q = (10.5 - 9.719) / 10 = 0.781 / 10 ≈ +0.078

A q-value of +0.078 falls in the band -0.014 < q ≤ +0.216, which is Zone B: visible under perfect atmospheric conditions.

Cross-checking against the Odeh V-value with the same inputs:

V = 10.5 - (7.1651 - 6.3226 × 0.350 + 0.7319 × 0.350² - 0.1018 × 0.350³) = 10.5 - (7.1651 - 2.2129 + 0.0897 - 0.00436) = 10.5 - 5.038 ≈ 5.46

A V of 5.46 falls in the range 2 ≤ V < 5.65, which Odeh classifies as visible with some effort and possibly requiring optical aid to first locate: consistent with Yallop Zone B. Both criteria agree. A two-degree drop in ARCV to 8.5 degrees would push q below -0.014 into Zone C, where binoculars are needed to initially acquire the crescent, illustrating how sharply sensitive the outcome is to small changes in geometry.

The Odeh Criterion (2004)

Yallop's model is excellent, but astronomy keeps accumulating data. Mohammad Shawkat Odeh, who founded the Islamic Crescents' Observation Project (ICOP) in 1998 under the International Astronomical Center, published a refined criterion in 2004 in Experimental Astronomy. It was derived from 737 observation records, about half of them gathered through ICOP, making it one of the most empirically grounded crescent models available.

Odeh's model produces a V-value that combines the arc of vision with the crescent width:

V = ARCV - (-0.1018·W³ + 0.7319·W² - 6.3226·W + 7.1651)

The V-value sorts visibility into four regions:

Because it draws on a larger, more recent dataset that includes systematic optical-aid sightings, the Odeh criterion sometimes places the boundary between "naked eye" and "optical aid" slightly differently from Yallop. Odeh also gives an empirical floor for what optical aid can achieve: a practical optical-aid (CCD or telescope) limit of about 6.4 degrees of elongation. Below that, even instruments struggle.

Hilal Vision computes both engines independently. Toggling between Yallop and Odeh on the same evening is one of the most instructive things you can do, especially for the marginal cases where the two models disagree, and you can compare them side by side on the moon dashboard.

The Danjon Limit: The Hard Floor

Beneath both criteria sits a physical boundary that no model and no instrument can argue away. André Danjon first reported the effect in 1932 and quantified it in 1936 in L'Astronomie: below a minimum elongation, no crescent has ever been reliably seen. He placed that minimum (the Danjon limit) at roughly 7 degrees of arc of light.

The empirical value is still debated, somewhere in the range of about 5 to 7.5 degrees. Fatoohi, Stephenson and Al-Dargazelli (1998), re-examining the historical record, derived a figure near 7.5 degrees for naked-eye sightings, while the optical-aid limit, as Odeh notes, sits closer to 6.4 degrees.

The cause of the limit remains an open question, and should be presented as a hypothesis rather than settled fact. Three competing explanations are discussed:

• Lunar topography and cusp foreshortening (Danjon's own hypothesis): at small elongations sunlight grazes the Moon's mountainous limb, mountains cast long shadows into the valleys behind them, and the continuous arc of the crescent fragments into disconnected specks of light, with the tips (cusps) foreshortened away first.
• Photometric brightness fall-off (Schaefer): the surface brightness of the crescent drops sharply toward the cusps because of the illumination geometry, so the visible arc shrinks faster than simple geometry predicts.
• Atmospheric seeing and contrast threshold: the crescent's already low contrast against a bright twilight sky collapses below the threshold the human visual system can detect.

Whatever the true mechanism, the practical consequence is firm: if the elongation is below roughly 7 degrees, a naked-eye sighting is essentially impossible. We treat this in much greater depth in our dedicated piece on the physics of the Danjon limit. It is also why even the most spectacular records sit so close to conjunction in elongation but cannot be matched by ordinary eyes: Thierry Legault's celebrated 2013 image was captured at essentially the instant of conjunction (an age near zero hours) at an elongation of about 4.4 degrees, using a tracked CCD setup in daylight, far removed from anything a human observer could perceive. The youngest binocular-aided crescent on record, by Mohsen Mirsaeed in 2002, was at about 11 hours and 40 minutes after conjunction.

Limitations of the Criteria

No visibility criterion is a crystal ball, and it is important to understand where they stop.

• They are statistical, not deterministic. The zone boundaries are fitted to past observation records. A crescent in Zone C may be seen by an exceptional observer at a high, dry site, and missed by a casual observer at sea level on the same night. The zones describe probabilities of detection, not guarantees.
• They assume a clear, transparent atmosphere. Neither q nor V contains a term for cloud, haze, dust or humidity. A flawless Zone A prediction is worthless under an overcast sky. Local weather and atmospheric extinction must be layered on top, which is why we treat atmospheric refraction and weather as a separate input.
• Observer and equipment vary. Visual acuity, experience, and whether binoculars are used all shift the effective threshold. The Yallop and Odeh bands encode "typical" observers and cannot capture every individual.
• They depend on accurate inputs. Errors in the assumed semi-diameter, in the refraction model, or in the topocentric correction propagate straight into q and W. Marginal cases (Zones C and D) are the most sensitive to these inputs.
• Garbage in, garbage out for the training data. Both models were calibrated on observation databases that inevitably contain a few erroneous claims. The Danjon limit is used precisely to filter out physically impossible reports before fitting, but no dataset is perfect.

This is exactly why crescent determination remains a dialogue between calculation and observation rather than a matter settled by either alone, a theme we explore in moon sighting versus calculation.

Why We Run Both Engines

By providing both the Yallop and the Odeh models side by side, Hilal Vision lets you analyse the sky the way a working astronomer would. Where the two engines agree, you can be confident. Where they disagree, you have found a genuinely marginal evening, the kind that sighting committees and researchers care most about. The app removes the guesswork and replaces it with transparent, reproducible mathematics, while being honest about the uncertainty that remains.

The next time you open the app to check whether the crescent will be visible from your city, remember the geometry running beneath the surface: the spherical triangle of ARCL, ARCV and DAZ; the crescent width W derived from the elongation; the topocentric parallax correction; and the q and V values, all solved in milliseconds. Try it for your own location on moonsighting.live, and if you want cloud overlays and the extended ICOP archive, the Pro tier adds those layers for serious observers.

References and Further Reading

• Yallop, B.D. (1997). A Method for Predicting the First Sighting of the New Crescent Moon. HM Nautical Almanac Office, NAO Technical Note No. 69.
• Odeh, M.Sh. (2004). "New Criterion for Lunar Crescent Visibility." Experimental Astronomy.
• Danjon, A. (1932, 1936). L'Astronomie. (The Danjon limit.)
• Fatoohi, L.J., Stephenson, F.R. & Al-Dargazelli, S.S. (1998). "The Danjon Limit of First Visibility of the Lunar Crescent." The Observatory.
• Kasten, F. & Young, A.T. (1989). Revised optical air-mass tables. Bennett, G.G. (1982); Saemundsson, T. (atmospheric refraction).
• The Islamic Crescents' Observation Project and International Astronomical Center: www.astronomycenter.net.

Clear skies and happy sighting.