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Extinction angle and heliacal events

<in the pictures:
the km values for the Visibility Range should be replaced with the
astronomical extinction coefficient values:

13 km = 0.45, 19 km = 0.35, 30 km = 0.15, 140 km =0.12>

Visibility

The extinction angle determines at what apparent altitude a celestial body becomes visible. The above picture has been made with information from Bradley Schaefer [2000]. A JavaScript and Excel Add-in Visual Basic (VB) program is available. The assumptions used within the Schaefer implementation can be seen here.

The thick blue line (winter solstice at average present  Irish conditions: 80%, 5° C) is the extinction angle depending on the Magnitude of the celestial body (moon and sun as much as possible under horizon). The thin lines provide the 1s boundaries (the purple text gives an idea of the astronomical extinction coefficient).
An astronomical extinction coefficient between 0.15 and 0.45 [-] looks representative.
The green-blue crosses are sightings in south England under favorable conditions from North [1996, page 34] (which compares to an astronomical extinction coefficient of 0.12 [-] at 10% RH, which compares indeed as good conditions). And the pink line is the rule of thumb from Thom [1976, page 15].
The red lines are the apparent altitudes which are visible through the roof box of Newgrange.
An example: In winter time Sirius (the brightest star: Magnitude -1.46) is visible when it is at an apparent altitude of  higher than appr. 2.3°.

Arcus Visionis and heliacal altitude

The minimum Arcus Visionis is the smallest unrefracted angle difference between the star (in general: celestial object) and the sun while the star is just visible on the heliacal (rise/set) day (as defined by Schaefer, [2000]).

In the below picture one can see the minimum Arcus Visionis as a function of the object's Magnitude (same conditions as above are used and variable astronomical extinction coefficient).

Arcus Visionis
The colored lines represent different astronomical extinction coefficients.

The object's helical (rise/set) altitude, the minimum Arcus Visionis and the Sun's altitude can be seen below (again for an astronomical extinction coefficient of 0.35 [continuous line] and its 1 sigma boundaries: being 0.15 and 0.45 [dotted lines]):
Heliacal rise/set angle
(the discontinuity in the heliacal/sun's altitude is due to night vision)

It is interesting to see that the Sun's altitude at the moment of heliacal altitude is quite independent on the astronomical extinction coefficient

Checking with other sources

The above graph also hold the Sun's altitude information gotten from Meeus ([1997], page 289 and 295; given as a yellow line). The purple line gives the default formula used in the program PLSV 3.1. Meeus information seems to agree with Schaefer's model (except when night vision starts to get into play: Magn >3). The default values of PLSV seems to be somewhat low, this might be because its default values for its 'Arcus Visionis' are comparable values of Ptolemy's/Schoch's Arcus Visionis, see below.

Another check is done with this article of Kolev [2001]:

Reijs' results
Kolev's observations
[2001]
Location
Star
Heliacal
event
Date
First time UT
[hh:mm]
Time visible
[min]
Arcus Visionis (opt) [o]
Ext. Coeff.
[-]
First time UT
[hh:mm]
Time visible [min]
Arcus Visionis [o]
Visibility
Varna Tashla, BG
procyon morning  first/heli. rise  2000/8/15
2:18 13 14.3 0.25
2:19 13 13.8*
good
Varna Tashla, BG
procyon morning  first/heli. rise  2000/8/16
2:25 3 15.7 0.33
2:21 3 ???
bad
Varna Tashla, BG
sirius morning  first/heli. rise  2000/8/17
2:38 12 10.8 0.29
2:32 12 10.7
bad
Varna, BG
procyon morning first/heli. rise  2000/8/18
2:26 2 17.6 0.43
2:33 ??? ???
bad
Varna, BG
sirius morning first/heli. rise  2000/8/19
2:32 24 12.5 0.36
2:32 25 12.5
average
* 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]).

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

Reijs' results
Kolev's observations
[2007]
Location
Star
Heliacal
event
Date
First time UT
[hh:mm]
Time visible
[min]
Arcus Visionis (opt) [o]
Ext. Coeff.
[-]
First time UT
[hh:mm]
Time visible [min]
Arcus Visionis [o]
Measured Ext. Coeff. [-]
Varna, BG
sirius evening first/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 Reijs' results are determined by using Schaefer's methodology; by varying the astronomical  extinction coefficient (Ext. coeff.) in such a way that a Date and Time visible are achieved as observed by Kolev. One can see that the First time of Reijs and Kolev are quite similar. The qualitative Visibility and quantitative Ext. coeff. have some correlation.
There looks to be a good fit with Kolev's practical observations and Reijs' implementation of Schaefer's methodology.

Relation between Arcus Visionis and heliacal altitude

A sample Arcus Visionis (say of ~12 degrees) on a specific day (on which the star will be firstly/lastly visible: heliacal day, in this case Aug. 31st, 2001) can be seen in below picture (assume an object's Magnitude of -1.46, aka Sirius):
Sample Arcus Visionis
On this heliacal day the object is visible when it is between ~4.5 and ~7.25 degrees apparent altitude (average astronomical extinction coefficient of 0.2 [-]; continuous blue line). The chance that a celestial object is only visible when it is precisely at its helical apparent altitude (5.5 degrees) is small, because then the actual Arcus Visionis must be precisely 11.5 degrees (minimum Arcus Visionis) at the start/end of the day.

Sensitivity analysis of the heliacal date

In the above one can see there is some inaccuracy, this section gives some more insight in this. A sensitivity analysis of the heliacal date is given for the most important parameters (parameters that are more or less out of control when theory forming was done in former times, less well documented observations in past or present and interpretating literature):
  • Latitude
    People using published heliacal dates could be anywhere in the world. I assumed a 1 sigma between 53 N and 33 N (average 43 N).
  • Total astronomical extinction coefficient (ktot)
    ktot can vary a lot. It is assumed that the variation (say 95%=2 sigma) is between 0.15 and 0.65. Thus 1 sigma is between 0.25 and 0.55 (average: 0.4).
  • Magnitude of the object
    Used here are the Magnitude changes of Mercury; between 2 and -1.5 (for 95% of the cases). So the 1 sigma is between 1.2 to -0.62 (average: 0.3)
  • Acuity of the observer
    Most people are between 20/10 and 10/20 (so I assume this is again 95%). The 1 sigma is thus around 20/15 (~1.33) and 15/20 (~0.75) (average 20/20=1)
  • Additional azimuth difference.
    The azimuth difference between Sun and object might change. A 1 sigma of 15 degrees has been used (this might be too large even) (average: 0).
If using the average values derived from the above minimum and maximum values and using it on Procyon (is close to the average Magnitude of Mercury but has not such a short heliacal periodicy), one gets a heliacal rise date of August 18, 2000. When changing the parameters independently, the following results are seen for the stars and most planets:
  • Latitude: August 13th (33 N) to August 24th (53 N): thus 1 sigma: ~5.5 days
  • ktot: August 21st (0.55) to August 15th (0.25): thus 1 sigma: ~3 days
  • Magnitude: August 21st (1.2) to August 15th (-0.62): thus 1 sigma: ~3 days
  • Acuity: August 20st (0.75) to August 16th (1.33): thus 1 sigma: ~2 days
  • Additional azimuth difference: August 17th (+15 deg) to August 18th (-15 deg): thus 1 sigma: ~0.5 days
  • Other parameters like (Age of person, Latitude, Eye height, Temperature and Air Pressure) have no large impact (compared with above)
So the resulting 1 sigma for the heliacal rise date is: 7 days!!!
If the latitude of the observation is compensated, the 1 sigma becomes lower, around 5 days
In a lot of cases (like stars and most planets) the error will be even smaller, because Magnitude is well known and Additional azimuth difference is close to zero/constant, thus more likely: 4 days.
If the weather conditions are very well documented (or known to be stable) and the acuity of the person is known;  the error (1 sigma) could be reduced even more, but not expected to be lower then 1 to 2 days.

Mean periodicy of heliacal rises

In Ingham [1969] the mean periodicy of Sirius' heliacal rise is being studied related to the Sothic cycle. Using the above visibility methodology of Schaefer, the periodicity of heliacal rise/set of Sirius and some other stars have been calculated, see below picture. The period of heliacal rise/set is defined as follows:
The time difference between two successive moments when the sun is at its optimum altitude for heliacal rise/set (which is altitude when the min. Arcus Visionis would have happened) and the celestial object is first visible after its conjunction with the Sun.

Heliacal periodicy

Comparing Ingham's results (his Table 3) with the Reijs' results for Sirius gives:
Middle
astronomical
year
Reijs' results
[Day]/[Solar day]
Ingham's results
[Solar day]
Table 3
-3498
365.25000/365.25042
365.25051
-2043
365.25035/365.25066
365.25085
-590
365.25104/365.25124
365.25126
861
365.25190/365.25199
365.25181

It is most likely that Ingham used Solar days as his definition (also looking at the publication date: 1969). Another difference is that Ingham used a theoretical min. Arcus Visionis, while Reijs' min. Arcus Visionis is using Schaefer's methodology.

Interestingly the mean periodicy of several stars is close to the Julian Year (365.25 Days, with Days of 86400 SI Sec), some people see this as a reason why Sirius played its important role in Egyptian calendar. As you can see Sirius is not the only one and furthermore would the Egyptians have been able to note this difference in periodicy (0.004%). Although: Sirius is of course the brightest star!

Literature and Arcus Visionis

Below are some values quoted from public sources (Ptolemy , Schoch and Meeus) and calculated by Reijs using Schaefer's methodology. Cursive values are for Arcus Visionis (object at zero altitude) and  normal font for the min. Arcus Visionis (as defined by Schaefer; unrefracted angle between Sun and object) [all in degree]:
Object Ptolemy
Handy Tables
Schoch
Meeus
[1997]

Reijs*
hr/hs
ar/cs
hr
ar
cs
hs
hr/hs hr
ar
cs
hs
Saturn
13

13


10
10
13.5
10.0
9.6
12.8
Jupiter
9

9.3


7.4
8
9.7
6.4
6.0
9.8
Mars
14.5

14.5


13.2
11
14.0
7.6
7.0
16.0
Venus
5
7
5.7
6
6
5.2
6
6.9
7.6
7.3
6.4
Mercury
12

13
9.5
10.5
11
-
14.8
10.2
11.3
16.1
hr: heliacal rise (or morning first)
ar: acronycal rise (or morning last)
cs: cosmical set (or evening first)
hs: heliacal set (or evening last)

*Reijs has calculated it for Babylon, astronomical extinction coefficient of 0.12, RH 40% and around 2007 CE.
hr (heliacal rise) and hs (heliacal set) have comparable values and also cs (cosmical set) and ar (acronycal rise) have comparable values; this because the Sun-object azimuth differences in both situations are comparable (as a general rule).

Some part of the differences in the above values is due to the difference in Arcus Visionis definition. For Mars (around 2007 CE), for at present an unknown reason, an Arcus Visionis of 13 degrees is equivalent with an Arcus Visionis of 14 degrees. At other dates Mars behaves like the others;-) For other planets and stars the difference is more around 0.1 to 0.4 degrees.

Different definitions of Arcus Visionis

In Ptolemy, Pickering [2002] and Ingham [1969], the unrefracted angle the Sun and the object at zero altitude, is defined as the Arcus Visionis: so that is a few minutes before the actual event takes place.
The Arcus Visionis is not always an actual event that can be witnessed; most celestial objects will not yet be visible at an altitude of precisely zero degrees.

It is important to realize this difference in definition. On this web page Arcus Visionis means Schaefer's definition.
The Arcus Visionis (non cursive text; Schaefer's definition) is more versatile when also incorporating the new moon visibility; where the altitude angle difference between Sun and Moon (object) is more obvious.

Meeus on the other hand does not use the term Arcus Visionis, he used two angles: the Sun's altitude and the object's apparent altitude at the moment of the heliacal event. Adding these together gives he Schaefer's Arcus Visionis (if using the object's altitude and not its apparent altitude!). PLSV is equivalent with Meeus and Schaefer, except that it calls the Sun's altitude; 'Arcus Visionis' (but it defaults its 'Arcus Visionis' to the values of Ptolemy's/Schoch's Arcus Visionis).
So it looks like there are several definitions of Arcus Visionis (the first four definitions can be calculated with the Excel Add-in):
  1. Schaefer:
    Arcus Visionis: unrefracted angle difference of Sun and object at the moment of the heliacal event. This definition is used on this web site. By the way; Schaefer calculates the Arcus Visionis from the Sun's altitude and the object's altitude.
  2. Meeus:
    The Sun's altitude and object's apparent altitude at the moment the object is at its heliacal event position.
  3. Ptolemy:
    Arcus Visionis: unrefracted angle difference of Sun and object at the moment the object is at zero altitude
  4. PLSV:
    The Sun's altitude (called 'Arcus Visionis', which is different then Ptolemy's Arcus Visionis and Schaefer's Arcus Visionis) at the moment the object is at its heliacal event position (its Critical altitude: the object's apparent altitude)
  5. Arc of Vision:
    Some people see Arc of Vision as just a translation of Arcus Visionis. But there is an other meaning used in literature:
    "The Arc of Vision of the Moon is the time, which the Moon needs to set from the moment of the sunset (Al-Batani [c. 890 CE]). This can also be known as Arc Appartionis." [Pers. comm: A. Belenkiy].

Magnitudes of planets

In general: A difference of 1 magnitude corresponds with a difference of 2.512 in brightness (a 5 magnitude difference is equal to a factor of 100 in brightness). In general celestial objects which are brighter than a magnitude of 6 can be visible sometime in the sky.

 The planets have the following maximum visible magnitudes (the actual magnitude is depending on phase of planet and/or ring position):
 
Planet
Magnitude (maximum)
Sun -27
Moon -13
Venus -4.7
Mars -2.9
Jupiter -2.8
Mercury -1.9
Saturn
0.7
Uranus 5.5
Neptune 7.7
Pluto 13

Casting of shadows

The moon brightness at low apparent altitudes is much lower than when high up the sky due to extinction. See below picture where the moon brightness varies with 100,000-1,000 less within the Newgrange roofbox range than high up:

The casting of shadows depends on the brightness of the celestial object and the amount of background light from other celestial objects.

The above picture represents if the moon light casts a shadow inside Newgrange on three dates: Remember that the above shadows will only be visible if the eyes are used to darkness (at least 30 minutes for scotopic vision) and make sure that no stray/artificial light can come through any non-essential hole/passage. It could be that Generation III nigh vision camera's can record the moon light.

From literature (Waugh, [1973, page 167]) it looks like that the moon still casts a shadows in open air when its phase is 7 days before (or after) full moon. Its magnitude is then around -10.
I have experience that with full moon at an apparent altitude of 1.5° and the sun at nautical twilight, I could discern a shadow in closed duct, aligned to the moon. If people have comparable experience, please let me know.
I got an e-mail reaction that the Venus (magn -4) shadow has been seen quiet faintly. This was at an ideal location: Apache Point, about 9000 feet above sea level in the mountains of New Mexico and the sun had set more than an hour earlier.

Under good conditions, of the planets only the moon, venus and the sun can cast a shadow, and of the stars: non.

Acknowledgments

I would like to thank the following people for their help and constructive feedback: Ari Belenkiy, Rumen Kolev, Dieter Koch, Jim Lowdermilk, Keith Pickering, Bradley Schaefer, Thomas Schmidt 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.


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Last major content related changes: March 17, 2007