Accuracy of magnitudes in pre-telescopic star catalogues
RReceived 26 April 2016; Revised 6 June 2016; Accepted 6 June 2016DOI: xxx/xxxx
ARTICLE TYPE
Accuracy of magnitudes in pre-telescopic star catalogues
Philipp Protte | Susanne M Hoffmann Physikalisch-Astronomische Fakultät,Friedrich-Schiller-UniversitätJena,Germany
Correspondence
Susanne M Hoffmann, Email:susanne.hoff[email protected]
Historical star magnitudes from catalogues by Ptolemy (137 AD), al-S. ¯uf¯ı (964) andTycho Brahe (1602/27) are converted to the Johnson V-mag scale and compared tomodern day values from the HIPPARCOS catalogue. The deviations (or “errors”)are tested for dependencies on three different observational influences. The relationbetween historical and modern magnitudes is found to be linear in all three cata-logues as it had previously been shown for the Almagest data by Hearnshaw (1999).A slight dependency on the colour index (B-V) is shown throughout the data sets andal-S. ¯uf¯ı’s as well as Brahe’s data also give fainter values for stars of lower culminationheight (indicating extinction). In all three catalogues, a star’s estimated magnitude isinfluenced by the brightness of its immediate surroundings. After correction for thethree effects, the remaining variance within the magnitude errors can be consideredas approximate accuracy of the pre-telescopic magnitude estimates. The final con-verted and corrected magnitudes are available via the Vizier catalogue access tool(Ochsenbein, Bauer, & Marcout, 2000).
KEYWORDS: magnitudes, history of astronomy, almagest, extinction, naked eye observation
Many processes in astronomy have long timescales, espe-cially questions on the evolution of stars (just recently, thetimescale of Betelgeuse’s supernova was publicly discussed).Further examples include close binary systems such as cat-aclysmic variables (CVs) and their nova eruption behaviour(S. M. Hoffmann, Vogt, & Protte, 2019; Shara et al., 2017;Vogt, Hoffmann, & Tappert, 2019), or even supernovae inclose binary systems which have the potential to eject runawaystars (Neuhäuser, Gießler, & Hambaryan, 2019). All thesequestions on the evolution of astronomical objects requirelong-term observations but our telescopic surveys only reachback for a few decades (in cases of CVs) or roughly two cen-turies (in cases of sunspot observations (Neuhäuser, Arlt, &Richter, 2018; Neuhäuser & Neuhäuser, 2016)). Aiming forconclusions on long-term evolution it is, thus, desirable to Sporadic telescopic observations have, of course, existed for a bit longer but systematic surveys have not been common practice from the early beginning on. include data from non-telescopic observations which couldpossibly provide a much longer baseline: Far Eastern tradition,for instance, recorded transient phenomena (such as novae,supernovae, and comets) more systematically than the Westernone. However, one of the biggest questions in these terms is thetransformation of any ancient (or old) description of the bright-ness of the phenomenon. Only in very few cases (e. g. 437,SN 1572) the historical records mention daylight visibility. Ina handful of cases (e. g. 1175, 1203, 1596, and 1603 accordingto Ho (1962), and (Xu, Pankenier, & Jiang, 2000, 129–146)),the description refers to the brightness giving Mars, Saturn, orbright stars like Capella ( 𝛼 Aur) and Antares ( 𝛼 Sco) as com-parison.Although we can look up the brightnesses of the planets andfixed stars in a modern star catalogue or model their bright-ness at a certain date with our knowledge on their variability,it appears worthwhile to study the accuracy of such historicalestimates.As commonly known, Argelander in the 19th century defineda clear method to estimate the magnitude of a given object by a r X i v : . [ a s t r o - ph . I M ] A ug AUTHOR ONE
ET AL comparing it to a couple of stars in the vicinity. The method(s)of earlier astronomers to derive the magnitude of a star or tran-sient object are yet unknown. As, therefore, the numbers inhistorical star catalogues are hardly reproducible, we try toderive a better understanding of their scattering, error bars, andtheir dependencies.
In particular, the visual appearance of a celestial point sourcedepends on many influences, e. g. the brightness of the back-ground, the local and temporal conditions of the atmosphereand the constitution of the observer’s eye. Currently, we can-not consider the observer’s eye and we are not even sure thatthe author of a text book really observed every data point onhis own and that not students or assistants were helping or eventaking over the measurement. The interplay of the human eye’slens and the atmospheric conditions cause optical effects suchas ( 𝑖 ) reddening of stars close to the horizon (extinction), ( 𝑖𝑖 )blurring of bright stars due to humidity or sandstorm, and ( 𝑖𝑖𝑖 )the impression of rays, horns, or fuzziness of bright objects(see APPENDIX A:) which might affect the estimate of themagnitude. The background brightness of the sky depends onthe density of stars in a particular area, the zodiacal light aswell as geophysical influences such as the omnipresent airglow(discovered by Ångström (1869)) and the presence or absenceof meteor showers (Siedentopf, 1959). As this work aims tomake historical star catalogues usable for modern research,our goal is to find an algorithm for how to deal with historicalmagnitudes. We assume that magnitudes for a star cataloguehave not been observed only one time by one person but crosschecked by the assistants of the historical book author. Aseven Ptolemy mentions the difficulties (and errors) of obser-vations close to the horizon, we assume that they estimatedmagnitudes at highest possible altitude for a given star. There-fore, the remaining influences to be considered in Section 3are the dependency of the human brightness estimation on thefollowing questions: • Redder stars appear fainter, so how does the colour of thestar influence the estimate? Can the historical numbers beimproved by application of a colour correction? • The atmosphere influences the appearance by extinction:Does a correction improve the conversion of historicalmagnitudes into modern ones? • Observational bias of the environment: How does thepresence of bright stars and a background of many faintstars (e. g. in the Milky Way) influence the historicalbrightness estimation?We analyze these questions in Section 3 by using the data ofthree historical star catalogues introduced in Section 2 which, of course, had already been analyzed by other scholars beforeus. Their results are, therefore, summarized in the followingsubsection.
The idea to represent different brightnesses by numbers goesback to Antiquity. Pliny the Elder’s ( +1 st century) wordssuggest that Hipparchus ( −2 nd century) might have mea-sured these magnitudes but this cannot be verified or falsified(S. Hoffmann, 2017, p. 92 and 194). The first surviving appear-ance of the magnitude scale is in Ptolemy’s Almagest fromthe nd century (Hearnshaw, 1996, p. 4) and from there itwas copied to the Arabic and Latin science culture but thenumbers always remained estimations. With the dawning ofelectric photometry, astrophotography and the necessity ofexact values from telescopic observations, the 19th centurytook some efforts to develop a mathematically exact scale.Pogson (1856)’s system finally prevailed but it differs fromthe ancient scale because mathematics in the meantime hadintroduced negative numbers and the zero. Additionally, anytype of logarithmic law can only approximate the human senseand it neglects personal influences of the observer (guessingerrors). There is no easy conversion from historical magnitudesbecause the new scale was used for the new star catalogues.Although the Almagest’s star catalogue has thoroughly beenanalysed and discussed with respect to its record of star posi-tions, there are only a handful (e.g. Hearnshaw, 1999; Schaefer,2013) of recent investigations into the star’s magnitudes. Evenfewer authors consider other pre-telescopic magnitude esti-mations, which are given most notably by al-S. ¯uf¯ı and TychoBrahe. Recently, Verbunt & van Gent (2010a, 2012a) releasedonline versions of the three catalogues by Ptolemy, Ulugh B¯egand Tycho Brahe with Ulugh B¯eg’s catalogue containing mag-nitude estimations that were adopted from al-S. ¯uf¯ı’s Book of thefixed stars .The computer-readable data makes it easier than ever to eval-uate and statistically compare the three catalogues. If thedata optimally converted and corrected for systematic devia-tions, it might be possible to utilise the ancient magnitudesfor investigations of stellar evolution and variability (as exam-ples for such endeavours see Mayer (1984) and Hertzog (1984)although the former’s results were later refuted by Hearnshaw(1999)). All the effects on magnitude estimates, mentioned atthe end of Section 1.1 have already been analysed during thesecond half of the 19th and the first half of the 20th century(for an elaborate example see Zinner, 1926) but there is no sys-tematic query of all three catalogues based on the conversionmethod, introduced by Hearnshaw (1999) while using moderncomputer-aided statistical procedures.
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ET AL We evaluated three star catalogues with measurements fromdifferent epochs and cultural backgrounds, beginning with theone, featured in Ptolemy’s
𝑀𝛼𝜃𝜂𝜇𝛼𝜏𝜄𝜅 ́𝜂 Σ ́𝜐𝜈𝜏𝛼𝜉𝜄𝜁 (engl.: Mathematical treatise , commonly known by its Arabic name
Almagest , 137 AD). This work contains the oldest extensivedata set of stellar brightness and Ptolemy also was the firstastronomer to verifiably make use of a numeric scale: the mag-nitudes. He assigned the brightest stars to the magnitude 1 andthe remaining ones into five gradually fainter classes, labelled2–6. For some stars he added qualifiers, saying a star was eitherslightly brighter or fainter than the given magnitude.The second catalogue is the one by Ulugh B¯eg from around1437 AD which contained the first independent, comprehen-sive position measurements in 1300 years, yet adopted (seeKnobel, 1917) its magnitude data from Abd al-Rahman al-S. ¯uf¯ı’s ’Book of fixed Stars’ (for a modern english translationsee Hafez, 2010), which he most likely composed around 964AD. in the city of Shiraz (Hafez, 2010, p.64). His list of starsis explicitly based on Ptolemy’s catalogue, containing almostthe same set of stars with positions, only corrected for pre-cession. However, al-S. ¯uf¯ı was only the second astronomer tosystematically assign magnitudes to all the entries in his cata-logue, using the same numerical scale, as Ptolemy. Al-S. ¯uf¯ı ’scatalogue served as an important source for many subsequentIslamic-Arabic astronomers who used his data or cited his texts(see Hafez, 2010, p.66 ff). One of those was Ulugh B¯eg , who,when he compiled his own star catalogue in 1437, adopted themagnitudes (and in 27 cases also the positions (Verbunt & vanGent, 2012b)) from the
Book of the fixed Stars .Lastly, we included Tycho Brahe’s star catalogue from1602/1627 which again consists of newly gathered data forpositions and magnitudes. Brahe was the first modern Euro-pean scholar to compile a comprehensive original star cata-logue. The results were first published as a 777-star cataloguein 1602 shortly after his death but a handwritten copy ofa more extensive catalogue had already been sent to sev-eral astronomers during the 1590’s. Finally, in 1627 JohannesKepler published an edition of the list containing 1004 starswhich was very similar to the manuscript version (Verbunt &van Gent, 2010a). While Tycho set new standards of precisionfor position measurements, he adopted Ptolemy’s magnitudescale with no finer graduation than those of his predecessors.As pointed out by most analyses of the different versions of hiscatalogue (e.g. Baily, 1843), the 1627 version even omits thebrighter-/fainter-qualifiers that were still included in the previ-ous release.Figure 1 shows the chronology and data transfer of the fourcatalogues.
FIGURE 1
Transfer of position and magnitude data betweenthe four ancient catalogues. Dashed arrows indicate an uncer-tain amount of influence, bold arrows show that data wascopied almost “word-for-word”.
For the historical magnitudes we used the data, given by Ver-bunt & van Gent (2010b, 2012b). Additional modern data (e.g.exact position, V-mag and colour index) have been taken fromthe VizieR release of the HIPPARCOS catalogue (ESA, 1997).Verbunt and van Gent used the translation of Ulugh B¯eg’s cat-alogue by Knobel (1917) for their digitalisation and the latter,knowing that Ulugh B¯eg adapted al-S. ¯uf¯ı’s magnitudes, enteredthem directly from al-S. ¯uf¯ı’s
Book of the fixed Stars to minimisetranslation errors. Therefore the catalogue designated as UlughB¯eg’s is actually a hybrid catalogue and as only the magnitudesare analysed, they are referred to as ’al-S. ¯uf¯ı’s magnitudes’ andabbreviated as 𝑚 𝑆 and the catalogue as 𝑆 .Brahes catalogue was released in different editions, where onlyhis 777-star list from 1602 includes qualifiers to specify mag-nitudes beyond 1-mag steps. The data given by Verbunt andvan Gent includes a total of 1007 stars, merged from all avail-able editions and the qualifiers were included in our analysiswhenever they were given.For the analysis we excluded double entries as well as starswithout safe modern identification or magnitude (e.g. for dou-ble stars). Furthermore, stars designated as “faint” or “neb-ulous” were left out and the two brightest stars (Sirius andCanopus) were excluded from most analyses as significant“outliers“. The used data sets are summarised in Table 1 ,where the last set (PSB) only includes stars that were existentin all three catalogues and had concordant modern identifica-tions. This set was used to examine the covariance betweenthe historical catalogues. Different magnitude designations areused throughout the analysis: 𝑚 HIP are the modern V-mag val-ues taken from the HIPPARCOS catalogue, while 𝑚 𝑜𝑙𝑑 and 𝑚 AUTHOR ONE
ET AL
Ptolemy m O l d Al-Sufi modern magnitudes m
HIP
Brahe
FIGURE 2
Relation between historical and modern magnitude scales for each star of the data subsets P, S and B. Although thescatter plot does not look particularly linear, the regression can be justified (see text) and is used to convert the historical valuesto the modern magnitude scale.designate the historical magnitudes before and after conver-sion, respectively. Additional indices (P,S,B) might be addedwhen talking about data from a certain author. Finally, wedefine 𝛿𝑚 = 𝑚 − 𝑚 HIP . TABLE 1
The analysed catalogues. Identifications taken fromVerbunt & van Gent (2010b, 2012b). PSB is the intersectionof P,S,B, containing only stars that are concordantly identifiedin all three catalogues.
Abbr. Included authors
P Ptolemy 990S Ulugh B¯eg 988B Brahe 937PSB Ptolemy+Ulugh B¯eg +Brahe 695
The historical magnitude values are not identical with modern,photometric V-band magnitudes. While today’s magnitudesof the stars in our reduced catalogue cover a continuous rangefrom ∼ 0 to ∼ 6 , in the pre-telescopic era they were based onestimated assignments into 6 discrete groups. Neverthelessintermediate steps between these groups were used by allthree authors as differently formulated ’qualifiers’ which indi-cate if the star is slightly brighter or fainter than the denotedmagnitude. The disparate definitions result in two problems: on the onehand, the data shows a different range for both magnitudes(roughly ↔ ). Therefore, a direct comparison ofboth values in form of a difference 𝑚 𝑜𝑙𝑑 − 𝑚 HIP will be biasedtowards showing large positive values for brighter stars. Tominimise any dependency of the difference on a star’s bright-ness, an adequate conversion formula for 𝑚 𝑜𝑙𝑑 is needed.The other problem, however, has to be tackled first: Theintermediate qualifiers, given by the historical observers, donot imply an exact value. Trying to convert the qualifiersinto numerical divergence, most previous authors added +0 . mag for the ’fainter-qualifier’ and −0 . mag for the’brighter-qualifier’, but also ±0 . mag and ±0 . mag havebeen applied. In an attempt to find the best approximation, wecompared the ’two-step-system’ ( ±0 . mag) with the ’one-step-system’ ( ±0 . mag).Looking at the average modern magnitude of each group ofstars with qualifiers (e.g. 2(f) – stars a little fainter than 2ndmag, or 3(b) – stars a little brighter than 3rd mag), the ’two-step-system’ is found to show several inconsistencies. Forexample, the Almagest’s 2(f)-stars which would be identifiedwith 𝑚 𝑃𝑜𝑙𝑑 = 2 . mag are fainter on average than the 3(b)-stars, identified with 𝑚 𝑃𝑜𝑙𝑑 = 2 . mag. The ’one-step-system’on the other hand, is consistent in almost all cases and is there-fore adopted for our analysis. The applied values of 𝑚 𝑜𝑙𝑑 and 𝑚 HIP are shown for each star in the scatter plots of Figure 2 .It might be added at this point that most of the following analy-sis was done with . mag-steps before the ’one-step-system’was chosen. The differences in the results were negligible inalmost all cases.The modern magnitude scale is logarithmic in regard to the UTHOR ONE
ET AL light flux, a fact that corresponds to (and historically derivesfrom) the logarithmic perception of brightness in the humaneye (Weber-Fechner-Law). Therefore the relation between 𝑚 𝑜𝑙𝑑 and 𝑚 HIP should be approximately linear which is notevident in any of the three sub-figures of Figure 2 , due tothe large scattering. But even averaging the 𝑚 HIP for each stepof 𝑚 𝑜𝑙𝑑 , as it has been done in many previous works (mostrecently Schaefer, 2013), does not yield a linear correlation,but rather implies a curved function. Thus, instead of tryingto find a consistent conversion formula from 𝑚 𝑜𝑙𝑑 to 𝑚 , mostauthors applied an empirical method: The modern averagesare immediately used as 𝑚 for every star within the respectivestep.Contrary to all previous (and some subsequent) stud-ies,Hearnshaw (1999) showed that the linear relation canindeed be found in Ptolemy’s data when switching the depen-dent with the independent variable. He argues that taking themean modern values for each historical magnitude is statis-tically invalid because the variable with larger uncertaintiesshould be averaged. A quick look at the brightest known vari-able stars shows that their variability (see Figure 3 ) is usuallymuch smaller than the resulting uncertainties of the historicmagnitudes (see Figure 9 ). Applying the same procedure as / mag FIGURE 3
Histogram showing the distribution of variabil-ity amplitudes among the 1,137 known variable stars of 𝑚 𝑚𝑎𝑥 , 𝑚 𝑚𝑖𝑛 < mag. The possible magnitude error, inducedby variability is small, compared to the overall uncertainties ofhistorical magnitudes in most cases.Hearnshaw (1999), we found linear relations for 𝑚 𝑃𝑜𝑙𝑑 and for 𝑚 𝑆𝑜𝑙𝑑 and 𝑚 𝐵𝑜𝑙𝑑 , as well which can be seen in Figure 4 where 𝑚 𝑜𝑙𝑑 is averaged for bins of 𝑚 HIP . It is therefore justified toconvert the 𝑚 𝑜𝑙𝑑 to a new variable called 𝑚 by applying a 𝑚 𝑚𝑎𝑥 , 𝑚 𝑚𝑖𝑛 < mag linear conversion formula: 𝑚 = 𝑎 ⋅ 𝑚 𝑜𝑙𝑑 + 𝑡 with 𝑎 = 1 𝑏 𝑡 = − 𝑡 ′ 𝑏 (1)Where 𝑏 and 𝑡 ′ are the regression coefficients from Figure 2 : 𝑏 𝑃 = 0 .
75 ± 0 . 𝑡 ′ 𝑃 = 0 .
92 ± 0 . 𝑏 𝑆 = 0 .
86 ± 0 . 𝑡 ′ 𝑆 = 0 .
69 ± 0 . (2) 𝑏 𝐵 = 0 .
93 ± 0 . 𝑡 ′ 𝐵 = 0 .
43 ± 0 . From 𝑏 𝑃 we can calculate 𝑎 𝑃 = 1 . +0 . . , meaning that onePtolemian magnitude corresponds to 1.33 modern magnitudes(concordant with 𝑎 𝑃 = 1 . , as found by Hearnshaw (1999)).In the same way, the values for the other two authors are: 𝑎 𝑆 =1 . +0 . . and 𝑎 𝐵 = 1 . +0 . . After converting the historicalmagnitudes to the modern scale, we can now define the errorvariable 𝛿𝑚 = 𝑚 − 𝑚 HIP for each star. m o l d P e r c en t age o f s t a r s i n t he b i n Mean(m old )% of stars in the binLinear regression
Ptolemy m o l d P e r c en t age o f s t a r s i n t he b i n Al-Sufi m o l d P e r c en t age o f s t a r s i n t he b i n m HIP
Brahe
FIGURE 4
Average historical magnitudes for modern-magnitude-bins of . mag in the data sets P, S, B. Error barsare the SEM where bin-size 𝑁 > and else the averagedSD of all other bins. The histograms in the background showthe relative number of stars within each bin. The regressionhere is only for visualisation of the linearity while the actualcoefficients are taken from Figure 2 . AUTHOR ONE
ET AL
In a first step, the distribution of the magnitude errors 𝛿𝑚 isanalysed for each of the three star catalogues P, S, B, as well asthe common catalogue PSB. Table 2 gives mean values andstandard deviations for both cases with the squared correlationcoefficients between the 𝛿𝑚 added for the shared data set. The TABLE 2
Mean values 𝜇 and std.dev. 𝜎 of the 𝛿𝑚 in thesingle-author catalogues (P, S, B) and for the shared catalogue(PSB), including squared correlation coefficients 𝑟 .[mag] P, S, B PSB 𝛿𝑚 𝑃 𝛿𝑚 𝑆 𝛿𝑚 𝐵 𝛿𝑚 𝑃 𝜇 ∶ 0 . 𝜇 ∶ −0 . 𝜎 ∶ 0 . 𝜎 ∶ 0 . 𝛿𝑚 𝑆 𝜇 ∶ −0 . 𝑟 ∶ 0 . 𝜇 ∶ −0 . 𝜎 ∶ 0 . 𝜎 ∶ 0 . 𝛿𝑚 𝐵 𝜇 ∶ 0 . 𝑟 ∶ 0 . 𝑟 ∶ 0 . 𝜇 ∶ 0 . 𝜎 ∶ 0 . 𝜎 ∶ 0 . mean values of 𝛿𝑚 for the three single catalogues come closeto zero but al-S. ¯uf¯ı’s data in the common list shows a slightlylarger offset. This might be due to a selection effect in theshared catalogue where certain stars were omitted. The stan-dard deviation shows similar values for Ptolemy’s and Brahe’scatalogues but a significantly slimmer scattering for al-S. ¯uf¯ı’smagnitude errors. Finally, the 𝑟 -values indicate a strong corre-lation between Ptolemy’s and al-S. ¯uf¯ı’s magnitude errors whileBrahe’s data seems to be largely independent. The correla-tion is visualised in the scatter plots in Figure 5 , includingcovariance-ellipses that contain ∼ 95% of the data points. The FIGURE 5
Correlation between the 𝛿𝑚 of the three cata-logues within data set PSB. 𝜎 -covariance ellipses are givenfor each scatter plot.correlation between Ptolemy and al-S. ¯uf¯ı should come as no surprise because al-S. ¯uf¯ı takes Ptolemy’s magnitude estimationas a basis for his own (see also Figure 1 ). He even gives literalreferences like: ”The fourth [star] [ … ] is much greater then[sic] 4th magnitude, but it was mentioned by Ptolemy as 4thmagnitude exactly.” (Hafez, 2010, p. 154). From these kind ofcomments it can be assumed that al-S. ¯uf¯ı only changed a star’smagnitude if he deemed it distinctly erroneous and thereforeleft a large fraction of them unchanged, causing this depen-dency.The distinctly weaker correlation between Ptolemy’s andBrahe’s data might as well be due to some of the latter’smagnitudes being influenced by the Almagest . The transmission spectrum of the modern V-band filter is quitesimilar but not identical with the spectral sensitivity functionof the human eye. In fact there are at least two such sensitiv-ity functions (one for daylight- or photopic vision and one fornight- or scotopic vision) and the V-band curve lies in betweenof them both. Generally, the photopic vision is employed underbrighter ambient light and means that the eyes’ cones are activeand we can perceive colour. In contrast, the scotopic vision isthe extreme case where our vision depends solely on the eyes’rods which can only differentiate between bright and dark butnot detect colour (e.g. Clauss & Clauss, 2018, p. 178). As therods are more sensitive to shorter wavelengths, the sensitivityfunction for the scotopic vision is shifted towards blue colours,making blue stars a little brighter than red stars of the same V-band magnitude. The upcoming analysis is restricted to starsof 𝑚 HIP > . , excluding those stars which are bright enoughto be seen with photopic vision (indicated by them having acolour to the naked eye) . In this case we would expect the cal-culated 𝛿𝑚 to be something like a colour index between theV-band filter and the human eye-“filter” for 𝑚 > . mag. Aplot of two colour indices (e.g. ( 𝐼 − 𝐽 ) against ( 𝐾 − 𝐿 )) showsan almost linear relation with a slope 𝜕 𝐵𝑉 that can be calculatedfrom the effective wavelengths of the four filters (Ballesteros,2012). We performed a linear regression analysis for plots of 𝛿𝑚 against the ( 𝐵 − 𝑉 )-values for the three catalogues.To better visualise the tendency among the data, averages of 𝛿𝑚 were calculated for ( 𝐵 − 𝑉 ) -bins of . mag. The resultcan be seen in the left column of Figure 6 . together with ascatter plot of the entire catalogues. A linear model was fittedto the scatter plots and the parameters are shown in Table 3 .From the slopes of the linear regression, it is possible to calcu-late the effective wavelength of the human eye (under the givenpremises). The three catalogues yield values from nm to nm with error bars of ±6 nm. Both, the increase of 𝛿𝑚 for The exact threshold of colour vision in terms of star magnitudes is not clearlydefined but from observational experience, 2.3 mag could be an approximate value.
UTHOR ONE
ET AL reddish stars as seen in Figure 6 , as well as the calculatedeffective wavelength falling short of the V-band filter ( 𝜆 𝑒𝑓𝑓 =548 nm), agree with the above assumption of predominantlyscotopic vision. Nevertheless, the calculated wavelength islonger than what would be expected for exclusively scotopicvision ( ∼ 507 nm, see CVRL, 1995). The calculated regres-sion coefficients make it possible to systematically adjust 𝑚 with regard to the colour index of each star (see section 4.1).Lastly, a similar analysis was attempted for the brighter stars,but given the small number of stars with 𝑚 < . mag and thelarge standard deviation of 𝛿𝑚 , the statistical analysis did notyield significant results for any of the three catalogues. TABLE 3
Parameters of the fitted models in Figure 6 . Forthe two linear models 𝜕 is the slope and 𝜃 the intercept. 𝑛 is thenumber of stars included in each model and 𝑅 the fraction ofvariance, explained by the model. For statistical testing of themodels, see 4.2.Colour Extinction Background 𝜕 𝐵𝑉 𝜃 𝐵𝑉 𝑘 𝑓𝑖𝑡 𝜕 𝛽 𝜃 𝛽 P 0.11 -0.06 -0.01 -0.37 -0.69S 0.14 -0.06 0.09 -0.41 -0.75B 0.14 -0.06 0.05 -0.30 -0.53 𝑛 𝑅 𝑛 𝑅 𝑛 𝑅 P 922 0.006 992 0.0003 990 0.024S 925 0.017 990 0.022 989 0.045B 886 0.011 938 0.016 936 0.016
Even though the effect of atmospheric extinction is obviousin its existence for everyone who has watched the (night-) skywith some attention, no discussion of the effect can be foundin any of the works containing the three catalogues. If theobservers had completely ignored the extinction, it would haveto be expected that stars with low culmination altitudes wereestimated too faint. Schaefer (2013) analyses the dependencyand comes to the conclusion that all three catalogues are insome way “corrected” for extinction .With the new conversion, a similar analysis is shown in themiddle column of Figure 6 , as a scatter plot of all stars,and as averaged 𝛿𝑚 -values for ◦ -bins of culmination altitude. That does not necessarily mean they were explicitly corrected by some for-mula or observational procedure but could also mean to just estimate slightlybrighter magnitudes for low standing stars. Trying to observe stars at their high-est position could also be considered such a correction and is presupposed for thefollowing analysis. δ m [ m ag ] P t o l e m y ( P ) -2-1012 δ m [ m ag ] a l - S u fi ( S ) -2-1012 δ m [ m ag ] B r ahe ( B ) -2-1012 (B-V) [mag] alt [°]
20 40 60 80 β [mag/sr] -2,5 -2 -1,5 FIGURE 6
The dependency of 𝛿𝑚 on colour index B-V (left) , maximum culmination altitude alt (middle) and back-ground brightness 𝛽 (right) within catalogues P, S and B.Grey scatter plots are the single stars and bold coloureddots are mean values of bins of the independent variablewith SEM-error bars. The bold lines show models, fit-ted to the scatter plots for each dependency. Stars with 𝑚 HIP < . mag were excluded from the colour-analysis.For the middle column, the dashed lines show expectedextinction coefficients of 𝑘 = 0 . and . mag/ 𝑋 .The 𝛽 -bins correspond to the levels of grey inFigs. 7 and 8 . All model parameters can be found in Table 3 . AUTHOR ONE
ET AL
Schaefer gives the following extinction function: 𝛿𝑚 = 𝑘 ⋅ [( sin( 𝑎𝑙𝑡 ) + 0 . ⋅ 𝑒 −11 ⋅ sin( 𝑎𝑙𝑡 ) ) −1 − 1 ] (3)With the extinction coefficient 𝑘 , given in magnitudes per air-mass 𝑋 and the horizontal altitude 𝑎𝑙𝑡 of the respective star. Inapplication to the historical catalogues, the altitude (or ratherthe culmination point of a star) can be calculated from the geo-graphic latitude 𝜙 of the observer and the declination 𝛿 of thestar at the time of observation. 𝑎𝑙𝑡 = 90⅄⁃ − | 𝛿 − 𝜙 | (4)The extinction curves are plotted within each sub-figure for 𝑘 = 0 . mag/ 𝑋 (as suggested by Schaefer, 2013), as well as 𝑘 = 0 . mag/ 𝑋 . Pickering (2002) assumes such a value for apre-industrial atmosphere. Additionally, models according toequation 3 were fitted to the data in Figure 6 with the resultingparameters 𝑘 𝑓𝑖𝑡 listed in Table 3 . Schaefer’s general result isreproduced with close to no (in fact even a slight but insignifi-cant negative) extinction effect showing in the Almagest’s data.In contrast, an effect is clearly visible in al-S. ¯uf¯ı’s magnitudesbut it still falls short of the expected intensity for Schaefer’sextinction coefficient. However, the data could almost agreewith the lower extinction coefficient of 𝑘 = 0 . mag/ 𝑋 . InBrahe’s magnitude estimations, a weak extinction effect canbe found but again, it falls way short of the plotted models.Although the effect is weaker than expected, it can be correctedfor at least in al-S. ¯uf¯ı’s and Brahe’s data, as a clear systematicdeviation can be found there.It should be noted that the models, fitted in Figure 6 arevery sensitive to single extreme 𝛿𝑚 at low altitudes, whichmight occur due to falsely identified stars or other sporadicerrors. Furthermore, it also seems to be highly controversialwhat extinction coefficient would have to be expected for apre-industrial atmosphere (see Hearnshaw, 1999; Pickering,2002; Schaefer, 2013) and lastly, the actual effect found ineach catalogue also depends on the exact method by whichthe magnitudes were estimated which can only be speculatedabout. The dependency of 𝛿𝑚 on the colour index, as well as the cul-mination altitude, are both effects that can be understood andmodelled in a (bio-)physical sense. However, there seem to befurther trends within the data, which can be found looking atthe spatial distribution of the 𝛿𝑚 . Figures 7 and 8 showmaps of all stars within catalogues P,S,B. The 𝛿𝑚 -values arerounded into 5 bins and the stars coloured accordingly. The Possibly, the worldwide 2020 Corona-lockdown might bring new insights intothis question. exact colour-scales can be found within both figures. Addi-tionally, the maps show a kind of “background brightness” orrather a summed brightness of stars per area, which is depictedby a graduated grey-scale. The actual area from which the fluxis summed are not the grey rectangular fields, but circles – orrather cones in actual 3D-space – around the centre of eachfield. These cones have a uniform radius of ◦ each, so thesummed flux can easily be converted to mag/sr. The grey fieldsare not of perfectly uniform solid angle but were necessary forthe visualisation, as they cover the whole projection withoutgaps or overlaps. The fields cover ◦ of latitude each and alongitude segment that corresponds best to ◦ of a great circlewhile still guaranteeing an integer amount of segments withinthe ◦ circle of latitude. For the summation, all stars withinthe HIPPARCOS catalogue (ESA, 1997), between 6 mag and10 mag were used (a total of 112,914 stars). Only faint starswere chosen for several reasons: • Assure that a star’s background is not primarily definedby the star itself. • Keep the differences in surface brightness small, even fora high spatial resolution (i.e. small grey fields). • The density of those dimmer stars corresponds well tothe perceived brightness of the actual night sky (e.g. theMilky Way (MW) is clearly visible)Looking at the maps, we can find areas within each cataloguewhere stars are predominantly estimated too faint (“red areas”).More specifically, we can make the following observations forcatalogues P an S:1. The similarity of catalogues P ans S can be found oncemore in the maps.2. Nevertheless, Al-S. ¯uf¯ı seems to have reworked many ofPtolemy’s most southern stars to fainter values.3. Both show “red areas” throughout the MW and especiallyaround the centre of the galaxy ( < 𝜆 < ◦ ∕ 0 <𝛽 < ◦ ).4. Like Ptolemy, Al-S. ¯uf¯ı still estimates many stars in thearea < 𝜆 < ◦ ∕ − 30 < 𝛽 < ◦ as too bright.However the visibility limit shifts away from those starsand towards the galaxy centre for al-S. ¯uf¯ı’s time.5. Al-S. ¯uf¯ı generally has fewer extreme mistakes, depictinghis lower standard deviation in 𝛿𝑚 .Points 2 and 4 mostly explain the stronger extinction effect inal-S. ¯uf¯ı’s data (see Figure 6 ) but can only partly (2.) be con-sidered to be really caused by extinction. Apart from that, itseems obvious that both authors show a tendency to estimatestars in bright areas too faint. UTHOR ONE
ET AL FIGURE 7
Plots of star catalogues P (top figure) and S (bottom figure) in ecliptical coordinates (equinox J2000). The stars’size is according to their 𝑚 HIP and colour according to their 𝛿𝑚 . Green stars were estimated too bright and red ones too faint (seescale). The pink lines mark the Milky Way ( ±10 ◦ of gal. lat.), the orange lines are the southern visibility limits of the respectivetime and place. The background depicts the summed brightness of stars in the area as grey-scale. More precisely, the flux ofall stars between mag < 𝑚 HIP < mag from the HIPPARCOS catalogue is summed and given as surface brightness of therespective area. For details on background-colouring, see text. There is a clear tendency of many too brightly estimated stars indarker areas and vice versa. . AUTHOR ONE
ET AL
FIGURE 8
Plots of star catalogue B in ecliptical coordinates (equinox J2000). For detailed description, see Figure 7Brahe’s map differs considerably from P and S, showing thefollowing notable features:1. The brightest parts of the MW are missing, due to Brahe’snorthern geographic latitude.2. “Red areas” can be found from the galaxy centre (onlypartly visible) along the visibility limit (VL) within <𝜆 < ◦ .3. However, other areas along the VL are not particularlyred.4. Another large “red area” can be found at < 𝜆 < ◦ ,roughly along the MW.5. Again, other parts of the MW do not show any cleartendency.It becomes obvious, where the extinction feature (see middlecolumn of Figure 6 ) stems from in Brahe’s case (2.) but itis remarkable that the effect is obvious within this area andcompletely vanishes for other longitudes. A dependency on thebackground brightness can also be found in some parts of themap. For all three catalogues it is very notable that effects ofbackground brightness and extinction are partly visible but cannot be the sole reason of every “red” or “green area”.Concerning the dependency of 𝛿𝑚 on the background bright-ness, one might want to explain it by the varying degree of dark-adaptions for differently bright areas. However, singlebright stars would have the strongest effect here and those areexcluded from the background brightness, shown in the maps .So the whole phenomenon seems to be of rather psychologicalnature which makes it harder to quantify theoretically. Never-theless, several authors (Hearnshaw, 1999; Zinner, 1926) havedescribed and analysed the dependency but mostly restrictedthemselves to a comparison between stars within and outsideof the Milky Way. Going a step further, we used the valueof background brightness from the maps (Figs. 7 ,8 ) to plotaverage 𝛿𝑚 for 10 bins of surface brightness. The mean val-ues of each bin are shown in the right column of Figure 6together with a scatter plot of all stars. The decline of 𝛿𝑚 fordarker backgrounds becomes clearly visible. As there is noavailable mathematical model to describe the expected depen-dency of 𝛿𝑚 on the background brightness, a linear regressionis the simplest approximation. The regression parameters (seeTable 3 ) can again be used to correct the values of 𝛿𝑚 for thedescribed effect. Of course, the correction can only be madeif the exact same background brightness values are calculatedfor every star which might be laborious for anyone trying tomake use of the correction formula (5). As an alternative, theaverage 𝛿𝑚 for stars within and without the Milky Way ( ±10 ◦ looking at the surroundings of the brightest stars, there is no clear trend in anyof the maps, either way. UTHOR ONE
ET AL TABLE 4
Average 𝛿𝑚 of stars within ( 𝛿𝑚 𝑀𝑊 ) and without( 𝛿𝑚 𝑛𝑜𝑡𝑀𝑊 ) the Milky Way.Cat. 𝛿𝑚 𝑀𝑊 𝛿𝑚 𝑛𝑜𝑡𝑀𝑊 P .
23 ± 0 .
06 −0 .
06 ± 0 . S .
24 ± 0 .
05 −0 .
07 ± 0 . B .
08 ± 0 .
06 0 .
01 ± 0 . of galactic lat.) is given in Table 4 and can be used for thecorrection formula (5) instead. The above analysis investigated different systematical depen-dencies within the magnitude data of the three historical cat-alogues. It was shown that stars are often estimated slightlyfainter, if . . . • they are red • they stand close to the southern visibility limit • they are seen in bright areas of the night sky. The effects were described quantitatively and can therefore becorrected. As the single corrective terms are small in com-parison to the overall variance in 𝛿𝑚 , they can be assumed tobe independent from one another and the correction takes theform of a simple additive parameter Δ 𝑖 for each of the threemodels which is subtracted from the initial converted magni-tude. This yields a new magnitude 𝑚 ∗ for each star which canbe considered the best approximation to the modern V-magscale. 𝑚 ∗ = 𝑚 𝑜𝑙𝑑 − 𝑡 ′ 𝑏 − Δ 𝐵 − 𝑉 − Δ 𝐸𝑥𝑡 − Δ 𝛽 (5)Where 𝑡 ′ and 𝑏 are given in equation (2). The single Δ 𝑖 arethen the respective models which were fitted to the data, or forthe background brightness it can also be the alternative modelbased on the position within or without the Milky Way. Δ 𝐵 − 𝑉 = 𝜕 𝐵 − 𝑉 ⋅ ( 𝐵 − 𝑉 ) + 𝜃 𝐵 − 𝑉 Δ 𝐸𝑥𝑡 = 𝑘 ⋅ [ (sin( 𝑎𝑙𝑡 ) + 0 . ⋅ 𝑒 −11 ⋅ sin( 𝑎𝑙𝑡 ) ) −1 − 1 ] Δ 𝛽 = 𝜕 𝛽 ⋅ 𝛽 + 𝜃 𝛽 or = 𝛿𝑚 𝑀𝑊 ∕ 𝑛𝑜𝑡𝑀𝑊 The empirical parameters 𝜕 𝐵 − 𝑉 , 𝜃 𝐵 − 𝑉 , 𝑘, 𝜕 𝛽 , 𝜃 𝛽 can be takenfrom Figure 6 and 𝛿𝑚 𝑀𝑊 ∕ 𝑛𝑜𝑡𝑀𝑊 is given in Table 4 . Looking at the three single correction models, it is conspicuousthat the variance, explained by the models is much lower thanthe residual variance induced by the wide scattering of the 𝛿𝑚 .In fact the 𝑅 -values (see Table 3 ) which express the fractionof the variance, explained by the models, remain at a few per-cent for all nine models. However, when testing the models forsignificance against a null hypothesis 𝐻 which predicts zerocorrelation, 𝐻 can be dismissed (meaning 𝐹 𝑛 > 𝐹 ,𝑛 −2 , . ) inalmost all cases on a
95 % or higher confidence level. This isdue to the high number of 𝛿𝑚 values and suggests that the cor-rections – however little variance they might explain – are infact significant. The only model, not showing statistical signif-icance is the extinction correction for the Almagest data, whichshowed a negative (and therefore nonsensical) value of 𝑘 𝑓𝑖𝑡 .Nevertheless, when applying the corrections, the low values TABLE 5
The models in Figure 6 were tested for statisticalsignificance against the hypothesis 𝐻 of no dependency. Theresulting parameters 𝐹 𝑛 were calculated from 𝑅 and 𝑛 (seeTable 3 ) and are given together with the confidence level (CL)on which 𝐻 can be rejected.Color Extinction Background 𝐹 𝑛 CL 𝐹 𝑛 CL 𝐹 𝑛 CLP . . – . . S . .
9% 22 . .
9% 46 . . B . . .
9% 15 . . 𝐹 ,𝑛 −2 , . . . = 3.9 / 6.7 / 10.9 for < 𝑛 < of 𝑅 lead to an almost negligible “improvement” of the 𝛿𝑚 ’sstandard deviations. Figure 9 shows the distribution of the 𝛿𝑚 before and after the correction formula was applied. So after all, what is the accuracy of pre-telescopic magnitudeestimations and how can they best be converted to their cor-responding V-mag values? From the three major catalogueswhich contain original magnitudes, Ptolemy’s Almagest andTycho Brahe’s data can be considered largely independent andboth show uncertainties of very similar size. Al-S. ¯uf¯ı’s estima-tions, on the other hand can be seen as an – in most instances –improved version of Ptolemy’s data which show a significantlyhigher accuracy. The distribution of magnitude errors 𝛿𝑚 in allthree catalogues follows almost Gaussian curves, so the dou-bled standard deviations 𝜎 can be considered error bars on a AUTHOR ONE
ET AL A b s o l u t e f r equen cy δ m P [mag] -3 -2 -1 0 1 2 3 initial σ = 0.79 magcorrected σ = 0.78 mag δ m S [mag] -3 -2 -1 0 1 2 3 initialcorrected σ = 0.64 magσ = 0.61 mag δ m B [mag] -3 -2 -1 0 1 2 3 initial σ = 0.76 magcorrected σ = 0.75 mag FIGURE 9
Distribution of the 𝛿𝑚 before and after the corrections, applied in section 3.2. The histograms of bin size . magshow approximate normal distributions for all three catalogues. The corrections result in only minor changes of the distributions.Overall, Al-S. ¯uf¯ı’s magnitude errors show the lowest variance.95% confidence level.Any magnitude, taken directly from one of the catalogues tobe used for studies of transient observations and longterm vari-abilities or even processes of stellar evolution, should firstbe converted before comparing them to modern V-mag val-ues. We recommend, adding the brighter- / fainter-qualifiers as . mag-steps to the original magnitudes and then employingformula (1) to attain the Johnson V-magnitude. The resultingvalues should be sufficient for most applications and come witherror bars of: 𝜎 𝑃 = 1 . mag , 𝜎 𝑆 = 1 . mag , 𝜎 𝐵 = 1 . magIn comparison, Hearnshaw (1999) calculates standard devia-tions between 0.41 and 0.72 mag for most groups of Ptolemianmagnitudes while Zinner (1926) gives values between 0.44 and0.60 mag as “mean errors” to adopt the corrected values, eventhough the error bars are not distinctly reduced by the correc-tion. Those magnitudes can either be attained by equation (5)or taken from the online catalogue, provided by the authors(see APPENDIX B:) Using the corrected values seems espe-cially necessary when analysing only certain groups of starswhich might otherwise be systematically biased. This couldfor example be red giants which all show high values of (B-V)or stars within a certain constellation which might all be in anespecially bright or dark part of the night sky. probably mean absolute errors which are always smaller than (or equal to) thestandard deviation. Both values are given for the historical magnitudes 𝑚 𝑃 where ourown standard deviation results in 𝜎 ′ 𝑃 = 0 . mag ∕1 .
33 = 0 . mag and therefore ingood concordance with the previous studies.In some cases it might be sensible Other than that, the analysis of dependencies on colour, extinc-tion and background brightness might also be used to inves-tigate otherwise unrelated questions like the extinction coef-ficient of the pre-industrial atmosphere or even the effectiveabsorption wavelength of the human eye (under naked eyeobservation conditions).As we conclude that the error bars of the magnitudes in his-torical catalogues are ∼ 1 . to ∼ 1 . mag, almost all (> 93 %)variabilities of the naked eye stars (as displayed in Fig. 3 ) arecovered by the error bars which makes it virtually impossibleto detect longterm variabilitiesIt should, however, be kept in mind that statistics mean lit-tle for a single data point. As we know, for particular cases,the ancient observers must have recognised changes in bright-ness of less than . mag. After all, it was possible to nakedeye observe the brightness drop of Betelgeuse ( 𝛼 Ori) in win-ter 2019/ 2020 for many laymen and there are hypothesesthat the variability of Algol ( 𝛽 Per) had been known in ancientEgypt (Jetsu et al., 2013). Cases like these are possible for indi-vidual stars which are in a region with appropriate naked eyecomparison stars. That is why, our statistical error bars shouldbe considered the general first step but for some handpickedindividual stars careful case studies appear worthwhile. Also, Aboriginal Australians seem to have discovered Betelgeuses’ variability(Schaefer, 2018)
UTHOR ONE
ET AL ACKNOWLEDGMENTS
P.P. thanks Prof. Bradley Schaefer, Dr. Rob van Gent and ProfF. Richard Stephenson for answering my requests and helpingme get a first grip of the topic during my Master Thesis.We furthermore thank our referee, Prof. John Hearnshaw, forhis recommendations and reassuring comments.S.H. thanks the Free State of Thuringia for the employment atthe Friedrich Schiller University of Jena, Germany.We thank Ralph Neuhäuser (AIU, Friedrich-Schiller-Universität Jena) for the initiative of investigating historicaldata and to create room for transdisciplinary projects.This research has thankfully made use of VizieRcatalogue access tool, CDS, Strasbourg, France(DOI:10.26093/cds/vizier, (Ochsenbein et al., 2000)), and theVSX variable star catalogue of the American Association ofVariable star Observers (AAVSO) (Watson, Henden, & Price,2006).
Author contributions
This analysis of pre-telescopic magnitudes was mostly writtenby PP in the context of his master thesis. SH developed the ideafrom the thesis towards this contribution and offered extensivesupport, advice and revision during the completion of the the-sis as well as the article. Additionally, SH contributed largeparts of the introduction and the final remarks.
How cite this article:
Protte, P., Hoffmann, S.M., (2020),Accuracy of magnitudes in pre-telescopic star catalogues, AN ,. APPENDIX A: OPTICAL INFLUENCES OFHUMAN VISION ON MAGNITUDEESTIMATES
The pictures in figure A1 show that a bright object (inthis case, Venus) can be described as ‘having rays’, ‘horned’,‘hairy’, or ‘fuzzy’. The photos were taken in central Europe(April 4th to 6th 2020) under normal clear weather conditions.The rays and horns of bright objects are not only an effect of theweather but are produced by the interplay of a lens (of the eyeas well as of camera optics) and its entrance pupil with an enter-ing wavefront. Passing through a lens with a limited entrancepupil, the wavefronts are described by the Zernike polynomials 𝑍 producing the known effects like astigmatism 𝑍 , coma 𝑍 ,the trefoil effect 𝑍 (three rays) , spherical aberration 𝑍 , andhigher orders of aberration in the perfectly spherical lens. Theeffect is caused by the limited size of the pupil (LÃşpez-Gil etal., 2007) and unevenness of the border increases the effect, as well as astigmatism of the lens itself. The irises of both, cameraand eye, are limitations of the pupil and the polygonal shapeof the mechanical iris of a camera lens as well as the musclesat the edge of the eye both increase such effects: The photosof these rays do in fact show roughly the same as what the eyesees.With atmospheric conditions of the desert or in tropical climate(with sandstorm or humidity) the atmospheric effects becomestronger and the bright point source is blurred; the beam oflight from the star does not enter the pupil parallelly and thecontraints for applying the Zernike polynomials directly arenot fullfilled perfectly anymore. The atmospheric influence caneven lead to less rays, simply showing the blurred Airy disksaround the bright object instead (rightmost picture, with cirrusclouds on April 10th). APPENDIX B: ONLINE-ONLY MATERIAL
We prepared the catalogues according to our suggestions inthe above work. The data files will be uploaded in CDS assoon as the paper is published. It will be available at CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsarc.u-strasbg.fr/viz-bin/qcat?J/AN
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