Checking with Kolev data

^* for Procyon it seems that Kolev has different ephemeris information then Reijs (also the time-altitude pair in his article does not map Reijs' ephemeris [Swiss Ephemeris]). This has been rectified in Kolev (2013).

A further check is done with this article of Kolev [May 2007]:

				Reijs' results				Kolev's observations [2007]
Location	Object	Heliacal event	Date	First time UT [hh:mm]	Time visible [min]	Topo ARCV (opt) [o]	AECRK**** [-]	First time UT [hh:mm]	Time visible [min]	AV [o]	AECRK**** [-]
Varna, BG	Sirius	evening last/heli. set	2000/5/10	17:45	16	12.6	0.40	17:45	???	12.0	0.42
Varna, BG	Sirius	morning first/heli. rise	2000/8/17	2:38	12	10.9	0.30	2:32	12	10.8	0.36
Rila, BG	Sirius	morning first/heli. rise	2004/8/16	2:57	13	10.9	0.29	3.04	13	10.8	???
Varna, BG	Sirius	morning  first/heli. rise	2006/8/19	2:33	23	12.1	0.33	2.24**	28	12.0	0.42
Seattle, USA	Sirius	morning  first/heli. rise	2003/8/19	12:28	15	8.2	0.16	12:28	16	8.2	0.16
Seattle, USA	Sirius	morning  first/heli. rise	2005/8/19	12:29	17	8.6	0.17	12:30	17	8.6	0.20

^** In the article there might be an error/typo (in the article 3.24 was stated, so 1 hour too late for a heliacal rise time of the star).
^**** The AEC_RK of Kolev is based on a slightly different Airmass function. So the AEC_RK is slightly smaller than AEC used in Schaefer's criterion.

A check has been done with several Mercury observations by Kolev ([2006]):

			Reijs' results	Kolev's observations [2006]
Observation	Location	Event	AEC****	Visibility
1	Seattle, USA	morning first	0.22	clear
2	Seattle, USA	morning last	0.19	clouds (clear before/after)
3	Seattle, USA	evening last	0.25	clouds (clear before/after)
4	Varna, BG	morning first	0.21	very clear***
5	Varna, BG	morning last	0.21	very clear
6	Varna, BG	morning last	0.20	very clear
7	Varna, BG	evening last	0.29	fair clear
8	Varna, BG	evening last	0.19	clouds? (clear before/after)

^*** Kolev states for this observation an AEC_RK of 0.46, which sounds very larger for his Visibility of very clear.

The Reijs' results are determined by using Schaefer's criterion; by varying the Astronomical  Extinction Coefficient (AEC) in such a way that the Date and Time visible are achieved as observed by Kolev. One can see that the First time of Reijs and Kolev are quite similar.
If Kolev did not observe the event (for instance due to clouds); he determined the Date with his own method, Reijs determined the average AEC from the lowest and highest AEC still given Kolev's Date.
The qualitative Visibility and quantitative AEC[_RK] have some correlation, so there looks to be a fit with Kolev's practical observations and Reijs' implementation of Schaefer's criterion.

Checking Curtis data

Curtis [1936] did an observation of Venus on September 5th, 1935 CE during inferior conjunction (so no heliacal event, but during the day). Schaefer's criterion can also predict such an event, of course.
Curtis witnessed Venus during day time (around midday). An animation using Schaefer's criterion can be seen here:


One clearly can see that Venus can be visible during the day around the time Curtis has also saw it. To map this with Schaefer's criterion, an acuity of 1.6 was needed; which is possible as this is slightly better than average acuity (=1.3).

Conclusions

2 parameter criterions

Most people use a two parameter criterions (like Fotheringham [1910, Geo based], Maunder [1911, Geo based], Schoch [1927, Geo based], Neugebauer [1930, Geo based], Indian [1966, Geo based], Bruin [1977, Geo based], Yallop [1998, Topo based], Caldwell&Laney [2001, App based], Moonsighting (since 2002, Topo based), Odeh [2004, Topo based], Hoffman [2005, Topo based], Qureshi [2010, Topo based], etc.) to predict visibility of celestial objects (remember: these are more or less calibrated for a certain latitude and (average/known/assumed?) meteorological situation).
A good paper looking at several two parameter criterions is from R. Hoffman (Observatory, 125, 156-168 [2005]), he distinguishes between Moon's illuminance and sky's darkness related parameters (a more elaborated view is given below).

An overview of different two parameter criterions is here:

Criterion	Year published	Valid locations	Parameter set	Best Time	Number of categories	Method	Derived from
Fotheringham	1910	Greece	(GeoARCV, DAZ)	GeoAltS@0	2	Emp.
Maunder	1911	Greece + West Europe	(GeoARCV, DAZ)	GeoAltS@0	2	Emp.
Schoch	1927	Babylon	(GeoARCV, DAZ)	GeoAltS@0	2	Emp.
Neugebauer	1929	Babylon	(GeoARCV, DAZ)	GeoAltS@0	2	Emp.	Schoch
Indian	1966	Babylon	(GeoARCV, DAZ)	GeoAltS@0	2	Emp.	Neugebauer
Bruin	1977	Babylon	(GeoARCV, GeoW)	GeoAltS@0	2	Theo.
Yallop	1998	Many	(GeoARCV, TopoW)	(TopoAltM@0*4+TopoAltS@0*5)/9	7 q-values	Emp.	Indian, Bruin
Caldwell&Laney	2001	Many	(AppARCL/3+AppARCV, DAZ)	AppAltS@0	3	Emp.
Moonsighting	2002	Many	(GeoARCV, TopoW)	(GeoAltM@0*4+GeoAltS@0*5)/9	7 q-values	Emp.	Yallop, different q-values
Odeh	2004	Many	Topocentric	(TopoAltM@TopoDip*4+TopoAltS@TopoDip*5)/9	4	Emp.	Yallop
Hoffman	2005	Many	(AppAltM-TopoAltS, TopoW)	(TopoAltM@0*6+TopoAltS@0*4)/10	4	Emp.	Yallop
Reijs	2007	Many	(Geo/TopoARCV, DAZ/TopoW)	TopoAltM@Maximum contrast	2	Theo.	Schaefer
Qureshi	2010	Many	(GeoARCV, TopoW)	(GeoAltM@0*4.3+GeoAltS@0*5)/9.3	6 s-values	Theo.	Odeh, Yallop, Schaefer

The values belonging to the categories of Yallop, Moonsighting and Qureshi are:

Yallop [1998, page 12]			Moonsighting (pers. comm. K. Shaukat, [2011])			Qureshi [2010, page 16]
Category	Description	q-values	Category	Description	q-values	Category	Description	s-values
A	Easily visible, ARCL >= 12	q>=0.216	A	Easily visible to the naked eye	q>=0.27	EV	Easily visible	s>=0.15
B	Visible under perfect conditions	0.216>q>=-0.014	B	Visible to the naked eye under perfect atmospheric conditions	0.27>q>=-0.024	VUPC	Visible under perfect conditions	0.15>s>=0.05
C	May need optical aid to find the crescent	-0.014>q>=-0.16	C	May need optical aid to find the crescent before it can be seen with naked eye	-0.024>q>=-0.212	MROA	May require optical aid to find crescent	0.05>s>=-0.06
D	Will need optical aid to find crescent	-0.16>q>=-0.232	D	Can only be seen with binoculars or a (small) conventional telescope	-0.212>q>=-0.48	ROA	Require optical aid	-0.06>s>=-0.16
E	Not visible with a telescope, ARCL <= 8.5	-0.232>q>=-0.293	E	Below the normal limit of detection with a (small) conventional telescope	-0.48>q>=-0.523
F	Not visible below Danjon limit, ARCL <= 8	q<-0.293	F	Not visible with even large observatory type telescopes	q<-0.523	I	Not visible with optical aid	s<-0.16

Below is a numeric comparison (Yallop's curve is converted from topocentric crescent width to a geocentric ARCV- and azimuth-differences at reference time of GeoAltS@0 and Caldwell&Laney's curve is converted from apparent altitudes to geocentric ARCV at reference time of AppAltS@0). Maunder(?), Schoch, Neugebauer, Indian, Bruin and Caldwell&Laney are close to Yallop_q=-0.014 (B category: Visible under perfect conditions).
Caldwell&Laney curve is so close to B Category, as they state ([2001], page 19):

Fotheringham looks to be close to a Yallop_q of 0.216 (A category: Easily visible) curve, which can be related perhaps to two quotes of Fotheringham and one comment of Maunder (text between <> is mine):

1.  Fotheringham ([1921, page  311):
   "... it is essential that the Moon should not be discovered either with a pointer or with any kind of optical glass, even if she is seen with the naked eye after being so discovered."
2. Fotheringham (The visibility of the lunar crescent, In The Venus tablets of Ammizaduga, 1928, Oxford Press, page 35)
   "This <Schoch's> formula implies a greater transparency of the air <for Babylonian observations> at all altitudes than I had deduced from the Athens observations."
3. Maunders (heard through pers. comm. R. Krauss, [2011])
   "<Fotheringham> based his dividing line upon the negative observations <aim to reduce Type II errors>, whereas it should have been based upon the positive. For the latter, if accepted, are definite and decisive, the former are not."
   

There is an interesting difference between Fotheringham and Maunder as they used almost the same data (the Greek data was at least common). Here is graph of Fotheringham's data [1910]:

These two parameter criterions are also very close to each other and don't really bring in new ideas, except perhaps a slightly different curve! Yallop's is in some way nice (based on the criterion of Indian [1966], which in its turn is based on Neugebauer/Schoch's) as it adds 7 categories to the whole process of visibility; so that is a step in a good direction. By the way: a similar methodology of categories can be utilised on any other two-parameter criterion (as implemented in ARCHAEOCOSMO).
Yallop also recognises that the criterion does not really work in certain conditions; like good meteorological conditions and large heights (Yallop [1998], page 12-13). And under such relatively normal conditions, more parameters are essential to explain so called outliners (while one should not see them as outliners)!

Yallop's C category

Experienced people (in the past and now) know where the celestial object will be in the sky and thus a strict optical aid (using lenses) is not essential. Any grid/alignment would be an 'optical aid' to find the object, IMHO. And perhaps that is also how Yallop allows for in C category (May need optical aid to find the crescent) (I have asked HMNAO). Of course weather conditions are essential in this C category definition (as is mentioned in the text of B category, but I think it is even more important in C category!).
So I would evaluated Yallop's definition of his C category may need any optical aid (and not only involving lenses) and stress the importance of the weather and geographic conditions.

Confining boundaries due to Sun and Moon position (First/last crescent window)

Beside the normal visibility due to physical, physiological and meteorological parameters (the boundary between the blue en pink areas), there is also a restriction due to the relative location of the Moon related to the Sun around first/last crescent moments. This is related to the speed and declination of these celestial bodies.
When one calculates the GeoARCV and DAZ for first crescent in the period -750 to 1220 (some 24000 events), one gets the following graph (using Schaefer's criterion for determining the GeoACRV and DAZ of the first crescent (blue) and it also incorporates these parameter values for the day before the first viability (pink)):
So first/old crescent events are not bounded only by the boundary between blue and pink area (first visibility due to physical, physiological and meteorological parameters), but also by this straight line on the right side (first visibility due to the constraining relative position of Moon and Sun around certain Equinox)!

The two straight lines are depending on the latitude (φ) of the observations, the Earth's obliquity (ε), the Moon's inclination (i) and astronomical refraction (this is thus different than depicted in Bruin [1977], page 338 Fig.6):

For the latitude of Athens the straight lines are as depicting here:

Meteorological parameters

It is recognized that a criterion to pre/postdict the visibility of a celestial objects will need more than just two parameters! It is also known that the meteorological parameters have a large influence on the first/last visibility date (Schaefer [1993], Fatoohi [1999], Hoffman [2005], Caldwell&Laney [2001], etc.). The issue is that most sources just leave it with that statement and sadly continue to base their theories on results from two parameter criterions, without properly evaluating/mentioning/incorporating meteorological (and geographic) parameters.

Main parameters in Schaefer's theory that are important for heliacal events with the Moon (day/photopic vision, average acuity and Sun determines sky's brightness), are:

Process	Main parameters
Astronomical Extinction coefficient (AEC)	Geographic (Eye Height, Latitude), Meteorological (Relative Humidity,Temperature), Season (Sun's Right Ascension)
Airmass between object/sky and observer	Viewing object's altitude (AltM), AEC
Object's brightness at observer	Object's Magnitude (Distance Object-Sun, Distance Object-Earth, Object's Phase angle [ARCL]), Airmass
Sky's brightness at observer	Sun's Magnitude, Sun-Object angle (ARCL), Airmass

A problem when evaluating old and recent sources of observations is that meteorological parameters are not noted:-(  And this is a major shortfall of these sources (certainly for recent sources as we now know it is essential to record).
There are a few databases that are an exception to this sad practice: Hoffman (provides temperature and RH) and Schaefer's database (added visibility later as estimates) and Kolev (measures visibility at time of observation). Kolev uses the best practice: he includes a measured estimation of the actual astronomical extinction coefficient (AEC)!

Due to these unknown meteorological parameters, it is very easy to map Schaefer's criterion (Topo based) on an actual observation, as the physics of visibility is so variable on these meteorological parameters.
Using Schaefer's criterion, probabilities can be linked to the possible heliacal date depending on e.g. AEC and/or Visual acuity, etc. by using Monte Carlo analysis. An example for Venus at Babylon is:
 
So postdicting an event is relative easy with Schaefer's criterion (with actually recorded or estimated meteorological parameters), but predicting an event is more difficult if one does not know the meteorological parameters values for the prediction.
The only correct way is thus to give a range of likely visibility dates!

Schaefer's criterion

Even if Schaefer's criterion is seen by some people as "considerably more optimistic than all the other criterions that have been proposed", the fact stays that Schaefer's criterion is the only one that allows to study the effects of meteorological and geographic parameters (which are essential for understanding heliacal events).
This shows already in the very interesting results seen in IFA, daytime observations, analysis of meteorological influences, form of visibility curve, variability of dates due to several parameters, shadow casting and post/prediction dates.

Checking positive and negative observations

So it is perhaps great to see that all positive observations (until now checked) can be reproduced with Schaefer's criterion. But let this not give me/us false hope;-)
Work is being done to compare false/true positive observations (where true positive observation: actually observed celestial objects is postdicted (X)) and false/true negative observations (where true negative observation: actually not observed celestial objects is postdicted (Y)) with the different criterions. Post/predicting true positive observations is relatively easy, but also being correct for true negative observations is more difficult.

So per criterion, the number of observations in each of the four below cells need to be determined and if one has multiple categories (which is very handy) a cluster analysis is needed. The best criterion will have the smallest percentage for P and Q (Type I and II errors):

	Actually observed (positive obs.)	Actually not observed (negative obs.)
correct (true) postdiction	X	Y
incorrect (false) postdiction	P (Type I error)	Q (Type II error)

Acknowledgements

I would like to thank the following people for their help and constructive feedback:  Roy Hoffman, Dieter Koch, Rumen Kolev, Rolf Krauss, Khalid Shaukat and all other unmentioned people. Any remaining errors in methodology or results are my responsibility of course!!! If you want to provide constructive feedback, let me know.

---

Disclaimer and Copyright
HomeUpSearchMail

---

Last major content related changes: Nov. 20, 2008