HomeUpSearchMail
NEW

Ptolemy's star catalogue and star calendar

Considering the translations

Considering the starting date of the calendar

Star visibility

• In Ptolemy times, sky would be under perfect dark sky conditions, as all light pollution would have been absent; stars as faint as +7.5magn might be visible with the naked eye at zenith (Clark, 1990, Fig. 2.8 page 16). M33 (magn=5.72) would have been visible with direct vision (Bortle, 2001, page 126).
• All 1025 (unique) Ptolemy stars (Pratt, 2015) are included in the sefstars.txt file for the Opensource Swiss Ephemeris.
  
• Ptolemy could measure ecliptic coordinates up to magnitude 6.5 stars (I assume using direct vision, correct?). The picture depicts the magnitude of Ptolemy's full star database compared to the present day magnitude (see also Fig. 12 Evans, 1987, p. 261).
  
  
• If we include (using ARCHAEOCOSMO package) a limiting visual magnitude (LVM) of +6 or +6.5 at zenith, the below picture shows which observations of Ptolemy could realistically happen (depending on the star's altitude at meridian transit):
  
  Some 16 stars are outside the LVM of +6 (average LVM) and AEC=0.35 (average Atmospheric Extinction Coefficient). So 98.4% is within the LVM range of +6.
  With a LVM of 6.7 (optimistic LVM) and AEC=0.25 (optimistic AEC) almost 100% is within range.
  

Star coordinates and magnitudes

• Scenario1: Coord&Magn=SIMBAD
• ScenarioA: Coord=SIMBAD, Magn=Almagest
• ScenarioB: Coord=Almagest, Magn=SIMBAD
• ScenarioC: Coord&Magn=Almagest

• Coordinate Conversions
  This web pages does the conversion in one step, although it does everything in degrees (not in hours for RA; in that case divide the decimal format by 15).
  

Some evaluation points

• It was tested if the mentioned entries in Ptolemy's calendar mapped (by changing the Atmosphere Extinction Coefficient (AEC)) with Schaefer's criterion. If not, it might be that: Ptolemy's entries are not observations; Schaefer's criterion is wrong; a combination; or otherwise.
• For any scenario some 35% of the entries can not be mapped on Schaefer's criterion (even using a slightly higher acuity [1.4] than the average 17-18 year person's acuity of 1.3+/-0.2 (Ohlsson and Villarreal, 2005)). An unrealistic AEC of smaller than 0.15 would be needed to map the two  (aka a very very very clear sky). There was only one entry that needed an unrealistic high AEC >0.75!
  
• To map the calendar entries on the Schaefer's criterion low AEC are prominent:
  
  During real atmospheric conditions, a more Gaussian distribution is expected; peaking higher than 0.15 (see for instance this distribution). Due to the difference in AEC (or |DAzi|) distribution the below formula might not be easily comparable to other formula in literature!!!
  
• The standard deviation of the date difference depends on the scenario:
  

  Scenario	Coord	Magnitude	Standard Deviation [Days]
  Scenario1	SIMBAD	SIMBAD	5.1
  ScenarioA	SIMBAD	Almagest	3.3
  ScenarioB	Almagest	SIMBAD	4.8
  ScenarioC	Almagest	Almagest	2.3

  
  
• The smallest standard deviation (2.3 days) of date differences happens with ScenarioC. And using the Magn_p of Almagest reduces the date difference the most.
  
• The Almagest data (ScenarioC) looks to have been used (mostly) in Ptolemy's star catalogue, instead of or beside real observations (Scenario1).
• Nickiforov (2017, pers. comm.) also says that Ptolemy's calendar was build mainly around a theoretical model instead of observations. In (Nickiforov, 2015, page 74) is stated that no real preference on model/observation can be given.
  
• The standard deviation of the date difference for non-mapped entries is not really depending on the climata. Except climata 15.5 hours has some outliners, mainly due to Rukbat and Mirfak. These two stars are close to the horizon for climata 15.5 hours and 58CE.
• Relation Magnp and TopoAV (regardless of |DAzi| and AEC) is:
  TopoAV(Magnp) = 8.0 + 2.7*Magnp [deg]   (1)
  <Schaefer's criterion has not been used to derive this formula>
  
  
  
  
• Nickiforov (2015, page 72) used another translation of Ptolemy's Phases (L'Abbe Halma, 1816), his relation between Magnp and TopoAV (called by him: arc of visibility) is:
  TopoAV = 9.8 + 2.2*Magnp [deg]   (2)
• The difference between formula (1) and (2) could be due to: differences in the entries (due to translation issues); and/or Nickiforov only used Magnp=1 and 2, while formula (1) is based on intermediate magnitudes (as given by Peters and Knobel (1915, page 120)).
  
• Relation |DAzi| and TopoAV is (|DAzi|<100 and regardless of Magnp and AEC):
  TopoAV(|DAzi|) = 14.5 - 0.065*|DAzi| [deg]   (3)
  <Schaefer's criterion has not been used to derive this formula>
  
• The TopoAV for the 4 heliacal events (MF, EL, AR and CS) seems to stabilising at around 6.25 degrees for |DAzi| from above 100 degrees.
  
  
  
  
• The stabalising of the TopoAV could be due to the Earth's shadow onto the atmosphere starting from |DAzi| of 90 degrees. Some scattering extends this 90 to some 100 degrees perhaps. Schaefer has a stabalising angle also around 100 deg for B_twi (Schaefer, formula 15c)
  
• Nickiforov (2015, page 73) used another translation of Ptolemy's Phases (L'Abbe Halma, 1816), his relation between |DAzi| and TopoAV (called by him: arc of visibility) is:
  TopoAV(Magnp=1) = 13.26 - 0.090*|DAzi| [deg]   (4)
  TopoAV(Magnp=2) = 15.40 - 0.096*|DAzi| [deg]   (5)
• The difference between formula (3) and (4)&(5) could be due to: differences in the entries (due to translation issues); and/or Nickiforov only used Magnp=1 and 2, while formula (3) is based on intermediate magnitudes (as given by Peters and Knobel (1915, page 120)).
• If the Excel's GRG Nonlinear Solver is used to determine the TopoAV dependency on mp and |DAzi| (upto 100deg) the following combined formula is the result (regardless of AEC):
  TopoAV(Magnp,|DAzi|) = 10.5 + 2.3*Magnp  - 0.051*|DAzi|  [deg]   (6)
  <Schaefer's criterion has not been used to derive this formula>
  
• Nickiforov (2015, page 74) gives as combined formula:
  TopoAV(Magnp,|DAzi|) = 9.8 + 2.2*Magnp  - 0.093*|DAzi|  [deg]   (7)
• The (small) difference between formula (6) and (7) could be due to: differences in the entries (due to translation issues); and/or Nickiforov only used Magnp=1 and 2, while formula (6) is based on intermediate magnitudes (as given by Peters and Knobel (1915. page 120)); and/or difference in the analysis method to determine the formula coefficients.
• If looking at ScenarioB, we can determine a combined formula for the entries that can be mapped on Schaefer's criterion (aka Ptolemy's calendar entries that could have been observed according to Schaefer's criterion).  This gives an idea of a three parameter behavior of Schaefer's criterion for TopoAV and Arcus Visionis:
  TopoAV(Magn,|DAzi|,AEC) = 8.0 + 1.8*Magn  - 0.024*|DAzi| + 17.7*AEC [deg]   (8)
  Arcus Visionis(Magn,|DAzi|,AEC) = 9.6 + 0.69*Magn - 0.016*|DAzi| - 3.6*AEC [deg]   (9)
  
  
  
• Looking at the above formulas (8) and (9): the Arcus Visionis (9) is much less depending on Magn, |DAzi| and AEC than the TopoAV (8). The Arcus Visionis is some five times less dependent on AEC than the TopoAV! This was also derived in another analysis.
• Looking at the above: the single parameter criterion of Schoch (1924, page 734) misses the significant influence of |DAzi| (assumed smaller than 25 degrees)  and AEC (assumed low and constant?). The heliacal angle derived by Schoch are tabled here.
  
• If we combine the (Schaefer's criterion) mapped calendar entries of Ptolemy and observation of Kolev the resulting graph, can be seen here.
• The AR&CS events mentioned in Ptolemy's calendar still need an evaluation. There are some 270 of them and they might give a good insight how there were observed/derived. But remember these entries could be calculations instead of observations.
  
• A teaser of the dependency of TopoAV on magnitude for AR&CS events, can be seen below:
  
  One can clearly see the stability of the TopoAV, which is around 6.25 degrees. Such a stability is not expected with real observations. See also here.
• The sequence of Phases (recorded for the stars entries in Genimus' calendar [Ptolemy, 1993]) determines the Object Type of the mentioned 30 stars (and 5 different latitudes). The distribution of Types mentioned in the Ptolemy's calendar can be seen in below figure (in total some 140 Type entries):
  
  'Undetermined' means that not enough entries were provided to determine any Type (ALH: Arisen and Laying Hidden and CP: Curtailed Passage). But most of the 'Undetermined' could be changed into 'likely (sub)Type' by deducting from other entries, given the below picture:
  
  
  
• In below picture one can see the stars (filled symbols) and their Phase sequence mentioned by Ptolemy, all fit nicely in the proper (sub)Type region.There are some sets of calendar entries that show that the Type of a star (open symbols: Menkalinan, Altair, Alpheratz, Betelgeuse, Alphard and Rukbat) changes depending on the climata (geographical latitude): in all cases it changed to ALHb for climata 14 or 13.5 hours (geo. lat=23.85). Two stars (Bellatrix and Procyon) are also very close to such behavior: 1 or 2 days.
  See below (AEC=0.15; climata 13.5h; and if a symbol is on the borderline, its Declination is outside the border, graph determined using ARCHAEOCOSMO):
  
  If the AEC would be smaller, the red band would be lowered and closer mapping the stars that have changing Types. This might be another hint that Ptolemy's entries were not observations but calculations.

Conclusion

References

Bortle, J.E., 2001. The Bortle Dark-Sky Scale. Sky & Telescope, https://web.archive.org/web/20130611180652/http://media.skyandtelescope.com/documents/BortleDarkSkyScale.pdf
Clark, R.N., Visual astronomy of the deep sky, Cambridge University Press, 1990
Geminus, Calendar or register attributed to Geminus, Trans. Robert Schmidt, The Golden Hind Press, 1993
Nickiforov, Michael G., 'Analysis of the calendar C. Ptolemy: Phases of the fixed stars', Bulgarian Astronomical Journal 20 (2014): 68-85.
Ohlsson, Josefin, and Villarreal Gerardo, 'Normal visual acuity in 17-18 year olds', Acta ophthalmol Scand. 83, no. 4 (2005): 487-91.
Pedersen, Olaf, 'The Almagest in translation', Journal of history of astronomy xviii (1987): 59-63.
Peters, Christian Heinrich Friedrich, and Knobel Edward Ball, Ptolemy's catalogue of stars: A revision of the Almagest, Washington: The Carnegie Institution, 1915.
Pratt, John P., 'The Ptolemy star catalog', 2015 http://www.johnpratt.com/items/astronomy/ptolemy_stars.html, accessed on 16 March, 2017.
Protte, P., Analysis and follow up observations of historical star catalogues: Featuring a reconstruction of the Ptolemaic armillary sphere, June 2019, Friedrich Schiller university Jena (not yet published)
Ptolemy, Claudius, Apparitions des fixes et annonces: Traduites du Grec de Ptolemy. Translated by M. L'Abbé Halma, Paris, 1816.
Ptolemy, Claudius, The phases of the fixed stars. Translated by Robert Schmidt. Edited by Robert Hand: The golden Hind press, 1993.
Schaefer, Brad E., 'Astronomy and the limits of vision', Archaeoastronomy: The journal of astronomy in culture XI (1993): 78-91.
Schoch, Carl. 1924. 'The 'Arcus Visionis' of the planets in the Babylonian observations', Monthly Notices of the Royal Astronomical Society, Vol 84: pp. 731-34.

Acknowledgements

Disclaimer and Copyright
HomeUpSearchMail