Another argument over the authorship of the star catalog examines the stars near the southern horizon. Since Hipparchus observed from Rhodes with a latitude of $\approx 37º$ and Ptolemy from Alexandria which is $\approx 31º$ N, this means that Ptolemy could have observed starts $5º$ further south that were never above the horizon of Hipparchus.
However, Delambre notes that there’s not a single star in the catalog that could not have been observed by Hipparchus at his latitude. So if Ptolemy was the originator of the catalog then, for some unknown reason, he declined to observe stars near the horizon1.
Thus, the inclusion or exclusion of stars cannot settle the matter. But astronomers have turned to other questions regarding stars near the southern horizon to approach it. So in this post, we’ll explore three papers on this subject.
Rawlins
In his $1982$ paper, Dennis Rawlins first approaches the question of authorship by examining the frequencies of stars of certain magnitudes being included in the catalog.
While Ptolemy claims that the catalog is a complete accounting of stars brighter than $6^{th}$ magnitude, this is obviously not true2. Thousands of stars brighter than this magnitude are excluded, becoming more likely to be missed the fainter than they get.
Below, I provide a graph based on Rawlins’ estimates on how likely stars of given magnitudes are to be included.
Now enter the phenomenon of atmospheric extinction. This is the dimming of the apparent brightness of stars as their light passes through the atmosphere due to scattering. The more atmosphere, the more the light is scattered and the dimmer the star appears. And since we are looking through more atmosphere near the horizon, the more these stars will be dimmed.
Since the same southern star would appear closer to the horizon for Hipparchus than it would for Ptolemy, this means that Hipparchus would suffer more dimming than Ptolemy would.
Because the brightness of the star is related to the probability of it making it into the catalog, this means that Hipparchus should have excluded more stars that would have been included without atmospheric extinction than Ptolemy would have. Thus, Rawlins proposes to attempt to calculate the apparent magnitude3 of these stars after extinction for both observers and see whose best fits the above probability distribution.
Doing so, Rawlins finds that it matches far better for Hipparchus than it did for Ptolemy. In other words, if Ptolemy were the observer, he should have observed far more stars in this region than he did as they wouldn’t have been dimmed as much.
Rawlins’ states his conclusion with considerable snark:
So much for Ptolemy’s observership.
Schaefer
In $2001$, Bradley Schaefer of the University of Texas gave a long rebuttal to Rawlins’ claims stating that Rawlins’ work had three fundamental problems, any of which could be fatal to his conclusions.
The first is that Rawlins only selected $30$ stars for his analysis. While this isn’t inherently a problem, so long as they are a representative selection of the possible set, it is possible that this could cause bias.
Second, Schaefer states that Rawlins vastly underestimated the scattering the atmosphere would cause. He states that the amount of atmospheric extinction Rawlins estimated
is better than the best-in-the-world high-altitude observatories. An atmosphere as clear as Rawlins has postulated is absurd for any sea-level site.
From what I can tell, Rawlins used an extinction constant of $0.15$. We’ll discuss Schaefer’s values here shortly.
Third, Schaefer criticizes the model Rawlins used for extinction, calling it “rather simplistic.”
To understand why this is an issue, let me reproduce a graph from much latter in Schaefer’s paper:
Here Schaefer is showing how the latitude is changed based on the the same using probability function for three different values of the extinction coefficient, $k$. The most likely latitude is where the rejection level is minimized. From this, we can clearly see that the lower extinction coefficients push the latitude predicted by this model northwards. Thus, Rawlins’ low extinction coefficient alone can explain the latitude derived.
Fortunately, Schaefer tells us, these issues are easily remedied.
First, Schaefer spends considerable effort trying to piece together a better value for the extinction coefficient. He first considers numerous modern observations from both sites, using over $27,000$ measurements. Using these, Schaefer finds that the extinction coefficient ranges fairly considerably, but is usually in the range of $0.27 – 0.37$.
The second method is rather brilliant in my opinion. Here, Schaefer looks at something known as the arcus visionis (the arc of vision). What this boils down to is that the refractive effects of atmosphere, an object will appear to rise above the horizon before it otherwise should if there were no atmosphere. Although the effect is different than atmospheric extinction, it is still caused by the same underlying situation: looking through a bunch of atmosphere. Thus, the refractive index can be determined from this. And we actually have information on the arcus visionis for numerous planets from Ptolemy himself. Using these, Schaefer determines an extinction coefficient of $0.25$ on average4.
Third, Schaefer tries to model atmospheric extinction using over $300$ sites around the world to determine how it should be having for various parameters such as the humidity, altitude, latitude, temperature, and time of year and then applies this model to the two sites.
Lastly, he uses a standard model known as GADS to estimate the constant.
In all cases, the extinction coefficient is significantly higher than given by Rawlins.
Thus, Schaefer adopts a higher value for the extinction coefficient than Rawlins.
But before jumping into applying this information to Ptolemy’s catalog, Schaefer attempts to build his reader’s confidence by validating the model using Tycho’s star catalog from the $16^{th}$ century as well as some of his own data.
Using it for Tycho’s catalog, he is able to determine that it was observed at a latitude of $55º \pm 1º$ which is quite consistent with Brahe’s observatory at Hven which has a latitude of $55.9º$.
For his own observations, I want to pause a moment to appreciate the methodology here. Specifically, Schaefer obviously appreciates the need to explore history in more than an academic setting. He has been working on building his own naked-eye star catalog from various dark sky sites around the world5. He appears to be less focused on the positional data, but more concerned with estimating brightnesses by eye to get a better understanding of the accuracy of such methods – something my project is sorely lacking but I have considered for inclusion.
I won’t go through the tests on each of his observations he gives, but will summarize by stating that his method was generally able to predict the latitude to a better degree of accuracy than the $5º$ that separates Rhodes and Alexandria6.
Schaefer finally turns to the star catalog in the Almagest. First he looks at the error in magnitude as reported by Ptolemy as compared with the expected magnitudes from his updated model and comes to a rather surprising conclusion.
First, let’s look at the data:
Here, the dots with error bars are the average error in magnitude for stars compared with their modern brightnesses (not accounting for extinction).
What we should expect is that, as the declination decreases (i.e., gets more southern) the error should increase due to the atmospheric extinction. By how much is given by the solid line for Alexandria and the dotted line for Rhodes.
Yet neither match. Schaefer takes this as clear evidence that the observer must have applied some sort of correction to account for extinction!
He states,
This conclusion might be surprising since no mention is made of extinction in Antiquity. Yet, the dimming of stars near the horizon is a trivial observation, and so the existence of extinction is easy to realize. Once realized, an observer who is quantifying a star’s brightness could easily introduce some sort of approximate correction. While there are not enough information to recover the details of the correction, the existence of the correction is proved by the lack of southern star magnitudes dimming as required by extinction.
However, the presence of such a correction would mean that we can no longer expect the above data to fit either curve to determine the best fit to the latitude.
But there are other tests we can apply. So next, Schaefer considers the limiting magnitude. For those not familiar with the term, this is basically a question of the faintest star visible to the instrument being used – in this case, the unaided eye.
Effectively, Schaefer is asking whether the way the brightnesses fall off due to extinction for the faintest stars better matches Rhodes or Alexandria.
Before doing that, Schaefer broke the stars up into sections of the sky. While he appears to have stared by dividing the sky into octants, he quickly shifts to discussing quadrants. Specifically, he notes that the fit for the first three quadrants7 is notably different than for the last one8. So we’ll start, as he does, by discussing those first three sections.
First, let’s look at a graph of the data vs the predictions for limiting magnitude for each locale:
Here, we can clearly see that the data strongly prefers the solid line (Alexandria) as the most southern stars should have been dimmed too much to have been visible from Rhodes (the dashed line).
Next, Schaefer applies the same test as proposed by Rawlins – looking at the distribution of likelihoods that stars are included or excluded based on the extinction of stars at the two locales. Again, Schaefer finds that Ptolemy is the preferred observer.
Schaefer asks the very fair question about how robust these results are. This is where the graph I showed above came in showing that over a wide range of plausible values for the extinction constant, Ptolemy is still the preferred observer for these quadrants.
Turning to the fourth quadrant, Schaefer finds that it is notably different than the previous ones:
In this case, the data clearly seems to favor Hipparchus as the observer with a latitude of $35.7º \pm 1.1º$. However, Schaefer offers several reasons this may he incorrect.
The first is that there are some stars in this section of sky whose identity is not well established. In particular,
four faint stars near Piscis Austrinus were identified as stars much farther south by Toomer9.
Using Toomer’s identifications alone changes the the best fit latitude to $34.7º \pm 1.0º$.
Similarly, the time that this quadrant of the sky would be observed would be “around summertime when the skies are the haziest.” In other words, it is not unreasonable to assume a higher extinction coefficient than for the other quadrants which will again push the best fit latitude lower. The values that reduce the latitude back to that of Ptolemy are easily in line with plausible values from what Schaefer determined in the beginning of his paper.
Thus,
a pro-Ptolemy advocate need only insist on normal extinction to conclude that all four quadrants have Ptolemy as the observer.
Still, for this quadrant, Schaefer rejects Ptolemy at an $85\%$ confidence level from this test.
To see how robust this conclusion is, he also performs the above test regarding limiting magnitude:
Again, the data seems to favor Hipparchus as the observer for this quadrant of sky, although the difference between the two is greatly reduced.
Pickering10
A final paper in this discussion comes from Keith Pickering from $2002$, this time criticizing Schaefer’s results.
The first issue he raises is that there are some stars definitively observed by Hipparchus in the Aratus Commentary that, based on Schaefer’s extinction models, would have almost certainly been below the limiting magnitude of Hipparchus at Rhodes.
He gives the specific example of γ Ara11 which, according to Schaefer, should have an apparent magnitude of $6.7$ which puts it up against the very limits of observation by an observer with near perfect eyesight under ideal conditions. Furthermore, it would be very unlikely that such an apparently faint star would be given inclusion in the first place12.
Pickering summarizes his argument quite succinctly:
any procedure that eliminates Hipparchos as the observer of the ASC13, based on a statistical analysis of the southern limit, will also eliminate Hipparchos as the observer of the Commentary, too: a known incorrect result.
That certainly seems damning.
Pickering notes that he gave the comment regarding γ Ara directly to Schaefer at a conference. In response, Schaefer evidently posted to a historical astronomy listserve attempting to address the issue.
Specifically, Schaefer claimed that there should be a $1.5\%$ chance of a star of such magnitude getting included. Pickering does not state whether Schaefer offered any justification for this calculation, but we can attempt to address it here by referring to the frequency function Rawlins gave above.
There, Rawlins gave the probability of a star of magnitude $6.5$ as $0.25\%$ and a star of magnitude $7$ as an order of magnitude less or $0.025\%$. Estimating linearly14 between the two I arrive at a probability of $0.16\%$.
However, this is for a single star. Given there should be multiple stars that could be included, we need to multiply this probability times the number of stars that could qualify.
Pickering states that Schaefer estimates $50$ stars that could qualify. Again, no evidence is provided for this estimate, but again, we can use the Yale Bright Star Catalog to estimate its accuracy15.
Filtering to stars with a magnitude of $6.5 – 7.0$ and with a declination between $-59º$ to $-49º$16, I find only $26$ such stars although my threshold for what constitutes “southern stars” may be set too low.
Thus, the possibility that one of these stars gets included I find to be about $4\%$17. Unlikely, but not impossible.
Pickering comes up with even lower numbers18 coming up with only $13$ stars and a probability of $0.08\%$ for stars of this magnitude, which I find unreasonably low, but we’ll return to that later.
As the source of the error for Schaefer’s analysis, Pickering first looks at the probability function Schaefer used to determine the likelihood of stars of a given magnitude being included, noting that, if applied to the Aratus Commentary, it would exclude Hipparchus as an author at a confidence level of $99\%$ with a best fit latitude of $30º$.
So where did this probability function go wrong?
Pickering notes that the issue may well be the way Schaefer derived the function in the first place. To do so, Schaefer used the magnitudes of the star before extinction. Yet when he then plugs stars into the equation, he uses the values after extinction. Pickering refers to this as a “colossal blunder which invalidates everything that follows”19.
He goes on to criticize the derivation after that, describing Schaefer’s methods as “inexplicable” as he tries to derive two parameters of the equation separately. Pickering proposes a much simpler method to do so.
With his method, Pickering finds limiting magnitudes $0.2$ magnitudes fainter than predicted by Schaefer and that the graphs of probability I showed above are much wider, reducing the uncertainty at which Hipparchus could be ruled out.
Another issue Pickering discusses is the oddity that Schaefer found with the fourth quadrant. Pickering notes that this region is the one that happens to be furthest from the galactic plane where stars are more sparse. Thus, to include more than a few stars at all, the observer would have had to include stars of fainter magnitude.
Similarly, Pickering notes that an observer would likely include more faint stars in regions that were of especial interest, such as the ecliptic. Indeed, if the limiting magnitudes are calculated based on the data in the catalog for constellations near the ecliptic, fainter stars are included, but only by about $0.3$ magnitudes which is similar to the spread near to far from the galactic plane.
Thus, these biases of intentionally going hunting for fainter stars in specific places can bias the derived probability function.
Lastly, Pickering does a sanity check on how the probability function behaves and notes that it is inversely symmetrical about the $50\%$ magnitude. In other words,
for a typical [limiting magnitude] of $5$, the changes of cataloging a magnitude $7$ star would be exactly the same as missing a magnitude $3$. But this cannot be true, since there is one magnitude $3$ star missing from the ASC (η Tau) out of about two hundred possibilities, while there is not a single magnitude $7$ star in the catalog out of several thousand possibilities. Therefore, the real probability function is slightly asymmetric from the $50\%$ magnitude.
In response, Pickering re-derives the probability function, selecting only stars near the ecliptic and galactic plane. In this derivation he uses an extinction coefficient of $0.182$ which is significantly closer to that of Rawlins than that of Schaefer and he then attempts to justify.
To do so, Pickering uses the magnitudes from multiple pre-industrial catalogs to compute extinction coefficients and finds that they are much closer to Rawlins’ derived values.
Going further, Pickering points out that there are indeed sea level sites such as Barrow, Alaska which have values in line with what Rawlins assumed.
As a result of all of this, Pickering rejects the high extinction coefficient value Schaefer assumes.
At this point, we should probably be starting to wonder what the results of these improved probability functions and extinction will be if the same methods are applied.
Before doing so, Pickering does a sanity check, again applying these techniques to known data to see if it can correctly predict things. Sadly, he finds that it rejects the correct latitude20 the majority of the time.
Thus, Pickering’s conclusion seems to be that this technique is unable to produce results, stating
the procedure is chaotic, i.e., there is a sensitive dependence upon initial conditions – those conditions being the chosen atmosphere, [the probability] function, and the magnitude limit. It is quite possible, when several of these factors work together, that the statistical procedure will reject the correct value at a statistically significant level. When, as in the case of Schaefer $2001$, we combine an incorrect [probability] function, an overly opaque atmosphere… an incorrect result is almost impossible to avoid.
Well, we tried.
Ultimately, I’m not entirely convinced that Schaefer’s extinction coefficients are inappropriately high. In particular, the demonstration of correctly calculating the latitude of Tycho’s observatory using observations that were similarly pre-industrial I find quite convincing.
However, I do agree with Pickering’s conclusions that the uncertainty in these matters is likely too high to make any firm determinations.
- There are plenty of reasons that spring to mind. For my own purposes, I often don’t observe anything under $10º$ in altitude as it becomes extremely difficult to use my quadrant. Worse, since the entire quadrant itself swings, pulling it that far from the center of mass means that it tilts the entire base column which leads to inaccurate readings. Perhaps similar problems plagued the observer with an armillary sphere. Another possibility is that Ptolemy was aware that the brightness of stars near the horizon was dimmed as is easy to see by watching a star throughout the night. Thus, it is entirely reasonable that they simply avoided stars too close to the horizon. Lastly, we could also postulate that, as Ptolemy often did, he was following Hipparchus’ lead and did not see fit to include stars that Hipparchus did not.
- A more complete accounting is given in the Yale Bright Star Catalog which includes nearly all stars greater than $7^{th}$ magnitude. This, of course, includes many stars that are not visible in Alexandria. However, filtering this to stars that would be at least $1º$ above the horizon in Alexandria (i.e., with a declination $>-54º$ in declination) and magnitude $6$ or greater, I find that there should be at least $4,500$ stars!
- Do note that I’m using the term “apparent magnitude” in a slightly unusual way here. Typically in astronomy, absolute magnitude is the brightness of a star at a distance of $10$ parsecs. The apparent magnitude is how bright it appears to be at whatever distance it’s truly at, but does not generally include atmospheric effects. Thus, my inclusion of them here changes the definition I’m using.
- In many respects, I find myself wanting to rely on this value the most as all other methods are under the burden of using the modern atmosphere after several centuries of change caused by industrialization which I do not see any account made for by Schaefer.
- He lists Chile, Connecticut, Massachusetts, Venezuela, Tunsia, and Bermuda.
- Because the position of the stars also depends on the epoch, this method can also be used to estimate the year in which the observations were made, but the errors are sufficiently large (as shown by Schaefer in the above tests on known data) that it would be unable to distinguish between the epochs of Hipparchus and Ptolemy. Hence, I do not discuss it further here despite Schaefer bringing it up frequently through the remainder of the paper.
- In right ascension instead of the ecliptic longitude we’ve been dealing with.
- Schaefer states that this isn’t particularly surprising. Previous authors have noted that different sections of the sky have different errors in ecliptic longitude.
- Schaefer adopted the identifications of Peters & Knobel from $1915$.
- Not this Pickering.
- Star $995$ in Ptolemy’s catalog.
- See the table of probabilities of inclusions for magnitudes from Rawlins, above.
- The “Ancient Star Catalog” – a term authors in this collection of papers use since the authorship is in question.
- The function is clearly not linear, this should be good enough for a sniff test.
- I have not calculated the positions of the stars for the ancient epochs, so take this with a grain of salt.
- I.e., stars within $10º$ of the horizon at Alexandria.
- This is using a binomial probability calculator with at least 1 star being included.
- Although he limits himself to the first three quadrants.
- It seems to me that Rawlins made an identical error that Pickering does not note.
- As well as epoch.