The fact, that the new lunar crescent appears shorter than 180⁰ in length, is known for centuries. It was Danjon who first gave an explanation for the phenomenon (Danjon, 1932 & 1936) and attributed it to the lunar terrain close to the cusps. McNally attributed a different reason with this phenomenon discarding Danjon’s hypothesis (Mcnally, 1983). McNally proved that the length of the shadows close to the lunar cusp and the departure of lunar surface from being perfect sphere couldn’t diminish the brightness of the regions of crescent close to cusps to the level that they become invisible. Therefore he attributed the length shortening of crescent to the “seeing affect” due to the turbulence of the atmosphere. Schaeffer rejected McNally’s explanation on the basis of his view that the shortening of the crescent length is simply because of the sharp decline of the brightness of the crescent close to the cusps (Schaeffer, 1991). McNally has also developed formulas for calculating the length of the crescent. Using the accurate model of Hapke (1984) for calculating the surface brightness of the crescent Schaeffer claims that Danjon’s collected observations and his own new data fits the model. Lately Sultan (2005) has attributed the shortening of length of crescent to the Blackwell Contrast Threshold (1946) and has developed a formula to calculate crescent’s length.

Most of the early description of the phenomena concentrated on relating it to the phase (or elongation) of the Moon. The description by Danjon was shown to be incorrect mathematically by McNally. However his over estimated limit on the minimum width of visible crescent (2 to 6 arc seconds) lead to very small values for Danjon Limit. The description due to McNally is logically sound so is that of Sultan and both resulted into techniques for calculating length of crescent. However both McNally and Sultan have not reported the application of their description on the recorded historical data found in literature (Yallop, 1998, Schaeffer 1991 etc.). According to Danjon, as described by Fatoohi et.al.        

                                          (1.1)

Where the arc PQ = α is the deficiency arc in Fig. 1 (Fatoohi et. Al.. 1998), a is the elongation and w is half the crescent length. It appears that Danjon used the Sine formulae by assuming spherical angle at Q to be right angle. McNally rejected Danjon’s argument and using four-part formula arrives at:

                                           (1.2)

Numerically, for small angles α and a there is only a marginal difference between the two results. Generally the elongation (a) can be calculated and the crescent length (2ω) is observed, so the two formulas can be used to find the deficiency arc. However, to calculate the crescent length none of these can be used. McNally develops a formula for angular separation φ from a cusp in terms of crescent width R as:

                             (1.3)

    where DR  is the minimum visible width measured in

disc. Therefore the length of the crescent he obtains is 180⁰ - 2φ.

Using the Blackwell threshold contrast Sultan (2005) arrives at the minimum visible width (in terms of diameter of the smallest equivalent Blackwell (1946) disc) of crescent at perigee and apogee at his location of observation. The formula that he developed for calculating the crescent length is:

                                                                    (1.4)

where r is the semi-diameter of the Moon, and

                                        (1.5)

with w is the diameter of the smallest visible equivalent Blackwell disc and W is central width of the crescent. Sultan considers minimum diameter of Blackwell disc to be 0.14 arc minutes when the Moon is near perigee and 0.16 arc minutes when the Moon is near apogee.

The present work is based on the observation of the last (old) crescent on February 26, 2006. During this observation that started from the beginning of morning twilight till well past sunrise, it was noticed that for this –48 hours Moon, more than 27 degrees away from the sun the crescent length started to decline with the rising sun. A similar observation on March 28^th when the age of Moon was around –33 hours and around 18.5 degrees away from the sun, the crescent length decreased more rapidly with decreasing contrast. Two days later new crescent with age 27.6 hours at 15 degrees away from the sun was observed till setting. Close to the horizon through thick humid atmosphere the crescent length was again observed to be decreasing. These observations clearly demonstrate that there is much more to be explored about the phenomenon of shortening of crescent length apart from Hapke’s accurate model that shows dependence of crescent length on elongation alone.

Therefore, in this work, to begin with we consider a simple geometrical model for the crescent. The model describes the phenomenon of shortening of length that depends on the actual semi-diameter of the Moon as well as the relative altitude (ARCV) of the crescent over local sky. This is derived from the single parameter (q-value) criterion of earliest visibility of crescent due to Yallop (1998) and is based on the fact that whenever the width (or brightness) of crescent close to cusp is below the minimum visible central width (or brightness) of the crescent the whole length of crescent with smaller width would not be visible. Applying our model on the recorded visibility and invisibility data available the length of crescent in each case is calculated. The calculated lengths of crescent are also compared with the observed lengths and with the lengths calculated using formulas (1.3) due to McNally and (1.4 & 1.5) due to Sultan. In case of using McNally’s formula ΔR is considered from the criterion used in this work and for Sultan’s formula the minimum width of Blackwell disc (w) is also considered to be the one we developed.

Let O be the centre of the lunar disc and the origin of coordinate system, x-axis be the direction of the Earth, y-axis is along OB and the z-axis along OC. C and D are the Cusps of the crescent. OS is the direction of the Sun and the angle between OS and the negative x-axis is the elongation E of the Moon from the Sun as seen from the Earth. As shown in the Fig. 1 the disc of the Moon as seen from the Earth is the circle ACBD that lies in the plane of the paper (yz-plane), whereas the part of the lunar surface exposed to the Sun is the hemisphere described by the semicircle CFD. Thus the illuminated part of the surface of the Moon that is exposed to view from the Earth is the region between the semicircle CFD and the semi circle CBD.

Fig: 1  Geometry of Crescent.

If the semi circle CFD that is above the plane of the paper (the octant with positive x and y) is projected on to the plane of the paper it forms a half ellipse. A point in the region between semi-circle CBD and CFD is represented by:

   (2.1)

 where r is the radius of the lunar disc, φ varies from 0⁰ (along z-axis) to 180⁰ (along negative z-axis) and θ varies from 90⁰ –E (plane of semi-circle DFC) to 90⁰ (plane of paper). If all these points are projected on the plane of the paper we obtain the crescent that can be seen from the Earth. The region is shown in Fig. 2 as the region enclosed between the half ellipse CF’D and the half circle CBD. The outer limb is described by a circle of radius equal to radius of the Moon given parametrically by:

                        (2.2)

Fig. 2 Width of Crescent

 The inner terminator CF’D is described by:

                      (2.3)

 which is an ellipse. The width of the crescent along its length at any point is then the difference between the y-coordinates for same values of z:        

                                                (2.4)

This width of the crescent is defined parallel to the plane containing the centers of the Sun, the Moon and the observer (segment JK in fig. 2). The width defined by McNally is in the radial direction (segment HK in fig. 2). The reason that we have defined the width in this way is that as the lunar phase increases JK  increases from zero at any j to rSinj (0⁰ to 180⁰ along OB) whereas McNally’s model gives the width W in radial direction so that at 90⁰ elongations ΔR is same as the semi-diameter R. According to our definition (2.4) that considers the latitudinal width describes the variation in width for all values of elongation. The definition (2.4) shows that the width at the cusps (  and ) geometrically vanishes. The width of the crescent at its center (j = 90⁰) is simply:

                                                     (2.5) 

This agrees with the definition of McNally:

                            (2.5a)

Which also yield the same central width (at φ = 90⁰) as given by (2.5).

The Table 1 shows the widths of crescent for various elongations against the displacement from the cusp to the center of crescent according to (2.4). This describes the width of complete Moon at every latitude of the lunar surface. Thereby for 90⁰ elongations the crescent “width” goes from around 77 arc seconds at 5⁰ from cusp to the maximum “width” of 882 arc seconds at 90⁰ from the cusp. McNally has reported this only up to elongation 90 ⁰ in table II (pp 426, 1983). It shows the maximum width of illuminated lunar surface to be 932 arc-sec for 90⁰ for every point from cusp to center of the crescent. Table 1 is generated for only the minimum semi-diameter of lunar disc when the Moon is at the greatest distance. Similar data can be generated for various semi-diameters up to the maximum possible (when the Moon is closest).

A vector normal to the point given by (2.1) is      and the unit vector in the direction of the Sun at the

same point is     . Therefore the solar flux received at any point of the illuminated surface of the Moon is given by:

      .                                      (3.1)

\Where F_m is the maximum solar flux at the point on the lunar surface where the Sun is at the zenith. In view of equation (3.1) as θ varies from 90⁰ – E to 90⁰ the flux received by lunar surface decreases sharply for values of θ close to 90⁰ – E. The total solar flux received by the visible crescent, integrated over the whole region     that defines the visible crescent, is then given by:

                                                 (3.2)

This total solar flux received by the crescent is the measure of the brightness of the crescent. It depends on the maximum possible solar flux F_m, received by the Moon that itself depend on the distance between the Sun and the Moon, the semi diameter r and the elongation E of the Moon from the Sun. Thus brightness and consequently the visibility of the crescent must depend on all these factors.

The actual brightness of the visible crescent depends on the albedo “a” (reflectivity of lunar surface) and on the distance R of the observer from the Moon. The intensity

Table 1: Width of crescent along its length from cusp to the center according to Formula (2.4) using smallest value 882 arc-sec for semi-diameter of lunar disc.

of light decreases as the square of the distance, therefore the actual brightness of the visible crescent is:

     or                       (3.3)

where S is the semi-diameter,     of the Moon and      is the phase of the Moon. The equations (3.3) show that

the brightness of the crescent varies as square of its semi-diameter. Across the length of the crescent both the width W and the brightness B varies, being greatest at the center and decreasing away from the centre. If (3.1) is integrated only over θ then the expression for brightness along a constant φ is:

                                        (3.4)

(2.4) gives the width of the crescent for fixed φ as:

                           (3.5) 

Mathematically both the brightness and the width vanish as φ approaches 0⁰ or 180⁰ and both these expressions may be re-written as:

                                        (3.6)

                   (3.7)           

where ψ varies from 0⁰ to 90⁰ along the length of the crescent from centre to a cusp respectively. The plot of the variations in the brightness depending on y and E only is shown in Fig. 3 is in agreement with the Fig No 2 (pp 270) of Schaeffer (1991) in the sense that both exhibit a sharp decline in the brightness close to the cusp. Fig. 4 shows the variations in the magnitude of the crescent along its length from center to the cusp for different elongation. Taking into consideration the semi-diameter (r/R), the maximum solar flux F_m on the lunar surface (at any point of time) and the albedo of the lunar surface more realistic brightness values may be obtained.

width of crescent is not only dependent on the Elongation as treated by many authors that have dealt with the problem of the length of lunar crescent and it’s shortening but it also depends upon the actual radius, semi-diameter, of the Moon that appeared in the expression above. The minimum semi-diameter when the Moon is farthest from us is 14.7 arc minutes and the maximum is 16.74 arc-minute. Thus for the same elongation E, φ = 90⁰, W_c  may be 13.9% as great as its minimum possible value and the total area of the lunar disc when closest may be as large as 30%  to that when it is farthest. Thus there ought to be significant variations in the width and the brightness of the crescent.

For the development of any model that describes the minimum possible width that can be visible through naked eye one requires to seek guidance from the actual observations. In the history of scientifically reported observation of the very young crescent Moon, the record is that due to Pierce on February 25, 1990 (reported by Schaefer and Yallop). The crescent he claims to have seen with naked eye was just 14.8 hours and its width was 0.18 arc minute. Amongst all the recorded observations the sighting of such a young and thin crescent was never reported. In the model that is developed in this work the lower limit of the width of visible crescent is considered to be 0.18 arc minute. However this minimum is not the absolute minimum for all crescents for all possible relative altitudes (ARCV). In this work we consider this minimum of 0.18 arc-minutes of crescent width when the relative azimuth DAZ of the Moon is zero and the relative altitude ARCV gives the q-value of -0.22 according to Yallop’s criterion:

 q = (ARCV - (11·8371 - 6·3226 W

+ 0·7319 W 2 - 0·1018 W 3 )) / 10                                     (4.1)

Fig. 3: Logarithmic Dependence of Brightness of Crescent along its length on distance from crescent’s center.

Fig. 4: Magnitude of Crescent along its length for various Elongation

Thus for q = -2.2,   

ARCV =9·6371 - 6·3226 W +

0·7319 W 2 - 0·1018 W 3                                                   (4.2)

 The minimum ARCV according to this criterion giving the q-value equal to -0.22 would yield a different lower limit on the visible width of the crescent. This is caused by different relative azimuths DAZ. For the least possible ARCV (zero) the width of the invisible crescent would be around 108 arc-seconds that occurs at a large value of DAZ. In view of this criterion the minimum width of visible crescent for any ARCV is termed W_m and as the crescent is just invisible for this width:

                                                          (4.3)

 Where W_c is the theoretical central width of the crescent given by (2.5) and      is the reduced width at angle ψ from the center of the width. W_m is the width reduced by the relative altitude ARCV. In order that some part of crescent is visible      at some value ψ = ψ_m.  Thus the effective visible width of the crescent for ψ ranging from 0⁰ to ψ_m is given by:

                                                    (4.4)

 As mentioned earlier (equation 3.1) the brightness of crescent falls sharply as θ approaches 90⁰ – E  the actual visible width of crescent at any value of ψ must be less than the geometric value of width given by equation (3.7). Therefore the “effective visible width” given by (4.4) is justified. Thus it is not only the length of crescent that shortens but the visible width of crescent has to diminish also. Whenever the crescent is invisible in view of (4.3) y _m has to vanish so that:

                                                           (4.5)

 In all other cases, i.e. whenever the crescent is visible its width at some angle y _m must vanish as the crescent is never seen a complete 180⁰ in length. Thus at ψ = ψ_m

                                                      (4.6)

 Therefore in (4.6) y _m is a measure of half the length of the crescent so that the total length of the crescent is given by:

                                                           (4.7)

 Thus the crescent length can be evaluated whenever the theoretical width W_c exceeds the minimum width W_m visible according to Yallop’s q-value criterion for the particular values of ARCV. In the Fig. 5 the segment ED or AC is the minimum width W_m at any ARCV invisible according to Yallop’s criterion. The segment AB is the theoretical width W_c at the center of the crescent given by (2.5) or (2.5a). At angular separation y _m from the center of crescent ED equals     . Therefore, the points on the outer limb of the crescent that has angular separation from center greater than      should not be visible. The visible crescent then extends from D to D’ and has length 2y _m . One should note that whenever W_m (minimum visible width according to Yallop’s criterion) is greater than W_c (theoretical width) (4.6) can not be used and the crescent is not visible, i.e. it has no length.

Fig. 5

The model developed in this work to compute the length of the crescent has been applied to a number of observations reported in literature (Schaeffer, 1984, Yallop, 1998). The results are for the crescent length against the elongation, are presented in Fig. 6. The chart shows that the functional relation between crescent length and the elongation is not smooth as reported by Schaeffer on the basis of Hapke’s model. The main reasons for this is that the crescent length has to be affected by the Earth-Moon distance as claimed above and the atmospheric turbulence close to horizon as claimed by McNally and Sultan. During this work the 70 observations of Danjon (mentioned by Schaeffer and Fatoohi et. al.) could not be accessed, however the pictorial data of crescent length is being generated at the Astronomical Observatory at University of Karachi. The observed crescent length from photographic records is given in Table 2. This includes some observations made by others during past few years and their pictures are available from www.icproj.org maintained by Odeh.

Fig. 6: Crescent Lengths vrs Variations due ARCV

 Table 2. Observed & Calculated Lengths of Crescents

The model developed in this work for both the brightness and the length of crescent is mainly geometric supplemented by the Yallop’s criterion for the earliest visibility of new lunar crescent. The model provides a simple method of calculating length of lunar crescent and takes into account the atmospheric affects indirectly through Yallop’s q-value criterion. Whenever W_c < W_m the crescent length is not calculated and the crescent was not seen according to recorded observation. Whenever Yallop’s criteria indicates requirement of optical aid (ROA = require optical aid and MROA = may require optical aid) the calculated crescent length was less than 100 degrees. In case of visibility under perfect condition (VUPC) as per q-value criterion the crescent length was more than 100 degree. In all cases of easy visibility (EV) the calculated crescent length varied from 120 degrees (for low values of elongation) to 170 degrees (for large values of elongation).

The model has been tested in two ways. First, for some of the recent observations whose photographic records are available the calculated and observed crescent lengths are found to be in close agreement shown as in Table 2. The calculated values are slightly greater than the observed values. In the calculations using formula due to Sultan the maximum diameter of Blackwell disc is taken as W_m as defined in this work (if the maximum diameter is considered to be from 0.14 arc minutes to 0.16 arc minutes the results obtained are much larger). The crescent lengths obtained are much greater than the observed values in general and in one case smaller than the observed value. In calculations using the formula of McNally ΔR = W_m and R = semi-diameter of lunar disc. The root mean square deviation of our calculated results (2.2 degrees) from the observed values is much smaller than those calculated using the formula of McNally (18.05 degrees). The root mean square deviation for the technique due to Sultan is much greater (53.3 degrees). In case of use of our formula the error is always positive in the sense that our results are just more than the observations. In case of McNally’s techniques the results are inconsistent in the sense that it has both positive and the negative errors.

The second and indirect test of the model is its comparison with the results of Danjon mentioned in Fatoohi et. al. If the minimum visible width W_m for various ARCV according to Yallop’s criterion is replaced by the minimum ever central visible width of 0.18 arc minutes that is equivalent to ignoring the atmospheric affect for lower ARCV then the relation between crescent length and elongation becomes smooth as presented by Schaefer (1991). The same is shown in Fig. 7. On the basis of the calculated lengths of crescent and its elongation using our model the Danjon deficiency arcs are calculated ignoring the affects of ARCV by formula given by Danjon and that given by McNally (shown in Fig. 8 and 9 respectively). These results are in close agreement with the Fig 1 in Fatoohi et al (that is a reproduction of Danjon’s fig 2).

 Fig. 7: Crescent Lengths against Elongation without variations due to ARCV

Fig. 8: Deficiency Arc against Elongation according to Danjon.

Fig. 9: Deficiency Arc against Elongation according to McNally.

The model for the calculation of crescent length may be used as the earliest visibility criterion much in the same way as Yallop’s criterion can be used. However, our emphasis was not to develop an alternate criterion for the same. This work was intended for a better understanding of the geometry of the lunar crescent and to develop a method for calculating its length.

1.        The author is grateful to the organizers of www.icoproj.org and in particular Mr. Muhammad Shaukat Odeh who have collected pictures of young crescents that were used after seeking permission for calculating the crescent lengths.

2.        The author specially acknowledges and express his gratitude to the office of Dean Faculty of Science, University of Karachi, who sponsored this research which a part of the project entitled “Integrated Software Development for Localized Study of Lunar Dynamics” approved and funded by the office of the Dean.

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