A few papers have compared the catalogued magnitudes with modern magnitudes, including LundmarkK., “Luminosities, colours, diameters, densities, masses of the stars”, Handbuch der Astrophysik, v (Berlin, 1932), 210–573; FujiwaraT.YamaokaH.MiyoshiS. J., “Survey of long-term variability of stars, I: Reliability of magnitudes in old star catalogues”, Astronomy & astrophysics, cdxvi (2004), 641–6. A few papers have sought evidence for variable stars, including HertzogK. P., “Supernova progenitors and Be stars: Stellar variability from a 21 century perspective”, Monthly notices of the Royal Astronomical Society, ccix (1984), 533–41; MayerP., “Secular brightening of supergiants”, Observatory, ciiii (1984), 77–80; HearnshawJ. B., “An analysis of Almagest magnitudes for the study of stellar evolution”, New astronomy reviews, xliii (1999), 403–10.
A review of the long history and arguments is in SchaeferB. E., “The great Ptolemy–Hipparchus dispute”, Sky & telescope, ciii (2002), February issue, 38–43.
4.
EvansJ., “On the origin of the Ptolemaic Star Catalogue: Part 2”, Journal for the history of astronomy, xviii (1987), 234–78 does discuss the reported magnitudes for the six most southerly stars in the context of the debate. He concluded that “On the basis of the magnitudes assigned to the southernmost stars of the catalogue, it appears more likely that the ancient observer was located at Alexandria than at Rhodes”. SchaeferB. E., “The latitude of the observer of the Almagest star catalogue”, Journal for the history of astronomy, xxxii (2001), 1–42, has used the modern magnitudes of the Almagest stars around the southern horizon as the key input, but this is not using the ancient magnitudes.
5.
KnobelE. B., Ulugh Beg's catalogue of stars (Washington, DC, 1917).
6.
BailyF., “The catalogues of Ptolemy, Ulugh Beigh, Tycho Brahe, Halley, Hevelius”, Memoirs of the Royal Astronomical Society, xiii (1843), 1–248.
7.
DreyerJ. L. E., Tycho Brahe (New York, 1890), 227.
8.
DreyerJ. L. E. (ed.), Tychonis Brahe Dani opera omnia (15 vols, Copenhagen, 1913–29), ii, 258–79.
9.
DreyerIn (ed.), Opera omnia (ref. 8), for example in vol. xii, pp. 103–5, 162–82, 226–30, and 231–65, we are given observed magnitudes as one digit notations next to star positions, but never are we given any commentary.
10.
Dreyer, Tycho Brahe (ref. 7), 227, 265. RawlinsD., “Tycho's star catalog: The first critical edition”, Dio, iii (1993), 3–106.
11.
Baily, “Catalogues” (ref. 6), 127–65. Dreyer (ed.), Opera omnia (ref. 8), 344–73. Rawlins, “Catalog” (ref. 10).
12.
BouguerP., Essai d'optique, sur la gradation de la lumière (Paris, 1729).
13.
de MairanJean-Jacques, “Eclaircissement sur le mémoire de la cause générale du froid en hiver, & de la chaleur en été”, Mémoires de l'Académie Royale des Sciences pour l'année1721 (Paris, 1723), 8–17.
14.
The recognition of extinction as a phenomenon is easy to see by any observer, because stars dim by many magnitudes as they get close to the horizon. As such, it is inevitable that many people (from prehistoric times till the Renaissance) independently came to an empirical realization that atmospheric extinction exists (i.e., that stars appear dim as they get near the horizon, for whatever reason). So it is surprising that there are no surviving records that recognize extinction as a phenomenon. I have very broad knowledge on similar topics in astronomical history, and I have consulted a number of astronomy historians with very broad and deep knowledge (including Gary Thompson, Andrew Young, Owen Gingerich, as well as the many readers of the HASTRO-L group), with null results. While it is effectively impossible to prove a negative (that no pre-1723 sources mention extinction), the lack of any old sources does suggest that formal knowledge of extinction was not widespread.
15.
Schaefer, “Latitude” (ref. 4).
16.
The early catalogues of magnitudes by William Herschel, John Herschel, and Friedrich W. A. Argelander did not have extinction corrections, but care was taken to avoid to avoid low altitudes. The first large-scale photometry catalogue explicitly to correct for extinction was in 1872 by Eduard Heis for his Atlas coeletis nova, where he consciously corrected his observed magnitudes for extinction. In 1872, Benjamin Gould made observations for the Uranometria Argentina by setting up secondary standard stars by making looking at them while the northern primary standards were at the same altitude.
17.
To use an everyday example, humans are quite good at estimating the height of people seen in the distance, where we unconsciously know to scale the estimated height (from the observed angular height) by the inverse of the estimated distance. This happens despite most humans having no idea of the existence of what astronomers call the ‘small-angle formula’. In a closer example, involving naked eye photometry, we have the everyday task of a driver on a highway at night viewing streetlights near the horizon. All humans naturally and intuitively will take the observed brightness and correct for the distance so as to estimate the luminosity and type of the light, even though most humans have never heard of the inverse-square-law for light. The point is that humans are all the time making such corrections to their perceived measurements, and this is often done unconsciously, so it is easy to think that the old observers could have made an intuitive correction for the dimming of stars near the horizon.
18.
The use of a traveller's report is plausible only for singular cases involving some famous star. The only possibilities for this are Achernar at the end of Eridanus and Canopus. Canopus (α Car) had a declination of −52.5° in the time of Ptolemy, so it would culminate with an altitude of 6.3° above the southern horizon of Alexandria. Canopus has a negative magnitude of −0.70, so with k = 0.25 mag/airmass, it will always appear dimmed by at least 1.88 mag, so it would never appear brighter than m = 1.18. For Ptolemy, it would be appropriate still to call it a first magnitude star as based on direct observation without correction. From my university campus in Baton Rouge Louisiana (λ = 30.4°), Canopus culminates 6.9° above the horizon, and I can always astound students by pointing to it and having them compare it with Rigel, even on the clearest of winter evenings. For Hipparchus as observing from Rhodes (λ = 36.4°), the culmination is at 0.9° above the southern horizon, and it actually starts to matter that the refraction correction raises this to 1.4°. For the best plausible extinction for sea level for an eastern Mediterranean site (0.23 mag/airmass), the dimming (with a zenith distance of 88.6° and X = 23 airmass) is 5.05 mag at culmination, so Canopus will never appear brighter than m = 4.35 mag. Thus it is that Canopus is certainly never a first magnitude star as seen by Hipparchus. Nevertheless, this is not a strong argument against a Hipparchan source for the Almagest magnitudes, because the magnitudes are already extinction corrected. Or maybe Hipparchus included Canopus simply due to travellers' reports.
19.
My simple physical model for the brightness of the dark nighttime sky (KrisciunasK.SchaeferB. E., “A model of the brightness of moonlight”, Publications of the Astronomical Society of the Pacific, ciii (1991), 1033–9, see Equations 2 and 3) gives the brightness 10° above the horizon to be 2.0 times the brightness near the zenith. A more detailed physical model (GarstangR. H., “Night-sky brightness at observatories and sites”, Publications of the Astronomical Society of the Pacific, ci (1989), 306–29) shows that the sky brightens from the zenith to 10° altitude by close to a factor of two, but then greatly darkens going to 0° altitude. Observations (HulburtE. O., “Night sky brightness measurements in latitudes below 45°”, Journal of the Optical Society of America, xxxix (1949), 211–15) show the dark and clear sky brightness peaks at around 10°–15° altitude with the peak being 50%–70% brighter than the zenith.
20.
The Milky Way region comes to a peak brightness close to a factor of 2 times the zenith sky brightness at high galactic latitudes. See Hulburt, “Night sky” (ref. 19), and AllenC. W., Astrophysical quantities, 3rd edn (London, 1973), 134. This means that the systematic biases in estimating magnitudes for stars in the Milky Way are essentially the same as for measuring stars near the horizon.
21.
Lundmark, “Luminosities” (ref. 1), 250, 254 and 256.
22.
SchaeferB. E., “Atmospheric extinction effects on stellar alignments”, Archaeoastronomy (supplement to Journal for the history of astronomy), no. 10 (1986), S32–42 gives a full physics model plus many observations of extinction angles.
23.
SchaeferB. E., “Astronomy and the limits of vision”, Vistas in astronomy, xxxvi (1993), 311–61, p. 317.
24.
Sites near Alexandria have seasonal variations of 0.10–0.20 mag/airmass and one-sigma variations of 0.06–0.08 mag/airmass. Sites near Isfahan have seasonal variations of 0.08–0.24 mag/airmass and one-sigma variations of 0.10–0.14 mag/airmass. Sites near Hven have seasonal variations of 0.04–0.18 mag/airmass and one-sigma variations of ∼0.11 mag/airmass.
25.
With the model from Schaefer (“Extinction effects” (ref. 22)), an observed extinction angle of, say, 10° could be interpreted by Ptolemy as magnitude 4 (V = 4.22 with k = 0.25 mag/airmass), >4 (V = 3.81 with k = 0.29 mag/airmass), <3 (V = 3.25 with k = 0.37 mag/airmass), or 3 (V = 3.11 with k = 0.39 mag/airmass). For an observed extinction angle of 5°, the interpreted magnitude can range for 2 to <3 (over five Almagest bins) for extinctions ranging from 0.23 mag/airmass to 0.35 mag/airmass and m values ranging from 1.74–3.25. I have a lot of experience at estimating the extinction on nights while I have simultaneously measured the k with stellar photometry, so I know that it is impossible visually to evaluate the k value to an accuracy of better than 0.15 mag/airmass or so. The point is that the observer could not distinguish the exact k value and that normal variations (night-to-night and seasonal) will lead to variations in the reported magnitude that spans 4–5 magnitude bins over which m – V will vary by over 1.0 magnitudes. This source of unavoidable observational error will make for a scatter in m – V that is greatly larger than the typical values (see Tables A1–A4 of the online edition). So the logic is that any use of the extinction angle to measure the magnitudes must lead to a greatly larger scatter in m – V than observed, then this was not the method for extinction correction used by any of the old observers.
26.
Schaefer, “Extinction effects” (ref. 22), Figures 1 and 2. I was a highly experienced and motivated observer who knew exactly where to look, so my uncertainties cannot be improved upon significantly.
27.
BruinF., “Atmospheric refraction and extinction near the horizon”, Archive for history of exact sciences, xxv (1981), 1–17, is the only work that I know of to look at the history of extinction. He opens with: “The theory of atmospheric extinction was developed only when the knowledge of the precise magnitudes of the stars became of interest, and again the extinction at low altitude was not important. This relatively late development one finds reflected in the books of Chauvenet (1891) and Wolf (1892) who devote, respectively, forty and twenty pages to atmospheric refraction but are silent about extinctions.”
28.
The historical sources provide no discussion or definition of what a magnitude is, and they never talk about how they measured their magnitudes. HearnshawJ. B., The measurement of starlight (Cambridge, 1996), 1, says “Ptolemy says almost nothing in the Almagest about how he defines a magnitude…. Compared with the detailed discussion on his positional observations and reductions, this lack of comment on the scale of magnitudes is perplexing”. Similarly, Tycho gives extensive details on how he measured the positions of the stars, but no word on how he measured the magnitudes. LundmarkK., “Luminosities” (ref. 1), 230, says “As regards the method followed at the estimation of the magnitudes we know practically nothing”. So it is hardly surprising that they do not mention extinction, much less how to correct for it.
29.
Bouguer, Essai (ref. 12).
30.
The utter lack of mentions of extinction is in sharp contrast to the situation for the similar phenomenon of atmospheric refraction. In a classic review of refraction history, MahanA. I., “Astronomical refraction — Some history and theories”, Applied optics, i (1962), 497–511, starts his abstract with “Astronomical refraction has had a long and fascinating history. Cleomedes (100 A.D.) and Ptolemy (200 A.D.) were aware of its existence and understood in a qualitative way some of its properties. Alhazen (1100 A.D.) quite correctly suggested that the flattening of the sun's disk near the horizon was due to astronomical refraction. Tycho Brahe in 1587, however, was the first to make direct measurements of the magnitude of the refraction…. Attempts to evaluate the ‘refraction integral’ for the concentric spherical shell model have led to the theories of Bessel, Bradley, Gylden, Ivory, Laplace, Mayer, Simpson, Young, and others”. Refraction was recognized by many pre-telescopic observers, it was measured extensively from Tycho on, many post-1700 scientists have modelled the effect, and historians have closely documented the case. So why hasn't extinction been similarly recognized?.
31.
CarlisleC. C., “Ancient astronomers: Smarter than we knew?”, Sky & telescope, cxxiii (2012), May issue, 18.
32.
Schaefer, Limits (ref. 23).
33.
Schaefer, Limits (ref. 23).
34.
Schaefer, Latitude (ref. 4).
35.
I have over two dozen papers reporting visual observations as tests of celestial visibility in support of getting modern ground-truth for what is easy and possible, and many of these involve massive observing campaigns. Examples of these papers include SchaeferB. E., “Heliacal rise phenomena”, Archaeoastronomy (supplement to Journal for the history of astronomy), no. 11 (1987), S19–33, SchaeferB. E., “The latitude and epoch for the origin of the astronomical lore of Eudoxus”, Journal for the history of astronomy, xxxv (2004), 161–223, and DoggettL. E.SchaeferB. E., “Lunar crescent visibility”, Icarus, cvii (1994), 388–403. Further examples are cited in refs 4, 19, 22 and 23.
