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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>

Extinction
        nagle calcuated by VR

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 and AEC=0.3) 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 (0.07).
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 0.3 [-] at 10% RH). 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. 1°.

Topocentric Arcus Visionis and heliacal altitude

The minimum Topocentric Arcus Visionis is the smallest unrefracted angle difference between the star (in general: celestial object) and the Sun while the celestial object is just visible on the heliacal (rise/set) day.

In the below picture one can see the minimum Topocentric 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 Topocentric 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.

Changeable extinction coefficient

The extinction coefficient is quite changeable; see these examples of the atmospheric visibility for Europe and Asia:

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 formula of PLSV seems to be provide somewhat low, this might be because its default values for its 'Arcus Visionis' are comparable interpolated values of Ptolemy's/Schoch's Arcus Visionis, see below.

More checks to benchmark Schaefer's criterion can be found on this web page.

Relation between Topocentric Arcus Visionis and heliacal altitude

A sample Topocentric 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 Topocentric Arcus Visionis must be precisely 11.5 degrees (minimum Topocentric Arcus Visionis) at the start/end of the day.

Sensitivity analysis

The below text is looking at heliacal events of stars and some planets. In case of for instance the Moon, different (smaller) sensitivity is seen.

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
  • Extinction coefficient (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, Longitude, 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.

Sensitivity analysis of the Object's/Sun's altitudes

The sensitivity due to several parameters has been analyzed on Object's/Sun's altitudes. The main players on the altitudes of the Sun and Object at a heliacal rise/set event are (the below formula are derived using ARCHAECOSMO functions):
  • Magnitude of the object
    Can be described as a 2nd order function:
    Object's apparent altitude = 0.1204*Magn2 + 0.7941*Magn + 5.59063
    Sun's altitude = -0.1504*Magn2 - 1.646*Magn - 8.1015
  • Extinction coefficient of the atmosphere
  • Can be described as a linear function:
    Delta Object's apparent altitude = 23.276*ktot - 7.1958
    Delta Sun's altitude = -2.1165*ktot + 0.602
  • Acuity of the observer's eye
    Best described as a linear function.
    Lets assume for now that the Observer's acuity is close to normal (=1).
  • Other parameters (Latitude, Longitude, Temperature, Age of person, Eye height, Air Pressure, Additional azimuth difference and declination of Object) are not that important when moderately changed.
  • Humidity is effecting the Object's altitude, but that is because the humidity determines the extinction coefficient also.
  • Errors in altitude due to above simplification formula:
    1 sigma error in Sun's altitude: 1.4 [deg]
    1 sigma error in Object's apparent altitude: 1.9 [deg]
  • The resulting Arcus Visionis (sum of above formula) compared with Schaefer's methodology gives a 1 sigma of 1.1 degrees over a Magnitude range of -3 to 4 and extinction coefficient range of 0.15 to 0.45.
The above formula don't represent accurately the effects of night vision, so larger errors occur due to that for Magnitude between 2 and 5. The Arcus Visionis is less variable on night vision.

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. topocentric Arcus Visionis (as defined by Reijs; 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 Topocentric 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 topocentric Arcus Visionis means the (topocentric) altitude difference between the Sun's and the objects centers. The topocentric Arcus Visionis (non cursive text; this website'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.

Schaefer uses the terms: Arc of Vision for the Moon and Arcus Visionis for stars. 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 interpolated values of Ptolemy's/Schoch's Arcus Visionis).
So it looks like there are several definitions of Arcus Visionis (the first five definitions can be calculated with the Excel Add-in):
  1. Reijs:
    Reijs calculates the Topocentric Arcus Visionis from the Sun's altitude and the Moon/star's (topocentric) altitude. This definition is used on this web site.
  2. Schaefer:
    Schaefer calculates the "Arcus visionis" from the Sun's altitude and the star's geocentric altitude, which is very close to the Topocentric Arcus Visionis of Reijs due to the neglectable parallax of Sun and stars. He calls the geocentric altitude angle difference between Sun and crescent Moon: "Arc of Vision", which will have a difference of around 1 degree with the Topocentric Arcus Visionis definition of Reijs (due to parallax).
    Yallop ([1998], page 1) is using also the geocentric altitude and azimuth of Sun and Moon for his ARCV. Further work on MoonWatch program can be found here.
  3. Meeus:
    The Sun's altitude and object's apparent altitude at the moment the object is at its heliacal event position.
  4. Ptolemy:
    Arcus Visionis: unrefracted angle difference of Sun and object at the moment the object is at zero altitude
  5. PLSV:
    The Sun's altitude (called 'Arcus Visionis', which is different then Ptolemy's Arcus Visionis and Reijs' Arcus Visionis) at the moment the object is at its heliacal event position (its Critical altitude: the object's apparent altitude)
  6. Arc of Vision:
    Some people see Arc of Vision as just a translation/kind-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].

Mean periodicy of stars

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 Morning First of Sirius and some other stars have been calculated (at latitude 30.05° and longitude 31.25°), see below picture. The period of Morning First is defined as follows:
The time difference between two successive moments when the sun is at its optimum altitude for heliacal rise (which is altitude when the min. Topocentric Arcus Visionis would have happened) and the celestial object is visible-on-first-day after its conjunction with the Sun.

Heliacal periodicy

Heliacal period with large periodicy values is when the star is close to a circumpolar path (Type C star), while the low periodicy values come when the star gets close to being always invisible near the horizon (Type I star).
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. Topocentric 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. Although: Sirius is of course the brightest star!

A simple method

A simple method for deriving the periodicy of star related Phases (like Morning First, Evening Last, etc.) might be possible by looking at the derivative of the Right Ascension (RA) with regard to the ecliptic longitude (dRA/dlong) . The change in ecliptic longitude is related to the precession (and proper motion) and thus the change in RA can be easily determine, using the conversion formula from ecliptic to equatorial coordinates (Duffett [1990], page 59-61).
The above idea comes from Chase (Chase A.B., Manning, H.P., The Rhind mathematical papyrus; Notes on Egyptian calendar, page 45-46, 1927).
Remark: The formula Chace provides, has an error (misses a cos(obliquity) term).

Below is a picture of dRA/dlong for four stars:
Periodicy of stars
This picture looks comparable to above periodicy picture. The curves are not fully the same, so some more study is needed to find a fully matching  simple formula for the periodicy.
There is one difficulty: I don't have yet the vertical axis;-), although the 2nd line (dRA/dlong=1) from the top is very likely equivalent to the sidereal year.
 
Below is a picture of Sirius' and Antares' dRA/dlong over a period close to the precession cycle (~26200 Years), which has an average of around 1:
Periodicy over precession
        cycle
The form of this picture changes considerable with the star's ecliptic latitude (the peak gets higher with larger ecliptic latitude) and the form shifts depending on the ecliptic longitude:

By the way; this behaviour is for any star event, so not only heliacal rises! So the above complex derivation, looking at helical rise (same way as Ingham [1969] does), might to be an overkill to get to the periodicy.

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 2.7 km 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