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

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 (k_tot)
  k_tot 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 (k_tot): 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 2^nd 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*k_tot - 7.1958
Delta Sun's altitude = -2.1165*k_tot + 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].