معیار یالوپ

6. Derivation of the q-test. The visibility test parameter q is based on the Indian method, which is defined discretely by Table 2 and continuously by inequality (3.6). The parameter q is calculated at the best time from the equation q = (ARCV − (11·8371 − 6·3226 W' + 0·7319 W' 2 − 0·1018 W' 3 )) / 10 (6.1) where W' is the topocentric width of the crescent, and q has been scaled by a factor of 10 to confine it roughly to the range –1 to +1. (Note the use of the topocentric width of the crescent.) The values of q have been calculated for the 295 observations referred to in section 5, and the results are listed in Table 4 in order of decreasing q. Table 4 has also been partitioned into six ranges of q. These ranges in the q-test have been calibrated empirically by comparing the visibility code Schaefer used for the 295 observational records, with a similar code derived from the calculated value of q. It has also been found necessary to use theoretical arguments to obtain some of the limiting values for q. Table 5 lists the six criteria by type, A to F, by range in q and by visibility code. Table 5: The q-test criteria. Criterion Range Remarks Visibility Code

(A) q > +0·216 Easily visible (ARCL ≥ 12°) V

(B) +0·216 ≥ q > −0·014 Visible under perfect conditions V(V)

(C) −0·014 ≥ q > −0·160 May need optical aid to find crescent V(F)

(D) −0·160 ≥ q > −0·232 Will need optical aid to find crescent I(V)

(E) −0·232 ≥ q > −0·293 Not visible with a telescope ARCL ≤ 8·5° I(I)

(F) −0·293 ≥ q Not visible, below Danjon limit, ARCL ≤ 8° I The limiting values of q were chosen for the six criteria A to F for the following reasons:

(A) A lower limit is required to separate observations that are trivial from those that have some element of difficulty. After some experimentation, it was found that the ideal situation ARCL = 12° and DAZ = 0° produces a sensible cut-off point, for which q = +0·216. To avoid ambiguities, the constant geocentric quantity W defined by equation (3.4), that was adopted by Bruin for the crescent width, was used to calculate q from equation (6.1) instead of W'. There are 166 examples in Table 4 when q exceeds this value, and in general it should be very easy to see the Moon in these cases, provided there is no obscuring cloud in the sky.

(B) From observers reports it has been found that, in general, q = 0 is close to the lower limit for first visibility under perfect atmospheric conditions at sea level, without requiring optical aid. Table 4 is used to set this lower limit for visibility more precisely. From inspection of Table 4, Tn69x.doc 12 of 14 2004/09/22 10:14 AM the significance of q = 0 can be seen, but q = –0·014 is another possible cut-off value. There are 68 cases in Table 4 with q in this range.

(C) Table 4 was used to find the cut-off point when optical aid is always needed to find the crescent moon by matching the q-test visibility code with Schaefer’s code. The rounded value of q = – 0·160 was chosen for the cut-off criterion. Entry number 44, the first entry in the next group, was ignored because it is false. In Table 4 there are 26 cases that satisfy this criterion.

(D) In this case Table 4 has too few entries from which to estimate a lower limit for q. The situation is made worse by the fact that where there is an entry, in most cases, the Moon was not seen even with optical aid. In fact it is rare for the crescent to be observed below an apparent elongation of about 7°·5, see Fatoohi et al (1998). Table 4 has 17 cases. This is the current limit below which it is not possible to see the thin crescent moon with a telescope. Allowing 1° for horizontal parallax of the Moon, and ignoring the effect of refraction, for an apparent elongation of 7°·5, ARCL = 8°·5. If DAZ = 0° this corresponds to a lower limit of q = –0·232. Without good finding telescopes and positional information, observers are unlikely to see the crescent below this limit. The only sighting that was seen both optically and visually near this limit was No. 278 for which q = –0·222. It is an important observation because it is the observation with the smallest elongation, see Table 6. If the Moon were observed near this elongation and it was also at or near perigee, Age would be about 12 hours.

(E) There is a theoretical cut-off point when the apparent elongation of the Moon from the Sun is 7°, known as the Danjon limit. This limit is obtained by extrapolating observations made at larger elongations. Allowing 1° for horizontal parallax of the Moon, and ignoring the effect of refraction, an apparent elongation of 7° is equivalent to ARCL = 8°. With ARCL = 8°and DAZ = 0° the corresponding lower limit on q is –0·293.

(F) In Table 4 there are only 17 cases in this range of q, but three of them (169, 194 and 195) contradict the q-test, in particular, 194 and 195 are anomalous observations. The main reason for the discrepancy must be due to the extremely clear atmosphere experienced on high altitude mountain sites. The elongations, however, are well above the Danjon limit, and since ARCV is about 4°, the observations were probably made using daylight vision. These observations show that the curve q = 0 needs modifying for high altitude observations. No 169 was made at ARCL = 12°·4, and width W' = 0'·39, which are both large. Since ARCV = 6°, it should have been possible to make the observation. This observation shows also that the curve q = 0 needs modifying when the atmospheric conditions are so perfect